Light Scattering and Photon Correlation Spectroscopy

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3. High Technology - Vol. 40

Light Scattering and Photon Correlation Spectroscopy

edited by

E. R. Pike Clerk Maxwell Professor of Theoretical Physics, King's College, London, U.K.

and

J. B. Abbiss Chief Scientist, Singular Systems, Irvine, California, U.S.A.

Springer Science+Business Media, B.V.

Proceedings of the NArO Advanced Research Workshop on Ught Scattering and Photon Correlation Spectroscopy Krakow, Poland August 26-30, 1996

A C.I.P. Catalogue record for this book is available from the Ubrary of Congress

ISBN 978-94-010-6355-5 ISBN 978-94-011-5586-1 (eBook) DOI 10.1007/978-94-011-5586-1

Printed an acid-free paper

AII Rights Reserved © 1997 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1997 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photo­copying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

DEDICATION

DR KLAUS SCHATZEL

*12.11.1952 t14.10.1994

Lehr8tuhIinhabel' (ler A" 011(1. AI aiel'ie eler Johannes Guienbe7'9 Universtiit Mainz

It seems fitting in this volume, devoted in large measure to recent de­velopments in field of photon correlation spectroscopy, to record, on behalf of all who knew him, the tragic and untimely death of one of the pioneers of the field and one of its most brilliant innovators. The two editors of this volume came to know Klaus Schatzel as a graduate student working with Eric Schiiltz Du-Bois on h~'drodynall1ic flow problems at Ki('l in the late 70s and eV(,1I h('tter when h(' came to England in 19;9/80 to spend a year at RSRE ~Ialv<'rll. His contrib1ltions to the dev('lopment of correla­tion t('chniques are well known and h(' was invariably an invited speaker at the series of international ronfer('nc('s held in this field. lie had wider interests, how('ver, and while at l'.lalvern, for example, worked on the phase structure function at the transition to turbulence and also on tracking of in­dividual particks in fl1lid flow systems. He was also an accomplished player of Charles vVheatstone's invention, the concertina, and made a number of records of folk mllSic with vario1ls groups.

IIis first digital rorrelator. built in the late iO's, was a softwar(' instru­ment powered by a ~OVA romput<'f which took several hours to accumulate a useable correlation funct.ion, and it is interesting to note that it is only in the last year or two, nearly 20 years later, that a sufficiently fast general purpose CPU chip has allo\',:e<1 a f1l1l software correlator of this type to be a practical proposition. Kla1ls q1lickly mastered the intricacies of clip­ping and random scaling, which Wef(' needed in those days to speed up the computations, and brought his own scholarly mind to the the question of why we used correlation functions rather than the structure functions more

v

vi

used by statisticians. His work on the comparison of these methods was deep and comprehensive. The situation turned out not to be very clear cut with only marginal advantages for one method or the other according to circumstances. On his return to G<'Tmany he acted as scientific consultant for the development of a commercial correlator, released in 1984, which allowed the measurement of the structure function as an option.

He kept up with and contributed to the later developments of geomet­rically increasing delay times for particle-sizing correlators and multi-bit computation as hardware became faster, and introduced the technique of "symmetric normalization", which is important for long sampling times. He went on to help develop "plug-in" single-board correlators for the PC and he was closely involved in the development of the special space flight correlator for the University of Maryland to be used in their micro-gravity experiment (ZENO). As described later in this volume, this was sent into orbit on March 4th 1994 on the Space Shuttle flight STS-62 and again in 1996. More recently he developC:'d a range of other techniques, including two-colour dynamic light scattering experiments and advanced equipment for electrophoresis and particle tracking.

Our community has lost a p<'Tson of great talent, both in theory and experiment, at the forefront of hi~ field; it has also lost a warm, kind, likC:'ahle and modest human being. In making this dedication we can only speculate as to what his future rant ributions might have been in his new and well-deserved position at tIl(' Unh·ersity of ~Ia.inz had hC:' remained with us.

TABLE OF CONTENTS

Dedication ............................................................. v Ta.ble of contents ...................................................... vii Preface ................................................................ xi Group photograph ................................................... xiv

COLLOIDS Experimental challenges in colloids P.N. PtUiey, P.N. Segre, a.p. Bc/m:nd, S.P. Meeker and lV. C.l(. Poon ..................................................... 1

Dynamic depolarized light scattering studies of anisotropic Brownian particles V. Dcgiorgio, R. Piazza, T. Bellini and F. A/antegaz::a ................. 7

Application of 111(' dvnamic light scattering method for investigation of colloidal stability II. V. l\'/yubin ......................................................... 2:3

Long-time dynamics of concentrated colloidal suspensions R. Kldn (Ahstract only) ........................................... :37

METHODS Suppression of multiple sca'ttering using a single beam cross­correlation method W. V. Meyer, D.S. emmell . ..I.E. Smart, T.W. Taylor' and P. Tin ..... :39

Theory of 1l11lltiple scattering suppression in cross-correlated light scattering employing a single laser beam J.A. Lock ............................................................. 51

viii

X-ray photon correlation spectroscopy S. Dierker ........................................................... 65

Polarisation fluctuations in radiation scattered by small particles E. J akeman .......................................................... 79

Surface light scattering spectroscopy lA. Mann, Jr. ......................................................... 97

Non-linearity of APDs at high count rates M. Gran, E. R. Pike and E. Pailharey ............................... 117

From speckles to modes: principles and applications of fiber optic dynamic light scattering. J. Ricka (Abstract only) .......................................... 129

POLYMERS, GELS, LIQUID CRYSTALS, MIXTURES Photon correlation sp<'ctroscopy of interactive polymer systems G. Fylas, K. C/u·is'<;0p0tlloll, S.ll. Arwstasiatlis, D. V[assopotllos and h'. l\"rafn .. ~o.<; ............................ ........................ 131

Local dynamics ill hranched polymers V. r,'appe and W. B1I7'C!u/1"([ ......................................... 141

Spontaneo1ls domain grO\vth in the one-phase region of a gel/mixture syst('m B..J. Frisken, A.E. Bailey and D.S. ('(/fwdl. ......................... 161

The shape, dim('nsion and organisation of maltodextrins gel fragments with and witho1lt associated phospholipids M.A.R.B. Castanho, M.JE. Prieto, D. Betbeder and N C. Santos . ........ 1 n

Dynamic ligllt scattering from block copolymers P Stepanek and T.P Lodge ............................................ 189

Particle diff1lsion and crystallisat ion in sllspensions of hard spheres W. tmu Mcgen. S.M. Underwood. J. Miiller, T.C. Mortensen, S.l. llenderson, J.L. llarl(/fui and P. Francis ......................... 209

ix

Use of light scattering to chara.cterize the polysaccharides of starch P. Roger and P Colonna ............................................ 225

STATISTICS AND DATA PROCESSING Spatial photon correlation and statistics of nonlinear processes in nonlinear waveguides M. Bertolotti, M. De Angelis, C. Sibilia and R. Horak ............... 231

PllOton correlation of correla.ted photons: experimental aspects of quantum cryptography and computation J. G. Rarity and P. R. Tapster ........................................ 247

New opto-elcctronic technologies for photon correlation experiments R. G. H'. Brown ...................................................... 263

Correlated, sUPNPosed and squeezed vibrational states J. J(tnsz~:y (tnd Z. Kis ................................................ 277

Structure and propNties of linear inverse problems J.B. Abbiss .......................................................... 29.5

New idea.s in data inversion in photon correlation spectroscopy E.R. Pike, B. McNally and P. Patin .................................... 313

Tempest in a teapot-surface and volume turbulence in a closed system WJ. Goldburg and C. Cheung (Abstract only) ...................... :32:3

SCATTERING BY DENSE MEDIA Diffusing photon correlation A .F. l(o,<;/~'o ........................................................ 325

Photon correlation spectroscopy of opaque fluids I. K. Yudin flud G. L. Ni~~ola('1l~'o ..................................... :341

Scattering of light ill inhomogellC'olls medium L.A. Zllbkotl aud V. P. Romanot' ..................................... 3.53

Observation of shear-induced gelation using light scattering imaging D. Pine (Abstract only) ........................................... 367

x

Single particle motion of hard-sphere-like polymer micronetwork colloids up to the colloid glass transition E. Bartsch, S. Kirsch, F Renth and H. Sillescu (Abstract only) ......... 369

Measurement of viscoelasticity of complex fluids by diffusing­wave spectroscopy D.A. Weitz (Abstract only) .......................................... 371

SPACE APPLICATIONS A pplying photon correlation spectroscopy in space A.E. S7nart ..........•.............................................. 373

lIard spheres in space: light scattering from colloidal crystals in microgravity Jixiang Zhu, P Chaikin, Min Ii, WB. Russel, R. Rogers, W Meyer. R.H. Ottewill and STS-73 Space Shuttle Crew (Abstract only) ......... 387

Zeno: Critical fluid light scattering in microgravity R. W. Gammon, J.N. ShallmEYEI'. M.E Briggs, 1I. Boukm'i ond D.A. Gent .......................................... :389

PHASE SEPARATION and CRITICAL PHENOMENA Static and dynamic light scattering in phase-separating systems S. V. I\'a::o/.:ov (lnd N.l. CIzf1'1101"(J •..••••.•••••••••••••••••••••••••••• 401

Shear induced displacenH'nt of the spinodal, and spinodal demixing kinetics lInd<'f shear .1.1\'.(,'. Dlzont ....................................................•.. 42:3

Spectral kinetic and correlation characteristics of inhomogeneous mixtures in the vicinity of the critical point of stratification A. D. A lcklzin, S. G. Ostapchcnlm. D. B. Svydka and D.l. Malyol'cn~'o ................................................ 441

List of participants .................................................. 4G1

Index ............................................................... 465

PREFACE

Since their inception more than 2.5 years ago, photon correlation techniques for the spatial, temporal or spectral analysis of fluctuating light fields have found an ever-widening range of applications. Using detectors which re­spond to single quanta of the radiation field, these methods are intrinsically digital in natnre and in many experimental situations offer a unique degree of accuracy and sensitivity, not only for the study of primary light sources themselves, but most particularly in the use of a laser-beam probe to study light scattering from pure fluids, macromolecular suspensions and laminar or turbulent flowing fluids and gases.

Following the earliest developments in laser scattering by dilute macro­nl01ecular suspensions, in , ... hich particle sizing was the main aim, and the use of photon correlation techniques for laser-Doppler studies of flow and tnrbuence. both of which areas were the subject of NATO ASls in Capri, Italy in 19;:3 and 19;6. significant advances have be('n made in recent years in many other areas. These were reflected in the topics covered in this NATO Advanced Research Workshop, which took place from August 2;th to 30th, 1!)!}6, at the Jagiellonian University, Krakow, Poland. These in­cluded ('xperimental techniques. statist.ics and data reduction, colloids and aggregation, polymers, gels, liquid crystals and mixtures, protein solutions, critical pllf'nomena and dense media.

Other notable r('c('nt developments discussed , ... ere in novel Meas such as in basic physics experiments on phase transitions in the microgravity environment provid('d by the space-shuttle missions, in quantum cryptogra­phy and in the applicatiOll of photon correlation at x-ray wavelengths using synchrotron sources.

Experimental techniques are continuously evolving, taking advantage of the miniaturisation and performance improvements provided by solid state lasers, fibre optics and avalanche photodiode detectors. Powerful new tech­niques of analysis from the lively field of inverse problems are being applied to data analysis. ~fuch progress is being made in the study of dense me­dia using index-matching techniques, two-colour systems and the theory of photon diffusion. Relationships between structural relaxation a.nd viscos­ity have been discovered in hard-sphere colloids a.nd tIle effects of small

xi

xii

amounts of polydispersity have been shown to have a significant influence on the dynamic structure factor. The complex process of crysta]]jsation of conoid solutions has been followed through the liquid-solid phase transition and different crystal structures from those which occur on earth appear in zero gravity.

A new bulk relaxation, characteristic of binary polymer blends near the macrophase separation temperature has been found and layer-height fluctu­ations at the surface has been studied by evanescent-wave dynamical light scattering. A universal function describing the conc(>ntration and tempera­ture dependence of the static and dynamic properties of binary separating mixtures has been obtained.

Details of these advances are to be found in the following pages. Unfor· tunately not all speakers v,'ere able to provide full manuscripts but at least an abstract of all the formal contributions is pu blished here, which will SNVE

to give a complete reference of the meeting for interested readers to pursue, The rec('nt orientation ofth(' ~ATO ARW programme towards colla bora.·

tive meetings with the Cooperation Partner countries was especially appre' ciated by our East(>rn European colleagues, with participants from Russia, Kazakhstan, Ukraine, Hungary, and the Czech Republic. Such a forum fot interchange of exp('rtise is clearly of value to all concemed. \Ve wish tc express om gratitude on bt'half of the pa.rticipants for the generous support of the NATO Scientific Affairs Division.

Although not reproduced in this volume, in a. working session highly ap­propriate to the intC'rnational character of the meeting, NASA's microgravity technology programs wen' described and the potential for intemational col­laboration in microgravity science and in the associated tedlllology and its commercialization were debated.

The organising comlllittC'e were: Co-dirC'ctors:

E. R. Pike, King's Collc-ge, London, J. K. Yudin, Oil and Gas Research Institut.e, Moscow, Russia.

Members: P. N. Pusey, Uni\'('rsity of Edinbmgh, UK. R. G. W. nrown, University of Nottingham, UK. M. Bertolotti, U ni\'('rsity of Home, Italy . . J. n. Ahhiss, Chief Scient.ist, Singular Systems, Irvine, CA, USA.

We were also advised by Prof Xicole Ostrowsky of the University of Nice who was unfortunately 110t able to attend the meeting.

xiii

Th(' meeting was held in th(' v('ry elegant Bohrzynski Saal of the .Jagiel­Ionian Univ('fsity and w(' are grateful to th(' U('ctor Prof Dr hal> AI('ksa.nder Koj and his staff for the V('fY ('fficient local organisation. Our thanks are also du(' to th(' lIIanag(,lIIent. and st.aff of th(' 1101.('1 Cracovia for th('ir t.ol('rance and hospit.alit.y. W(' a.r(' indehted a.s w('11 to Dr lIanna. Ma.usch, Dir('clor of the Polish Cultural Institute in London and to Mr S('w<'fyn Chom('t for advice and assistance and t.o Miss Elizaheth Lake for h('lp with the typing and prE'pa.ration of thE' material for this volume.

E. R. Pike .J. B. Ahhiss

April 1 !)!)i

Th(' conf('f('Ij(~(' was also g('n<'fOllsly support('d hy D('partnl('nt. of the Navy Grant. NOOO\'1-!)()-1-002:1issu('d hy tIl(' US Offic(' of Naval R('search Europ('an Offi(·('. The Unit.ed St.at.es ha.s a royalty-free license throughout the world in all copyrightahll' lIIat('fial contained herein. The cont('nt of the information pu hlished dol'S not. nl'cl'ssarily reneet the position or the poliey of the United States GOV('flllllent and no official endorsement should be inferred.

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EXPERIMENTAL CHALLENGES IN COLLOIDS·

P.N. PUSEY, P.N. 'SEGRE, O.P. BEHREND, S.P. MEEKER and W.C.K. POON Department of Physics and Astronomy The University of Edinburgh Mayfield Road. Edinburgh EH9 3JZ. UK

Abstract. Recent experimental studies, largely by dynamic light scattering (DLS), of suspensions in a liquid of colloidal particles which interact like hard spheres have yielded two unexpected findings whose interpretation challenges theory. First, the rate of structural relaxation (defined below) of the suspensions shows the same dependence on suspension concentration as the inverse of their zero-shear-rate viscosity. Second, the intennediate scattering functions, measured by DLS, show an interesting scaling property which suggests that structural relaxation is controlled by self diffusion of the particles.

Detailed descriptions of this work have been published recently. Thus this paper will give only a very brief summary which directs the reader to the literature. A more detailed, but still concise, summary is given in Ref. [1].

1. Background

Dynamic light scattering measures the nonnalised intennediate scattering function f( k, t) of the suspension

f(k, t). S(k, t) S(k,O) ,

where the intennediate scattering function s(k, t) is given by

*Reprinted from Theoretical Challenges in the Dynamics of Complex Fluids,

ed. T.C.B. McLeish (Dordrecht, Kluwer, 1997)

E. R. Pike and J. B. Abbiss (eds.), Light Scattering and Photon Correlation Spectroscopy, 1--6. @ 1997 Kluwer Academic Publishers.

(I)

2

(2)

and

S(k,O) ... s(k) (3)

where s(k) is the static structure factor. Here N is the number of particles, k is the

scattering vector and 'j (t) the position of particle j at time t. As can be seen from its

definition, in general the intermediate scattering function measures a collective motion of the particles.

In concentrated suspensions, where the fraction ¢ of the suspension's volume which is occupied by the particles may be 0.5 or larger, the static structure factor resembles that of simple atomic liquids, showing a pronounced diffraction peak at 2n/k - 2R, where R

is the particles' radius. The dominant structure in the suspension, which gives rise to this peak, is the short-ranged ordering, or cage, of particles surrounding a given particle.

In a dilute suspension, where interactions between the particles can be neglected, the

intermediate scattering function takes the simple form J(k, t)- exp (-Do12t ), where Do

is the free-particle (Stokes-Einstein) diffusion coefficient. In a concentrated suspension, due to both direct and hydrodynamic interactions between the particles, J(k, t) has a

more complicated dependence on 12t, the slowest decay being found at the peak of S(k). Furthermore J(k,t) decays via a two-stage process: an initial exponential decay,

(4)

where T Ii is the "structural relaxation time", essentially the lifetime of a particle's cage

of neighbours; and a second, slower, approximately exponential decay at long times,

(5)

3

2. Experimental

The suspensions consisted of sterically-stabilised particles of poly-methylmethacrylate in cis-decalin [see e.g. 2]. Due to a slight difference between the refractive indices of the particles and liquid, these sample were slightly opaque. The "two-colour dynamic light scattering" technique [3] was used to suppress multiple scattering, and thereby select only the strong single scattering.

3. Short-time diffusion

The short-time diffusion coefficients Ds(k) describe the average motions of the particles

over distances small compared to their radius and reflect both direct and hydrodynamic interactions between the particles. Extensive measurements were made of Ds(k) as

functions of both scattering vector k and suspension concentration'" [2]. These wen: compared with the predictions of theory and computer simulation, good agreement being found with the latter [2].

4. Long-time diffusion

The long-time diffusion coefficients DL (k) describe motions of the particles over

distances comparable to, or larger than. their radius. There is no satisfactory theory to date of long-time diffusion. The new results outlined below may provide insights which will stimulate theoreticalldevelopmcnts.

As noted above, in a dense fluidlike assembly of hard spheres the dominant structure, which gives rise to the main peak in S(k) at k = km' is the cage of particles

surrounding a given particle. Thus it can be argued that the long-time decay of f(knr , t), the intermediate scattering function measured at k - km' reflects the dominant

structural relaxation of the system so that DL (km ) is a measure of the rate of structural

relaxation. By comparing measurements of DL (km ) with measurements of the zero­

shear-rate viscosity 11 of the suspensions [4] we have found that the rate of structural

relaxation shows the same dependence on suspension concentration as the inverse of the viscosity over the whole range 0 < '" < 0.50, i.e. that

(6)

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Figurt J. Relative viscosity TlITlo (open circles) and inverse rate 0' structural relaxation Dol DL (k .. ) (filled circles) versus volume fraction ~

of suspensions of PMMA spheres (from (4). q. v. for an explanation of the data points represented by squares in the inset).

where 110 is the viscosity of the liquid in which the particles are suspended (see Fig. I).

While one would certainly expect these two quantities to show similar dependences on concentration - the processes of simple shear flow and structural rearrangement ooth involve the relative motions of neighbouring particles· the apparent identity found experimentally is surprising and remains to be explained by theory.

S. Scaling of the intermediate scattering functions

During an attempt to understand better the mechanism of structural relaxation we made a second surprising discovery [5]. This was that for kR> 2.7 . a range of s(;allering vector k which encompasses most of the strong variation of structure faclor S( k)

including the main peak. plots of Inf(k,I)/Ds(k)k 2 against I, measured at different

values of k, lay on a master curve. As can be seen from Eq. (4), this way of ploning

the data ensures that they superimpose at shon times (since In f( k,I)/ Ds (k)k 2 = -I ,

for 1« r R ) . What is surprising is the additional superimposition of data at

intermediate and long times. r ~ r R (see Fig. 2). As noted in [5], this finding implies

that. for kR> 2.7 • the intermediate structure factor can be written

(7)

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FiKure 2. (a) Logarithm of normali~ed intennediate ~cattering functions Inf(k.l) versus time I for a PMMA suspension of volume fraction ~ =

0.465 for different value.~ of kR as indicaled. (b) Same dau plotted as

In l(k.1)/ Ds(k)l;2 v~us I. showing scaling for kR > 2.5 (from [5».

5

where (6r2(1)) is the mean-square displacement of a single particle. suggesting that

structural relaxation is controlled by self diffusion. Although previous work [6] has suggested a connection between structural relaxation and self diffusion. the detailed scaling implied by Eq. (7) awaits a full theoretical explanation.

6. References

I. Pusey. P.N .• Segre. P.N .• Behrend. C .P .. Meeker. S.P. and Poon. W.C.K. (1996) Dynamics of concentrated colloidal suspensions. Physico A. in press.

2. Segre. P.N .. Behrend. a.p. and Pusey. P.N. (1995) Shon-time Brownian mOl ion in colloidal suspensions: Experiment and simulation. Phy .•. Rei'. E 52. 5070·8:\.

6

3. Segre, P.N., van Megen, W., Pusey, P.N., Schlitzel, K. and Peters, W. (1995) Two-colour dynamic light scattering,}. Mod. Opl. 42,1929-52.

4. Segre, P.N., Meeker, S.P., Pusey, P.N. and Poon, W.C.K. (1995) Viscosity and structural relaxation in suspensions of hard-sphere colloids, Phys. Rev. Lett. 75, 958-61.

5. Segre, P.N. and Pusey, P.N. (J 996) Scaling of the dynamic scattering function of concentrated colloidal suspensions, Ph),s. Rev. Letl. 77,771-4.

6. de Schepper, 1M., Cohen, E.G.D., Pusey, P.N. and Lekkerkerker, H.N.W. (1990) Analogies between the dyanamics of concentrated charged colloidal suspensions and dense atomic liquids, Physica A 164, 12-27.

DYNAMIC DEPOLARISED LIGHT SCATTERING STUDIES OF ANISOTROPIC BROWNIAN PARTICLES

VITTORIO DEGIORGIO. ROBERTO PIAZZA AND TOMMASO BELLINI Diparlimelllo di Elellrol/ica, UI/iversilo di Pm'ia, 27100 Pm'ia,llaly

AND

FRANCESCO MANTEGAZZA ISlilUlo eli Sciel/;e Farmacologiche, Ullh'ersila eli MilaI/o, 20133 MilaI/o, Ilaly

Abstract. The rotational correlation function of anisotropic colloidal particles can be measured by observing the fluctuations of the depolarized intensity scattered in the forward direction, We present a theoretical treatment of forward depolarized light scattering which compares homodyne and heterodyne detection of the dynamics of the depolarized field. and describes the statistical properties of the depolarized forward scattered field. showing that the relation connecting the intensity correlation function to the field correlation function is different from the stand1l"d Siegert relation, Our calculations indicate the import;lJlce of the insertion of a quarter-wave plate between the transmilled beam and the depolarized forward scattered field. In order to illustrate the theoretical calculations. we present experimental results concerning the mc.1surement of the rotational diffusion coefficient in dispersions of fluorinated polymer colloids.

1 Introduction

In recent years dynamic light scattering (DLS) has been widely used for the inves­tigation of the Brownian motion of interacting colloidal particles [1,2,3]. Both the

7

E. R. Pike andJ. B. Abbiss (eds.), Light Scattering and Photon Correlation Spectroscopy, 7-21. © 1997 Kluwer Academic Publishers.

8

collective and single-particle translational diffusion coefficients of hard spherical col­loids have been measured as functions of the particle volume fraction <P. The results have been compared with calculations based on the generalized Smoluchowski equa­tion including hydrodynamic interactions. Some years ago, the theory was extended to include rotational diffusion [-1]. The study of rotational diffusion by dynamic light scattering is possible only if the colloidal particles present some anisotropy, either intrinsic or due to a non-spherical shape. The anisotropy gives rise to a depolarized component in the scattered field. so that the rotational diffusion coefficient is more conveniently extracted from the experiment by measuring the autocorrelation func­tion of the depolarized component. In fact, the decay time of the autocorrelation function of the depolarized field scattered at a non-zero scattering angle B is related to both the translational and the orientational Brownian motion of the particles [.5]. However, if one has to rely on form anisotropy, the experiment becomes fea­sible only with large particles possessing a strongly non-spherical shape, \lihereas theoretical studies to date have dealt only with spherical particles. Some years ago, by using DDLS from spherical colloids which present an intrinsic optical anisotropy due to a partially crystalline internal structure [6. 7], the first measurement of the concentration dependence of the rotational diffusion coefficient of spherical Bro\v­nian particles \'"as performed [~l. :\lore recently. the investigation was extended to concentrated samples up to \"alues of <P which correspond to the coexistence region between colloidal fluid and colloidal crystal [:3].

In this paper ,,"e discuss in some detail the particular case of fOI'\'"ard scattering (8 = 0). in which the decay of the depolarized correlation function is determined only by the rotational diffusion of the particles. The first experiment of dynamic forward scattering \\"as performed b~' Wada et al. [9] on solutions of tobacco mosaic virus (T:"1\,). The measurement \vas carried out by blocking the transmitted beam ""ith a crossed polarizer (anal~"zer) \\"hich ideally selects onl~' that part of the fOI'\\"ard scattered light \\"hich is polarized orthogonally to the incident beam. Subsequently. several other experiments of zero-angle depolarized light-scattering \\"ere performed on solutions of biological macromolecules [10, 11]. solutions of synthetic polymers [12, 1:3]. and dispersions of polydisperse fluorinated polymer colloids [1-1]. but the method \\"as ne\"er analyzed in depth. In a real experiment. it has to be taken into account that the fOf\yard scattered beam is superposed to the much more intense transmitted beam. Since the analyzer always leaks a portion of the transmitted beam, one can raise the question whether such a contribution can act as a local oscillator. In some experiments [9, 11. 12, 13] the heterodyne correlation function was obtained. whereas -others [10. 1-1] present homodyne correlation functions. In the case in which the analyzer is perfectly crossed with the polarization of the in­cident beam, only the non-ideality of the optical components could give rise to a horizontally polarized component of the reference beam which might act as a local oscillator for the heterodyne measurement. A very simple method of generating in a controlled way a component of the reference beam having the same polarization as

9

the signal is that of slightly offsetting the analyzer with respect to the crossed posi­tion. However, as discussed in our recent article presenting a systematic treatment of the forward scattering method [15], the offset of the analyzer is not a sufficient condition for generating an interference term. In fact, it can be shown [16] that the phase of the scattered field is shifted by r. /2 with respect to the incident field. A simple way to obtain an interference signal is that of inserting a quarter-wave plate between the scattering cell and the analyzer in order to compensate the r. /2 phase shift bet\\'een the transmitted field and the depolarized scattered field [1.'>]. It is interesting to note that the final sequence of optical elements is exactly the same employed in an electric birefringence apparatus (see, for instance, Ref.[11, 18]). The forward scattering experiment can be seen as an electric birefringence experiment at zero electric field in \"hich the spontaneous fluctuations of birefringence are in­vestigated.

The organization of the article is the following. The next Section compares homodyne and heterodyne detection of the dynamics of the depolarized field. and describes in a simple and intuitive way the heterodyne configuration proposed in Ref.[l.5]. Section 3 treats the statistical properties of the depolarized forward scat­tered field. and sho\,'s that the relation connecting the intensity correlation function to the field correlation function is different from the standard Siegert relation \\'hich applies for scattering at a non-zero scattering angle. Section -1 describes the prop­erties of fluorinated polymer colloids [,], and presents some experimental results concerning the orientational d~'namics of concentrated dispersions of hard spheres.

2 Forward Scattering

Consider an incident plane waYe with real amplitude Eo, linearly polarized in the vertical direction. The wave goes through a scattering medium consisting of a dispersion of colloidal particles. We consider a dilute dispersion of monodisperse anisotropic particles \\'hich are characterized by a polarizability tensor Q:. We assume cylindrical symmetry for Q: along a main optical axis, and call aI, a1 and a3 the diagonal components of the polarizability tensor in the particle-fixed frame. The average polarizability of the particle is: a = (a3 + 2ad/a. I call ;3 the anisotropy of the particle polarizability, .3 = a3 - a1.

We assume that the particles can be treated as Rayleigh scatterers, so that the approach outlined in Chapter i of Berne and Pecora [.5] can be used. The scattered field will contain a vertically polarized contribution E\'I/ and a horizontally polarized term EVH. Under the assumption that the particles behave as independent scat­terers and that multiple scattering effects are negligible, the first-order correlation functions of the two contributions are:

(1)

10

(2)

where 10 and A are, respectively, intensity and wavelength of the incident laser beam, ns is the index of refraction of the solvent, N is the number of particles per unit volume, d is the pathlength in the scattering cell, As is the cross-section of the laser beam, Db and Do are the translational and rotational diffusion coefficients of the individual particle, and k = (4"l1s1 A)sin(8/2) is the modulus of the scattering vector.

We consider, for sake of simplicity, ellipsoidal particles having a geometric sym­metry axis coincident with the symmetry axis of the intrinsic optical anisotropy. 'Ve call a,a,b the lengths of the semiaxes of the ellipsoid. The components of a' are expressed as [16]:

2 (2 2) T ns I1p1 - ns

a1 = l·p 2 '2 2) ns + L1(l1 p1 - I1s

2(. 2 2) r I1s I1p3 - ns

03 = l·p '2 2 2 ns + L3(np3 - I1s)

(-!)

where 1:; is the particle volume, I1p1 and np3 are. respectively, the index of refraction of the particle for polarization perpendicular and parallel to the symmetry axis. L1 and L3 are kno\vn functions of the ratio bla.

In usual cases. 0 » J. so that the time-dependence of Gt!~·(J, ... t) is essentially controlled bv translational diffusion. If \ve are interested in the orientational motion. "'e have to ~easure the intensity correlation function Gt;h(k. t) at different scatter­ing angles and extrapolate the value of the first cumulant at k = O. An example of such a procedure is giYen in Ref.[3] where the dependence of the rotational diffu­sion coefficient on the volume fraction <I> was studied for colloidal hard spheres. In principle. the simplest thing would be to measure the correlation function of EFH

at k = 0, as first proposed by ',"ada et al. [9]. This "'ould require to block the transmitted beam "'ith a crossed polarizer (an:tlyzer). In practice. considering that h· H < < I", even by using the best available polarizers, the portion of the transmit­ted beam which leaks through is never smaller than I~'H' "'ada et al. assume that the presence of some direct laser light on the detector can be exploited to perform the measurement in the reference-beam (often called heterodyne) configuration. How­ever, this is not generally true for the following reasons: i) the field acting as local oscillator must present the same polarization as EVH, ii) even if the polarization is the same, there is no mixing if the two fields are in quadrature. Concerning point i), a simple method to control the strength of the local oscillator is to set the analyzer at an angle ir /2 - 8 with the vertical axis. Concerning point ii), it turns out that the phase of the field scattered by a collection of many particles is shifted by ir /2 with respect to that of the incident beam [16]. Assuming that the analyzer is not completely crossed \\'ith the incident polarization, the total field Edt) arriving on

11

the detector can be written as the superposition of two contributions which have the same polarization, but present a relative phase-shift of 7r /2. If 8 is small:

(,j)

Note that the vertically polarized transmitted beam contains also a contribution coming from depolarized scattering. HO\\'ever, such a contribution represents a small random addition to a large deterministic term, and, consequently, is neglected in our treatment. The total intensity on the detector is:

(6)

We see from Eq.6 that there is rio interference term. The intensity correlation function is gh'en by:

(I)

Eq.(7) shows that the laser contribution gives to the measured correlation func­tion an incoherent background \\'hich reduces the visibility of the signal correlation function.

:\. simple method to perform heterodyning was proposed recently by my group [I.'}]. It consists in inserting before the analyzer a quarter-wave plate, with fast axis along the vertical direction. The effect of the quarter-wave plate is to compensate the 7r /2 phase shift bet\\'een the transmitted field and the forward scattered field. I call E2(t) the total field impinging on the detector in presence of the quarter-\\'a\,e plate. The total intensity on the detector becomes:

(8)

The intensity correlation function is now gi\'en by:

(9)

Vnder the assumption that < h'H >« la, the third term at right-hand side of Eq.(9) can be neglected. so that the only significant term is the one containing the first-order correlation function.

In a real experiment. partial mixing can be observed even in absence of the quarter-wave plate because of the non-ideality of the used optical components, such as the presence of stress-induced birefringence in the cell windmvs. The discussion presented above makes·clear that it is not sufficient to generate a horizontally po­larized reference beam. but the non-ideality of the optics should also Y1elci a phase difference of the reference field with respect to the scattered field which is different from 7r /2. In the case in which heterodyne arises from the non-ideality of the optics, no control of the reference signal is possible, so that it may not be known whether the

12

experiment is performed in homodyne or heterodyne or in an intermediate situation [Ii)] .

At this point it is interesting to comment the previous measurements of the dy­namics of depolarized forward scattering. The experimen~s of Refs. [9, 10, 12, 13, 14] use two crossed polarizers (8 = 0) without quarter-wave plate. Wada et al. [9], Han and Yu [12], and Crosby et al. [13] have recorded a heterodyne signal which was attributed to the presence of a su bstantial amount of stray depolarized light. Schurr and Schmitz [10] have been able to perform homodyne measurements by using a very long pathlength to raise the scattered intensity of T:-IV solutions relative to the level of the transmitted direct beam. Russo et al. [14] have also performed a ho­modyne measurement by using polydisperse fluorinated polymer colloids. Thomas and Fletcher [11] have inserted a quarter-wave plate before the scattering cell with the aim of introducing some compensation for the birefringence of lenses and cell windows. They observe a heterodyne signal, again attributed to the presence of a de­polarized leakage background much greater than the forward-scattered depolarized signal.

It should be noted that most of the difficulties encountered in standard hetero­dyne experiments are not present in the forward depolarized scattering configuration we have proposed and tested. In fact the signal-to-reference ratio is easily controlled by acting on the direction 8 of the analyzer. Furthermore. the coherence area for fOf\,·ard scattering coincides \\·ith the incident beam cross-section and does not de­pend on the length of the optical path inside the scattering cell: this implies that a good o\"E'rJap of the wavefronts of signal and reference field is automatically ensured. and that one can use a long pathlength. Clearly, the mechanical stability of the optical components is much less critical than in a standard heterodyne experiment.

3 Statistics of the Forward Scattered Field

We assume that the incident field is linearly polarized in the vertical direction. and that the scattered field is observed in the horizontal plane. The scattering volume contains :\1 anisotropic particles having random orientation. The orientation of the j-th particle is characterized by the t\VO angles c5j and 9j. The first is the angle formed by the particle axis with the x axis (vertical axis), the second is the angle formed by the projection of the particle axis onto the yz plane with the y axis. Because of the particle anisotropy the total scattered field is the superposition of two terms: the first is a vertically polarized contribution with amplitude proportional to the number of particles and to the optical mismatch bet\'·een particle and solvent, the second is a depolarized contribution (depolarized means that this contribution contains both vertically and horizontally polarized components) with amplitude proportional to the internal particle anisotropy /3. The magnitude and sign of the depolarized contribution due to the j-th particle depend on the particle orientation. I call Evv and EF H, respectively, the vertically polarized and the horizontally

13

polarized components of the scattered field. Here We focus the attention on EVH which can be expressed as a superposition of single- particle contributions:

M E\'H(k, t) = Eoeiw(t-Ljc) L Aj(t)eik.1' It) (10)

j=l

where L is the distance bet\\'een detector and scattering "olume, I'j(t) is the position of the j-th particle at time t, and .-'lj(t) is the scattering amplitude. If the particles can be treated as anisotropic Rayleigh scatterers, .-'lj(t) is given by [5]:

(11)

where c is a real constant \\'hich depends on various experimental parameters. Note that .4j (t) is time-dependent because of the rotational motion of the anisotropic particle which makes 6j and OJ time-dependent.

We calculate now the intensity correlation function of E\'H which is defined as:

(12)

By using Eq.l0. and assuming that translational motion is decoupled from ro­tational motion [7. :3]. the result is:

.11 G~~1(t) = I; L < ...J.j(O),-lm(O)...J.n(t).4p(t) >< f ik.[,· {O)-1' {O)+1' {t)-1' (tl] > (13)

jmnp

We assume no\\". for sake of simplicity. that the motion of each particle is un­correlated with the motion of the other particles. :\'oting that < ...J.j(O) >= 0, < fik.1' to) >= 0.< ...J.j(O)...J.n(t) >= 0 if jl1. < f ik .[,· {O)l' (tl] >= 0 if jll, and so on, Eq,1:3 reduces to:

Gt~1(t) = 1;.11 < ...J.J(O).4](t) > +1;.11(.11 - 1) < .4J(0) >< ...J.~(t) > + 1;.11(.11 - 1) < .4j(0)...J.j(t) >2 1 < eik.[1' {O)+1' (tl] > 12-f:'

1;.11(.11 - 1) < Aj(O)...J.j(t) >2 1 < eik.[1' {O)-1' {tl] > 12 (1-1)

Since ~'1 is a very large number, the first term at right-hand side of Eq,l-1 can be neglected. In a stanqard light scattering experiment () is different from O. so that the third term at right-hand side of Eq.l-1 becomes equal to 0 becau~ i~contains the average of a complex number \\'ith random phase. Therefore, for k :; 0, Eq.14 becomes the very \\'ell known Siegert relation which applies to a complex Gaussian random process:

( 1.5)

14

where gVH is the normalized correlation function of the scattered field EVH:

( 16)

The situation is different in the forward scattering case because the forward scattered field presents amplitude fluctuations, but no phase fluctuation, and is. therefore, a real Gaussian process. If k = 0, the third term at right-hand side of Eq.14 is different from 0 and coincides with the fourth term. Therefore, instead of Eq.1.5, We obtain [1.5, 19]:

(17)

The discontinuity found bet\\"een the behavior at k = 0 and at k "lOis associated to the plane "'ave approach. A more realistic description should consider an incident beam with a Gaussian intensity profile. Such an approach, as discussed in Ref.(19], is capable of describing the transition from the zero-angle behavior to the finite angle behavior. Such transition occurs in a very narrow range of scattering angles corresponding to the divergence angle of the excitation beam.

4 Experilnental Results

Generally speaking, the study of the dynamics of the fluctuations of the depolarized scattered intensity is made easier by maximizing the ratio p =< hH > 1< hT >. For Rayleigh scatterers. p can be derived by recalling that < In' >= G~:((t = 0),

< IVH >= Gt:1(t = 0). and using Eqs.l and 2. One obtains:

p = ,- 2 ' '32 .... )0 T"', (18)

In the case of particles made of isotropic material, depolarized scattered light can arise only if the shape is non-spherical. As an example. \\'e take an ellipsoidal Rayleigh scatterer with a symmetry axis, having semiaxes a, a, b. A plot of p as a function of the semiaxes ratio bla is presented in Fig.1 for two different values of the optical mismatch against the solvent. IIp - ns. As expected. p = 0 for the isotropic sphere. \"ote that even rather elongated ellipsoids present quite small values of p. For instance, if \ve take np - I1s = 0.17, the value of p corresponding to bla = 10 is 2.3 X 10-3 and decreases to -1 x 10-4 • if the optical mismatch is reduced to IIp - 11s = 0.17. If the particle is made of anisotropic materiaL p can be large also for spheres, and takes its maximum value, 0.7.5, at index matching. \"ote that, rigorously speaking. the index-matching condition for ellipsoids possessing an intrinsic anisotropy depends on the axial ratio, but the effect is rather small.

The method discussed in the previous Section was tested by using aqueous dispersions of colloidal particles of tetrafluoroethylene copolymerized with perflu-

3'10.3 r-----,,-----"T---.---r-----,

o 10.0

b/a

15.0

j 20.0

15

Figure 1: The ratio p between Iv Hand Ivv plotted as a function of the axial ratio bin for an ellipsoidal Rayleigh scatterer which presents only form anisotropy (n p."3 - 1/.p 1 = O. The full line refers to the case np - na = 0.17, and the dotted line to the case np - ns = 0.07.

orol11et.hylvinylether UvIFA), prepared and kindly donated to us by Ausimont, ~I'li­

lano, Italy. The la.tex is obtained by a dispersion polymerization process in presence of an anionic surfactant (7). By a careful control of the nucleation steps, the process yields fairly Illonodisperse spherical particles (standard deviation in volume below .5%). MFA particles are partially crystalline. Their internal structure is probably a conglomerate of some tens of microcrystallites dispersed in an amorphous matrix [7J. Each cryst.allit.e is a folded ribbon of polymer chains packed in a regular crys­talline structure. The crystallinity is about 30%, with a chain folding length of the order of 50 nl11. The latex particles bear a negative surface charge which is due in part to adsorbed surfactant and in part to the end groups of the polymer chains (Ouorina.ted carboxyl ions) generated by the decomposition of the initiator.

The used particles have a radius of 110 nm, an average index of refraction 71. p = 1.3.52, and an intrinsic anisotropy D-np ::::::: 0.5 x 10-2 • They are dispersed in an index-matched solvent (18% by weight urea-water mixture) at a volume fraction of 2 .. 5%. 100 mM NaCI was added to the dispersion in order to screen the electrostatic interparticle interactions.

Note that the particles are too large to be considered Rayleigh scatterers. However, they satisfy the Rayleigh-Debye (also called Rayleigh- Gans) approxi­mation. Indeed, for particles made of isotropic material the condition of validity of the Ra.yleigh-Debye approximation is: (411"/>.)R(np - ns) « 1, where>. is the wavelength of light and R is the size of the particle. In the case of anisotropic pa.rticles, there is an additional condition for the validity of the approximation: (4rrj)..)RD-np « 1. One should therefore expect that p at index-matching takes

16

r-

,.--

I Laser ~ ~ Scattering

~ +- *" f+- Correlator r-cell

'--

D "-

PI 1J4 P2 Computer I--

Figure 2: Scheme of the experimental set-up for the forward-scattering measurement in the heterodyne configuration. PI and P2 are polarizers, ),,/4. is a quarter-wave pla.te, D is a phot.odetector.

the value 0.7.5, were it not for the fact that the particles are polycrystalline and contain amorphous regions occupying a volume fraction which might fluctuate from particle to particle. As a consequence, the MFA particles are optically polydisperse. The effect of optical polydispersity is that p at index-matching takes for the MFA particles a value of 0 .. 50 - 0 .. 5.5 instea.d of 0.75 [7].

In order to illustrate the theoretical considerations developed in the previous Sec­tion, we discuss an experimental comparison between the standard dynamic light scattering technique and the heterodyne forward scattering technique. The descrip­tion of the a.ppal'atus used for the standard depolarized li~ht-scatterin~ measurement can be found in Ref. [3] which presents a detailed study of the Brownian dynamics of these particles by intensity correlation measurements at non-zero scattering angles in a wide range of volume fractions. The heterodyne forward scattering measurement was performed by using the optical set-up schematized in Fig.2. The experiment is described in some detail in Ref.[20]. The components are: a low power He-Ne laser, two Glan-Thompson polarizers having an extinction ratio better than lO-i , a mica quarter-wave plate, and a cylindrical scattering cell with a 10 mm pathlength and very low residua.! stress-induced birefringence. The analyzer was offset from extinc­tion by a. small angle which was trimmed to give a signal-to-reference ratio between 10-3 and 10-2 • Despite the fact that only a very small fraction of the incident beam could reach the detector, the pow~r of the laser beam had still to be attenuated be­fore entering the scattering cell down to about 100 J'W in order to avoid excessive count rates. taking into account that count rates are large, detection was made by using a fast H5783P Hamamatsu metal package photomultiplier and a high-speed discriminator. The accumulation time for the forward heterodyne measurement was

"5 0.1 'to • ~ \ .... ~ ... . "

\ . . 0.0] L-_________ --:. ___ ~__'

10J lC'" 1<Y 1()4

J (l-ls)

17

Figure 3: Normalized time-dependent part of the heterodyne correlation function IG~11 measured in the forward-scattering configuration (full dots) compared to the normalized field-correlation fu nction extracted from a homodyne intensity correla.­tion measurement at a scattering angle of 1.5 0 (open dots). Both measurements refer to r-.·IFA suspensions a.t a particle volume fra.ction <P = 0.1.

a.bou t a factor of three short.er than for the homodyne measu rement at 15 o. For both homodyne and heterodyne experiments the correla.tion function was measured by a Brookha.\:en 819000 multi-tau correlator.

The depolarized field correlation functions obtained with t.he two different tech­niques by lIsing a !\lFA sample at a volume fraction <P = 0.10 are shown in Fig.3. The homodyne measurement is performed at a. scattering angle of 15 o. At such a small scattering angle, the contribution due to translational diffusion is so slow that t.he deca.y of the correlation funct.ion is controlled only by rot.ationa.l dilTusion. One can see that the two curves are perfectly superposed. The logarithmic time-scale choscn for the plot puts evidence 011 the long-time behavior: it secllls that the het­erodyne measurement is less noisy than the homodyne at long times. This might be connected with the fact the correlation function is calculated from the homodyne data through a square-root operation.

The measurements of the rotational correlation function were performed at var­ious volume fractions up to tile region in which the hard-sphere colloidal crystal is formed [3,20]. As discussed in Ref. [3], the shape of the correlation function departs considerably from exponential behavior when <P > 0.1. We have derived from the first cumulant the short-time rotational diffusion coefficient D'S as a function of <P. I present in Fig.4 the experimental results obtained with the two techniques. The obtained data clearly indicate that the two techniques yield results of comparable quality. The forward scattering measurement is much quicker and requires a simpler set-up.

18

<1>

Figure 4: <.I.>-dependence of the short-time rotational self-diffusion coefficient D'S di­vided by the rotational diffusion coefficient of the independent particle Do. Full dots and open dots refer, respect ively, to the heterodyne forward scattering mea­surement and to the homodyne non-zero scattering angle experiment. Full line and broken line a.re, respectively, the theoretical results of Refs.[3] and [21]. Stars show numerica.l simulation results of Ref.[22] .

FigA also shows the theoretical predictions [3 , 21) which are expressed in terms of a series expansion of D'S in powers of <.I.> truncated at the quadratic term, and the numerical simula.tion results obta.ined by a Stokesian-dynamics method [22) . The comparison between theory and experiment is very satisfactory, as discussed in more detail elsewhere [3 , 20) . It should be mentioned that a very recent paper by Wat.zl<l\\·ck and Niigele [23J present.s a calculation of the short-time rotational self­diffusion co('fficif'nt. as a function of <I> for chargcd spherical part.icles with variolls amounts of added electrolyte. The results reported in [23] fully justify the as­sumption used in the interpretation of the experimental results [3 , 20] that charged part.icles at the ionic strength of 100 mM beha.ve as hard spheres.

5 Conclusions

By observing the fluctuations of the depolarized intensity scattered in the forward d irection by a dispersion of anisotropic particles the measurement of the rotational correlation function can be performed directly without invoking any decoupling ap­proximation which enables to separa.te the orientational correlation function from the tra.nsla.tiona.1 one. We have described the sta.tistical properties of the forward scattered depolarized field, and we have discussed the conditions under which het­erodyne detection can be performed in a controlled and reliable way. The theoretical treatment shows that it is the important to insert a quarter-wave plate between the

19

scattering cell and the analyzer. The role of the quarter-wave plate is to generate a heterodyne signal bet"'een the transmitted beam and the depolarized forward scat­tered field. In practical cases, the stress-induced birefringence of the cell windows can give heterodyne effects similar to those produced by the quarter-wave plate, but it might be difficult to decide whether the measured correlation function is fully heterodyne or rather it contains both homodyne and heterodyne contributions. It should be noted that our treatment is based on the assumption that the incident beam is a plane ,,"ave. Some features of the problem are better described by using an approach based on Gaussian beams, such as the approach developed by Ricka [2.1].

In order to illustrate the theoretical calculations, we have also reported some experimental results obtained \yith dispersions of fluorinated polymer colloids in a wide range of volume fractions. The observed rotational dynamics is in good agree­ment with the theoretical calculations of the effect of hydrodynamic interactions on the rotational beha\"ior of colloidal hard spheres.

20

References

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[2] P. N. Pusey, inLiquids, Fru::ing and Glass Transition, edited by.J. P. Hansen, D. Levesque and J. Zinn-Justin (~orth-Holland, Amsterdam, 1991).

[:3] V. Degiorgio, R. Piazza, and R. B .. Jones, Phys. Rev. E 52, 2707 (1995).

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[.5] B .. J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, :.'\ew York, 1976).

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[7] V. Degiorgio, R. Piazza. T. Bellini, and ~\L Visca, Adv. Colloid Interface Sci. 48, 61 (199-1).

[8] R. Piazza, V. Degiorgio. :\1. Corti. and .J. Stavans. Phys. Rev. B. 42, -188·5 (1992).

[9] A. Wada. N. Suda. T. Tsuda. and I\:. Soda, .J. Chern. Phys 50, :31 (1969).

[10] J. ?-.I. Schurr and I\:. S. Schmitz. Biopolymers 12, 1021 (1973).

[ll] J. C. Thomas and G. C. Fletcher. Biopolymers 18, 1333 (1979).

[12] C. C. Han and H.\"tl .. J. Chem. Phys 61. 26.50 (1974).

[1:3] C. R. Crosb~' III, ~. C. Ford .Jr. F.E. Karasz. and K. H. Langley .. J. Chem. Phys 75, 4298 (1981).

[14] P. S. Russo, ?-.I. .J. Saunders. 1. :\1. DeLong. S. KuehL I\:. H. Langley. and R. 'V. Detenbeck, Analytica Chimica Acth 189. 69 (1986). .

[1.5] V. Degiorgio. T. Bellini. R. Piazza. and F. :\Iantegazza, Physica A (1996)

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21

APPLICATION OF THE DYNAMIC LIGHT SCATTERING

METHOD FOR INVESTIGATION OF COLLOIDAL STABILITY

Abstract.

V.V. KLYUI31N Lebcdev Researchy Institute of Synthetic Rubber Gapsal'skaja u/.l 198035 Sankt-Pctel'sburfJ Russia

The development. or the dyna.mic light sca.ttering method (Photon Cor­rf'lation S pf'rt roscopy) has a.llowed liS to a.d va.n ce considf'rably ex peri men tal research in t.h!' physics or dispersed systellls and to provide th!' answers to lIIany prohlf'lIIs. In t.his work t.he processes of the coagllla.tion of polYlller colloids consisting of charged particles were investigated by this method a.nd also hy t.h<' tllrhidit.y Illethod. Ollr investigations have shown c1ea.r1y tltat t.ltf'l"!' is 110 illdllctioll IH'riod ill t.lte IlI'O("('SS of latl'x P11'cl.rolyl.l' COa.)!;II­Ia.tioll. Tit!' Illaxillllllll on t.h!' tllrhidity cllrv(' is not, a r('sldt. of th(' fa.rt. that t.he roaglliatioll proCf'SS stops hut a consequence or the dirrt'rent cha.ra.cter or th!' sca.tt.('I"i n)!; law.

23

E. R. Pike and J. B. Abbiss (eds.), Light Scanering and Photon Correlation Spectroscopy, 23-35. © 1997 Kluwer Academic Publishers.

24

APPLICATION OF THE DYNAMIC LIGHT SCATTERING METHOD FOR

INVESTIGATION OF COLLOIDAL STABILITY.

Many problems of the dispersed systems physics may be successfully solved by means of

light scattering experiments. The investigation of Rayleigh spectral structure of light

scattering permits us to study various transformations occurring in colloidal systems. It

should be noted that development of dynamic light scattering method has allowed to

advance essentially the experimental researches in the physics of dispersed systems and to

provide answers to many problems concerning the building-up processes, long-duration

storage and change of the colloidal dispersion under the action of various factors.

The problem of stability of hydrophobic dispersed systems placed in varying external

conditions is of great interest. The large liquid-solid interface in these systems leads to

creation of significant free energy which is in turn the reason for their instability. The

trend to reduce a value of free energy causes reduction of a degree of dispersion of these

systems. However in practice many hydrophobic colloids conserve their stability for a long

time. It occurs because except of hydrophobic interactions there arc forces preventing the

process of phase division. These forces are determined by Coulomb interactions or

sometimes by interactions of nonelectrostatic nature. For hydrophobic colloids with a

surface-bound charge their stability is explained by DL VO theory (Derjagin-Landau­

Vervey-Overbeek theory) [Derjagin, B. V., Landau, L.D. (1945), Vervey, E.1. W., Overbeck,

J.Th.G. (1948)1. The theory takes into account counteracting forces connected with

existence of double electrical layers on one hand and Van der Waals attractive forces on the

other.

25

The colloidal hydrophobic dispersions can be tolerant for a long time but can

transform to a biphase state in a few minutes. A measure of stability of such systems is a

speed of coagulation which can change in wide range. Therefore it should be considered

the kinetics of coagulation process if you want to discuss of stability of dispersion systems.

The way of coagulation in time depends on two factors: thermal diffusion of colloidal

particles and their interactions by collisions. H there is a strong power repulsive barrier

between particles they will disperse every time after collisions. H a repulsive barrier does

not exist and only attractive forces act then every collision of colloidal particles will result

in their strong connection. In this case the speed of coagulation will be determined only by

Brownian movement of particles. M.Smolukhovsky called such type of coagulation fast

coagulation and he formulated many laws of this process.

The process offast coagulation is important as an application of Brownian movement.

However a process of slow coagulation is more informative. The difference between the

slow and fast coagulation is that in the case of slow process colloidal particles have a

repulsive barrier but its value is some more then the energy of particle's thermal

movement. In this case not all collision of particles will result in their coagulation. To

describe a difference between slow and fast coagulation M.Smolukhovsky suggested to

introduce a coefficient of slowing down the process but the way of slow coagulation process

he considered similar to the fast one. Unfortunately, up to now, there is not any theory

connecting a coefficient of slowing down of coagulation with the real physical values (such

as size of aggregates, potential of a double layer, concentration of electrolyte and so on).

In the light of this problems the experimental studies permit to check theoretical

conclusions and to improve numerous laws of the process. For a long time optical

methods based on the measurements of transmitted light are widely used to investigate the

coagulation of dispersed system. The measurement of colloid's turbidity allowed to obtain

various characteristics of the coagulation process however it leads to forming an incorrect

notion of dynamics of colloidal coagulation. Thus the analysis of turbidity changes of

polymer latexes under the coagulation by electrolyte solutions led to the conclusion about

existence of the induction period in course of which the process of aggregation stops. The

26

reason for such conclusion is that the turbidity changes do not describe accurately the

changes of structure of coagulable latexes since the turbidity depends on the size of

scaUerers in a very complicated manner especially in the case ()f polydispersed samples.

This problem has to be investigated by using a method which is capable of providing

direct information about the size changes of aggregates in the course of the process.

Dynamic light scattering method is the most suitable for this purpose. The method has high

accuracy and sensitivity to scatterers of large sizes. Therefore it can be used for

investigation of dynamics of colloidal aggregation.

Now I would like to tell you about some experiments which were carried out by the

dynamic light scattering method. These experiments have allowed to establish the correct

law of the size changes of the colloidal particle aggregates in the electrolyte coagulation

process.

In studies of the aggregation of latex particles in electrolytes a peculiar time

dependence of the sample's turbidity in the course of the process was obtained. On the first

picture you can see a characteristic curve describing this dependence. It is a curve number

1 of this picture. The similar curves were obtained in many experiments. The description of

results of these experiments can be found in several publications in Colloidal Journal

[Aleksandrova, E.M. et aI., 1964, Lebedeva. N.N., Sthastukhina, N.N., 1979, Naman, R.E.,

Kiseleva, O.G., 1986, Naman, R.E. et aI., 1961).

0,15

0,10

0,05

a

Figure 1.

d mkm

- 7

- 6

- 5

- 4

- 3

- 2

- 1

t, hour

27

Latex is a good system in which process of coagulation is described by DL VO theory

well enough. Latex is a water dispersion of polymer spherical particles. On a surface of

these particles there is a negative charge caused by ionic groups of the polymerization

initiators or surface - active substance.

From a curve 1 picture 1 you can see that optical density of a sample grows quickly at

an initial stage of coagulation process but then it remains constant for two or three hours.

Then the turbidity of a sample begins to decrease. Such character of turbidity change of the

coagulative latex dispersion set researchers to make the following conclusion. They thought

that in the beginning of the process there was fast growth of aggregate's sizes but then the

growth stopped. Then induction period came and the aggregation process stopped.

Induction period corresponds to the flat top of a turbidity curve. This period lasts about two

or two point five hours. When induction period was finished the coagulation process

restarted again. In the system there was the further growth of aggregates which precipitated

from colloid when they had reached critical sizes.

Thus a point of view of researchers, who deal with turbidity, on the changes of the

aggregate's sizes in coagulation process, is possible to represent graphically as it is shown

on a curve 2 picture 1. In the beginning the aggregate's sizes are increased quickly, then

during the induction period they remain constant, but after finishing induction period the

aggregate's sizes begin to increase again. Such charatter of change of the aggregate's sizes

in the coagulation process causes confusion. It is impossible to understand why the

aggregation stops for a while in the middle of the process. Just this incomprehension has

induced us to investigate process of growth of aggregate's sizes by means of the dynamic

scattering method.

The method of dynamic scattering gives the direct information about hydrodynamic

sizes both primary particles and aggregates arising in coagulation process. For initial latex

particles the hydrodynamic sizes coinCide practically with their geometrical sizes. For latex

units the dynamic scattering method allows to define the sizes of equivalent spheres,

volumes of which are equal to volumes of the units. Even in case of asymmetric aggregates

28

the dynamic scattering method enables to define their effective sizes and to register changes

of structure of these units in the coagulation process.

On the following figure 2 the distributions of units are shown for one of samples of

coagulative latex in various moment of time [Klyubin, V.V. et aI., 1990). The initial latex

sample investigated here has narrow distribution of particles with an average diameter d =

54 nm and relative half-width of distribution L\d/d = 0.21. The distribution of initial latex

particles is shown on top histogram of the figure 2. Following histograms give

representation about changing of a dispersive structure of units in the coagulation process.

Second histogram of this figure shows distribution of a mass share of units on the sizes 30

minutes after the beginning of the coagulation process. Third histogram - distribution of

units after 2 hours, and bottom histogram - distribution of units 4 hours after the beginning

of coagulation process. From these figure it is seen that an average diameter of units

increases in the coagulation process and relative width of distribution increases too. This

fact points to constant growth of the units' sizes.

W,ma 40 30 20 10

initiat latex

o~--~----~----~----+-----~----~ 40 30 20 10

3 minutes I O~~-r~--~-----r--~+-----~-----+

40 30 20 10

2 hours

O~---r~--~-----r----+-----~-----+

40 30 20 10 0L-~ __ ~ __ ~ __ -L __ ~~~

100 200 300 400 500 600 d, nm

Figure 2.

If we take the average units' sizes from these measurements and show their changing in

the coagulation process we would obtain a curve presented on a figure 1. It is a curve

29

number 3. It is evident from this curve that there is no induction period. There is no interval

of time when the growth of the units' sizes stops. We see that the sizes of units increase

monotonously over the course of all coagulation process. In the beginning of the process

units grow rather quickly, but then the growth rate is reduced. It is connected with a fact

that the units' sizes increases proportionally to a root cubic from number of particles

involved in the structure of aggregates.

Thus the measurements [Klyubin, v.v. et aI., 1988, Klyubin, V.V. et aI., 1990] carried

out with the help of a turbidity method and a dynamic scattering method give an

inconsistent picture of the process of growth of the units' sizes in a coagulating latex

dispersion. What Is the reason for this fact? Let's try to understand this discrepancy more

attentive.

Using the Smolukhovsky's theory it is possible to calculate a change of the aggregate's

sizes in the coagulation process. From this theory it follows that the number of particles

(both initial particles and their aggregates) in coagulation process decreases under the next

law: ( 1)

where No is number of particles in a sample before the beginning of coagulation and Nm

is number of aggregates consisting of m initial particles. The Smolukhovsky's theory gives

the following expression for number of aggregates Nm:

(2)

Value m is named as the order of aggregation. It is defined by the number of initial

particles which are included in a structure of given aggregate. Value to is equal to

1 expression (3) t =----

o 4trRDN o

(3)

Value to is named as characteristic time of coagulation. It is determined as a time for

which the number of aggregates in volume of a sample decreases twice. The value D

included in this expression represents the diffusion coefficient and R is the distance of

30

centre to centre interacting particles in aggregates. In the first approximation this distance

is possible to be equal to a diameter of interacting particles d.

Using the Stoks-Einstein equation it is possible to write a following expression:

DR=Dd= K8T 37rTl

(4)

where T and " are temperature and viscosity of dispersive environment, and Ks is

Boltzmann constant. If the water at room temperature is a dispersive environment than

characteristic time of coagulation can be calculated under the formula (5):

(5)

The formula (1) is obtained by Smolukhovsky in the assumption that in each moment in

a sample there is the complete set of aggregates beginning from units consisting of two

particles up to units consisting of infinite number of particles. But in a real situation the

degree of aggregation m is always finite. Let M (large) indicates a number of particles

included in the largest aggregate, then tills value may be calculated under condition that

there is only one largest aggregate in system at any moment of time (NM =1). From this

condition we can obtain that maximum degree of aggregation M at the moment of time t

can be calculated under the formula (6):

where function <p of't is (7): r

(]J(r)=­l+r

And't represents relative time of coagulation:

(6)

(7)

(8)

For a case of a limited degree of aggregation the complete number of aggregates in the

system can be calculated by following expression (9):

31

The final sum in this expression is equal to the relation (10):

(10)

Using the expression (10) it is possible to rewrite expression for the complete number of

aggregates Nl; as follows:

N -N -2 2( ) <pM(r)-l_ No (1 M(» l: - . r <p r· - --. -<p r (11)

o <p( r) - 1 1 + r

TIns formula for M approaching to infinity turns into the formula (1) which was

obtained by Smolukhovsky for the number of particles in the coagulating sample.

So we have obtained an expression which allows to calculate a change of the number of

particles Nt in a coagulation process. Now it is possible to calculate a change of the units'

sizes in this process. The dynamic light scattering method measures an average weight

diameter which is equal on definition to: (12)

In this formula dm and Nm are diameter and number of units consisting of m initial

particles. It is possible to estimate diameters of aggregates containing m initial particles

with the help of the following expression (13): (13)

in which do is diameter of initial particles.

If expressions (2), (11) and (13) are substituted in (12) we shall obtain the following

expression (14):

(14)

The sum in expression (14) can be calculated as follows:

Im<pm-I(r) = ~(I<pm(r)l = M<pM+I(r) - (M + 11<pM (r) + 1 (15)

m=1 dq:> m=1 ') (q:>C'r) -1)

Substituting (15) in (14) we obtain the average size of units dw:

32

d _ d 1- rpM (r)- MrpM (r)(1- rp(r)) W- 3----'--.:.....:---'---'--'--'----'----'-~

o (1- rp( r) )(1- rp( r)) (16)

The dependence of the average size of units which was obtained by means of

calculations carried out within the framework of the Smolukhovsky's theory is graphically

shown on a figure 3.

Figllre3.

t On the initial stage of coagulation process (at small value r = -) the average

to

diameter is linear with the time of coagulation:

(17)

When "C is much more then unity ("C » 1) the average diameter of aggregates is

proportional to a root cubic of coagulation time:

(18)

Thus calculations carried out within the framework of the Smolukhovsky's theory show

that the character of change of the average units' sizes qualitatively agrees with dependence

obtained experimentally by the dynamic light scattering method.

Now I want to show why the turbidity is not a monotonic function of sizes but a curve

has shape with a maximum even for samples in which the sizes of aggregates grow

monotonously. We shall try to calculate a form of a turbidity curve. The turbidity X of dilute

dispersion of scattering particles can be determined by means of expression (19):

33

(19)

in which constant a" is detennined by a relative index of refraction nOT = Ilk/no (Ilk - is

the index of refraction of a scattering particle, no - is the index of refraction of a dispersive

environment). The value an is defined by the following expression (20):

3 n~r -1 an =-tr

4 n;r + 2 (20)

The fonnula (19) contains also diameters of scattering particles d and their number N

and the function F (z) which depends on the value:

rather complicated.

d z = 8tr-

A (21)

Value z is detennined by the relation of a diameter of scattering particles d to a

wavelength of stimulating light A.. The function F (z) for Rayleigh scatterers has the

following form:

o tr- 4

F(z)=-z 54

For large scatterers the function F(z) has another fonn:

2 tr 0

F(z) = -z-2

(22)

(23)

Taking into account expression (19), (22) and (23) it is possible to write down that for

Rayleigh scatterers the turbidity is proportional to product of number of scatterers on their

diameter in the sixth degree:

(24)

For large particles the turbidity is proportional to product of number of particles on a

diameter in the fourth degree:

(25)

34

It is possible to consider that at early stages of development of coagulation process the

turbidity changes under the law (24). In this case its changes from time occurs under the

linear law:

(26)

When the units' sizes become rather large it is necessary to use dependence (25) for

turbidity calculation. If we substitute in this expression a dependence of number of

scatterers (9) and their diameter (14) on time and to go to a limit for or much more then

unity (or » 1) we shall obtain that for large times the turbidity will change under the law

(27):

X ~No .d:.3 r(lg~;) 7 (27)

The character of the turbidity change which was calculated by means of expression (27)

for several value No and do = 100 nm is shown on a figure 4.

x

Figure 4.

do = 100 nm 4

Ig"(

N -3 0, em

1 - 2 108

2 - 2 109

3_21010

4-21011

35

From this figure you can see that the turbidity increases in the beginning of the

coagulation process and than the turbidity reaches a maximum and begins to decrease.

Thus the calculations carried out show that in a case of electrolyte coagulation of

charged colloidal particles the process of the growth of the units' sizes goes monotonously.

In a course of tlus process there is no induction period when the coagulation stops. Our

investigation, performed using both the turbidity and the dynamic light scattering methods,

have shown clearly that there is no the induction period in the process of latex electrolyte

coagulation. This induction period was a consequence of incorrect processing of the

experimental data. The maximum on the turbidity curve is not a result of fact that

coagulation process stops but it is a consequence of different character of scattering law. For

the small particles we may use Rayley law of scattering, but for the large aggregates we

must take into account another type of dependencies.

References.

\. A1eksandrova, E.M., Shits, L.A, Tjurikova, 0.0. (1964) About kinetics of coagulation of dioxide titanium

hydrosols, Colloid J. 26, 645-646.

2. Derjagin, 8. V., Landau, L.D. (1945) Theory of the ~1ability of charged lyophobic sols and charged particles. J.

Experimental and Theoretical Physics 15,663-681.

3. Klyubin, V.V., Kruglova, LA, Sokolov, V.A (1990) Possibilities of the dynamic light scattering method for

investigation of the coagulation process dynamics of colloid systems, Colloid J. 52, 358-365.

4. Klyubin, V.V., Kruglova, L.A, Sokolov, VA (1988) Investigation of the coagulation of latexes by dectrolytes

by using the dynamic light scattering method, Colloid J. 50, 864-872.

5. K1yubin, V.V., Kruglova, L.A, Gurari, V.E. (1990) Some specific features of electrolytic coagulation of

concentrated latexes, CollOid J. 52, 46-53.

6. Lebedeva, N.N., Sthastukhina, N.N. (1979) Influence of concentration of synthetic latexes on kinetics of

electrolyte coagulation, Colloid J. 41, 798-801.

7. Naman, R.E., Ljashenko, OA, Kirdeeva, AP. (1961) Research of stability and coagulation of synthetic latexes,

ColloidJ.23, 732-738.

8. Naman, R.E., Kiseleva, 0.0. (1986) To the characterization of induction periods of coagulation of synthetic

latexes by electrolytes and freezing, ColloidJ.48, 931-935.

9. Vervey, EJ.W., Overbeek, J.Th.O. (1948) Theory of the stability of lyophobic colloids. Academic Press, New

York.

LONG-TIME DYNAMICS OF CONCENTRATED COLLOIDAL

SUSPENSIONS

Abstract.

RUDOLF KLEIN Fakulliit fiir Physik, Univcrsitiit J( onstanz, PostIlleh 5560, D-78F14 J(on,<;tanz, Germany

The relaxat.ion of densit.y fluctuations in charge-stahilized colloidal sus­pensions is cha.ra.derized by the dynamic structure factor S(k, t), which can be mea.sllfcd hy dynamic light scattering. In concentrated suspensions S( k, t) decays non-exponentially, due to memory effects. A suitable mea­slife of the overall decay of S(k, t) is the reduced memory function ~(k), which can be determined experimentally from the time-integral of S(k, t) and the first cumulant. Formally exact results for ~(k) can be obtained from the many-hody Smoluchowski equation, but for its evaluation one has t.o int.roduce approximations. Earlier calculations of ~(k), based on a particular form of the mode-coupling approximation, were found to be in disagrccment with experimental results. In particular, it was predicted for llIonodisperse suspensions that ~(k -+ 0) = 0, whereas experimental data ext.rapola.t.e t.o finite va.llIes of ~(O). By extending the previous the-ory to polydisperse suspe-nsions it will be shown that small amounts of polydis­persity give- rise t.o finite values of the non-exponentiality factor ~M(O), of the ll1('a'<;lIre-d dynamic structllre factor S!II(k, l). These effects arise from incohe-rent olH'-pa.rtide contrihutions to "!II(k, t) in the polydisperse case, which dominate S!II(k, l) for strongly correlated suspensions in the long­wavelengt.h limit.. De-tails can he found in G Na.gele, P Baur and R Klein, Physira A231, 119-61,1996, and P Bam G N~i.gele and It Kldn, Phys. Rct). E53, 622/1-62:17, 19%

37

E. R. Pike andJ. B. Abbiss (eds.), Light Scattering and Plwton Correlation Spectroscopy, 37. @ 1997 KlllWer Academic Publishers.

SUPPRESSION OF MULTIPLE SCATTERING USING A SINGLE BEAM CROSS-CORRELATION METHOD

Abstract

WILLIAM V. MEYER NASA LeRC / Ohio Aerospace Institute, MS. 105-1 21000 Brookpark Road, Cleveland, Ohio 44135-3191

DAVID S. CANNELL Department of Physics, UC Santa Barbara Santa Barbara, California 93106-9530

ANTHONY E. SMART 2857 Europa Drive, Costa Mesa California 92626-3525

THOMAS W. TAYLOR Department of Physics, Cleveland State University Cleveland, Ohio 44115-2440

PADETHA TIN NASA LeRC / Ohio Aerospace Institute, MS. 105-1 21000 Brookpark Road, Cleveland, Ohio 44135-3191

We present a simple, single beam, laser light scattering technique which discriminates against multiple scattering in turbid media using cross-correlation of the scattered intensity at slightly different spatial positions. Experimental results obtained at a scattering angle of 90° for colloidal suspensions of various concentrations show that the technique yields information on particle diameter, even for samples which are visually opaque.

39

E. R. Pike and J. B. Abbiss (eds.), Light Scattering and Photon Correlation Spectroscopy, 39-50. © 1997 Kluwer Academic Publishers.

40

1. Introduction

Beginning with the advent of the laser, dynamic light scattering has proved to be an invaluable technique for determining the dynamic properties of a variety of systems. It has been used to study order-parameter dynamics near the critical points of both pure fluids and binary mixtures. It provides the most accurate method known of determining

. the diffusion coefficient of macromolecules such as proteins and polymers, as well as that of colloidal particles. Because the radius of a spherical particle is determined very simply from knowledge of its diffusion coefficient, dynamic light scattering has become the method of choice for measuring the size of colloidal particles [11.

The method is not without problems however. In particular, for strongly scattering samples, not all of the scattered light originates as the result of single scattering. Under these conditions, it can be difficult to interpret data in any reliable way. One exception to this situation occurs in the limit of extremely strongly scattering samples, where the photons may be treated as diffusing throughout the sample[2-4l. In this case it has proved possible to deduce useful information regarding short time scale dynamics of the process giving rise to the scattering.

For dynamic light scattering in the intermediate regime where high order scattering is significant but the diffusing photon limit has not been reached, the most effective experimental method yet devised for dealing with multiple scattering is the cross­correlation technique invented by Phillies[S.6l, and extended by others[7-9l. Phillies' method relies on the fact that in order for the incident light to be scattered in a particular direction, the wave vectors of the incident and scattered light must be coupled by that of the dielectric constant fluctuation responsible for the scattering, in a Bragg-like relation, kj + q = ks. Here kj is the wave vector of the incident light, ks is that of the scattered light, and q is the wave vector of the fluctuation responsible for the scattering.

Because of the Bragg condition, two different beam-detector combinations can be aligned !!o as to simultaneously collect light which has been scattered by the same fluctuation. Of course, the two detectors also collect light which has been multiply scattered. This results in detector signals ia(t) and ib(t) which arise from both single and multiple scattering contributions. The single scattering contributions are strongly correlated with each other at all times, while the multiple scattering components are only weakly correlated. Measuring· the temporal cross-correlation function of the two detector outputs < ia('t) ib(O) >, then provides the same information as would be obtained in the single scattering limit using a conventional single-beam, single-detector arrangement to measure < i('r)i(O) > .

Although feasible, the cross-correlation technique summarized above is expensive and difficult to align in practice. As a possible alternative, we have begun to explore the possibility of using a single-beam, two-detector system. The basic idea hinges on the

41

observation that single scattering originates exclusively from the region of primary illumination, typically the small cross-section of a focused laser beam. Multiple scattering however arises from a broader 'halo' surrounding the incident beam, and thus appears to come from a significantly larger source.

Because scattered light arises from a spatially incoherent source, its spatial coherence properties are determined by the apparent dimensions of the source as viewed by the detector (van Cittert-Zemike theorem[IOl). Consequently singly scattered light gives rise to correlated patches (time-dependent speckle) which are typically much larger in one dimension than they are in the other. They are large in the dimension in which the source appears small, namely transverse to the incident beam, and small in the dimension in which the source appears large, that is parallel to the incident beam. Multiply scattered light also gives rise to time-dependent speckle, but since the smallest source dimension for multiple scattering is larger than that for single scattering, the mUltiple scattering speckles are smaller than the single scattering speckles in the direction transverse to the incident beam.

We have demonstrated experimentally that this idea, regarding the relative sizes of speckle arising from single versus multiple scattering, can be used to discriminate single from multiple scattering. This was accomplished by using two optical fibers leading to two separate detectors to collect scattered light at two closely spaced positions. We used fibers which were single-mode for 632.8 nm to collect scattered light at 514.5 nm. However single mode behavior is not necessary for the results reported below because each fiber only conveys light to a detector, with the fiber core acting as a small physical aperture stop. The fibers were not terminated with lenses, and thus served simply to collect light at two small regions. The two fibers were placed sufficiently close together to lie within one speckle of the singly scattered light, but not within one speckle of the multiply scattered light. Under these conditions we found that the cross-correlation function of the two detector outputs yielded useful data for the diameter of colloidal spheres, even when sample concentrations were so large as to render the 10 mm diameter samples visually opaque.

2. Experimental Method and Data Analysis

The optical arrangement used to test the feasibility of single beam spatial cross­correlation is sketched in figure 1. The samples were contained in cylindrical glass cells (test tubes) with inner and outer diameters of 10.1 mm and 11.6 mm, respectively. The sample cells were placed on the axis of a cylindrical glass vat having inner and outer diameters of 80 mm and 84.6 mm, respectively, and containing water to approximately match the refractive index of the sample. The two glass vessels thus had only a very small effect on the optical behavior. The incident laser beam, with a wavelength of 514.5 nm, was focused to an e·2 diameter of 88 ~m in the center of the sample, crossing it horizontally, perpendicular to the vertical axis of the system, as

42

shown in figure I. Entry to the vat was through a small polished flat area (not shown) to preserve circular symmetry.

Incident Beam

Sample Cell

Index­Matching Vat

To APD #1

To APD #2

Figure I Geometry witb bare-ended fibers to cross-correlate single-scattering speckle from turbid media

The cores of two optical fibers were separated vertically by about 250 J.UD and placed near the horizontal plane containing the incident beam. The fibers were polished but without special optical termination. Each fiber collected light which had been scattered at 90 0 relative to the incident beam, and the fiber faces were positioned 170 mm from the system axis. This position was chosen because it is at this distance from the system axis that most of the collimated light leaving the sample in the horizontal plane at a given scattering angle is brought to an approximate line focus by refraction at the outer surface of the vat. In practice the experimental configuration was established simply by removing the collection optics from the arm of a Brookhaven Instruments goniometer and replacing it by a small block through which the polished ends of the fibers protruded. The detectors were two SPCM-AQ 141-FL single photon counting silicon avalanche photodiode modules obtained from EG&G Optoelectronics, Canada.

43

The measured correlation functions were fitted using the two-cumulant expansion

(1)

Here B is a delay-time independent baseline, A an amplitude corresponding to the extrapolated intercept with the zero-time axis, and the frrst cumulant K) is related to the translational diffusion coefficient Dr by

(2)

(3)

is the wave vector of the fluctuation responsible for the scattering, with n being the index of refraction of the solvent, A.o the wavelength of the incoming laser beam in vacuum, and e the scattering angle. The radius of the particle, R, is calculated from

(4)

where kB is Boltzmann's constant, T is the absolute temperature, and Tt is the viscosity. R is the hydrodynamic radius.

3. Results

Measurements were made using polystyrene latex spheres suspended in distilled water at concentrations from 0.0017 wttJlo to 1.00 wttJlo. Initially we used conventional optics to collect light scattered at 90 0 from two dilute (0.005 and 0.01 wttJlo) suspensions of these spheres to verify their diameter independently. The measurements resulted in indistinguishable exponential intensity-intensity correlation functions from which we deduced the particle diameter to be 107 nm.

We replaced the normal" collection optics of the goniometer by the pait:~of.fibers to collect light scattered at 90 0 from each of the various suspensions. We'began by measuring a series of single-detector auto-correlation functions for samples of various concentrations, with each measurement being accumulated for 300 seconds. For dilute suspensions, ranging from 0.0017 wttJlo up to about 0.05 wttJlo, we found the normalized correlation functions, G(,;) , to be accurately exponential, as exemplified by the uppermost two curves in figure 2, which presents G(,;)/B - 1 versus delay time, ,; , on a

44

semilog plot, for several of the samples. In preparing this plot we normalized each measured correlation function by dividing it by the baseline, B, determined from the totals of the counts received and processed during the run. As we studied suspensions of increasing concentration, deviations from purely exponential decay became more and more apparent as shown by the data for higher concentrations. The rapid initial decay followed by a much more slowly decaying tail evident in the auto-correlation functions is noteworthy for the most concentrated samples. Auto-correlation functions with this general shape are commonly observed from strongly scattering samples approaching the diffusing photon limitI2-41. Although not apparent from figure 2, the curves have an apparent intercept with the y-axis which is almost unity for the most dilute sample and reduces to about 0.3 for the 1 wt«Ilo concentration.

§ ._ 0.1

1 § .~

] o (,) o "S « "'0

~

0.01

lE-3

:a lE-4

~ lE-5~--~--~--~--~~--~--~--~--~----~~

o 200 400 600 800 1000 Delay Time (microseconds)

Figure 2 Normalized auto-correlation functions for increasing concentrations of 107 nm diameter polystyrene spheres in water

To quantify the turbidity of our samples we used a power meter to measure the fraction of the incident beam power transmitted through the 10.1 mm diameter samples. After transmission though a sample of pure water, the beam passed through a 1 mm diameter aperture and was roughly collimated by a lens. A second lens about 3 m from the sample focused the beam through a 0.5 mm diameter aperture onto the detector of a power meter. This optical geometry was a precaution to minimize the scattered, rather

45

than purely forward transmitted, light reaching the detector. The results in figure 3 show that samples with concentrations in excess of about 0.2 wt«'1o, transmit an unscattered fraction of the incident beam of less than 0.6%, and this fraction reduces more steeply still at higher concentrations. We could not make a reliable measurement of attenuation factors in excess of about 106, where the data lie increasingly above the result expected from Mie calculations. This may arise from collecting some scattered light despite the precautions outlined above .

• • • • • .£ • c.. 0.1 § • en

~ 0.01

• 0 ..... lE-3 0 s:: 0

"1 . til lE-4 ~ til ·s til

~ lE-5 l!-

E-< '"C • til lE-6 ~ 0 ~ •

lE-7 lE-3 O.oI 0.1

Concentration (wt %)

Figure 3 Measured forward transmission through 10.1 mm of water containing increasing concentrations of polystyrene spheres

Because of the extremely strong scattering exhibited by samples with concentrations in excess of about 0.1 %, it is not surprising that the auto-correlation functions reported in figure 2 deviate strongly from single exponential decay. This is a direct consequence of collecting light which has been scattered more than once before leaving the sample. Each such scattering process significantly broadens the spectrum of the statiered . light, producing a rapid initial decay of the measured correlation function.

To determine the extent to which spatial cross-correlation might be useful in reducing the unwanted effects of multiple scattering for certain dynamic light scattering experiments, we measured the cross-correlation function of the pulse streams from both

46

detectors in the geometry described above for the samples for which we had previously measured individual auto-correlation functions. Representative results are presented in figure 4, which shows a semilog plot of cross-correlation functions versus delay time, normalized in the same way described above for the auto-correlation measurements. As expected, dilute samples gave accurately exponential cross-correlation functions, as demonstrated by their linearity. The main point however, is that the high-concentration samples exhibit the same linear behavior and slope as those with low-concentration. Thus the present data demonstrate that the simple artifice of spatial cross-correlation is adequate to permit useful dynamic light scattering measurements, even for samples which scatter so strongly that the probability of a photon traversing the sample without being scattered is only about one in 106• The data presented in figure 4 required increasingly long times for the more concentrated samples; about one hour was used for the 0.5 wr'1o and an overnight run often hours for the 1.0 wr'1o sample.

= 0.1 0

'1 ~

= 0 0.01 .... "tU U t: 0 0 I

lE-3 <Il <Il

I: u -0

CI) N

1 lE-4

0 Z

lE-5 0 200 400 600 800 1000

Delay Time (microseconds)

Figure 4 Normalized cross-correlation functions for increasing concentration of 107 nm diameter polystyrene spheres in water

We note that in the low concentration limit, the intercept of G(t) with the t = 0 axis for the cross-correlation functions did not exceed 0.42 whereas that for auto-correlation functions approached unity. Even in the absence of multiple scattering, the product of the scattered intensity at two different points (cross-correlation intercept) is always less

47

than the square of the intensity at a given point (auto-correlation intercept) because the speckle field comprising the scattered light has less than perfect spatial correlation.

The intercept value for the measured cross-correlation functions, is not however constant, but decreases strongly with increasing sample concentration, falling from about 0.42 for the dilute samples to about 2.5 x 10.3 for the 0.5 wt<'10 sample and to only about 7 x 104 for the 1.0 wt<'10 sample. This is also to be expected; mUltiple scattering contributes to the light collected by each fiber, and thus to the baseline. As the data show however, multiple scattering contributes very little to the cross-correlation signal. Reasoning in this manner leads one to conclude that of the light scattered by the 0.5 wt<'10 sample and collected by either fiber, only about "';0.002510.42 = 0.08 was singly scattered, and thus about 92% was multiple scattering. One obvious consequence is that to preserve measurement accuracy, the assessment of the baseline becomes more critical as the intercept reduces and requires the longer observation times discussed above.

120

o

IE-3

o • ~ • •

o •

o

• Diameter from Cross-correlation

o Diameter from Auto-correlation (I)

t:. Diameter from Auto-correlation (2)

0.01

• t:. o

0.1

•

o

Concentration (weight %)

• •

Figure 5 Apparent diameters of 107 nm polystyrene spheres in water measured at increasing concentration (0.2% looks like milk). The notation (1) and (2) refer to the different detectors.

48

To provide a direct comparison between the ability of cross-correlation and auto­correlation measurements to determine particle size for turbid suspensions, we fit both types of data using the two cumulant expansion given in equation (1). We then deduced the particle diameter from the frrst cumulant via equations (2) to (4) and present the results in figure 5. Clearly the results obtained by auto-correlation measurements (open symbols) begin to deviate systematically from the proper value as the sample concentration exceeds about 0.05 wt'lo. This deviation becomes dramatically more significant as concentration increases further. The measurements obtained using cross-correlation (solid symbols) show hardly any such tendency.

4. Discussion

We have presented data showing that spatial cross-correlation may be used to discriminate single from multiple scattering in making dynamic light scattering measurements from which to infer particle diameter. These data should be considered more as a proof of concept than as a demonstration of what may be possible. Nevertheless they do show that useful results can be obtained using remarkably simple apparatus, even under conditions where the samples scatter sufficiently strongly to have the physical appearance of milk. There is certainly no special reason to choose a 90 0

scattering angle for cross correlation, and we have verified that the method works over a range of scattering angles and for particle diameters ranging from 40 nm to 200 nm. In passing, we note that the alignment of the fibers is quite tolerant of the vertical plane position, goniometer angle and radial placement, but more critical with respect to rotation of the fiber retention assembly to place the fiber cores in the vertical direction aligned with the speckle elongation. Even this is easy compared with earlier techniques.

A further aspect of our collection geometry bears comment. Bare optical fibers used as collection optics do not limit the spatial region from which the scattering is collected as narrowly as do conventional collection optics. Instead the fibers accept all light within their numerical aperture, typically about 0.1 for single-mode fibers. Although both conventional optics and the arrangement described here convey light scattered at a given angle to a detector, it is usual for conventional collection optics also to restrict the physical length and height of the region from which light may be observed. Thus well designed conventional optics can discriminate against multiple scattering somewhat better than bare fibers, for this reason alone. Nevertheless, it is well recognized that for samples which transmit less than 10 % of the incident beam, significant multiple scattering is collected even by conventional optical arrangements, with concomitant distortion of measured auto-correlation functions. Thus the present fmding that spatial cross-correlation, even with non-discriminatory collection optics, can give reliable correlation functions for samples transmitting less than one part in a million of the incident beam, must be regarded as significant.

49

Under conditions of strong attenuation, as with the more concentrated samples, our optical geometry collects the single scattered light preferentially from where the beam enters the cell, rather than as conventionally from the cell center. This happens because our geometry collects light scattered at 90 0 to the beam, regardless of where along the beam path the scattering occurs, except for some small effect of optical aberrations in the cylindrical sample container and vat. For highly turbid samples, this overwhelmingly favors single scattering events for which the total distance traveled before and after scattering in the sample is minimal. For example, with an attenuation of 106 for a 10 mm path, the attenuation for a 1 mm path is only a factor of 4, making the contribution from such short path processes dominant.

In addition to restricting the observed scattering region with an optical stop or stops, another obvious improvement on our geometry might be to use a much more strongly focused incident beam. This increases the size of single-scattering speckles and allows the advantage of greater separation of the fiber cores. This arrangement should also increase the size of the region from which multiple scattering can arise, thus reducing the size of the multiple scattering speckle and its contribution to the cross-correlation function.

Rather than being viewed as unfortunate, the dramatic reduction of the signal amplitude-to-baseline ratio for the cross-correlation function as multiple scattering becomes significant might be used to determine the ratio of single to multiple scattering, at least in principle. The square root of the amplitude to baseline ratio is proportional to the ratio of the single scattering to the total scattered power, provided the fibers are far enough apart to yield negligible cross-correlation from multiple scattering. Combining the cross-correlation signal amplitude-to-baseline ratio with a direct measurement of the total scattering should allow measurements of the single scattering cross-section, even for highly turbid media.

Acknowledgments

The authors would like to thank James A. Lock of Cleveland State University for providing a theoretical analysis of the phenomena described here. His analysis is the subject of a another paper °in this volume. We would also like to thank Joanne Walton for assistance in the observations. This research was supported by the Microgravity and Science Applications Division of NASA.

50

References

1. See, for example, Berne, B. J. and Pecora, R. (1976) Dynamic Light Scattering Wiley, N.Y.

2. Ivanov, D. Y. and Kostko, A. F. (1983) "Spectrum o/multiply quasi-elastically scattered light' Opt. Spektrosk. 55, 950-953.

3. Maret, G. and Wolf, P. E. (1987) "Multiple Light Scatteringfrom Disordered Media. The Effect o/Brownian Motion o/Scatterers" Z. Phys. B-Cond. Matt. 65, 409-413.

4. Pine, D. J., Weitz, D. A., Chaikin, P. M., and Herbolzheimer, E. (1988) "Diffusing Wave Spectroscopy" Phys. Rev. Lett, 60, 1134.

5. Phillies, G. D. J. (1981) "Suppression o/multiple-scattering effects in quasielastic­light-scattering spectroscopy by homodyne cross-correlation techniques" J. Chern. Phys. 74, 260-262.

6. Phillies, G. D. J. (1981) "Experimental demonstration o/multiple-scattering suppression in quasielastic-light-scattering spectroscopy by homodyne coincidence techniques" Phys. Rev. A 24,1939-1943.

7. Dhont, J. K. G. and de Kruif, C. G. (1983) "Scattered light intensity cross correlation. I. Theory" J. Chern. Phys. 79, 1658-1563.

8. Mos, H. J., Pathmamanoharan, C., Dhont, J. K. G., and de Kruif, C. G. (1986) "Scattered light intensity cross correlation. II. Experimental' J. Chern. Phys. 84, 45-49.

9. Schlltzel, K. (1991) "Suppression o/multiple scattering by photon cross­correlation techniques" J. Mod. Opt. 38,1849-1865.

10. See, for example, Born, M. and Wolf, E. (1980) Principles o/Optics 6th ed. pp 508-516 (Pergamon, London)

THEORY OF MULTIPLE SCATTERING SUPPRESSION IN CROSS-CORRELATED LIGHT SCATTERING EMPLOYING A SINGLE LASER BEAM

Abstract

James A. Lock Physics Department, Cleveland State University Cleveland,OH 44115, U.S.A.

In previous systems for measuring cross-correlated light scattering by seed particles suspended in a liquid with multiple scattering suppression, the particles have been illuminated with two laser beams. We illustrate that multiple scattering suppression should also occur in cross-correlation for a system employing a single laser beam and two closely-spaced detectors. We calculate the single scattering, double scattering, and single scattering-double scattering cross term contributions to the intensity cross-correlation function for this system. We find that the two cross terms are of opposite sign, greatly reducing their contribution when added together. The amplitude of the double scattering term may be greatly diminished by judicious detector spacing because the spatial coherence area in the detector plane for double scattering is much smaller in one direction than that for single scattering.

1. Introduction

During the last three decades, dynamic light scattering has become the nonintrusive measurement technique of choice for studying the behavior of small particles suspended in a liquid. Single scattering dominates when the suspension is dilute. If the particles are noninteracting, the autocorrelation function of the scattered light decays exponentially with time [1]. In dense suspensions, however, multiple scattering dominates, causing the time dependence of the autocorrelation function to become complicated. This complication has been dealt with in a number of ways. For semi-dilute systems where multiple scattering first becomes significant, various investigators have calculated and compensated for the effects of double scattering [2-5]. For dense suspensions in the deep multiple scattering regime, others have employed photon diffusion [6.7] and radiative transfer [8] models. Still others have iIIumlnated the sample with two laser beams and cross-correlated the scattered light recorded by a pair of detectors [9-15]. L"nder certain circumstances this greatly suppresses the contribution of multiple scattering relative to that of single scattering.

A common concern shared by two-beam two-detector cross-correlation systems is that they are difficult to align. Thus there is motivation to find simpler cross-correlation .geometries that also suppress multiple scattering. In this paper we examine a new cross-correlation

51

E. R. Pike and J. B. Abbiss (eds.J. Light Scattering and Photon Correlation Spectroscopy. 51-64. © 1997 Kluwer Academic Publi3hers.

52

geometry that should enable one to straightforwardly measure the particle diffusion coefficient in dense suspensions and which is easier to align than existing two-beam two-detector systems. Our system consists of a single laser beam and two closely-spaced detectors. We show in the context of scattering theory that for so-called wide field of view detectors, multiply scattered light ceases being cross-correlated for small detector spacings while singly scattered light ceases being cross-correlated only aunuch larger detector spacings. Thus if two wide field of view detectors are placed beyond the double scattering threshold separation but within the single scattering threshold separation, the multiple scattering contribution to the cross-correlation function should be strongly suppressed with respect to the single scattering contribution. Our calculation parallels the treatments of [9,11,15] for the two-beam two-detector systems. The experimental verification of the suppression for this new geometry is presented in a companion paper [16].

2. Nota,tion and ~eometry

We consider the following general scattering situation. A chamber of volume V and with transparent walls is filled with a liquid of refractive index nL• Suspended in the liquid are N monodisperse dielectric spherical seed particles of refractive index np and radius a. The average number density of the particles is p=NN, their volume fraction is <I>=41ta3p/3, and their relative refractive index in the liquid is n=n/nL• A monochromatic laser beam of wavelength l and angular frequency w passes through the chamber and is scattered by the particles. The electric field of the unscattered portion of the beam at the position 1 in the chamber at the time t is

(1)

~

where Eo is the incident field strength, B(r) is a dimensionless function describing the transverse beam profile, L is the distance the beam has traveled through the chamber to the position -;, Lscall

A is the single scattering mean free path [7], uint is the beam polarization direction, and the wave number of the light in the liquid is

~

k= I kine I = 21tnJl (2)

~

If the beam either converges or diverges in the chamber, the incident wave vector kine and the polarization direction ~inc are also functions of 1.

The beam of EqJ 1) is scattered by the seed particles. The scattered light is incident on a detector at the position R.t in the far-zone. The relative sensitivity of the detector to light having its last interaction with a seed particle at the location 1 is D(f). The center of the intersection volume of B(~) and D~) is taken to be the origin of coordinates. The electric field at the time T and at location R in the chamber due to scattering by particle j at the position ~ at the earlier time tj , in the far-zone limit, is

-t -t Escatt (R, T)

~

S3

(3)

where AMie is th.E far-zone Lorenz-Mie scattering amplitude [17], Lj is the distance from particle j to the position R, the scattered wave vector ksc," is

~ the scattered momentum transfer K is

~ ~ -t

K = ksc.lI-kinc

with

-t IKI = 2k sin(6/2)

(4)

(5)

(6)

and where 6 is the scattering angle. Only some of the seed particles in the chamber are illuminated by the laser beam, and only some of these in tum are in the field of view of the detector. The volume of the laser beam within the chamber, the volume of the field of view of the detector within the chamber, and the intersection of these two volumes are denoted by Y I' Y 2'

and Yo respectively. The average number of particles in these volumes is N1, N2, and No. We assume that the incident laser beam has been focused by a spherical lens upbeam from the chamber so that the chamber lies entirely within the length of the beam focal waist. As a result, itc is independent of~ and light scattered at the angle 6 by each particle in the chamber has the same scattered momentum transfer.

Our goal is to estimate the relative amounts of light that are singly and doubly scattered by the seed particles and to determine the conditions under which the amount of double scattering is suppressed. When double scattering is suppressed for our cross-correlation geometry, we infer that multiple scattering in general is suppressed [15]. We make a number of approximations and idealizations in order to simplify our analysis. For example, we ignore the attenuation of both the laser beam and the scattered light in the suspension (i.e. L<<Lscall in Eq.( I) and Lj«Lscall in Eq.(3». We consider a top-~at laser beam profile of radius Ry• We assume point detectors whose sensitivity profile D(r) is a cone whose radius is Rfov»Ry at the position of the laser beam (see Fig. I ). We call Rfov»Ry the wide field of view detector condition. In order that a wide field of view detector record singly scattered light having only one value of the momentum transfer, we must place the scattering cell (usually a test tube containing the liquid and suspension) in a large coaxial cylindrical index matching vat and either place a bare monomode optical fiber [18,19] (i.e. our approximation to a point detector) on the vat paraxial focal line [16] or place a monomode fiber equiped with a GRIN lens far from the vat [20]. We ignor~both the polarization and angular dependence of the scattered light, i.e. we consider AMie ' and thus the electric field, to be scalars. In the Rayleigh scattering ka«1 limit we take

54

(7)

These approximations allow us to analytically calculate an order of magnitude estimate of the relative amounts of single and double scattering. In a separate paper [21] we relax many of the above assumptions and calculate both the single and double scattering cross-correlation functions taking into account the polarization and angular dependence of the Rayleigh scattering amplitude.

3. Single and Double Scattering by a Collection of Particles

Consider light reaching the detector at the time T produced by single scattering of the laser beam by seed particles in the chamber. Since the participating particles must be both illuminated by the laser beam and in the field of view of the detector, the single scattered electric field at the detector under the assumptions mentioned in the previous section is

where the subscripts d and I label the detector and denote single scattering respectively. We assume that the motion of the seed particles is random. The probability that particle j is at the position

.... .... given that it was at the position rj at the time tj is P(Aj,1:). Singly scattered ligiU,reaching the detector at the time T +1: averaged over all the possible particle displacements Aj is then

..., E A 'kR . T .... ~... .... .~ ... ( ) 'Ed1 (Rd ,T+l) = _O_e' d-'"' e-i",tS'(K,l)LB(r.)D(r.)e-n.rj tj

kRd 1=1 J J

(8)

(9)

-p (10) where the scattering structure factor S(K;t) is defined as

..., -.",!~

S(K,'t)= J d3Aj P(Aj,'t) e'K.Aj (II)

If the particles are noninteracting and their motion is diffusive, then [I]

(12)

55

where D is the diffusion coefficient of the seed particles in the liquid.

Consider the case of double scattering where the incident beam scatters from particle j at the position ~ at the time tj and travels the distance rjC to the position ~ at the time tc where it scatters from particle Q. The doubly scattered light then propagates to the detector in the far-zone without further interaction with the seed particles. For this situation. particle j must be illuminated by the incident beam and particle Q must be within the field of view of the detector. The double scattered electric field reaching the detector at the time T is then

2 N N EoA ikR -i"'T~ ~

= --e d L L kRd j=l 1=1

..., (13) where the intennediate wave vector kj' is given by

(14)

For doubly scattered ~glJ! reachin~ tlJ,e detector at the time T +t we assume that particles j and Q

were at the positions rj+d/r) and rc+d,(t) with probabilities p(lj.t) and p(ilc.t) when the two scatterings occurred. We assume that dj«rjc and dc«rjc so that the intennediate wave vector from any of the possible positions of the first scattering to any of the possible positions of the second scattering is well-approximated by Eq.(l4). The double scatt~d lig~t reaching the detector at T +t averaged over all the possible particle displacements d j and d, is then

(15)

4. Electric Field Cross-Correlation Functions

In this section we consider a single incident laser beam and two closely-spaced detectors ex and Jl in the scattering far-zone. The angle between the detectors is (, (typically about 0.09°) and their fields of view overlap almost perfectly. i.e.

~ ~ ~

Dc> (r) =DB (r) =D(r) (16)

S6

We consider the electric field cross-correlation functions m

C ( 1) = f dT E; ( T+ 1 ) E~ ( T) (17)

and m

d(l) f dT E" ( T+l ) E~ ( T) (18)

where the electric field at each detector is the sum of single and double scattering contributions. The correlation function c("t) is dominant for autocorrelation [1] (i.e. 0=0), and d("t) is dominant for cross-correlation employing two counter-propagating beams and two oppositely-placed detectors [9,11].

The T integral in Eqs.( 17,18) is theoretically modeled by the ensemble average of the integrands, i.e. the average over all the locations where particles j and ~ can be at the times t and tH when the scatterings occured [9,11,15]. Averaging over the particle locations at t+"t, given the locations at t, was performed in the previous section and produced the scattering structure factors. We now average over the locations of the particles at time t. For single scattering, substitution of Eqs.(8, 10) into Eq.(I7) gives

(19)

where I denotes single scattering, -+ -t" -+~ E = kscatt - kscatt

-+ lEI 2k sin (5/2) (20)

-+ and with k"·P ",all being the scattered wave vector from the origin to detector IX or p. The autocorrelation strength factor is

57

~21)

and the dimensionless and normalized single scattering geometrical factor [11] is

(22)

In obtaining Eq.( 19). we ignored the contribution of the particle pair correlation function. Thus we limit our treatment to volume fractions cj> $1 0-1 where the pair correlation function is near unity and we do not consider cricical opalescence near phase transitions of the system [5]. The single scattering volume Vo in our one-beam two-detector cross-correlation geometry is a cylinder of radius Ry and length 2Rz oriented along the z axis. This models a focused laser beam in the vicinity of its focal waist. The wide field of view detectors ex and ~ are placed far down the x axis and are stacked on top of each other so as to have different y coordinates as is shown in Fig.!.

.. ---

y LASER BEAM

\ z

._ _ _ _ _ _ A -__ t-' --~

, ------I __ • -: .'

: c::==t a. .. :--(", ... -..

2Rrov

x

Fig. I A laser beam of radius Ry propagates through the liquid containing the suspended particles_ The radius of the field of view of the detectors ex

and ~ is RIO,'

58

The length 2Rz of the single scattering volume is the smaller of either the diameter of the scattering chamber or 2Rfov ' If we ignore the effects of beam attenuation in the scattering cell, then

(23)

where ~ lies in the yz plane and makes an angle ~ with the z axis. For the geometry of Fig.I, we have ~=90°.

By a derivation similar to that which led to Eq.( 19), the single scattering contribution to the electric field cross-correlation function d(1:) is

(24)

~

The geometrical factor Go'(2Kavc) greatly suppresses d(1:) with respect to c(1:) for our one-beam two-detector geometry. For 8=90° scattering and with A=0.5145Ilm, Ry=41llm, and ~=5.2mm which correspond to the experimental parameters of [16], we obtain Go·(2K.v~= 10.9• On ~e other hand, if the detectors a and ~ are placed sufficiently close together, we have E=O and Go(E)= I for c(1:) in Eq.(l9). Thus we no longer consider the electric field correlation function d(1:).

For the double scattering ensemble average we obtain

(25) Similarly, for the single scattering-double scattering cross terms we obtain

59

( ) E2 A 3 2 ""." ... (26)

'" 0 P iWT ~ a 3 2 '" -t 1E. r j C12 (1)= E CX1 (1)Eo2 (O) =. e S(K ,1) r d r.IB(r.)1 D (r)e

" k 2 R; J Va J J J

and

... ... ... D(r)'" ~ ...... _. _" .... r d 3 f S(k cx -k ) S(k -k ) 2 (kj( kscaeel.rj(

X J 1 r jr--- scatt jr' ""[ jC inc'""[ e V, krjf

(27)

Equation (i 9) suggests an interesting physical interpretation of the electric field cross­correlation function. The single scattering geometrical factor Go( E) of Eq.(23) for arbitrary ~ may be written as [19]

2J1 (osinVO~Oh) 1 osin~/O~Oh (28)

where oco/ and 0COhz have the following meanings. If the single scattering volume were replaced by a light source with the same dimensions, the spatial coherence area of the source appearing in the detector plane far down the x axis has the angular diameter

(29)

in the y direction and

OZCOh = JlkRz (30)

in the z direction. Thus single scattered light from within the volume Vo is strongly cross­correlated if the two detectors lie within the same spatial coherence area of the analogous light source. This spatial coherence argument also applies to the region over which double scattering occurs. This will be examined in the next section.

60

5. The Intensity Cross-Correlation Function

In this section we consider the intensity correlation function ~

C(l) = fdTIc:x(T+l)I~(T)

which may be approximated by the ensemble average

where each electric field has both single and double scattering contributions. Using Wick's theorem to expand Eq.(32) in products of electric field correlation functions [II], we obtain

where the first term in Eq.(33) is the product of the average intensity at each detector,

and

(31 )

(32)

(33)

(34)

(35)

In obtaining Eq.(33) we assumed that the average number of particles within the scattering volume is independent of time. As was seen in the previous section, we neglect Z·Z with respect toWW.

The suppression of double scattering, and by inference the suppression of multiple scattering in general, for a single focused laser beam passing through the scattering chamber and two closely-spaced wide field of view detectors may be seen as follows. In the double scattering and cross term contributions to Eq.(33) we assume that each point ~ within the beam is

61

surrounded by an identical isotropic environment so that we can perform analytically the d\( integral in Eqs.(25-27) in spherical coordinates out to some radius R.y< which is comparable to Rz• Further. since we are interested in order of magnitude results only. we temporarily consider the t=O limit of the electric field correlation functions so as to avoid having to integrate products of structures factors in the d\, integral. Even for t>O. S is of order unity. and so ignoring its momentum transfer dependence will not change the order of magnitude of our results [15]. With these approximations along with the assumptions R.ve«Lsc," and kR.ve»1 we obtain

(36)

and

(37)

where

p =

(38)

artd

Q =

(39)

The double scattering suppression factor F(l)kR.v.) is

1 OkJR.~ d sinu 6kR u-u-

ave 0 (40)

The suppression factor satisfies F(l)kR.v.)= I for I)kR.v.«1 (e.g. autocorrelation is 1)=0). and F(l)kR.v.)= 1t/2I)kR.v• for x»1 (e.g. cross-correlation employing two closely-spaced detectors). The double scattering spatial coherence area is modeled in Eq.(37) by the product Go( €)F( I)kR.v.)' For Rayleigh scattering with A given by Eq.(7). we have

62

(41)

and

Q 3( n 2 -1) 2 (ka) 3<t>kR F(okR ) n 2+2 ave ave

(42)

In the limit of high sample turbidity where L"'.II<R.ve, the expressions for P and Q are no longer given by Eqs.(38,39) but assume different forms [21].

Consider polystyrene latex spheres in water for which n= 1.2, and A=O.5145J.1m and R.v.,,4.5mm corresponding to the experiment of [16]. We assume that ka and <I> are sufficiently small so that Rave«L",.n. For this case, the two single scattering-double scattering cross term contributions to Eq.(35) nearlYcCompletely cancel. The double scattering contribution of Eq.(25) to autocorrelation becomes larger than the single scattering contribution for <I>~2x10-l when ka=I.O, and for <I>~3xI0·3 when ka=O.5. For cross-correlation, if the two detectors are placed so that 5::; IIkRy, the single scattered light is strongly correlated (i.e. Go(l) " 1.0) while the double scattering suppression factor is

rtR F(okR ) ,,-_Y "0.015 (43)

ave R ave

which is comparable to the double scattering suppression calculated by Schatzel for the two-color two-detector system [15]. The suppression occurs because double scattering is present over a relatively large distance in the y direction (i.e. 2R..e), thus producing a smaller far-zone spatial coherence length in the y direction than for single scattering which occurs over the much smaller y distance 2Ry• The single and double scattering spatial coherence lengths in the z direction are comparable since single and double scattering are both present over the distance 2Rz as seen by the detectors. Thus if the y coordinates of the detectors a. and P lie within the same single scattering spatial coherence area but lie on two different smaller multiple scattering coherence areas, single scattering should be strongly cross-correlated while multiple scattering should not be.

6. Conclusions

Interest in dynamic light scattering is not confined only to suspensions at low volume fractions, but extends to higher volume fraction systems as well. The laser-sample-detector geometry of

63

autocorrelation experiments is relatively easy to align. But the interpretation of the autocorrelogram at high volume fractions is complicated because of the distortion produced by multiple scattering. The strong suppression of multiple scattering for certain cross-correlation geometries is an attractive alternative to autocorrelation. Many laser-sample-detector geometries have been discovered for which suppression occurs. As a result, emphasis has been placed on finding new suppression geometries for which the apparatus is relatively easy to align. Our one­beam two-detector geometry is a promising candidate for such a system. The real test of the usefulness of this .system, however, is the ease of its alignment, the rate at which and the level to which noise in the cross-correlogram is decreased, and the simplicity of analysis of the cross­correlogram. These issues are addressed in [16].

Acknowledgement

The author thanks Mr. William V. Meyer of the Ohio Aerospace Institute and the NASA Lewis Research Center for suggesting this problem.

References

I. Clark, N .A., Lunacek, J .H., and Benedek, G.B. (1970) A study of Brownian motion using light scattering, Am. J. Phys. 38,575-585.

2. Sorenson, C.M., Mockler, R.c., and O'Sullivan, Wol. (1976) Depolarized correlation function of light double scattered from a system of Brownian particles, Phys. Rev. A14, 1520-1532.

3. Sorenson, C.M., Mockler, R.C., and O'Sullivan, Wol. (1978) Multiple scattering from a system of Brownian particles, Phys. Rev. A17, 2030-2035.

4. Dhont, 1.K.G. (1985) Multiple Rayleigh-Gans-Debye scattering in colloidal systems: dynamic light scattering, Physica 129A, 374-394.

5. Bailey A.E., and Cannell D.S. (1994) Practical method for calculation of multiple light scattering Phys. Rev. E50, 4853-4864.

6. Pine, Dol., Weitz, D.A., Chaikin P.M., and Herbolzheimer, E. (1988) Diffusing-wave spectroscopy, Phys. Rev. Lett. 60, 1134-1137.

7. Weitz D.A., and Pine Dol. (1993) Diffusing-wave spectroscopy, in Brown, W. (ed.) Dynamic Light Scattering: The Method and Some Applications, Clarendon, Oxford, pp.652-720.

64

8. Ackerson, B.1., Dougherty, R.L., Reguigui, N.M., and Nobbmann, U. (1992) Correlation transfer: application of radiative transfer solution methods to photon correlation problems, Joum. Therrnophys. Heat Trans. 6, 577-588.

9. Phillies, G.DJ. (1981) Suppression of multiple scattering effects in quasielastic light scattering by homodyne cross-correlation techniques, J. Chern. Phys. 74, 260-262.

10. Phillies, G.DJ. (1981) Experimental demonstration of multiple-scattering suppression in quasielastic light scattering spectroscopy by homodyne coincidence techniques, Phys. Rev. A24, 1939-1943.

11. Dhont J.K.G., and de Kruif, e.G. (1983) Scattered light intensity cross-correlation. 1. Theory, J. Chern. Phys. 79, 1658-1663.

12. Mos, H.J., Pathmamanoharan, c., Dhont, J.K.G., and de Kruif, C.G. (1986) Scattered light intensity cross correlation. II. Experimental. J. Chern. Phys. 84,45-49.

13. Drewel, M., Ahrens, J., and Podschus, U. (1990) Decorrelation of multiple scattering for an arbitrary scattering angle, J. Opt. Soc. Arn. A7, 206-210.

14. Schatzel, K., Drewel, M., and Ahrens, J. (1990) Suppression of multiple scattering in photon correlation spectroscopy, J. Phys.: Condens. Matter 2, SA393-SA398.

15. Schatzel, K. (1991) Suppression of multiple scattering by photon cross-correlation techniques, Journ. Mod. Opt. 38, 1849-1865.

16. Meyer, W.V., Cannell, D.S., Smart, A.E., Taylor, T.W., and Tin, P. (1997) Suppression of multiple scattering using a single beam cross-correlation method, in Pike, E.R., (ed.) Light Scattering and Photon Correlation Spectroscopy, Kluwer, Dordrecht, this volume.

17. van de Hulst, H.e. (1981) Light Scattering by Small Particles, Dover, New York, pp.122-123.

18. Brown, R.G. (1987) Dynamic light scattering using monomode optical fibers, Appl. Opt. 26, 4846-4851.

19. Ricka, J. (1993) Dynamic light scattering with single-mode and multimode receivers, Appl. Opt. 32,2860-2875.

20. Ackerson, BJ. (1996) Phys. Dept., Oklahoma State Univ., personal communication.

21. Lock, lA. (submitted) Theory of multiple scattering suppression in cross-correlated light scattering employing a single laser beam, submitted to Appl. Opt.

X-RAY PHOTON CORRELATION SPECTROSCOPY

Abstract.

S. DIERKI~R Dqmrtmcnt of Physics, llnillc7·.r;;ity of Michigan, A,m A "b07', Mirllig(lU, USA

The new field of x-ray photon correlation spectroscopy (XPCS) offers an unprecedented opportunity to extend the range of length scales over which a material's low frequency dynamics can be studied down to interatomic spacings. The ctitical decelopment which has now made XPCS a feasible technique is the high brightness of insertion devices at second and third generat.ion synchrot.ron sources. In this talk, I will describe the principles of the XPCS technique and how it is practiced, as well as its potential use for a. va.ri('t.y of import.ant prohlems in the low frequency dynamics of condcnscd maHer systems, such as complex rJuids, glasses, surfa.ces and metallic alloys. Illustrations will be drawn from our recent results on using XPCS to study the nrownia.n motion of a gold colloid as well as recent results of others.

65

E. R. Pike and J. B. Abbiss (eds.), Light Scanering and Photon Corre1ation Spectroscopy, 65-78. © 1997 Kluwer Academic Publishers.

66

X-RAY pnOTON CORRELATION SI~ECTROSCOPY

I. Intmduction

Visible photon correlation spectroscopy (peS) has been an indispensable technique for studying the long wavelength hydrodynamics of fluids, including simple liquids, liquid mixtures, liquid crystals, polymers, and colloids. Nevertheless, it has a number of impor­tant limitations. First. it can only be upplied to reasonably transparent materials. Although the recently developed technique of Diffusing Wave Spectroscopy can be used to study strongly multiply scattering materials. and either very thin sample cells or back scattering geometries can be used to study materials for which the absorption length is small but still much larger than the wavelength, visible pes cannot be applied to truly opaque materials such as metallic solids, etc~ Secondly, since visible light couples primarily to the polar­ization fluctuations in a material, it cannot be used to study optically isotropic fluctuations, sllch us. to give just one example, bond orientational fluctuations in hexatic B liquid crystals. Finally. and most importantly. visible pes can only probe long wavelength ex­citatiuns. i.e .. those having waveveetors less than - 4 x 10.3 A·I. due to the - 0.5 ~m wavelength of visible light.

The principles of PCS arc independent of the wavelength of the scattering radiation. Indeed. the post-detection signal processing concepts underlying PCS were first applied to microwaves in radar detection. Thus. there is no fundmnental barrier to performing PCS measurements with coherent x-rays, with wavelength of - I "A, rather than visible light. If feasible. this would enable us to study the .fllortlellgth scale. slow dYllamics of condensed matter systems. One could study excitations with fluctuation times, 't, ranging from ~secs to - 103 seconds and having wavevectors ranging from 10-3 A-I on the low end, which overlaps the upper range of visible PCS, all the way up to several A-Ion the upper end, corresponding to wavelengths comparable to interatomic spacings. Since hard x-rays are quite penetrating, one could study truly opaque materials such as solids. Finally, since charge scattering dominates polarization scattering in the x-ray region (i.e., the A·p term in the radiation-matter interaction Hamiltonian is negligible compared to the A 2 tenn, where A is the vector potential and p is the electron momentum), one could study optically iso­tropic excitations. Note that this last point implies that the contrast for scattering x-rays from excitations in a material is in general quite different than that for scattering visible light.

Although the ability in principle to perform x-ray PCS (XPCS) measurements has

67

been recognized for several decades, the low flux of coherent x-rays available until quite recently precluded its practical implementation. With the advent of insertion devices at second and third generation synchrotron sources, this situation is very rapidly changing. Several experiments have now been done which demonstrate the practical feasibility of XPCS and it has aroused a great deal of interest. Several forthcoming technical enhance­ments give great promise that the technique will find wide application and have a similarly immense scientific impact as visible PCS has had.

While visible PCS and XPCS have many principles in common, they also have a number of major technical and scientific differences. Many of the intuitions gained from long experience with visible PCS are quite misleading with XPCS. On the other hand, scientists with an x-ray scattering background are often unfamiliar with the ideas of co­herence and PCS. One of the goals of this paper is to provide a brief review of XPCS which might make it more accessible to scientists with either background and who might like to perform XPCS measurements. After the introduction in the first section, the second section discusses the scientific opportunities which motivate the development of this new technique. The third section discusses various aspects of x-ray sources which are relevant to performing PCS measurements with coherent x-rays. Section four provides a brief re­view of current applications of XPCS.

2. Scientific Opportunities

The scientific opportunities enabled by XPCS stem primarily from its ability to ex­tend the range of wavevectors for which we might study the low frequency excitation of materials. The significance of this ability is illustrated by considering the variety of tech­niques which we have at our disposal for studying excitations of condensed matter systems as a function of the range of energies and momenta they cover. As shown in Figure 1, the visible light scattering techniques of Raman scattering, Brillouin scattering, and visible PCS do a good job of covering the range of full range of excitation energies at small momenta. At larger momenta, the long standing technique of inelastic neutron scattering (including neutron spin echo) as well as the more recent technique of inelastic x-ray scat­tering, cover much of the energy range from micro-electron-volts to several electron volts. However, there has not previously been any technique with which we might study low energy, large momenta excitations. It is precisely this void which XPCS can fill. It is one of the recurring themes of physics that the advent of techniques which dramatically extend our ability to study the world inevitably result in new scientific discoveries.

The initial reaction of many condensed matter scientists, especially those with light scattering backgrounds, is that low frequency and short wavelength are contradictory. This is because the phenomena most often studied with visible PCS have relatively large trans­port coefficients. Since excitations with low frequencies tend to have small restoring forces, their dynamics tend to be dominated by damping, i.e., they are usually diffusive modes. Thus, it is of interest to consider the range of diffusion coefficient which might be probed with XPCS. The dispersion relation for a diffusive excitation is simple given by 't = 1I(Dq2), where 't is the relaxation time, D is the diffusion coefficient, and q is the wavevector. Figure 2 shows such dispersion curves over the range of relaxation times and

68

~v~=:~'~ i2.,~catterin~

Visible Brillouin

Scattering

Figure 1. The region of wavevector-energy space covered by various techniques is indicated. XPCS would provide a unique probe in a previously inaccessible region.

wave vector accessible to XPCS. We see that with XPCS, diffusion coefficients with values ranging from 10-4 cm2/sec all the way down to 10,19 cm2/sec could be measured. This encompasses the range measurable with visible PCS and in addition, the lower limit is some 6 to 7 orders of magnitude lower than that measurable with visible PCS. While we are certainly unaccustomed to thinking about excitations with diffusion coefficients near the lower end of this range (perhaps due to our inability to study them!) there are in fact systems for which this is the appropriate regime.

In the next 6 subsections, I describe 6 broad classes of problems which represent areas of scientific opportunity for XPCS.

2. 1. STUDIES OF THE DYNAMIC STRUCTURE FACTOR OF LIQUIDS ON LENGTH SCALES DOWN TO THE INTERATOMIC SPACING

In this limit, the conventional assumptions of hydrodynamics break down: the liq­uid is no longer a continuum, nor is it in thermodynamic eqUilibrium. Thus, one might test ideas of non-equilibrium statistical mechanics. Mode coupling theory, in various forms, has been developed to cope with this situation. It has been tested in visible PeS measure­ments on a number of experiments on colloidal systems, where the lattice constant can be

69

104

103

102

- 101 (fl

l:J C 0 10° u Q) (/) --- -1 Q) 10 E i= 10-2 C 0

:;:::::; 10-3 ctI

X ctI Q)

10-4 a:

10-5

10-6

10-7

10-4 10-3 10-2 10-1 10° 101

-1 Momentum, q (A )

Figure 2. Dispersion curves for diffusive excitations over the region of wavevector, q, and relaxation time, 't, accessible by XPCS. The values of the diffusion coefficient, D, for each curve are indicated, in units of cm2/sec.

increased to be comparable to the wavelength of light, and also in a number of glassy systems, but only at long wavelengths. The ability to study the wider range of materials afforded by XPCS would provide additional tests of it, particularly in the crucial regime of short length scales. Candidate systems include:

• Colloidal systems • Moderate molecular weight simple liquids • Liquid crystals

- Smectic layer fluctuations - Nematic-Smectic A transition - Hexatic bond orientational fluctuations

• Polymers - Polymer liquid crystals - Block copolymers

70

2.2. STUDIES OF SOLID ALLOY PHASE TRANSITIONS

Visible PCS studies of binary phase transitions in fluids have provided some of our best tests of the theory of critical phenomena. This class of problems represents the analog of those studies in solid alloys. Additionally, since, in contrast to fluids, solids support a shear strain, some of these materials should show strong effects due to composition-strain coupling. Simulations suggest that this will lead to large shifts in transition temperature and spatial anisotropy. Examples for this class include:

• Spinodal decomposition/nucleation and growth studies in solid alloys. • Order-disorder transitions. XPCS experiments have already been performed on

one transition of this type, the B2-D03 transition in both CU3Aul,2 and Fe3Ae.

2.3. STUDIES OF THE DYNAMICS OF MOVING DOMAIN WALLS IN INCOM­MENSURATE SYSTEMS

Candidates:

• Ferroelectrics • Charge Density Wave systems • X-ray Doppler Velocimetry of domain walls moving in response to an applied

field. An example is sliding CDWs. • Spiral order in magnets and magnetic domain walls in general, including random

field magnets. • Adsorbates on surfaces.

2.4. STUDIES OF SURFACE DYNAMICS

Candidates: • Roughening/facetting transitions on single crystal surfaces • Pattern formation dynamics accompanying surface chemical reactions, including

during in-situ crystal growth

2.5. STUDIES OF THE INTERNAL CONFORMATIONAL DYNAMICS OF POLY­MER MOLECULES

At present, these can be studies with visible PCS only for very large molecular weight molecules. XPCS would permit theories of reptation and Rouse dynamics in poly­mer melts to be examined in new regimes of space and time.

2.6. STUDIES OF THE TEMPERATURE DEPENDENCE OF THE DYNAMICS OF SHORT RANGE DENSITY FLUCTUATIONS IN LIQUIDS UNDERGOING A GLASS TRANSITION

These could be probed at shorter length scales than previously possible. One might also be able to study the temperature dependence of the dynamics of short range orienta­tional, or configurational, fluctuations at a glass transition.

71

3. X-ray Sources

3.1. SYNCHROTRONS AND INSERTION DEVICES

The first observations of speckle were made by Exner in 1877 with a white light source. However, it took almost a century and the advent of visible laser in the early 1960's before visible light sources with sufficient coherence and intensity were available for speckle and PCS to become practical techniques.

Just as with visible PCS, it was the very low flux of coherent x-rays available with previous sources which, until recently, precluded its application as a practical technique. The critical development which has now made XPCS feasible is the use of synchrotron radiation sources, particularly involving insertion devices such as undulators instead of bending magnets. In the last several decades, this has brought about an exponential growth in the attainable hard x-ray coherent flux, as shown in Figure 3. Synchrotrons have evolved through three generations. In the first generation machines, such as the Stanford Synchrotron Radiation Laboratory (SSRL), the use of hard x-rays for condensed matter experiments was parasitic to the machines primary purpose for High Energy Physics. The second generation of machines, such at the National Synchrotron Light Source (NSLS), were the first to be constructed as dedicated sources for condensed matter experiments. These machines primarily use bending magnets for producing x-rays. The most recent, third generation, machines, such as the European Synchrotron Radiation Facility (ESRF) and the Advanced Photon Source (APS) are also dedicated to condensed matter experi­ments and are the first to be designed to use insertion devices, such as wigglers and undulators for x-ray production. These insertion devices, particularly undulators, produce much more collimated x-ray beams which leads to substantially greater coherent x-ray flux.

Many technical aspects of performing synchrotron x-ray experiments are complete­ly different than those encountered in visible light scattering experiments. Windows, monochromators, mirrors, and apertures are all completely different to name just a few items. Due to space constraints this article cannot go into all of these issues. Instead, we will simply consider the available coherent x-ray flux and the coherence requirements in an XPCS measurement.

3.2. COHERENT X-RAY FLUX

The far field spatial profile of the output intensity of a synchrotron undulator source having rms source dimensions 0"1i and 0"1] can be expressed as a function of the transverse coordinates, ~ and 11, in a plane perpendicular to the average propagation direction as:

I(~, 1]) = ~ e -(2:? I t~) (1) 27r0'{0'f/

Eq. (1) is normalized so that II I(~;rl)d~dl1 = Fo, the total flux from the source. For exam­ple, the source dimensions an undulator at the APS are 0"1i = 325 Jlm and 0"1] = 86 Jlffi.

The normalized coherence factor for the electric field amplitude, JlA, in a plane a distance L from the source is given from the Van Citert-Zernike theorem by the normalized

72

1012

APS or ESRF Undulator

-~ 109 al ~ 0 T"""

0

~ 106 Q) SSRL Wiggler

~ en NSLS Bending Magnet t: 0 -0

SSRL Bending Magnet ..t: a.

103 ->< ::J U. -t: Q) .... Q)

10° ..t: 0 ()

X-ray Tube

10-3

1900 1940 1980

YEAR

Figure 3. Evolution of the coherent flux produced by various hard x-ray sources during the last century.

Fourier transform of I(x,y), i.e.

(2a)

(2b)

Eq. (2a) expresses IlA in terms of wavevectors qx and qy, while Eq. (2b) expresses IlA in terms of the transverse coordinates x,y in a plane a distance L from the source. Note that both the source distribution and the coherence factor are separable in x and y for a gaussian

source. The transverse coherence lengths, Ix and ly, can be detennined from ~A to be +00

73

Ix = J IILA(X)1 2dx (3a) -00

>'L (3b)

2,,;:;ruf,

and similarly for ly. For a typical APS beamline, L = 35 m, so at A. = 1.5 A (8 keY), we have Ix = 4.6 ~m and ly = 17. ~m.

The longitudinal coherence length, lcoh, of the source is detennined by its spectrum. If the full width half maximum of the spectrum is fl.A., then leoh is

leoh = >.(:>.) (4)

The radiation emitted by an undulator is spectrally bunched at a fundamental wavelength which is essentially the Lorentz contracted period of the undulator magnet structure and its harmonics. The bandwidth of the fundamental, which is - I %, can be isolated by using a mirror as a low pass filter, giving lcoh - 100 A. If a greater longitudinal coherence length is required, a silicon crystal monochromator can be used, giving leoh - I ~m.

The coherent flux, Fe, from a synchrotron source can be calculated from the inser­tion device brightness, B, as

(5)

where 21tcr~crTl is the area of the source, lx/L and 1.JL are the collection angles in the x and y directions, respectively, and (fl.')JA.) is the monochromaticity of the x-rays. If a square aperture of size Ix 2 is used to collimate the beam instead of a rectangular aperture of size lxly, then Fe is reduced by a factor of crTl/cr~.

Using the brightness of an APS undulator as an example, the expected coherent flux for a square collimating aperture of size Lx 2 is - 2 x lOl l photons/sec if a mirror is used to isolate the undulator fundamental. With a silicon monochromator, the expected coherent flux would be - 2 x 109 photons/sec.

3.3. COHERENCE REQUIREMENTS

As discussed in the last section, there is an important trade-off to be made between coherent flux and x-ray coherence, so it is important to tailor the photons to the needs of the experiment. Usually, the transverse dimensions of the beam are limited to sizes of order the transverse coherence lengths calculated above in order to have good transverse coherence. In addition, in order to see speckle the maximum path length difference in­curred by the scattered photons, PLD, must be of the order of or less than the longitudinal coherence length. lcoh. lcoh is only of order 102 - 104 A, much less than the value of meters typical for visible lasers. However, this turns out to be sufficient for experiments involving hard x-rays.

There are two contributions to the path length difference incurred by the scattered photons. One contribution comes from the finite thickness of the sample. In a reflection,

74

Longitudinal Coherence Condition

Penetration Depth, 0

, +) -~ ,

Thickness, h

D

BRAGG

, /,/.-i- ~ -,

LAUE

" 11 ~

~,.

l~h \1/2 .f h I

I t COh ) \ D

~ ) ,

(.IC~h )

Figure 4. Schematic illustration of the limitations placed on range of scattering angles for which the longitudinal coherence condition is satisfied. Limitations due to both the sample penetration depth or thickness and also the beam diameter are illustrated for both Bragg and Laue scattering geometries.

or Bragg, geometry, PLD depends on the scattering angle, S, and the absorption length in the sample, 0, through the relation PLD = 2 0 sin2S. In a transmission, or Laue, geometry, PLD is given by 2 h sinS tanS, where h is the sample thickness. In a Laue experiment, h is usually - 0 and the small value of lcoh limits one to working at small angles. So, these two expression are essentially identical. In a symmetric Bragg geometry, this contribution to PLD limits the coherent scattering angles to lie in a range away from the plane of the surface, whereas in a Laue geometry, it limits to coherent scattering angles to a range about the forward direction, as shown in Fig. 4.

A second contribution to the PLD comes from the finite diameter of the beam, and is given by D sinS, where D is the transverse dimension of the beam in the scattering plane. This contribution limits the coherent scattering angles to a range centered about the spec­ular beam direction, as also shown in Fig. 4.

It is useful to estimate from these expressions the maximum wave vector transfer, qrnax, that one can achieve in an XPCS experiment under the coherence conditions de­scribed above. For small S, qrnax "" (21t/A.) sinS, and sinS can be estimated from the requirement that lcoh > PLD. A rather general expression can be derived for the limitation on qrnax due to the finite absorption length in a material of charge density p, resulting in

75

qmax = 0.014(pA.leoh)l12 kl. For example, for a sample with the charge density of gold and "I "1"1 2" 4" with A. = 1.5 A , qrnax - 0.6 A or 10 A- for leoh = 10 A or 10 A, respectively. The

contribution due to the finite beam diameter limits qrnax to qrnax = (27t1A.)(leohlD). For a typical beam diameter of 5 11m, this limits qrnax in a Laue experiment to qrnax - 0.008 kl or 0.6 kl for leoh = 102 A or 104 A, respectively. In a Bragg experiment, it limits q to a smalJ range about the specular beam. This means that q's as large as 0.6 kl can be accessed under coherent conditions, even for leoh - 102 A. Thus, XPCS can indeed potentialJy probe slow dynamics on interatomic length scales.

4. Examples

The first observation of speckle using coherent x-rays was made by Sutton, et. al.,1 on the wiggler beamline X25 at the NSLS. They observed Fraunhofer diffraction of 8 keY x-rays passing through a circular pinhole aperture. They also observed 1 a static speckle pattern modulating the (001) Bragg peak in CU3Au, a result of randomly arranged an­tiphase domains. They later studied2 the time dependent changes in the speckle intensity due to nonequilibrium domain growth following a quench from the disordered to the or­dered phase of CU3Au. Subsequently, they obtained some evidence3 for observation of equilibrium critical fluctuations in Fe3AI with XPCS. A CCD area detector or linear Po­sition Sensitive Detector was used in some of their work.

Although the reduced brilliance of bending magnets makes them unsuitable for most XPCS measurements, a bending magnet was used by Cai, et. al,4 on beamline X6B at NSLS, to study the static speckle patterns from gold-coated films of symmetric diblock copolymer films. Their measurements benefitted from the high reflectivity at grazing in­cidence of the gold coated films and from the use of a CCD area detector.

Pindak, et. al,5 have attempted to use coherent x-rays on X25 to study the dynamics of moving CDWs in the one-dimensional conductor, Ko.3Mo03. So far, they have observed speckle modulating the CDW superlattice peak in KO.3Mo03 and observed changes in the speckle pattern when the sample was field cooled. Since CDW's are a displacive phe­nomena, the intensity of the CDW superlattice peaks scales with q2 and only for 29 values > 300 are they strong enough to study. This makes the experiment even harder, as a Si(220) monochromator is required in order to produce the longer leoh, which results in a weaker coherent intensity.

An experiment carried out at on beamline X25 NSLS Dierker, et. al} demonstrated the ability of XPCS to make equilibrium dynamic measurements on a highly disordered material. They used coherent x-rays to measure the Brownian motion of gold colloid par­ticles diffusing in the viscous liquid glycerol. At first glance, this experiment's prospects for success might seem small, since the scattering typical of highly disord~re(i materials, such as liquids or glasses, is notoriously weak. To overcome this, they used the fact that the scattering in the forward direction is strongly enhanced for the large colloid particles. They were able to use XPCS to measure the diffusion coefficient for Brownian motion of gold colloid particles dispersed in glycerol over the range I x 10-3 kl < q < I X 10-2 kl. This extends far beyond the upper q range of visible light scattering, for which qmax - 4 x I (r.1 k I. In addition, since the colloids studied had a gold volume fraction of - I %, they

76

were completely opaque to visible light, and thus could not be studied with visible PCS. They also realized a dramatic increase in collection efficiency by using a CCD detector to measure the scattering from - 105 pixels simultaneously. Since the dynamics only depend on the magnitude of q, the autocorrelation functions measured in all of the pixels in a band of q values can be averaged together. This is equivalent to performing an ensemble aver­age over the pixels as well as a time average for each pixel and reduces the time needed to measure the correlation functions with good statistics by the number of pixels averaged over. With - 104 pixels in a radial band of width 10% in the average q of the band, the reduction is substantial. A schematic illustration of their experimental setup is shown in Fig. 5.

CCO Area Detector at a distance A = 1.25 m from the sample

Coherent Flux = 4x107 ph/sec

Crystal W/B4C Multilayer Monochromator LIE I E = 0.015

coh = t.. ( A.I LIt.. ) = 100

Speckle Diffraction Pattern Temperature Controlled Gold Collioid on 512 x 768 CCD Pixel Array _----::s:-a-m---'p'---l~e'\in~a-v-ac-u-u-m-C-h-a-m-b-e-r---"

.... ~;;.,,;;.~;;.,,;;.~ - (Coolant)

Figure 5. Schematic illustration of the experimental setup used by Dierker, et. al} to study Brownian motion of gold colloid particles diffusing in glycerol.

Dierker, et. al.,6 recorded 1920 images at the rate of one image per second for 32 minutes, of the intensity in a 400 x 400 pixel region on the CCD (see Fig. 5). The time autocorrelation function of each pixel was then calculated and the ensemble average of the correlation functions calculated. The results for two separate 900 arcs corresponding to q's of 3.3 x 10·3 kl and 5.5 x 10-3 kl, with widths of 10% of their average q's and containing 1750 and 5000 pixels, respectively, are shown in Fig. 6 along with fits of single exponential relaxations and their characteristic decay times. The large signal to noise ratio of the data

0.9

0.8

0.7

0.6

~ 05

'" 0.4

0.3

0 .2

0 .1

(b) Q = 3.3 X 10-3

200 l-lm diam. Gold Wire Beam Stop

5000 Pixel Sub-Array

(b) 't = 43.1 seconds

.' • (a) 't = 24.1 seconds ... "#~. ::. ", •••••

50 100 t (seconds)

77

150

Figure 6. Autocqrrelation functions of colloid scattering intensity as collected with a CCD camera for two different wave vectors along with single exponential fits and relaxation times, as indicated. The inset schematically depicts the scattering "halo", the beam stop, and the subarrays which were ensemble averaged over. Curve (b) is offset by 0.15 for clarity. From Ref. 6.

can leave no doubt that they correspond to dynamic x-ray scattering from equilibrium colloid concentration fluctuations. The results are in good agreement with expectations based on the viscosity of glycerol. A deviation of the relaxation rates from a precise q2 dependence was observed. In subsequent measurements, this was found to result from charge interactions between the colloid spheres, resulting in a liquid like structure factor for the colloid and suppression of the dynamics near the peak of S(q).

5. Conclusion

What is the future for XPCS measurements? The work reviewed here clearly shows that XPCS is a feasible technique and can be expected to have wide application to certain classes of systems, especially complex fluids and surfaces. This is particularly true given several significant forthcoming enhancements, such as optimization of the use of area de­tectors for making ensemble measurements and the introduction of more high brightness x-ray sources. With these enhancements, XPCS studies of even atomic liquids will be very feasible, and XPCS will realize its full potential as a unique and important new technique.

6. Acknowledgments

The author has benefitted from many enjoyable and illuminating conversations on this subject with colleagues too numerous to mention by name. However, special thanks

78

are due to Lonny Berman, Eric Dufresne, and Ron Pindak.

7. References

1. Sutton, M., Mochrie, S. G. J., Greytak, T., Nagler, S. E., Berman, L. E., Held, G. A., and Stephenson, G. B. (1991) Nature 352, 608. 2. Dufresne, E., Bruning, R, Sutton, M., Stephenson, G. B., Rodricks, B., Thompson, C., and Nagler, S. E. (1992) NSLS Annual Report, BNL 52371,381. 3. Brauer, S., Stephenson, G. B., Sutton, M., Bruning, R, Dufresne, E., Mochrie, S. G. J., GrUbel, Als-Nielsen, J. and Abernathy, D. L. (1995) Phys. Rev. Lett. 74,2010. 4. Z. H. Cai, B. Lai, W. B. Yun, I. McNulty, K. G. Huang, and T. P. Russel (1994) Phys. Rev. Lett. 73, 82. 5. Pindak, R, Fleming, R. M., Robinson, I. K., and Dierker, S. B., (1992) NSLS Annual Report, BNL 52371, 381. 6. Dierker, S. B., Pindak, R, Fleming, R, Robinson, I. K., and Berman, L. E. (1995) Phys. Rev. Lett. 75, 452.

POLARISATION FLUCTUATIONS IN RADIATION

SCATTERED BY SMALL PARTICLES

Abstract.

E. JAKEMAN Department oj Electr'ical and Electronic Engineer'ing, University oj Nottingham, Unillel'sity Park, Nolling/wUl NG72nD, United Kingdom

The occurrence of polarisation fluctuations in the intensity of scattered radiation is elucidated through a study of scattering by small particles. Sys­tems considered include freely tumbling non-spherical particles and dipoles ncar a dielectric intcrface. Statistical properties Qf the scattered radiation are calculated for Gaussian anel nOll-Gaussian configurations a.nd tile results IIS('<I to model scal.tering by two-scale rOllgh surfaces.

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E. R. Pike 0IId J. B. Abbiss (eds.), Light Scattering 0IId Plwton Correlation Spectroscopy, 79-95. @ 1997 Kluwer Academic Publishers.

80

POLARISATION FLUCTUATIONS IN RADIATION

SCATTERED BY SMALL PARTICLES

1. Introd uction

The research to be reviewed in this paper has polarisation as its common theme. However, it derives from a number of different programmes of work. These include studies of non-Gaussian polarisation fluctuations in scattered radiation, polarised scattering by rough surfaces, and ir emission polarisation. The first mentioned programme is relevant to many environmental sensing problems but currently its principal objective is the enumeration of objects which are not resolved by the illuminating radiation pattern. The second mentioned programme relates to the long- standing problem of scattering by two-scale rough surfaces, especially backscatter at grazing incidence. This is particularly reI event to maritime radar systems whose performance is limited by the presence of unwanted returns from the sea surface. The polarisation properties of microwave sea-echo have remained largely unexplained despite over fifty years research I The third programme referred to above concerns the ability to discriminate between man-made objects and their environment by exploiting differences in their infra-red emission polarisation signatures. In all of these programmes simple particle scattering models have provided new physical insights and suggested new areas of work and potential applications. I choose to develop the present review mainly through discussion of the more recent

81

work on non-Gaussian statistics, although applications to surface scattering will also be covered briefly.

In the next section a brief review of scalar non-Gaussian scattering theory will be given, including reference to more recent developments. An extension of the theory to include polarisation fluctuations will be presented in section 3 together with a summary of predicted behaviour for the case of scattering by an assembly of spheroidal particles. Section 4 will briefly consider the case of scattering by particles near an interface and demonstrate how enhanced backscattering can be generated by multipath effects. In section 5 it will be shown how scattering by a two-scale surface can be modelled using particles to represent the small scale structure riding on large-scale undulations. Concluding remarks appear in section 6.

2. Non-Gaussian Scattering

It is well known that when a variable is the sum of a large number of independent random contributions then, by virtue of the Central Limit Theorem of classical statistics, it will be Gaussian distributed irrespective of the nature of the individual contributions. When sound or a polarised component of electromagnetic radiation is scattered by a large number of objects, or by many independent scattering centres on a single object, then the scattered field may be represented in a similar way by the sum of a large number of random phasors. This random walk picture predicts that the othogonal components of the resultant vector are independent Gaussian variates. In the optical case, the brightness of the scattered intensity pattern then exhibits a speckled appearence and has a contrast of unity, provided the scatterers are distributed over a region which is much larger than the wavelength. When only a few scatterers are present, however, the Central Limit Theorem cannot be invoked and the statistics of the scattered wave fielQ is no longer Gaussian. Important insights into the behaviour in this regime can be obtained by considering the predictions of the random walk model with a finite number of steps[ 1].

82

In the scalar version of the random walk model, the scattered wave field is represented by the sum of N random phasors: .

N

E = Ian exp(ilPn) (1) n=!

where the amplitudes, a, and uniformly distributed phases, lP, are assumed to be statistically independent variables. The probability density of the

intensity, 1= IEI2 , was given early in the century by a number of authors

including Lord Rayleigh[2] and may be expressed in terms of an integral over averaged Bessel functions as follows

co N

PN(I) = t f uduJo(u.JI)I1 (Jo(uan )) (2) o n=!

This quantity can be evaluated numerically for the case when the amplitudes are identical, by converting the integral into a sum. Results for small values of N are shown in figure 1. The shape of the curves is clearly a useful signature of the number of scatterers present,at least for values below N=6. Above this value the distribution rapidly approaches the exponential shape expected in the Gaussian limit.

6

o 2 4 6 8 I 10

Figure J. Distributions of intensity for a small number, N, of scatterers

In some situations of interest the number of scattering centres may vary with time. For example, particles in suspension will diffuse in and out of an illuminating laser beam leading to a Poisson distribution of N. Averaging

83

equation (2) over these fluctuations and assuming the individual amplitudes are statistically identical leads to the formula

00

(PN (/») = t f uduJo (u.Ji) exp[ -N(l- (Jo(ua»))] (3) o

This has also been evaluated previously for various values of the mean

number ofscatterers, N [3]. Because of its analytical simplicity, the normalised second intensity

moment has often been considered to be a useful measure of deviation from Gaussian statistics and therefore of the number of scatterers present[l]. This quantity can be calculated from equation (3) without further approximation and takes the form

(/2)/(/)2 =2+(a4 )/N(a2)2 (4)

which reduces to the Gaussian value of 2 when the mean number of scatterers is large. It is clear, however, that being an average over the probability density (3) it cannot provide the same degree of sensitivity to scatterer number as the full distribution. Recent work has therefore sought to identifY other characteristics of the full distribution which might provide a simple but more reliable measure of this number. One promising possibility is the behaviour near /=0. Calculations suggest that this exhibits a remarkably similar variation with mean scatterer number irrespective of the choice of number or amplitude fluctuation models[ 4]. A generalisation of two approximations devised by Rayleigh[2] and Berry[5] can be used to demonstrate this behaviour. In equation (3) set

exp[ -N(1- (Jo(ua)))] == exp(-N) + [1- eXP(-N)]exp( -Nu2(a

2) J x

[1- exp( - N)]

( N2u4(a2/ exp(-N) Nu 4 (a 4 ) J

x 1- + (5) 32[1- exp(-N)]2 64[1- exp(-N)]

This is valid for all values of N when u is sufficiently small and for all

values ofu when N is sufficiently large. The integral can now be evaluated exactly and leads to the result

84

(I)(PN (0))=[1-eXP(-N)]3(1+ ia4

) 2J 2N(a 2 )

(6)

which depends only on the mean number of scatterers and the normalised fourth moment of their amplitude fluctuations. Plots for various values of this latter parameter are shown in figure 2 and provide an alternative measure of deviation from Gaussian statistics.

(I)(PN 0))

InN

-2 -1 1 2 3 Figure 2. Probability of zero intensity for various degrees of amplitude fluctuation

3. Polarisation Effects

Polarisation can be accommodated in the random walk model via the properties of the individual scattering amplitudes, a. Mueller matrices provide a general description of the transformation from the initial to the final polarisation state, with Stokes parameters being the measurable intensity descriptors used to characterise the scattered field. For simplicity, in the present paper consideration will be confined to the statistics of the linearly polarised intensity fluctuations. Some of the quantities calculated cannot be measured directly but can be deduced from measurements of fluctuations in the Stokes parameters. Formulae for the statistics of the Stokes parameters are given elsewhere[ 6].

85

The simplest polarisation sensitive statistic, apart from the mean intensity of the scattered field, is the cross correlation function of linearly polarised intensities

(7)

where

(8)

Here, the labels i,J refer to the initial to final polarisation configuration. For example with initially s-polarised radiation, the s-polarised scattered intensity, i=[s-+s], might be cross-correlated with the p-polarised intensity,j=[s-+p] and so on. Apart from the labels, formulae (7) and (8) ressemble the scalar results given previously, with the g and f terms being Gaussian and non-Gaussian contributions respectively.

In order to develop some insight into the possible significance of polarisation fluctuations, it is instructive to consider the predictions of the above formulae for the case of non-spherical Rayleigh scatterers governed by the dipole approximation

(9)

where e defines the polarisation state of the field scattered by a single particle with polarizability tensor a, and E is the linearly polarised

incident field. For example, in the case of a spheroid, with its orientation parameterised by the angles ¢ and If! and polarisability tensor elements a ij = O,i::l;j, a)) =a),a 22 =a 33 =a 2 , the [s-+s] amplitude IS given

by[7]

ass = EJ(a) - a 2 ) sin2 If! cos2 ¢ + a 2 (10)

The statistical properties of the s-copolar intensity fluctuations due to variations in the orientations of the particles can be calculated directly from this result. Thus if the particles are randomly oriented[7]

p(ass ) = [2JEs(a) - a 2 )(ass - a 2Es)rl a 2Es <ass < alEs (11)

= 0 otherwise

86

and the equivalent non-Gaussian enhancement factor f appearing in equations (7) and (8) depends only on the polarisability ratio r = a) / a 2

f -_ 5 35r4 + 40r 3 + 48r 2 + 64r + 128 7 (3r2 +4r +8)2

(12)

The single particle distribution extends over a finite range and exhibits an infinite asymptote. This ch~racteristic structure shows up clearly in the full

distribution (3) for a varying number of particles provided that N is small, because the dominant single particle term contributes via a delta-function. The behaviour of f with polarisability ratio for s-copolar scattering, figure 3, is also typical in being the same for two values of r. This characteristic can also be seen in the graphs for p-copolar scattering shown in figure 4, although there is an additional variation with scattering angle e in this case. Nevertheless, it is clear that particle shape information is contained in the polarisation fluctuations when the number of scatterers is small.

-3 -2

2.5

2.25

2

1.75

1.5

-1

f

In r

1 2 3 4 Figure 3. s-copolar non-Gaussian enhancement factor versus polarisability ratio.

87

Figure4. p-copoJar non-Gaussian enhancement factor.

Although it is possible in principle to determine the magnitude of the non-Gaussian (/) term in equation (7) experimentally if N is small, number fluctuations and low return intensities may well make measurements significantly more difficult than in the Gaussian, large N, regime. Unfortunately, the Gaussian ( g ) factor in equation (7) is zero for most polarisation configurations and carries no information. Calculations show, however, that in a few situations g is a function of the polarisation ratio and could therefore provide information on particle shape. Cross­correlation of p- with s-copolar intensities is such a situation plotted in figure 5. In practice, such a quantity is not measurable and it would be necessary to cross-correlate the appropriate combination of Stokes parameters for a particular incident field configuration.

88

Figure 5. Gaussian fluctuation parameter.

4. Particles on Surfaces

In the last section the polarisation fluctuation properties of radiation scattered by small non-spherical particles was discussed. In the case of a freely diffusing population of spherical Rayleigh scatterers, no polarisation fluctuations would be generated, of course. However, if the particles move in the neighourhood of an interface between two media with differing refractive indices, then polarisation dependent effects occur even for the case of spherical Rayleigh scatterers due to the presence of image scatterers in the the second medium. This effectively creates a dumbell shaped composite particle with dipole strength related to the particle­interface separation. Treatment of the perfectly reflecting planar surface case is fairly straight forward using the standard method of images. The more general planar surface case requires more care. Videen and co-

89

workers[8] describe an elegant approach to the derivation of the field in the Fraunhaufer region above or below the interface.

In this approach the scattered field is represented as the sum of four contributions. The first is the radiation scattered by the particle out of the directly incident beam. The second is the radiation scattered by the particle out of the beam which has been reflected from the surface (reflection coefficients Rs,Rp). The third is the radiation scattered out of the direct

beam by the particle but then rescattered by the surface and the fourth contribution is the radiation scattered by the surface both before and after scattering by the particle. The results obtained in the plane of incidence for spherical particles in the dipole approximation are

where /(B) = 2M cosB, i,s label the incident and scattered directions with respect to the normal to the surface, k is the radiation wave number and d the distance of the particle from the interface. When d-XJ, these formulae reduce to results previously obtained from exact solutions of Maxwell's equations for radiating dipoles on an interface[9]. In the latter calculations it was assumed that the field at the interface generated by the incident plane wave was not perturbed by the presence of the dipoles which simply reradiated the surface field in the presence of the interface.

Formulae (12) and (13) could be generalised to non-spherical Rayleigh scatterers and to directions out of the plane of incidence. Results have

been obtained for the case d-XJ [9]and an application will be discussed in the next section. In the case of spherical particles above an interface one

parameter which could vary is the distance d. This model could be used as an approximate representation for particles of various sizes resting on an

interface, for example, or it could be a straightforward representation of a distribution of scattering centres near a surface. In any case, it is

instructive to compare a plot of the scattered intensity versus angle when d is fixed (figure 6) with that for a situation where an average is carried out over a distribution of values of d (Perfect conductivity is assumed in these

90

examples). Figure 7 shows p-copolarised scattering assuming that

-1.5 -1 -0.5

J 3.5

3

2.5

2

1.5

0.5 1 1.5 Figure 6. p-copolar scattering by particles above an interface, incident angle 30°, d-iI.

logJ

0.5

0.45

-0.5 0.5 1 Figure 7. As fig.6 but with a gamma distribution of particle-interface separations.

d is gamma-distributed with mean equal to the wavelength and relative variance of about 3. Note that the interference fringes visible in figure 6 have averaged out but that in addition to a strong forward scattering peak, a significant maximum in the backscattering direction is predicted. This is a manifestation of the multiple scattering phenomenon known as enhanced backscattering which arises from the constructive interference of radiation

91

which has traversed reversible paths between the source, scatterer and detector[10]. In the present configuration it arises from multipath effects, ie from the existence of additional paths involving the surface, rather than multiple scattering between the particles themselves. It may be significant that early experimental work on scattering from particles on a surface generated intensity distributions remarkably similar to figure 7[ 11]. The shape and size of the enhanced back scattering peak carries information about the distribution of the distance d, which is equivalent to the size distribution of the particles in one interpretation given above.

5. Two-,cale surface scattering model

The scattering of electromagnetic radiation and sound waves from rough surfaces is a problem of longstanding interest with a wide range of applications at all wavelengths throughout science and engineering. For many years the principal theoretical options for tackling surface scattering problems were perturbation theories, suitable for surfaces with height fluctuations which are small compared to the radiation wavelength, and physical optics or Kirchhoff approximations, suitable for smoothly varying surfaces with low slopes and large radii of curvature. As computer power has increased during the last decade, considerable progress has been made in probing the range of validity of these methods using numerical techniques and it has become clear that the remaining problems in the subject lie in the area of polarisation effects at high incident angles. At optical frequencies, very little data exists near grazing incidence since pulsed operation is required to discriminate against unwanted scattering from sample edges etc. However, there is a large and increasing body of radar data on this regime which exhibits polarisation effects not explained by conventional theories. Microwave sea echo has presented a particularly interesting challenge over the years since it might be expected that the sea surface can be modelled as a random process but characterised by more than one scale of roughness. Indeed, two- scale surface scattering models were developed more than thirty years ago[12] and have been refined by many authors over the intervening years. However, their predictions for

92

grazing incidence geometries have failed to agree with observed trends in experimental data. A generalisation of the results given in the last section can be used to develop further insight into this intractable problem[9, l3].

A simple two-scale scattering model can be constructed from the geometry considered earlier as follows. It will be assumed that the larger scale ocean structures (gravity waves, swell) are smoothly varying so that the physical optics approximation can be used to determine the surface field exited by the incident radiation. If the tangent plane approximation is applied, then the large scale structure is effectively treated as a collection of flat facets. The small scale roughness (capilliary waves) which are comparable to or smaller than the radar wavelength are modelled as small non-spherical particles (dipoles) embedded in the surface. These do not perturb the incident field but re-radiate in the presence of the ocean-air boundary. As indicated in the last section, this approach requires a generalization of the calculations carried out previously for Rayleigh scattering in the plane of incidence by spherical particles above a planar interface, to the case of Rayleigh scattering by particles of arbitrary shape above an interface tilted out of the plane of incidence. The final predictions are obtained by averaging over particle orientation, spatial distribution and surface tilt[ 13].

Comparison of the predictions of this model with the experimental data of Daley et al(for references see [9,13]) is shown in figures 8 and 9. Figure 8 shows a plot of the measured ratio of vertical (P) to horizontal (s) copolarised intensities in the backscattering direction for two sea states, compared with the predictions of Rayleigh-Rice (perturbation) theory (a) assuming infinite conductivity for the sea, (b)assuming the measured complex relative dielectric constant of sea water, 49-i37 and with the present theory (c )assuming a Gaussian slope model with modest mean square slope (d,e)scaling the rms slope with scattering angle to take account of shadowing of lower slopes near grazing incidence. Figure 9 shows the fluctuation parameter (v= 1 If where f is defined as in equation (7)), for the observed non-Gaussian, vertically copolarised intensity fluctuations in another set of experiments reported by Ward[14], compared with the predictions of the present theory. The parameter values leading to (d,e) of figure 8 were used for the theoretical curves, which appear to bound the data.

-.s .e -~ p

(al 10S

• :a

102

j • V :a .......

1 101 v

I I

100

I 0 10 20 SO 40 50 60 70 SO 8D

8D

Figure 8. Theoretical prediction of particle model with experimental data (d:5-8ft waves, e:16-25ft waves)

1.0

0.1

0.8

0.7

0.8

0.5

004

0.8

0.2

0.1

0.0 0 10 20 30 40 50 60 70 80

eo

93

90

Figure 9. Theoretical predictions of particle model with the angle dependence of the shape parameter v=l(ffound by Ward [14]

94

6. Concluding Remarks

In this paper a brief review has been given of polarisation fluctuations in the context of particle scattering systems. It has been shown that simple dipole models continue to provide useful insight and the early theoretical predictions suggest that polarisation fluctuations in both Gaussian and non-Gaussian scattering configurations could provide a useful source of information regarding particle number, shape and size. Further theoretical work is required to generalise existing results for scattering by particles near an interface and to establish the magnitude of polarisation fluctuations in the case of larger particles. Correlation properties of the fluctuations are also worthy of investigation. The most pressing need, however, is for experimental measurements.

Acknowledgements

The work reported here was supported by the Defence Research Agency, Malvern, UK, and I would like to thank colleagues both there and at Nottingham University for stimulating and encouraging my continuing interest in this area of research.

7. References

1. J akeman, E. (1984) Speckle statistics with a small number of scatterers,Opt Eng. 23, 453-461.

2. Lord Rayleigh, (1919) On the problem of random vibrations and of random flights in one, two or three dimensions, Phil.Mag. 37,321-

347. 3. Pusey, P.N. (1977) Photon Correlation Spectroscopy and Velocimetry

eds. H.Z. Cummins and E.R. Pike, Plenum Press, New York. 4. Watts, T.R., Hopcrafi, K.I. and Faulkner,T.R. (1997) Single

measurements on probability density functions and their use in non­Gaussian light scattering, JPhys.A. in the press.

5. Berry, M.V. (1973) Phil. Trans. Roy. Soc. 273,611-654.

6. Bates, AP., Hopcrafi, K. and Jakeman, E. (1997) Non-Gaussian fluctuations of Stokes parameters characterising scattering by small particles, to be published.

7. Jakeman, E. (1995) Polarisation characteristics of non-Gaussian scattering by small particles, Waves in Random Media 5 427-442.

8. Videen, G., Wolfe, W.L. and Bickel, W.S. (1992) Light scattering Mueller matrix for a surface contaminated by a single particle in the Rayleigh limit, Opt. Eng. 31, 341-349.

9. Jakeman, E. (1994) Scattering by particles on an interface, J.Phys.D 27, 198-210.

1O.Kuga, Y. and Ishimaru, A (1984) Retroreflection from a dense distribution of spherical particles, J.Opt.Soc.Am.A 1 831-839.

95

1 1. Moreno, F., Gonzalez, F., Saiz, lM., Valle, PJ. and Jordan, D.L. (1993) Experimental study of copolarised light scattering by spherical metallic particles on conducting flat substrates, J.Opt.Soc.Am.A 10 141-149.

12.Wright, lW. (1966) Backscattering from capilliary waves with applications to sea clutter, IEEE Trans. Antenn. Propag. 14 749-754.

13 . Jakeman, E. (1994) A particle model for scattering by two-scale rough surfaces, PIERS 1994 Kluwer Academic Publishers (CD-ROM).

14.Ward, K.D. (1982) A radar sea clutter model and its application to performance assessment, lEE Conference Publication 216 (Radar '82) 203, Institution of Electrical Engineers.

SURFACE LIGHT SCATTERING SPECTROSCOPY

Abstract

J. ADIN MANN, JR. DeWI1·tment 0/ Chemical Engineering, Cll.qe Western Reserve University, Clevelllnd, Oil 44106-7217, USA

The grating method of surface light scattering spectroscopy provides a reference beam for homodyne detection of scattered light that can be used to measure accurately the response function of a free liquid interface. This enables the determination of interfacial tension and other parameters that include volume phase viscosity or visco-elastic parameters of monomolecular films spread at the interface or produced by reactions. The focus of this essay is on the measurement science required to construct a compact, robust instrument for the determination of the spectrum of surface fluctuations accurately. In addition various applications are mentioned that include the classical monolayer work and the extension to properties of high temperature melts.

1. Introduction

Classical surface chemistry has a long history dating· back to "spilling oil on troubled waters." The reader can obtain some feeling for the scope of the field from Adamson's (1990) book. A fundamental measurement is that of the surface tension or interfacial tension between two fluids. A short introduction to surface thermodynamics will serve as background for the discussion of interfacial light scattering spectroscopy.

Following Hansen's (1962) development, the excess free energy can be written in two ways so as to obey the phase rule. The Gibbs-Duhem equation for the flat interface with surrounding phases suggests that the independent intensive variables are T, p, {J.Ii}. However, the phase rule for such a system requires two fewer independent variables. Hansen and Cahn (1979) showed how any two of T, p, and the set of Chemical potentials {~;} can be eliminated rigorously from consideration. Firstly, consider the liquid-vapor interface with adsorption; the system is composed of multiple components but only two

phases. Then the surface excess free energy F, can be written as

97

E. R. Pike and J. B. Abbiss (eds.), Light Scattering and Photon Correlation Spectroscopy, 97-115. @ 1997 Kluwer Academic Publishers.

98

c dF = -SdT + L Ili dn~ + ydA (1)

i=2

where S is the excess entropy, {n~} is the set of excess mole numbers and y is the surface tension (specific excess free energy). The Gibbs-Duhem equation is

c o = sdT + L fidlli + dy (2)

i=2

where the excess is fj = n~ / A and s = S / A is the excess entropy per unit area. Note that neither the .pressure nor the chemical potential of the major component, Ill' appear as intensive, independent variables. Secondly, consider the liquid/liquid interface. Following Hansen (1962), it is convenient to set the excess mole numbers of the two majority components (solvents in phase 1 and 2) to zero. Thus, the Gibbs-Duhem equation becomes

c-0= sdT - rdp + L f,df.J, + dy (3)

1=3

where by choice of convention the independent variables are T, p, {Ill' 114' ... ' Ild. Note that the two majority components, Ill' 112, are not considered as independent variables. The coefficient t is an excess volume per unit area first defined by Hansen. See Turkevich and Mann (1990), Hansen (1962) and Calm (1979) for many additional details.

These formulas suggest that the determination of the specific surface free energy

(:~) = y is fundamental to understanding the chemistry and physics of systems of

interfaces. Moreover. composition temperature and pressure must be taken into account systematically and these formulas suggest what must be done for multi component systems of two phases. The extension to multiple phases is direct. The implicit assumption built into the Hansen-Calm theory of surface thermodynamics is that the interface is flat. The extension to curved surfaces is wickedly difficult. Fortunately, the surface· spectroscopy discussed herein requires the analysis of capillary waves of only very small amplitudes and relatively large wave lengths, C/AriPPle <10-4; the Hansen - Calm theory is sufficient.

Certain interfacial systems exist such that ultra-thin films occupy the interfacial region and remain out of equilibrium with the surrounding phases; the chemical potentials of the components that make up the film are not uniform in each phase as is required in the derivation of equations (1, 2, 3). Examples include spread monomolecular films of lipids. liquid crystal molecules and various polymers (Adamson. 1990; Ulman, 1991). Another class of systems that are not yet well studied include high temperature melts such as liquid metals and metal oxides. In particular. liquid metals suffer

99

segregation of components from the liquid phase to the surface as well as reactions with components of the vapor phase. There are only a few careful studies of oxide formation on liquid metals using ultra-high vacuum techniques and the methodology of surface science. See for example Ricci and Passerone (1993) and citations therein.

In each of these examples, the determination of the surface tension is important. But it is of more interest to study the fluctuation spectra of the ultra-thin films directly. Surface light scattering spectroscopy is a useful tool for this purpose but the theory for interpretation is more complex and will only be outlined herein. See Mann (1985) and Langevin (1992) and references therein.

The the appropriate generalization of equation (1) for constant temperature and composition is

dF = AoL (iijduij

ij

(4)

where (iij is the surface tension tensor and l1;j is the surface strain tensor. Again (iij =

8F / Ouij as expected from similar considerations for 3D materials (Landau and Lifshitz,

1970). In general for small strain, the constitutive equation for insoluble mono layers can be written as

(5)

where x = Yo - y. Yo is the surface tension of the monolayer free interface and y the surface tension with the monolayer at some known density r. Recognize that most often the reference state for the monolayer is some surface pressure that corresponds to the surface density of a compressed film. The stress-free state (x = 0) is usually not a convenient reference state in this situation. Equation (5) simplifies as a result of symmetry considerations (Mann er al., 1987); two coefficients are required for the isotropic and hexagonal classes. three for square, four for rectangle, and five for the oblique two­dimensional classes.

The spectrum determined by light scattered from interfaces is that of very small amplitude capillary waves generated by thermal fluctuations in the volume phases that couple to the surface. Consider the surface to be represented by C; = f(x,y) with the support plane defined by <C,> = 0, indeed, C; is a stochastic variable. The root mean

square elevation J < C;2 > can be determined experimentally to be roughly .3nm for

water. See, for example, Mann (1985) for a discussion in the context of surface chemistry. The spectrometer to be described determines the correlation function

< C;q (O)C;q (t) > where C; is expanded as a Fourier series to be described later. (equation

(20».

100

For constant temperature and composition equation (1) is M' = y6.A and

(6)

[n view of the Fourier series representation of S, a standard calculation using the equipartition theorem yields

(7)

where the higher order tenns including a bending contribution, and for sufficiently thin films, e.g. soap films. a contribution from the free energy of dispersion forces. Since the

irradiance of the scattered light scales with < s~ >, we expect that the signal to noise

ratio will be favorable for high temperature, low surface tension and small wave numbers.

These tluctuations have been studied experimentally in the range of 200 cm·· to about 2000 em-I using light scattering methods. Reflectivity in the x-ray region has been studied at much larger wave numbers to yield interfacial thickness. These points have been summarized by Mann (1985). Also see the fine compilation-of papers edited by Langevin (1992).

The optical set-up will be outlined in the next section along with a summary of the analysis of operation.

2. Analysis of the Transmission System

The optical arrangement discussed in this section originated with a paper by Lading, Mann and Edwards (1988, 1989) referred to herein as LME where we analyzed a very general arrangement shown in Figure (2) of LME (1989). A previous paper by Edwards, Sirohi. Mann, Shih and Landing (1982) contains fonnulas that will be quoted. The spectrometer is designed using the tools of Fourier optics as outlined. for example, by Goodman (1968). It is easy to design spectrometers that stay within the approximations required for validity ofGoodman's methodology.

I believe that a major improvement that is emerging with implementation of Lading's ideas is the miniaturization of the entire optical system. It is now clear that this can be done without sacrificing precision and accuracy of the results. Now that bright, diode lasers are becoming available, the optical signal to noise ratio is satisfactory. Modern optical fiber technology can also be used to advantage. See Meyer, Tin, et al. (1994) and Meyer. Tschamuter, et al., (1994).

101

There are two modes of operation that are useful in practice: detection of the transmitted. diffracted laser beam. or the detection of the diffracted beam reflected at the interface. Recognize that the same analysis scheme works for either mode. The implementation, however, requires consideration of surface slosh due to environmental noise. These practical problems will be discussed later.

The optical arrangement is as follows:

x --- Laser

"J" --- Grating, g _- LI

I ____ L2

I Image Plane: u(-)

~ u(+)=gsu(-)

'(>- To Correlator

• ~ Detection Plane

L3

Figure I. Outline of a transmission system. A diverging lens can be used down stream of L3 so as to funher separate the first-order beam from the zero-order beam. Diaphragms may be necessary between L I and L2 and after L3 to block stray light.

It is assumed that the deviation of light beams away from the optical axis is small (paraxial approximation). Moreover, the instrument is adjusted so that each object is at the focal plane of the appropriate lens. A complex coefficient is a constant under these conditions and will be ignored in equation (8), Goodman (1968). Under these conditions the scalar field at the Fourier plane, lit. is

+00 .(21t _ ) _ f f -I U xr "0

uf(xf,Yr) = uo(xo,yo)e dxodyo (8)

and is in the form of a Fourier transform, Uo (xo' Yo) is the field on the object plane (e.g. dcf 27t

the grating) and q = H xr is the spatial frequency (wave number) on the Fourier plane;

A. is the wavelength of light, f the focal length of the transforming lens and Xf the position on the focal plane. The z-axis is the optical axis indicated in Figure (1). In short hand

(9)

We start the analysis by considering the field u(·) just before the fluctuating interface. Note that Ll and L2 do sequential Fourier transforms of the optical field to form an image of the grating on the surface. Just in front of the surface

102

U(-l = Uo g' (10)

where Uo is the Gaussian field at the grating so that

(11)

where r = <Ix - 51.0 12) on the image plane, CYg is the Gaussian beam width measured at the

grating and M the magnification factor. I assume that the input beam at the grating is well collimated so that the phase front at the surface has a large radius of curvature. The more general case has been analyzed by Meyer et al. (1997).

Remark: The gratingfonction g' on the image plane may be modified by a spatial filter placed at the Fourier plane of Ll. The image formed on the interface located at the image plane of Ll plus L2 may be magnified by M

The electric field transmission function for the grating is taken to be g defined as

iap cos(kg xl g = goe (12)

where the grating oscillations are along the x axis perpendicular to the optical axis. If the grating is a pure phase-grating go is a constant in the grating plane. More generally

~ = ~o (1 + TJ(x, y)) (13)

so that if ~ = 0 and TJ = a" cos (kg x), g is a perfect amplitude grating.

Remark: It is easy to produce amplitude gratings of sufficiently small efficiencies (aA is small) for configuring the optical train of the spectrometer. However. the intensity at the Fourier plane of Ll is distributed into many orders of diffracted beams. In contrast, we have been working with high-quality phase gratings for which only the first order beam is used; the second order beam is much less intense and may not be observable. The higher order beams are not observed. In the remainder of the paper we will assume that the phase grating of the instrument follows formula (/2) with 1'/ = 0 and that the only orders

are 0, = I. The functional form ofu(') for 0 < ~ < < I is

(14)

103

so that three spots will be observed on the Fourier plane of L3 (the detection plane). From the view point of the classical analysis of homodyne operation. the first term of

equation (14). gouoo e 2(Mag )2 , is that of the zero order beam which generates the scattered kc

±i-x light from the surface. The two terms involving e M represent the two first order beams on the focal plane of L3 that act as reference beams for the scattered light. Here we follow the analysis in LME.

Consider the rippling interface of the liquid to be a grating represented by

gs = gos(l + ,,(x, y, t»eikA.t;(x,y,t) (15)

where k. = (2;), A. is the wave length of the incident light, I; (x,y,t) is the ripple

amplitude and is a random variable, 11 (x,y,t) represents the variation of transmission ( or reflection) in the surface. Since k.1; < < 21t and Tl ~ 0, the time varying grating of the interface is

gs = &'s (1 + i kA, ~ (x, y, t» (16)

Remark: Indeed, recognize that surface fluctuations driven by thermal fluctuations are extremely small. The free surface formed between two phases of a one-component liquid is isotropic of uniform composition so that 11 (x,y, t) = 0 everywhere. However, a

therma/ly generatedfield of random ripples covers the surface so that r; (x,y,t) is not zero. When the spectrometer is run in reflection mode a factor 2 r; (x,y,t) ohtains to account for the path difference between transmission and reflection, and -gos obtains. In addition, for most liquids the reflection coeffiCient is small so that only a fraction of the incident light (- 4% for water) will be collected by L3 onto its Fourier plane for detection. However, the basic formulas do not change so that the analysis of the reflection system fo/lowsfrom the analysis of the transmission system discussed herein. See Meyer (1997).

For transmission, the field following the surface is

(17)

which gives

104

(18)

On the Fourier plane of L3 that is also the detection plane

(19)

and uses an obvious notation for the Fourier transforms of the four terms in equation (18).

We expand ~ (x,y,t) as a Fourier cosine series so as to keep the representation real:

~(x,y) = LSq(t)cos(q ·(x -Xq(t))) ii

(20)

where the amplitude. 1;cj (t) ,and the phase, x ij (t), terms are random variables of zero

mean and uncorrelated. see LME and Edwards, et al (1982).

Remark: Consider the phase function tp(q, t) = - q. Xq (t) where Xq (t) points to a

place and time where a thermal impulse changes the local curvature of the interface and 21t

generates a wave with amplitude 1;q and wavelength -. Formula (20) can be regarded q

as representing a rough surface that changes in time and space with a spectrum of waves propagating in random directions generated by pressure fluctuations.

Consider ii" the zero order spot on the detector plane and ii), the first order spot.

, 2 2 2 • _ 2 (M )2 -Z(Mag) (Qx+Qy)

U, - 1t cr g e (21)

which is a Gaussian beam centered on the optical axis and will be blocked. However,

(22)

where Q" Qy are computed on the Fourier plane of L3 and represents a Gaussian beam k

displaced from the optical axis by ~. The form of equation l22) is the kernel of the

instrument function, equation (23). Notice that the argument of the exponential (equation

105

k 22) can be written in terms of dimensionless groups by factoring out ~. Then the

sharpness of resolving q is determined by kgcrg measured at the phase grating; the magnification factor does not determine wave number resolution. Define Ng = kgcrg that can be considered as the number of fringes projected on the interface.

Each cosine term in equation (20) can be written as a sum of two complex exponentials as was done for the stationary phase grating, equation (12) and (14). Only capillary waves that diffract beams to match the reference beam will modulate the intensity detected by the photodiode. The optical system thereby selects a range of ripplon wave numbers for analysis. This operation is specified in equation (23).

Integration over the detection plane (the zero order spot, u1 ' is blocked along with one of the first order spots) and averaging the correlation of the photo-current gives

(23)

A group of coefficients that scale the result have been dropped . e.g. g." g.,., u.., that are not essential for the purpose of this paper. The amplitude-correlation function is

~(q;r) = < C;q (t )C;q (t + t) >. Consult Edwards et af. (1982), Appendix B, for the

calculation of the expected value of the surface fluctuations used to get (23).

Remark: In a practical spectrometer. the pinhole on the detector plane accepts only the 1st order heam. In practice. the aperture must be larger than the radius of the I/e2 point of the Gaussian beam. However. it must not be so large as to admit the fringe of the zero order beam or the 2nd order beam (which is of small intensity or zero).

3. Notes on Implementation of a Transmission System

Several implementations of the transmission and reflection systems have been constructed during a joint project between NASA-Lewis Research Center and CWRU. Additional detail of the instruments is in the process of being published Meyer (1997). A few remarks are appropriate.

Noise free laser operation is critical, obviously, since homodyne detection is used. For example argon ion lasers can have substantial noise at mUltiples of line frequency that can be difficult to reduce by signal conditioning. In recent years we have used laser diode modules that can be run from well-conditioned power supplies or by battery. However, the irradiance of the beam is small enough at roughly 10-20 mW that the optical signal to noise ratio suffers.

106

We find that a set of Spindler-Hoyer Microbench components allows the construction of the system shown in Figure I easily and accurately. The lenses Ll and L2 have focal lengths of 60 mm while L3 is of 120 mrn. Obviously other combinations may also project a sharp image of the grating onto the surface of the liquid, the magnification need not be unity. Note that the optical path can be made even smaller and suitable for operating on small tables in chemical laboratories.

We recommend the use of a phase grating (or a set of phase gratings) and have had excellent results with the grating produced by RiS0 (Lading, 1996). Note that the gratings must be relatively inefficient but the self-beat term must be "overpowered" by the reference beam but not so intense that signal to noise ratio is lost. The Ris0 gratings used by us allow a selection of grating efficiencies to optimize the signal to noise ratio.

The detector acts as a second noise source if not carefully controlled. Our group uses photo diodes exclusively and will often run by battery power. The output of the photodiode. conditioned by a pre-amplifier followed by a battery-driven, band-pass amplifier made by EG&G. is the input to the correlator. Note that the detector and laser components are sufficiently small that the size of the spectrometer is determined by the optical system and can be folded if necessary.

In our laboratory the conditioned signal is analyzed by a Brookhaven Instruments BI8050 correlator card that provides for an analog input and computes the correlograms including the zero channel. or we use a home built correlator designed by Edwards. His correlator allows for the possibility that slosh will drive the reflected beam off of the face of the detector. When this condition is detected, the processing interrupts, and the corrupted data string is purged. The correlator restarts when the beam is again positioned in the detector. The efficiency of the correlator is affected but the correlograms are accurate. Software controls the correlator and displays the correlogram as it builds over time by the averaging process. Data tiles are generated for further processing by a suite of programs that tits correlogram data to various models, corrects for the instrument function, transforms correlograms to power spectra and computes estimates of various parameters such as the surface tension and various surface visco-elastic coefficients.

The coupling of building vibration to the surface can easily destroy the signal. This problem is minimized but not eliminated by working in transmission mode. The depth of field of the imaging system is sufficient that small variations of the elevation will not affect the correlogram. As mentioned, the concern is that sloshing may deflect the reference beam outside the sensitive area of the detector. This is a major problem with operation in reflection mode. Recently Meyer, Lock, et aI., (1997) have demonstrated an optical arrangement that minimizes slosh by controlling the optical lever effect.

107

4. Interpretation of Surface Light Scattering Correlograms

The raw data obtained from an experiment is a digitized representation of the autocorrelation function of the filtered photocurrent and is referred to as a correlogram. {R, (n.6t)} , n = 0, 1,2, ... The interest is in the quantities that can be extracted from R, that characterize the material coefficients that describe the dynamics of the interface. Refinement of optical and electronic processing provides {R,(n6t)} of high precision which sharpens the possibility of estimating material properties of interfaces.

Obviously, the surface tension is one such parameter as was discussed in the introduction. Other constitutive parameters are the densities and viscosities of surrounding phases within micrometers Df the interface and the visco-elastic parameters that represent the dynamics of the interface. How many of these parameters can be determined from correlograms? How precisely and accurately can they be determined? The following remarks are offered to answer or point to answers to these questions.

Consider a representative correlogram taken with the instrument patterned after Figure (1), see section (3). Data was collected on pure water in a Langmuir trough located in a class 100 clean room on the eighth floor of a research-classroom building. The trough and spectrometer were mounted on an optical table. The water was surfactant free and at a temperature of20 °C. One representative data set is shown in Figure (2).

to

I

~ 0

..

.. •

1\ 'f\. " II II v

. .-

I I : I , I : I

, , I

! , ! ,

: I I I I ,

0.- 0- 0 __ ........... 1

Figure 2. The correlogram and residual to the fit of experimental data for water with equation (24). T =20 'C. IFT=72.S mN/m. The uncorrected center-frequency was 14.89StS Hz and the width 1.1 SOt8 Hz. The Edwards el al. (1982) correction gave 14.911 Hz and 695 Hz. The grating constant was 496.0 I/cm and ka = 22. In the fit. the correlation coefficient between (&) and r was 0.0008.

! A I I I

...... I 'I : I

.. - I : ! I \ I I 11 \ i I

G(I) . ......

...... ) \ : ! I •• 0001

w/ . '-:...... I

I l I .... ""'" • 10000 ZOOCIO lOOOO ~ .~

F~u8ncy ( Hz )

Figure J. The FFT of the correlograrn shown in Figure 2. The high·pass filter was set at 2 KHz and the low-pass tilter at 50 KHz.

Remark 1: Under certain conditions, representing the correlogram hy

(24)

108

is a reasonable approximation. In fact. under conditions that can be specified. OJq can be determined with a precision estimated to be a few Hz out of 15kHz. The estimated error

of 1(/ is often better than 1 %. See Figures (2. 3).

A condition for this approximation can be constructed using a simple form of the dispersion equations. In fact, the simplest dispersion equation was derived in the last century by Kelvin (see Landau and Lifshitz, 1960; Levich 1962) and may be derived from accurate dispersion equations (Mann, 1984) by taking the low viscosity limit. Approximately

J.l 2 r = 2-q q p

where J.l is the volume viscosity of the liquid phase and p is the density.

(25a,b)

The correction to formula (25a) can be small since the dimensionless group

pro 2 defined by Hansen and Mann (1964), y I = -)-, can be calculated accurately and is

yq usually within 5% of YI = I. However, rq is not well estimated by formula (25b) and should not be used to estimate the viscosity coefficient J.l of the liquid phase. However, Katyl and Ingard (1968) defined a dimensionless group, Y, that is useful in deciding when one may use the dispersion equation method for estimating visco-elastic parameters. By definition.

(26)

where OJo and ro are computed by equations (25 a, b). The desired condition follows from Y.

When Y » 1, the capillary wave spectrum is approximated quite well by the Lorentzian function which amounts to the Fourier transform of equation (24). For example, for water at 72 mN/m and ripples at 500 Hz, Y = 349, See Figure (4). The capillary wave spectrum deviates from the Lorentzian approximation noticeably for Y < 100. However. the center frequency deviation is smaller and will start shifting significantly when Y < 20. The surface tension may be estimated using a Lorentzian fit with reasonable accuracy when 20<Y. But, the width is poorly determined for Y < 100 and it is reasonable to argue that even for Y » I the fit should involve the accurate spectrum function. Notice that Y is related to the capillary number, Nc., which is a ratio of characteristic speeds and is often cited in scaling of the dynamics of liquid interfaces.

Notice that Y... has units of velocity so that for capillary ripple dispersion, J.l

These numbers. YI and Y, provide a useful guide for experimental design.

N =_1_. ~ 2.JY

' .' t-----.---i--\-----------i

•.• I--------- i'----Ir-.,..---------I

Gll) • .• f---------,f ---------\--,---------l

Figure 4. C6mparison of the response spectrum with the Lorentzian computed for water. lIT = 72.5 mN/m.

109

Ur---~--~----------~~

u~------~----------~~

~ u , • i .,

" r '" 4.S

., I ! i .... • I i 1 I • ... ... ... ....

n_

Figure 5. The inverse FIT of the computed spcc:trum in Figure 4. Note the shape of the residuals of the fit by equation (24).

Figures (4,5) illustrate several of these points. The same q ( = 496.0 Vcm) and water properties ( at 20°C ) that were recorded for Figure (2) were used to generate the spectrum. G,( (j)). See Langevin (1992) for specific fonnulas for the spectrwn function for pure liquids. The comparison of the spectrum function and the Lorentzian is shown in Figure (4). The numbers Wq and rq used in the Lorentzian function are solutions ofa

general dispersion equation for capillary ripples (Edwards et ai., 1982; Mann, 1984; Mann and Edwards. 1984; Langevin, 1992). The numerical table for Gc; was transformed by FFT to the correlogram shown in Figure (5). Note the signature of the residuals at short times and compare to Figure (2). Indeed, the correction for the deviation from the Lorentzian fit is rather small for water at room temperature. Moreover, the center frequency and width match the corrected numbers found in the fit of equation (24) to the raw data shown in Figure (2). The correction procedure derived in Edwards, et al. (1982) is consistent.

Remark 2: Given a pure interface. instrumental broadening and external noise can lead

to suffiCient error in determining rq so that the estimate of viscosity. for example. can be in error by a factor 2 or more. These factors must be controlled.

Earnshaw (1997) has written a good review about the design considerations necessary to build a reliable instrument. Also see Mann and Edwards (1984) and Langevin (1992) for details and different perspectives.

While it is hard to isolate a liquid surface totally from environmental vibration that causes slosh. normal isolation techniques are often sufficient. In fact the long wavelength ripples O l-riPPle > -lmrn) will not perturb the capillary wave spectrum in any practical sense since thi~ is still the regime of linear response. where mode-mode coupling is insignificant. However, the optical lever effect when the incident;ecam is off the normal can be devastating. Modest laboratory noise can cause the reference beam to flicker in and out of the active area of the photo detector (see Section 3). There are several ways to handle this problem: mechanical and acoustic isolation. nonnal incidence

110

operation, a eat's eye optical system and beam steering along with interruptible data collection. The Newport active isolation table (Evis table) has worked well under difficult conditions. At this writing, Newport is bringing out a new system for active control. However. even when environmental noise is well controlled. ~(t) is distorted by the instrument functional defined by equation (23).

Remark 3: So far as I know, all surface light scattering spectrometers operate within the limitations of the Fourier transform analysis used herein so that equation (23) is an accurate representation of the instrumental distortion of R( by the optical system. Recognize that equation (23) presents an inverse problem since RI is determined experimentally but R( is desired Regardless of the details of the optical arrangement, it has the form of a Fredholm integral equation of the first kind similar in form to the Fredholm integral equation for determining the particle size distribution function from the volume light scattering spectrum. The inverse problem is ill-conditioned in both cases. In a sense. the surface scattering problem is somewhat more tractable since there are accurate models for capillary ripple spectra and the distribution function is relatively simple.

One approximation to the inverse problem was given by Edwards et al. (1982) wherein corrections to the frequency and time-damping coefficients determined from fitting the correlogram are computed. However, in a number of practical situations where Ng < 20 the algorithm loses reliability. It is possible to design the optical system to have Ng > 20. Such a design, however, tends to compromise the optical signal to noise ratio. We have designed and are implementing a non-linear least squares algorithm based on evaluating equation (23) rapidly with an integration formula based on Hermite polynomials. Thus the correlogram data is fit to the functional that contains the surface visco-elastic constitutive coefficients through the spectrum of 1\ as well as Ng as parameters.

Earnshaw (1997) points out that equation (24) can be modified by adding a phase term and a Gaussian term so that

(27)

This fitting function is more flexible, reduces the residuals in the fit. and is a useful trial function, but. in my opinion, is ad hoc in view of equation (23) and Edwards et al. (1982). Wang, e/ al. (1994) fit their measured spectra to a Voigt function following the work of Martin and Puerta (1981). They find reasonable fits, indeed. the Voigt function is the convolution between a Lorentzian and a Gaussian. We expect to report on the quantitative comparison of these methods shortly.

III

Remark 4: The theory of surface fluctuations and surface response functions is rich in results and well enough defined that accurate correlation functions are available or can be derived for most experimental situations. The raw data must be accurate.

See Langevin (1992) and papers therein, Mann ( 1975, 1984, 1985 ), Mann, et af. (1987), O'Brien. et af. (1986), and Edwards, et af. (1982). However, all of this theory reduces to detennining experimentally the surface tension to about 0.1 % for a wide variety of systems and conditions. To go beyond this important but comparatively modest result, the correlograms must be of high precision. A sensitivity analysis (Mann, 1984) showed that the fit to produce {J)q' rq must be accurate to about 1: 1 04 or better and I: I 03 or better ,respectively so as to obtain modest precision (10%) in detennining the surface viscosity. It is now possible to obtain this level of precision and perhaps better.

It is clear from the structure of the complete dispersion equations that only two parameters can be detennined if only O}q ,rq are detennined. However, the typical

correlogram {R,(~t)} may have n> 128 so that the number of degrees of freedom seem to justify extracting more than two parameters. Sensitivity analysis suggests that this may be a false expectation. Also, notice how well the Lorentzian fit actually works; the additional infonnation needed to detennine more than two coefficients is, effectively, in the residuals.

There is another question that must be asked now that {R,(~t)} can be obtained with satisfactory precision. How accurately do hardware correlators process stochastic signals? How can this be detennined experimentally by independent measurements. While it easy to measure perfonnance approximately but it is not so easy to detennine how well {R,(~t)} is detennined to the accuracy required for surface light scattering spectroscopy as discussed above.

5. Outline of a few Applications.

There are many applications for the methodology of surface light scattering spectroscopy. The traditional ones are covered in Langevin' s (1992) book compiled a few years ago. Certainly the study of dynamics of monolayer systems is a leading example. Lading, el af. (1989) suggest several extensions of the methodology: in-plane fluctuations of excess may be observable and high-amplitude fluctuations, O<g>O.OI, can be studied. In addition, there are new but challenging opportunities for careful work.

Langmuir films undergo phase transitions when compressed ( Adamson, 1992). The early work on mono layers ( Adamson, 1990; Gaines. 1996 ) suffered from a lack of methods to visualize the morphology of ultra-thin films in situ. Brewister angle microscopy (BAM) was invented by Henon and Meunier (1991) and Honig and Mobius (1991) to allow observation of monolayer structure in detail. Examples are provided by the papers of Mann, E. K., et af. (1995) and de Mul and Mann (1994). One can now detennine easily whether a Langmuir film is homogeneous or composed of domains. See

112

Wang, et al. (1994) for a beautiful example of the use of surface spectroscopy and in a separate experiment BAM. Tseng, Mann, Tin, and Meyer (1997) have taken the next step. We have demonstrated that the surface light scattering spectrometer and BAM can be integrated in the sense that the footprint of BAM and SLSS can be made to overlap on the interface. Then the morphology of the monolayer is known at the precise location of the scattering field. BAM will report if there are well ordered domains on the surface that can be considered as two-dimensional crystals. It is then possible to determine the fluctuation spectra in such domains as a function of their orientation. Recall Figure (1). It is only necessary to rotate the grating to study the spectrum as a function of orientation. At this time, the minimum size of the SLSS footprint is determined by kgO"g, kIM and the intensity of the .incident beam. If kgO" g' is too small, the correction procedure is suspect even though signal to noise is improved. IftyM is too large, > -2000 l/cm, the signal to noise ratio is small, see equation (7). Higher intensity of the zero-order beam improves signal to noise. At the present time a footprint of one millimeter is about optimum. This is sufficient resolution at this time and improvements are anticipated.

The image system is small enough and sufficiently robust that a number of new investigations are possible. For example, the surface properties of liquid metals can be studied in ultra-high vacuum chambers. Window ports of optical quality oriented for normal incidence can be arranged easily. Initial bench-top experiments have been encouraging; high quality correlograms were obtained on liquid gallium (Meyer, et al .• 1994) with a reflection system. Our group showed that it was possible work at temperatures up to at least 2000°C in a crude experiment with liquid copper. We expect that operation at much higher temperatures will be possible with the use of optical band­pass filters. The black body radiation that does get through the filters will be uncorrelated and add to the noise only. An interesting topic will be the study of the reacting interface.

In summary, surface light scattering spectroscopy has rich possibilities as a fundamental tool of surface chemistry but will be most effective in combination with other surface science techniques.

6. Acknowledgments

I wish to thank L. Lading and R. V. Edwards for many conversations about the implementation of surface light scattering spectroscopy. I wish to thank 1. Earnshaw for a number interesting conversations about SLSS and preprints of papers from his group. Senior NASA Lewis Research Center collaborators include P. Tin and W. V. Meyer and J. Lock, Cleveland State University. I wish to thank C.-J. Tseng for the use of his data shown in Figure (2). Support for this work was provided by Code UG (Microgravity Science and Applications Division) at NASA Headquarters.

113

7. References

Adamson. A.W .. (1990) Physical Chemistry of Surfaces. 51h edition, Wiley, New York.

Cahn, J.W., (1979) Thermodynamics of Solid and Fluid Surfaces in Interfacial Segregation. papers presented at a seminar of the Materials Division of the Am. Soc. Of Metals, October 22-23,1977. ed. By W.C. lohnson, J. Blakely, Metals Park, OH.

de Mul, M.N.G. and Mann, l.A., (1994) Multilayer Formation in Thin Films of Thermotropic Liquid Crystals at the Air-Water Interface. Langmuir 10, 2311

Earnshaw, J.C. (1997) Surface Light Scattering: A Methodological Review. scheduled for publication in the feature issue Photon Correlation and Scattering in Applied Optics.

Edwards, R.V., Siroki, R.S., Mann, l.A., Shigh, L.B. and Lading L., (1982) Surface Fluctuation Scattering Using Grating Heterodyne Spectroscopy, Applied Optics 21, 3555.

Goodman. J. W., (1968) Introduction to Fourier Optics. McGraw-Hill, New York

Hansen, R.S., (1962) Thermodynamics of Interfaces Between Condensed Phases. J. Phys. Chern. 66,410.

Hansen. R.S. and Mann, l.A., (1964) Propagation Characteristics of Capillary Ripples. L The Theory of Velocity Dispersion and Amplitude Attenuation of Plane Capillary Waves on Viscoelastic Films. l. Applied Phys. 35 (1) 152.

Henon, S, and Meunier, J. (1991) Microscope at the Brewster Angle: Direct Observation of First-Order Phase Transitions in Monolayers, Rev. Sci. Inst. 62(4), 936.

Honig, D. and Mobius, D. (1992) Rejlectometry at Brewster and Brewster Angle Microscopy at the Air-water Interface, Thin Solid Films 2101211(112),64.

Katyl, R.H. and Ingard, U. (1968) Scattering of Light by Thermal Ripplons. Phys. Rev. Lett. 20, 248-249.

Lading, L., (1996) Phase and amplitude gratings were produced by the Optical Diagnostics and Information Processing group, RIS0, Denmark.

Lading, L.. Mann, l.A., and Edwards, R.V., (1989) Analysis of a Surface - Scattering Spectrometer. l. Opt. Am. A 6(11), 1692.

Lading, L., Mann, l.A. and Edwards, R.V., (1988) Photon Statistics of Light Scattered By a Liquid Gas Interface, in Proceedings of the OSA Topical Meeting on Photon Correlation Techniques and Applications." May 31-lune 2. 1988, Washington,

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Optical Soc. of America.

Landau. L.D., and Lifeshitz. E.M., (1970) Theory of Elasticity. Pergamon Press, New York.

Landau. L. and Lifshitz. E.M. (1960) Fluid Mechanics. Addison-Wesley, Reading.

Langevin, D., (1992) Light Scattering by Liquid Surfaces and Complementary Techniques. Surface Science Series Vol. 41, Marcel Dekker, Inc., New York.

Levich. YO. (1962) Physicochemical Hydrodynamics, Preutice_Hall. Inc., Englewood Cliffs, NJ.

Mann, E. K., Henon, S., Langevin, D. and Meunier, J., (1995) Hydrodynamics ofDmain Relaxation in a Polymer Monolayer. Physical Review E, 51 (6), 5708.

Mann. 1.A., Tjatjopoulos, O. T., Azzam, M-O.J., Boggs, K.E., Robinson, K.M. and Sanders, J.N., (1987) Pre-Llangmuir-Blodgett Monolayers. Thin Solid Films, 152.

Mann, J.A. (1985), Dynamics. Structure, and Function of Interfacial Regions. Langmuir 1, 10.

Mann. l.A. and Edwards. R.V., (1984) Surface Fluctuation Spectroscopy: Comments on Experimental Techniques and Capillary Ripple Theory. Rev. Sci. lnstrum 55 (5), 727. Also see the companion article by L.B. Shih bid, 55,716 (1984).

Mann, 1.A. (1984) Dynamic Surface Tension and Capillary Waves. Surface and Colloid Science 13, 145 Also see Digital-Computer-Oriented Numerical Analysis in Surface Chemistry. ibid page 213.

Mann, 1.A. and Porzio, K.C., (1975) Capillarity: The Physical Nature of Fluid-Fluid Interfaces Including the Problem of Biomembrane Structures. in International Review of Sci., Phys. Chern. Ser 2, Vol 7, Surface Chemistry and Coil., M. Kerker (ed.), Butterworths, London.

Martin. P. and Puerta, 1., (1981) Generalized Lorentzian Approximations for the Voigt Line Shape. Applied Optics 20(2),259. Ibid .. 20(22),3923.

Meyer, W.V., Lock, J.A., Cheung, H.M., Taylor, T.W., Tin, P., and Mann, J.A., (1997) A Hybrid Refection - Transmission Surface Light Scattering Instrument with Reduced Sensitivity to Surface Sloshing, 1. Appl. Opt. Submitted. Preprints are available from the authors.

Meyer, W.V., Tin, P., Mann, 1.A., Cheung, H.M., Rogers, R.B. and Lading, L., A Preview of a Modular Surface Light Scattering Instrument with Autotracking Optics,

International Symposium on Space Optics, April 18 - 22,1994, Oarmisch­Partenkirchen. FRO, EOS/SPIE Joint Venture, paper 2210-29.

O'Brien, K.C., Mann, J.A. and Lando, lB., (1986) Mechanical Testing of Monolayers: Substrate Decoupling in the Long-Wave Approximation, Langmuir, 2, 338.

115

Ricci, E. and Passerone, (1993) Review: Surface Tension and Its Relations with Adsorption, Vaporization and Surface Reactivity of Liquid Metals, Materials Science and Engineering A161, 31-40.

Turkevich, L.A; and Mann, lA., (1990) Pressure Dependence of the Interfacial Tension Between Fluid Phases 1. Formalism and Applications to Simple Fluids, Langmuir 6, 445-456.

Ulman, A. (1991), An Introduction to Ultrathin Organic Films: From Langmuir­Blodgett to Self Assembly, Academic Press, Boston.

Wang, Q., Feder, A. and Mazur, E., (1994) Capillary Wave Damping in Heterogeneous Monolayers, J. Phys. Chern. 98,12720.

NON-LINEARITY OF APDS AT HIGH COUNT RATES

M. GRAN AND E. R. PIKE

Physics Dept., King's College, London, UK

AND

E. PAILHAREY

University of Paris Sud, Paris, France

Abstract. Several APD photon-counting systems have been studied for linearity

of response as a function of light intensity using a controlled LED light source. The effectiveness of the APD as a photon detector at high rates is limited by various types of non-linear response above 100 kcounts/sec. Two types of APD non-linearity, and the way each distorts a reference pulse, will be discussed. "Undercharged" APDs can cause distortion at quite low counts rates, and dead-time effects can cause distortion at count rates which approach the inverse recharge time.

1. Introduction

Avalanche photodiodes (APDs) have long held promise as a replacement for photomultiplier tubes (PMTs) in photon counting experiments, PMTs are excellent tools for photon counting: they have acceptable photon de­tection efficiency, short dead time, and linear detection rates over many orders of magnitude up to Mcounts/sec. Under optimal conditions, APD photon-counting systems have demonstrated probability of detection val­ues from 1.5 to 3 times that of PMTs. (Robinson and Metscher, 1987) However, they have been limited to lower count rates, because of increas­ing non-linearity at high rates. For passive systems, an upper limit of 100 kcounts/sec. has often been mentioned. For active systems, count rates above 3.5 Mcounts/sec. have been said to cause thermal problems with APDs (Brown et al., 1987), although this is not universally confirmed.

117

E. R. Pike and J. B. Abbiss (eds.), Light Scanering and Photon Correlation Spectroscopy, 117-127. © 1997 Kluwer Academic Publishers.

118

In revisiting these upper limits, we have investigated how APD photon­counting systems distort a reference light signal from a fast-response LED driven by a square-wave voltage source. When "on" the stream of photons from the LED is assumed to obey poissonian statistics and hold a constant mean count rate. Therefore, an ideal photon detector should measure a constant mean count rate as well.

With bright pulses pushing our laboratory APD photon counting sys­tems beyond their conventional limits, both active and passive exhibited non-linear behaviour. Instead of detecting a steady stream of photons obey­ing poissonian statistics for the duration of each pulse, the APD systems erroneously reported pulses that had different mean count rates for different parts of the pulse.

Two different types of behaviour were observed when detecting signals in this way. A decaying count rate was observed with the passive systems, and an oscillatory count rate was observed with the active systems. Investigation has led us to hypothesise two models which account for these behaviours.

2. Nomenclature

Symbol

APD VBR

VR

VAPD

RB Res Pd (J

TABLE 1. Nomenclature

Description

avalanche photodiode APD breakdown voltage Driving voltage of APD system Internal voltage of the APD Ballast resistance in a passive quench circuit Current sensing resistance in a passive quench circuit Photon detection efficiency Mean count rate

3. Basic APD Systems and Electronics

The standard method of photon counting using an APD is to connect it in reverse-bias to a driving voltage source that exceeds the APD's break­down voltage (VBR). In this state, the absorption of a photon by the APD will cause it to avalanche, to allow current to pass through in the re­verse bias direction, a process that will continue indefinitely unless the avalanche is quenched. To quench the avalanche, the driving voltage VR must be brought below VBR. The methods by which the quenching is ac-

119

complished are broadly classified as active or passive. Passive systems use the reverse-bias current to stop the avalanching by placing a resistor or other combination of passive components in series with the APD. The in­crease in current causes a voltage drop across the resistor, quenching the avalanche. Active systems attempt to quench more quickly by sensing the moment which avalanching begins, and then bringing VR down below VBR and back up again as quickly as possible.

In either case, once the voltage across the APD has been reduced as a result of the quenching process, there will be a finite time during which the APD must recharge. APDs have a capacitance, and therefore the voltage to which an APD is charged will lag the driving voltage VR.

Most APD photon-counting systems then have electronics that take the raw APD output pulses above a certain threshold voltage or current, and generate clean digital pulses in response. The raw APD output pulse is often low voltage or of unsuitable shape to act as a trigger pulse on its own.

These three components, an APD, its quenching circuit, and the output pulse generator, constitute an APD photon-counting system.

4. Experimental setup

To generate the light pulses, a LeCroy 9210 pulse generator was used to drive a GaAsP communications-grade LED in series with a resistor. Set up in this way, the rise time and decay time of the LED and pulse genera;t-or was measured to be less than 30 ns. The LED/pulse generator system was capable of producing a 3V - 5V square wave with 25% - 50% duty cycle and periods from 2 Hz to MHz.

The LED was placed at one end of light-tight black paper tube and the APD was placed at the other end, facing the LED. The output of the APD was connected to a triggered pulse generator, which was set to output pulses, whenever an input pulse of more than the threshold voltage was detected.

These standard pulses were input to a Malvern 7032 correlator, which computed the cross-correlation ofthe photon-detection pulses with a "delta­function" trigger signal emitted by the pulse generator at the beginning of each pulse. The output of the correlator, when run in this way, is a histogram, where the first bin holds the number of detections between 0 and tl.T after the trigger, the second holds the number of detections from tl.T to 2tl.T, and so on, where tl.T can be from 1000 ms to .5 J.LS for this correlator.

120

5. Simple Passive Systems

The quenching circuit used on passive systems can be as simple as a ballast resistor placed in series with the APD. When the APD avalanches, the current through the resistor brings a voltage drop of I R quenching the avalanche. The ballast resistor can be placed before or after the APD, with different results.

5.1. PASSIVE SYSTEM 1

5.1.1. Setup We present here the experimental results from one of the passive quenched APD systems. The experimental arrangement is shown in figure 1.

Ro = 220 kQ

Output

Res = 1.2kQ

Figure 1. Passive circuit # 1

Assuming VR is in excess of break down, then before the absorption of a photon, no current is flowing through the resistors, making the voltage across the APD to be equal to - YR. After the absorption of a photon, current flows through the resistors, reducing the VAPD to VR-I(RB+Rcs), which quenches th~ avalanche. (APDs have a stray capacitance of 2pF.) The smaller resistor converts current into voltage from a low impedance for ease of measurement.

5.1.2. Properties of the APD itself Photon detection efficiency. The photon detection efficiency (Pd) of the APD is the percentage of absorbed photons that cause appreciable avalanche. Pd is zero when the driving voltage less than the breakdown voltage, and increases as VAPD - VER, within the operating range. Brown et al.

121

(1986) measured the Pd of a similar system as 7.5% ± 1%, when operated at -lOoe and 3V in excess of breakdown. Lightstone et al. (1989), from RCA, a manufacturer of APDs, claimed photon detection efficiencies in excess of 50%.

5.1.3. Properties of the APD and its quenching circuit Charging status. Because of the capacitance of the system, there will be a finite time required after avalanching for the APD to recharge to VR. In this case, we expect a recharge time of a few microseconds ( 2pF APD capacitance, 220kS1 series resistance, VR - \lBR = 3-5V). In this circuit, the output signal-measures the charging status of the APD.

\loutput ~ (VAPD - \lR) i~S' (1)

The following figure shows the output during a typical event.

I~

Figure 2. Output from Passive #1

The voltage at the output drops quickly after a photon absorption, and then returns to VR with a rise time characteristic of the circuit.

Dead time. Immediately after the absorption of a photon, during the avalanche and some of the re-charging phase, the APD circuit will not be sensitive to photons, beca,use it will not have charged sufficiently above breakdown for the Pd to be non-zero. The time between the absorption of the photon and the recharge of the APD to just above the break­down voltage is the dead time of the circuit, and is characteristic of the APD/quenching-circuit system for a given VR and temperature.

122

5.1.4. Normal functioning Figure 3 is correlogram of a delta function emitted before the start of each pulse from the LED with the photon detection events of the APD system, and can be understood as the relative probability density of photon detec­tion events occurring during the LED pulse. In this case the correlelogram would be "square", just as the pulse is "square". Therefore, the APD has equal probability of detecting a photon at any time when the LED is on, as one would expect from a poissonian photon source.

5.1.5. Failure functioning As the LED becomes brighter, and the mean count rate of photons in­creases, the correlelograms become less and less square, showing a much higher probability of detecting a photon at the beginning of the pulse than at the middle or end of the pulse.

This effect can be seen in the correlograms in fig. 4. When counting at the slow (144 kHz) rate, in which each cycle the LED caused the APD to detect an average of 0.9 photons, the correlogram shows that the APD has detected approx. equal numbers of photons in each bin. At the high count rate (576 kHz), during each pulse, the APD will detect an average of 3.3 photons. The probability density for photon detection is clearly much higher at the beginning of the pulse in this case.

Explanation for the failure. The explanation of this effect is that the APD is detecting photons faster than its recharge rate. It has already been noted above that there is a period of time between the APD re-charging to just above breakdown and the APD reaching YR. The first photon of the pulse will always be detected by a fully charged APD. If the mean time between photon absorptions is less that the time necessary to recharge the APD fully to VR, then the lower voltage beyond breakdown of the APD will result in lower Pd. Therefore, the shape of the correlogram can be explained by the first photon being detected at a higher Pd than the remaining photons in each pulse.

Using this system to count photons at rates faster than the full recharge rate of the APD system would make the interpretation of the signal difficult.

Looking back on fig. 2, it can be seen that the second photon detection occurs with a less charged APD than the first.

5.2. PASSIVE SYSTEM 2

A passive system can also be constructed by swapping the APD and ballast resistor. While the circuit is similar to the other passive circuit, the shape of the output pulse is quite different, and gives no informatio~ about the

450E+3

400E+3

350E+3

300E+3

-E 250E+3 ::J

8200E+3

150E+3

100E+3

50E+3

Passive #1

------ . ------------,

~"'""'6-144kHZ --301 kHz

I ~576kHz

OOOE+O ~~~-------+---~~~ 2 2.5 3 3.5

IJS

4 4.5 __ . 5

Figure 3. Passive System #1, correlogram, failure

123

recharge status of the APD. The pulse width, in- this case, is only the time necessary to quench the avalanche. In general, one cannot presume the output pulse of an unknown system gives the charging status of the APD.

The figure shows what the output pulses of fig. 2 would look like had they been output from this circuit.

This small modification does not significantly change the way in which the system behaves or fails.

6. Active systems

There are various designs of active quenching circuits in existence today, including commercially available units.

6.1. THE PROTEIN SOLUTIONS SYSTEM

6.1.1. Properties of an active APD system Dead Time. In the system testes, which was a commercial unit manu­factured by Protein Solutions Inc. of Charlottesville, VA, the output pulse

124

.. ;.

FiyUl'c 4, Output from Passive #2

generation electronics give a digital pulse of a finite width to signal the absorption of a photon by the APD. While this output pulse is on, no information about further detections by the APD can be given. (Because this dead time is caused by the pulse-generation electronics, it is not equiv­alent to the APD dead time. The pulse dead time should be greater.) The maximum registered count rate is then given by inverse pulse width .

6.1.2. Failure Functioning With count rates that approach the inverse output pulse width, the cor­relogram of the "square" input pulses of the APD is distorted, showing a probability distribution which has a damped oscillation as a function of time. The correlogram is shown in figure 5.

Explanation for failure. Given an arbitrary system where, when a photon in absorbed, a digital output pulse is generated and during the time of the output pulse no new output pulses will be given, this "dead time" will effect the shape of the o~tput pulses.

The concept that dead time reduces count rates is certainly not new, but, it is considered in a little more detail here since it can produce inter­esting results in out test system.

The argument is as follbws: imagine a photon counting system which, at time t=O, begins to be illuminated by a stream of photons. The photons have a constant mean rate of absorption and have a poisson ian distribution. For poissonian system, the probability of detecting x photons in a time tlt is

125

Active '1

2160000

2140000

2120000 .....

~ ~

2100000

!!

V g 2080000 RM@ u

2060000

2040000

2020000

2000000 0 0.5 1.5 2 2.5 3

liS

Figure 5. Correlogram of Active System #1

(2)

from which it follows that the probability of paving detected the first photon at a time T after the photon beam is first turned on is

PI (T) = 1 - e-uT 'tiT;::: 0, (3)

and the probability density of detecting the first photon between time T and time T + dT is

dPI(T) -uT dT = ue (4)

After detecting 'a photon, the circuit will not respond to any photon absorptions until after a certain dead time (D) has passed, after which time, poissonian statistics will take over again. Knowing this, the probability of having detected a second phpton by time T is

P2 (T) J: dPl~tu- D) ,Pl(T - u)du

- 1 - e-u(T-D) .[1 + u(T - D)] 'tiT ;::: D (5)

126

From this, the probability of detecting the second photon from time T to time T + dT is

dP2 = -q(T-D) 2(T _ D) dT e .(7 (6)

Similarly, it can be shown that the probability of having detected the nth photon by time T is

n-l m(T D)m Pn(T) = 1- e-q(T-nD). L (7 -,n ,

m=O m.

and the probability of detecting the nth photon between time T a.nd T + dT

dPn n (T - nD)n-l q(T-nD) --=(7 e dT (n - I)!

(7)

A graph of this probability density function with (7 = 13M If z and D = 50 ns is displayed in the figure below.

Dcad TIOIC Ji11cr (lhcorclical)

2.00E-02

1.90E-02 \ . 1.80E-02

1.70E-02

I.GOE·02

1.50E-02

1.40E·02

1.30E.02 , ______________________ 1

O.OOE+ 2.00E· 4.00E· (;.oOE· 11.001., · 1.001.,· 1.20[ · IAOE· I.GOE· I.llOE- 2.00E-00 Oil Oil 02 Oil 07 07 07 07 07 07

Figure 6. Theoretical Dead-Time Jitter

127

The thermal heating problem. In the correlograms that we made with the Protein Solutions system, we did not detect the type of fading count rate that Brown et al. {1987} attributed to APD warming. Even at 4 times the maximum rate reported by the authors, the count rate remained reasonably constant ignoring dead-time effects.

7. Conclusion

As the problem of designing better APD quenching circuits is resolved, the count rates of the active systems will continue to increase. For some applications, the old techniques of dead-time correction will still have to be applied to APDs.

Passive APD systems are simple to constructed,but in designing a pas­sive circuit, it is necessary to recall that it is the recharge time, not the dead time, that limits the maximum count rate that may not be exceeded if linearity is to be maintained.

8. Acknowledgements

This material is based upon work supported in part by the US Army Re­search Office under grant No DAAH04-95-i-0240

References

Brown, R.G.W., ffidley, K.D., and Rarity, J.G. 1986 Characterisation ofsilicon avalanche photodiodes for photon correlation measurements. 1: Passive quenching, Applied Op­tics Vol. 25 No. 22, pp. 4122-4126

Brown, R.G.W., Jones, R., Rarity, J.G., and ffidley, K.D. 1987 Characterisation of silicon avalanche photodiodes for photon correlation measurements. 2: Active quenching, Applied Optics Vol. 26 No. 12, pp. 2383-2389

Kaneda, T., Hirobumi, T., MatsumQto, H., and Yamaoka, T.1976 Avalanche buildup time of silicon reach-through photodiodes, Journal of Applied Physics Vol. 47 No. 11, pp. 4960-4963

Lightstone, A., MacGregor, D., Mcintyre, R., Trottier, C., and Webb, P. 1989 Pho­ton counting modules using RCA silicon avanche photodiodes, Electronic Engineer­ing October 1989, pp. 37-45

Robinson, D.L., and Metscher, B.D. 1987 Photon detection with cooled avalanche pho­todiodes, Appl. Phys. Lett. Vol. 51 No. 19, pp. 1493-1494

FROM SPECKLES TO MODES: PRINCIPLES AND APPLIC­

ATIONS OF FIBER OPTIC DYNAMIC LIGHT SCATTERING

Abstract.

J. RICKA Institute oj Applied Physics, University oj Bern, Sidlerstrasse 5, Bern ell-30 12, Switzerland

The instrumentation of Dynamic Light Scattering (DLS) is undergoing a small revolution: the classical pair of pinholes, whose role is to select a single speckle, is being replaced by single-mode fibers that select a single mode. This improves the performance of the apparatus substantially, but the improvement is also accompanied by a sad loss: the visible (and beau­tiful) boiling speckle pattern is replaced by the seemingly abstract notion of receiver mode. In my lecture I will introduce the principles of mode­selective optical receivers, discuss the relation of modes with speckles and show that it is worthwhile to accept this slight change of the point of view. The gain is a considerable simplification of the instrumentation as well as of the theoretical reasoning. This enables the researcher to tackle, more easily, difficult experimental prohlems such as the in vivo DLS in human eye vitreous or the motion of colloidal particles in opaque porous media.

129

E. R. Pike and J. B. Abbiss (eds.), Light Scanering and Photon Correlation Spectroscopy, 129. © 1997 Kluwer Academic Publishers.

PHOTON CORRELATION SPECTROSCOPY OF INTERACTIVE POLYMER SYSTEMS

G. FYTAS, K. CHRiSSOPOULOU, S. H. ANASTASIADIS,t D. VLASSOPOULOS and K. KARATASOS Foundationfor Research and Technology-Hellas Institute of Electronic Structure and Laser P.O. Box 1527, 7lIIOHeraklion, Crete, Greece

ABSTRACT: We present two recent applications of dynamic light scattering to study order parameter fluctuations in chemically dissimilar polymer systems: (i) Diblock copolymer solutions near the disorder to order transition with wavevectors q- q * at the maximum of the structure factor Seq*), (ii) Binary polymer blends near the macrophase separation temperature in the low q limit.

1. Dynamic Structure Factor of Diblock Copolymer Solutions

Diblock copolymers AB are thermodynamically single component systems

with interacting chemically dissimilar blocks that unlike binary polymer blends, cannot

macrophase separate. Instead these systems can self assemble at low temperature below

the T ODT and/or concentration above <j>ODT in a common solvent depending on the

overall polymerization index N=NA+NB• The transition (ODT) from a disordered to an

ordered state is clearly manifested in the static structure factor Seq). Since the most

probable composition fluctuations 'l'q(t) occur a finite q= q * -O(KI) (R is the size of the

AB chain), Seq) peaks at q* and for a monodisperse diblock vanishes in the

thermodynamic limit q ~O [1,2].

Nowadays the dynamic structure factor S(q,t) can be best studied by photon

correlation spectroscopy (PCS) mainly due to its broad time range provided q * falls

within the range of the (low) light scattering q's. Hence, synthesis of very high

t Also at University of Crete, Physics Department, 71003 Heraklion, Crete, Greece

131

E. R. Pilce andJ. B. Abbiss (eds.), Light Scattering and Photon Corre1ation Spectroscopy, 131-140. @ 1997 Kluwer Academic Publishers.

132

molecular weight AB polymers is a crucial requirement. At such high N, the

measurements of S(q,t) near ODT should be performed in solution even for the case of

weakly unfavorable interactions (x.) between A and B segments; it is the product xN that

controls the phase morphology. For low molecular weight AB melts [3,4] near TOOT,

S(q,t) can be studied only at low q/q* values i.e. far from the maximum of S(q). For

symmetric AB copolymers within the mean-field approximation [2]

(1)

where the function F(x) is a combination of the Debye correlation functions for

unperturbed Gaussian coils depending on x=(qRi. S(q) shows a maximum at x* for

which F(x *)=2XN.

Figure 1 shows the autocorrelation function of the polarized light scattering 2

intensity (Gvv(q,t)-1)/{ oc IS(q,t)1 ({ is a measure of the coherence area) for a 7.1 wt%

solution of styrene-isoprene diblock copolymer (SI-IM) (Table 1) in the toluene for

qR=1.35 at 20°C.

TABLE 1: Molecular characteristics

Sample Mw(10,,} Wps N fps <POOT

SI-53 [5] 0.172 0.53 2042 0.50 0.195

SI-IM [6] 1.04 0.44 12477 0.41 0.067

The spectrum of relaxation times L(1m) (upper inset of Fig. 1), obtained from the

inversion of the experimental Gvv(q,t), displays a main peak at 't-0.02 s. This mode

corresponds to the main relaxation of the composition fluctuations in disorder diblocks

relaxing via chain reptation; for the SI-IM1toluene system, the mode is probed near the

maximum of the static structure factor and, thus, it exhibits the highest intensity, 11 [6].

The fastest weak peak of L(lm) at 't-3xl0-s s, on the other hand, is due to the total

concentration fluctuations decaying via cooperative diffusion and hence it is not diblock

copolymer specific. Its intensity Icoop relative to that of the main mode, IcooplIl -gIN

(g_<p-5/4 is the number of monomers per blob) is expectedly small for high N's even in

the absence of thermodynamic effects. The intermediate process in L(1m) at 't-l 0-3 S can

be attributed to composition fluctuations relaxing via Rouse-like chain conformation

133

motions in times shorter than the main chain reptation process [6]. This mode can be

resolved for sufficiently high N and observed by pes due to the refractive index

contrast between the two blocks, i.e. it is invisible in high MW homopolymers. No

additional process is observed in the L(ln!) of SI-IM.

40

1.0 30

20

10 """~

C\I ..-... -~ 0--...-

C/) 0 .5

x=O.38

t(s)

Figure 1. Concentration correlation functions at 23°C for 7.1 wt% SI-IMItoluene at X=(qR)2=1.82 (0) and

for 18.7 wt% SI-53/toluene at x=O.38 (0). The insets show the distribution of relaxation times L(lnt)

multiplied by the total polarized intensity normalized to that of toluene.

The behavior is modified for low MW AB with qR<1 . For 18.7 wt% SI-53

(Table 1) solution in toluene, the coherent scattering function also shows three peak

structure, however, the assignment is different. While the fast process corresponds, like

before, to the cooperative diffusion, the intermediate peak is now the diblock copolymer

mode due to chain reptation. The third peak in L(lm) of SI-53 corresponds to the chain

self diffusion detectable due to chemistry imperfections leading to small but finite

composition polydispersity in block copolymers [7,8]. This is analog to the incoherent

scattering from size polydispersity of colloidal particles [9]. The polydispersity process

is not observed in SI-IM since II dominates the scattering for q/q*::1 whereas lpoly is

134

more evident at q ~ O. The slowest weak process for SI-53/toluene is attributed to the

long range density fluctuations [10]. Thus, the resolution of the relaxation processes in a

dynamic light scattering experiment enables the assignment of the different

contributions to the total static structure factor facilitating the comparison with

theoretical predictions.

Z ~ ....

10" ",

0.20 or;9C:/b .!!!. "! C

~ I] 0.15 Z

L.,-

10"

0.0 0.5 1.0

0.10 qR

0.05

I] I] I]

1.5

c c

c

C

1]1]

2.0 C

c c

0.00 L...-_ ....... _..D..lIollol...::or:JJ=---'c __ -'" __ ....... __ ..L-_--''--_--I

0.0 0.5 1.0 1.5 2.0

qR

Figure 2. Intensity of the main internal copolymer mode, II, normalized with the total number of segments

and the volume fraction of the polymer in the solution for the SI-IMltoluene (0) and SI-53/toluene (0) as a

function of qR. The insets shows the relaxation rate plotted according to Eq. (4) version qR.

Figure 2 shows the intensity of the copolymer specific mode reduced with the

concentration and number of segments, Il(q)/<pN, for the two SI copolymers covering

different qR ranges. The net S(q) (free from the polydispersity contribution) becomes

vanishingly small at low qR in accordance with the prediction of Eq. 1 for monodisperse

diblocb.

2 2 S(q ~ 0) oc q R (2)

135

For the SI-IM, the q. appears to fall into the highest q-range of the PCS; the Seq) of

even higher MW SI clearly shifts [11] to lower q. For q<q., S(q,t) supports the

theoretical prediction :

S(q,t) - S(q)exP[-f1(q)t] (3)

In the low qR limit, the collective thermal decay rate ft(q) oc q2/S(q) becomes q­

independent and equals the inverse of the longest chain relaxation time 'rio i.e. scales

with NJN3 for entangled chains (Ne is the mean number of monomers between

entanglements). For entangled copolymer solutions in the low qR limit

f:. 3(1-V)/(3v-l)N / N 3 1 lP e (4)

where the scaling exponent v=O.59 for good solvent. The effect of the proximity to the

ODT in S(q,t) is expected for q=q •. The dramatic increase of S(q·) (Eq. 1) due to the

unfavorable thermodynamic interactions should lead to significant slowing down of

ft(q=q*). As shown in the inset of Figure 2 for the SI-IM, the decay rate of long

wavelength composition fluctuations is about three times times faster than that of 'l'q •.

For the shorter SI-53, ft-qO at low qR values even near CPODT.

2. Interaction-Induced Anisotropic Light Scattering from Polymer Blends

The static structure factor of binary polymer blends AlB assumes its maximum

value S(O) in the thermodynamic limit q=O and hence S(q,t) is best studied by PCS [12].

For blends in the one phase region far above the glass transition (Tg), S(q,t) decays

exponentially with fc=Dq2 where D is the interdiffusion coefficient; only very close to

the critical point, Ts, (Ising re~ime) where ql;»1 (~is the correlation length), r_q3 [13].

In the mean field approximation, S(O) oc s·t where s=1-T/T.

Such polymer mixtures near and above Ts are isotropic on macroscopic scales,

and single light scattering from composition fluctuations is also isotropic. Due to the

chain-like nature of macromolecules, it is conceivable, however, that instant snap-shots

of the spatial order parameter (composition) fluctuations of O@ can be both

inhomogeneous.and anisotropic, i.e. not exactly spherical; light scattering from these

transient anisotropic domains will be depolarized. Anisotropic scattering could also

136

arise from fluctuation-induced segmental orientational correlations as recently

considered for short diblocks [14] near ODT; this is in fact weak (IvH-N3f2).

The presence of dynamic depolarized light scattering from critical polymer

blends already 20-30 K above Ts was recently revealed in five chemically different

systems AlB. In the examined temperature region near Ts, all polymer mixtures exhibit

single isotropic scattering. Figure 3 shows the polarized (VV) and depolarized (VH)

intensity correlation functions ICvv(q,t)12=(G(q,t) - 1)1( a polystyrene (PS) I poly(methyl

phenyl siloxane) (PMPS) blend (Ts=IOO°C) with PS volume fraction fps = 0.5, at 120°C

and q=0.027 nm·I.The ICvv(q,t)12 exhibits two relaxation processes: a fast, which is the

well-known interdiffusion [12], and a slow which will be discussed at the end of this

paper. The two relaxation processes are not related with the fast segmental orientation

dynamics of dense polymers near Tg, since Tg (= -3°C) of this blend is rather low. While

the fast process is present in all examined blends near Ts, the slow relaxation can be

hardly seen in blends far above Tg [12].

1.0

0.8

~ /""'00. .... cj<

0.6 "-' U 104 10~ ,~ 1~

't (s)

0.4

0.2

0.0 10-6 1()-4 10.2 100 102

t (s)

Figure 3. Correlation functions in the VV(O) and VH(~) geometries for a PSIPMPS blend with fps=0.5. at

120°C and q=O.027 nm· l . The inset shows the corresponding distributions or relaxation times.

137

The fast process of GVH(q,t) shows the following characteristic features: (i)

CVH(q,t)=[(GVH(q,t)-I)I(]ll2 exhibits non exponential shape, exp[-(ft)p] with fEO.8, (ii)

the rate r exhibits distinctly weaker q-dependence than rc of S(q,t) with a finite

intercept at q ~ 0, (iii) r exhibits a critical slowing down, (iv) it is an induced process

with intensity IVH oc e-2 whereas Ivy-e- t and, (v) IVH is essentially not related to

permanent segmental optical anisotropy and observed under single VV scattering

conditions. A theoretical account of this new VH process in critical blends is based on

fluctuations induced second order scattering, which, while insignificant for the strong

single VV scattering, provides the dominant contribution to the dynamic VH scattering

intensity. A quantitative assessment of dynamic double scattering from composition

fluctuations in a symmetric near critical blend (fA=O.5, NA=NB=N) leads to [15] :

C~~(ko.k, t) = Iok~A 4 J d 3Q S(k - Q, t) S(Q - ko' t)

x [f(Q)f* (Q) + f(Q)f*(k o + k - Q)] (5)

where f(Q)=41t (Qeo) (Qe) I [Q2-(ko+ilI.,i], eo and e are the unit vectors for the electric

field polarization of incident and scattered light, ko and k are the corresponding wave

vectors, A= a nI a cp the refractive index increment and 10 is the intensity of the incident

beam. In the derivation of Eq. 5, it was assumed that the size of the scattering volume L

is finite, i.e., koL» 1. While no optical anisotropy is explicitly assumed, formally the

form anisotropy of the 3D patterns of composition fluctuations is reflected in the

structure factors (in Eq. 5) which are anisotropic with respect to the orientation of Q (for

fixed ko and k). Near the critical point, S(q,t)-S(0)/(I+q2~2) exp(-Dq2t) and, for ko~«l,

C ~~ (9,t) (Eq. 5) at a scattering angle 9 can be approximately written as:

C~~(O, t)::: I~~ exp (- 4Dk~t) sinh(4Dk~t COS(OI2») 14Dk~t cos(OI2) (6)

where the double scattering contribution to the VH intensity is:

(7)

The strong molecular weight dependence in eq. (7) is important since it makes

essentially the effect polymer-specific. Further, eq. (6) can readily accommodate the

138

non-exponential decays observed in the experiment, and predicts weaker q-dependence

for the decay rate f VH of C(9,t) compared to fc(=Dq2) of S(q,t). The ratio

( )112.2 f If - l-cos(0/2) Ism (0/2) VH c

(8)

decreases from about 2 at 45° to 0.9 at 145° (see inserts of Fig. 1 in Ref. 15); at 9=90°,

f vHlfc:1 in agreement with the correlation functions of Fig. 3.

As to the slow process of Fig. 3, it is manifested in both isotropic (VV) and

anisotropic (VH) scattering; it decays exponentially with a diffusive (_q2) relaxation

rate (fs), its intensity (ls(q)) increases with increasing q and shows no critical slowing

down of fs and Is, in clear contrast to the behavior of S(q,t) and C~~ (9,t) [16,17]. As

already mentioned, for polymer blends well above Tg both Gvv(q,t) and GVH(q,t) (near

Ts) exhibit only a single relaxation process identified as S(q,t) (interdiffusion) and

C<2) (9,t) (fluctuation induced), respectively. We therefore relate the slow process of VH

Fig. 3 to "long range density fluctuations" observed in one-component glass forming

systems [10]. However, the present findings suggest that slow process in a mixed glass

is related to non isotropic fluctuations [17].

Acknowledgment

We are grateful to N. Hadjichristidis, M. Xenidou, K. Adachi, A. N. Semenov,

A. Likhtrnan, N. Boudenne, G. Meier, and G. Fleischer for their contribution to this

work. Part of this research was sponsored by the European Union (HCM-CT-920009,

BRE2-CT-94061O), by NATO's Scientific Affairs Division in the framework of the

Science for Stability Programme, and by the Greek Secretariat of Research and

Technology.

References

1. Bates, F.S. and Fredrickson, G.H. (1990) Block copolymer Thermodynamics, Annu. Rev. Phys. Chern.

41,525-557

139

2. Leibler, L. (1990) Theory of Microphase Separation in Block Copolymers, Macromolecules 13, 1602-

1617.

3. Anastasiadis, S.H., Fytas, G., Vogt, S. and Fischer, E.W (1993) Breathing and Composition Pattern

Relaxation in "Homogeneous" Diblock Copolymers, Phys. Rev. Lett. 70, 2415-2418; (1994) Dynamics

of Composition Fluctuations in Diblock Copolymer Melts above the Ordering Transition,

Macromolecules 27, 4335-4343.

4. Stepanek, P. and Lodge, T.P. (1996) Dynamic light Scattering from Block Copolymer Melts near the

Order-Disorder Transition, Macromolecules 29, 1244-1251.

5. Anastasiadis, S.H., Chrissopoulou, K., Fytas, G., Appel, M., Fleischer, G., Adachi, K. and Gallot, Y.

(1996) Self Diffusivity of Diblock CopolYfi!Crs in Solutions in Neutral Good Solvents, Acta Polymerica

47, 250-264.

6. Boudenne, N., Anastasiadis, S.H., Fytas, G., Xenidou, M., Hadjichristidis, N., Semenov, A.N. and

Fleischer, G. (1996) Thermodynamic Effects on Internal Relaxation in Diblock Copolymers, Phys. Rev.

Lett. 77, 506-509.

7. Jian, T., Anastasiadis, S.H., Semenov, A.N., Fytas, G., Adachi, K. and Kotaka, T. (1994) Dynamics of

Composition Fluctuations in Diblock Copolymer Solutions Far from and Near to the Ordering

Transition, Macromolecules 27, 4762-4773; Pan, C., Mancer, W., liu, Z., Lodge, T.P., Stepanek, P., von

Meerwall, E.D. and Watanabe, H. (1995) Dynamic light Scattering from Dilute, Semidilute, and

Concentrated Block Copolymer Solutions, Macromolecules 28,1643-1653.

8. Jian, T., Anastasiadis, S.H., Semenov, A.N., Fleischer, G. and Vilesov, A.D. (1995) Interdiffusion and

Composition Polydispersity in Diblock Copolymers above the Ordering Transition, Macromolecules 28,

2439-2449.

9. Pusey, P.N, Tough, RJ.A. (1985) Dynamic light Scattering, in R Pecora (ed.), Plenum NY, pp. 85-179;

Baur, P., Nagele and Klein, R (1996) Nonexponential relaxation of Density Fluctuations in Charge­

Stabilized Colloids, Phys. Rev. E 53,6224-6237.

10. Fischer, E.W. (1993) light Scattering and Dielectric Studies on Glass Forming Uquids, Physica A 201,

183-206.

11. Chrissopoulou, K., Anastasiadis, S.H. and Fytas, G. (1996) Unpublished data.

12. Meier, G. Fytas, G., Momper, B. and Fleischer, G. (1993) Interdiffusion in a Homogeneous Polymer

Blend Far Above its Glass Transition Temperature, Macromolecules 26, 5310-5315 and references

herein.

13. Stepanek, P., Lodge, T.P., Kedrovoski, C. and Bates, F.S. (1991) Critical Dynamics of Polymer Blends,

J. Chem. Phys. 94, 8289-8301; Meier, G., Momper, B. and Fischer, E.W. (1992) Critical Behavior in a

Binary Polymer Blend as Studied by Static and Dynamic light Scattering, J. Chem PhyS: 97,. 5884-

5897.

14. Jian, T., Semenov, A.N., Anastasiadis, S.H., Fytas, G., Yeh, F., Chu, B., Vogt, S. Wang, F., Roovers,

J.E.L. (1994) Composition Fluctuation Induced Depolarized Rayleigh Scattering form Diblock

Copolymer Melts, 1. Chem. Phys. 100, 3286-3296.

140

15. Fytas, G., Vlassopoulos, D., Meier, G., Likhtman, A. and Semenov, A.N. Fluctuation Induced

Anisotropic Pattern Relaxation in Critical Polymer Blends, (1996) Phys. Rev. Lett. 76, 3586-3589.

16. Meier, G., Vlassopoulos, D. and Fytas, G. (1995) Phase Separation and Glass Transition Intervention in

a Polymer Blend, Europhys. Lett. 30, 325-330.

17. Karatasos, K., Meier, G., Vlassopoulos, D. and Fytas, G. (1996) Unpublished data.

LOCAL DYNAMICS IN BRANCHED POLYMERS

Abstract.

v. TRAPPE AND W. BURCHARD Institute of Macromqlecular Chemistry, University of Freiburg, Sonnenstr. 5, D-79104 Freiburg, Germany

The static and dynamic properties of four statistically branched systems at good solvent conditions have been investigated by light scattering. The q-dependence is analysed in terms of the particle scattering factor, the normalised apparent diffusion coefficient and the reduced first cumulant. The reduced first cnmulant proved to be a sensitive measure of the internal mobility, and the information on the internal structure of the polymer can be deduced from this quantity. Also an interdependence of the static and dynamic quantities was recognised, which appears to be characteristic for the density of the systems.

141

E. R. Pike andJ. B. Abbiss (ells.), Light Scattering and Photon Corre1ation Spectroscopy, 141-160. @ 1997 Kluwer Academic Publishers.

142

LOCAL DYNAMICS IN BRANCHED POLYMERS

1. Introduction

The flexibility of macromolecules is a property that is widely investigated by

rheological techniques. The resulting quantities describe relevant material properties,

but they are in conclusion only empirical values which are a result of a sum of

microscopic effects. Several attempts have been made to measure the microscopic

relaxation processes directly, where the time resolved techniques in fluorescence

spectroscopy and NMR-spectroscopy have been probably the most successful methods.

Here, however, only the tail of the very fast relaxations of the spectrum is measured

which gives insight into the motions of fairly short chain sections. For conclusions on

the macromolecules one has to rely on theories. Dynamic light scattering has the

advantage of giving molecular response from fluctuations in the larger time domain up

to about 103s. The dominant process is, of course, the translational diffusion of the

particles. but if the molecules have a dimension in the range of the wavelength of the

light then the internal motions contribute significantly. Such internal motions are

defined as motions with respect to the centre of mass and are fully independent of the

particles position in space, i.e. independent of the magnitude of the scattering vector q.

In a macromolecule a large number of internal motions are realised which are highly

coupled to each other, resulting in a rather complex spectrum of relaxation times. The

interpretation of dynamic LS data becomes even more difficult since the relaxation

time spectrum is not directly registered, but what is measured is a time correlation

function which is a sum of all possible relaxation processes with their amplitude

factors. These are now q-dependent even though that the corresponding relaxation

times are not. Thus the time correlation function is in a fairly intransparent manner

dependent on q and on t. For this reason it became imperative to perform detailed

143

calculations on a highly simplified model. This is presented by the spring-bead model

for linear chains that had been treated by Rousel and Zimm2 and applied to the

interpretation of viscoelastic measurements of linear chains. Pecora3-' extended these

calculations to the derivation of the dynamic structure factor S(q,t) and the time

correlation function

gl(q,t) = S(q,t)/S(q) (1)

in which S(q) is the common static structure factor of the individual macromolecule.

Pecora's general result is very complex and immediately called for simplifications to

make the theoretical equation applicable to interpretation of experiments. There have

been three principally different approaches which may be outlined in the following.

This outline results in a number of equations which may be used by the reader as a

collection of formulae, but which we wish to quote here, since it will help to follow the

discussion and the interpretation of experimental findings.

1.1. GENERAL FEATURES OF THE DYNAMIC STRUCTURE FACTOR

1.1.1. The slowest relaxations

P~ora himself focused his attention mainly on the slowest relaxation processes and

arrived after appropriate approximations at the series of eq (2)

where the q-dependent weighting factors Sj(q) fulfil the condition

n S(q) = So(q) + LSj(q)

j=l (3)

In solution the motions of the individual segments are coupled to each other through

the spring forces and in addition via hydrodynamic interactions. For Zimm-dynamics

the relaxation times scale with 'tj == 't}/P/2 and under neglect of the hydrodynamic

interaction (Rouse) with 'tj = 'tI/P. When q is increased one obtains in both cases

lI'tj »Dq2 and the translational motions become negligible.

There have been several attempts to determine the first few relaxation times. Of these

experiments the recent one by Wu et al. 6 may be quoted here.

144

1.1.2. The infinitely long chain

De Gennes7,8 and Dubois-Violette8 considered the limit of infinitely long chains and

expressed the various sums as integrals. They found:

gl (q, t) = j exp{ -u-(tlt*)mh( u.(tlt*)-m)} du o

(4)

(5)

The exponent m has a value of 2/3 for chains with strong hydrodynamic interactions

(Zimm) and 112 for chains without hydrodynamic interactions (Rouse). t* is a charac­

teristic mean relaxation time and was shown to decay with

t* _q-2/m (6)

Eq (4) is obviously no longer a Laplace integral. At large delay times it reduces to a

stretched exponential

gl(q,t) t»t*) exp(-b(t/t*)m) (7)

and a closer inspection of the integral reveals that the whole TCF can be expressed as a

universal function of tlt*

gl (q, t) = exp(f(t 1 t*») (8)

The essential feature of the so called shape function f(tlt*) is that the q-dependence of

gl(q,t) no longer appears explicitly but is fully included in the characteristic time t*.

Akcasu et al.9 identified t* as being the inverse first cumulant, such that for infinitely

long chain one can write

(9)

in which the first cumulant r is the initial slope of the logarithmic TCF. The relevant

property of eq (9) is: whatever q-value may be chosen, a universal curve results for

gl(q,t) when the delay time is mUltiplied by the first cumulant.- In practice, of course,

the chains are not infinitely long, and at ql\ < 1 the translational diffusion shows

influence, that soon becomes dominant (ql\« 1). However, according to an

145

estimation by Pecora the influence of translational motion becomes negligible when

qRg > 2, which has been confirmed experimentally in several laboratories.

l.l.3. The first cumulant

Since the first cumulant of the field TCF was recognised as the decisive scaling

parameter, AkcasulO wondered whether this important quantity may be derived by a

completely different approach in which the specific dynamic model is introduced at a

point as late as possible. Such an approach was developed by Bixon11 and Zwanzig12

and is known nowadays as the Zwanzig12 -Mori 13 projection operator technique.

Accordingly the field TCF of dynamic LS can be expressed as

where

g (q t) = (p*p{t») 1, (p*P)

(p*e-t.fp)

(p*p)

N P = LexP(iqrj)

j=l

(10)

(11)

and p* being the conjugate complex to p. p(t) is defined in the same manner, but now

in terms of rj<t) which describe the positions of the j-th segment at some times t later.

The so far not specified operator of has the effect of transforming a static property into

a time dependent one. The projection operator technique proves that a solution of

eq (10) is obtained from the generalised differential equation of motion

(12)

with r- (p*.,tp) __ c)Ing1(Q,t)/

- (p*p) - ()t t~ (13)

To find a complete solution for gl(q,t) the memory function has to be taken into

account. This, however, is a very complex integral and can only in rare cases be

expressed by a simple, still non-linear function (for details see ref. 14). Because of the

immense difficulties in derivi!,g the memory function and solving the non-linear

differential equation, the memory term is mostly neglected which means, of course,

abandoning a correct description of the time correlation function gl(q,t) at long delay

times. However, r(q) can be calculated from eq (13), but to this end the operator of has

to be specified. For the spring and bead model the operator is obtained from Ki kwood' al'sed diffus' . 15 16 r s gener 1 Ion equation. '

146

(14)

'I' o( {r}) denotes the multi variant equilibrium distribution of the segments in space

(the curly brackets denote the whole set of coordinates of the N segments in the chain).

Finally the diffusion tensor is given as

(15)

with the Oseen Tensor

(16)

that describes the hydrodynamic interactions among the segments. Inserting eq (11)

and eq (14) into eq (13) one finally obtains

~*(g·~jk -g)eXP(iqRjk») r(q)= J N N

:r.:r.(exp(iqRjk») j k

(17)

Concerning the required averaging procedure Akcasu et al.17 introduced the following

approximations.

- the preaverage approximation

(~.Djk .~)eXP(iqRjk») =q2(Djk)(exp(iqRjk») (18)

- the uniform expansion approximation for a coil in a good solvent. Assuming

Gaussian statistics one obtains

where (Rjk)=b2Ij-krV is the mean square distance of the segment k to the

segment j, and b is the effective bond length.

147

With

(20)

they finally obtained a function describing the q-dependence of the first cumulant over

the whole range of q.17 Since we are mainly interested in the high q-regime, only the

result for the asymptotic region at large ql\ is presented.

r*( ) = r(q) '110 = r[(I-v)/2v] q q3 kT 67Ut1/2r[1I2v]

(21)

with nx) being the gamma function. For the dynamics at O-conditions (v = 1/2) this

reduces to r*(oo) = 1/611: and becomes r*(oo) = 0.071 for the dynamics at good solvent

conditions.

2. Results and Discussion

A theoretical treatment of the dynamics of branched polymers is at present not or only

partly available. The main difficulties in the description of statistically branched

structures consist in the development of large size distributions and the realisation of

numerous isomeric structures. In this contribution we attempt to give an empirical

approach to this problem by discussing the q-dependent behaviour in static and

dynamic light scattering of four different systems at infinite dilution. These systems are

shortly introduced here with respect to branching mechanism and topology to enable a

discussion of the experimental results in terms of polydispersity and structural density

at different length scales. With scheme 1 an overview is given.

Three of the systems under investigation are randomly branched systems, which are

fairly well described by the three dimensional percolation theory,18,19 i.e. the

crosslinked polycyanurate (c-PCyan), the crosslinked polyester chain (c-PEst-chain)

and the crosslinked (end-linked) polystyrene star (c-PS-Star). On the other hand, the

amylopectin is a biopolymer with areas of high branching density (see Scheme 1, areas

surrounded by a dotted circle) which are linked via linear sections. Amylopectin is a nonrandomly, hyper-branched material and can be described by the ABC-model. 20-23

The different models will not be discussed in detail here, and only the results are

mentioned, which are relevant for the further discussion. One of these results is that

the polydispersity ratio MwfMn increase linearly with ~ for the randomly branched

148

Randomly Branched

Aa - Polycyanurale C-PCyan

-1 MO= 4179 mol

Crosslinked Polyester Chains c-PEst-chain

·1 Me,:: 15 000 9 mol (chain)

Crosslinked Polystyrene Stars CoPS-star

-1 Mo :: 65 000 9 mol (star)

Nonrandomly Branched

Amylopectin

-1 M :: (6000-aOOO) 9 mol o

------------------------'

SCHEME 1: Overview of the investigated systems. The indicated Mo corresponds to the molar

mass of the unimer. The unimer of each system is surrounded by a box. The dotted circle in the

amylopectin structure encompass the areas of high branching density, which on a sufficiently

large length scale can be treated as a single branching point with a functionality of 4 or 5.

149

structures (percolation), whereas for the amylopectin Mv/Mn increase with Mw 112

(ABC-model).

A preliminary ordering of the investigated structures with respect to their branching

density is given by the following two characteristics.

(1) The size of the basic unit, i.e. the unimer, decreases in order: c-PCyan

(Mo = 417 g1mol) < amylopectin (Mo == 6000-8000 g1mol) < c-PEst-chain (Mo == 15000 g1mol) < c-PS-star (Mo == 65000 g1mol). Evidently, the spacing between two

branching points increases with increasing unimer size.

(2) The functionality of the branching points increases in order: c-PCyan, c-PS-star (3)

< c-PEst-chain (4) < amylopectin (4-5). A functionality of 4 to 5 results for the

amylopectins, since on a sufficiently large length scale the amylopectin appears as a

system of partially end-linked 4-arm and 5-arm stars.

Samples of different weight average molar mass were obtained as follows: for the c­

PCyan system by stopping the reaction at different extent of branching, for the c-PEst­

chain and c-PS-star systems by varying the ratio of crosslinker per unimer (chain, star)

and leading the reaction to full conversion, for the amylopectin by degradation to

different degrees.

In Figure 1 the data obtained by static light scattering are shown for various samples of

the c-PEst-chain system and the amylopectin in a Kratky representation. A common

static behaviour is found for the various samples of the randomly branched c-PEst­

chain system, in spite of the large variation in the distribution width of the different

samples (Mv/Mn - Mw). This observation is due to the fact that randomly branched

clusters are fractals in a twofold manner. First, each single cluster is a fractal of

dimension d.w and second, the cluster ensembles at different extent of branching are

self-similar to each other. Both self similarities result in the ensemble fractal

dimension dtr,e of the non fractionated sample, which is connected to dtr by dtr,e = dn-(3-1) where 't is the expOnent describing the power law decay in the number

distribution of the clusters. Within the limit of experimental error the data of the c­

PCyan and c-PS-star systems exhibit the same behaviour as found for the c-PEst-chain

system. This fact is worth emphasising, because the unimers of these systems are very

dissimilar to each other. In the c-PCyan system the unimer is a low molecular weight

monomer, it is small, rigid and disclike in shape. In the c-PEst-chain system the basic

unit is a flexible, narrowly distributed chain, that is more than 20 times larger than the

c-PCyan unimer, and in the c-PS-star system the star unimer has long, flexible arms

and is more than 80 times larger than the c-PCyan unimer. The resulting clusters differ

in the spacing of their branching points and in the functionality of the branching points

(3 in c-PCyan and c-PS-star; 4 in c-PEst-chain). The static scattering <is apparently not

150

6

5 N

'"' 4 Ol

a::: tT

3 '-.J

'"' tT 2 '-.J

a..

0

3

N

'"' 2 Ol

a::: tT

'-"

'"' tT '-.J

a..

0

a. c-PEst-chain

0 2 4 6

b. amylopectin

0 2 4

B

e.. R _ 40nm g

o Rg- 67nm

V Rg- 119nm

o R,-245nm

¢ R,-341nm

10 12

R,-71nm

14

_____ e"" ___ _

--------- R -110nm

----- - - - - - - - - - -g-. - - - - -

6 B

FIGURE 1: Kratky representation of the static LS-data obtained for various samples of the c­

PEst-chain system and the amylopectin. The c-PEst-chain system is shown as representative

example for the randomly branched systems.

affected by these topological details and seems to be purely dominated by the simple

concept of co~tivity in the ensemble of randomly branched clusters.

In contrast, the various degraded amylopectins do not exhibit a common behaviour in

static light scattering (see Figure lb), which indicates that the cluster ensembles of the

different samples are here not self-similar to each other.24

E c:

" .r. 0::::

DI 0::::

E c:

" .r. 0::::

DI 0::::

o. c-PEst-choin ..... :!:.j ..... .' .,A.'

....... ~:.;;..: '-'-' -=-==--:! •• ""p. •••• ". - 0.58+0.02

..... •• .I!r. h -.: .'

". - 0.511:1:0.03 " •• ' ... A· , ....... .... I ... :k' tS

... ~::: ... . •• ·~Lr·· .. '

102

101

100

102

10 1

b. amylopectin .e= ....... . ... ::::. ·····_:::::0'·-1

•• , •• 0- .

"II, - 0 .... 0±0.02 ....... :!:::o;· _ 0 .... 7+0.02 '.' "h -I .... - 0. •• , . •••••• .-0'"

•• ' • m'-O

.. , •• r:fJ"

100 ~~~~~ __ ~~~~ __ ~-w~w-~~~~~~

10'"

-1 Mw / 9 mol

151

FIGURE 2: Variation of the radius of gyration Rg and the hydrodynamic radius Rb as a function

of the weight average molecular weight Mw for the randomly branched c-PEst-chain system and

the nonrandomly branched amylopectin.

In Figure 2 the increase of the radius of gyration l\ and the hydrodynamic radius Rtt with Mw are shown for the c-PEst-chain system and the amylopectins, where again the

c-PEst-chain system is regarded as a characteristic example for the randomly branched

systems. The main difference between the two branching classes consists in the parallel

development of the two radii for the randomly branched systems, whereas for the

amylopectins the hydrodynamic radius increases faster with Mw than the radius of

gyration. The latter is expected for structures with increasing density, since the ratio

P=R/Rh (22)

is lower for compact structures (hard sphere: P = 0.78) than for open conformations

(linear chain under good solvent conditions: l7 P = l.86). In Table 1 the p-parameter of

the investigated systems are listed.

152

TABLE 1: The p-parameter (p = R/Rtt) of the investigated systems.

system

c-PS-star c-PCyan

c-PEst-chain

amylopectin

functionality of the branching point

3 3 4

4-5

p comment

1.94 ± 0.10 1.80 ± 0.05 independentof~

1.45 ± 0.05

1.8 ~ 1.0 dependent on ~

The p-parameter is affected by the polydispersity of a system and can therefore not be

regarded as a reliable structural parameter for the investigated, highly polydisperse

systems. Nevertheless, the experiments demonstrate that p decrease with increasing

functionality of the branching point of the systems which is, as already pointed out, a

possibility of ordering the various systems in the sense of local density.

For the further discussion it is essential to emphasise the parallel development of ~

and Rtt with Mw for the randomly branched systems, resulting in an equivalence of the

normalisation when using ~ and Rtt as scaling parameter. In other words, if the data

of the various samples of a system exhibit universal behaviour when plotted versus qRg they will do the same when qRtt is chosen as dimensionless length scale. This is not

expected for the nonrandomly branched amylopectins (decreasing p with increasing

Mw), and it turns out that the q-dependent dynamic LS data of the various amylopectin

samples scales with qRtt but not with qRg.

In Figure 3 the normalised apparent diffusion coefficients Dapp(q)/Dz are plotted versus

qRtt for the four systems under discussion, where Dapp(q) == r(q)/q2, and Dz is

translational diffusion coefficient obtained by extrapolation of Dapp(q) to q = O. The

apparent diffusion coefficients of the c-PCyan, the c-PEst-chain and the amylopectin

do not increase linearly with q at high 'QRtt, as predicted and experimentally found25-33

for monodisperse, linear systems in the Zimm-limit, but follow in the high qRtt-regime

power laws of D app( q) - qO.80±0.05 and D app( q) - QQ' 76±O.05, respectively. In contrast,

for the c-PS-star system a linear q-dependence ofDapp(q) - ql.OO±O.05 is found. In terms

of the first cumulant the asymptotic power-law varies in these examples between req) - q2.76 and req) _ q3.

v c-PS-star t::. c-PEst-chain • c-PCyon o amylopectin

FIGURE 3: Dapp(q)/Dz as a function of qRtt for the four systems indicated.

'"' {T -* L

0.10

0.05

0.00 o

v c-PS-atar

t::. c-PEat-chain

• c-PCyan

o amylopectin

:!!!JII-.IJii#1l' .................................... ........ .

Stokes- Einstei n

2 4 6 8

153

0.082

0.045

FIGURE 4: Variation of P( q) as a function of qRtt for the four systems indicated. The bold

line represents the translational behaviour of hard spheres. The dotted line corresponds to the

plateau values found experimentally for linear systems under a (v = 112; P( 00) == 0.045)27,30

and good solvent conditions (v = 3/5; P( 00) == 0.062)?9,32

154

When plotting the data as r*(q) versus qRtt (Figure 4) the differences in behaviour are more clearly disclosed. It may be noted that r*(q) (eq 21) is a dimensionless quantity and includes no polymer specific normalisation. When. as expected for linear chains in the Zimm-limit, a q3 dependence of the first cumulant i~ reached, r*(q) becomes a constant, r*(oo). The value of this constant is predicted (see eq (21» to depend only on the solvent quality for linear systems, i.e. the v-parameter. For the branched systems c­PCyan, c-PEst-chain (r(q) - q2.S) and amylopectin (r(q) - q2.76) the r*(q)-values, of course, decrease continuously, and only for the copS-star system (r(q) - q3) a constant r*( 00) is found. The really interesting point here is the observation of clear differences in the absolute values of r*(q) at high q for the different systems. The r*(oo)-value found for the copS-star system corresponds to that obtained for linear chains in good solvents (r*(oo) = 0.062 ± 0.003, ref. 29 and 32). The r*(q)-values of the c-PCyan and the c-PEst-chain drop slightly below the values found for linear systems under e conditions (r*(oo) = 0.045 ± 0.003, ref. 27 and 30), and for the amylopectin the r*(q)­values drop even further to r*(q) == 0.03. As already mentioned, a decrease ofr*(oo) is expected with decreasing v for linear systems. The parameter v, however, can be understood either as a parameter describing the solvent quality or as the inverse of the fractal dimension. The present four branched systems were investigated under good solvent conditions. Thus the differing r*(q)-values at large qRtt cannot be attributed to differences in excluded volume interactions, but must be correlated to structural

differences. In Table 2 the fractal dimensions of the branched systems are summarised.

TABLE 2: Fractal dimensions of the investigated systems in comparison to these of linear,

monodisperse systems at 9 and good solvent conditions. The index /r denotes the fractal

dimension of the single cluster and the index/r,e the fractal dimension of the cluster ensemble.

For the percolation systems the polydispersity correction for ~,e is indicated in the last column.

Systems drr,e drr = drr.t!(3Jt)

't = 2.2 (percolation)

linear chains (good solvent conditions) 1.67 1.67

linear chains (9 conditions) 2.00 2.00

c-PCyan 1.74 ± 0.02 2.18 ±0.03

c-PEst-chain 1.72±0.06 2.15 ±0.08

CoPS-star 1.82 ± 0.07 2.28 ± 0.08

amylopectin 2.3±0.2

155

One realise, that dtr.e of the amylopectins is considerably larger than those of the

randomly branched systems and, consistent with eq 21, the lowest r*(q)-values are

found for this system. For the three randomly branched systems basically the same

fractal dimension is obtained, which for the single cluster is around dtr = 2.2 ± 0.1 and

corresponds to the fractal dimension of a fairly swollen cluster. For the c-PCyan system

and the c-PEst-chain system the r*(q)-values drop slightly below the asymptotically

found r*( 00 )-value of linear chains at a conditions, which is consistent with the fact

that the fractal dimension of the clusters at good solvent conditions is similar to that of

linear chains at a conditions (dtr = 2). In a tentative interpretation we conclude that the

internal motions are determined by sections of the clusters that include branching­

points guaranteeing self-similarity to the complete branched clusters (dtr == 2.2). The

fact that the r*( q)-values of the c-PS-star system reach the plateau of linear systems at

good solvent conditions indicates that here linear sections of the clusters dominate the

internal dynamics, which agrees with the large spacing between the branching points

(largest unimer size) in this system.

Clearly, the absolute values of r*(q) at high q are correlated to the internal density of

the system and could be used as a valuable parameter for the characterisation of

PQlymers. It is worth recalling that r*(q) is independent of polymer specific quantities,

i.e. the size of the polymer or the friction coefficient of the segment. Experimentally, a

problem consists only in the proper determination of the first cumulant from the

correlation function obtained at high q~. Because of the strongly non exponential

behaviour in this region, this is not a trivial problem.

It remains to find an explanation for the q-dependence of r(q) that deviates from q3.

Two possibilities have to be considered. (1) The q-dependence around q2.8 is due to polydispersity. The internal motions of the

large clusters are superimposed on the translational motion of the smaller clusters.

These show a q2-dependence and will thus reduce the q3-dependence. The

superposition-regime is not "seen" for the c-PS-star system, because the samples of this

system have a much lower polydispersity-ratio than the samples of the same molar

mass of the other systems (c-PCyan,. c-PEst-chain and amylopectin).

(2) The q2.8-dependence is the result of a hindered internal mobility in branched

systems. The q3-dependence of the :first cumulant predicted for linear systems may not

necessarily hold for densely branched systems since the constraint exerted by branching points on the fairly short linear sections between two branching points may

have a non negligible influence.

156

N o

" ..--. 0-........

a. a. c

o

L1 c-PEst-chain

• c-PCyan

o amylopectin

FIGURE 5: DapplDz as a fimction IlPz(q) for the systems: c-PEst-chain, c-PCyan and amylopectin. The nearly straight line can be described by power laws (eq 23) with exponents of n = 0.54 ± 0.01, n = 0.46 ± 0.01 and n = 0.40 ± 0.01, respectively.

As final item the correlation between the particle scattering factor and Dapp(q) may be discussed. Figure 5 shows plots of the normalised apparent diffusion coefficients versus the inverse form factor for the c-PCyan system, the c-PEst-chain system and the amylopectins. Interestingly the data obtained for the different samples of the various systems form master curves, which appear to be characteristic for the different systems. Even for the amylopectin samples, such master curve is found, although the form factors exhibit no universal behaviour when expressed as a function of ql\ (see Figure Ib). Nearly straight lines are found in the double logarithmic plots, which

allows to express the dynamic LS data in terms of the static structure factor.

(23)

This means that the transition from initial to asymptotic behaviour in dynamic and static light scattering occurs in the same q region and progress in a similar manner. For static light scattering it is known18,19 that the q-dependence is determined by the

structure and polydispersity of the system. The observed behaviour seems to imply that,

157

at least for the length scales probed by light scattering, these structure and polydisper­

sity characteristics have a similar influence on the q-dependence in dynamic LS.

A further result is that for Dapp(q)/Dz and Pz(q) are suitable scaling functions for each other, which is in agreement with predictions derived previously.34

However, the meaning of the exponent n remains to be evaluated. For the c-PEst-chain

system, the c-PCyan system and the amylopectins the exponents are n = 0.54 ± 0.01,

n = 0.46 ± 0.01 and n = 0.40 ± 0.01, respectively. In the context of the already

discussed findings and the knowledge of the structures, the decrease in n seems to

correlate with the increase in the absolute density. Of all systems discussed here the

amylopectin has the highest density. This correlates with the lowest p, the highest

fractal dimension and the lowest asymptotic r*(q)-value. A differentiation between the

c-PCyan-system and c-PEst-chain-system seems to be difficult because of the same

behaviour in static LS, the same fractal dimension and nearly the same behaviour of

r*(q) as a function of qRtt. In contradiction to the structural understanding of these

two systems, the p-parameters would indicate that c-PCyan is less dense than c-PEst­

chain. Of course c-PCyan, have a lower functionality of the branching point than c­

PEst -chain, but the spacing between two branching points will be much larger in the c­

P~st-chain system than in c-PCyan system. The result in Figure 5 is therefore in a

better agreement with the more intuitive ordering of the system, where c-PCyan is

more dense than c-PEst -chain.

The scaling of the whole TCF in terms of r-t has been considered in two separate

papers.35,36 There it was shown that a universal sl}ape function is indeed found for the

TCFs obtained at qRg> 3. This function exhibits a shape similar to that predicted for

linear chains with hydrodynamic interactions. However, this statement is implicated with some uncertainties in the determination of the first cumulant. Another problem

arises from the large polydispersity of the investigated systems. These points are only

mentioned here and will not further be considered. We refer to the original papers.

3. Conclusion

The present experiments with four branched systems revealed that branching has an

influence on the internal motions if the branching density is high. At high q-values the

first cumulant r(q) exhibits approximately a q2.8 dependence for the systems with high

branching density and a q3 dependence for the system with the largest repeating unit,

i.e. largest distance between two branching points. From an analysis of the asymptotic

power law behaviour alone it remains unclear whether the motions of the branching

158

units influence the behaviour or whether simply the very broad size distribution causes

the observed effects. However, the absolute values of the reduced cumulant r*(q) == r( q)llJ( q3kn at high q-values proved to be a sensitive measure of the internal

mobility. By comparing the r*(q) values from the branched systems with those from

linear systems under e and good solvent conditions the nature of the local dynamics in

the various systems can be classified: for the system with the largest repeating unit the

local dynamics of linear sections in the clusters are registered, whereas for the systems

with smaller repeating units, i.e. higher branching densities, the local dynamics of

branched sections are probed. Although an asymptotically constant value of r*(q) is

not obtained for the higher branched systems, a correlation between the asymptotic

r*(q)-values and the fractal dimension is clearly perceived.

An interdependence of the first cumulant and the particle scattering function has been

demonstrated by plots of Dapp(q)/Dz versus Pz(q). It can be approximated by a power

law, where the resulting exponent seems to be characteristic for the absolute density of

the investigated systems. However, a clear physical assignment of these exponents has

not yet been possible, since the existing theory only formally supports these findings.

4. Acknowledgement

The data for the amylopectin, the c-PS-star system and the c-PCyan system were taken

from the PhD-Theses of Gabriela Galinsky and Max Weissmillier and from

unpublished results of JOrg Bauer, Fraunhofer Institute Teltow. We greatly appreciate

their contributions and thank them for their help.

5. References

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18. Stauffer, D., Cognilio, A, and Adam, M. (1982) Gelation and Critical Phenomena, Adv.

Polym. Sci. 44, 103-158.

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20. Flory, P.J. (1953) Principles in Polymer Chemistry Cbapt 9, Cornell University Press,

Ithaca and London.

21. Erlander, S. and French, D. (1956) A Statistical Model for Amylopectin and Glycogen. The

Condensation of A-R-Bt_l' Units, J. Polym. Sci. 20, 7-28.

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Macromolecules 10, 919-927.

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Biopolymers, Adv. Polym. Sci. 48, 1-124.

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schaften der Starke, Freibmg.

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Solvent by Raleigh Scattering Experiments, Macromolecules 10, 1229-1237.

26. Han, C. C. and Ak:casu, A. Z. (1981) Dynamic Light Scattering of Dilute Polymer Solutions

in the Nonasymptotic q Region, Macromolecules 14, 1080-1084.

27. Tsunashima, Y., Nemoto, N., and Kurata, M. (1983) Dynamic Light Scattering Studies of

Polymer Solutions. 2. Translational Diffusion and Intramolecular Motions of Polystyrene in

Dilute Solutions at 9 Temperature, Macromolecules 16, 1184-1188.

28. Nemoto, N., Makita, Y., Tsunashima, Y., and Kurata, M. (1984) Dynamic Light Scattering

Studies of Polymer Solutions. 3. Translational Diffusion and Internal Motions of High

Molecular Weight Polystyrenes in Benzene at Infinite Dilution, Macromolecules 17, 425-

430.

29. Tsunashima, Y., Hirata, M., Nemoto, N., and Kurata, M. (1987) Dynamic Light Scattering

Studies of Polymer Solutions. 5. Universal Behavior of Highly Swollen Chains at Infmite

Dilution Observed for Polyisoprenes in Cyclohexane, Macromolecules 20, 1992-1999.

30. Tsunashima, Y., Hirata, M., Nemoto, N., Kajiwara, K., and Kurata, M. (1987) Dynamic

Light Scattering Studies of Polymer Solutions. 6. Polyisoprenes in a 9-solvent, 1,4-Dioxan,

Macromolecules 20, 2862-2866.

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Fluctuations in Polymer Solutions,Phys. Rev. Len. 56/1,61-64.

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Tetrahydrofuran at Intennediate Scattering Vectors, Macromolecules 21,3015-3022.

33. Bhatt, M., Jamieson, A. M., and Petschek, R. G. (1989) Static and Dynamic Scaling

Relationships in the Light Scattering Properties of Polystyrene in Good Solvents,

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in Static and Dynamic Light Scattering from Randomly Branched Systems, Macromolecules.

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molecules.

SPONTANEOUS DOMAIN GROWTH IN THE ONE-PHASE

REGION OF A GEL/MIXTURE SYSTEM

BARBARA J. FRISKEN Department of Physics Simon Fraser University Burnaby BC V5A 1S6

ARTHUR E. BAILEY Scitech Instruments Inc. North Vancouver BC V'lJ 2S5

AND

DAVID S. CANNELL Department of Physics University of California Santa Barbara CA 93106

Abstract. We present results of experiments designed to investigate a novel domain state occurring in a binary fluid mixture when imbibed in dilute silica gel. This state occurs near the coexistence curve of the pure binary mixture (in the absence of gel), but above the coexistence curve of the bi­nary mixture/gel system. The growth rate of the domains is proportional to t 1/ 3 and most measurements are consistent with dynamic scaling. How­ever, there is an additional k-independent response which is observed at very early times following the quench, which is unlike the behavior of the pure system.

1. Introduction

Interest in the effect of disorder on critical systems was greatly stimulated by a paper of fmry and Ma (Imry and Ma, 1975) in which it was argued that a 3D-Ising system in the presence of a weak random field should still achieve long range order at low temperatures. Experimental realization of such a

161

E. R. Pike andJ. B. Abbiss (eds.), Light Scattering and Photon Correlation Spectroscopy, 161-171. © 1997 Kluwer Academic Publishers.

162

system became feasible only when Fishman and Aharony (Fishman et at., 1979) showed that a uniform magnetic field applied to a doped uniaxial antiferromagnet resulted in local random fields. The most well-known ex­amples of physical systems which are consistent with this model are several diluted anti-ferromagnets (Belanger and Young, 1991) which show behavior consistent with the random-exchange Ising model when cooled in zero field, and with the random-field Ising model when cooled in an external field.

The effect of disorder on fluid systems has mainly been tested by placing either single component fluids or two component mixtures of fluids into porous media. The porous media can have two basic geometries - either interconnected pores such as Vycor glass (Dierker and Wiltzius, 1987; Aliev et al., 1993; Lin et al., 1994), or interconnected strands such as gels (Maher et al., 1984; Wong and Chan, 1990; Frisken et al., 1992). Behavior of systems in these two geometries appears to be quite different, where the behavior in pores appears to be dominated by confinement effects (Monette et al., 1992). Further complications can occur if the gels are flexible, rat~er than rigid (Xia and Maher, 1987).

There are some similarities between the fluid/gel systems and random­field systems. The surface of the gel prefers one component of a fluid mix­ture, or the fluid phase of a single component fluid. Thus the gel can be modeled as a quenched field which is generally randomly ordered in space, although there can be some short length scale correlations (Brochard and de Gennes, 1983).

We have been involved in a series of experiments focusing on confining two binary fluid mixtures, lutidine/water (LW) and isobutyric acid/water (IBAW), to dilute silica gels. These gels are easy to make and characterize, are rigid, and do not change structure significantly in the presence of the binary mixture.

Studies at temperatures in what would be the one-phase region of the pure system reveal large changes in the amount of light or number of neu­trons scattered by the system, but small changes in the structure (Frisken et al., 1995a; Frisken et al., 1995b). These results were interpreted in terms of preferential adsorption of one component of the fluid mixture by the silica. As one component becomes preferentially attracted to the silica-rich regions of the sample, the remaining fluid becomes depleted in that component. By comparing the scattering of gel/mixture samples at different temperatures to the scattering of comparable gels containing only water, it was possible to estimate the concentration of the non-adsorbed fluid. Results for various properties of the fluid, the diffusion coefficient in particular, were consis­tent with the critical behavior being determined by the concentration of the non-adsorbed fluid rather than the average concentration of the sample.

In these experiments, it was also discovered that the behavior following

163

a small temperature drop changed abruptly close to the coexistence curve of the pure system. Instead of a gradual change of the intensity following a temperature change, the scattering increased dramatically and then de­cayed slowly, becoming quite dependent on sample history. This behavior was interpreted as indicating that the samples had entered the coexistence curve of the gel-mixture system and that the large overshoots were indica­tions of phase separation. However, the samples cleared only very slowly, sometimes over the course of several weeks. Even when samples did clear it was not obvious that macroscopic phase separation had occurred as samples did not generally show a meniscus.

More recent experiments (Zhuang et al., 1996) involving measurement of the concentration of IBAW mixtures in silica gel as the concentration of an external reservoir was changed revealed a true coexistence region which is greatly altered from that of the pure system. The new coexistence region is considerably narrower, lies below and to one side of that of the pure system. In these respects it is very similar to coexistence regions observed in systems consisting of single-component fluids, He and N2 , in dilute aero­gel (Wong and Chan, 1990; Wong et al., 1993) and SF6 in controlled pore glasses (Thommes and Findenegg , 1994). However, the discovery of this coexistence region means that the observations described above which were interpreted as phase separation really are the hallmark of a third state in the gel-mixture system which lies between a homogeneous one-phase region and the region of true two-phase coexistence.

The experiments described below were designed to explore this region by doing quenches into it and studying changes in structure at early times and long length scales.

2. Experiment

The silica gels were made by polymerization of a silica precursor (tetram­ethylorthosilicate) in water by a two-step process (Avnir and Kaufman, 1987; Cabane et al., 1990). The gels used in these experiments were about 2% silica by volume. They are easily characterized by scattering experi­ments (Ferri et al., 1991). At length scales smaller than the gel correlation length ~G the gel scatters like a mass fractal of fractal dimension D f. At length scales larger than ~G scattering from the gel is independent of scat­tering wavevector k indicating that there are fluctuations of equal ampli­tude on all larger length scales. Gel samples were made in a circular slab contained between two thin glass windows separated by about 2.0 mm.

The binary mixture used in these experiments was isobutyric acid/water (IBAW). The pure system (in the absence of gel) has a critical temperature of 26.67°C and a critical concentration of 38.8 wt. %. Mixtures of various

164

! t

Figure 1. Cross-sectional view of the cylindrical apparatus. The gel-mixture sample is contained in the dotted region. The arrows indicate the flow direction of the water used to control the sample temperature. The sample is bounded by 0.25 mm thick flexible windows in the axial direction, and both the sample and the water in the adjacent chambers are quenched together to minimize axial heat flow. The lens maps light scattered at a given angle onto a circle in a plane containing concentric rings of optical fibers leading to detectors D1 - D 24 •

concentrations were diffused into the gels by placing mixture in contact with the gel structure (after one of the glass windows had been removed) and waiting for several days. The mixtures were changed twice a day to make sure that the external concentration remained constant. After the samples had a chance to equilibrate, the external mixture was removed and the window replaced so that the average concentration of the sample would remain constant during the measurement. The average concentrations of the fluid portion of the samples were determined to be 22, 24, 31, and 35 wt% IBA by gas chromatography.

Light scattered at small angles, which yields information about larger structure in the sample, was measured in an apparatus specifically designed for these type of measurements (Bailey and Cannell, 1993). As shown in Fig. 1, light from a He-Ne laser was incident along the cylindrical axis of the sample. Light scattered by the sample was detected in the focal plane of a lens glued to the exit window of the sample chamber. The detector consisted of concentric rings of optical fibers, with each ring corresponding to scattering at a particular angle () and hence scattering wavevector k = 4~n sin () /2, where n is the refractive index of the sample and A is the

165

IQ4

1000 ...... i 100 8 () 10 ......

...... 1 ..lI: ...... rn 0.1

0.01 • ------------.--------------.--0.001 •

100 (b) 35 wt." IBA

~ I

8 c

0 10 ...... ...... ..lI: ...... rn

1

a IQ4 2xlQ4 3xlQ4

k (em-1)

Figure 2. Structure factor at various times following a quench to 0.5 °C below TO for the (a) 24 wt% and (b) 35 wt% IBA samples. The magnitude of the scattering from the pre-quench systems is shown as dashed lines.

wavelength of the incident light in vacuum. The sample was separated from the adjacent water reservoir by two thin windows which flexed sufficiently so as to maintain the sample pressure equal to that of the reservoirs. The pressure of the reservoir was controlled by an external piston. The sample chamber was located in the center of a cylindrical temperature controlled housing. This kept the average temperature of the sample chamber constant to ± 15 11K. To change the temperature of the sample quickly enough to be able to perform an effective quench into the domain state, the pressure of the sample was dropped. This produced a fast, uniform increase in critical temperature (-3.74 mK/psi (Morrison and Knobler, 1976)) and an overall temperature decrease (+0.34 mk/psi (Wong and Knobler, 1978)) for both the sample and the adjacent water reservoirs; the net result of both effects being a variation of T - Tc with pressure of d(T - Tc)/dP = +4.08 mk/psi. A typical quench of 150 psi (and corresponding (T - Tc) change of 0.51 0C) could be accomplished in approximately 15 msec.

166

We have used this apparatus to investigate the abrupt changes in scat­tering observed previously (Frisken et ai., 1995a) near the coexistence curve of the pure system. Overshoots in the scattered intensity were observed on decreasing the temperature of the three lowest concentration samples be­Iowa certain value of temperature T*. Overshoots in the intensity of the high concentration sample were only observed on pressure quenching the sample. Results for quenches to temperatures 0.5 °C below T* are shown for two samples in Fig. 2. Figures 2a and 2b show data typical of lower con­centration samples and the sample of highest concentration, respectively. Data taken at various times after the quench show the growth of a peak in the structure factor whose location km moves to smaller and smaller k as time following the quench increases. The existence of a peak in the struc­ture factor is consistent with fluctuations in concentration occurring on one dominant length scale in the system. We associate the growth of these fluc­tuations with growth of domains of different concentration in the sample. The amplitude of the peak S(km ) grew with time, while the scattering at k higher than km initially increased and then decreased to approach a value above that of the prequench data. The highest concentration sample gen­erally showed much smaller increases in the scattered intensity, and much faster decreases in intensity following the initial overshoot.

Figure 3 shows the change in km as a function of time following the quench for quenches to various depths below T* for the samples shown in Fig. 2. Figure 3 shows that km decreases in time in a manner consistent with km ex: r 1/ 3 as is generally observed in off-critical quenches of the pure system (Wong and Knobler, 1978). In other words, no evidence of domain pinning or logarithmic slowing down was observed. The dashed line in Fig. 3 shows the approximate location of data for the pure system. The data show that the dominant length scale for domain size in the gel/mixture system is about a factor of two bigger than it is in the pure system.

Figure 4 shows the change in S(km ) as a function of time following quenches to various depths below T* for the samples shown in Fig. 2. The data for the lower concentration sam pIe is consistent with the peak height growing linearly with time for all of the quenches studied. This behav­ior is also observed in off-critical quenches of pure system samples (Wong and Knobler, 1978). The highest concentration sample did not show S(km )

increasing linearly with time for any quench depth. We believe that this observation and the fact that intensity overshoots on cooling were not ob­served in this sample are due to the fact that the high concentration sample did not actually enter the new domain state.

Scattering following quenches ofthe low concentration samples is consis­tent with late stage scaling theories for pure mixtures (Binder and Stauffer,

167

5xl0· (a) 24 wt.% IBA o O.t·c

<> D 0.2·C

i 8 E 8 0 @ ......, g

E 10· g .!o:: fl

6 8 6 0

D

5xl0· (b) 35 wt.% IBA o O.t·C

6 () D 0.2·C

,-... I

()

E <> 0 D () ......,

D

10· <> E 0 D

.!o:: 0 0 0 a

0 ~ 0

6

1 10 100

time (s)

Figure 3. Peak position k m as a function of time for various quench depths below T* for the (a) 24 and (b) 35 wt% IBA samples. The solid line has a slope of -1/3 and a magnitude comparable to that measured in pure IBAW mixtures for off-critical quenches (Wong and Knobler, 1978).

1974; Furukawa, 1984) which predict

(1)

where F(x) is a universal scaling function. Figure 5 shows results of multi­plying the quench data by k~ax and plotting as a function k/kmax . There are some deviations at early and late times but in general the data for the three low concentration samples is consistent with scaling theories. The de­viations at late times on the high-k side of the peak are due to multiple scattering. The data for the highest concentration sample never scales.

The data have one feature in the early stages which is very different from that of pure binary mixtures. In pure binary mixtures undergoing spinodal decomposition or nucleation, the intensity at each k grows at a k-dependent growth rate determined by the diffusive time scale (D k2)-1. Figure 6 shows increases in intensity at several scattering wavevectors lying

168

(a) 24 wt."- IBA

1000 o O.t'C ~ D O.2'C I 60.5'C 8 100 o 1.0'C 0 -- 10 &

.!II: -I'll 1

1000 (b) 35 wt."- IBA o O.t'C

~ D 0.2'C I 6 O.5'C <> 8 100 o 1.0'C t.. 0 t.. - t.. - t.. E

.!II: 10 t.. 0 - 0 I'll t..

t.. 8

B 8 0 t..

0 t.. 0

1 1 10 100

time (s)

Figure 4. Peak amplitude S(km ) as a function of time for various quench depths below T" for the (a) 24 and (b) 35 wt% IBA samples. The solid lines have a slope of 1.0. Note that S(km ) evolves nearly linearly in time at all quench depths for the lower concentration sample, but the exponent appears to depend on quench depth for the 35 wt% sample.

within the accessible range for quenches in the 24 wt% and 35 wt% sam­ples. During the early stages of the quench, the intensity increased more or less simultaneously at all accessible k. The growth rate is k-independent but does depend strongly on concentration. We have checked for various experimental explanations of these particular observations and believe that they are not due to problems with the transmitted intensity, multiple scat­tering, or irreversible increases in the stray light. Similar behavior has been reported recently by J .C. Lee in a simulation of spinodal decomposition in dilute gels (Lee, 1994).

3, Summary and Conclusions

We have investigated large intensity overshoots observed on temperature­quenching a binary mixture/gel system into a region which is close to the coexistence curve of the pure, gel-free, system but far above the coexistence

-... I

S 0

~ 0 ......

.. s

...:.: -...:.: -en

-... I

S 0 ., 0 ......

.. 13 ...:.: -...:.: -en

2.0

~

1.0

0.0

0.5

0.0 0 2

(a) 24 wt.% IBA

01.0 s

02.0 s

A 5.0 s

010.0 s

v20.0 s

.50.0 s

.100.0 s

00.5 s 01.0 s I> 2.0 s 05.0 s v 10.0 s .20.0 9

.50.0 9

.100.0 s

.200.0 s

6

169

Figure 5. Graphical test of the dynamic scaling hypothesis for the quenches shown in Fig. 2. Except at 1.0 s, the 24 wt% IBA sample appears to obey scaling, while the 35 wt% IBA sample never does.

curve of the binary mixture/gel system. Small angle light scattering experi­ments confirm that scattering from low concentration samples is consistent with growth of domains of different concentration, which grow both on times comparable to and with a power law consistent with those observed in the pure system. The domains quickly grow to length scales many orders of magnitude greater than the dominant length scale of the gel. There is no evidence for domain pinning or logarithmic growth of domains within the first few hundred seconds following a quench as might be expected for do­main growth in the presence of disorder. We do not believe that the highest concentration sample studied actually enters the domain state. As well as scattering characteristic of domain growth, the gel/mixture system shows evidence of a k-independent increase in scattering intensity which occurs quickly, and remains unexplained.

170

12

10 ~

I 8 e () - 6 -.!II: 4 -rI.I

2

0

2

-i e () --.!II: - 1 rI.I

-50

(a) 24 wt.% lBA

2965 em-I 5500 em-I 10520 em-I 20550 em-I 30570 em-I

o

(b)

500

o 50 100

time (ms)

1000

150 200

Figure 6. Growth of intensity as a function of time for selected wavevectors for a) the quench shown in Fig. 2a and b) the quench shown in Fig. 2b. Note the difference in the time-scales shown.

References

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THE SHAPE, DIMENSION AND ORGANISATION OF MALTODEXTRINS GEL FRAGMENTS WITH AND WITHOUT ASSOCIATED PHOSPHOLIPIDS

Abstract.

MIGUEL A. R. B. CASTANHO·,2., MANUEL 1. E. PRIETO·, DIDIER BETBEDER3, NUNO C. ~ANTOS·,2 1. Centro de Quimica-Fisica Molecular, Complero 1, 1. S. 1:, 1096 Lisboa Coder, Portugal; 2. Dep de Quimica e Bioquimica, Facllidade de Ciencias da Universidade de Lisboa, }.aijicio C1, Campo Grande, 1700 Lisboa, Portugal; 3. Biovector Therapeutics, S. A., Chemin du Chelle Vert, 31676 Labege Ceder, FraTIce.

Charged maltodextrins gel fragments prepared on a Rannie homogeniser were studied by light scattering spectroscopy techniques and transmission electron microscopy. These fragments associate with phospholipids. The fragments/lipid association is named a Biovector and can be used as a drug delivery system or in vaccine formulations.

Light scattering reveals that the fragments are spheres with Rh ~ 50 nm. Under some experimental conditions (namely, higher ionic strenb>1hs) P = R/Rh < 0.775, suggesting that a density gradient exists at the gel fragments. This is typical of the so called "microgels" (spatially confined gels of microscopic size). The fragments' shape and dimensions were confirmed by scanning electron microscopy.

Neutral gel fragments have a strong tendency towards large scale self-association. The addition of phospholipids to the gel fragments results in the formation of

concentric bilayers around an internal core. These cores are believed to. be the gel fragments.

The Biovectors may be used to entrap hydrophilic or hydrophobic drugs, in the core or lipid bilayers respectively. Implantation of antigenic molecules on the surface might lead to new vaccine formulations.

1. Introduction

The quest for the ideal drug delivery system (DDS) dates back from at least the beginning of this century, when Paul Ehrlich developed his idea of a "magic bullet" [I). The ideal DDS has the main goal of carrying the desired drug to a target tissue, preventing

173

E. R. Pike and J. B. Abbiss (eds.). Light Scattering and Photon Correlation Spectroscopy, 173-187. © 1997 Kluwer Academic Publishers.

174

undesirable side-effects and giving rise to the possibility of using significantly smaller drug amounts. Thus, the DDS should increase the residence time of the drug within the vascular system and extravascular areas, and permit the drug to reach intracellular sites [2].

Liposomes (phospholipid vesicles), the most classic of all the DDS, represent an improvement compared to the free drugs but still suffer from a number of faults [3]. Modified more stable liposomes have been proposed (e. g., [1]). Among them, some were stabilised with polysaccharides (e. g., [4,5]). Recently, a somewhat different approach was carried out: a maltodextrins gel was fragmented in a Rannie homogeniser and the gel fragments used as a matrix to stabilise lipid emulsions. The polysaccharide lipid particles were named Biovectors, and the first tests in vitro and in vivo were encouraging [3,6-8].

The aim of this work is to characterise the shape and dimension of the maltodextrins gel fragments both with and without associated phospholipid. The organization of the phospholipid (forming bilayers or not? surrounding the polysaccharide fragments or homogeneously spread inside them?) will also be addressed. The organization of the phospholipids in Biovectors is an important issue regarding their applications in immunology. The ability for protein insertion in such a way that the antigenic activity is retained, would enable the Biovectors to be used in vaccine formulations. This work follows a sequence of works devoted to similar questions [9,10].

2. Materials and Methods

2.1. APPARATUS

A standard multiangle laser light scattering apparatus from Brookhaven Instruments Inc. (USA), model 2030AT was used. Light from a He-Ne laser (632.8 nm, 35 mW, Spectraphysics, model 127) was scattered by samples placed in a cylindrical cell immersed in a decalin bath with temperature control by water circulation (21 ± O.S°C). A 128 channels autocorrelator was used to compute dynamic light scattering (DLS) data, yielding up to three different sampling times. The last six channels are used for baseline

calculation. The instrument setup was tested using standard latex beads (Duke Sci. Co.,

USA) of 100 nm and the specified variance of2 nm was observed. Scanning electron microscopy studies were carried out in a JEOL JSM-840

Scanning Microscope (Tokyo, Japan). Sample gold coating was made in a Bio-Rad SEM

coating system. The transmission electron microscopy study was carried out using a JEOL 100 SX

Electron Microscope, operated at 60 kY.

175

2.2. GEL FRAGMENTS AND BIOVECTORS PREPARATION

Biovectors are synthesized according to the procedures described elsewhere [3,11]. In

general tenns, the synthesis of the polysaccharide gel fragments consists of the following steps: The reaction of maltodextrins with the cross-linking agent (phosphoryl oxychloride

for negative Biovectors, or epichlorydrin for neutral and positive Biovectors) is carried

out under vigorous stirring in a NaOH solution at controlled temperature. The positive

Biovectors are cationised with glycidyltrimethylammonium chloride (hydroxycholine). After several hours, the gel fonned is neutralised and disrupted by extrusion in a high

pressure homogeniser (Rannie, APV, France). The size of the particles obtained

(polysaccharide gel fragments) depends on the pressure of the homogenization. The

polysaccharide gel fragments solutions were sterilized by filtration through 0.2 /lm filters (Nalgene, Polylabo, France) and stored at 4°C.

2.3. SAMPLE PREPARATION

All the material used for preparation of light scattering samples was treated with chromo sulfuric mixture and thoroughly rinsed with distilled water that had been

previously filtered through 0.2 /lm cellulose nitrate membranes (MFS, Dublin, California). The chromosulfuric mixture removes lipid and other traces that retain dust from material wall surfaces. Samples were placed in a syringe and filtered through

Millipore Millex-HV 0.22 /lm disposable filter units. Retention of polysaccharide by

Durapore membranes was not detected. Filtration was carried out directly into the cylindrical light scattering cells. To remove any remaining "dust" particles from the light

path an additional mild centrifugation (1300 g for 45 minutes) was perfonned to sediment it in the bottom of the cell. Afterwards cells were handled with extreme care. Solutions were prepared in phosphate buffer, Dulbecco's phosphate buffered saline (PBS) pH 7.4, or in aqueous solution of NaCI.

Scanning Electron Microscopy was used as a complement to the light scattering results. Solutions were placed over a polycarbonate 0.1 IJlIl filter, and allowed to dry in air. The drying process, even at low humidity conditions, allows the presence of a thin water film covering the molecules, and has no influence on their apparent shape (e. g., [12]). Samples were then covered with a gold coating.

Transmission electron microscopy samples were placed over copper grids, covered

with a Fonnvar® membrane, both purchased from Sigma, and dried at room temperature.

The negative staining was obtained with phosphotungstic acid solutions (1%) at different

pH values.

176

2.4. DATA ACQUISITION AND TREATMENT

Static light scattering measurements were computed according to the Zimrn method [13]. In case of polydispersed samples, the average radius of gyration calculated by this method is in fact the square root of weight average of the squared radius of gyration [14]:

~w.R2. I.J I g,l

I <R2 > = l-,-i -=-__ V g w ~.

i.Jwi i

(1)

For the sake of simplicity, <R.g2>w!12 will be referred to as R.g. The Rayleigh ratio was calculated relative to benzene, using a Rayleigh ratio for benzene of 1l.8 x 10.Q cm"! [15]

The dnldc (i. e., the refractive index increment with concentration) value for most polysaccharides is fairly constant and equal to 0.15 cm3g"! [16].

In terms of dynamic light scattering, the intensity correlation function was related to the field correlation function by the Siegert relation (e. g., [17]). Two kinds of data analyses were performed in this work: the cumulants method, proposed by Koppel [18] and Provencher's CONTIN [19]. Both methods lead to similar results. The intensity averaged (z averaged) diffusion coefficients, D, were calculated from the first moment of the distributions and extrapolated to zero angle and infinite dilution to prevent intra and interparticle effects. The doubly extrapolated value, Do, is related to the hydrodynamic radius, Rt" by the Stokes-Einstein relationship. From the Stokes-Einstein equation, the reciprocal of the z-average of the reciprocal of the hydrodynamic radius «Rt,-l>z-l) is obtained. For the sake of simplicity it will be referred to as Rt..

Experimental measurements were carried out at angles ranging from 450 to 1500

and concentrations ranging from 6 to 120 mg dm-3 . Measurements were made at increasing ionic strengths (J from 26.6 to 800.0 roM), so any evidence concerning polyelectrolyte behaviour of polysaccharide fragments should appear as different DLS parameters.

3. Results and discussion

3.1. THE MAL TODEXTRINS GEL FRAGMENTS

Some of the polysaccharide gel fragments used in this work have several positively or negatively charged groups attached. Therefore, chemically they can be considered polyelectrolytes. However, typical polyelectrolyte behaviour in terms of light scattering is not always detected. Such typical behavior includes two diffusion modes. The fast mode

177

is attributed to a coupled diffusion of polyions and counterions. The slow mode has a pronounced angular dependence [20]. Both positively and negatively charged gel fragments present unimodal relaxation rate distributions. Therefore, a typical polyelectrolyte behaviour is not observed.

The relaxation rate distributions were converted in size (RJ.) distributions. The main features of these distributions are identical for positively and negatively charged fragments: they are unimodal and moderately polydispersed (Figure 1).

'0 Q) 0.8 .!!.! iii E 0.6 .... 0 oS

b 0.4

C) ~ 0.2

0

2 4 6

In(RJnm) Figure J. Size distribution of negatively charged L __ .J and positively charged gel

fragments L-.). Both distributions were obtained from CONTIN. The measurements

angles and concentrations were 9 = 1200 and c = 10 mg dm·3 respectively for the negative

gel fragments, and 9 = 1500 and c = 84 mg dm-3 respectively for the positive gel fragments.

Adapted from [9] and [10).

Figure 2. Representations of the structure scattering factor, vs. qRg, for negatively charged

maltodextrins gel fragments (0) and for three model geometries: hard sphere L __ --.J, random coil ( ) and rod L ___ _ .J. Adapted from [9].

178

The first approach to the shape of the polysaccharide gel fragments was carried out by means of the structure scattering factor. The experimental structure scattering factor was compared with the expectations for some model geometries [21]. Figure 2 clearly indicates that the spherical geometry is more suitable to describe our data than any other geometry.

However, the geometry dependent parameter p = ~ does not always present the limit value expected for hard spheres (0.775 for monodispersed samples), as presented in Table 1. Slightly larger values can be attributed to polydispersity but the smaller ones cannot be attributed to homogeneous spheres.

TABLE 1. Data regarding positively and negatively charged maltodextrins gel fragments in

terms of the hydrodynamic radius (R0 and radius of gyration (Rg). Ric and dr are the

internal core radius of the particles (step function model approach) and the fractal

dimension of the microgel. respectively.

Charge Solvent ~/nm Rg/nm p=RJ& R.c I nm dr + phosphate buffer 53.8 45.5 0.84

PBS 4S.6 33.6 0.69 .. B.4 I.S

water 100.3 SO.7 O.SO

NaCl150 mM 69.7 45.S 0.66 59.1 1.5

When the ionic strength of the solutions increases, Rg and ~ decrease. Probably, a shielding of the charged groups occurs leading to polymer contraction. Upon contraction, p decreases to a value lower than the value expected for a hard sphere. A density gradient in the scattering particle (higher in the center and lower in the surface) would explain such results. A non-uniform contraction occurs upon the increase in the ionic strength. Other examples of nonuniform density particles include micro gels (spatially confined gels of microscopic size) of polyvinylacetate [22] and l3-casein micelles [23]. For a review see, e. g., [24].

The p values obtained with charged gel fragments can be conceptually explained under some theoretical model frameworks:

a) Soft sphere model Soft spheres are polymers of regularly branched chains. When some shells of branching are completed, the overall shape of the polymer is spherical (Fig. 3). The molecule, however, cannot be considered a hard sphere because the boundary is not very well defined and because the chains are flexible, not rigid. Values of 0.98 < P < 1.5 were predicted [25].

Figure 3. Schematic representation of a soft sphere (regularly branched polymer of a

definite functionality). Adapted from [25].

b) Star-like polymers of many arms

179

A star-like polymer of many arms in a good solvent is also expected to approach the p value for a rigid sphere [26]. These polymers have a compact core and the outer endings of the flexible arms tend to behave as coils. As the number of arms increases, the density profile approaches the uniform density of a sphere, this being more quickly attained in a bad solvent [27]. According to Douglas et al [26], the sphere-like behavior of stars starts to set in for functionality ;::: 20.

10

8

..,. CT 6 a::-

4

2

0 2 4 6

q2Rg2

Figure 4. Representations of the reciprocal of the structure scattering factor vs. f/R,,2, for

negatively charged maltodextrins gel fragments (0) and for three model structures: hard

sphere l. ___ _ .J, monodisperse star-like polymer ( ) and polydisperse star-like

polymer L __ -->. Adapted from [9].

Figure 4 shows the reciprocal of the form scattering factor for an infinite functionality of star-like polymers with monodisperse and polydisperse arms [25]. The hard sphere limit for infinite functionality is clear. The experimental data is also shown. An intermediate behaviour between a hard sphere and the star with infinite monodispersed arms is apparent. However it is also reasonable to speculate that the

180

p-I(q) experimental values are overestimated due to excess scattering. Aggregates or even remaining "dust" traces can eventually be responsible for such excess scattering. The effect would be to shift the experimental data upwards, towards the hard sphere theoretical expectations. In any case, in qualitative tenns, the results are in agreement with the above proposed structure for the gel fragments.

Although different, these two models and the concept of microgel rely on. molecular organisations with the following basic characteristics: i) compact packing of (sub)chains; ii) fonnation of tridimensional meshes. These, we believe, are the main features ofmaltodextrins gel fragments, ensuring their resistance and stability.

For a better perception of the density profile of the gel fragments we have approximated the density decay to two simple (however, crude) functions:

A

B Density

R

Figure 5. Simplified model for a non-homogeneous sphere. An inner core is surrounded by

an infinitely thin outer shell, the space between being empty (A). In terms of density profile

along the radius (8), this is represented by a step function. The step-function is a crude

approximation to the sigmoidal-like shape profile suggested by Schmidt et al. [28].

a) Step-function This model considers a homogeneous spherical inner core surrounded by an infinitely thin outer shell of the same material. The space between is empty (Figure 5). Considering that p = RgI~ is known, the inner core radius is simply,

(2)

181

The closer Ric is to ~, the more the sphere approaches the homogeneous limit. The calculated R;c values are presented in Table 1.

The step-function is a crude approximation to the sigmoidal-like shape profile suggested by Schmidt et at. [28]. A closer approximation would be e. g.:

p'(r) a -1

(3)

(p'(r) is the relative density at distance r from the center of the sphere and a and b are constants). However, more than one (a., b) solutions are possible for each p= Rg/R!.. Some extra infonnation would be needed to find the "real" solution among them.

b) Exponential function

An exponential function of the type (k, where r is the radius and k is a constant, can also be considered. This is the case expected for instance in the context of the fractal theory (e. g., [29]). As is demonstrated in Appendix I, the fractal dimension, dc, can be obtained from p. The de value can be used to get an idea of the gel fragments structure. Calculated de values are also presented in Table I .

The exponential function is also a somewhat crude approximation since it predicts an infinite density at the center (r = 0). This singularity lacks physical meaning.

The SEM micrographs confinn the spherical overall shape of the fragments and the size polydispersity (results not shown). An extended aggregation is clear when neutral gel fragments are used (Figure 6), probably due to the absence of repulsive electrostatic

A 8

Figure 6. Neutral maltodextrins gel fragments micrographs obtained by scanning electron

microscopy. Large aggregation (as in the example of figure A) and. in some cases, the

reconstitution of a dendrite-like gel (8) are detected. The scale bar represents I J.l111..

Adapted from [9).

182

interactions. In the most critical cases, the macroscopic gel seems to be reconstituted in a dendrite-like way (Figure 6b). This prevented the use ofJight scattering techniques to study the neutral gel fragments.

3.2. BIOVECTORS: GEL FRAGMENfSILIPID INTERACTIONS

Using the preparation method described in the Materials and Methods section, phospholipids and maltodextrins gel fragments do associate. This was demonstrated by centrifugation experiences on a sucrose gradient (results not shown). In terms of light scattering, it makes a big difference whether or not the lipid is homogeneously distributed through the porous structure of the gel fragments. In case that the lipid is surrounding the gel fragments, the simplest scattering model that can be used regarding data analysis is the coated sphere model (Figure 7). This model consists of an internal homogeneous sphere surrounded by a concentric shell of a different refractive index.

Figure 7. Coated sphere model. An inner homogeneous sphere of radius Ro and refractive

index lip; is surrounded by a concentric outer shell of thickness r = R., - Ro. where R., is the

outer radius of the shell, and Dp,o is its refractive index. The refractive index of the medium

is 110. Adapted from [10].

We used TEM to study lipid location and molecular organisation in Biovectors. Figure 8 clearly shows that the lipid forms concentric shells, surrounding a core. We believe this core is the gel fragment. Whether or not the core includes lipid is unknown. The shells number depends on the polyshaccharide/lipid ratio and can be as low as one [30). The concentric lipid shells thickness is in agreement with the thickness of lipidic bilayers (e. g., [31]). MoreOver, lipophilic fluorescent probe DPH (diphenylbexatriene) anisotropy measurements detect the typical phase transition of DPPC bilayers from the gel to the liquid crystal phase (R:41°C; results not shown). Our results are in agreement with a very recent paper [32] in which it is proposed that vesicles (e. g. from Egg Yolk Phosphatidylcholines) are excluded from polymer (e. g. dextran) networks.

Light scattering data analysis is currently under study.

Figure 8. Transmission electron microscopy photo of positive biovectors with DPPC,

negative stained with phosphotungstic acid 1% (w/v) pH 7 (x240000). Adapted from (10).

4. Conclusions

183

The gel-like nature of Biovectors is an important factor in their use as drug delivery systems and in vaccine formulations. The molecular organization (bilayer) and location of

the phospholipids is also very relevant. It enables the insertion of antigenic proteins that might trigger surface recognition phenomena.

Acknowledgements

This work was supported by Project Eureka - PUEMISlERC/93 and Projec~S/CI

SAUI144/95 (J. N. I. C. T., Portugal). M. A. R. B. C. acknowledges the helpful discussions and comments of Dr. Chaikin (Princeton Univ., N. 1., USA) and Dr. Bartsch (Univ. Mainz, Germany) during the NATO ARW on Light Scattering and Photon Correlation Spectroscopy (Krakow, 1996). N. C. S. acknowledges a grant from 1. N. I. C. T. (Portugal). The authors acknowledge the collaboration of Dr. Antonio Pedro Alves de Matos in the transmission electron microscopy studies, and the facilities provided by the Department of Pathologic Anatomy of the Curry Cabral Hospital (Lisbon, Portugal)

in the use of their microscopy equipment. The authors also acknowledge helpful

discussions by Dr. 1. Abbiss (Singular Systems, CA, USA)

Appendix I

By definition the square of the radius of gyration of a sphere is:

184

I p'(r)r2dV

J p'(r)dV (AI. I)

where p'(r) is the radius, r, dependence of the density and V is the volume. In spherical coordinates, for a sphere of radius &:

If p'(r) decays exponentially, then:

lIlt. p'(r)r4dr

lIlt. p'(r)r2dr

p'(r) oc r"k

(AI.2)

(AI.3)

where k is a constant. This is the kind of behaviour expected in the fractal theory framework (e. g., [29]). In this case:

rclr p'(r) oc -d =rclr-d

r (Al.4)

where de is the fractal dimension and d is the Euclidean dimension of the space where the fractal is placed. Hence,

2 dr p = d +2

f

(AI.5)

In the case that the sphere is homogenous, de = 3 and therefore p2 =3/5, as expected. Moreover, p' = constant, as it would obviously be expected.

References

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7. Prieur, E., Betbeder, D., Niedergang, F., Major, M., Alcover, A, Davignon, J-L. and Davrinche, C. (1996) Combination of Human Cytomegalovirus recombinant immediate-early' protein (IE 1 ) with 80 om cationic supramolecular bovectors: Protection from proteolysis and potentiation of presentation to CD4+ T cell clones in vitro., VaCCine, 14, 511-520.

8. CastignoUes, N., Morgeaux, S., Gontier-lallet, C., Samain, D., Betbeder, D. and Perrin, P. (1996) A new family of carriers (biovectors) enhances the immunogenicity of rabies antigens, Vaccine Res., in press.

9. Santos, N.C., Prieto, M., Morna-Gomes, A, Betbeder, D. and Castanho, M.A.R.B. (1996) Structural characterization (shape and dimensions) and stability of polysaccharidellipid nanoparticles, Biopolymers in press.

10. Santos, N.C., Sousa, AM.A, Betbeder, D., Prieto, M. and Castanho, MAR.B. (1996) Structural characterization of organized systems of polysaccharides and phospholipids by light scattering spectroscopy and electron microscopy, submitted for publication.

11. Peyrot, M., Sautereau, AM., Rabanel, J.M., Nguyen, F., Tocanne, J.F. and Samain, D. (1994) Supramolecular biovectors (SMBV): a new family ofnanoparticulate drug delivery systems. Synthesis and structural characterization, Int. J. Pharm. 102,25-33.

12. Kirby, AR., Gunning, AP., Morris, V.J. and Ridout, M.l (1995) Observation of the

helical structure of the bacterial polysaccharide acetan by atomic force microscopy, Biophys. J. 68, 360-363.

13. Zimm, B.H. (1948) Apparatus and methods for measurement and interpretation of

the angular variation of light scattering; Preliminary results on polystyrene solutions., J. Chem. Phys. 16, 1099-1116.

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14. Santos, N.C. and Castanho, M.A.R.B. (1996) Teaching light scattering spectroscopy: The dimension and shape of Tobacco Mosaic Virus, Biophys. J. 71, 1641-1650.

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16. Ruglin, M.B. (1989) Specific Refractive Index Increments of Polymers in Dilute Solutions, in I Brandrup and E.H. Immergut (Eds.), Polymer Handbook, John Wiley & Sons, New York, pp VIII409-484.

17. Berne, B.J. and Pecora, R. (1990) Dynamic Light Scattering, Robert E. Krieger Pub. Co., Malabar.

18. Koppel, D.E. (1972) Analysis of macromolecular polydispersity in intensity correlation spectroscopy: The method ofCumulants, J. Chem. Phys. 57,4814-4820.

19. Provencher, S.W. (1982) A constrained regularization method for inverting data represented by linear algebraic or integral equations, Com put. Phys. Commun. 27, 213-227.

20. Sedlak, M. and Amis, E.I (1992) Concentration and molecular weight regime diagram of salt-free polyelectrolyte solutions as studied by light scattering, J. Chem. Phys. 96, 826-834.

21. Schmitz, K.S. (1990) An Introduction to Light Scattering by Macromolecules, Academic Press, NY.

22. Burchard, W. and Schmidt, M. (1979) The diffusion coefficient of branched polyvinylacetates and of polyvinylacetate microgels measured by quasielastic light scattering, Ber. Bunsenges. Phys. Chem. 83, 388-391.

23. Burchard, W. (1992) Static and Dynamic Light Scattering Approaches to Structure Determination of Biopolymers, in S.E. Harding, D.B. Sattelle and V.A. Bloomfield (eds.), Laser Light Scattering in Biochemistry, Royal Society of Chemistry, Cambridge, pp 3-22.

24. Antonietti, M., Bremser, W. and Schmidt, M. (1990) Microgels: Model polymers for the cross-linked state, Macromolecules 23,3796-3805.

25. Burchard, W., Kajiwara, K. and Nerger, D. (1982) Static and dynamic scattering behavior of regularly branched chains: A model of soft-sphere microgels, J. Polym.

Sci.20,157-171. 26. Douglas, IF., Roovers, I and Freed, K.F. (1990) Characterization of branching

architecture through "universal" ratios of polymer solution properties, Macromolecules 23,4168-4180.

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28. Schmidt, M., Nerger, D. and Burchard W. (1979) Quasi-elastic light scattering from branched polymers: 1. Polyvinylacetate and polyvinylacetate-microgels prepared by emulsion polymerization, Polymer 20, 585-588.

187

29. Feder, 1. (1990) Fractals, Plenum Press, NY. 30. Betbeder, D. and Major, M. (1996) Unpublished results. 31. Seddom, 1.M. and Templer, R.H. (1995) Polymorphism of Lipid-Water Systems, in

R. Lipowsky and E. Sackmann (eds.), Strocture and Dynamics of Membranes. From Cells to Vesicles., Elsevier, Amsterdam, pp 97-160.

32. Meyuhas, D., Nir, S. and Lichtenberg, D. (1996) Aggregation of phospholipid vesicles by water-soluble polymers, Biophys. J. 71,2602-2612.

DYNAMIC LIGHT SCATTERING FROM BLOCK COPOLYMERS

Abstract.

PETR STEPANEK Institute of Macromolecular Chemistry, Academy of Sciences of the Czech Republic, Heyrovskho n.2, 162 06 Prague 6, Czech Republic

AND

TIMOTHY P. LODGE Department of Chemistry, University of Minnesota 207, Pleasant St. SE, Minneapolis, MN 55455, USA

Theory and experiment of static and dynamic light scattering from block copolymer solutions and melts are investigated including birefrin­gence properties. In different block copolymer liquids, five different modes are identified: internal mode, cluster mode, cooperative diffusion mode, heterogeneity mode and one as yet undefined X-mode. Experimental obser­vations are satisfactorily compared with theoretical predictions.

189

E. R. Pike CIIId J. B. Abbiss (etb.), Ught Sctmering CIIId Photon Correilltion Spectroscopy, 189-207. @) 1997 KlwIu ACfIIkmic Publisher,.

190

DYNAMIC LIGHT SCATTERING FROM BLOCK COPOLYMERS

1. Introduction

Block copolymer solutions and melts exhibit rich structural and dynamic properties. The order-disorder transition (ODT) induced by the thermodynamic interaction between blocks, the variety of morphologies in the ordered state, micellization resulting from solvent selectivity, and strong concentration fluctuations in the disordered state near the ODT render the dynamics of block copolymer liquids both complicated and interesting.

Dynamic light scattering (DLS) is a powerful tool for the study of dynamics of polymer liquids. DLS senses the spontaneous concentration fluctuations in the system, with an intensity related to the corresponding optical contrast. This is particularly useful for block copolymers since the presence of different chemical units (two monomers and, in solution, a solvent) can serve to 'label' multiple modes of relaxation. Consequently, a single measurement by DLS may resolve several relaxation modes.

This contribution is concerned only with symmetric diblock copolymers either in a melt or in solution with a neutral good solvent Thus all aspects of solvent selectivity and micellization are not considered. It is supposed that the interaction between the solvent and each of the two monomers is equal and that it is only the interaction between the two monomers that leads to various structural and dynamical properties of the block copolymers, both in solution and in the melt.

2. Theory of dynamic light scattering from disordered block copolymer liquids

The current description of the dynamics of disordered block copolymers is founded on two complementary theoretical approaches. The fiISt, due to Benmouna et al[l],[2], is based on the linear response fonnalism for multicomponent systems and utilizes a dynamical mean-field (random-phase approximation) assumption; Two dynamic modes are predicted for block copolymer solutions: a cooperative diffusion mode, reflecting relaxation of

191

fluctuations in concentration (as is also observed in homopolymer solutions) and an internal chain mode arising from the relative translational motion of the centers of gravity of the two blocks. This theory makes several simplifying assumptions: a) the monomers of both blocks are characterized by the same friction constant Ci b) the two blocks have the same radius of gyration R,.; c) the two blocks have the same number of monomers with the same volume.

The second theory takes into account the fact that, experimentally, assumption c) can never be satisfied; all real polymers exhibit heterogeneity in composition f, i.e., fluctuations in f from chain to chain. This permits fluctuations in relative block concentration of arbitrarily long wavelength, and generates in block copolymer solutions a third mode, the heterogeneity mode, govemed by the translational diffusion of the whole chains. This mode is analogous to the mode generated by polydispersity in scattering power, as described by Pusey et al.[3] for hard spheres which differ in optical scattering power, but it is important to recognize that it is due to a distribution of composition, not of molecular weight The presence of excess scattered intensity due to compositional heterogeneity was first noticed[4] in the 1950s.

The dynamic structure factor for a solution of a diblock copolymer in a neutral good solvent can thus be written in the following form:

~q,f) = A~xp(-rl f) + Acexp(-r c f) + Aaexp(-rH f) (1) ~q,O)

where the subscripts I, C, H denote the internal, cooperative, and heterogeneity modes, respectively. This relation does assume that the 3 modes are uncoupled. As will be detailed below, block copolymer melts and solutions differ in that the cooperative mode does not exist in the melt, and that the amplitudes and relaxation rates of the modes in solution depend on concentration.

2.1. BLOCK COPOLYMER MELTS

In a monodisperse melt, a fluctuation in composition can be created only by shifting a center of mass of block A relative to the center of mass of block B. The corresponding relaxation process is characterized by a decay rate r, which, for qR,.«1 and to within a constant of order unity, is

(2)

where 't" 1 is the longest viscoelastic relaxation time of the chain. The amplitude of the internal mode is given by

(3 )

where nA and nB are the refractive indices of the blocks A and B, N is the degree of polymerization and v is the segment volume.

This internal mode is the only dynamic process predicted by the Benmouna theory for a block copolymer melt which is monodisperse both in composition and in overall molecular weight; however, real polymers always

192

exhibit some polydispeISity in molecular weight and some heterogeneity in composition. It is not trivial to measure the compositional heterogeneity in a block copolymer directly[5], but for anionically synthesized, narrow molecular weight distribution samples, Jian et a1. [6], [41] masie the reasonable assumption that the molecular weight polydispersities of the two blocks are statistically independent. The heterogeneity factor

1C = <NA><N-? - <NAN g>2 ( 4 )

Nl«NA + N )2>

with N=<NA>+<NB> can then be expressed in terms of the polydispeISity in overall molecular weight,

1) = M.J~ - 1

as

The heterogeneity mode is then characterized by a decay rate

fR = DJIl2

(5 )

(6)

(7)

where the diffusion coefficient, DR , of the heterogeneity mode is related to the self-diffusion coefficient, D., of the block copolymer by

DR = Dj.1-2XNK) (8)

where X is the Flory-Huggins interaction parameter. The amplitude of the heterogeneity mode is then predicted[2] to be

(9 )

Comparing Eq.(3) and Eq.(9) it can be found that the amplitude of the heterogeneity mode is stronger than that of the internal mode, AR > AI when

1) > (qR/ 6

(10)

Taking a ~ 0.1 as a typical value for anionically synthesized polymem, then for a typical magnitude of the scattering vector q - 2.75 x lOS cm-1 the heterogeneity mode dominates for Rz < 30 nm which corresponds to Mw ~ 300 000 for a typical copolymer.

193

2.2. BLOCK COPOLYMER SOLUTIONS

The presence of a neutral solvent introduces two main differences into the dynamic light scattering from disordered block copolymer systems: a) the amplitudes and decay rates of the internal and heterogeneity modes have to be renormalized owing to the presence of the solvent, and b) an additional cooperative diffusion mode will exist. If g is the number of monomeric units per blob[7] on a copolymer chain and <I> the volume fraction of the copolymer in solution, the following. renormaliza tions will account for the presence of solvent[7]:

y - cpgv, N N- Z=-, (11) g

where the last relation takes into account that the mean number of contacts between neighboring blobs in a good solvent is of order ft. The best estimate is[8] z = -0.17. Then XN should be replaced by

xenZ" XN cp(1-Zi/(3V-l) .. XN cp1.S3 (12)

where we have used the value of the Flory exponent v = 0.59. Then Eqs. (8) and (9) for the diffusion coefficient and amplitude of the heterogeneity mode are replaced by

(13 )

(14)

The heterogeneity diffusion coefficient behaves essentially as the self-diffusion coefficient, D .. since the correction in the parenthesis in Eq.(13) is very small: for typical values X = 0.01, N = 103, <I> = 0.1 and MJMn = 1.1, we obtain DH smaller than D. by 1.5%.

Similarly, the properties of the internal mode for a disordered solution of a diblock copolymer as calculated using the original theory of Benmouna et al.[1],[9] are:

(15)

where x=l for dilute solutions and x=0.77 (using v = 0.59) for semidilute solutions, and ri1 = "C l(<I>,N). Although "C 1(<I>,N) can be predicted on the basis of

194

the Rouse, Zimm, or reptation models, one does not generally observe a clear Rouse-like or reptation-like regime in semidilute solutions, particularly for moderate molecular weight polymers. For comparison of theory with experiment, it is thus probably more appropriate to determine '1(<1» directly via viscoelastic or flow birefringence measurements or to estimate its concentration dependence on the basis of established empirical relations such as[10]

(16)

where [11] is the intrinsic viscosity and a is a parameter of order 0.4. The cooperative mode in copolymer solutions is essentially identical to

that observed in homopolymer solutions, and is thus, at least to fIrst approximation, independent of the copolymer nature of the chain[1,2]. In dilute solution,

whereas in semidilute solution

r c = q2~ _ cpO.7S

6m7s~

(17)

(18)

where ~ is the correlation length and 118 is the solvent viscosity. The amplitude scales as[11]

(19 )

in dilute solution, and

(20 )

in semidilute solutions, where <n> is the average refractive index of the copolymer and ns is the solvent refractive index. This mode reflects mutual diffusion of polymer and solvent, with amplitude and rate increasing with <I> in the dilute regime, but with amplitude decreasing with <I> in the semidilute regime. A very important practical consequence of Eq.(20) is that the amplitude of the cooperative mode can be varied by varying the refractive index ns of the solvent In particular, by an appropriate selection of the solvent and measurement temperature, a "zero average contrast" condition can be achieved, where Ac = O. This can be of great ac;lvantage in masking the usually strong contribution from cooperative diffusion, if one wishes to study the internal and heterogeneity modes.

195

3. Experimental results

3.1 POLARIZED DYNAMIC UGHT SCATIERING FROM DmLOCK COPOLYMER MELTS

Only recently has there been much experimental activity in this area. Suitable copolymers consist of two blocks having sufficiently different refractive indices, but that at the same time are reasonably compatible so that their ODT exists in an accessible temperature range for interesting molecular weights. The glass transition temperatures of both blocks also should be sufficiently below the interesting temperature range so that the fluctuations in density[12] do not obscure the dynamics of composition fluctuations. Several diblock copolymer materials have been examined: poly(ethyl-methylsiloxane-b-dimethylsiloxane) [pEMS-PDMS] [20], poly(ethylenepropylene-b-ethylethylene) [PEP-PEE] [ 19], poly(ethylene-b-ethylethylene) [PE-PEE][19], and poly(styrene-b-isoprene) [pS­PI][13], poly(styrene-b-butadiene) [PS-PB][14], poly(vinylcyclohexane-b­ethylene) [pVCH-PE][19], poly(vinylcyclohexane-b-ethylethylene) [PVCH­PEE][ 19], poly( dimethylsiloxane-b-ethylenepropylene) [PDMS-PEP][ 1 S].

Polarized dynamic light scattering from these systems reveals complex correlation curves from which, using inverse Laplace transformation[l6][17] (ILT), several components of the spectrum of relaxation times can be extracted. In all cases examined so far a very slow mode has been observed, which is not predicted by theory; this mode resembles that found in glass-forming liquids, in certain semidilute solutions, and in plasticized polymers[24], which is variously ascribed to clusters or long-range density fluctuations. We shall first describe this mode, which is apparently unrelated to the theoretical predictions listed above. Figure 1 shows representative correlation ftmctions obtained on several polyolefin block copolymers melts[18],[19].

1.25

/ PEE·PE-8. 18o"C

PEp·PEE·2. 202"C

g 0.75

0 c "';- PVCH·PE. 27o"C E

N Cl 0.25

log t.1lS

Figure 1. COITe1ation functions obtained on block copolymer melts for the materia1s and temperatures indicated in the legend. For better comparison, the correlation functions are normalized to have an intercept of 1; t is correlator delay time.

196

1.0 -

0.8 -

1: O.G ~

"" 0.4

0.2

0.0 i O~ 102

Figure 2. Correlation functions obtained on the block copolymer of PDMS-PEMS (M,. - 95 (00) at the temperatures indicated; after ref.[20].

Figure 2 reproduces[20] similar data for the siloxane copolymers. A decay corresponding to a "cluster" mode is always observed at delay times comparable to, or longer than, 1 s. The relative amplitude of this process varies for the various systems, and may depend on temperature: it is almost negligible at 120 0 C for the PDMS-PEMS copolymer in Figure 2, whereas it is dominant for the PE-PEE copolymer in Figure 1, so much so that no other dynamics can be observed for that particular correlation curve. The amplitude of this mode may vary with time over time scales of the order of weeks. This process is diffusive, since the relation r = Dq'- is satisfied for both cases shown.

If we assume (without rigorous justification) that this cluster diffusion obeys the Stokes-Einstein relation,

DcJ=~ fYTrrlRh (21)

where TJ is the viscosity of the medium in which the "cluster" moves, we can estimate the corresponding hydrodynamic size, ~ : it is approximately constant in the disordered phase, and of the order of 100 nm. Below the ODT, Rb represents only an apparent size, since the concept of a single value of the macroscopic viscosity is problematic.

The exact origin, structure and properties of the clusters observed by dynamic light scattering have not yet been fully established. They can, however, be compared to other theoretical and experimental work. Fried and Binder[21] have predicted the existence of a large scale mesostructure in the disordered phase resulting in a partial compositional segregation of the block copolymer due to stretching of the polymer chains. Stiihn et al.[22],[23]. have claimed, based on NMR and dielectric normal mode spectroscopy, that for SI block copolymers crossing the ODT from the ordered phase does not lead to homogeneous mixing of both blocks, and that local order remains even above the ODT over an extent of ca. 70 run. These observations and possible explanations of clusters are related to the block copolymer nature of the material.

197

It is, however, important to remember that clusters, possibly generated by a different physical mechanism, are also observed in other homopolymer systems and· even in low-molecular-weight glass-forming systems. For example, an extensive static and dynamic light scattering study has been recently presented by Fischer et al.[24] on poly(methyl-p-tolylsiloxane) over a wide temperature range. Large-scale heterogeneities have also been inferred[2S] from NMR measurements on a polyethylene melt above its crystallization temperature.

In the remainder of this ~ction on block copolymer melts we shall ignore the presence of this, not yet well understood, cluster mode. As we have seen in the theory section above, the two main dynamic processes predicted for a block copolymer melt are the internal relaxation and the heterogeneity mode. Figure 3 shows a spectrum of relaxation times obtained by ILT on a block copolymer melt of PEP-PEE-2, in the disordered phase. The component corresponding to the cluster mode was subtracted from the correlation function

Figure 3. Spectnun of relaxation times obtained on the copolymer PEP-PEE-2 (M. ~ SO 000) at ISO·C and a scattering angle of 90· by inverse Laplace transformation from the corresponding correlation fuDction shown in Figure 1. The dominant decay correspondiDg to the cluster mode has been subtracted from the i(t) prior to performing the inversion. Relaxation times corresp<mdiDg to the internal mode 1'1 as determined from rheology and to self-diffusion as determined from forced Rayleigh scattering are indicated.

before performing the final IL T; the subtraction technique is described in more detail in ref. 17. This subtraction is essential as attempts to recover weak components from a correlation function in the presence of a strong decay time yield unreliable and unstable results.

The arrow marked t 1 on Figure 3 indicates the relaxation time of the internal mode as determined from dynamic rheology. The arrow marked Ds corresponds to a decay time that would be generated by self-diffusion at that particular scattering angle. Values of Ds on the same system were measured by forced Rayleigh scattering[26]. Reasonable agreement is observed in Figure 3,

198

allowing us to assign the fastest component to the internal mode and the slowest component to the heterogeneity mode; the correction dUe to the quantity K in Eq.(13) is negligible, and thus DH - Ds. The intermediate component in the distribution in Figure 3 cannot be explained at the present time. Figure 4 shows a more extensive comparison, as a function of temperature, of the internal and heterogeneity modes with 'r 1 and Ds. Almost quantitative agreement is observed over the common temperature range.

10-6

10-8

J!. 00000

E 10-10 O~Q,O~~ 0 d· ci

10-12 • • • 0

• 10.14 •

0 50 100 150 200 250

T,oe

Figure 4. Temperature dependence for the copolymer PEP-PBE-2 of the diffusion coefficients of self-diffusion (D), DLS heterogeneity mode (e), DLS internal mode CO) and intemal mode deteIIItined by rheology (full line) . Fa the purpose of this diagram, the decay rates of the angle­independent internal mode were replaced by r/q'-.

The molecular weight dependence of the relaxation rates of the internal and heterogeneity modes has been tested on the PDMS-PEMS copolymets[20). Figure 5 displays the dependence of r I on N for a series of entangled PDMS­PEMS copolymets; within experimental error the data points follow the N-3

dependence predicted by the reptation model. The diffusion coefficient of the heterogeneity mode is, except for a

generally negligible correction, numerically equal to the self-diffusion coefficient, Ds . In the entangled regime the self-diffusion coefficient is predicted[7] to behave as

(22)

The experimental data presented in Figure 6 are somewhat scattered but still follow reasonably well the predicted dependence.

..---.. -I rn

'-"

5.2

~4.2 bD o .....

(a)

o ~3

3.2 L--_-l-_--1 __ ....L-_~ _ __J

2.7 2.8 2.9 3.0 3.1 log N

199

Figure 5. Dependence of the decay rate r. of the internal mode on polymerization index, N, for PDM5-PEMS symmetric block copolymers, at 80· C.

-7.0

-7.5

-'" -8.0 N

E ~-8.5 9-~ -9.'0

• ..... ................

.. ~... . '. " ..

(a)

-9.5 ':2-.. .. ~ ......

o ~ ..... -10.0

2.4 2.6 2.8 3.0 3.2 logN

Figure 6. Dependence of the diffusion coeff"u:ient of the heterogeneity mode, DR, on the polymerization index, N, for PDM5-PEMS block copolymers. at 80"C. The slope -2 is the prediction from reptation theory, see Eq.(22).

3.2. POLARIZED DYNAMIC UGHT SCAITERING FROM BLOCK COPOLYMER SOLUTIONS

3.2.1. Neutral Solvent with General Refractive Index Polarized dynamic light scattering from block copolymer solutions has been the subject of more activity in recent years than the case of block copolymer melts. As we have seen above, the presence of solvent in the system introduces an additional mode, the cooperative mode equivalent to that observed in homopolymer solutions, and it also introduces one more variable, the

200

concentration of polymer in the solution. Figure 7 shows typical polarized correlation functions obtained[ll] on a solution of poly(styrene-isoprene)

0.6

Sl(170-170) 0.5

0.'

"7 a 0.3 be

0.2

0. 1

2

log l. ~s

Figure 7. Correlation functions of polarized scattered light obtained on solutions in toluene of a PS-PI symmetric block copolymer (M .. - 340 0000), at 30· C, at a scattering angle of 90· and at the indicated weight fractions, w; from ref. [1 1]

SJ(l7()"J10) n 0,.0.01).1.'6 / \ cJr:· .O . .!7

• II. • 1 ~"'-'o-J

(b)

i\ ... 0.0169 !'\J \ cIt- · I.O --' \...-

\\' .0,.307 .... · .O.O'H6 cle· _".0 ck- .',4

10' 10' 10' 10' 10' 10' 0' 10' 10' 10' 10' 10' r.s'l r ·1 .'

Figure 8. Distribution functions of relaxation times obtained on toluene solutions of a PS-Pl block copolymer, at 30·C, at scattering angle of 90° , and at the weight fractions indicated: a) M,. - 35 000 , b) M. - 340 000; from ref.[UJ

201

symmetric diblock copolymer (total molecular weight 340 000) in toluene, at the indicated concentrations. As the concentration increases, the complexity of the correlation function also increases. The corresponding distributions of decay times are displayed in Figure 8, for two samples with total molecular weight differing by a factor of 10, M", = 35 000 and M", 340 000. The two modes observed in Figure 8a are both diffusive and can be identified as the cooperative (fast) and heterogeneity (slow) modes.

Figure 9 compares the diffusion coefficient obtained by pulsed field gradient NMR to that of the heterogeneity mode. A very good agreement is observed, confirming that for anionically synthesized polymers the magnitude

10·'

51(16-19) m mil uP HI

u,.' m m HI Il W

• • II:! c· 10·' 0

{ • u ~c

ci c. 10~ IB Dc

C C o. •

I~ • D,(NMRJ •

IO~'

0.001 U.OI ·0.1 W

Figure 9. Concentration dependences of Dc and DR obtained by polarized DLS and Os. obtained by NMR for toluene solutions of a PS-PI copolymer (M,. - 3S (00). at 30·C and at a scattering angle of 90"; from ref.[S]

of the term with K in the parenthesis in Eq.(8) is less than the experimental error of DH and Dc. The cooperative diffusion coefficient (upper curve in Figure 9) behaves as in homopolymer solutions, and can be described by a relationship Dc - <1>0.7 for concentrations larger than the overlap concentration, independent Qf the molecular weight

Figure 8b shows that for larger molecular weights an additional third mode exists in the distribution of relaxation times. As can be seen in Figure 10, the decay rate of this mode is almost independent of angle, which leads to its assignment as the internal mode. The internal mode is, however, difficult to resolve, since its relaxation time is very close to that of the cooperative mode.

The magnitude of r I can be estimated from r H as follows; since the Zimm, Rouse and reptation theories all predict that[27]

(23)

~ of the copolymer used in Figure 8b is ca. 21 nm, and thus r JI' H - 40, which

202

is in good agreement with the positions of the peaks in Figure Sb. The accuracy of the determination of the amplitude is not sufficient to permit a quantitative evaluation of the angular dependence of AI> according to Eq.(15). We shall,

3103

1103

SI(17O-170)

w=0.0489

4 10'0 8 10'0

q2. cm-2

•

• fe·s- I

1.5 10'

110'

Sill"

0 1.210"

Figure 10. Angle dependence of the decay rate for the three modes observed in FtgIlre Sb, for a toluene solution of PS-PI copolymer (M,. = 340 000) with weight fraction w = 0.049. The decay rate of the internal mode r l was divided by 6 to bring it onto the scale of the heterogeneity mode; from ref.[ll]

however, show in the next section that the internal mode can be comfortably resolved and studied when a lzero-average contrast' solvent is used

In the last years dynamic light scattering measurements have been reported on a variety of copolymer systems. The characteIS of the modes observed and the tentative interpretations advanced varied considerably. We feel, however, that it is now possible to reconcile all these findings.

BoISali et al.[S] examined a poly(styrene-b-methylmethacrylate)(pS­PMMA) diblock copolymer (Mw = 7.S x lOS) in toluene for c '" 5c*. Two modes were clearly evident, and both decay rates were roughly ¢-dependent; in addition, the faster mode was comparable with Dc for a polystyrene homopol,rmer solution. The interpretation of the slower mode was uncertain, because It was diffusive, and thus did not fit the character of the expected internal mode. However, we can now say that its magnitude (ca. 7 x 10-8 cm2/s) is close to that expected for translational diffusion. Thus Dc and DH were probably observed in this work, while r I was presumably unresolvable because it overlaps Dc for this concentration and, moreover, Dc and DH were only a factor of five apart.

Duval and cowOrkeIS presented two studies of a PS-PMMA copolymer in toluene, one[2S] with Mw = 6.4 x lOS and the other[29] with Mw = 3.4 x lOS. The former study involved one concentration, c = 1.6c* , and two modes were seen. The slower mode was diffusive, and numerically consistent with Dc . The faster mode was attributed to rI . The latter study included a range of semidilute

203

solutions, and two diffusive modes were observed. The faster was again Dc , but the slower was attributed to "superstructures". However, its magnitude and concentration dependence are consistent with DH , and this seems the more likely interpretation. The higher Mw and lower concentration in the former study presumably account for the observation of r I and not r H • These workers also noted the presence of an even slower mode that disappeared with time, and that was apparent at higher concentrations.

Kanak and Podesva[30] examined a poly(styrene-b-isoprene) (pS-pn diblock (Mw = 1.8 x lOS) in 1,2-diphenylethylene; four concentrations up to c = 8c' were employed. They noted two modes, and attributed the faster to r I and the slower to Dc , on the basis of the q-dependence of the decay rates. However, the magnitude of the slower mode (3-6 x 10.8 cm2/s) is at least an order of magnitude too low for Dc , and is therefore more likely to reflect DH , even though it increased slightly with concentration. The appearance of a still slower mode was noted, particularly at high concentrations.

Balsara et al.[3l] examined a PS-PI diblock with Mw = 3.1 X 104 in toluene; five rather high concentrations were employed (w = 0.08 - 0.60). As a consequence of this wide concentration range, these were the first DLS measurements to traverse the ODT. Two modes were consistently seen, the faster mode diffusive and the slower having r close to q3-dependent A comparison with forced Rayleigh scattering and NMR measurements led the authors to the suggestion that the diffusive mode corresponded to translational diffusion, a then novel speculation that now has both theoretical and experimental support. Interestingly, the diffusivity was apparently unaffected by the ODT, in agreement with more recent results in melts[26]. The slower mode was attributed to incipient grains, especially as it was found to slow down dramatically near the ODT. It is worth noting that these solutions were measured within a few days of their preparation. Consistent with some of the other results above, this slow mode may reflect a non-equilibrium state. The absence of r I and rein this work is a consequence of the low molecular weight and high concentrations involved.

Tsunashima et al.[32] examined a PS-PMMA diblock (Mw = 1.5 x 10~ in very dilute benzene solution. Two modes were seen, the slower clearly identified as Dc and the faster possibly as r I ; the latter interpretation was uncertain because a q-independent regime was not detected as q .... O. However, as the authors pointed out, the asymmetry of the sample (fx,s = 0.39) can compromise the simple assignment of modes in the mean-field theory.

Konak et al.[33] also examined a PS-PMMA diblock in toluene, but of low molecular weight (Mw = 8 x 104) and of higher concentrations (w = 0.005 ,. 0.04). The faster mode observed was consistent with D6 the slower mode, which was also diffusive, was ascribed to aggregation induced by crystallization of stereoregular sequences of the PMMA chain. It should be, however, pointed out that the magnitude of this diffusion coefficient (4 x 10.8 cm2/s) is comparable to DH•

Jian et al.[4l] examined the DLS properties of two PS-PI copolymers in toluene. One sample was asymmetric, the other symmetric with total molecular weight of 1.6 x lOS. Three modes were found, and attributed to rc, rIb and rio An additional slow mode was found consistently, and was interpreted as being due either to clusters of copolymer chains, or to long-range density fluctuations.

204

3.2.2. Neutral, Zero-average-contrast Solvent The amplitude Ac of the cooperative mode observed in semidilute polymer or copolymer solutions is given by Eq.(19), showing, as we have discussed above, that Ac = 0 if the average refractive index of the copolymer is equal to that of the solvent. Using such a 'zero-average-contrast' · (ZAC) solvent masks the cooperative diffusion, and renders much easier the examination of other, weaker dynamic processes in the solution. This approach was previously used with a similar aim in a solution of two different homopolymers in a ZAC solvent(34],(35].

Liu et al.(36] reponed the first DLS experiments of this kind on a solution of a diblock copolymer. A symmetric styrene-isoprene diblock copolymer was used, having the total molecular weight Mw = 340 000 (the same as for data shown in Figure 8b above) dissolved in a mixed solvent of toluene and l-chloronaphthalene. Differential refractometer measurements showed that ZAC conditions are achieved for this polymer-solvent system at a weight fraction of toluene in the mixed solvent of Wlol = 0.48. Figure 11 shows the distributions of relaxation times obtained at various angles on a typical solution under ZAC conditions. Although the same polymer is used as in Figure 8b, only two modes

12u· .'\ 1\ . -

'lIl'

f\f\ c 15 ... I-

J\j\ 60'

..... , ............. .. p .... \ __ I'

10' 10' 10' 10' 10' 10' IU'

r · 1 • S

Figure 11. Angle dependence of the distributions of relaxation rates for a solution of PS·Pl copolymer (Mw - 340 000) at a weight fraction w - 0 .002 and temperantre 30°C, in a mixed solvent toluene/l·chloronaphthalene in the ZAC conditions; from ref.[36]

are seen: the cooperative mode is not present. In Figure II the decay rate of the faster mode is clearly angle-independent and corresponds to the internal

205

relaxation of the copolymer chain, whereas the decay rate of the slower mode depends on angle and was shown[36] to be diffusive.

Figure 12 shows the concentration dependence of the diffusion coefficient of the heterogeneity mode and of the relaxation rate of the internal mode, presented as rJtf. Also shown are measurements of Ds in chloroform by PFG-NMR, confirming that DH '" Ds CDs has been scaled by the ratio of solvent viscosities). The concentration dependence of the internal mode has been modelled with Eq.(16), a semi-empirical expression[37],[38] that has been shown

ur'

10"

10"

'" ~ E U 10.7

~I - exp(ac['1])

-----.. --------rfJ / ---.. :-...............

II', , • .. iI. . 0

10"

~. ." De .. ..

.1

• 10"

• Inlernal (Iftq')

o 0, (NMR)

1

~

,OOT

c

Figure 12. Concentration dependence of OH' Os from PFG-NMR and f'Jq2 (at 9(n for PS-PI copolymer (M" = 340 000) in the ZAC solvent. The vertical line denotes the order-disorder transition, and the dashed line a fit of f'I to the exponential relation given in Eq.( 16); from ref.[36]

JO'

10'

~ <.J

<.-10'

'" <. ~ JOl -'" , e

102

JOI 0.001

a =90·

• Au 0 AI

om c

1 .' 1

1

•• 01

•

0.1

Figure 13. Concentration dependence of the absolute amplitudes of the heterogeneity and internal modes for the PS-PI copolymer (M ... = 340 000) in the ZAC solvent; from ref.[36]

206

to describe oscillatory birefringence, viscoelasticity, and dielectric relaxation data very well [39]. The value of IX obtained by the fit shown in Figure 12 is 0.46, in excellent agreement with values between 0.4 and 0.5 obtained for polystyrene (M> 105) in Aroclor [10], [39], [40].

The intensities of the internal and heterogeneity modes are plotted as a function of concentration in Figure 13. In dilute solution, Al > AH, but then AH supersedes Al near c' '" 0.01. The prediction for dilute solutions is [36] that both Al and AH should increase linearly with c, but only AH follows this prediction. In the semidilute regime, Jian et al.[41] predict that Al scales as CO.75 which is not observed either. Above c '" 0.05, both Al and AH increase very strongly (note the logarithmic axes on Figure 13). Eq.(9) includes the possibility of a divergence at high concentration, i.e. 1-21(XNc1.53 ~ 0, if the heterogeneity factor l( is sufficiently large. However, given that XNC1.53 -10 at COT and l( - 0.02 for this block copolymer, Eq.(9) cannot describe the strength of the observed increase in AH• This extra intensity is more likely to correspond to the onset of large­amplitude fluctuations in composition as the COT is approached (c .. 0.13).

4. Conclusion

The aim of this contribution was to describe basic, yet exciting features of block copolymer liquids as observed by dynamic light scattering. These properties are often quite complex, and one can only imagine that triblock and multi block copolymers will be even more intricate. Some recent results [42] on asymmetric styrene-isoprene copolymers give some examples of this.

Acknowledgement

We aclmowledge the support of this work by the National Science Foundation through the grant CHE-9203173 from the NSF Division of International Programs (a US-Czech Cooperative Research award) and through the grant 94/203/0817 of the Grant Agency of the Czech Republic.

References

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Sci. 16, 517 . 5. W. Bushuk and H. Benoit, (1958). Can. J. Chem. 36, 1616 6. Jian, T., Anastasiadis, S. H., Semenov, A. N., Fytas, G., Fleischer, G., Vilesov, A. D.,

(1995). Macromolecules, 28, 2439 7. de Gennes, P. G. (1979), Scaling concepts in polymer physics. Cornell University Press,

Ithaca, New York 8. Joanny,1. F., Leibler, L., Ball, R. (1984). J. Chem. Phys., 81, 4640 9. Borsali, R., Fischer, E. W., Benmouna, M. (1991). Phys. Rev. A, 45, 5732 10. Lodge, T. P., Schrag, J. L. (1982). Macromolecules, 15, 1376 11. Pan, C., Maurer, W., Liu, Z., Lodge, T. P., Stepanek, P., von Meerwall, E. D.,

Watanabe, H. (1995). Macromolecules, 28, 1643 12. see, e.g., Giebel, L., Meier, G., Fytas, G., Fischer, E. W. (1992). 1. Polym. Sci.: Part

B: Polym. Phys., 30, 1291

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13. Hoffmann, A., Koch, T., Stuhn, B. (1993), Macromolecules, 26, 7288 14. Papadakis, C., Brown, W., Johnsen, R., Pa!selt, D., A1mdal, K. (1996), J. Chem. Phys.,

104(4), 1611 IS. Stepanek, P., Lodge, T. (1996), Proceedings o/the 6th Polymer Federation Symposium

on Polymeric Materials, Aghia Pelaghia, Greece, Oct. 7-11 16. Provencher, S. W. (1979), MaJcromoL Chem., 180, 201 17. Stepanek, P. (1993), chapter 4 in W. Brown (ed), Dynamic light scattering: the method

and some applications, Clarendon Press, Oxford 18. Stepanek P., Lodge, T. P., Bates, F. S. (1993). Polym. Prepr. (Am. Chem. Soc., Div.

Polym. Chem.) 35(1), 616 19. Stepanek, P., Lodge, T. P. (1996). Macromolecules, 29, 1244 20. Vogt, S., Anastasiadis, S. H., Fytas, G., FISCher, E. W. (1994), Macromolecules, 27,

1429 21. Fried, H., Binder, K. (1991). Europhysics Letters, 16(3),737, 22. Stuhn, B., Stickel, F. (1992). Macromolecules, 25, 5306 73. Fleischer, G., Pujara, F., Stuhn, B. (1994). Macromolecules, 27, 3335 24. Kanaya, T., Patkowski, A., FIScher, E. W., Seils, J., Glaser, H., Kaji, K. (1994). Acta

Polym., 45, 137 25. Bremner, T., Rudin, A. (1992). J. Polym. Sci:Part B: Polym. Phys., 30, 1247 26. Dalvi, M. C., Lodge, T. P. (1994). Macromolecules, 27, 3487 27. Doi, M., Edwards, S. F. (1986) The theory 0/ polymer dynamics, Clarendon Press,

Oxford 28. Duval, M., Haida, H., Lingelser, J. P., Gallot, Y. (1991). Macromolecules, 24, 6867 29. Haida, H., Lingelset, 1. P., Gallot, Y., Duval, M. (1991). MakromoL Chem., 192, 2701 30. Kanak, C., Podesva, J. (1991). Macromolecules, 24, 6502 31. Balsara, N. P., Stepanek, P., Lodge, T. P., Tirrell, M., (1991). Macromolecules, 24, 6227 32. Tsunashima, Y., Kawamata, Y. (1993). Macromolecules, 26, 4899 33. Kanak, C., Vlcek, P., Bansil, R., (1993). Macromolecules, 26, 3717 34. Giebel, L., Becmouna, M., Borsali, R., FIScher, E. W. (1993). Macromolecules, 26,

2433 35. Benmouna, M., FISCher, E. W., Ewen, B., Duval, M. (1992). J. Polym. Sci.: Part B:

Polym. Phys., 30, 1lS7 36. Liu, Z., Pan, C., Lodge, T. P., Stepanek, P. (1995). Macromolecules, 28, 3221 37. Adler, R. S., Freed, K. F. (1978). Macromolecules, 11, lOS8 38 Muthukumar, M., Freed, K. F. (1978). Macromolecules 11, 843 39. Amelar, S., Eastman, C. E., Morris, R. L., Schmeltzly, M. A., Lodge, T. P., von

MeerwaIl, E. D. (1991). Macromolecules, 24, 350S 40. Martel, C. J. T., (1983) Faraday Symp. Chem. Soc., 18, 173 41. Jim, T., Anastasiadis, S. H., Semenov, A. N., Fytas, G., Adachi, K., Kotaka, T. (1994).

Macromolecules, 27,4762 42. Z. Liu, K. Kobayashi, and T. P. Lodge, submitted to J. Polym. Sci., Polym. Phys. Ed

PARTICLE DIFFUSION AND CRYSTALLISATION IN SUSPEN­

SIONS OF HARD SPHERES

Abstract.

W. VAN MEGEN, S. M. UNDERWOOD, J. MULLER T. C. MORTENSEN, S.l. HENDERSON, J. L. HARLAND AND P. FRANCIS Department of Applied Physics, Royal Melbourne Institute of Technology, Melbourne, Victoria 3000, Australia

Hard sphere colloidal susl)('nsions composed of polymer particles in a non-aqueous solvent are studied using light scattering techniques. Single particle motions in the metastable fluid are obtained from measurement of the incoherent intermediate scattering function. The emergence of crys­ta.lline order is observed by monitoring the development of the main Bragg reflection.

209

E. R. Pike andJ. B. Abbiss (ells.), Light Scattering and Photon Correlation Spectroscopy, 209--223. © 1997 Kluwer Academic Publishers.

210

PARTICLE DIFFUSION AND CRYSTALLISATION INSUSPEN­

SIONS OF HARD SPHERES

1. Introduction

Despite· many decades of intense investigation there is still not a thorough

understanding of the microscopic mechanisms by which a liquid solidifies as it is

cooled. In the absence of seeding, a liquid can be cooled below its frcezing

temperature without crystallisation. The difficulty for the experimentalist is that, once

crystallisation is initiated, macroscopic crystals grow very quickly (1). Resolution of

the microscopic mechanisms by which solidification takes place has proved to be

virtually impossible. A related problem is that simple liquids, those of inert atoms or

simple molecules, cannot be cooled quickly enough to attain the glass state.

The classical view of the mechanism of solidification is that it occurs through the

formation of clearly defined spherical nuclei by thennally driven density fluctuations

[2]. The energy barrier posed by competing bulk and surface contributions results in

211

the survival of only those nuclei that exceed a minimum (critical) radius.

Interpretation of experiments in terms of this model is difficult because heterogeneous

nucleation on impurities and container walls tends to dominate and because any latent

heat produced is slow to dissipate [2].

The di1:fusive motions of near micrometer sized colloidal particles suspended in

liquid are roughly ten orders of magnitude slower than those of atoms [3,4]. For

suspensions of particles with narrow size and shape distributions, this translates into

crystallisation times that are easily accessible experimentally [5,6]. Furthermore, the

slow crystallisation rates combined with the weak lattice forces of colloidal crystals

allows them to be easily shear melted, giving ready access to the metastable fluid and

glass states [4]. Other important properties that distinguish suspensions from atomic

liquids, are that the continuous suspending fluid acts as an effective heat sink and

typical laboratory samples are free from macroscopic impurities so that heterogeneous

nucleation may be avoided.

Although the thermodynamic properties and spatial distribution functions are

equivalent for suspensions and atomic systems having the same interaction potential,

there is no reason to expect a strict analogy of the dynamical properties. After all, the

smallest relevant motions of atoms and colloidal particles are, respectively, ballistic

and diffusive. Nonetheless, studies of particle motions [3] and crystal growth [7,8]

suggest that the results of experiments with model colloids have implications that

transcend colloid science.

The spatial and temporal scales typically encountered in suspensions of near

micrometer sized particles allow their structure and particle motions to be studied by

time-averaged and dynamic light scattering, the optical analogues of X-ray

crystallography and dynamic neutron scattering respectively. Suitable selection of the

refractive indices of particles and suspending liquid allows multiple scattering to be

minimized and also, in a manner analogous to the exploitation of isotopes in neutron

scattering, measurement of the coherent and incoherent intermediate scattering

functions [9] and partial structure factors.

212

In this paper we describe the ctystallisation kinetics, measured by the growth of the

main inter-layer Bragg reflection, and the single particle motions, obtained from

measurement of the incoherent intermediate scattering functions. We use suspensions

of polymer particles whose equilibrium phase behaviour mimics that expected for the

perfect hard sphere system [IOJ. Thus we infer that the interaction among the particles

is hard-sphere-like. As the particle concentration is increased, the slowing of large­

scale particle diffusion in the metastable shear-melted col1oidal :fluid scales with the

reduction of the nucleation rate. The cessation of large-scale particle diffusion and

nucleation occur at the same concentration [l1J, which we refer to as the glass

transition (GT) concentration. Ctystal growth, seeded by secondaty nuclei, persists up

to random close packing, the concentration where all particle motion ceases.

2. Experimental Methods

2.1 SAMPLE DESCRIPTION

The cores of the particles used in the various experiments consist of silica, poly-methyl

methactylate (PMMA) or copolymer of methyl methacrylate and tetra:fluoroethyl

aetylate (P(MMA!fFEA» [12,13J. In all cases the particles were coated with a thin

(-100m) layer of poly-12-hydroxystearic acid, which provides stability against

irreversible coagulation. Particle radii (R) and polydispersities, obtained by dynamic

light scattering on dilute suspensions, along with other details pertaining to the

suspensions are listed in Table I.

Suspension

II

III

TABLE 1. Properties of colloidal particles.

Particle Core Radius(nm) PolydispersitY

P(MMAfI'FEA) 200±2 7%

PMMA 201±2 S%

Silica 200±2 3%

Liquid

cil-decalin

decaIinICS2 cil-decalin

213

Freezing and melting concentrations were respectively identified as the

concentration where Ctystallisation first occurred and that where the suspension

became fully Ctystalline [10,12,14]. Equating the observed freezing concentration with

the value, +r= 0.494, known for the :freezing value of hard spheres [15], gave a

multiplicative factor with which concentrations of all samples were converted to

effective hard sphere volume fractions, +. Significantly, the observed effective hard

sphere volume fraction at melting, +m~ 0.55 agrees with that expected for perfect hard

spheres [15]. The glass transition was located at +~ 0.58 [4].

Samples were prepared in a metastable fluid state by shear melting (tumbling) prior

to measurements. Dynamic light scattering (DLS) measurements were performed

before crystallisation. For the Bragg scattering measurements elapsed times were

measured from the time shear melting ceased. Time is measured in units of the

Brownian time, 'ta - R2/Do ' where R is the particle radius and Do is the :free particle

diffilsion coefficient, giving a dimensionless time, 't = tI'ta.

2.2 DYNAMIC LIGHI' SCATTERING

Samples were prepared with a mixture of particles of types I and m, in a number ratio

of 100:1. Importantly, the particles have the same radius and surface coating but

different refractive indices. For this system of dynamically identical but optically

contrasting particles the particle scattering amplitudes, bj(q), and their positions, rj('t),

are statistically independent so that the intermediate scattering function (lSF), FM(q, 't),

obtained in a standard DLS experiment, can be expressed as [3,9],

(1)

where

N bm = N-1l:bj . (2)

j=1

214

F(q;t) and F.(q, "t) are the coherent and incoherent ISFs for wavevector q and delay

time "t. Note also that F.(q,O)=1 and F(q,O)=S(q), the static structure factor. The

incoherent ISF, the main quantity of interest here, can be written as

(3)

where <&-2("t» is the particle mean squared displacement (MSD). One can define a

(time dependent) 4iffusioncoefticient by the expression,

(4)

Eq. (3) assumes that non-Gaussian terms in the particle displacement, which are

successively weighted by amplitudes proportional to [<&-2("t)>q2]2, are negligible [16].

To obtain the incoherent ISF from the measured ISF (Eq. (I» it is necessary to

attain the optical condition where the average scattering amplitude, b, vanishes. This

was determined by vaIying the temperature until the main Bragg reflection, located at

the scattering vector '1m, of a crystallised suspension containing the mixture of polymer

and silica particles described above, was no longer visible. Accordingly, we found

b ('Im>=O at a temperature of Tm=20.4°C for the wavelength )..= 514.5 nm of the

argon ion laser. This optical match condition applies at all other scattering vectors

provided that the scattering by the sample at Tm for the wavelength ).. is entirely due to

the silica (tracer) particles and that these particles can be regarded as optically uniform

spheres in the Rayleigh-Debye limit. These assumptions were vindicated by our

observation that for a given sample the initial decay rate on fM(q,"t) varied in

proportion to q2 [16].

In this work measurements were performed at a small wavevector, given by qR=1.3

(compared with CJ.nRIw3.4). This allows a reasonable range ofMSDs, <&-2("t)~3R2, to

215

be obtained from F .. as approximated by Eq. (3), without interference from possible

non-Gaussian effects.

For samples with volume fractions below the glass transition DLS measurements

were performed with an AL V two-colour multiple-scattering-suppression spectrometer

[17]. Results indicated that multiple scattering effects were negligible, certainly for the

low wavevectors discussed here. Therefore, samples with volume fractions above the

GT were studied by conventional DLS, using a technique that permits the effects of

non-ergodicity of samples in the glass phase to be taken into account in a fairly

straightforward manner [4,18].

2.3 BRAGG ANGLE SCATTERING

A diode array camera mounted on a goniometer arm was used to record the intensity

scattered from a volume illuminated by the collimated beam of a diode laser.

Intensities spanning the main Bragg reflection from the growing close-packed planes

were divided by the particle form factor, b"2( q), and corrected for scattering from the

remaining fluid [6]. From the resulting Cl)'stal structure factors, Se(q;'t), (such as those

shown in Fig. I) the following quantities were calculated. First, the total integrated

area under the peak, X, is proportional to the fraction of Cl)'stal present. Second, the

peak width. 4q, is related to a crystal length scale L by

L=27tK1Aq, (5)

where K=I.155 is the Scherrer constant for a Cl)'stal of cubic shape. Third, assuming

the Cl)'stal structure to be close-packed, the crystal volume fraction, 'e, was obtained

from the peak position, 'Ln, of the inter-layer reflection by

(6)

216

3.3 3.4 3.5 qR

3.6 3.7

Figure 1. Crystal structure factors, Sc(q;,;) plotted against reduced scattering vector, successive 1rIcea have been displaced vertically for clarity. This graph shows the evolution of the main s.-agig peak for a sample in the coexistence region, +=O.S37.

3. Results and Discussion

3.1 DYNAMIC LIGHT SCATIERING

The particle MSDs as a functions of the delay time are displayed in double logarithmic

plots in Fig. 2. Diffusive motions occur where these results follow straight lines of unit

slope (Eq. (4». This occurs over the whole range of accessible times only for a very

dilute sample, where the particle motion is characterised by the free particle diffusion

coefficient, Do' At higher volume fractions diffusive motion is seen at short times,

before appreciable change to the particle configuration (<Ar2>;:SO.OlR2), and at long

times, during which many (statistically independent) changes of configuration are

required for the particles to perform large-scale excursions (<ar2»R2). The

increasing subdiffusive region that connects the diffusive regions at short and long-

217

times is indicative of the increasing persistence of a particle's environment of

neighbours.

1

o

-1 A -... (\1--2 .... <:I v

-3

-5 o

-2 4 6

Figure 2. Log of the particle mean square displacement versus log of the delay time. Volume fractions are indicated for highlighted data.

For the highest volume fractions, above cFO.580, the second diffusive region is not

approached. In these cases, the particles are trapped within their cages of nearest

neighbours over the course of the measurement (5000s) and the MSD settles to an

almost constant value at long times.

Coefficients Ds=D(T-+O) and DI=D(T~), respectively characterising the short and

long-time particle diffusion, are shown as functions of the volume fraction in Fig. 3. It

is evident that DI('~.58)-+O. Accordingly, the volume fraction ~.58 is

identified as the GT volume fraction '8' the concentration where large-scale particle

diffusion is arrested.

218

0

-1

-2 \

C \ 0 ... g» -3

-5

0.0 0.2 ~ 0.4 0.6

Figure 3. Short- and long-time particle diffusion coefficients (open and closed symbols respectively) as fimcions ofvolume fraction. Data shown as diamonds is taken from reference [9J The solid lines show tbeotetical predictions (see text for details).

As the GT is approached the long-time diffusion coefficient scales with the

function (1-+I,g)2.6, which represents the relaxation rate predicted by mode-coupling

theory (MCf) for the a.-process in the hard-sphere fluid. Previous work [3] has shown

that the long-time relaxation rates of concentration fluctuations of all wavelengths

scale in this same manner. However, while large-scale diffusion arrests at 'g' local or

small-scale diffusion persists up to random close packing, +r=O.64. Experimental

results for D. shown in Fig. 3 scale with the function (1-+I+r)J.2 [19].

3.2BRAGG SCATTERING

Fig. 1 shows the crystal structure factor around the interlayer reflection for a sample,

prepared from suspension n, in the equilibrium coexistence region (with +=0.537)

parameterised by reduced time, 'to As discussed in Sec. n, we calculate from these data

0.5

0.0

>< -0.5 o ....

0) o - -1.0

-1.5

-2.0

3

.. 0

• o •

5

10910 't

• .- 0.537

.. .-0.553

6

Figure 4. Amount of crysta1 (calculated from the area under the Bragg peak) as a function of reduced time. The volume &actions indicated on the figure refer to bigbligbted data.

0.64

0.62 •• o 0 00

.. 0

0.60 "0

0"0 u -e-

0.58

0.56

0.54

3

0 0

0 0 0

4

0 0

0

5

10910 't

.-0.537

."'0.553

.- 0.565

6

Figure 5. CryIItaI volume &action (calculated from the position of the Bragg peak) as a function of reduced time. The volume &actions indicated on the figure refer to bigbligbted data.

219

220

the fraction of aystal, X, the average linear aystal size, L, and the aystal volume

fraction, +c. The initial stage of rapid increase in X (Fig. 4), which has been associated with

nucleation and growth [6,7], crosses over, at a time 'to' to a region that shows a much

slower rate of increase. The latter is associated with ripening and improvement in the

quality of the aystal phase as it relaxes towards equilibrium. Expansion of the lattice

during aystallisation is indicated by the shift of the Bragg peak to smaller scattering

vectors. The consequent reduction in aystal volume fraction is shown in Fig. S.

The maximum number of aystals or nucleation site density was calculated from,

N_ =max[X('t)/L('t)3]. Assuming that the conversion of fluid to aystal is complete at

the crossover, we obtain the average nucleation rate density, R.va =Nma/'tc' shown in

Fig. 6. Also shown in Fig. 6 is the classical nucleation rate density for hard spheres.

(7)

In the calculation of~ ... the difference in chemical potentials, AJ.L, between the aystal

and fluid phases was obtained from equations of state of the hard-sphere system and

the (dimensionless) diffusivity was obtained from D=D{Do=(I-+t+l·6, which, as

shown in Fig. 2, provides a reasonable fit to the measured large-scale single particle

diffusion coefficients over the range of volume fractions spanned by these

aystallisation experiments. The fluid-solid surface tension, y, was varied within the

limits set by current computer simulation estimates [19] and the dynamical factor, A.

was treated as a parameter to obtain the fit to experimental data seen in Fig. 6.

According to classical theory, the reduction in the nucleation barrier, expressed by

the quantity in the exponent in Eq. (7), is largely responsible for the sharp increase in

~ seen in Fig. 6, as the melting volume fraction is approached. Beyond the melting

point, R.v. decreases in a manner that follows D) and the data suggests both these

quantities converge to zero as the GT is approached. At the same time, the maximum

nucleation site density, N_, remains almost constant

-3

- ---- --8 -4

Il: ~ E

0 -5 z ..-

0 01 .Q

-12 d) £

-6

-16 -7

0.52 0.54 0.56 0.58

~ Figure 6. Circles show the average nucleation rate density as a function of volume fraction, filled cirles &om reference [6] and open circles from reference [20]. The solid line is the classical nucleation rate density (Eq. (7». the short-dashed line is the nucleation barrier expressed by the exponent in Eq. (7) and the long-dasbed line is the dimensionless di1fusivity. Triangles show maximum number of crysta1s (nucleation site density).

221

While crystallisation by homogeneous nucleation stops at 'g' crystal growth seeded

by secondary nuclei, such as cell walls or the meniscus, continues to take place to some

extent in colloidal glasses with volume fractions up to close-packing. It seems,

therefore, that crystal growth requires only the small-scale particle motions

characterized by D s.

4. Concluding Remarks

The lengthening subdiffusive crossover regime seen in the MSD data (Fig. 2) indicates

that as the volume fraction is raised, particles are becoming progressively more

hindered in their attempts to diffuse long distances. As the GT volume fraction is

approached. D. -. 0 and particles are effectively confined to their neighbour cages.

222

The average nucleation rate density, ~ scales with the long-time particle diffusion

coefficient and the observation that both these quantities converge to zero at the GT

supports the assertion that nucleation requires large-scale diffusion. Saturation of the

nucleation site density with increasing • may imply that crystals are being nucleated at

sites separated by distances comparable to their size. This proposition is supported by

the crystal sizes measured from the width of the Bragg peaks.

The classical nucleation rate density for hard spheres compares well with the

observed average nucleation rate density. This agreement was obtained using Eq. (7)

with a dynamical factor of A = 100 (some 8 orders of magnitude smaller than the value

typically found for molecular systems). However, this accord with the classical theory

may be somewhat fortuitous, given the sensitive dependence of Rm. on microscopic

thermodynamic quantities (A~ and y) and the observation of accelerated nucleation in

these systems which is more compatible with strongly coupled nucleation events [6].

~. References

1. Kelton, K.F. (1991). CryItalNuclcatiClllinUquidsIDdG~SoUdStQtePhy,. 45. 75-177

2. Freukel, J. (1946)KIINlic TMory ofLlqtllrh, Oxford University Press. Oxford.

3. Pusey, P.N. (1991) Colloidal suspensioal. in J.P. lIaDsea, D. LeveIque IIId J. Zinn-Justin (eels.). Liquid"

Fruzing and the GlaI, Tran,tlion. North-HoUand, AmsterdIm, pp. 765-942.

4. van Megea, W. (1995) Ci)ltaUilatiClllIIId 1he Glus TrmmtiCIIl in SUlpCllliOJll of Hard Colloidal SphenI.

Tran.rport Theory and Stat Phy,. 24, 1017-1051.

5. SdIItzeI, K. IIId AI:k-. B.J. (1993) Deasity FluctuatiOJll cUring crysta1IizatiClll of colloids. Phy,. Rev. E

41, 3766-3777

6. Harlmd, J.1.., Hendenoa, S.I., UodenwocI, S.M. IIId van Megea, W. (1995) ObservatiClll of Accelerated

Nucleation in Dease Colloidal Fluids of Hard Sphere Partic1es.Phy,. RtN. Letter,7!, 3572-3575.

7. Aastueo, D.J.W .. Clarlt, N.A, Cotter, 1..K. IIId AcIcenon, BJ. (1986) NucleatiClll and Growth of Colloidal

CrystaII. Phy,. Rev. Letter, !7, 1733-1736.

223

8. WOrth, M .• Schwarz, J .• Culis, F .• Leiderer. P. and Palberg. T. (1995) Growth kinetics of Body Centered

Cubic Colloidal Crysta1s. Phys. Rev. E 52. 6415-6423.

9. van Megen, W. and Underwood, S.M .• (1989) Tracer diffusion in concentrated colloidal dispersions. m.

Mean squared displacements and self-di1fusion coefficients, J. Chem. Phys. 91. 552·559.

10. Pusey. P.N. and van Megen, W. (1986) Phase behaviour in concentrated suspensions of nearly bard colloidal

spheres, Nature (London) 320. 340-342.

11. van Megen, W. and Underwood, S.M. (1993) Change in crystallisation mechanism at the glass 1ransition of

colloidal spheres, Nature (London) 362. 616-617.

12. Underwood, S.M .• Taylor. J.R. and van Megen, W. (1994) Sterically stabilised colloidal particles as model

bard spheres, lAngmuir 10. 35S0·3SS4.

13. Underwood, S.M. and van Megen, W. (1996) Refractive Index Variation in Nonaqueous Sterica1ly

Stabilized Copolymer Particles, Colloid Polym. Sc. (to appear)

14. Paulin, S.E. and Adcerson, B.J. (1990) Observation ofa Phase Transition in the Sedimentation Velocity of

Hard Spheres, Phya. Rev. Letters 64. 2663·2666.

IS. Hoover. W.O. and Ree. F.H. (1968) Melting Transition and Communal Entropy for Hard Spheres, J. Chern.

Phya. 49.3609·3617

16. van Megen, W. and Underwood, S.M. (1988) Tracer diffusion in concentrated colloidal dispersions. II. Non­

Gaussian effects, J. Chern. Phys. 88. 7841·7846

17. Segre. P.N .• van Megen, W .• Pusey. P.N .• SchAtzel, K and Peters, W. (199S) Two-color dynamic light

scattering. J. Mod. Optics 42. 1929·19S2.

18. Pusey. P.N. and van Megen, W. (1989) Dynamic light scattering by non-ergodic media,Physica A 157. 70S·

741.

19. van Duijneveldt, J.S. and Lekkerkerker. H.N.W. (199S) Crystallization in Colloidal Suspensions, in van der

Eerden, J.P. and Bruinsma, O.D.L. (eels.) Science and Technology of Crystal Growth. Kluwer Academic.

Dordrecht

20. He. Y .• Adcerson, B.J .• van Megen, W .• Underwood, S.M. and SchAtzel, K (1996) Dynamics of

Crystallization in Hard·Sphere Suspensions, Phys. Rev. E 54. 1·12.

USE OF LIGHT SCATTERING TO CHARACTERIZE THE

POLYSACCHARIDES OF STARCH

Abstract.

PHILIPPE ROGER AND PAUL COLONNA Institut National de la Recherche Agronomique, rue de la Geraudiere, BP1627, 44316 NANTES, Cedex 03, France

Our group is involved in research studies based upon the relation be­tween macromolecular features and functional properties of starch and starchy products. Starch is a mixture of mainly two polysaccharides: amy­lose, the essentially linear component, and amylopectin, the other more branched component. Light scattering techniques have been used to find these macromolecular features. In particular, static (SLS) and dynamic (DLS) light scattering techniques and coupling between size exclusion chro­matography and multi-angle-laser-Iight-scattering (SEC-MALLS) have been applied. A review of some recent results obtained by these techniques from our laboratory are presented, with in the order of complexity: - the behaviour of synthetic amylose in different aqueous solvents; - the molecular weight distribution of amylose fractions extracted from starch; - wllat can be obtained or expected when amylopectin is studied by LSj - the macromolecular features of starch with the influence of the amylose content.

1. Synthetic amylose

The behaviour of amylose in different aqueous solvents has been studied first by using synthetic amylose fractions. Synthetic amylose fractions have several advantages: the possibility to get some rather monodisperse frac­tions, a better solubility in water as compared to amylose directly obtained from starch and the fact that these fractions are strictly linear. The syn-

225

E. R. Pike andJ. B. Abbiss (eds.), Light Scattering and Plwton Correlation Spectroscopy, 225-229. @ 1997 Kluwer Academic Publishers.

226

thetic amylose fractions were enzymatically prepared using maltohexaose as starter, glucose-I-phosphate as agent of polymerization and phosphory­lase as the enzyme. SEC-MALLS results [I] show that the polydispersity (P = Mto/Mn) increases from 1.002 up to 1.175 with a weight average molar mass (Mw) in the range 2.4x105 - 1.2xl06 g/mol. So to obtain the exponent a in the power-law relationship between the radius of gyration and the molar ma.'iS, weight averages have been used. Then a = 0.62 in KOIl O.IM and a = 0.51 in pure water. So amylose can be considered as an ex­panded random coil using KOII O.IM as solvent and a random coil in theta condition using water as solvent. The expansion of amylose in KOH O.IM and the colla,psed theta condition can be observed on a plot of Rg2 / M to

vs. M wO,5 [1]: the influence of M 111 on flg 2 / M to is negligible for water but increases linearly for KOII O.IM due to the excluded volume effect. These kinds of plot allow the determina.tion of the persistence length, 11), both in a good or a thet,a solvent [2], which give a very low value for 11) = 1.5 - 1.6nm, characterising a fully flexible chain in solution. These results obtained by SEC-MALLS confirm some previous results ohta.ined hy SLS a.nd DLS [2].

2. Amylose fractions directly extracted from starch

As the macromolecular features of synthetic amylose are fully understood, let us turn to the more difficult case of amylose fractions directly extra.cted from starch. A key problem in the analysis of gelling polysaccharides is the presence of so called super-aggregates or microgels which give rise to erroneously high molar masses. In the case of amylose the problem is even more tricky a.'i, amylose being extracted directly from starch, traces of re­maining amylopectin with molar masses known to be higher than 108g/mol can influence greatly the macromolecula,r features [3]. In order to get rid of the high molar mass population, ultracentrifugation has been used suc­cessfully on amylose solutions in KOH O.IM [4], [5]. DLS was used in this case to a.<;sess the disa.ppearance of the slow diffusing motion. This method ha.'i heen applied to study the dependence of the macromolecular features of amylose of five different botanical origins [4]. It has been shown that the .1\1", (between 3Ax 105 and 1.05xl06 9 /mol), flg (between 44.5 and 89.9 nlll) and Rh (between 17 . .5 and 29.3 nm) were botanical origin dependent but in a.l1 tllC ca.ses, the exponent in the power-law relationship M to - Rgo were in the sa.llIe range «(\' = 0.61 - 0.68) characterizing an expanded random wil. So a,lIlyloRe fractions directly extracted from starch ha.ve the same he­ha,viour a~ st.rictly linear a.myloses. That means tha.t even if the extracted a.mylose cont.ain branches, they Ilave no influence on their behaviour in so­lution. The influence of the temperature of leaching on the macromolecular features of a.mylose was studied in the sa.me way [5]. Results show clearly

227

that M wand Rg both increase as the temperature increases from 65 to 95° C. For the temperatures higher than 850 C, the distribution of molecular weight clearly indicates the presence of two populations. The distribution has been deconvolved and M w obtained for each peak.

3. Amylopectin

The LS study of amylopectin is quite different as compared to amylose. In the ca..,e of amylopectin huge molar masses and sizes have to be considered. In the range of scattering vector (q) commonly used, the assumption of low qRg values is no longer valid and extrapola.tion to zero scattering angle has to be considered with caution. However recent advances have been made in the way of the dispersion procedure of amylopectin using high temperature wit.h microwave [6], [7] or autoclave treatment [8], [9]. Then results from different laboratories on waxy maize starch seem to indicate some consistency with Mw in the range 8xl07 - 3 x 108 g/mol and Rg from 234 - 320 nm. In SLS a downturn curvature is observed for the q dependence of tIle Rayleigh rat.io, which can be explained by the high polydispersity of amylopectin. When SEC- MALLS is used, LS data are available all along the output of the chromatography columns. For the majority of the profile, an upturn is observed which indicates that the chromatography system is efficient to separat.e amylopectin components. For the remaining part of the population with still a downturn curvature, some questions remain: is this part of the population degraded or is it another kind of amylopectin?

4. Starch

For starch, a mixture of two components has to be considered. That is why a study involving starches with different amylose content (respectively 0, 28.5, 52.5 and 65.8%) has been investigated in order to see how the LS results are infltH'nced by this parameter [7]. Kratky plots which have been used recently for studying the macromolecular structure of starch [8], show a difference hetween high and low amylose starches in the high qRg range [7]. However this method presents the disadvantage of still using a macro­molecular charact.eristic obtained by extrapolation. The results obtained below present. the advantage of using only the raw data of either SLS or DLS without ha7.ardous extrapolations to zero scattering vector. A fractal dimension (dJ) can be obtained directly from the slope of the asymptote at high q values on the log-log plot of R(q) vs q. This fractal dimension showed effectively dependence upon the amylose content. A marked dif­ference occurs het.ween the two low (dJ = 2.20 and 2.28) and the two high (df = 1.69 and 1.55) a,mylose content starches. The values of (lJ for amylopectin-rich starches are between the fractal dimensions predicted by

228

percolation theory which are dJ = 2.5 for non-swollen crosslinked clusters and d I = 2.0 for fully swollen clusters [1], whereas the· values obtained for amylose rich starch (dJ = 1.69 and 1..56) are characteristic of linear flexible structures. Using DLS and taking into account the high qllg range, the significant measuring parameters are the exponcnt 0 in the powcr law be­tween the first-order relaxation time r and the scattering vector q, r _ qOt, and the pre factor r*= rTJo/( q3J( BT) obtaincd asymptotically in the high-q ra.nge. Experimental 0 values incrca.c:;c from 2.80 up to 2.89 with the amy­lose cont.ent but never reach the value of 3 expccted for a fully flexible chain [11]. A lower a value is cxplained in the literature by a loss of inter­na.! nexibility which apply well to our results as nexibility is expected to be lower as the branching density increases i.e. when the amylose conti/nt decrea.c:;es. The prcfactor r*= r."o/(q3J(BT), which is expected to take the values 0.05 - 0.06 and 0.045 for nexible chains in good and theta-solvents respectively [12], is found here to increase from 0.037 up to 0.055 with the amylose cont.ent. The lower r* value obtained for high amylopectin content starches is explained in the same way as o. The conclusion of this study is tha.t the results obtained by LS have to be considered separately depend­ing whether the amylose content is high (> 50%) or low « 30%). This informa.t.ion is part.kularly important for norma'! sta.rches where the amy­lose content. exceed ra.rely :JO%. In this case, the macromolecular features arc mainly dependent on those of amylopectin. If it is possible to do SLS and DLS on the crude product i.e. without preliminary separation of starch macromolecules, the int.erest. of such studies is restricted. To obtain clearly the features of amylopectin and amylose separately, a powerful method of fractionat.ion of starch has to be used. That seems to be achieved, at lea.<;t partly, recent.ly using SEC-MALLS [13] or Sedimentation FFF-MALLS [9]. Dut still some prohlems occur due to peaks overlapping in the first ca.<;e and due to a lack of separation below 50 nm ill the other ca.<;e. Some cross­now FFF experimcnts a.re now in progress to try to improve the separation hetween starch macromolecules.

References

1. Roger, P., Axclos, M.A.V. and Colonna, P. {mannscript in preparation} Influence of solvent on the macromolecular features of linear chains of a1pha-glucans.

2. Roger, P. and Colonna, P. {1992} The innuence of chain length on the hydrodynamic behaviour of amylose, Carbohydrate Research, 227, 73-83.

3. Roger, P. and Colonna, P. (1993) Evidence of the presence of large aggregates con­taminat.ing amylm,e solution", Carbohydrate Polymers, 21,83-89

4. Roger, P., Tran, V., Lesec, J. and Colonna, P. (1996) Isolation and characterisation of single <:hain amylose .. J. of Cereal Sci. 24,241-262.

ri. Roger, 1'. and Colonna, P. (1996) Molecular weight. di"trihntion of amylo"e fradiolll; oht.ained hy aqneon" leaching of <:orn "t.arch, Int. . .I. or Biological Macromolecules I!I, 51-61.

229

6. Fishman, M.L. and IIoagland, P.D. (1994) Characterization of starches dissolved in water by microwave heating in a high pressure vessel. Carbohydrate Polymers 23, 175-183.

7. Roger, P., 8ello-Prez, L.A. and Colonna, P. (manuscript in preparation) Laser light scattering of high amylose and high amylopectin materials in aqueous solution.

8. Aberle, T., 8urchard, W., Vorweg, W., and Radosta, S. {1994} Conformational con­tributions of amylose and amylopectin to the structural properties of starches from various sources. Starch/Strke 46, 329-335.

9. Hanselmann, R., Burchard, W~, Ehrat, M., and Widmer, II.M. (1996) Structural properties of fractionated starch polymers and their dependence on the dissolution process. Macromolecules 29, 3277-3282.

10. Stauffer, D., Conoglio, A., and Adam, M. (1982) Gelation and critical phenomena Adv. Pol. Sci. 44, 103-158.

11. Brown, W. and Nicola, T. (1993) Dynamic Light Scatt.ering. The Method and Some Applications. W. 8rown, ed. Clarendon Press: Oxford.

12. Trappe, V., Weissmuller, M, and 8urchard, W. (1996) Hydrodynamics of linear and non-linear polymer systems. Proceedings of the present Workshop.

13. Bello-Perez, L.A., Roger, P., Colonna, P., and Baud, B. Macromolecular features of starches determined hy aqueous high-perrormance size exclusion chromatography J. or Cereal Sci. Suhmiu.ed ror puhlical.ion.

SPATIAL PHOTON CORRELATION AND STATISTICS OF

NONLINEAR PROCESSES IN NONLINEAR WAVEGUIDES

Abstract.

M. BERTOLOTTI, M. DE ANGELIS, C. SIBILIA Dipartimento di Energetica, Universita eli Roma "La Sapienza", Via Scarpa 16, 00161 Roma, ItaZy

AND

R HORAK Department of OJ)tics, Palacky University, 17 Listopaelu 50, 772077 Olomouc, Czech Republic

Spatial features of propagation in nonlinear planar waveguides are dis­cussed. Squeezing and bunching in spatial frequency are demonstrated to occur and the possibility of detecting squeezing via a spatial correlation measurement is discussed.

231 E. R. Pilu! andJ. B. Abbiss (eds.), Light SCIlIIering and Photon Correlation Spectroscopy, 231-246. @ 1997 Kluwer Academic Publishers.

232

SPATIAL PHOTON CORRELATION AND STATISTICS OF NONLINEAR PROCESSES IN NONLINEAR WAVEGUIDES

1. Introduction

It is well known that the spatial structure of the radiation field in planes orthogonal to the direction of propagation can display phenomena of spontaneous pattern formation and transformation similar to those met in hydrodynamics, or nonlinear chemical reactions. With respect to these cases , the optical systems present the unique feature of displaying interesting quantum effects. Even if the number of papers which study spatial aspects of quantum optical phenomena is still limited, recently there is a growing interest in the analysis of quantum effects in nonlinear optical patterns [Lugiato et al. 1995], mainly under conditions of confined propagation. A different class of transverse phenomena occurs during the free propagation through nonlinear materials , whose geometry is such that the field evolution can be reduced to a "20" problem. This can be realized in a waveguide geometry, where the field confinement into the transverse dimension is given by the refractive index step of the waveguide. In this geometry it is possible to study transverse quantum phenomena in the presence of third or quadratic nonlinear materials. When the propagation in a third order nonlinear waveguide is studied I Sibilia et al. 1994 I quantum effects in the space domain put into evidence the presence of quantum noise , spatial squeezing and bunching. In particular one of these effects shows a maximum at some spatial frequency and each of them is related to different propagation regimes, such as diffraction free propagation or modulational instabilty . A similar behaviour has been found in propagation in a quadratic nonlinear waveguide [De Angelis et al.], when a second harmonic generation process occurs . In this case an intersting condition of interaction can be realized

233

between pump and second harmonic beams: the so called " cascading " effect. That is the fields during the propagation undergo an intensity dependent self - phase modulation , as the one occurring in a third order nonlinear material. This effect depends on the phase mismatch and on the input intensity values ; it can be realized when there is a low conversion regIme The problem is the detection of these quantum effects in space . The spatial squeezing properties can be made evident by spatial cross correlation measurements. Since the field is squeezed just along some directions ( i.e. suitable spatial frequencies) a cross correlation between two detectors as a fi.mction of angular frequencies is able to reveal the phenomenon. A discussion about the cross correlation scheme of detection of quantum fluctuations will be presented , and the propagation conditions necessary to obtain transverse quantum phenomena in space domain, will be discussed.

2 Clasical Propagation in a Nonlinear Waveguide

A wave equation for the electric field E is usually used to solve problems of nonlinear optics. If one wants to have complete electromagnetic description of the problem , the magnetic field H has to be calculated using the Maxwell equations. The nonlinear wave equation is written as

(1)

where V is the nabla operator, au= ~, J-lo is the permittivity and P at

is the polarization . - -Under monochromatic approximation for the vectors E and P the wave equation can be written as

We are interested to study the propagation in a nonlinear planar waveguide which is made by a nonlinear thin film surrounded by two linear semi­spaces with the plane of the film perpendicular to the y-direction . Let

assume a decomposition of the field E given by

234

E(x,y,z,t) =x Ex(x,z,t) ux(y) + y Ey(x,z,t) Uy(y)+

Z Ez(x,z,t)uz(Y) (3)

where x, y, z are unit vectors and u j ( y) are mode functions of the

linear waveguide satisfying the equation

(4)

and the corresponding boundaries conditions for j= x,y,z .The quantity k is

k2 2 0)2 2 2 = 0) /JOE = - = kon

v2 (5)

where n is the effective refractive index of the waveguide. The quantites P j are the magnitudes of the vectors of propagation in a linear waveguide

corresponding to different modes ( Pj are not components of the

vector J3 ) We suppose that the field propagates in the z- direction; then

Ux ( y) is the mode function of TE field , and uy ( y) and Uz ( y) are mode functions of TM fields. In the following we suppose to have TE fields only. Considering a third order nonlinear material the wave equation assumes the following simplified expression

d~ Ex + d'1zEx + 2iOJeJlodt Ex + p2 Ex = - xl Exl2 Ex (6) where X is a nonlinear coefficient , which takes into account also the transverse mode profile. When paraxial and SVEA approximation are taken into account, eq. (6) becomes [Horak et al. 1994]

i 8Q = _.! 82Q -IQI2 Q 8Z 28X2

(7)

where Z=Bz X=Bx 't = t-~z Q- E ~ X " ~v' - x 2~2

The time variable plays the role of a parameter which affects the behaviour of the field by means of boundary values of Q only . From this point of view our problem is reduced to a space "propagation" (time indipendent) problem only, even if we study a pulse propagation. A solution is now searched in the form of a strong plane wave

Qy(Z,T) = Qo(T)exP[iIQo(T)r Z] (8)

235

where Qs(Z=O, r) = 9:>( r) . We assume the field Q to be a strong plane wave with a small perturbation superimposed to it

Q(X,Z) = Qs(Z) +q(X,Z) = Qs(z)[l +d(X,Z)] (9)

where Iql«IQJ=I~ which is equivalent to the relation Idl«l. Substituting eq.(9) into eq. (7) and linearizing this equation we obtain

idz = - ~ dxx _1~12 (d + d*) (10)

where * denotes complex conjugated quantities and subscripts denote derivatives. Performing the complex conjugated of eq. (10) one obtains an equations for d*. These equations are linear and can be solved using the Fourier trasform

1 00 .

d(X,Z) = ~f c( cr,Z)e-IC1X dcr v2n -00

(11)

The transformed equations give an equation system for c( (J, Z) , whose solution is placed into eq.(11). An inverse Fourier trasform we obtain the solution for d

00 00

d(X,Z> = f A(u-X,Z)d(u,O)dut-f N(u-X,Z)t/ (u,O)d

where 1 00

M(u,Z) = - f Jl(cr,Z)cos(cru)dcr no

1 00

N(u,Z) = - J v( cr,Z)cos( cru)dcr no

(12)

(13)

The solution has the integral form and can generally be evalutated in a numerical way only. Finally this solution inserted into eq.(9) provides the form of the field propagation inside the nonlinear stucture . One particular solution can be realized if we assume that a strong plane wave is incident on a nonlinear planar waveguide in the z-direction. The perturbation is produc.ed by two weak plane waves with directions of propagation forming an angle with the z-direction, given by the relation. [ Horak et al. 1994 ]

sina = ±Ll (14) The resulting distibution of the "intensity" IQI2 in the waveguide is then

(15)

236

Some crucial spatial frequencies, linked to the spatial periodicity of the input perturbation, appear from the linear stability analysis of the linearized system of equations governing the propagation. The maximum of instability takes place when the spatial frequency reaches the value

a= .J21~1. This is the well kown Benjiamin-Feir or modulational instability [Newell and Moloney 1992]. Another crucial spatial frequency is found when we search conditions which lead to a diffraction free pattern of the perturbed field Q. The general condition for a diffraction free field is

This is realized when

BIQ(X,Z)f = 0 BZ

(16)

cr = 0 ; cr = ±21~1 (17) The existence of these two groups of spatial frequencies is very important by the point of view of the consequences on quantum noise , as discussed in the next paragraphs.

3. Second Order Nonlinear Optical Processes

Considerations very similar to the ones presented in the previous paragraph can made for the case of second order nonlinear processes, such as a second harmonic generation . A second harmonic generation process is described by the following equations

2ikJ8zEJ+8xxEJ +T}EJ+ xE;E2e-iAkz =0

2ik28zE2 +8xx E2 +11' E2 + XEJeiAkz =0 (18)

where E J , E 2 are the field of the first and second harmonic respectively, k1' k2 are the field propagation constants . al~ng the z­

direction, 11,11' are defmed by = el ~ k{, 77' = 4el.- ki, £1, £2

are the dielectric constants for first and second harmonic, X is the nonlinear coupling constant for the nonlinear process, 11k = 2k1 - k2 is the phase mismatch. We have assumed a TE polarization along y direction of each of the monochromatic field involved during the interaction for a type I quadratic material [ Buryak and Kivshar 1995 ]. A suitable change of variables can be considered:

u = ~(kf- k~el)' V= x~kf- k~el 2kl

= elk~ -kf , 17'= 4(e2k~ -ki) So that we have

iOuA + 0wA- A + A*Be-il1kz ~ 0,

i k2 ° B+o B -aB+ A2e+ il1kz = 0 k1 u w ,

237

(19)

(20)

ki-k~e2 where 0< = 2 2 Under the hypotesis that

kl -koel

aIBI» Ii k2 0uB+owB I , we can simplify the second eq. ( 20 ) , so k1

to obtain an expression for B to be substituted in the ftrst eq. ( 20 ) . When this is made a "cubic behavior" of the quadratic nonlinear material occurs, and we fmally obtain an equation for the the pump fteld

(21)

which reduces to a NLSE when I A r / a> > 1. In this way , after a suitable change of symbols, we can apply the same considerations presented in the previous paragraph, i.e. it is possible to fmd spatial frequencies for which modulational instability and diffraction free propagation occur for the pump fteld , when a superposition of a plane wave and a spatial harmonic perturbation is considered. Under conditions for which a " cascading" effect is realized during a second harmonic generation process, we can treat the pump evolution in a similar way, in the space domain, as we did in a third order nonlinear material. In what follows, looking at the quantum aspects of the propagation, we describe in an unifted way both interactions.

238

4. Quantum Description

The quantum description of the field can be performed introducing th corresponding quantum operators which fulfilll the following equatio obtained from the classical equation (7), in which losses and fluctuation are added through a coefficient a, and the Langevin term r(X,Z) Sibilia et. al. 1994 ] .

. "1 A "+ A A • " "

z8ZQ+ 2 8XX Q+Q QQ+zrOQ=r(X,Z). (22) A

Q is the quantum-field operator and subscripts stand for derivatives; y

is the loss term given by Yo = a/2r/. The term iy 0 becomes equal to

and the Langevin term is zero if the previous equation describes the purr evolution in a quadratic nonlinear material. Because it is not possible to measure the field in exactly one space- tin point, we have to defme the suitable commutation relations for the t1 operators Q and Q+. Assuming that the quantum character is given 1 small quantum fluctuations (perturbations) of the classical field Q, i.e.

Q=Q+q, (23) where Q is a solution of eq. (23), it follows that the perturbation opera1 q bears the full quantum character, and only the quantum fluctuations ~ affected by damping and Langevin forces. The commutation relations a

[q(X, Z), q+(X', Z)] = 8(X - X') / Lz ' (2,

[q(X,Z),q(X' ,Z')] = [q+(X,Z),q+(X' ~Z')] = 0 . (2:

Substituting (23) into (22) we arrive at the following linearized equati(

i8zq+ ~ 8xxq+2IQI2q+Q2q+ +iroq =r(X,Z) . (2( A

Let's assume that the c-number part Q of the operator Q has the form ()

plane wave [Sibilia et a1. 1994] , where QO = IQolexp(iqJO) is a const

and let us introduce for q(X,Z) the Fourier transform

exp{i (IQoI2 Z + <po)} . q(X,Z) = ..n;; eAc(cr,Z)e-zcrXdcr (27

where cr are spatial frequencies. Studying a monochromatic behavi< the spatial frequency cr represents the angle between the Z-direction i

239 A

the direction of propagation of a field described by the operator Q . The general solution for c is given by

c( u,Z) = p( u,Z)cO( u)+ v( u,Z)ct( -0')+

+ f~[V( u,Z')r( u,Z') - T( O',Z')r+ (-u,Z') ]dZ', (28)

in which the terms under the integral sign take into account losses. Eq. (27) enables us to have the solution of the quantum part of the field given by the following expression

q(X,Z) = qhom(X,Z)+q L (X,Z) =

exp[{IQoI2 Z + 9'0 )] . = .ffi f~~ (Chom( O',Z)+cL (O',Z))e-ZuX du.

(29) where q hom is the solution obtained in the absence of losses and q L is the part due to the presence of the Langevin term.

5. Moments

To investigate the quantum aspects of the field during the propagation let

us introduce the quadrature operators Ql and Q2 [Sibilia et al. 1994 ]

1~1 ~ ~1(Q~~~~· (30)

Q2 -2/Q Q ),

in which we have supposed e = 0, where the status of the field is a vacuum coherent state. We have for the variances

((~.2 - (~.2))} (( ... ~:})= ![(.W)+(q+q)±«(qq)+(q+q+)lj. (31)

where q(X,Z) is given by eq. (27). After some algebra we have:

(( ... ~i) = JO"S~ (d .Z)dd. (32)

where 81,2 (cr' , Z) is the power spatial spectrum of the quantum noise

240

, L SI,2 (a' ,Z) = SI,2 (a' ,Z)+ SI,2 (a' ,Z), (33)

The lossless term SI.2 ( cr, Z) is given by

SI,2 ( a,Z) = 21;r[1,u( a,Z)12 +Il{ a,Z~2 ±~,u( a,Z)IIl{ a,Z~ cof oS{ a,Z)] J. (34)

with 2

.9(a,Z)=2IQol Z+2<PO+arg[,u(a,Z)]+arg[ l-\a,Z)] (35)

For the loss term S~2(cr,Z) a very complicated expression has been

found and its contribution has been discussed by Sibilia et at. 1994. A

We may derive also the fourth order moment of the field operator Q

((t-2Q2) -( (t-Q) 2 =2R~Q*2( q'q')}+ 21Ql (q+'q') -4RJ{Q* (q')}

(36) Bunching can be found if the expression (36) results greater than zero. After some algebra we obtain:

(Q+2Q2)_(Q+Q)2 = faB(a',Z)da', (37)

where B( a,Z) = Bhom (a,Z) + B L (a,Z), (38)

and Bhom ( cr , Z) is the 10ss1ess term given by the following expression:

Figure 1 Spectral squeezing versus spatial frequency and propagation distance

241

(39) with

tjJ( a, Z) = arg[.u( a, Z) ] + arg[ It{ a, Z) ] . (40)

the loss term BL is discussed in ref. [Sibilia et al. 1994] The results derived in this section refer to the spatial distribution of a plane wave with a small superimposed spatial perturbation. The presence of losses in the nonlinear medium reduces the magnitude of the squeezing, as expected ( see Sibilia et al. 1994 ). In fig. 1 the spatial evolution of the quantum noise represented by S', defmed in eq. (41), in one of its quadrature components, is plotted as a function of the spatial frequency during propagation in the Z-direction, where initial conditions

for the input field are qJO = O,/QO/ = 0.5. In this case a third order

nonlinear material has been considered. By increasing the propagation distance, S' shows some distribution , as a function of cr, which has a distinct minimum at cr min == 0 .6. This is the spatial frequency for which modulational instability occurs in the classical propagation ~ In fig. 2 the bunching in spatial domain is presented for a third order material. B shows a peak at the diffraction-free frequency [ C.Sibilia et al. 1994] .

Figure 2 Spectral bunching versus spatial frequency and propagation distance

242

LO:l(S I(C.Z) )

z c.

G a

A) ~) Figure 3 Spectral squeezing versus spatial frequency and propagation distance for

different k vales A) 6k =0 • 9) 6k = 40

Figs.3 show the log of the squeezing spectrum S J for different values of the mismatch l1k between pump and generated beam, as a function of the propagation length in the space frequency domain for a second order nonlinearity. As in the previous case the squeezing is very pronounced for the spatial frequency for which in the classical propagation there is modulation instability [Horak et a1. 1994 ] The main difference is that now a largest squeezing is obtained . It increases as a function of the propagation distance and of mismatch. In fig 4 the presence of bunching is shown for different mismatch values . The presence of buncing is for a spatial frequency different from the one for which squeezing occurs, as discussed for the Kerr case [C.Sibilia et a1. 1994 J. The increase of mismatch increases the amount ofhullChilig.

6. Detection of Spatial Frequency Components of Squeezed Light

Ou et al 1987 have proposed a method for detecting a squeezed state by coincidence cOWlting or cross correlation of signals from two photodetectors. In the spectrum of the photocurrent, squeezing is manifested by the appearance of positive cross correlations, whereas nonsqueezed or classical light always gives rise to negative cross correlations, and this is an unmistakable signature of squeezing . A variation of an autocorrelation measurement has been proposed by Potasek and Yurke 1988. This method can be extended to detect the spatial-frequency components of the squeezed field propagating through the NL waveguide. Since the field is squeezed just along some directions

243

(i.e. for some spatial-frequencies cr) a cross-correlation between two detectors as a function of angular frequencies is able to put into evidence the phenomenon. With reference to fig. 5, consider to put an array of photodiodes in the focal plane of a lens in front of the waveguide output plane. Two photodiodes detecting the symmetric spatial frequencies cr and -cr are activated and the currents from the two diodes are cross­correlated electronically as a function of cr, or which is the same, as a function of which pair of diodes is activated. For the photocurrents we have [Ou et al. 1987 ]:

Figure 4 Spectral bunching versus spatial frequency and popagation distance for a)~=O,~=40 b)

(I1,2(X»)=TJL: dp8(X-p)(Q~2(X,Z)QI,2(X,Z»), (41)

where TJ is the detector trasmission function, 8(X) is its spatial A

impulsive response, Q is the field operator and the detectors are identified by subscripts "1,2". Let us suppose that the angular band­width (field of view) of the two detectors is .1cr and that they are symmetrically shifted from the Z-direction, i.e. diode" I" is positioned at cr = cr c while diode "2" is positioned at cr = -cr c (cr c is the spatial frequency corresponding to the particular direction along which we want to detect the field), as it is shown in fig. 5. For a field in a vacuum state we obtain the following expressions for the photocurrents:

244

(/1(X)) = JIQol2 +_1 r C+:ill{o',Z)12do'}, "1 2n- O"C-""2

(/2 (Xl) = ~IQi + 2~C:~~Ho',Z~2dol (42)

If Aa is quite narrow, we may assume that Iv(a,Z)12 does not change on

this range, and takes the value it has at the center band frequency (i.e. at a = a c). Therefore

(11 (X)) = (/2(X)) = ~ICJoI2 + 2~1l{ CTc,Z)12 ACT}. (43)

For the cross correlation function we have [Sibilia et al. 1994, Ou et al. 1987] after some tedious calculations, considering a lossless case for the sake of simplicity,

(/1 (X)/2(X + p)) =

_ ( ( ))( ( )) Iql [ I.d: CTc,Z~ J.tI )]]., )2/ sirf k - 11 X 12 X +-;- 1+I~CTc,Z~CO,,,,D;;,Z t-\.D;;,Z ·IT

(44) Aa

where k = -po Normalizing with respect to the product of the two 2

intensities, we arrive at the following expression:

(l,(xl/,(x + p l) _ ( l,u( CTc ' Z)I [.d )]) sin2 k (/1(X))(/2 (X)) -1- 1+Il{CTc,Z)ICOSV'\CTc'Z k2 '

(45) In fig. 6 we have plotted eq. (45) for two different frequencies, ( for a

third order nonlinear material) ac = .J2IQol (i.e. the squeezed spatial

frequency) and ac = 41QoI (a nonsqueezed Spatial frequency) as a

function of k (i. e. the distance between the photodiodes ). As we can see, the spatial frequency corresponding to the squeezed component of the field shows up a positive cross correlation, while the nonsqueezed spatial frequency shows up a negative cross correlation.

OUT

Oiodes-orray

Nlmedlum

~ ~ \ /

\ I

Electronic cross~orrelator

Figure 5 Detection scheme for the spatial correlation properties

Conclusions

245

We have studied the nonclassical effects in the transvers space domain, when the spatial propagation of a pump field through a medium governed by a quadratic or a third order interaction, can be still described by a spatial nonlinear Schroedinger equation. In the case of a quadratic material this occurs when the cascading effect is taken into accout ; in other words when the conversion efficiency between the pump and the generated field is very low. We have found quantum effects associated to the transverse spatial distribution during propagation, such as squeezing and bunching. In both cases the spatial squeezing magnitude is pronounced at that spatial frequency corresponding to the frequency at wich a modulational instability occurs in the classical propagation of the same field distribution . Also bunching is found . In the case of a quadratic nonlinear material ,both these quantum effects depend on the phase mismatch occurring during the generation process.

1

• -1

-2

-s

-, -10

J "\

a=4lQol

\ a=.J2IQol

f

10 20

k - axis

Figure 6 Spatial correlation properties for two different frequencies

246

References

Lugiato L., Grynberg 0.-1995, Europhys. Lett.29, 675 - 677 Sibilia C., Schiavone V.,Bertolotti M., Horak R., Perina J - 1994, JOSA B 11, 1364 - 1370 De Angelis M., Sibilia C., Bertolotti M., Horak R., submitted for public. Horak R., Bertolotti M., Sibilia C., 1994 J.ofMod. Optics 41,1615 -1621 Newell A.C, Moloney J.V. ,1992" Nonlinear Optics" Redwod City ,

C A Addison Wesley Buryak A.V. ,Kivshar Y.S. , 1995 ,Phys. Lett A 197,497499 Ou Y. ,Hong O.C., Mandel L. - 1987, Phys.Rev.A 36,192 -199

Potasek M.J, Yurke B. - 1988, Phys.Rev.A 38 , 1335 1342

PHOTON CORRELATION OF CORRELATED PHOTONS.

Experimental aspects of quantum cryptogrophy and computation

J.G.RARITY AND P.R.TAPSTER

Defence Research Agency St Andrews Rd, Malvern, Worcestershire WR14 3PS UK

Abstract. We introduce the concept of encoding information on single photons using interference. and describe progress towards use of this in secure key sharing (quantum cryptography). Both one- and two-photon interference based quantum cryptography schemes are described. System performance is limited by the efficiency of single photon tounting detec­tors at communications wavelengths. Finally we measure the intensity fluc­tuations of pulsed spontaneously emitted light and show that narrow band filters increase the measured second moment in line with thermal light.

1. Introduction

The bit is clearly the fundamental unit of information in computing and communication. In conventional communication systems these bits are rep­resented by macroscopic pulses of electrical current and, more recently, light. One advantage of light over electrons is the ease with which its wave­like properties can be exploited. However light can be ascribed both a wave frequency v and an indivisible quantum (or photon) energy hv (h= Planck's constant) and the wavelike properties show up as interference effects even at the single quantum level:Here we introduce experiments which demon­strate this wave-particle duality. These experiments have led us to develop secure key sharing schemes known generally as quantum cryptography [1]. Single bits are associated with the detection of single quanta while the in­formation is encoded using interference effects. Such systems should have security guaranteed by the laws of physics rather than by computational complexity.

247

E. R. Pike and J. B. Abbiss (efis.), Light Scattering and Photon Correlation Spectroscopy, 247-262 © 1997 Crown Copyright.

248

We begin this paper (section 2) with a review of single photon interfer­ence effects and simple ways of encoding information into a single quanta. We go on (section 3) to review two approaches to secure key sharing: one using weak coherent pulses and the other exploiting non-local pair-photon interference. In section 4 we describe work on components for cryptog­raphy systems including single photon counting detectors for communi­cations wavelengths and development of single photon sources using corre­lated photon pairs creat~d in parametric downconversion. Finally to link us to the main theme of this volume, that of photon correlation spectroscopy, we discuss recent experimental measurements of the photon statistics and photocount autocorrelation function (g(2)(r)) of our pulsed pair photon light source (section 5). We find photon bunching as expected from a chaotic light source when viewed through a narrow band filter.

2. One-photon interference and encoding

The first evidence of one photon interference came in the experiments of G I Taylor[2] in 1909. Taylor used extremely faint light sources in a standard Young's slit interference apparatus (see figure 1). In his experiments the source emitted much less energy than a single quantum (hll) in the time it took for light to pass from source to screen. Thus the probability of more than one 'photon' being between source and screen at anyone time was extremely low, yet after integrating long enough, interference fringes still appeared at the detector (a sensitive photographic plate). This was the first experimental evidence supporting Dirac's famous statement 'Each photon then interferes only with itself'[3].

A Mach-Zehnder interferometer can be used to sort single input photons into two channels as shown schematically in figure lb. Each arm of the in­terferometer is equal in length apart from a small phase shift ,po In the quantum description of the experiment one associates a probability am­plitude 1/.../2 for passage of the photon via a particular arm (a or b) of the interferometer and add amplitudes at the detectors including the phase shift. This leads to a probability of detection in the detector (DO) given by

1 PDO = 2(1 + cos,p) (1)

for each single photon input into the interferometer. A corresponding equation with the plus sign replaced by a minus holds for the other detector (D1) so that the total probability of detection remains unity. In the real world the detection quantum efficiency is always less than unity and conventionally written as 1] in this paper. Choosing a phase ,p = 0 sends all pulses to DO while ,p = 11' sends all pulses to D1. A simple encoding scheme involves switching 4> between zero and 11' and interpreting a detection in DO as a zero

.. )

b.J

c.}

-===t:::::=:=-il !NiENSIIr

~ Week 5eur~e

95 ;l ~~--~\~~---HO--~\~

Week ?c.;ise:

on

:5

I \J

J'

?oier'se: U Sc~:'"::e pes

DC

249

Figure 1. a.) Young's slit experiment carried out by Taylor using extremely weak light sources; b.) Mach-Zehnder interferometer for encoding information on faint light pulses. Key: BS bea.msplitter, q, phase shifter, DO,Dl photon counting detectors; c.) Polarisa­tion as an interferometric encoding scheme. Key: ~/2 half-wave plate, PBS polarising beamsplitter.

and D 1 as a 1. It is important to note that when the phase lies between these two extremes, at 7r /2, no information is encoded because a photon entering the interferometer will have a 50% chance of turning up in either output (PDO = PDt = 1/2).

Information can also be coded on a single photon simply by using two orthogonal polarisations such as vertical and horizontal and a polarising beamsplitter to separate '1' s and 'O's (see figure lc). When the plane of polarisation is 0° or 90°, the input photon exits into a definite detector but for +/-45° polarisation there is again a 50% chance of finding the photon in either output channel.

250

3. Secure Key sharing using one-photon interference

The above has identified two interferometric coding methods which could be used to send information at a one photon per bit level (if the interferometer were loss free and stable). In most communications applications losses are large and most photons (ie bits) would not arrive. This is not a problem if a.ll we want to do is establish identical random numbers at two remote locations because we can simply send a random series of bits then communicate back from the remote location the times at which photons have arrived. Only those bits that arrive are then incorporated into the 'key'. Here one sees the first level of security of such a system. Photons are indivisible objects thus if an eavesdropper picks off a sma.ll percentage of photons and measures them they will not reach the receiver and not be included in the key. However a subtle eaves-dropper can measure the photons then create copies to reinject into the communication channel thus breaching this security.

To guarantee absolute security the phase is split into two parts <Pa at the sender and <Pb at the receiver with <P = <Pa - <Pb. The probability of a weak pulse leading to a detection in the '1' or '0' outputs is then given by

(2)

which now includes O!/ < 1 the lumped transmission losses, 1/ the detection efficiency and m < 1 the mean number of photons per pulse. Note that now <Pa - <Pb = 0, (1\") can be used to direct gated pulses to the '0',('1') channel. Absolute security is guaranteed by randomly changing the receiver phase between <Pb = 0 and <Pb = 1\"/2 which corresponds to switching between two 'non-orthogonal measurement bases'. The transmitter phase is randomly selected from <Pa = 0,1\" and <Pa = 1\"/2, 31\" /2 coding 'O's and '1 'so As seen above, 100% correlation (error free detection) only occurs when the phase difference is 0 or 1\" thus sender and receiver must communicate and discard a.ll received pulses where the phase difference was 1\"/2 or 31\"/2. After a.ll uncorrelated bits are discarded the transmitter and receiver are left with near identical random bit strings to be used as a key. An eavesdropper must now guess which receiver phase was used <Pb = 0 or 1\"/2. He will choose wrongly 50% of the time and 25% of reinjected photons will turn up at the wrong output because the sum phase is 1\"/2 or 31\"/2. The error rate is estimated by openly comparing a fraction of the key bits which are then discarded. If a large number of errors are detected the sender and receiver must assume that their key security has been compromised and restart the key exchange.

PULSED LIGHT SOURCE

IITTENUATOR (TO 0.1 PHOTONS/PULSE)

251

RECEIVtR

'1' chonnel

Figure 2. Faint pulse key sharing system using time delay interferometers and a weak pulsed light source. High visibility interference between short-long and long-short events can be selected by gating detection using a reference pulse from the source. Active mod­ulators are used to set up the protocol described in the text.

4. Demonstration experiments

4.1. WEAK COHERENT SOURCES

A large Mach-Zehnder interferometer would. not be stable and the phase would drift rapidly. This problem is avoided by using two time division Mach-Zehnder interferometers connected in series as shown in figure 2 [4]. A light source emitting short pulses of light is attenuated until the average photon number per pulse is about 0.1. This ensures a low probability for de­tecting more than one photon in the system at anyone time. The pulses are split in time in the first interferometer which has a path length difference much long~r than the pulse width and the two pulses so formed propagate to the receiving interferometer. When the leading pulse takes the longer path and the trailing pulse takes the shorter path the pulses arrive simulta­neously at the final beamsplitter and interference is seen. Events where the pulse takes the long (or short) path in both interferometers will not show interference but can be gated out as they will arrive later (or earlier) when compared to a timing reference originating in the source. This is then equi­valent to a long (lOkm) Mach-Zehnder interferometer but the pulses pass through the same environment. Phase drift arises only in the time division sections which can be thermally isolated or actively controlled.

In the experimental test of this time-division interferometry interference

252

ALICE

Figure 3. Pair photon interference apparatus configured for quantum cryptographic key sharing. See text for description

visibilities greater than 0.95 corresponding to error rates below 3% with de­tection rates of a few kilo-counts per second (Kcps) at ranges of 10km were measured[4, 5]. These experiments were carried out using standard telecom­munications fibres and interferometers operating at 1.3ILm with Germanium APD's operated in Geiger mode[6] as detectors. Later work at British Tele­com has demonstrated that key sharing at a few hundred bits per second can be performed on a similar system[7] and in multi-user networks[8]. Other groups have made similar demonstrations using polarisation as the coding variable rather than interferometric phase[9, 10, 11, 12].

4.2. A PAIR-PHOTON KEY SHARING SCHEME

We have also shown that non-local quantum correlations of pair-photons could be used in a secure key sharing scheme[13]. A simplified apparatus is shown in figure 3.

A non-linear crystal is pumped by a short wavelength laser to pro­duce downconverted photon pairs satisfying energy and momentum conser­vation in the crystal. Apertures AA and AB select pair beams and lenses L couple these beams into single mode optical fibres. The receivers Alice and Bob each view the photons at the outputs '0' and '1' of identical out­of-balance Mach-Zehnder interferometers using photon counting detectors. Energy conservation in the process of downconversion ensures a time un­certainty in the creation of the photon pair making pairs passing down

253

the long arms of the interferometers indistinguishable from pairs taking the short routes. Interference in the rate of detection of pair (coincidence) events thus occurs but no interference in the singles rate is seen. The pair detection probability is given by

(3)

where i and j are detector indices 0 or land ifJa.,b are small phase shifts in the otherwise identic3.I interferometers. This shows 100% correlation of output whenever ifJa. + ifJb = ml'. Alice and Bob can establish an identical key simply by noting the bit value of their detected photon whenever they detect in coincidence. Security can be guaranteed by randomly switching the phase in Alice's interferometer between 0 and 11'/2 and that in Bob's between 0 and -11'/2. On comparing phases after the measurement Alice and Bob can validate bits only when the sum phase is zero and discard all other bits. Eavesdroppers are again detected by testing for errors on a subset of key bits.

We recently tested the feasibility of such a system out to a range of 4km[14]. The coincidence probabilities P(i,j)a.b (equation 3 above) were measured as a function of phase. The sinusoidal variation due to interference effects were seen but with visibility limited to 82% (due to detector dark noise primarily). This is not high enough for key sharing but it does confirm that there is no local classical phase associated with individual pair photon emissions in agreement with the predictions of quantum mechanics [15] .

5. Components for quantum cryptography systems

Data rates in a typical faint pulse system with a range L km can be es­timated from equation 2 multiplied by the repetition rate of the pulsed source (typically r=100MHz)

R = '71O-0.1aLmr /2 (4)

The primary obstacle to demonstration of these systems over any distance is transmission loss Q dB /km, the efficiency of detection '7 and the mean number of photons per pulse m. Efficient photon counting detectors are needed with sufficient time resolution to allow short gate times t <: 1/r where 1/r is the time between pulses sent. A short gate time can also be used to suppress dark count rates using gated detection.

5.1. TRANSMISSION LOSSES AND DETECTORS

Near infra-red (600-900nm wavelength) photon counting using solid state silicon avalanche diodes has been studied for some time[16, 17]. The devices

254

are biased beyond their normal breakdown voltage and break down when­ever a photo-event excites an electron into the conduction band. Passive quenching of the breakdown occurs when a large resistor is placed in se­ries with the device. The breakdown current produces a voltage across the resistor which drops the bias below that required for breakdown. Recent ex­periments with these devices have shown they can achieve sub-nanosecond time resolution[14] and quantum efficiency greater than 70%[18]. Fibre losses at these wavelengths are high at 2dB /Km thus fibre systems based on this technology will be limited in range to a few kilometers. This is however the preferred detector for free space applications and laboratory demonstrations.

Fibre losses drop to 0.3dB/km at 1.3JLm and to below 0.2dB/Km at 1.55JLm which would allow much longer propagation paths. Alas, single photon counting detector technology at these wavelengths is not as well developed. We have studied the performance of various avalanche diodes biased beyond breakdown to operate in photon counting mode at 1.3JLm[6]. We find that dark counting rates dominate at room temperature and cool the devices to 77K. To date, only selected Germanium avalanche diodes have shown significant quantum efficiency at bias levels where the dark count is low. Typical performance for a commercially available device (NEC 5103P) operated at 0.25 volts beyond breakdown (25.32V) is 8% quantum efficiency and timing jitter 550ps. The dark count at this operating voltage is less than 12Kcps while afterpulsing probability (due to carriers trapped during the avalanche) is 10% after 1JLs. This last statistic forms the primary limitation of these devices as it implies that at least 10% of all counted pulses will be spurious. In a quantum cryptography system the pulses arrive at fixed times and gated detection with gate widths as low as 1ns can be used. Thus spurious counts due to afterpulses and dark counts can be suppressed by a factor equal to rt.

Recent work has investigated InGaAs avalanche detectors for use at 1.55JLm where fibre losses drop to 0.2dB/Km. Device performance has been measured at temperatures between 17K and room temperature and opti­mum operation temperature appears to be around 175K. As in the Ger­manium case a passive quenching arrangement with a series 33k resistor is used. Pulses from the detector were very noisy and performance varied significantly from one detector to another. Average pulse amplitudes after amplification varied from 0.05V at breakdown to 0.3V at 0~3V above break­down. Pulse widths were around 6ns FWHM but one expects pulse jitter to be much lower than this. The breakdown voltage varies from 40 to 70V, increasing approximately linearly with temperature.

Figures 4a,b show quantum efficiency, afterpulsing and dark counts as a function of voltage above breakdown for a selected NEC device (NDL5501P1)

255

II'

OUllllhUIl

n.s

1 ~ lIit ~IC:HlCy l'1..~

---~-----------(~L.u-~-==o.~, ~:""'--(..L,.~-:------;o-::.):-~--;;n': .. (--

VolI"!je "bovtl brP'"kc1nwn V

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<:: U ~o · / -

~ / -c. o

----------------------n · 1--,:::::::~-----.--~----'r--~--:-':~:-~-~O.5

00 01 02 OJ 0

11)-

Vol13ge above hreakllown (V)

M1T-----------------------------------~

15 -

IU·

GO

.ot-~--~~--~~--~~--~-.--~~~~_: 1.2 1.3 , 4 , . ~ 1.6 10

W3velenglh (microns)

Figure 4. Performance of the NEC5S01Pl InGaAs avalanche photodiode operating in photon counting mode at 17S°K. The device is biased beyond breakdown and quenched with a 33KO series resistor (see inset). a) Quantum efficiency and afterpulsing as a func­tion of bias voltagej b) Dark counts as a function of bias voltagej c) Quantum efficiency as a function of wa.velength (O.27V beyond breakdown).

256

operating at 175K and measured using a 1.3J.'m LED. Other unselected de­vices showed much poorer performance. Detailed characterisation methods are described in full in earlier work on Ge APD's [6).The efficiency at a variety of wavelengths was then measured using ct bright tungsten bulb and interference filters and is shown in figure 4c. The light-induced photocurrent when operating below breakdown (30V bias) allowed a rough calibration of the intensity of the filtered bulb based on the manufactures published analogue efficiency (device gain of unity). This was compared to the count­ing rate when the device was biased above breakdown operating in Geiger mode. The result indicates that if efficiency is 2% at 1.3J.'m it drops to 1.45% at 1.55J.'m.

Clearly the dark count is the limiting factor for these devices. The se­lected device showed the lowest dark count, allowing operation at 175K. At O.3V above breakdown, the dark count is below 20kHz, afterpulsing is 8% but the efficiency is only 1.45% at 1.55J.'m. These results may usefully be compared to those for germanium APDs. When operated at 17k and O.2V above breakdown, the dark count is 8kHz, afterpulsing is 7% (with a 1J.'s dead time) and the photon counting quantum efficiency is 7%. We can compare the performance ofthese devices in a cryptography application by comparing bit rates at 1.55J.'m with InGaAs detectors to those at 1.3J.'m using Germanium from

RlnGaAIl = T/InGaAIl lO-(a1.5s-a 1.3)L/IO

RGe 1JGe (5)

When one considers the efficiencies quoted above and the figures Cl1.3 = O.35dB/km and Cl1.55 = O.2dB/km it is clear that the longer wavelength system will support a higher bit rate at distances over 50km. However in a system operating at r=100MHz and m ~ 0.1 to guarantee security we expect a bit rate of order 7kHz and a dark rate in a 1ns gate of order 2000Hz. This leads to an error rate which is too high to be corrected by existing protocols [19].

From these results it is obvious that InGaAs device performance is still not able to improve on G~rmanium cooled to 17K. Further research into the design and manufacture of InGaAs detectors optimised for photon counting is essential for long range fibre (100km) applications.

5.2. IMPROVING SOURCES

Classical light shows Poisson statistics with standard deviation equal to the square root of the mean. This produces a very broad distribution when the mean number of photons per pulse m is close to unity. To ensure a low probability of detecting more than one photon per pulse (which could be

257

eavesdropped in principle) m must be much less than unity (of order 0.1). We would like to produce a source that emits photons one and only one at a time. A possible solution is to use a source that creates photons in pairs as in the process of parametric down conversion. Detection of one photon of a pair can be used to gate detection of the other. Demonstration experiments showing optical and electronic gating of single photons were performed some time ago in our laboratory [20] using bulk non-linear crystals. Although single photon states wer.e detected the counting rates were extremely low due to the poor detectors and other losses in the system. More recently we have been working on a pulsed source [21] where the single photon state is confined to within a 130fs time window by the duration of the pumping pulse (a doubled mode locked Ti-sapphire laser). In this experiment (figure 5) we again used a bulk crystal and then launched the pair photon beams into single mode fibres as would be required for a quantum cryptography source. In the experiment we count photons in fibre coupled detectors with efficiencies of order 1/=40%. However the singles rates in the detectors are only of order S=5 kilocounts per second while the gated single photon source rate is of order G=l kilo count per second.

The gated rate G can be expressed as

(6)

where r is the pulse repetition rate of the pump laser (100Mhz here), Pis the number of pair photons created in the single mode per pulse, and a is the loss due to mode matching in the fibres and other filter edge effects. Similarly the singles rates are S = a1/Pr. From this we see that a = 0.5 and that the gated rate is reduced by a factor of 4 by mode matching losses. At present there is a 20% probability that the single photon pulse contains a single photon and the probability of two photons in the pulse is vanishingly small. This is a slight improvement on an attenuated classical source but the overall emission rate of 1 kilo count per second is far too low.

Second order non-linearity has recently been demonstrated in poled op­tical fibres [22]. Creation Qf photon pairs within suitably poled optical fibre could increase the effective number of photons per gate up to 0.7 which is limited by the maximum detection efficiency of the gating channel, which would be designed to operated at short wavelengths.

Another sources of single photons has recently been suggested based on the fluorescence of a single dye molecule in a micro cavity [23]. Here the sin­gle photon nature is guaranteed by the fact that a single atom/molecule can only emit one photon per excitation. A single excitation per pulse is guaran­teed if the atom is excited by a pulse shorter than the times cales associated with excitation and spontaneous emission. The spontaneous emission into

258

IDoubler [ __ .• m •••

i I I

IMLi ! Loser L

Time Interval Analyser

Figure 5. Pulsed pair photon source and statistics measurement. A doubled mode-locked Ti-Sapphire laser (407.5nm) illuminates a BBO crystal where parametric downconversion takes place. Pair photon beams are emitted in to the gating (g) and the measurement (c and d) modes. To measure the fluctuation statistics of the ungated beam a single mode is selected from the downcanverted light by coupling it into a single mode optical fibre with a microscope objective (L). The statistics are estimated by measuring g(2) across a fibre beamsplitter using a time interval analyser set up as a sparse correlator. Detectors are silicon avalanche photodiodes operating in Geiger mode.

a single mode can be enhanced in the presence of a high finesse cavity and in principle efficiently launched into single mode waveguides.

6. Measuring the Photon Statistics of Pulsed Parametric Fluo­rescence

One of the first things we learn in the area of photon statistics and photon counting is that thermal sources tend to be chaotic and show Gaussian field statistics. Introductory analyses take the example of an assembly of excited atoms with linewidth dominated by collision broadening[24]. In this semi­classical analysis, each atom emits a single frequency with random phase and phases change randomly at each collision. The sum of a.ll fields emitted has a random amplitude which is Gaussian distributed and an intensity distribution with negative exponential statistics. The second order inten­sity correlation function g(2) rises at short times to an intercept of 2 (the normalised second moment of the negative exponential distribution). Of course this is what we see every time we make a standard photon corre­lation spectroscopy measurement on a colloidal sample in the laboratory.

However the first evidence of these light fluctuations were obtained in

259

the seminal experiments of Ha.nbury Brown and Twiss [25] who detected the intensity fluctuations in the light from a mercury arc lamp. Such fluc­tuations occur on picosecond timescales well below the resolution of photo­multipliers of the time. Integrated over the detection time windows of a few nanoseconds the measured effects were extremely small (within 10-3

of unity). To avoid distortion of the measurements by shot and amplifier noise in their measured photocurrents the experiment was set up with two photomultipliers viewing the source via a 50/50 beamsplitter. The duration and difficulty of the original Ha.nbury Brown a.nd Twiss experiment has dis­couraged later researchers from making intensity statistics measurements on sources with bandwidths greater than that of the detectors. In fact ma.ny people use the filtered tungsten bulb as a source of consta.nt intensity light suitable for testing photon correlation equipment for spurious fluctuation effects such as detector after-pulsing. Beyond T =50ns g(2)( T) == 1 to a very good approximation due to temporal averaging of the Gaussian field fluctuations which occur on femtosecond timescales.

As mentioned above we are developing a pulsed gated single photon source using a parametric down conversion crystal pumped by a mode locked laser with pulse length around 130fs (see figure 5). This pulse length corre­sponds to a natural bandwidth around 10nm typically that of off-the-shelf interference filters used to select the down converted beams. We are thus interested in the intensity statistics of this source both gated and ungated. For narrow filter widths we expect pulse to pulse fluctuations reflecting the Gaussian field statistics. With wider widths there will be several fluc­tuations within the duration of each pulse a.nd pulse to pulse fluctuations will be suppressed by time averaging.

In the experiment various narrow band interference filters were added to limit the bandwidth of the downconverted light. Figure 6 shows the histogram of time differences between photon detections in the two de­tectors in the case of a narrow filter width (2.2nm). The detector time resolution (less tha.n 0.5ns) is able to resolve the IOns pulse period but not the sub-picosecond pulse width. Since the correlation time induced by the narrowest filter (about 0.5ps) is much smaller than the pulse period, there are no correlations between one pulse a.nd a.ny other pulse. The peak at 145ns corresponds to zero delay at the detectors. Because of the bunching there is a.n enhanced probability of seeing a detection in both detectors in the zero delay peak. The ratio between the zero delay peak and the non­zero peaks thus provides a.n estimate of the second moment g(2)(0) of the integrated pulse intensity. A plot of g(2)(0) as a function of filter width is shown in figure 7. The theoretical curve is calculated assuming a Gaussia.n filter shape a.nd a.n unfiltered (Gaussia.n) pulse length of 130fs [26].

260

600

.00

200

1J 1 0 ... ~ " . . -

so 11K! ISO 2tJO 250 llul<iy (ns~

Figure 6. Typical cross correlation measured using the time interval analyser. Due to the pulsed nature of the source we see a comb like correlation function. The zero time peak arbitrarily shifted to 145ns delay rises above the other peaks due to the pulse to pulse fluctuations. Correlations between pulses are assumed negligible and we interpret the average background as representing gC2)(T) == 1 thus allowing us to normalise the g(2)(0)

z.O ,.....-,_=-_-. --

1.0 -~-.-•• - ..... '-' .• - ... --'" ......... -'-'-'--'.'-~--' I m ~

FWHMlnm)

Figure 7. g(2)(O) as a function of filter width for pulsed spontaneous emission.

7. Summary

We have introduced the ~oncept of encoding information on a single photon using quantum interference phenomena. The first application of this has been quantum cryptography, a method for sharing random numbers at two remote locations with absolute security. The eventual implementation of quantum cryptography systems (from office to office for example) is depen-

261

dent primarily on the availability of a transparent optical route between the two parties. Some development is also required to improve photon counting detector efficiencies at the longer communication wavelengths (1.55J.Lm). Detector packaging is also problematical as the present generation of Ge based photon counting detectors can only operate a liquid nitrogen tem­peratures.

The pair photon based quantum cryptography system raises a philo­sophical question. Does· the key exist before measurement? Interestingly enough, quantum mechanics tells us that the answer to the question is no. With a system based on pair photons (or paired particles in general) we could in principle send out keys and store them securely in quantum form. Unfortunately the longest storage time of a photon based system would be a some tens of microseconds in a fibre delay loop.

In the final section of the paper we have described a measurement of light bunching from a spontaneous emission source. Normally these inten­sity fluctuations would be too rapid to see using the limited time resolution of existing fast correlators and detectors. However here the spontaneous emission is confined to short pulses. When we filter the light to have a coherence time longer than the initial pulse duration the pulse to pulse intensity fluctuations reflect the Gaussian field statistics of the light.

References

1. For a recent review of quantum cryptography see: J. Modern Opt. 41 December 1994. 2. Taylor G.!. (1909), Proc. Cambridge Philos. Soc. 15, 114 (1909). 3. Dirac P.A.M. (1930), The Principles of Quantum Mechanic." Oxford University

Press, Oxford, p9. 4. Townsend P.D. Rarity J.G. and Tapster P.R. (1993), Electron. Lett. 29, 634. 5. Townsend P.O. Rarity J.G. and Tapster P.R. (1993), Electron. Lett. 29, 1291-1292. 6. Owens P.C.M. Rarity J.G. Tapster P.R. Knight D. and Townsend P.O. (1994),

Applied Optics 33, 6895-690l. 7. Townsend P.D. (1994), Electron. Lett. 30, 809-811. 8. Townsend P.D. (1997), Nature 385, 47-49. 9. C H Bennett, F Bessette, G Brassard, L Salvail and J Smolin (1992), J Cryptology,

5,3. . 10. Muller A., Breguet J. and Gisin N. (1993), Europhy.,. Lett. 23, 383-388. 11. Franson J.D. and Jacobs B.C. (1995), Electron. Lett. 30, 809-811. 12. Muller A., Zbinden H. and Gisin N. (1996), Europhy.,. Lett. 33, 335. 13. Ekert A.K. Rarity J.G. Tapster P.R. and Palma M. (1992) Phys. Rev. Letts. 69,

1293. 14. Tapster P.R. Rarity J.G. and Owens P.C.M. (1994), Phy.,. Rev. Letts. 73, 1923. 15. Bell J.S. (1964), Physics, 1, 195. 16. Brown R.G.W. Rarity J.G. and Ridley K.D. (1986), Applied Optic., 25, 4122. 17. Brown R.G.W. Jones R. Rarity J.G. and Ridley K.D. (1987), Applied Optic., 26,

2383. 18. Kwiat P.G. Steinberg A.M and Chiao R.Y. (1993), Phy.,. Rev. A. 48, R867. 19. Bennett C.H. Brassard G. and Robert J-M. (1988), SIAM Journal on Computing,

17,210.

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20. Rarity J.G. Tapster P.R and Jakeman E. (1987) Opt. Commun. 62 201. 21. Rarity J.G. Tapster P.R. and Loudon R. (1997) Los Alamos Preprint Server quant-

ph/9702032. . 22. Kazansky P.G. Dong L. and P St J Russell (1994), Opt. Lett. 19, 701. 23. Marrocco M. and De Martini F. (1994) in Quantum Interferometry eds De Martini

F. et al, World Scientific. 24. Loudon R. The Quantum Theory of Light, Oxford University Press (1973) ch 5. 25. Hanbury Brown R. and Twiss R.Q. (1956) Nature 177 27 26. Tapster P.R. Rarity J.G. in preparation.

NEW OPTO-ELECTRONIC TECHNOWGIES FOR PHOTON CORRELATION EXPERIMENTS

1. Abstract

RG. W. BROWN c/o Department 0/ Electrical and Electronic Engineering University o/Nottingham Nottingham NG7 2RD U.K.

New technology has always been at the centre of advances in photon correlation techniques and their applications. In this paper we look briefly at the technologies that have been used in the past two decades, but focus mainly on a wide variety of developments in opto-electronic device research that will provide new technologies for photon correlation experiments in the next few decades.

2. Introduction: a review of past technologies

This is an unusual paper as it attempts some 'cIystal ball' gazing on how various opto-electronic devices currently being researched and developed might have impact on the field of photon correlation experimentation in years to come. By 'opto­electronic' we mean devices essentially electronic in nature but involving light, ie, laser diodes and solid-state lasers, optical fibres, phase and frequency shifting elements, and semiconductor and small photo-multiplier detectors. Additionally, we shall look at new developments in electronic chip capabilities, integrated opto­electronic structures and some of the more futuristic ideas that might come to fruition.

We begin by reviewing the technologies that provided the original development of photon correlation techniques and their subsequent miniaturisation.

From about 1970 to 1985 the standard technologies for photon correlation experiments were light sources such as Argon or Krypton ion-lasers or Helium-Neon lasers (fairly bulky and inefficient gas lasers typically the size of a human arm or leg), bulky optical components such as beam-splitters and lenses of few-many centimetre dimensions (together with their associated bulky mechanical mounts) and

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E. R. Pilce and J. B. Abbiss (eds.), Light Scattering and Photon Correlation Spectroscopy, 263-276. @ 1 m Kluwer Academic Publishers.

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photon-counting photo-multiplier detectors (also the size of a human fore-arm and only few-percent efficient) [1]. In photon correlation laser veloclmetry, frequency- or phase-shifting was accomplished through Pockel's or Bragg cell modulation of the laser light [2]. The photon correlators were built firstly from TTL circuitIy and later CMOS and ECL electronic chips [3] to increase their speed, ie, to reduce their sample times. From about the mid-I980s, plug-in electronic correlator cards for desk-top computers and computer software correlators were developed to reduce the size of the processing electronics to something more portable [4].

Stringent requirements for the performance of each and every aspect of the experimental equipment were recognised early in the development of the subject, minimal oscillations in the laser source, near-perfection in the quality of the laser beam(s), minjmal after-pulsing in the photon detector output and careful control of the pulses in the correlation process so as not to miss one or double-count (de­randomisation) [5]. In photon correlation, 'clipping' of the signal was initially a major issue because of the computational speed of the electronics [6]. Nowadays the electronics is so fast that full correlation is usually performed.

A significant shift in the·direction of technology development for photon correlation experiments became possible from the mid-I980s. A series of investigations showed that in certain experiments it was feasJ.ble to replace the traditional gas laser sources with semiconductor laser diodes [7] (later in the I990s also by small solid-state lasers), to replace bulk optical components with single-mode optical fibres and micro-optics [8], and to replace large photo-multiplier tubes by avalanche photo­diodes and miniature photo-multipliers [9]. These new technologies showed potential advantages in overall system performance such as size, weight, efficiency and signal-to-noise ratio. Various palm-of-the-hand sized, miniaturised photon correlation spectroscopy [10] and velocimetry [11] experiments were built and tested to demonstrate these advantages, fully described elsewhere [12]. These technologies were taken-up, particularly for space-borne experiments [13] and rugged velocimeters [14].

We move now to review and speculate on the technologies that will generate new, third and later generation photon correlation experiments in the 2IIt century.

3. New laser diodes and solid state lasers

Semiconductor laser diodes [15] used in photon correlation experiments to date operate typically at 780nm to 850nm wavelength and with output powers up to >50mW. The principles of their operation are well known, electron-hole recombination in the active region of a p-njunction in a double-heterostructure geometry made from GaAsI AlGaAs. This style of laser has been well developed for optical data storage applications. Carefully stabilised operation of this type of laser was shown to create a light source adequate for photon correlation measurements.

Factorial moments and excess-correlation properties can be excellent, closely corresponding to the ideal coherent source with Poisson statistical output [7].

In the mid-1990s, new semiconductor laser diode wavelengths have become available from devices fabricated from AlGaInP/GaInP materials, driven by the needs ofDVD (digital versatile disc) optical data stomge. Lasers opemting at 650nm red wavelength and with 5m W output power are already available and power levels in excess of 20 - 40 m W are expected to be commercially available before the year 2000. Lasers ofless power at 635nm are also under development. The optical properties of these new laser beams have to be close to diffraction-limited single mode to satisfy the stringent needs of the optical disc application. This makes them potentially ideal for photon correlation experiment applications.

Another area of research at present is the explomtion of blue and green wavelength laser opemtion using two different materials systems, ll-VI materials ZnMgSSelZnCdSe [16] and ill-V materials GaNIInGaN/AlGaN [17].

The leading research team at the present time in the ll-VI activity is probably Sony Corpomtion who have demonstmted CW and pulsed laser diode opemtion with device lifetimes exceeding 100 hours in the research labomtory. Many other University and industrial research teams are also active in this pursuit. At present the principal problem in the realisation of a commercial technology for this type of laser is the elimination of dislocations and their propagation into the active region of the laser thereby destroying the laser opemtion. If such very serious problems can be overcome in new growth and processing procedures then a new class of laser diode may emerge with output wavelengths in approximately the region 440nm - 470nm.

The ill-V activity is also showing great promise for creation of UV, blue and green laser diodes. The leading research team at the present time is probably Nichia Chemical Industries Ltd in Japan, led by Dr Shuji Nakamum. Nichia have already commercialised intensely bright, > 1 candela output light emitting diodes at blue and green wavelengths. Furthermore they have demonstmted pulsed laser diode opemtion at 390nm - 440nm wavelengths (typically 420nm) at 50% duty-cycle with an avemge output power of 57m W. An InGaN multiple-quantuID-well geometry is used. Less than lOOmA threshold current and less than 10V opemtion, together with lifetimes exceeding 24 hours have been achieved in the research labomtory. However, the ill-V approach is not without it problems in development to a commercial technology. As with the ll-VI activity, threading dislocation density will have to be reduced dmmatica11y through new growth and fabrication procedures, also a suitable substrate material (eg, bulk GaN) for laser opemtion must be secured.

The technological capabilities of both the ll-VI and ill-V approaches is developing at such a pace for the optical data stomge application that we confidently expect to see one of these technologies emerge with a blue laser suitable for photon correlation

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experiments before the year 2000. Ultra-violet opemtion may follow from the ill-V material system (GaN) because of the very wide band-gaps that might become possible. Such short wavelengths would be useful in photon correlation spectroscopy (dynamic light scattering) to give a wider range of scattering vectors, 'q' values, than are currently obtainable by other laser sources [3]. Wavelengths as short as 350nm might become possible in years to come.

Another style of laser diode development is active in the research labomtory today -vertical cavity surface emitting laser (VCSEL) diode arrays [19]. The basic principles of this subject are now well understood and many different laser armys have been fabricated, mostly at telecommunication wavelengths around l~m -1.5~m. Arrays of laser greater than 8x8 in size have been fabricated. In the next few years we can confidently expect larger armys to emerge and at shorter wavelengths such as 780nm (AlGaAs) and 650nm (AlGaInP) [20] driven by potential application to fast laser printers. Perhaps ZnSe and GaN system VCSELs will emerge with blue and shorter wavelengths.

The value of VCSEL technology to photon correlation experiments may arise from the laser's perfect circular mode output which allows efficient coupling to optical fibre arrays. Multiple measurement volume experiments in dynamic light scattering and laser velocimetry may emerge, spatial and/or temporal cross-correlations between measurement volume fluctuations becoming easier.

More esoteric and further-from-realisation research in the laser diode field is also under study, eg, photonic band-gap [21] and quantum wire/dot [22] laser studies. These are discussed later in section 9, beyond the year 2000. We should also note the first tests recently of an organic electro-luminescent laser [23]; perhaps plastic lasers will have an impact at some point in the future?

4. New modulators and frequency shifters

In the field of photon correlation laser Doppler velocimetry (LDV) [24], it has been traditional to employ either Pockel (Phase) or Bmgg cell (frequency) electro- or acousto-optic techniques (respectively) to remove the problem of Doppler ambiguity inherent in highly turbulent flow directional fluctuations. With the introduction of new technologies, often involving miniaturised devices, it may be necessary to find alternative means of phase or frequency shifting. A number of current research activities suggest themselves but their effective realisation in this new application is by no means certain yet.

On initial inspection the field of asymmetric Fabry-Perot modulation (AFPM) [25] explored as a potential modulation technique for telecommunications applications may seem appropriate for exploitation as a new LDV modulation scheme. However, AFPM is based on modulation of light intensity through change of the position of

the absorption edge in a semiconductor Fabry-Perot cavity, thus it performs the incorrect functionality for LDV.

An early form of frequency shifting in LDV involved rotating diffraction gratings, and it is interesting to note recent new developments in rotating micro-motor gratings of only 500IJ.m diameter by Mehregany and colleagues at the Case Western Reserve University [26]. This technology is at least on the correct scale for employment in miniaturised LOV optical systems, but grating rotation speeds and frequency-shift magnitudes are yet to be determined.

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An alternate means of mechanical frequency/phase shifting is to oscillate a mirror placed to reflect a laser beam [27]. With the recent advent of creation of micro­mirrors of IOlJ.m-50lJ.m dimensions fabricated in large arrays in silicon technology for projection-TV display and optical sensor applications, again we have the potential for a technology transfer into LOY. Vibration frequencies in the MHz range have already been demonstrated, consistent with the normal needs ofLOV, and in arrays who size is compatible with miniaturised LOV instruments. We expect therefore to see developments of this technology and its application in photon correlation experiments in the near future.

Another recent opto-electronic development has been that of quasi-phase-matched (QPM) frequency doubling [28] for the creation of say blue laser light from red laser diodes (eg, 860nm wavelength converted to 430nm). A typical low-voltage material to achieve this is LiTa03. Recently a patent has been filed showing how such QPM geometries can be developed through the use of sigilals applied to electrodes placed above and below the alternating refractive index grating (domain-inverted) characteristic of such operation [29].

Finally, with the increased use of optical fibres in LOV and dynamic light scattering it would be useful to find a technique suitable for frequency or phase shifting of laser light within the fibre. Users of fibres know from experience the immense disadvantages of having to extract light from an optical fibre to change it in some way and then to have to re-Iaunch the light back into a fibre. The penalties in mechanics needed and light-loss are severe. In recent years various optical fibre frequency shifting approaches have been proposed and demonstrated. Typical of these is the work of Pannell et al [30] who have demonstrated the application of flexure-waves to create multi-MHz frequency shifts. Unfortunately, to date, such methods have only conv6rted up to about 20% of the light within the fibre, a figure that must be increased closer to 100% for practical applications.

5. New optical fibre geometries

Single-mode optical fibres, in particular, have found widespread use in recent years in new optical architectures for both dynamic light scattering [8] and LO V [31]

268

applications. The special advantages of increased signal-to-noise ratio and maximum photon correlation function intercept (together with improved accuracy in estimates derived through correlation function inversion) are being used in new optical instruments for both research and industrial applications.

We will not review the various published schemes here, but point only to another advantage of single-mode optical fibre geometries not yet fully researched for potential experimental applications.

In the original paper and patent about single-mode optical fibre dynamic light scattering [8], Brown noted that possible experimental advantage could be gained through the use of optical fibre beam-splitters (couplers) to create multiple measurement volumes both in dynamic light scattering and LDV. In particular, heterodynelhomodyne arrangements become remarkably easy to create, with virtually no alignment mechanics being required. in stark contrast to the situation where conventional optical elements are used. Simple geometries for the creation of single scattering-angle heterodyne, multiple scattering-angle heterodyne and multi­sample multiple-scattering-angle heterodyne measurements were proposed. see Figure I below. Heterodyning can be particularly advantageous in dynamic light scattering.

LASER BEAM WAIST INSIOE SUSPE:NSION

LAUNCH / . LENS / 1

~O ",... LASER - 1",\' T BEAM (c)

@ • ,,) ) DETECTOR COUPLER

Figure I. Heterodyne and multi-sample monomode fibre schemes for dynamic light scattering. (a) single angle heterodyne, (b) two-to-N angle heterodyne and (c) multi-sample, multi-angle heterodyne.

Optical fibres offer great potential for performing new light scattering experiments involving the cross-correlation of temporal intensity or number fluctuations in

different spatial locations or at different scattering vectors. It is hoped that this potential will come to be realised with this new technology in years to come.

Optical fibre frequency shifting was addressed in the previous section (4), optical fibre correlators will be addressed in a following section (7).

In passing, we note the recent proposals for use of single mode optical fibres for use in optical aperture synthesis astronomy [32], building upon the techniques of intensity fluctuation interferometry that provided the earliest roots of photon correlation techniques. Significant technological developments in this closely related research area can also be expected in future years, expanding on the use already in optical synthesis astronomy of photon counting avalanche photo-diodes following their introduction into the photon correlation field [33].

We move now to consider these and other technology developments in photon counting detectors, the essential heart of photon correlation techniques.

6. New photon counting detectors

The traditional use of photon counting photo-multipliers continues in much experimental apparatus currently used for photon correlation experiments. Whilst most of these devices are physically quite large and require kilo-volt supplies to activate their dynode chains, smaller 'finger' -sized photon counting photo­multiplier tubes from Hamamatsu Corporation in Japan [34] have also been found suitable for photon correlation experiments in recent years.

An alternative technology to the photo-multiplier has also been successfully introduced to photon correlation experiments in the last decade, albeit requiring very careful electronic and thermal control for successful operation. Semiconductor avalanche photo-diodes (APDs) have been developed for a number of decades for telecommunications purposes. Brown et al showed [9] their potential value to photon correlation experiments in a series of experiments in the late 1980s. Commercial supply (by RCA, now EG&G Optoelectronics [35]) of photon correlation APD assemblies soon followed.

The interest in these new detectors arose because they are so small (typically millimetre dimensions when packaged), they can have large quantum efficiencies, operationally exceeding 20% at certain wavelengths, and with appropriate Peltier cooling can achieve dark-noise-count levels suitable for photon correlation applications, a few hundred per second. Further more, with carefully adjusted electronics drive circuitry their after-pulsing properties can also be suitable low (typically better that 0.04%) for creation of negligibly -distorted photon correlation functions. Despite these advantages it must be understood just how careful the experimentalist must be to create carefully the exacting thennal and electronic

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stability criteria necessary to achieve suitable quality of operation for photon correlation use of APDs. Thermal stabilisation to better than 1120 degree Centigrade, and voltage bias stable to a few milli-volts is essential. Under active­quenching operation, APD photon counting rates can exceed a few MHz.

In recent years an interesting new development in APD technology has occurred which will be of great value in photon correlation experiments, particularly quantum optics experiments exploring fragile non-classical light. The super-lattice structure APD has been introduced by Capasso and his colleagues at Bell Laboratories [36]. In this new architecture, in place ofa fairly simple p-njunction, a separate absorption, grading and multiplication region (SAGM) is introduced into the junction multiplication area, in the form of a chirped super-lattice. This has the effect of enhancing the electron ionisation rate relative to the holes and achieving ionisation ratios closer to those of silicon but in GaAsI AIGaAs. Consequently quantum efficiencies exceeding 90% have been demonstrated with anti-reflection coated super-lattice APDs, and dark currents less than 1 nA. Peltier-cooled detectors of this type hold considerable promise for photon correlation application if and when they become commercially available.

7. New correlator chips and designs

Developments in electronic correlator technology have been at the heart of most advances in the applications of photon correlation techniques. Correlators have developed on two fronts in the main, firstly the use of faster electronic chips to allow calculation of the full correlation function instead of the 'clipped' correlation function, avoiding restrictions to Gaussian signals and allowing construction of more accurate correlograms. TTL, CMOS and ECL correlators were constructed through the 1970s and 1980s with sample-time (L) capabilities being progressively reduced from SOns (Malvern K7023) to IOns (Malvern K7026) [37] and, in sparse correlation form to Sns (Spectron Correlex [38]). Photon correlators from Brookhaven Instruments, USA and AL V GmbH, Germany have also become prominent [18].

Full correlation function calculation is now standard in today's electronic photon correlators. Additionally in the field of photon correlation spectroscopy, (pCS, also termed dynamic light scattering) it is usual today to use not a linear spacing of sample times but instead a non-liner spacing of the sample times to construct the photon correlation function. This makes good use of the many advances in the information-theoretic understanding of the PCS Laplace transform situation gained from many years of research by Pike and his colleagues [39].

Although this chapter is devoted primarily to new opto-electronic technologies, it is useful to look briefly at the future potential for photon correlator developments. One

of these is the opto-electronic possibility of hybrid electronics-with-optical-fibre photon correlator construction to achieve even shorter sample-time capability [40].

The process of creating photon correlation functions is essentially that of forming a signal delay section (usually an electronic shift register), multiplication of the prompt signal with that stored in the delayed signal store (usually by a set of electronic AND gates) and an averaging operation (usually summation into electronic counters to store the accumulated correlation coefficients). Recently Jackson and his colleagues [40-42] have demonstrated that fibre optic photon correlator construction can eliminate clock-routing by use of a fibre-optic tree structure to create the delayed signals subsequently needed for calculation. This is achieved through the use of a series of fibre-couplers (beamsplitters) followed by fibre-optic delay lines (different lengths of fibre, eg, IOns delay = - 200 em of glass fibre core, etc). Jackson et al further showed [41] that the AND operation can be achieved through the use of discriminated photo-diodes (double gate GaAs MOSFETs) at GHz rates and that recently available GHz counters from the microwave community can be used to accumulate the correlation coefficients [42] and feed a desktop computer for further correlogram analysis. Thus the Kent University group have just demonstrated a 160 delay-time photon correlator with linearly spaced sample times ofO.5ns, an order of magnitude reduction in sample­time compared to the fastest previous all-electronic photon correlator. This is a notable achievement in hybrid optoelectronic-electronic correlator construction; the sustainability of commercial development of such a photon correlator system has yet to be proven.

We should not move on from any discussion of electronic correlators with mentioning recent and near-future developments in components. New, fast electronic chips are appearing in the market-place with increasing frequency. Amongst these, for example, are the Motorola DSP56300 [43] and Texas Instruments 1MS320C8X [44] chips. Here we have purpose-built digital signal processing chips that can be configured into photon correlator geometries. The 1MS320 chip has already been used in sparse-matrix correlator construction, where the data rate is very slow compared to the sample-time frequency, thus time-tagged events are correlated instead of computing the full correlation fwiction which would involve mainly AND operations on zeroes. Chips of the Motorola and Texas varieties already operate with single clock cycle instructions at 80 MHz, or 80 MIPS (million instructions per second). In 1997 we expect to see 100 MIPS devices commercially available, thus photon correlators close to these speeds will be constructed on electronic cards that can be inserted into desktop computers, for easy use and transportation.

Before moving away from the correlator discussion, we note in passing the current 'bottleneck' in triple-correlation [45] computation in optical aperture synthesis astronomy, where it is used for generalised closure-phase operation. Vast amounts of computation are necessary to re-create pictures from the photon counting signals in

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optical aperture synthesis. We confidently expect the use of these new 100 MIPS (and faster) devices to create new, dedicated hardware, photon triple-correlator systems to improve the computational situation in this form of astronomy.

8. Integrated optical systems?

With so much new and extremely small opto-electronic technology now being developed it seems sensible to explore if it is possible to combine some or all of these components together on a single substrate to create opto-electronic or photonic integrated circuits (OEICs or PICs [46]) for photon correlation experiments. Already in the opto-electronics community the concepts of OEICs and PICs are well developed. Laser diodes, optical waveguides, modulators and detectors have already been integrated on the same substrate for telecommunications applications.

An interesting early example of this concept applied to the light scattering field was published back in 1988 and involved a super-miniature laser Doppler velocimeter [47]. Here an integrated optical wave-guide assembly was created together with serrodyne modulation on a L~ substrate. Laser light was coupled into the wave­guides on the substrate, and Doppler-shifted light from the moving object of interest was coupled off the substrate into an APD detector and spectrum analyser. Today, with suitable thermal and electrical engineering, it is possible to place the laser diode and APD onto the same substrate as the wave-guides. Such a hybrid super­miniature LDV would find wide-spread use inside or on the surface of models being tested for their fluid dynamic performance.

Today's capability of being able to integrate active wave-guide devices, semiconductor lasers and detectors, modulators, amplifiers, micro-lenses (including geodesic lenses), reflectors, couplers, filters, switches, wave-guides and external fibres offers great opportunities for inventive new photon correlation optical system design and construction.

9. Beyond the year 2000

There are a variety of long range, more speculative research areas currently in progress that may lead to new opto-electronic devices that could have an impact on the technology used in photon correlation experiments in the 21 It century. Amongst these we note quantum wire and quantum. dot lasers, photonic bandgap lasers and detectors, ultra-violet emission semiconductor laser diodes and solar power supplies. It may also, in the extremely distant future, become possible to form photon correlation functions by quantum correlation processing at the atomic level, a form of direct quantum. computation.

On the applications front, we can expect to see credit card sized optical systems for PCS and LOV in the near future, made from today's opto-electronic devices, then OEIC super-miniature systems as discussed previously in section 8.

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Returning to the start of this speculation, we note the recent successes in the creation of quantum wire and quantum dot lasers [22,48]. It is currently thought that such quantum confinement structures placed in the active region of semiconductor laser diodes offers the potential of reducing the threshold current and increasing the power efficiency of such laser if the fabrication techniques can be improved to a point which does not introduce so much damage to the semiconductor materials that the recombination efficiency improvements expected with wires and dots is not thereby lost. Large arrays of high quality wires and dots can already be made in the research labomtory, but engineering this to production scale processing will take a considerable amount of more effort in future years. An extra feature of the use of quantum confinement structures to improve laser diodes is that colour (wavelength) variability occurs through choice of wire or dot dimension, for the same material composition. Thus with (Ga, AI) Aso.sNo.2 one might achieve red light at -1.8eV with a 20nm diameter dot, green light at -2.3eV with a 7nm diameter dot and blue light at -2.8eV for a 5nm diameter dot.

Photonic band-gap devices are in fashion as a research topic at present [21,49]. The potential advantage that is being pursued is that of perhaps being able to create good single mode micro-cavitiy emitters and detectors. It may prove eventually possible to create high efficiency emitters through strong coupling of spontaneous emission into a single mdiation mode, perhaps reducing lasing threshold to a very small value indeed. Current experiments on resonant cavity LEOs are of present note; applications to pmcticallasing seem to be a distant prospect, but if this is achieved then the excellent spatial mode properties that are expected could be of value in photon correlation optical systems.

UV semiconductor lasers are most likely at present to be created from GaN and related materials, with useful extra potential q-mnge (scattering vector) for photon correlation spectrocopy. APD detectors can be optimised for peak detection response in the blueIUV region by change of geometry and doping, thus a new UV lasing capability might be exploited in photon correlation experiments fairly quickly after the UV achievement.

With the miniaturisation of all the opto-electronic devices used in photon correlation experiments, the increases in efficiency and reductions in thresholds expected, it may become possible to power such experiments using high efficiency solar power cells, replacing electronic power supplies and batteries. Current silicon solar cells are about 24% efficient at best, but multi-quantum well designs [50] opemting at around 40% efficiency might one day be realised, and benefit super-miniaturised optical instrumentation used with photon correlation.

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The current fashion for discussing quantum computation [51,52], however remote the possibility seems now, suggests that it might be fun to think of performing the photon correlation calculation itself in a quantum computation comprising some arrangement of atoms as yet unknown 1 Direct formation of photon correlation functions from the quantum excitations themselves? Certainly not before the year 20001

In a more realistic mood, we should expect to see ever decreasing size in photon correlation optical systems, from current ·coke can' sizes to credit~d size and on to few-square-mi11imetre OBIC chips. This is how the new opto-electronic technologies will impact the photon correlation experimentalist in the next couple of decades.

10. Concluding remarks

In this paper we have reviewed a wide range of current and future research areas in opto-electronics and considered their potential for use in future photon correlation experiments. Following a first genemtion of conventional gas laser, bulk optics and photo-multiplier tube optical systems followed by a second genemtion of miniaturised optical systems based on semiconductor laser diodes, optical fibres and micro-optics and semiconductor avalanche photo-diode of mini-PMr detectors, we are now poised to see the development of a third genemtion of opto-electronic devices for use in photon correlation experiments.

This third genemtion of opto-electronic devices will include higher power, shorter wavelength (blueJUV) laser diodes and arrays of lasers, novel grating and mirror modulators, optical fibre experiment geometries such as heterodyne multi-sample­volume arrays, super-lattice detectors ofvery high quantum efficiency and the combination of these technologies with wave-guides into OBICs for photon correlation experiments. In years to come, well into the 21 at century we might see new devices based on quantum wires and dots and photonic bandgaps coming into play.

Coupling all this new opto-electronics with the mpid developments in multi-IOO­MIPS electronic chip technologies (and developments in electronic and optical data stomge), there is a most challenging and interesting time ahead for the photon correlation experimentalist in devising new measurements to get the best advantage from what is av8ilable. Now is the time to engineer these new devices and systems and plan new experiments.

11. Acknowledgements

The author is grateful to Professor E R Pike FRS for useful comments on a draft of this paper.

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linearly birefringent monomode fibre, Electronics Letters Zl, 847-848. 31. Chao, RK. Y., Jones, J.D.C. and Jackson, D.A. (1985) A compact all-optical fibre Doppler difference

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CORRELATED, SUPERPOSED AND SQUEEZED VIBRATIONAL STATES

J. JANSZKY AND Z. KIS

Research Laboratory for Crystal Physics Hungarian Academy of Sciences P.O. Box 132, H-1502 Budapest, Hungary

Abstract. Controlling the vibrational state of molecules with tailored light pulses are of special interest of today's research in molecular physics. In this paper the vibrational state of a polyatomic molecule excited with a trans­form limited light pulse is calculated analytically. In a diatomic molecule the vibrational analogue of the optical Schrodinger cat state can be real­ized in the experiment of double pulses. In polyatomic molecules a finite exciting pulse results in an entangled vibrational state. The magnitude of the entanglement is described by the von Neumann entropy associated with the vibrational modes. It is shown that in the case of a very short excita­tion, the change of the geometrical configuration of the vibrionic potential surface may also lead to entanglement.

1. Introduction

In quantum optics the most classical state of a monocromatic light field is the coherent state. This state exhibits similar properties as light considered classically, for large amplitudes the quantum nature disappears. Nonclas­sical states of the electromagnetic field are widely investigated domain of quantum optics. Research activities concern with both preparation and de­tection of nonclassical quantum states of light field. One manifestation of the quantum nature of light is the quantum noise. A pair of canonical con­jugate operators A and B can not be measured together with arbitrary precision: the Heisenberg's uncertainty principle says that

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E. R. Pike and J. B. Abbiss (eds.J, Light Scattering and Photon Correlation Spectroscopy, 277-294. @ 1997 Kluwer Academic Publishers.

(1)

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Either ~A or ~B can be as small as one likes, as long as the product satisfies the uncertainty principle. The states which have this feature are known as squeezed states [1]. The physical background of squeezing is the quantum mechanical superposition. Superposition of steady state field states with appropriate phase factors may reduce the variance of a variable through quantum interference. Quantum interference is a purely quantum mechan­ical phenomena without classical analogue. The most important types of squeezed states are quadrature, amplitude and phase squeezed states. Field quadrature operators are defined by the relation

A X +iy a = v'2fi' (2)

Quadrature squeezing means that either ~x2 < n/2 or ~y2 < n/2. In case of amplitude squeezing the variance of the amplitude of the state is smaller than that of a coherent state with the same mean photon number. Phase squeezed states are states with phase variance less than the phase variance of a coherent state. These states are not only theoretical results but they are realised experimentally as well. The possibility of application of these field states in optical communication and data processing are also in the focal point of today's research.

Squeezed states of light field are usually considered as a state of one field mode, thoughmultimode squeezed states are also defined and ver­ified experimentally. Entangled states of light field requires at least two field modes. Entangled states are states of composite quantum mechanical systems that cannot be written as products of single system states [2]. Two­particle entangled states have been known since the beginning of quantum mechanics. These states playa particularly important role in the study of the Einstein-Podolsky-Rosen (EPR) paradox [3] and in the test of Bell's in­equalities [4]. One of the most common system where entanglement arises is photon pairs generated by optical parametric down-conversion. An other widely investigated system is dressed atoms, where atomic states and the surrounding photon field form a compound quantum mechanical system.

Recent progress in ultrafast optics [5, 6] opened new possibilities in the control and observation of real-time molecular dynamics. There are both theoretical [7-17] and experimental [18-27] investigations ofthese problems. In recent papers we have shown for diatomic molecules that depending on the change of the geometrical configuration of the potential curves and the characteristics of the light pulse, a Franck-Condon transition [28] may result in vibrational quadrature or amplitude squeezed states with different properties [29]. The possibility of preparing other nonclassical vibrational states in diatomic molecules has also been investigated.

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In this paper we review the preparation of nonclassical vibrational states in polyatomic molecules. A displacement of the equilibrium nuclear dis­tances due to the excitation of the electronic system leads to· vibrations of the nuclei. The change of the vibrational frequencies also results in vi­brations even if there is no displacement of the equilibrium positions. We determine the time-evolution operator and the emerging vibrational state of a polyatomic molecule in a Franck-Condon transition when the electronic system is excited with a transform limited, weak light pulse. The poten­tial surfaces are approximated with general harmonic potentials. We show that depending on the duration of the excitation pulse and the parame­ters of the transformation of the nuclear potential the vibrational state will be vibrational Schrodinger-cat state, amplitude squeezed state, quadrature squeezed state, and vibrational entangled state. The von Neumann entropy of the states is introduced as a measure of the entanglement. An essen­tial consequence of the entanglement is that there exist observables of the different vibrational modes that exhibit correlations.

2. Model Hamiltonian

The vibrational Hamiltonian of a polyatomic molecule consists of 3N - 6 harmonic oscillator Hamiltonians if the molecule is nonlinear and 3N - 5 if the molecule is linear, where N is the number of the nuclei in the molecule. Let us consider an N-dimensional system described by the Hamiltonian

(3)

where Pn and lin are the momenta and normal coordinate associated with the nth vibrational mode, and Wn is the frequency of the vibration. First we summarize briefly how the vibrational state of the molecule can be found after a sudden change of the parameters describing the nuclear potential. We follow the derivation described in [30]. In general both the equilibrium distances and the harmonic force constants are altered. The new Hamilto­nian has the following form

N N N N H~ - 1 ~ ~~ 1 ~ ~ ~ ~ ~ f ~ / - 2 L..J Pn + 2 L..J L..J unmqnqm + L..J nqn·

n=l n=l m=l n=l (4)

Here Pn and lin are the same dynamical variables as in Eq. (3). It is advan­tageous to rewrite the final Hamiltonian in vector and matrix notation

(5)

280

A new coordinate system can be introduced by means of the linear trans­formation

Af SA d q = q+ , (6)

in which the Hamiltonian Eq. (4) is diagonal. Here S describes a pure ro­tation and d is associated with the displacement of the normal coordinate system. The variables associated with the new coordinate system are de­noted by an apostrophe.

The connection between the vectors d and f can be expressed as d = A-lSf, where the matrix A = diag{w~2} contains the vibrational frequen­cies of the molecule in the new normal coordinate system. In the end the final Hamiltonian reads

N HA 1 ,,\;,u.'2 + f 2 Af2) r:r

1= '2 L...J\¥n Wn qn - no, n=l

(7)

where Ho = dAd. Ho arises from the translation of the equilibrium dis­tances.

There is a unitary transformation denoted by E which connects both the initial and final Hamiltonian of the system and the state vector in the initial and in the final coordinate system [30]:

(8)

This unitary transformation in the Hilbert space is equivalent to the linear transformation of the normal coordinates in Eq. (6).

Let us consider the problem of the Franck-Condon transition in a poly­atomic molecule excited with a transform limited light pulse described by the electric field

E(t) = Eo exp( _u2t2 /2) cos(Ot) , (9)

where Eo is the maximal amplitude and 0 is the mean frequency of the pulse. The duration of the pulse is proportional to u-l . This light pulse is transform limited, i. e., the product of the half-width of the pulse and that of its Fourier-transformed counterpart is minimal. Assume that the dipole electronic transition takes place between the ground I g) and the excited I e) electronic levels. To meet this requirement the spectral bandwidth of the pulse should be narrow enough to avoid overlapping with a higher electronic level. This condition keep a lower limit to the duration of the exciting pulse. In the rotating wave approximation the vibrational Hamiltonian for the excited electronic level including the resonant interaction with the external classical field is

iII = iII + V(t), (10)

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where iI, is the Hamiltonian in Eq. (7), and the interaction potential V(t) is defined by

V(t) = ~Eo exp( _u2t 2 /2)[e- intdge I g)(e I +eintdeg I e)(g I). (11)

Here I g)(e I and I e)(g I are electronic state creation and destruction operators, dge is the electronic dipole matrix element.

Suppose that the molecule initially is in the ground electronic and vi­brational state I O}i I g}. An exciting pulse whose duration is much shorter than the period of the vibrations arrives at t = O. Then the time-evolution operator U(t) associated with the vibrations of the nuclei in the excited electronic level can be obtained in the form

(12)

Et transforms the ground vibrational state from the initial coordinate sys­tem to the final one then the operator exp( -iiI,t) evolves the state in the upper level. Though the vibrational coordinates are independent of each other the vibrations in the constituent modes start to evolve at the same time.

For a finite exciting pulse assuming weak electric field, one can calculate the time-evolution operator of the vibrational system using the first order time-dependent perturbation theory. It is found that this operator is the convolution ofthe weak interaction potential V(t) in Eq. (11) and the time­evolution operator U(t) in Eq. (12):

T(t) = ~Eodeg [00 d" exp( _u2,,2 /2)e i (n-W eg )rU(t - "), (13)

where nWeg is the energy difference between the electronic levels. After the exciting pulse has passed, i. e., for the time t » u-1 the integral can be evaluated. Finally, we obtain the operator T(t) in the form

(14)

where'Y = n - Weg - Ho, and N is an unimportant integration constant. The non-unitarity of the operator results from the perturbative method. The emerging vibrational state in the upper level is

(15)

This state is unnormalized since T(t) is non-unitary. We note that Eq. (15) is valid for any polyatomic molecule since we did not exploit the dimen­sionality of the system during the derivation. The quantity ,('I/J(t) I 'I/J(t)),

282

is the probability of finding the molecule in the excited state after the pulse has passed.

3. Diatomic molecules

Let us consider diatomic molecules i. e. when N = 1 in Eq. (3). In Eq. (4) the matrix {unm } is replaced by a single frequency w,2. In this case the operator E describes the displacement of the equilibryum nuclear distance and the change of the vibrational frequency from w in the ground electronic state to w' in the excited electronic level. The time evolution operator Eq. (14) is

(16)

where

E = exp(a[at - aD exp (~[a2 - at2 ]) , a = /¥;,d, 1 w A = "2 In w' . (17)

From right to left the exponential operator with parameter A is a quadra­ture squeezing operator corresponding to the change of the vibrational fre­quency, while the exponential operator with parameter a is a displacement operator.

Let us assume that the vibrational frequency does not change during the transition which results in the vanishing of the squeezing term in E. The emerging vibrational state, depending on the pulse duration, corresponds to several of the most important states in quantum optics. In the case of extremely short pulses the wave function is a usual coherent state

1

w(q') = f(q' I Et I O)i = (:,J 4 exp( -~(q' - d)2), (18)

while in the opposite limit of long pulses it is the n-phonon number state [29]. Between these limiting cases it is close to a quadrature squeezed min­imal uncertainty state, or, for longer pulses it is the banana-like amplitude squeezed state, which also appears to be an approximate number-phase in­telligent state associated with Pegg-Barnett phase operator formalism [32].

If the vibrational frequency changes during the transitions the half­width of the wave packet is different from that of the coherent state of the upper level:

(19)

The half-width of this wave packet spread and contract periodically in time with the period time half of the vibrational period [1]. This state is

283

a quadrature squeezed coherent state. When d = 0 we have a quadrature squeezed vacuum state.

Let us now consider two identical Gaussian shaped pulses following each other by an interval Tl

E(t) = Eoe- tl22 (t+¥-)2 cos (O(t+ 7t)) +Eoe- u:(t-¥-)2 cos (O(t -7t - ¢)),

here ¢ is a possible additional phase difference between the subpulses.

(20)

The vibrational state produced by such a twin pulse excitation has the form

It is assumed, that the vibrational frequency does not change during the transition.

To investigate the quantum properties of the superposition state of Eq. (21) it is convenient to consider its Wigner function

For extremely short pulses we have coharent superposition states which are the vibrational analogue of the so called optical.s~.hrod1!!g~!:_~~t~/states. The Wigner function and the time dependence of the absolute square of the wave function are shown on Fig. 1 and Fig. 2 correspondingly. The Wigner function consists of two bells of the superposed coherent states and an interference fringe between them. If the coherent states are far away the fringe has a lot of well-pronounced peaks. On the contrary, if the coherent states are near enough the fringe has only few peaks. In this case the fringe can partially merge with the bells and, depending on the phase between the component states, may decrease the uncertainty of one of the quadratures i; = (a + at )/../2 or y = i(at - 0,)/../2 below the vacuum level.

For long, strongly overlapping pulses Eq. (21) leads to the phonon num­ber state.

In the intermediate case of short (typically femtosecond) pulses the Wigner function of this state consists of symmetrically situated squeezed (for shorter exciting subpulses) or banana-like shapes revolving clockwise with the vibrational frequency along a circle which corresponds to the dis­placement parameter O!. A wavelike fringe pattern, according to the inter­ference, spins between them with the same w frequency .

284

Figure 1. The Wigner function of the Schrodinger-cat state. The prominent fringe structure is the manifestation of the quantum interference between the coherent states.

Figure 2. The absolute square of the wave function II{I(q'W of the Schrodinger-cat state.

285

A

B

Figure 3. The schematic terms for creation of a chemical superposition state. First by double pulse excitation one prepares a vibrational superposition state on level e. At some mgment of its separation by some secondary pulse(s) one can transfer the molecule into molecule A represented by the upper left term and simultanously into molecule B shown as the upper right term, creating this way a chemical "Schrodinger cat" state.

For small coupling constants (a '" w or less) the bell-like shapes are not resolved even for ultrashort pulses. In this case the quantum interference results not in oscillations but in narrowing in the spatial distribution in some intervals of time in every half a period of the vibration.

By a secondary excitation using probe pulse(s) one may transfer the vibrational cat state into chemical cat state creating quantum superposi­tions of different molecules. In Ref. [21] an Na2 molecule was excited by a short laser 'pulse. Applying a second laser pulse the state was excited once more. Depending on the time delay between the two successive pulses the resulting state was a molecule on another excited level or dissociated frag­ments. We suggest an experiment in which double pulse primary excitation leads to a Schrodinger-cat vibrational state in level e. Applying a third pulse when the two parts of the Schrodinger-cat state are the furthest from each other one obtains a superposition of the molecule with its fragments. This chemical cat state can lead us very near to the original paradox of Schrodinger. Let us suppose that this molecular superposition is superposi­tion of the undamaged form of a virus's DNA with a denaturalized variant of the same virus. The resulting" Schrodinger virus state" would be, in fact, a quantum mechanical superposition of a "living" and a "dead" virus.

286

4. Polyatomic molecules

For the sake of simplicity we shall consider a two-dimensional system. This model describes nonlinear XY2 molecules performing totally symmetric vi­brations. The vibrational state in Eq. (15) for a two-dimensional system can be found by inserting a complete basis set between the exponent part and tt in the time-evolution operator T(t) of Eq. (14):

= NI:n' n' Fn"n' e l' 2 1 2

Here the matrix elements

(nj WI +n2w2-'Y)2 2u2 (23)

F~'n' = J(n~,n~ I tt 100}i, (24) 1 2

can be determined by the recurrence formulas in Ref. [30]. The final vibrational state Eq. (23) depends on the duration of the

exciting pulse u-1 and the parameters of the transformation of the nuclear potential, that appear in the matrix elements F~, n' in Eq. (24). When the

1 2

duration of the pulse is very short the time evolution operator T(t) in Eq. (14) reduces to U(t) in Eq. (12). In this case the properties of the state vector Eq. (23) are more apparent in the coordinate representation. Using Eq. (6) and the wave function of the ground vibrational state we find that the vibrational wave function in the excited electronic level is

1

\IT(qL q~) = J(qL q~ I tt I OO}i = (~~~~)"4 e-~q' Lq'+dLq'-~dLd, (25)

h L-1 (WICos2X+W2sin2x ~(w2-wt)sin2x ) were - Ii ~(W2 -wl) sin2X W2Cos2x+wlsin2x .

It can be seen from Eqs. (23) and (25) that in general the emerging vibra­tional state is an entangled state. Entanglement means that the vibrational state of the molecule can not be separated to product of states of the con­stituent vibrational modes. Though the wave function of the whole system exists, the partial systems have only density matrices. A finite exciting pulse results in an entangled state because in Eq. (23) the exponential term can not be factorized when u is finite. There is an obvious exception, that we exclude from our consideration, when the operator t transforms the vibra­tional vacuum state in the ground electronic level to the vacuum state of the excited electronic level. In this case the resulting state is the vacuum for arbitrary duration of the pulse. A very short excitation can result in an en­tangled or non-entangled vibrational state depending on the configuration of the system.

287

For a systematic analysis of the arising state vector one can exploit the fact that the operator E, describing the axes switching in the Hilbert space, is a product of elementary transformations as translation, rotation, and dilatation:

A A At A A

E = StS>..,SxS>... (26)

Here St is the displacement operator

(27)

where ai is the phonon annihilation operator in the ith mode, and di is the ith component of the displacement d of the normal coordinate system in Eq. (6). The operator S>.. and S>.., describe the dilation of the normal axes, which is due to the change of the vibrational frequencies

(28)

where (29)

The operator Sx is the operator of rotation

Sx = exp[x(a! a2 - aIa~)], (30)

where X is the angle of rotation of the normal coordinate system in Eq. (6). Let us first determine the conditions when the vibrational state is not

entangled. As we mentioned before, the shortness of the exciting pulse is a primary requirement. The wave function in Eq. (25) can only be separated to the product of two independent wave functions of the two constituent vibrational modes if the matrix L is diagonal. This case arises when either X = 0 or WI = W2, i. e., either the normal coordinate system does not turn during the transition or the ground state vibrational potential has spherical symmetry.

As an example for a non entangled state induced by a very short pulse, let us consider the transition when there is only translation and no rotation and no dilatation of the normal coordinates (d =f:. 0, X = 0, Wi = wD. In this case the operator E = St and the matrix L is diagonal. From Eq. (25) we find that the wave function is separable and the emerging vibrational state is a two-dimensional coherent state

(31)

288

If the ith vibrational mode is in a coherent state the time-dependent average values of the normal coordinate q~ and the momenta p~ operators follow the classical behavior of the corresponding classical variables [31]. The fluctua­tions of q~ and p~ minimize the Heisenberg's uncertainty relation and they are time-independent:

(32)

The last property implies that a coherent wave packet rotates without spreading in phase space of the system. The resulting state is also a two­mode coherent state if WI = W2, X =I 0, and the vibrational frequencies do not change. When there is not translation (d = 0) in these transitions, the resulting state is the vibrational vacuum state.

If the vibrational frequencies change in the previous transitions the emerging vibrational wave function is also the product of two Gaussian wave packets in Eq. (31). But the half-width of these wave packets are different from that of the coherent state of the upper level:

(33)

The half-width of these wave packets spread and contract periodically in time with the period time half of the vibrational period [1]. These states are quadrature squeezed coherent states. When d = 0 we have two independent quadrature squeezed vacuum states in each vibrational modes.

There exists geometrical configurations when a short exciting pulse leads to an entangled state. This is the case when WI =I W2 and X =I 0 in Eq. (25) and d is arbitrary, that is, the normal coordinate system turns dur­ing the transition and the ground state potential is non-spherical. Then the off-diagonal elements of the matrix L do not vanish and the resulting vibrational wave function is given in Eq. (25).

If a system is in an entangled state, the partial systems are in mixture states though the whole system can be in a pure state. One can use the von Neumann entropy as a measure of the purity of the state of a vibrational mode:

(34)

where Pi is the density matrix of the ith mode. S(Pi) = 0 for a pure state. In our case the origin of the non-purity is the entanglement of the vibrational modes. The &um of the two entropies provides a good measure for the degree of the entanglement

o ~ S(pt) + S(fJ2). (35)

TABLE 1. The von Neumann entropy associated with one of the vibrational mode of an entangled two-dimensional vibrational state for different angles of rotation X. The excitation is very fast compared with the vibrational periods. The ground state vibrational frequencies are not equal WI ::I W2, the vibrational frequencies do not change during the transition, and the translation d is zero.

Si 0.0030 0.0075 0.0166 0.0241 0.0322 0.04545

289

Deeper entanglement results in larger entropies. The density matrix of the whole system is P =1 \lI)(\lI I. The von Neumann entropies S(Pi) of the two partial systems are equal to each other for every possible transitions. This assertion can be proved with the aid of the Schmidt decomposition of the state vector of the whole system. The Schmidt theorem says that 'if a pair of correlated quantum systems is in a pure state 1 \lI), it is always possible to find a preferred basis such that 1 \lI) becomes a sum of bi-orthogonal terms' [2]. In the Schmidt basis the density matrix of a pure state 1 \lI) is

P =1 \lI)(\lI 1= L MiMj 1 Ui)(Uj 1 ® 1 Vi)(Vj 1 . (36) i,j

The reduced density matrices of the two subsystems are

j j

These two density matrices have the same eigenvalues, so the values of the entropies S (pd and S (P2) are equal.

In our previous example the rotation of the non-spherical potential leads to an entangled vibrational state. In table 1 we show the von Neumann entropy as a function of the rotation angle X of the normal coordinate system. It can be seen that the larger is the angle of rotation the more entangled the state is.

Now we turn to the problem of finite exciting pulse. In this case the vibrational state in Eq. (23) can not be separated to the product of two vibrational states, so the state is entangled independently of the change of the geometrical configuration. In geometrical configurations where a short exciting pulse,leads to a non entangled state, a finite pulse results in an entangled state. As an example, let us consider the two-mode vibrational coherent state in Eq. (31), which is induced by a very short exciting light

290

TABLE 2. The von Neumann entropy associated with one of the vibrational mode of an entangled two-dimensional vi­brational state for exciting pulses with different duration. The duration of the pulse T = 11.- 1 is in the unit of the vibrational period in the ground state. The ground state vibrational fre­quencies are equal WI = W2, the vibrational frequencies do not change during the transition. A short exciting pulse would lead to a non-entangled state, but the finiteness of the exciting pulse causes entanglement.

T 0.05 0.1 0.2 0.5 1 10

Si 8.9E-5 9.3E-4 8.9E-3 0.1 0.394 0.811

pulse. For further treatment it is convenient to rewrite this state in the Fock basis:

n~ n~

I 'l/J(t)f} = e-gU2-gV2 L 91 ,~2, I I n~n~}, (38) n~,n~ Vn1.n2.

where 9i = (wU21i)~di. Now let the duration of the pulse be finite. In this case the role of the quadratic part in the exponent in Eq. (23) becomes important. In table 2 the von Neumann entropy is presented as a function of the duration of the exciting pulse when the potentials are spherical and the vibrational frequencies do not change during the transition. It can be seen, that a longer pulse leads to deeper entanglement.

If the duration of the exciting pulse is large compared with the vibra­tional periods then only those terms will survive the Gaussian cutoff in the expansion Eq. (38) for which 'Y = wini + w~n~. This is the so called CW limit, well known in the literature. The emerging vibrational states are de­termined by the solution of this equation for given vibrational frequencies and photon energy in the exciting light ('Y = w-weg-Ho). Ifthe ratio of the vibrational frequencies are not that of small integer numbers there is only one solution. In this case there is no entanglement, the modes are separable the vibrational state of the molecule is the product of two eigenstates. In case of small integers' ratio of the vibrational frequencies multiple solutions exists for appropriate values of 'Y. Now the arising vibrational state is en­tangled. In a degenerate case the two vibrational frequencies are the same. For the coherent state of Eq. (38) the act of the quadratic operator yields

(39)

291

The degree of entanglement can also be measured by the von Neumann en­tropy of the constituent vibrational modes. Carrying out the partial tracing of the total vibrational density matrix, one finds

A "f ('Y) n~ "f-n~ 1 '} ( , 1 Pi = Ln~=O n~ Pi Pk ni ni , i,k = 1,2 k =I i, where

. g2

PI = g~~g~' (40)

The entropy associated with the density matrix Eq. (40) can be written in the form

"f

S(Pi) = - L Wn In W n ,

n=O ( 'Y) n "f-n

Wn = n PIP2 . (41)

Here we omitted the index i since the quantities Wn are the same for both of the two subsystems. In table 2 the von Neumann entropy can be calculated with the aid of this furmula when the duration of the pulse is large compared with the vibrational period.

We have described two types of processes which lead to an entangled vibrational state in a polyatomic molecule: in the first case the change of the geometrical configuration during a sudden electronic transition leads to entanglement while in the second case entanglement comes from the finite excitation process. There is a difference between these states that appears in the joint phonon number distribution. The joint phonon number distributions for a state induced by a short exciting pulse can be obtained from the matrix elements of the operator tt:

P(n~,n~) =1 (n~,n~ I tt 100) 12

This formula yields for the coherent state in Eq. (38):

(42)

(43)

The plot of this distribution function can be seen in Fig. 4(a). In the case of an entangled state induced by a short exciting pulse, the joint phonon number distribution can be calculated with the help of the recurrence for­mulae in Ref. [30]. The result is shown in Fig. 4(b). For the finite exciting pulse the joint phonon number distribution associated with the state Eq. (39) is a binomial one

(44)

292

Figure 4. The joint phonon number distribution of two-mode vibrational states: (top left a) for a two-mode coherent state, (top right b) for an entangled state induced by a very short light pulse, (bottom c) for an entangled state induced by a finite pulse.

Figure 4(c) shows the plot of this distribution function. Comparing Figs. 4(a), 4(b) , and 4(c) it can be seen that if the vibrational state is an en­tangled state, the joint phonon number distribution function is narrower in a certain direction compared with that of a coherent state. The differ­ence between entanglement resulting from the change of the geometry and the finite excitation process is, that while in the first case the distribution function situated along a n~ '" n2 line, in the second case the distribution function is perpendicular to the n~ '" n2 line. It can be said that the states are correlated or anticorrelated.

293

5. Conclusion

We have shown, that in analogy to the widely investigated nonclassical states of light nonclassical vibrational states can be prepared in polyatomic molecules.

This work was supported' by the National Scientific Research Fund (OTKA) of Hungary under Contracts Nos. F019232, T017386, and T020202.

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13. M. Grubele and A. H. Zewail (1993) Femtosecond wave packet spectroscopy: Co­herences, the potential, and structural determination J. Chern. Phys., 98, pp. 883

14. U. Peskin and N. Moiseyev (1993) The solution of the time-dependent Schrodinger equation by the (t, t') method: Theory, computational algorythm and applications J. Chern. Phys., 99, pp. 4590

15. J. L. Krause, R. M. Whitnell, K. R. Wilson, and YJ. Van (1993) Optical control of molecular dynamics: Molecular cannons, refiectrons, and wave-packet focusers J. Chern. Phys., 99, pp. 6562

16. Guanhua Yao and Robert E. Wyatt (1994) Stationary approaches for solving the Schrodinger equation with time-dependent Hamiltonians J. Chern. Phys., 101, pp.1904

17. B. M. Garraway and K.-A. Suominen (1995) Wave-packet dynamics: new physics and chemistry in femto-time Rep. Pray. Phys., 58, 365

18. R. M. Bowman, M. Dantus, and A. H. Zewail (1989) Femtosecond transition-state spectroscopy of iodine: from strongly bound to repulsive surface dynamics, Chern.

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STRUCTURE AND PROPERTIES OF LINEAR

INVERSE PROBLEMS

Abstract.

JOHN B. ABBISS Singular Systems, 19451 Sierra Raton Road, Irvine, California, USA

The properties of inverse problems in which experimental data are re­lated to the object of study by a Fredholm integral equation of the first kind are reviewed. A singular function analysis reveals the ill-posedness inherent in such problems. Regularisation as a means of overcoming the difficulty in obtaining a meaningful solution is discussed. The discretised case is con­sidered and methods of solution based on singular value decomposition are presented and illustrated.

295

E. R. Pike and J. B. Abbiss (eds.), Light Scottering and Photon Correlation Spectroscopy, 295--311. @ 1997 Kluwer Academic Publishers.

296

STRUCTURE AND PROPERTIES OF LINEAR

INVERSE PROBLEMS

1. Introduction

The methods of photon correlation spectroscopy are now of importance not only in purely scientific research, as amply demonstrated by the contributions to this workshop, but also in a range of technological applications. For example, the manufacture of many industrial products, such as paints, ceramics and pharmaceuticals, requires a knowledge of particle sizes in the sub-micron· range, and the determination of size distributions by photon correlation spectroscopy is now a well-cstablished technique. The efficient extraction of this information Jrom the experimental data also provides an interesting example of an extensive class of inverse problems in which a signal or image g(y) is related to the object of interest f(x) by a linear Fredholm integral equation of the first kind. In this introductory review, we summarize the common properties of these problems and discuss methods, based on regularisation theory. for obtaining optimal solutions in the presence of noise. For simplicity we consider the one-dimensional case. The extension to two or three dimensions involves some numerical complication, but no new principles.

2. Nllturc of the Problem

The peculiar difficulties associated with finding g , given f and the kernel K , stem from the characteristics of the integral operator in a Fredholm equation of the first kind, where the unknown function appears only in the integrand:

b

I K(x,y) f(x)dx = g(y) , (1) Q

Thus f(x) is to be found as the solution of equation (1). The difficulties include the strongly smoothing nature of the operator, as a result of which roughness in f is significantly reduced and any discontinuities eliminated. As a consequence, we can show that very small perturbations in g can correspond, in the inverse sense, to very

297

large perturbations in the estimate off. The Riemann-Lebesgue lemma states that, if IKI is integrable with respect to x on the interval [a, b) for all y ,

b

lim IK(x,y)sinnxdx=O, c5.y5.d, 11-+""0

and hence the Fourier components off are transmitted ever more weakly to g as their frequency increases. Thus a small high-frequency perturbation in g , caused most commonly in practice by experimental noise, can result in a vastly disproportionate change in the estimate, which can be displaced arbitrarily far from! Conversely, for any given level of noise, objects which differ from one another only by their content at sufficiently high frequencies will correspond to indistinguishable images.

The phenomenon of discontinuous dependence of the estimate on the data identifies the problem as "ill-posed", in contradistinction to the concept of the well-posed problem, due originally to Hadamard. If a problem is to qualify as well-posed, the solution must exist, be unique and exhibit continuous data-dependence. (Later we shall be concerned exclusively with discretized problems, for which the existence and uniqueness of computed solutions can be assumed. For these problems, the integral operator becomes a matrix and ill-posedness reduces to ill-conditioning.) In recent decades, in parallel with the evolution of ever-greater power and availability of computing resources, ill-posed problems have been found to occur abundantly in physics and engineering, typically in remote-sensing applications - for example, in optics, radar, sonar, seismic exploration and tomography - as well as in many other theoretical and applied fields, including astronomy, probability theory and nuclear fusion research. Theoretical developments in the severely ill-posed problem of determining particle-size distributions from quasi-elastic· light scattering measurements, which involves the inversion of a Laplace transform, are discussed in Professor Pike's contribution to this volume. A number of examples of ill-posed problems associated with Fredholm equations of the first kind are discussed in some detail in [l].

J. Solutions by Singular Function Analysis

An explicit formula for the solution to equation (1) can be derived by means of a singular function analysis [2], which can also be used to investigate the effects of noise. We expand the kernel in terms of the real and positive singular values 0; and the functions uj (x) and Vj (y) , orthonormal sequences in object and image spaces respectively:

"" K(x,y) = 1: 0; uj (x)vj (y). j=O

(2)

298

b

Denote the integral operator from object to image spaces, I dx K(x,y) , by K. Its a

• adjoint K, which maps from image to object spaces, is defined through the

• equivalence of the inner products in these spaces: <Ku, v>=<u,K v> , integration between the appropriate limits replacing the summation process of finite-dimensional inner products. (For a discussion of integral equations in a Hilbert space setting, see,

• d _

for example, [3]). Here K = I dy K(x,y) , the bar denoting complex conjugation, and c

it follows from equation (2) that the sequences (u;(x)} and (v;(y)} are related by •

Ku; =(7; V; and Kv;=C1'j u;.

The object and image possess unique expansions in terms of the singular functions:

(II)

f(x) = }:; J; uj (x) j=O

(3)

and (II)

g(y) = }:; gj Vj (y) , (4) j=O

where, as a consequence of the orthogonality of the U; and Vj,

b

J; = I f(x) Uj (x) dx (5) a

and d

g; = I g(y) v;(y) dy . (6) c

The u;{x) and Vj (y), which are characteristic of the integral operator, can be regarded

as fundamental units of information in object and image spaces, into which the object and image are partitioned. It is straightforward to establish a connection between the coefficients f; and g; , and thus derive a formula for recovering f. Substituting equations (3) and (4) in equation (1), we find

(II) (II)

}:;u, /; v.(y) = ~ g. v.(y) , j=O I I I j=O I I

and we must have (7)

The object function can therefore be reconstructed from the image coefficients gj as

299

(8)

and in principle perfect reconstruction is possible. In practice, of course, the image will be perturbed by the effects of noise. Let us represent this noise as an additive function r(y). Expanding the noisy imageg' in tenns of the Vj (y) , we find for the new image coefficients

where • d

Ii = J r(y) Vj (y) dy c

and the estimate of the object becomes

_ ao rj I(x) = I(x) + ~ - uj(x) .

j=O Oi (9)

The ordered singular values typically tend rapidly to zero, and since in general the coefficients rj are random and finite, the sum will diverge and the reconstruction can depart arbitrarily far from the true object. This is again a manifestation of the ill-posed nature of the problem.

4. The Prolate Spheroidal Wave-Functions and Analytic Continuation

To make these ideas more specific, we consider the problem of recovering an improved estimate of an object from its noisy blurred image. In the course of their research into band-limited functions (Le., those functions whose Fourier transforms J(OJ) are zero for IOJI ~ OJo' the cut-off frequency), Slepian and Pollak [4] discovered the remarkable properties of the prolate spheroidal wave-functions, which are band-limited and form complete orthogonal sets over both the infinite real axis and a finite segment of it. In addition to solving the communication theol)' problems with which Slepian and Pollak were concerned, these functions also provide a means of reconstructing the image formed by a coherent ope-dimensional shift-invariant optical system. In this case, equation (1) takes the form

1 sin[Q(x- y)] I(x)dx = g(y) . -x 1r(x- y)

For convenience we have taken [a,b] = [-X,x). Then the range of y is [-X,x) for the geometrical image and (~,<Xl) for the infinite image. Q is the angular cut-off

300

frequency of the system. The eigenfunctions of the operator in this equation are the prolate spheroidal wave-functions II'i (c,X) , ;=0,},2, ... , where c is the space-bandwidth

product OX. The corresponding eigenvalues l; are also defined in terms of these functions [4]. The orthogonality properties of the II'i are:

and

x I II'i(X) V' j(x) dx = l;t5ij

-x

CD

I II'i(X) V' j(x) dx = t5ij -CD

where t5ij = I if ; = j, = 0 otherwise.

The singular values and singular functions can now be defined. From the relationships

between the Ui and Vi, given above, we find 0'; =,[T;, u;(x) = II'i(X)/,[T; and v; (y) = 11'; (y) . Finally, from equation (8), the recipe for reconstructing f becomes

f(x) = ;~ [ V/;~~) Ig(y) '1'; (y) dy 1 ' -XS;Xs;x.

The property of the object by means of which frequency components in f, completely lost in transmission through the optics, can be recovered is that, since the object is of finite extent, its Fourier transform is an entire function [5]. Consequently, the whole spectrum can be derived from any part of it by the process of analytic continuation. The prolate spheroidal wave-functions provide the mechanism for doing this: we find the coefficients in the expansion using the known segment and compute the spectrum elsewhere from the same formula. The estimate of f which results is said to be "super-resolved", since the conventional Rayleigh resolution limit associated with the cut-off at 0 has been exceeded. Although the stage of analytic continuation is obscured in the brief derivation given above, the same result could have been obtained by expanding the spectrum ofthe image, with support over [-0,0], directly in terms of the prolate spheroidal wave-functions, calculating the spectrum outside this region by means of the expansion, and finally Fourier-transforming back to object space [6]. Again, of course, noise will limit the extent to which the lost frequency components can be recovered.

5. Regularisation of an III-Posed Problem

As noted before, noise has a destructive effect on the computation. The behaviour of the A.; in the example above is characterised by a rapid decay from almost one to nearly zero for values of the index near c, the space-bandwidth product, and only small

301

amounts of noise, or even computer round-off error, can render the results meaningless. These effects are endemic in the class of inverse problems considered here, and some method for controlling them is essential in practise. Regularisation theory, a branch of modern mathematics which has undergone rapid evolution since the early sixties and now finds application in many areas of physics, offers a rigorous basis for the required techniques. We describe one particular technique of regularisation, the Tikhonov-Miller method [7,8,9], from among the many that have been developed or proposed.

Since an exact and plausible solution cannot be found, the basic strategy is to modify the nature of the inverse problem in such a way that only solutions which fit the data to some specified level and possess an aCceptable degree of smoothness are possible. A suitable measure of the departure of the image of an estimate 7 from the actual data is

provided by the norm IIK7 -g'llo in the Hilbert space G of images. Thus we select

only those solutions for which this quantity is bounded by some suitably small number & , and from this set choose the member with minimum norm. In order to reject solutions characterised by large and unphysical fluctuations, we impose the second

constraint, that the norm, in object space F. of the estimate itself, 11J1IF ' also has an

upper bound, E say. (This requirement on the norm of 7 constitutes zero-order Tikhonov regularisation - higher-order regularisation is accomplished by imposing similar constraints on derivatives of 7.) The effect is to restrain the instability

inherent in the problem by limiting excursions in the reconstruction, thus exercising a smoothing effect. We therefore require that

IIK7 -g'llo < & and 11711F < E .

It is convenient to combine these inequalities quadratically (which relaxes the constraints by a factor of J2 at most) and seek to minimise the functional

IIK7-g'II~+aIl711~ ,

where we have written a in place of & 2/ E2. Note that a , which is termed the

regularisation para~eter, will usually be a small quantity. The estimate fa minimising the functional is the solution of the normal equation [8J

(K·K +a 1)fa = K·g'

Since K·K is symmetric, its eigenvalues are non-negative, and the eigenvalues of

(K·K +al) are strictly positive. Hence (K·K +al) is a positive definite

operator. Therefore its inverse. together with fa , always exists. Inserting singular-

302

function expansions for fa and g' and using the stated properties of the operators, we find

which is to be compared with equation (8). We see that the effect of the constraints is to make contributions to the sum progressively smaller as the order of the index increases. The regularisation parameter acts as a filter for the singular values, and if its value is chosen correctly a reconstruction will be obtained in which the resolution achieved is optimal for the given noise environment. Methods are available for estimating this optimum value [10,11] from the data themselves. It can in fact be shown [12] that for white noise of spectral density & and for "white" objects (i.e., objects, admittedly rarely encountered, whose coefficients as given by equation (5) are uncorrelated) having spectral density E , the minimum is achieved precisely when a =

& 2/ E2, the ratio of noise power to signal power. Although more general relationships between the regularisation parameter and the signal-to-noise ratio do not seem to be known, the optimal a is usually found to be in the same neighbourhood. The degree of super-resolution which can actually be achieved in a particular case is a strong function of the space-bandwidth product and the signal-to-noise ratio [13,14]. For typical optical imaging systems and noise levels, the object must have strictly limited support if significant super-resolution is to be achieved. In practise, this means that the geometrical image will usually lie well inside the main lobe of the point spread function.

An alternative form of regularisation which yields very similar results is truncation of the set of singular values at an order which will depend on the noise in the data [15]. In the case that object and noise are white-noise processes, it can be shown [13] that only those singular functions corresponding to singular values for which 0; ~ & / E will contribute usefully to the reconstruction. Again, some experimentation may be necessary to find the optimum truncation point.

It is important to note that regularisation always provides an approximate solution to an ill-posed problem. The problem as initially stated is replaced by one which yields solutions possessing the desired properties of stability and smoothness, and these solutions can never be completely consistent with the original data.

6. Matrix Formulations

In practise, experimental results are usually obtained as discrete data and the problem is posed in terms of a finite-dimensional matrix A rather than an integral operator. Equation (1) then becomes

Af=g,

303

(10)

where for convenience we continue to use the symbols f and g for object and image.

Generally, f E en, g E em and A E emxn . We shall term A the imaging matrix.

The least-squares approach to solving equation (10) for noisy data g' is to minimise the

quantity IlAl-g't ' now a Euclidean norm, leading to the normal equations:

AHAf=AHg'

where AH is the Hermitian conjugate of A. As in the infinite-dimensional case, this constraint is insufficient to alleviate the inherent instability; ill-posedness in the

original problem is now manifest as a near-singularity in A , and hence also in A H A .

We again regularise the problem by constraining both the norm of the estimate, 11111 F '

and the norm of the distance of its image from the data, obtaining as the minimiser

(11)

Forming AHA and directly computing (AHA +al)-IAH entails numerical inaccuracy and is not recommended. The procedure we choose for obtaining fa while also

revealing the structure of A is singular value decomposition (SVD), the linear­algebraic analogue of the singular function analysis described above. For any matrix A we can write [16]

where

and

U consists of the n orthonormalised eigenvectors corresponding to the n largest

eigenvalues of AA H, and V consists of the orthonormalised eigenvectors of A H A . The OJ , the singular values of A , are the non-negative square roots of the eigenvalues of

A H A . They are assumed to have been arranged in descending order of magnitude. A zero singular value indicates that A is singular, the SVD in fact having been originally developed as a technique for revealing matrix rank. Substituting the SVD expansion

in equation (11), we obtain, since (AHA +al) is positive definite and therefore invertible,

(12)

where

304

( 001 Un J l:a = diag 2 , ••••••• , 2 • 001 +a Un +a

(13)

Thus we see again in the matrix representation that the regularisation parameter a counteracts instability by controlling the influence of the smaller singular values. A measure of the inherent instability associated with the inverse of a matrix is its condition number, defined in the 2-norm (which is most convenient for a least-squares problem) as the ratio of the largest singular value to the smallest. The condition number before regularisation is 001 I un ' and it can be shown [17] that for the

regularised problem it becomes ~( 0012 + a ) 1(00; + a) . Typically, 0012 » a » 00; , and the condition number will generally improve by a factor of about ra I un .

For large matrices, which occur commonly for optical images, the SVD procedure can be prohibitively expensive in both time and memory requirements. The number of operations involved is approximately 4m2n+8mn2+9n3 for an mxn matrix [18]. For an NxN image reconstructed into an object field of the same size, the imaging matrix is N2xN2 , and the SVD would involve 21~ operations. For a 128x128 image, the number of operations would be about 9x 1013 • However, in the special case that image and object fields are of the same size and sampled at the same intervals, the imaging matrix can be ex-panded into circulant form by inserting appropriately-positioned additional columns, and the computation dramatically accelerated. (The image is padded with groups of zeros, where necessary, to keep the arithmetic of the reconstruction unchanged.) We can now exploit a fundamental theorem of matrix algebra, that the Fourier transform diagonalizes a circulant [19], and reduce the essentials of the computation to a pair of fast Fourier transforms followed by a vector­vector multiplication and finally an inverse fast Fourier transform. The number of operations has now been reduced to 0(3Nlog2N), which has the value 0.7xl06 for a 128x 128 image. The nth -order Fourier matrix takes the form [19]

1 1 1 1

1 W w2 n-I W

1 w2 4 w2(n-I) W

H 1 F = .In

where w = exp(i 21l' I n) .

305

If A is a circulant, the decomposition A = FH AF exists, where A = diag(A.1, ••••• A.n) •

Inserting in equation (11), we obtain

where

The A.j are given by a fast Fourier transform of the first column of A , and the reconstructed image can then be computed as

which requires only one-dimensional FFfs. It should be emphasised that the modification-to-circulant procedure is not an approximation, and leads to results which are close to those obtained by direct SVD of the imaging matrix, at least in conditions of moderate noise. The two methods are intimately related - the Fourier transform diagonalisation is in fact the SVD in disguise.

7. An D1ustrative Example

To demonstrate some characteristic features of object restoration using the SVD, we take the object shown in Figure lea), consisting of an l8-by-18 array of ones and zeros. The 8-by-8 Gaussian point spread function is displayed in Figure l(b). To create the image, the object and point spread function are scanned column-wise into vectors. Each column of the imaging matrix A is constructed from the vectorised point spread function, and it can be shown that the overall structure of A consists of Toeplitz sub­matrices arranged in Toeplitz order (i.e., A is block-Toeplitz with Toeplitz blocks); see, for example, [20]. This structure can be seen in Figure 1 (c), which, for clarity, displays only the first 100 rows and 50 columns of A , whose dimensions are 625x324 in this example. The product of matrix A and the vectorised object generates the blurred image in column-stacked vector form. After reshaping, the image appears as in Figure led).

Singular value decomposition of A produces the orthogonal matrices U and V containing the left and right singular vectors, respectively, of A. The first twenty of each of these two sets of vectors are displayed in grey-level form in ·Figures 2(a) and (b). The right and left singular vectors generate the object and image coefficients respectively. The singular values of A are shown in Figure 2(c). The ratio of the largest to the smallest, the 2-norm condition number of this matrix, is about lOS. The

unregularised deconvolution matrix obtained from the ·SVD of A, Vl:a U T of equation

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309

(12) with a set to zero, is shown in Figure 2(d). Although some symmetry remains, most of the structure evident in A has been lost. Normally-distributed random noise was added to the image. The signal-to-noise ratio, defined for the present purposes as the ratio of the mean image pixel value to the standard deviation of the noise, was set at 250. The result of applying the unregularised deconvolution matrix to the noisy image is displayed in Figure 3(a), in which there is no discernible information. An approximately optimum reconstruction with a set to 5xl0-4 is shown in Figure 3 (b). The recovery of detail is least successful in the region of the essentially uncorrelated pixels forming the letter X. A fast FFf -based reconstruction was also perfonned by expanding A to circulant fonn - Figure 3(c) is to be compared with Figure 1 (c). The result is shown in Figure 3(d). The reconstruction is close to that obtained by SVD, although perhaps slightly noisier. Because of the larger support imposed by the circulant-matrix requirement, some signal energy can leak into the area surrounding the true object, and will appear as noise.

All computations were perfonned in the MA TLAB environment. On a personal computer equipped with a Pentium 166MHz processor, the SVD of A (625x324 elements) took about 50 seconds. The matrix multiplications involved in computing the reconstruction occupied a further 4.3 seconds. By contrast, the complete calculation based on the circulant fonn of A and FFfs took approximately 600 milliseconds. Note that only the first column of the circulant is needed. Of course, if multiple images are to be processed and the experimental conditions do not change, the SVD need be computed only once.

8. Conclusions

We have seen that a singular function analysis, and its finite-dimensional analogue, the SVD, provide fundamental tools for assessing the information content of experimental data whenever a Fredholm integral equation of the first kind describes the relationship of the data to the object under study. Examples in the fields of light scattering and photon correlation spectroscopy arise in connection with the determination of particle-size distributions from Fraunhofer diffraction patterns, by dynamic light scattering studies and from measurements of turbidity as a function of wavelength. Singular function analysis of such inverse problems shows clearly what can and cannot be expected of the data, in a way that no other methods do. For singular values below the noise level, the corresponding components of the object are irretrievably corrupted and, without additional a priori information, set an absolute limit to the extent to which instrumental degradation can be overcome.

The practical difficulty in ~pplying the ~VD in its full rigour to two-dimensional image data is the associated computational burden, for all but quite small images. However, provided that the image is to be reconstructed into a similarly-sized and structured object field, the imaging matrix can be expanded into circulant fonn, and the power of the fast Fourier transfonn exploited.

310

9. References

1. Wing, G.M. and Zahrt, J.D. (1991) A Primer on Integral Equations of the First Kind: The Problem of Deconvolution and Unfolding, SIAM, Philadelphia.

2. Smithies, F. (1958) Integral Equations, Cambridge Tract No. 49, Cambridge University Press, Cambridge, England.

3. Moiseiwitsch, B.L. (1977) Integral Equations, Longman, London.

4. Slepian, D., and Pollak, H.O. (1961) Prolate spheroidal wave functions, Fourier analysis and uncertainty - I, Bell System Tech. J., 40, 43-63.

5. Boas, RP. (1954) Entire Functions, Academic Press, London and New York.

6. Frieden, B.R (1967) Band-unlimited reconstruction of optical objects and spectra, J. Opt. Soc. Amer., 57, 1013-1019.

7. Tikhonov, A.N. and Arsenin, V.Y. (1977) Solutions of Ill-Posed Problems, V. H. Winston & Sons, Washington DC.

8. Miller, K. (1970) Least-squares methods for ill-posed problems with a prescribed bound, SIAM J. Math. Anal., 1, 52-74.

9. Groetsch, C.W. (1984) The Theory of Tikhonov Regularisation for Fredholm Equations of the First Kind, Pitman Publishing, London.

10. Wahba, G. (1977) Practical approximate solutions to linear operator equations when the data are noisy, SIAM J. Numer. Anal., 14,651-667.

11. Hansen, P.C. (1990) Analysis of discrete ill-posed problems by means of the L­curve, Mathematics and Computer Science Division Preprint MCS-PI57-0690, Argonne National Laboratory, University of Chicago.

12. Abbiss, J.B., Defrise, M., De Mol, C. and Dhadwal, H.S. (1983) Regularized iterative and noniterative procedures for object restoration in the presence of noise: an error analysis, J. Opt. Soc. Amer., 73, 1470-1475.

13. Bertero, M. and Pike, E.R (1982) Resolution in diffraction-limited imaging, a singular-value analysis: I. The case of coherent illumination, Optica Acta, 29, 727-746.

14. Bertero, M., Boccacci, P. and Pike, E.R. (1982) Resolution in diffraction-limited imaging, a singular-value analysis: II. The case of incoherent illumination, Optica Acta, 29, 1599-1611.

311

15. Hansen, P.C. (1987) The truncated SVD as a method for regularization, BIT, 27, 534-553.

16. Golub, G.H. and Reinsch, C. (1970) Singular value decomposition and least squares solutions, Numer. Math., 14,403-420.

17. Hammerlin, G. and Hoffmann, K-H. (1991) Numerical Mathematics, Springer­Verlag, New York.

18. Golub, G.H. and Van Loan, C.F. (1989) Matrix Computations, 2nd edn., The Johns Hopkins University Press, Baltimore and London.

19. Davis, P.J. (1979) Circulant Matrices, John Wiley & Sons, New York.

20. Pratt, W.K. (1991) Digital Image Processing, 2nd edn., John Wiley & Sons, New York.

NEW IDEAS IN DATA INVERSION IN PHOTON CORRELATION SPECTROSCOPY

E R PIKE AND B MCNALLY Physics Department, King's College, Strand, London, UK

AND

P PATIN Sematech S.A.R.L. 39 Chemin de Terron, Nice, France

Abstract. The data-inversion problem in photon correlation spectroscopy, in particular the calculation of the particle-size distribution for a dilute solution of polydisperse scatterers, is well-known to be difficult due to its ill-conditioned nature. This is compounded by the fact that the second­order correlation function (of intensity fluctuations) is measured while the inversion needs to be carried out on the first-order correlations (of scattered field amplitude). Stimulated by some recent work of Ross, Dhadwal and Suh, in which they inverted directly the measured data, we have taken their idea further and present some new reduction procedures which bear further investigation.

1. Introduction

The reduction of the measured Glauber second-order correlation function G(2)(T) of light fields scattered fom dilute solutions of macromolecules, in order to recover particle-size distributions (Photon Correlation Spec­troscopy), is, in general, a.n ill-conditioned problem whose solutions depend sensitively on noise in the data.

In the monodisperse case the spectrum of the intensity fluctuations of the quasi-elastically scattered light has a Lorentzian form with linewidth inversely proportional to the linear translational diffusion coefficient of the particles, r. The photon correlation function is of exponential form with decay time Tc equal to r /2 and a simple straight-line fit to a logarithmic plot suffices to find this quantity. The radius of an equivalent spherical

313

E. R. Pike and J. B. Abbiss (eds.), Light Scattering and Photon Correlation Spectroscopy, 313-322 © 1997 Kluwer Academic PubUshers.

314

particle can be deduced by using the Stokes-Einstein relation, f = 6~Tr' where k is Boltzmann's constant, T is the absolute temperature, ." is the viscosity of the medium and r is the particle radius.

For polydisperse suspensions, however, things are much less straightfor­ward. Conventional data reduction involves using the Siegert [1] relation g(2)(T) = 1+ \g(I)(T)\2 for the normalised second-order correlation function g(2)(T) = G(2) (T)/G(2)(00) to deduce the normalised first-order correlation function g(I). In this method the value of G(2) (00) or "background" must be estimated and subtracted from the data and the square root taken to give the first-order correlation function, which is the Laplace transform of the distribution, p(f). The inversion of this transform is a so-called "ill-posed problem", in principle insoluble, and has had to be approached by various ad-hoc methods. It turned out that such phenomena had been known since the work of Hadamard in the early years of this century [2]. Hadamard, in fact, had noted the existence of problems for which the solution depended extremely sensitively on the data, thus, in the presence of noise the solution can vary wildly from one realisation to the next; he assumed wrongly that they could not occur in real scientific problems but were merely mathemat­ical curiosities and called them "incorrect" or "ill-posed". The application to physical problems began to be realised much later [3], [4], [5]. To obtain sensible solutions suitable precautions must be taken. These precautions come under the name of regularisation techniques, the most well-known of which is perhaps that due to Tikhonov [6] in 1963, which restricts the possible solutions by imposing a bound on its £(2) norm or "energy".

We discussed the reasons for this sensitivity and the underlying theory of such linear integral equations (Fredholm equations of the first kind) in the companion OSA conference to this one in Capri last week [7]. The next few paragraphs are taken verbatim from that paper and summarise the problems of data inversion in general.

"In fact, it is rather surprising that Hadamard could have been so dismissive of practical occurrences of these problems since we now find them in science almost everywhere we turn. In my group, for example, we have not only applied mod­ern inverse-problem techniques to light scattering and imaging but have recently also worked on such problems in the fields of high-temperature superconductivity, quark-gluon plasmas in the early universe, and the mass spectrum of the funda­mental particles.

In these various fields we can find two main classes of inversion algorithm. The first is that of linear methods, using various orthogonal decompositions with a large variety of smoothing windows, and the second that of non-linear methods (which are usually computationally slower) employing itera~ions of many types and invariably incorporating constraints, for example positivity of the solution, its low-order moments and even various measures of its so-called entropy. We include mathematical programming, both linear and quadratic, in this latter class.

315

Unfortunately, the literature is now vast and unruly in that many contributions put up even more new ad hoc recipes, which, of course, contain that "magic" ingredient which is what everybody has been searching for to provide the ultimate solution. Examples are usually provided demonstrating the power of this magic in selected cases.

Most methods use the least-squares measure of fit in L(2) spaces but more re­cently, so-called "global variation" methods use an L(l) norm (the modulus of the difference rather than the square) either on the function or on its slope distribu­tion. Recent work using this latter technique has produced some really spectacular results which even surprised (for a while!) your eternally skeptic authors [8].

In spite of such undoubted progress in tailoring the recovery method to the expected form of the solution, the sad truth is, however, that there is no way to recover uniquely the missing components in the data and that every example of such a method can always be countered by an opposite example where the magic utterly fails.

We can therefore agree with Hadamard that these inverse problems are, in a strong sense, "incorrect", and that there are many solutions (in fact an infinite number) which fit the measured data to within the noise errors and which are only distinguished by the method chosen to perform the inversion" .

2. A new method for pes

In the present paper we will substantiate the above conclusions. We are not concerned particularly with yet more "shoot-outs" between recipes for inversion of a Fredholm equation of the first kind but take up some re­cent work of Ross, Dhadwal and Suh [9] ip which they pointed out that there is a second route to the solution of the pes problem which seems to have been missed in all previous methods. Using this alternative method to reduce experimental data from monodisperse suspensions of polystyrene latex spheres, they achieved much higher accuracies in the calculated par­ticle radius, up to a factor of 10! Such a claim obviously needs to be taken seriously. This is apparently not so much a new "magic" recipe for inversion problems hut indicates that the solution, in this particular problem, can be found by inverting before taking the square root of the data. This Laplace inversion can still be done using one's own favourite method. In ref [5] a non-linear, positivity constrained, iterative implementation of Tikhonov's (ref [6]) method was chosen.

The new method uses the fahltung theorem for the Laplace transform as follows.

= 1000 p(x) exp(-xr) dx 100 p(y)exp(-yr)dy

316

which, by the change of variable x + y = z

= ioo 1000 p(y)p(z - y) exp( -zr) dydz = 1000 p(2)(z) exp( -zr) dz

where p(2)(Z) = I: p(y)p(z - y) dy

ie the inverse Laplace transform of the data minus the background is the convolution of the distribution with itself. To obtain this result we have extended the limits of the integrals by using the facts that p(y) = 0 for y < 0 and p(z - y) = 0 for z < y. The authors of [9] now find the dis­tribution function by deconvolving p(2). They accomplished this, in fact, by a non-linear regressive fitting of a six-point histogram for p(r). They are not explicit, however, about the treatment of the background. One pre­sumes that some effort was made to guess it and subtract it from the data before the Laplace inversion was performed. Let us assume that the resid­ual error in the background estimate is € whichever method is used. Since the estimate of g(2) typically will fall monotonically to € while that of g(1)

will fall monotonically to €1/2 the relative residual error in the transformed data will be much less in the first case than in the second. For example a background in error by one per cent in the estimate of g(2) turns into a background error in the estimate of g(l) of ten percent. One also gains a bit more since the support of the transform in the first case is twice that of the second which aids accuracy.

To investigate further we note that the deconvolution can be done by using the fahltung theorem for Laplace transforms, namely,

£, I: p(y)p(z - y) dy = £,p(y)£,p(y)

where £, denotes the Laplace transform. Thus:

We see that the particle-size distribution is the inverse Laplace transform of the square root of the Laplace transform of the inverse Laplace transform of the data minus its estimated background! We seem to have gone around in a circle since, by cancelling out the two transforms under the square root, this is nothing but the original method! The fact that the square root of the background error comes back into the calculation again merely shows that the deconvolution of p(2) is also an ill-conditioned problem. Since, however, the method by which one performs the deconvolution should not obviously affect the result we have failed to find the advantage claimed for the new

317

method. One can hardly dispute the experimental results and we can only suppose that the circle has been broken implicitly after the direct inversion of the data by ignoring the part due to the background, .c(-l)(€) outside the estimated support of the convolution.

To test the method further we have written a computer programme to simulate particle-size data which consists of a small number of narrow peaks with selectable positions, amplitudes and widths. The photon corre­lation function can have selected values of G(2(O) and an added noise level. The programme then reduces this data using both our previous favourite method (singular system analysis (SSA) discussed in ref [4]) and the new method of direct inversion of the data, again using SSA, followed by the deconvolution described above, implemented in this programme by the fast Fourier transform. The SSA method is linear and uses a precomputed sin­gular system of the kernel.

Since, in our implementation, both of these methods are fast we have also used their outputs as inputs to some non-linear, least-squares methods including Levenberg-Marquardt iteration with positivity constraints and downhill simplex. The idea here is to try to "fine-tune" the linear fits by a'positivity constrained non-linear fit by starting close to the solution so that the number of iterations will be drastically reduced and, hopefully, spurious minima in the X2 surface, the bane of non-linear fitting algorithms, will be avoided. To overcome this latter problem we would also like to try the Metropolis (simulated annealing) method but have not accomplished this yet. Regularisation in all cases is achieved by discretisation, using the abscissae values chosen by the SSA algorithm.

This work is still in progress but it seems already clear that direct inversion of the data does indeed improve the reconstruction significantly in most of the cases we have tried. Typical examples are shown in figure 1.

It is also clear that the non-linear methods do suffer from severe spurious-minimum problems since their outputs always depend on whether the new or old inverse methods or indeed any other methods, even those of other non-linear methods, are used to find a starting point. We should emphasise that in the absence of spurious minima the result using the non­linear methods should be independent of where the starting point is chosen, save for the amount of time taken to reach it. This behaviour, therefore, indicates that the X2 surface has shallow spurious local minima in abun­dance like a crater-strewn moonscape, which as we stated above, is no more than we should expect for a problem with an infinity of solutions within the noise level, even when constrained to be positive.

318

Figures

0.5

o 0.01

0.2

o 0.01

I!\

- " / r\ ~

0.1 1 10

(a) The SSA inversion (original method)

1\ ,. / \

\ \

1\ 0.1 1 10

(b) The SSA inversion (new direct method)

Figure 1. An example of the difference between solutions obtained by (a) Taking the square root of the data minus an estimated background followed by Laplace inversion. (b) Taking the Laplace inversion of the data minus an estimated background followed by Fourier deconvolution.

319

3. Indeterminacy of positive solutions

To illustrate this statement graphically, we give an example of an inverse problem in optical imaging [10], which we also used in ref [7]. In this example the kernel of the first-kind Fredholm integral equation is the point-spread function of an incoherently illuminated square aperture and has the form:

K( ') = {Sin[c(x - xl)]}2 X,X (') 7rX-X

(1)

with c = 20.0, which produced an image with an effective support on [-1.5,1.5]. The following numerical experiment was performed. A positive object was constructed, the form of which will be irrelevent to the dis­cussion, using the first 58 singular functions of the kernel, and the image computed. The first 12 of these functions were taken to be transmitted accu­rately. The unknown coefficients of functions 13 to 58 were then calculated by quadratic programming in two ways: both constrained to have the cor­rect coefficients for the first 12 functions in the singular-function expansion, both constrained to be positive, but one to have the minimum L(2Lnorm and the other to have the minimum L(ILnorm (to a close approximation) of all such solutions. These two procedures produced significantly different reconstructions. (The existence of unique solutions to both of these con­strained minimisation problems has been proved recently by Borwein and Lewis [11].)

The L{1Lnorm-minimised solution was then taken and put though the same imaging process as the original object. In the reconstruction, a further constrained L(I) minimisation regenerated the original object exactly while the L(2Lnorm-minimised reconstruction was very poor.

When the same process was repeated for the L(2Lminimised object, the L(2) reconstruction reproduced the original object exactly, while the L(I)

reconstruction was very poor. These results can be seen in figure 2. Both of the reconstructions have

enforced positivity and both fit the data to exactly the same degree. How we arrived at them can now be forgotten, they both have the same band­limited images and there is no way to tell whether either was the real object. In fact any mixture of the two, going from a times one plus 1 - a times the other for 0 ~ a ~ 1, as is shown in figure 3, is also equally possible. Most probably none of all these solutions was the real object since, of all the objects with the same band-limited image, even the infinite set of figure 3 is of very small measure.

The moral is that there are many ways to guess what you don't know and we have no information in any particular case whether an arbitrary positive object is anywhere near either of these minimum-norm solutions.

320

C\.oIoJD~-

Singular Value Spectrum / Recorded Data

Ilg - J( 1£(1) 112 = 1.25 * 10-11 ! IIg - J( 1£(2) 112 = 1.27 * 10-11

u­" .

Transform of reconstruction

KiL,.,l

Positive L(1) reconstruction

"r--~~-~~~-...,= .. ~_----'_

TSVD reconstruction to U5

1 L(2)

Positive L(2) reconstruction

Figure 2. Process showing the route taken in calculating first the TSVD reconstruction (with unwanted negative regions) and then the positive L(2) and L(1) minimum norm reconstructions. Finally, the transforms of both of these reconstructions are shown, and a very small X2 value for each is noted.

2.5 -

2

1.5

0.5

-0.5

o

.... ", . .tIr----~--~--~-~-

...... '

£(1) norm versus a

150

Reconstructions -

£(2) norm versus a

1.67 .. ... 1.33 -_.-

1 _._.-0.667 _._.-0.333 ..... .

1.0

321

Figure 3. The 3D graph shows a subset of the infinity of solutions that minimise the cost function {Cl' * L(1) + (1 ~ Cl') * L(2)} as Cl' is varied from 0.999 to 1.0. This small

range of Cl' was chosen because the tiny numerical change in the L(1) norm meant that for 0.0 :::; Cl' :::; 11.999 the solution produced was almost identical to the minimum L(2)

solution. It is stressed that all these solutions fit the data equally well to within the noise level.

322

The positivity constraint, unfortunately, hardly improves the situation, in­deed it could make things worse since it takes a lot of computing time to give a false confidence in an apparent wide-band solution which is only, in fact, known to within the band limit.

4. Conclusions

The fastest and most promising technique for pes seems, at the moment, to be to use the new direct-inversion method to find a starting point for the Levenberg-Marquardt multi-variate constrained non-linear fit, but so many different distributions fit the data to the same order of accuracy that the extra non-linear ''fine-tuning'' iterations may not be worth the effort. Linear and quadratic programming methods methods take a great deal of time to no apparent advantage. We strongly suspect that even if the Metropolis algorithm were used to find the minimum of all the minima, the result would depend sensitively on each particular noise realisation and no real advantage would accrue, but we shall test this in future work.

5. Acknowledgements

This material is based upon work supported in part by the US Army Re­search Office under grant No DAAH04-95-i-0280

References

1. A J F Siegert, MIT Rad. Lab. Report No 465, 1943 2. J Hadamard, Princeton Univ. Bulletin, 13,49-52, 1902 3. F John, Amer. J. Math., 63, 141-154, 1941 4. AN Tikhonov, Dokl. Akad. Nauk SSSR 39,195,1943 5. C Pucci, Atti Acad. Naz. Lincei Rend., 18,473-77, 1955 6. AN Tikhonov, Dokl. Akad. Nauk SSSR 153, 49,1963 7. E R Pike and B McNally, Appl. Optics, Special Issue, Oct 1997 8. A H Lettington and Q H Hong, J. Modern Optics, 42, 1367, 1995 9. D A Ross, H S Dhadwal and K Suh, SPIE Proceedings Orlando, April 1994

10. G de Villiers, B McNally and E R Pike, to be published 11. J M Borwein and A S Lewis, Mathematical Programming, 57,49, 1992

TEMPEST IN A TEAPOT - SURFACE AND VOLUME

TURBULENCE IN A CLOSED SYSTEM

Abstract.

WALTER I. GOLDBURG AND CECIL CHEUNG Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA

Turbulence can be generated by oscillating a grid in a volume of water. The turbulent flow thus created has minimal mean shear and mean flow. We have studied the motion of small floating particles on the surface of such a turbulent fluid, as well as the velocity fluctuation in the volume of the fluid. For the surface motion, the standard deviation of the velocity difference of a pair of particles separated by a distance R can be fitted by loge Rj Ro). That is in contrast to the volume measurements where the data are fitted by a power law. Nevertheless, our result is not explainable by present theories.

323

E. R. Pike aM J. B. Abbiss (etis.), Light Scattering aM Photon Correilltion Spectroscopy, 323. © 1997 Kluwer Acodemic Publishers.

DIFFUSING PHOTON CORRELATION

Abstract.

A. F. KOSTKO St. Petersburg State Academy of Refrigeration and Food Technologies, Lomonosov str. 9, St. Petersburg 1 9 1 002, Russia

A review of general principles, recent achievements and some peculiar­ities and applications of diffusing photon correlation (diffusing-wave spec­troscopy) is presented, including results of the first papers published before 1987. Attention is given to the unsolved problems of light transport in strongly scattering (milk-like) media. Tests of the photon diffusion model are discussed.

325

E. R. Pike and J. B. Abbiss (eds.J, Light Scattering and Photon Correlation Spectroscopy, 325-340. © 1997 Kluwer Academic Publishers.

326

DIFFUSING PHOTON CORRELATION

1. Introduction

The traditional use of the photon correlation technique deals with mea­surements under the conditio~s of single scattering (Cummins and Pike, eds., 1974). Then a scattering medium remains transparent for light and its diameter is far less than the mean free path of a photon. In the opposite limit of high turbidities, two limit cases can be considered: 1) very high absorption (e. g., coffee, oil) and 2) very low absorption (e. g., milk, latex). The second case of a turbid medium with low absorption corresponds to high multiplicities of scattering. Such kind of turbid media may be called strongly scattering media or milk-like media. It is a new interesting object for photoll correlation techniques.

Our group of researchers from St. Petersburg began developing the diffusion approach to the photon correlation 15 years ago (Ivanov and Kostko, 1983; Ivanov et al. 1985a, 1985b, 1986; Pavlov, 1988). During first 5 years our group was alone, but from 1987, after the well-known paper by Maret and Wolf, numerous publications on the theme have appeared. Now this field of researches is well known as "diffusing-wave spectroscopy" thanks to th<> paper of the group of american authors (Pine et al., 198X).

One of the aims of this report is not only to review the achievements of diffusing photon correlation but also to show that a photon diffusion process in milk-lik<> media is more complicated than it is described by usual theories. Tha.t r.omplir.ar.y is revealing in all our experiments. In the papers of other authors tl1is problem is p.ot considered and the cause of such situation is not dea.r now. Pf'fha.ps it is bound with the essential differences between our experimental geometry and usual geometries of diffusing-wa.ve spectroscopy

327

(we prefer to use a narrow laser beam but not one widened by a lens). But it is only one of possible hypotheses which need to be discussed.

2. First experiments and their theoretical background

The first paper on diffusing photon correlation had been published by our group in 1983. Since that time, we use the narrow beam geometry cor­responding to common single-scattering measurements (see Figure 1). A cylindrical cell is filled with a strongly scattering medium which is a latex usually. The medium is illuminated by a narrow laser beam directed along the radius and focused on the medium boundary. Such geometry, which is ordinary for single scattering, will be called further as the cylindrical geom­etry. Of course, in the single scattering case the laser beam is focused at the axis of the cylinder. But it is the only and insignificant difference between the geometries. In substance, we have first inserted a milk-like medium in­to a common photon correlation spectrometer. Then we have found that the autocorrelation function of emerging light can be measured even under such conditions and can be characterized by the first cumulant r m, which may be interpreted also as the spectrum halfwidth of multiply scattered light. The results are shown in Figure 2. The measured values for multiple scattering are one or two orders of magnitude larger than single scattering values which are too small for plotting in the same graph. In addition, r m

shows a decrease contrary to the case of single scattering.

Figure 1. The experimental setup.

20

rm. kHz *

15 * *

* * * 10

* 2 * * ++ + + + *

5 + * ++ + + + + + * + + :3 + + + + + :t :j:

Figure 2. First cumulant r m of the autocorrelation function of multiple scat­tering in latexes: 1 - R = 280 nm, I = 0.27 mm ( r(900) = 290 Hz )j 2 - R = 770 nm, I = 0.21 mmj 3 - R = 770 nm, I = 0.24 mm.

328

To explain these hitherto unknown facts, we developed the approach based upon the theory of light transport through turbid media. A descrip­tion of the transport theory may be found in the well-known book by Ishi­maru (1978). The light transport theory is assuming that the individual scattering events are independent, the photons move along random-walk trajectories and all interference effects are negligible. In 1975 Ishimaru de­veloped the transport equation for the temporal autocorrelation function (see also Chap. 14 in (Ishimaru, 1978)). This equation contains the prin­ciple that the autocorrelation functions of single scattering events must be multiplied together to obtain the autocorrelation function for one of the trajectories. The transport equation for temporal autocorrelation function is more complicated than for cintensity and, therefore, to get any solution one must deal with essential simplifications.

In the case of the scattering on Brownian particles we can get the au­tocorrelation function for one of the trajectories:

N N

gY)(T) = Ii exp( -riT) = exp ( - t riT ) = exp( -N/fT). (1) i=l i=l

Here r is the single scattering value weighted by the form-factor and aver­aged over the scattering angle of single scattering.

Using a similar approach, Sorensen, Mockler and O'Sullivan in 1977 (see also (Sorensen et al., 1978)) suggested the formula

(2)

where

(3)

Here N is the mean multiplicity of scattering, it is derived by averaging over all trajectories of photon random-walks.

Equation (2) is valid only for Rayleigh scattering, when the particle size is far less than the wavelength. For the general case of Mie scattering, we have corrected (Ivanov and Kostko, 1983; Ivanov et al., 1985a) this equation in the form:

(4)

This Equation (4) is following from Equation (1). In our cylindrical geometry the mean multiplicity of scattering N is

varied with varying the angle <p between incident and analyzed beams. It is worth noting that in the case of multiple scattering this angle <p is not the scattering angle and it may be called the angle of observation. The mean multiplicity N is bound with the distance between the point of

329

entrance of the coherent beam and the point of exit of multiply scattered light. When the angle <p increases, this distance decreases and, therefore, the mean multiplicity N also decreases. It explains the observed angular dependence of r m'

In milk-like media N is far more than unity. So, the values of r m must exceed greatly the single-scattering values. Thus, we can observe the dye namics of scatterers in a time scale shifted towards small times. It is an interesting characteristic feature of diffusing photon correlation.

If we obtain the value of N, we can study the dynamics of scatterers by measuring r m nearly as in the case of single scattering.

Serious difficulties may appear when one try to estimate N. The ap­proach used by Sorensen et al. in 1978 (see also (Boe and Lohne, 1978)) is incorrect in the cases of strongly scattering media. In the paper (Ivanov and Kostko, 1983) we suggested for the first time to estimate N using a simplified model of photon diffusion. It permitted us to obtain the qual­itative agreement between the theory and the data. Later we formulated (Ivanov et al., 1985b) the improved diffusion approach to calculation of the mean multiplicity N.

The diffusion theory is valid under the conditions:

D» l* = --------::== 1 - Wocos ()

(5)

where D is the diameter of the scattering sample, l is the mean free path of photons, l* is the transport mean free path, and Wo is the albedo of single scattering. In our experiments with nonabsorbing media this albedo is equal to unity (Wo = 1). For big particles l* may be an order of magnitude longer than l.

The mean multiplicity is determined as

JnJ(r,n)dn N = J J(r,n)dn . (6)

Here J( r, n) is the distribution of light intensity over the multiplicities u. This distribution may be found from the following diffusion equation

(7)

with the boundary conditions

a! J +0.7l* or = O. (8)

This Equation (8) is related to the points on the medium boundary only.

330

The more rigorous but more complicated diffusion equation has been derived by Pavlov (1988) from the transport equation by Ishimaru. But the solutions of two diffusion equations become different only at the tail of the autocorrelation function. Since we are interested in the first cumulant only, the diffusion equation by Pavlov will not be considered here.

It is clear, that N is dependent upon the concentration of scatterers. So, the diffusion photon correlation technique permits to extract the infor­mation about the particle size and the concentration from the measured values of r m •

This concept had make possible for us to get the patent of USSR (Ivanov et at., 1986) on the method of particle sizing and measurement of particle concentration by photon correlation technique under the diffusive regime conditions. One year after, the well-known paper by Maret and Wolf (1987) was published. They studied photon correlation of light backscattered from a slab of a strongly scattering medium. This paper became the new push to intensive and numerous researches on diffusing photon correlation.

3. The slab geometry in diffusing photon correlation

Maret and Wolf (1987) used a diffusion approach for the interpretation of data, which is similar to our one (Ivanov and Kostko, 1983). The paper (Pine et at., 1988) gave the name "Diffusing-wave spectroscopy" to this new field of photon correlation. It started a sequence of numerous publications on the theme, in which the slab geometry was used. Due to these papers the slab geometry became well-known. In this geometry the scattering sample is a slab between two parallel planes. It is illuminated by a plane wave. One must use a wide laser beam for it. In such geometry the diffusion equation has the exact solution in elementary functions. '

When light has passed through a slab, the mean multiplicity has the following value (being the leading term for L --+ 00):

where L is the slab thickness.

1 L2 N=--

2 11* (9)

The slab geometry leads to a simple interpretation of data. It makes possible to analyze the whole shape of the autocorrelation function but not only the initial slope defining the first cumulant.

The slab geometry appeared to be convenient to develop numerous ap­plications. The detailed review of them may be found in (Weitz et at., 1993) and they will not be discussed here. Only two of them will be mentioned. Among the most significant results, achieved using the slab geometry, one must select the new physical possibilities of studying Brownian motion at

331

small times (Weitz et al., 1989) and of studying bubble dynamics III a foam (Durian et al., 1991).

The first possibility is based on the time-scale shift towards small times of the autocorrelation function of multiply scattered light. It takes place due to high values of the mean multiplicity in diffusion photon correlation experiments.

In the case of a foam the analyzed light beam consists of photons, which have traveled along different random-walk trajectories in the whole sample volume. These photons bring the averaged information about the dynamics of all bubbles in the volume. It is impossible in the case of single scattering.

The slab geometry has some disadvantages. One must have a powerful laser to use the slab geometry because the laser beam is widened as a rule. Its power is spread upon the plane. Also, it is not convenient for use if the scattering regime is changing from single- to multiple-scattering during one sequence of experiments. As an example, we may note the study of critical opalescence. The finite area of illuminated boundary of the slab causes some error. The mean multiplicity in the slab geometry can't be varied easy since it is fixed by the slab thickness.

Moreover, when the wide laser beam passes through a slab with a turbid medium, it is not easy to recognize visually whether the scattering regime is really diffusive or not. In contrast to this, a trace of the narrow laser beam passed through the medium is the true signal of a mean scattering multiplicity which is insufficient to apply the diffusing photon model. It is very important for any person who deals with diffusing photon correlation.

4. Tests of the diffusion model

In this section the closer experimental examination of the used diffusion model will be considered. It will be shown that these tests reveal some unsolved problems in the existing today photon diffusion theories.

In the cylindrical geometry applied by us, a such simple form ~or N as in Equation (9) is incorrect. The cylindrical shape of the medium boundary makes it impossible to solve the diffusion equation. However, we can obtain some results without solving the equation. We can get the concentration dependence of the m~an multiplicity of scattering (N) in the form:

(10)

where c is the concentration of scatterers. The function F( rp) is a geometric factor which can be determined in experiment, but its theoretical value is unknown. According to usual concept, this factor must not depend upon the properties of the strongly scattering media.

332

From Equations (4) and (10) we have

(11)

where the function A( <p) is bound with the particle size and the wavelength. But it is not dependent upon the particle concentration.

However, the nature of photon diffusion in milk-like media appears to be more complicated than it may be expected. Even in the first experiments we observed the evidence that the form (11) is not exact. The data required to add a new term to the form (11) as it is shown in Equation (12):

(12)

But to be sure enough, the experimental accuracy had to be improved. We have performed the special experiment using a latex with small

particles to avoid the effects of particle settling. We carefully took into account the distortion of the autocorrelation function by afterpulses. The results of our investigation were published in (Ivanov et at., 1989). Up to now it is the single paper concerning the additional term r o.

The data are shown in Figure 3. The values of r m correspond to three concentrations of the same latex. These suspensions were produced by a consequence of dilutions. So, the ratios of the concentrations are known with good accuracy. Owing to it, new interesting effects may be observed.

According to Equation (11), one of these angular dependencies in Fig­ure 3 may be transformed to the other by multiplying by the square of the corresponding concentration ratio. If we plot the values of one of these dependencies on the abscissa and the other two are plotted as ordinates, we must have the straight lines crossing the origin.

Let us turn to Figure 4. The middle values from Figure 3 are plotted as the abscissa and two others - as the ordinates. The dashed lines have the slopes following from the diffusion equation. They are equal to the square of the concentration ratios. We can see the shift of the data from these lines. So, Equation (11) is not valid. The solid lines correspond to Equation (12), fe) is a fitting parameter. The term r o is nearly an order of magnitude larger than the single scattering value ([(90°) = 920 Hz).

The value of Cn has the same trivial temperature dependence as the single-scattering value r (Ivanov and Kostko, 1983). It is determined by the temperature dependence of the viscosity. The main term in Equation (12) has the same temperature dependence since it is simply proportional to the absolute temperature divided by the viscosity as the single-scattering value [( 900 ) is. Thus we can see that the additional term in Equation (12) must have the same dependence on the viscosity as the main term.

80

1m. kHz ......

'" ** '" SO ...

* ... 0

E>

* I;) ~O

+++ * * +++ 0

* + + * E>

20 ++ * ++ ... 0

++t

0 0 50 100 HiO

~, grad

Figure 3. The angular dependences of the first cumulant (f m) of the autocorrelation function for light multiply scattered from aqueous suspensions of 160 nm diameter polystyrene spheres at t = 20 DC. Volume concentrations (c) are: ((;)) Cl = 4.4 10-3 ,

(*) C2 = 3.2 10-3 , (+) C3 = 2.2 10-3 •

333

80

Iml' 1m3, I

kHz I

so

'10

7/ ,,/ 20

0 0 20 '10 60 80

ro rm2t kHz

Figure 4. The same data as in Figure 3, plotted to test Equations (11) and (12).

So, this analysis of the data has evidenced that the additional term fo in Equation (12) exists. It is a constant within the studied range of concentrations. Up to now its nature is not clear.

One may ask: why it is important to study the additional term'! When the mean multiplicity is very high, the general term greatly exceeds the additional one and we can neglect it. But we must take into account that in such conditions the intensity of multiply scattered light is low. So, the measurement of an autocorrelation function may be too long, especially when a low-power laser is used. That's why the diffusion photon correlation measurements are more convenient when the mean 1l1ultiplicity is not too high. However, in such conditions the additional term in Equation (12) is not negligible.

The existence of the additional term is an experimental fact. But in usual theories it is not appearing. This leads to the question: what equation from our list requires the improvement. Various possible answers may exist. Two of them will be mentioned here.

If we suppose that from the diffusion theory such additional term cannot be obtained, then it must be added to the right side of Equation (4) as a non-diffusion term. This point of view we were supported in (Ivanov ft al., 19>-;9).

But recently we received some experimental results (Kostko and Pavlov, 1996a, 1996b) which give us the arguments to change Equation (10) describ-

334

ing the concentration dependence of the mean multiplicity N. Let us very briefly discuss them.

We tried to test the diffusion approximation in a more simple case of the spherical samples placed in a common apparatus for studying the light scattering (see Figure 1). In pursuing this aim, however, it is not neces­sary to measure the temporal autocorrelation function by the technique of photon correlation. The complete information about the validity of the dif­fusion approximation for light transfer in a strongly scattering medium can be derived from thorough measurements of the light intensity distribution on the scattering medium boundary. Such experiments can be more precise and reliable.

In the spherical geometry we can easily get an analytical solution of diffusion equation for the multiply scattered light intensity. This diffusion equation for the case of a point source within a non absorbing medium (Po is the power of the source positioned at the point fo) may be written as:

(13)

with the boundary conditions for a spheric sample

II = 0 , r=Rm (14)

where Rm is the radius of the expanded boundary: Rm = Dm/2 and

Dm = D + 1.41*. (15)

It describes the complete intensity' of the light which diffuses from a point source. We measured the distribution of the scattered light intensity on the spheric sample boundary (Kostko and Pavlov, 1996a, 1996b). This distribution we compared with the analytical one. The well-known solution can be presented in a linearized form vs cos <p - see Equation (16):

(16)

6 = ~ (17) Dm-~

Here ~ is the distance between the expanded boundary and the position of the point source.

Such linear dependence of the data, shown in Figure 6, demonstrates that the diffusion approximation is valid quite well. In the insert we can see that the straight line don't cross the point minus unity. It is caused by

1~ ----------------------- -----

0.0

0.8 O,,[Zl ~ -1.2 -1.0 -0.8 'L 0.6

~ 0."

D=21.5 mm

cos r:p

Figure 5. Multiply scattered light inten­sity for a spherical cell. Solid straight line fits the data.

1.0 .,..... ........ !I-iIo-.=;j.F;.::-:;:-~;-y -~ :. --:- :. - - - - - ; : *~,: : 1*' :

0.8 ---------r ---------i-- ----\--j , , , , , , ~ ~O.6 --------- ~---------~---- ----

<0., o ()

~M

, , :0:021.5 mm: , , ,

_________ L _________ L ________ I , , , , , , , , , , , , 0.2 ---------~---------~------ - : , , , , , , , , ,

335

Figure 6. Multiply scattered light intensity for a spherical cell. Dashed curve - t::,. = 1.7 1* , solid curve -t::,. = 2.8 mIll = 5.11*.

nonzero value of 8. It corresponds to some nonzero dipping of the effective point-like source of photons, which appears inside the med'ium when a nar­row laser beam penetrates into it . Such dipping of the effective source in the standard model is usually assumed as about one transport mean free path 1*. When the mixed boundary conditions (see Equation (8)) are sim­plified to the form used here, the expanded boundary is taken into account. Adding the distance (0.71*) between the real and the expanded boundaries, we should expect the value ~ = 1. 71* (see the comment to Equation (17)). However, in Figure 6 the dashed curve, obtained with ~ = 1.71*, runs far away from the data (for <p > 100°). Indeed, at the angle <p = 165° the measured intensity value is four times less than the value calculated with the standard ~ = 1.71* = 0.9 mm. The solid lines in these two figures correspond to another value of ~ considered as the fitting parameter.

Contrary to the standard diffusion model, our result :

~ = 5.31* ± 0.7l*. (18)

The position of the pointlike effective diffusing-photon source appears to be unexpectedly deep, and this phenomenon has no theoretical explanation today. One may suspect that the internal reflection (Zhu et al., 1991) on the medium boundary may be the cause of this phenomenon. Indeed, the standard value of the distance 0.71* between the real and the expanded boundaries (as well as the factor 0.71* in Equation (8)) is dependent upon the indexes of refraction of the scattering and the surrounding media. But our calculation (closely following (Zhu et al., 1991) and in the agreement with the result of (Durian, 1994), shown there in the line 3 in Table 1) give

336

the value (0.71 ± 0.02) 1* for this distance. We obtain that the standard diffusion model for our experimental conditions corresponds to the value ~ = 1. 71*. So we cannot explain the phenomenon of the deep position of the pointlike effective diffusing-photon source by the internal reflection.

It is worth to mention that the power of the used He-Ne laser was sufficiently low (about 10 m W) and it could not cause the detectable heating of the sample or any other nonlinear effect. In our opinion, in the studied phenomenon the initial stage of the light beam randomization demonstrates its complicacy.

We think that the existence of the additional term in Equation (12) may find its explanation by the observed phenomenon. But it is too early to discuss it now.

We have developed a method of determination of the effective source dipping by the intensity measurements. It can give the useful additional information for diffusing photon correlation. Recently Solov'ev and Pavlov have obtained for the spherical geometry the exact solution (in elliptic in­tegrals) for the initial slope of the autocorrelation function. The geometric factor in this solution appears to be quite sensitive to the value of the ef­fective source dipping. Using the intensity measurements together with the photon correlation, we expect to exploit the spherical geometry successful­ly. This geometry may become one of the most convenient ones for diffusing photon correlation.

Such geometries as the cylindrical and the spherical ones with narrow laser beam and goniometer are convenient for: 1) easy varying the mean multiplicity of scattering; 2) performing experiments with the transition from single to multiple scattering; 3) using low-power lasers and common single-scattering apparatus. These advantages are absent in the case of the slab geometry.

5. Specific peculiarities of the metod

The measurements of the diffusing photon correlation have some specific peculiarities. In such measurements the signal to noise ratio is far less than in the case of single scattering. Therefore the storing time is more long than usual and the problem of the afterpulses must be considered more carefully. It takes place due to the simultaneous decrease of the intensity and of the correlation time when the multiplicity of scattering is increasing.

There are two main methods to solve the problem of afterpulses. One can try to improve measured autocorrelation function "in first few channels of the correlator. But this way is not so simple as it seems and may pro­duce essential errors. It requires a control of a man and must not be fulfilled automatically by a computer only. The second method is to split the scat-

337

tered beam and to measure the crosscorrelation function. The result will be very good, but the apparatus complicacy and the loss of intensity will be a trouble.

6. Some applications

In addition we can briefly illustrate abilities of diffusing photon correlation by the examples of critical opalescence and of particle motion monitoring in evaporated latex suspensions.

As the critical point is approached, the scattering multiplicity continu­ously grows, unavoidably reaching very high values. When the cell diameter is not too small, the diffusion regime of light propagation is reached . We calculated (Ivanov et at., 1985a) the temperature dependence of f m for this case and measured it (Ivanov et al., 1993) in the experiment with the aniline-cyclohexane critical mixture. The results of the calculation and the experiment are shown in Figures 7 and 8. The data in Figure 8 demonstrate the essential growth of f m due to the continuous increase of the scattering multiplicity as the critical point is approached. This growth of f m predicted by our calculation was observed for the first time.

20

rm. r 15

. I

10 I

(

(

5 /

I f

--------------5 -4 -3 -2

'g(T/T.-l)

Figure 7. Dimensionless first cumu­lant of the autocorrelation function cal­culated for critical opalescence. Dashed line - r for single scattering (Kawasaki), solid line - r m for photon diffusion (our calculation (Ivanov et al., 1985a)).

8.0

r. kHz

5.0

... 0 ..

2.0

O.O_+5.5~~_5r.O~~_4T'.5~~_4r.O~_"'3r-.-.S~_~,o~_'12.5 Ig(T/Tc-l )

Figure 8. First cumulant of the autocor­relation function for critical mixture ani­line-cyclohexane. Data (dots) in compar­ison with calculation : dashed line - r for single scattering (Kawasaki), solid line - r m for photon diffusion.

It is worth to note that the values of fm on the plateau in Figure 8, ac­cording to our theoretical analysis, must be quite sensitive to the combina­tion ofthe pair of small critical exponents (of the anomalous dimensionality and of the viscosity). Our analysis, also, leads to another interesting result.

338

20 40 80 100

t. min

Figure 9. Diffusing photon correlation in the evaporated polystyrene latex suspension drop. 1 - first cumulant of the autocorrelation function of the diffusive scattered light. 2 - backscattered light intensity, 3 - transmited light intensity, 4 - volume con~en­tration.

The value of r m in the immediate vicinity of the critical point must be proportional to the wavelength powered by minus seven! It becomes clear if we take into account the exponent -4 for r and double exponent -2 for turbidity.

Another application of diffusing photon correlation was produced by us in the experiment with an evaporated latex suspension. During more than an hour, every five minutes we measured simultaneously the autocorrelation functions and the mass of the evaporated latex drop, which was placed on the glass plate on the analytical balance pan. The linearly polarized laser beam, directed vertically, was focused on the boundary of the drop and was illuminating it from below. The narrow beam of backscattered light was analyzed in crossed polarization. The intensity and the autocorrela­tion function of light in this beam were measured . The transmitted light intensity was measured too.

The results are shown in Figure 9. From the beginning, the first cumu­lant of the autocorrelation function don't change significantly during a long time. It is an evidence of the continuation of particle motion. But when the concentration reaches the value of 0.74 corresponding to close packing, the value of the first cumulant of the autocorrelation function falls down. So, the motion of particles stops immediately and the latex becomes transpar­ent for light (see curve 3 for the intensity of light passing through the drop). To our opinion, it is the evidence of the liquid-crystal transition in the latex suspension during its evaporation.

339

7. Summary

The diffusing photon correlation is rapidly developed now as the extension of the traditional dynamic light scattering to the regime of high multiplic­ities of scattering. The history of the method is somewhat longer than it is usually described in the most of recent papers. The general principles of diffusing photon correlation (or diffusing-wave spectroscopy) and the results of corresponding experiments were published before 1987. The sig­nificant results in application of diffusing photon correlation were received in recent years using the slab geometry. New physical possibilities of the method were found. But there are existing other convenient geometries be­ing useful for diffusing photon correlation. The geometries with a narrow laser beam and a goniometer have some advantages. The development of the diffusing photon correlation method for such geometries leads us to the evidence of new features of the photon diffusion. 80, the new phenomenon of the deep position ofthe pointlike effective diffusing-photon source, which appears inside the medium when a narrow laser beam penetrates into it, was discovered recently. The explanation of obtained experimental results requires a further development of the theory.

Acknowledgements

Thanks are due to the members of our research group D.Yu. Ivanov, V.A. Pavlov, 8.8. Proshkin, and A.V. 8010v'ev who are my collaborators in the works reviewed here. I am specially grateful to V.A. Pavlov for useful discussions and help.

References

BfIle, A. and Lohne, O. (1978) Dynamical properties of multiply scattered light from independent Brownian particles, Phys. Rev. A 17, 2023-2029.

Cummins, H.Z. and Pike, E.R., eds. (1974 ) Photon Correlation and Light Beating Spec­troscopy, Plenum Press, New York.

Durian, D.J., Weitz, D.A., and Pine, D.J. (1991) Multiple light-scattering probes offoam structure and dynamics, Science 252, 686-688.

Durian, D.J. (1994) Influence of boundary reflection and refraction on diffusive photon transport, Phys. Rev. E 50, 857-866.

Ishimaru, A. (1975) Correlation functions of a wave in a random distribution of stationary and moving scatterers, Radio Sci. 10, 45-52.

Ishimaru, A. (1978) Wave Propagation and Scattering in Random Media, Academic Press, New York.

Ivanov, D.Yu. and Kostko, A.F. (1983) Spectrum of multiply quasi-elastically scattered light, Opt. Spektrosk. 55, 950-953 [Opt. Spectrosc. 55, 573-575].

Ivanov, D.Ytl., Kostko, A.F., and Pavlov, V.A. (1985a) Behavior of the width of the multiple-light-scattering spectrum in the immediate vicinity of the critical point, DAN SSSRI. 282, 568-571 [Sov. Phys. Dokl. 30, 397-,-399J.

Ivanov, D.Yu., Kostko, A.F., and Pavlov, V.A. (19815b) Revealing the effects of the dif-

340

fusion approximation in spectra of multiple scattering of laser radiation, in The III All-Union Conference on the Propagation of Laser Radiation in Disperse Media, part II, Institute of Exper. Meteorology, Obninsk (USSR), pp. 71-74.

Ivanov, D.Yu., Kostko, A.F., Pavlov, V.A., and Ryzhov, V.A. (1986) The method of determination of. the mean size and concentration of suspended particles, Patent USSR No. 1990599 (priority from 11 November 1986).

Ivanov, D.Yu., Kostko, A.F., and Pavlov, V.A. (1989) Multiple scattering spectra in strongly scattering media: Diffusion and non-diffusion contributions to spectrum halfwidth, Phys. Lett. A 138, 339-342.

Ivanov, D.Yu., Kostko, A.F., and Proshkin, S.S. (1993) Critical opalescence investigation in the binary mixture aniline-cyclohexane by dynamic multiple light scattering, in 19-th European Conference on Thermophysical Properties, Lisboa, Portugal, pp. 377-378.

Kostko, A.F. and Pavlov, V.A. (1996a) Deep position of the effective diffusive source in diffusing-wave spectroscopy, Izv. Russ. Akad. Nauk, Ser. Fiz. 60(3), 162-167.

Kostko, A.F. and Pavlov, V.A. (1996b) Location of the effective diffusing-photon source in a strongly scattering medium, in Photon Correlation and Scattering, 1996 OSA Technical Digest Series 14, Optical Society of America, Washington, pp. 71-73.

Maret, G. and Wolf, P.E. (1987) Multiple light scattering from disordered media: The effect of Brownian motion of scatterers, Z. Phys. B 65, 409-413.

Pavlov, V.A. (1988) The diffusion equation in the theory of multiple-scattering spectra, Opt. Spektrosk. 64, 828-831 [Opt. Spectrosc. 64, 493-495].

Pine, D.J., Weitz, D.A., Chaikin, P.M., and Herbolzheimer, E. (1988) Diffusing-wave spectroscopy, Phys. Rev. Lett. 60, 1134-1137.

Sorensen, C.M., Mockler, R.C., and O'Sullivan, W.J. (1977) Autocorrelation spectroscopy studies of single and multiple scattered light from a critical fluid mixture, Phys. Rev. A 16, 365-376.

Sorensen, C.M., Mockler, R.C., and O'Sullivan, W.J. (1978) Multiple scattering from a system of Brownian particles, Phys. Rev. A 17, 2030-2035.

Weitz, D.A., Pine, D.J., Pusey, P.N., and Tough, R.J.A. (1989) Nondiffusive Brownian motion studied by diffusing-wave spectroscopy, Phys. Rev. Lett. 63,1747-1750.

Weitz, D.A., Zhu, J.X., Durian, D.J., Gang, H., and Pine, D.J. (1993) Diffusing-wave spectroscopy: The technique and some applications, Phys. Scr. T49, 610-621.

Zhu, J.X., Pine, D.J., and Weitz, D.A. (1991) Internal reflection of diffusive light in random media, Phys. Rev. A 44, 3948-3959.

PHOTON CORRELATION SPECTROSCOPY OF

OPAQUE FLUIDS

I. K. YUDIN AND G. L. NIKOLAENKO Oil and Gas Research Institute of Russian Academy of Sciences, Leninsky prospect, 63/2, 117917 Moscow, Russia

Abstract. A new approach for polydisperse particle sizing in opaque fluids, using

the technique of photon correlation spectroscopy, is proposed. Some model opaque dispersions have been measured. The aggregation kinetics of asphal­tene in solvent/precipitant mixtures have been studied. The mechanism of this aggregation was clarified. The colloid-like structure of crude oil was proved.

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E. R. PiIr.e and J. B. Abbiss (eds.), Light Scattering and Photon Correlation Spectroscopy, 341-352 @ 1997 Kluwer Academic Publishers.

342

PHOTON CORRELATION SPECTROSCOPY OF

OPAQUE FLUIDS

1. Introduction

Nowadays the photon correlation spectroscopy (PCS) is a routine experiment-'ll method to mec'lsure disperse particle size in various light transparent systems. For a number of PCS applications there arc problems of the correct result interpretation arising because of the following reasons, namely: interaction of polydisperse particles, crossover of a chaotic Brownian motion and macroscopic flow, mUltiple light scattering, opacity of disperse systems. Some novel experimental approaches also can be required. In particular, it is necessary to solve both technique and interpretation problems for particle sizing in opaque liquid dispersions. The disperse analysis of such systems is necessary in various fields: oil recovery and refining, motor oil testing, dye and paint production, and so on.

If the multiple scattering is negligible the extinction of light passed through liquid is described by Bouguer - Lambert law:

(1)

where 1,.1; are passed and incident light intensities correspondingly, h is the dist.'lnce, which light passed in liquid, aul is the extinction coefficient [4]. This coefficient consists of both sc.1ttering asCIII and absorbing aobs terms:

(2)

We will consider the liquid disperse systems with aobs » ascol> 0, which we define as opaque flUids. The theoretical description of static and dynamic light scattering in such systems is rather difficult problem because of the complex refractive index and Ihe non-ordinary angular dependence of the scattered light intensity [4,7,10). Besides, there arc some obvious experimental problems for light scattering study of opaque fluids. It is

343

necessary to take into consideration that the problem of the particle sizing in opaque fluids is simplified for the dynamic light scattering in respect to the static light scattering. It is interesting to consider the transformation of the basic PCS· relationships for opaque fluids. The mean particle radius in the PCS of transparent fluids is calculated from the following expressions [6]:

I kbT 2 -=--q Tc 31rTJR

21T1l . 0 q=-2sm-

A 2

(3)

(4)

where Tc is the correlation time, kb is Boltzmann constant, T is the temperature, 7] is the shear viscosity of the liquid, A is the wavelength of the laser light, n is the refractive

index of liquid, 0 is the scattering angle. For many opaque fluids refractive index is complex and its effective value depends

from the angle of incidence of laser beam on the surface of the cell. The actual scattering angle depends on this value too. The real refractive index n one should be replaced by complex value m in the Fresnel relations [5], it causes essential complications. The complex refractive index can be introduced in the following manner:

m=n-iK (5)

where the real term n defines the speed of light in an absorbing medium and K is responsible for the attenuation of the light in the process of its propagation. The refractive angle in the medium becomes a complex value after the substitution (5) in the usual refraction law. Nevertheless, one can obtain the refraction law for transparent/absorbing media in the real form [9]:

It is important to note that the refractive index is not constant: nip depends on the angle of the beam incidence. The abSC}rbing index Kip depends on this angle too:

(7)

344

I':or opaque fluids the formula (3) is transformed into the following one, which can be

written briefly as:

(8)

where the scattering vector q(n. A; 8, rp} is the explicit function of pointed variables. The depth of the light penetration into the sample is less than lmm for opaque fluids under the study. That is why the usual geometry of the experiment with the scattering volume in the ceritre of the cell is not suitable for such fluids. It is useful to calculate the scattering vector for the following geometry of the experiment: the scattering volume is in the neighbourhood of the plate wall of the cell, the laser beam is incident on this plate at angle 'P, the scattered light is observed under angle tP through the same wall. In this case

(9)

where n", , n~ are the refractive indicies (6) of incident and scattered light in absorbing medium~ X ,~ - actual refracted angles for incident and scattered light in absorbing medium, correspondingly. In the general case the expression for qz in the expanded form is sufficiently awkward. But it has more simplified form for some optical schemes of the experiment.

So, two bound problems should be solved for particle size measuring in opaque fluids. At first, it is necessary to use the special experimental approach allowing to observe a scattered light. The second problem is connected with the accurate data interpretation taking into account the geometry of the experiment and complex optical properties of medium under study.

The opaque disperse fluids can be classified schematically by three types:

(a) transparent liquid + absorbing particles~ (b) absorbing liquid + non-absorbing particles; (c) absorbing liquid + absorbing particles.

The first type is the most suitable for PCS study because in this case the scattered light propagates in a transparent liquid and the usual interpretation (3), (4) can be used. In this case the problem of opaqueness is connect with the choice of the suitable experimental technique only. The second and the third types of fluids can be combined in the framework of PCS method. In this case one should use both special experimental idea and special mathematics approach taking into account the inhomogeneity of the

345

refracted light waves propagating in the medium having the complex refractive index. We have solved both of these problems. Proposed approach allows to measure particle size in all types of opaque fluids. Moreover, the refractive and absorbing indicies of the medium can be found using this approach.

2. Technique and objects

The following experimental approaches for the investigation of opaque systems had been mostly taken, namely: optical schemes with light fibres,. special thin sample cells and back-scattering geometry (1801). We tested these methods and found them to be rather complex in application and requiring some essential modifications of the setup. For opaque fluids we have proposed a simple suitable geometry shown in Figure I that doesn't require a setup modification and utilises a standard rectangular sample cell [I). It is placed at the angle of 45° with respect to the incident laser beam. The optical axis of the photodetecting system is normal to the front plane of the sample cell. The scattered light is collected from the region of the laser beam entry to the sample. The X­Y positioner allows one to change the position of the scattering volume in accordance to the opacity of the sample and other experimental conditions.

Sample cell

Opaque media

Scattering volume

. '.:4 Reflected beams Scattered II&ht

Figure 1. The geometry of the experiment with the angle of the laser beam incidence qr4So.

The number of testing measurements using the model opaque liquid systems have verified an adequacy of this experimental approach. Dispersions of light absorbing particles (aniline dye) in transparent liquid have been tested. In this case it is enough to

346

use the well-known refractive index of a transparent solvent and the usual fonnula (3), (4). The results are represented in Figure 2 .

:s.

. 8r-----------------------------------------------------------------------,

.6 -. '"' • - -• - • ,-\ T ransparen1 dispersion

-

.2 -

I I I I 0.0 L-___ L-__ ---'L-__ ---l ___ ---l ___ ---l

0.00 .OS .10 .15 .20 .25

Coocentration, WOA.

Figure 2. The measured size of anilin~ dye particles in the wide range of concentrations.

\I Scattered Debt

Sample cell

Opaque media

Scattering volume

Laser beam

Figure 3. The normal geometry with the scattering volume in the comer of the cell.

347

The data interpretation becomes more complicated for opaque fluids of the second type.

In this case you should use the formula (8), with n and K. These values can be measured by special refractometer for opaque media. As a rule, such measurements are very complex or even impossible. We propose the following plan of the experiment for opaque fluids with unknown·n and K. Two measurements should be carried out for the same sample using two schemes (Figures 1, 3). In addition, the extinction coefficient aut is found directly from the measurement in the thin cell. It is important, that at the normal incidence of the laser beam and at the normal scattering angle (Figure 3) the expressions (6), (7) are transformed into

(n,)p=a = n • (K,)p=a = K. (10)

These values are the main optical indices of the opaque medium. The expression (8) is transformed into usual one (3). So, There are the following data

"rei =/(n.K). "rc2=/(n) • (11)

Using this data set it is possible to calculate the true particle size, n and K • From other ~d, n and K of unknown opaque liquid can be found by measuring the size of the testing particles added to this liquid (e.g. polystyrene latex) without· extinction measuring. We have carried out a number of testing experiments for non-absorbing particles of polystyrene latex in absorbing solvents (water-based ink), which confirm the suitability of this approach.

It is important to point out, that numerous fudustrial opaque objects are the first type fluids, in particularly, the main part of colloids. Nevertheless, the test of the type of an opaque fluid consists of the comparison of the results of the measurements carried out for two presented schemes. For many application it is enough to measUre the behaviour of particle size, but not the true particle -size. In accordance to our collaboration with oil industry research centres a number of problems have been studied:

(a) motor oiJ, degradation during engine exploitation; (b) self-emulsifying process of crude oil-water microemulsion; (c) testing colloidal properties of crude oil and petroleum products; (d) stability and aggregation kinetics of the crude oil heavy fraction (asphaltenes).

For our experiments we used a number of setups including a unique complex installation for. phase transition investigations [2, 8] and our simple commercial devices. Two models of our single-board correlators have been used: the UNICOR-M (8-bit, high speed, 72 channels) and the UNICOR-SP (l6-bit, middle speed, 255

348

channels) opetating in a multiple-tau time scale in dual correlator mode. These devices are extremely appropriate instruments for study of the fast kinetics processes accompanied by the particle growth in a wide range of sizes. The two presented schemes have been used. The first scheme (cp = 45°) is more suitable for the strong opaque fluids in respect to the second scheme. It allows to study the samples with small depth of light penetration. The laser beam was attenuated by optical neutral glass to avoid the thermal lens effect.

3. Results and discussion.

The most attractive results have been obtained for asphaltene solutions. These results are directly connected With the general problem of enhancing the oil recovery and decreasing the viscosity of the crude oil. The complex colloid-like structures contained in crude oil define its viscoelasticity. Asphaltenes and resins are the principal surfactant components of the heavy fraction of crude oil, which cause the formation of these structures. The reliable way of the colloid properties examination is the study of the aggregation kinetics [3]. We have investigated the asphaltene aggregation process in toluenelheptane mixtures using PCS. The natural asphaltenes of various origin as well as the same ones with the chemically modified metal-porphyrin kernels have been studied.

In our experiment we used toluene and n-heptane of chromatographic grade. The concentration of asphaltenes in toluene varied from I to 10 gil. Thus we studied asphaltene solutions above and below the CMC which is about 4 gil. Heptane served as a precipitant an~ was added gradually. The preparation of the samples was carried out in a dry nitrogen atmosphere. The process of aggregation was not observed at low concentration of heptane, and started no sooner the concentration of heptane had reached a certain threshold value. We observed and registered the appearance of asphaltene aggregates and their growth in time. The threshold concentration of heptane depends on the origin and the concentration of the asphaltene and varies from 50 to 52 % vol. The rate of aggregation increases dramatically with the increasing of the heptane concentration over the threshold value. The particle size measurements were terminated when the sedimentation process began dominating.

The classic DLVO theory predicts two possible mechanisms of colloid particle aggregation, namely: fast and slow. The difference between them is in the way the particles stick together. The fast aggregation is governed only by the diffusion process of the particles. Every contact of the particles results in their coupling. This is the diffusion limited aggregation (DLA). In this case the particle growth is described by the following equation:

(12)

349

where R is the radius of aggregates, A - an effective aggregation constant, t - the time, and d, is the fractal dimension. The slow aggregation is called the reaction limited aggregation (RLA). Unlike the DLA in this case the contact of particles won't result in their sticking. The aggregation takes place for the particles interacting in a special manner (chemical, steric, etc.). The RLA aggregation proceeds slowly in the comparison with the DLA one. The RLA kinetics is described by the following formula:

R = Roell ~ (13)

where Ro denotes the initial radius of the particle, Tis the characteristic time. At asphaltene concentrations below the CMC the DLA process solely is observed

[3]. Above the CMC the RLA process occurs in the initial stage of particle growth. Apparently, until the asphaltene concentration below the CMC the micelles don't appear and consequently the interaction between micelle-like particles is absent. Under these conditions the nascent asphaltene structures can stick together in any manner. This one we call the DLA process. There is a restriction for sticking of the particles together above the CMC in the initial stage of aggregation. Therefore we observe the RLA process. As the particle size grows these restrictions vanish and the aggregation mechanism goes to the DLA These specific features of the asphaltene aggregation ab.ove the CMC are shown in Figure 4.

6~------------------------------------~

Crossover behaviour 2

RIA

.o~~~~--~~~--~~~~~~~~--~

o 200 400

Time,min

Figure. 4. Crossover behaviour of asphaltene aggregation. Experimental points correspond to the aggregation at the threshold heptane

conoentration 52 %

600

350

It was found out that the character of aggregation for all measured heptane concentrations is alike to described above. The experimental curves for lOgll asphaltene concentration are represented on Figure 5. One can see that the characteristic time of the aggregation dramatically decreases with the heptane concentration increase.

8r---------------------------------------,

6

::l

2

100 200 300 400 500

Time, min

Figure 5. Aggregation kinetics of 1 % asphaItene solution.

0.1%

4

::l

f 3 1%

~ 2

500

Time,min

Figure 6. UniversaJ behaviour of aggregation for 0.1% and 1 % asphaItene concentrations.

351

The kinetic curves for definite asphaltene concentration can be fitted by one universal function (Figure 6). All the experimental data were normalised to curves with the threshold concentrations.

It is known that the aggregation kinetics in classic colloids can be described by the Smolukhowsky equation. In this case the following relationship is valid for the characteristic time of the process:

U lnToc~

k"T (14)

where UIJUIJ< is the potential barrier height, kb is the Boltzmann constant and T is the temperature. Proceeding. from the fitting of the experimental data we obtained the relation between the characteristic time and the heptane concentration. Linear law presented on Figure 7 allows to consider this function in the following manner:

(15)

where CeNu is a parameter, C is the concentration of heptane, Co is the threshold concentration of heptane.

.. .s

6~--~~------------------------------,

s

4

3

2

4.0 4.1

inC

4.2 4.3

Figure 7. Relation between the characteristic time and the concentration of heptane

Our experimental results show that the aggregation rate of the modified asphaltenes is half as much as the aggregation rate of the natural ones. Alsq, we have carried out the

352

independent investigation of viscosity of the natural crude oils and of the same oils that were affected by the extra-ligands. The viscosity of modified oils decreases by 20-35% in comparison with the same parameter of the natural ones. So, the rate of the asphaltene aggregation can be applied for the identification' or the rheological properties of crude oil. Thus, the mechanism of the asphaltene aggregation was clarified, and the way of its description by the fundamental Smolukhowsky approach was suggested. Asphaltene solutions and crude oils proved to be typical colloid systems.

References

1. Anisimov, M.A, Dmitrieva, lA, and Yudin, I.K (1988) Submicron Particle Sizing in Opaque Strong Absorbing Media by PCS. J. AppL Spectr. (Russian), 49, 144, pp. 28-31

2. Anisimov, M.A, Konev, S.A, Labko, V.l, Nikolaenk.o, O.L., Olefirenk.o, 0.1., and Yudin, lK (1987) Light­

Scattering Study ofThennotropic Liquid Crystals and Micellar Solutions, MoL Cryst. Liq. Cryst., Vol. 146,

pp.421-434.

3. Anisimov, M.A, Yudin, lK" Nikitin, V.V., Nikolaenk.o, O.L., Chemoutsan, AI., Toulhoat, H., Frat, D., and

Brio1ant, Y. (1995) Asphaltene aggregation in hydrocarbon solutions studied by Photon Correlation Spectroscopy, J. Phys. Chem., Vol. 99, No 23,9576-9580.

4. Bohren, C.F. and Huffinan, D.R. (1983)Absorption and Scattering o/Light by Small Particks, John Willey &. Sons, New York.

5. Born, M. and ~olf, E. (1964)Principks o/Optics, Pergamon Press, Oxford. 6. Cununings, H.Z. and Pike, E.R. (1974) Photon Correlation and Light Beating Spectroscopy, Plenum Press,

New York.

7. Hulst, Van de, H.C. (1957) Light Scattering by Small Particles, John Willey &. Sons, New York. 8. Konev, S.A, Labko, V.I., Nikitin, V.V., Nikolaenk.o, O.L. and Yudin, I.K (1988), The Light-Scattering

Photometer for Precise Phase Transition Study, Proceedings 0/ Third Int'l Symp. on Modem Optics, Vol. II, pp.408-412.

9. Prishivalk.o, AP. (1963) Reflection 0/ Light from Absorbing Media (Rllssian), Al:ademy of Sciences of BSSR; Minsk.

10. Shifrin, KS. (1951) Light Scattering in Turbid Medium (Russian), Moscow.

SCATTERING OF LIGHT IN AN INHOMOGENEOUS MEDIUM

Abstract.

L. A. ZUBKOV AND V. P. ROMANOV Physics Department, St. Petersburg State University, St. Petersburg 198904, Russia

Light scattering in a critical mixture near a phase transition point is considered. The contributions of double and multiple scattering are taken into account. A method for eliminating these inputs from the total inten­sity is described. This method includes' both experimental measurements of polarized and depolarized components and theoretical calculations for the real experimental geometry. The critical parameters in a strongly opalesc­ing system are found. The temporal correlation function of double light scattering is analyzed.

353

E. R. Pike and J. B. Abbiss (eds.), Ught ScoItering and Photon Correlation Spectroscopy, 353-366. @ 1997 Kluwu Academic Publishers.

354

SCATTERING OF LIGHT IN INHOMOGENEOUS MEDIUM

1. Introduction

The methods of light scattering are effective for determination of the sys­tem parameters (Fabelinskii, 1968). These methods have the number of advantages in comparison with the other experimental methods. First of all, the system remain practically unperturbed by the incident light at the optical measurements. It allows to get an information about parameters of the system and to investigate its behaviour in a very weak external fields. Besides, there is the possibility to conserve external conditions for a long time and as a result to collect a large set of the experimental data.

This type of experiments allows to study a very wide range of character­istic times from 10-12 sec to several hundreds of seconds. In the most cases an information about parameters of media we obtain from measurements of the single scattering of light. In this case the measured values and the parameters of media are connected by the most simple way. However there are the scatterings of higher orders, which limits the possibilities of the methods of light scattering. It becomes essential, when the length of free path of a photon is comparable to the characteristic size of the scattered system. In this case the measured values are the sum of the contributions of the scatterings of various orders. Therefore there is the problem of an isolation of the single scattering from the total intensity.

This problem has become essential at study of the phase transitions of the second order and the critical phenomena, where realization of measure­ments with the extreme possible acouracy was required (Anisimov, 1987; Stanley, 1970). As the system approaches to the point of the phase transi­tion the fluctuations of the order parameter increase. It results to increasing of the single light scattering intensity and to increasing of the contribution of multiple scatterings.

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By now some methods of the account of the multiple scattering are developed (Kuz'min et ai., 1994; Adzhemyan et ai. 1980, 1982; Bray and Chang, 1975; Schoeter et ai., 1983; Drewel et ai., 1990; Kawase et al., 1994). In the present work we put the special emphasis the cases, where the con­tribution of the multiple scattering is less or of the order of the single one. It means, that we consider the opalescencing systems, which remain trans­parent at the thickness about centimeters. The developed approaching is inapplicable to opaque systems such as the concentrated suspensions, where radiation transfer equation is valid. For such systems new methods, based on the study of coherent and correlation effects in the multiply scattering, recently develop essentially.

2. The theory of multiply scattering of light

We consider the optically inhomogeneous medium. The propagation and the scattering of light in such media is described by the Maxwell equations. We suppose that the magnetic permittivity J.L = 1, and the relation between an electrical induction vector and an electrical field intensity is linear and local

DCr, t) = c:( f, t)E( f, t), (1)

where the permittivity c: is supposed to be a scalar. The value c:(f,t) can be presented in the form

c:(f,t) = C:o + 6c(f,t), (2)

where C:o =< c:(f, t) > is the average permittivity, 6c:(f, t) is the deviation from the average value.

If we consider the system in external field Eo( f, t), it is convenient from the system of Maxwell equations to pass to the equivalent integral equation

where Tij is the dipole propagator. Its frequent Fourier-transform is equal to

1 [(wn) 2 (j2] e iw;:r

Tij(fw) = 4uo -;;- + or/Jrj -r- (4)

The dipole propagator is i-th component of an electrical field radiated by the point-like dipole of frequency wand polarization j, being in a point r = O. The intensity of the scattered light is proportional to the average product of fields E and E* in a point of the observation. The solution of the integral equation (3) by iteration leads to presentation of the total

356

intensity in the form of expansion in orders of 6e fluctuations. This series describes the single-, double- etc scatterings.

Firstly, we consider the integral intensity of the light scattering. The intensity of the single scattering of light for polarized component is written as

I v - 10 R e-U(Ll +L2)VG(q;;'\ V(I) - R2 ac I). (5)

This formula corresponds to the traditional setup of the light scattering experiment, widely used in the correlation spectroscopy also. In this geom­etry an incident and scattered light lay in horizontal plane and have the vertical polarizations. Distance from scattering volume up to a point of ob­servation R is much greater than the size of the scattering system. Value Rac is the constant of the scattering, (1 is the extinction, Ll and L2 are the distances travelled by the incident and the scattered ·light in media. It is supposed, that the illuminated and observated beams have the form of the thin cylinders with diameters much less than the sizes of a scattering sys­tem, V is the volume, from which the scattered ·light enters the detecting device, G( if) is the Fourier-image of correlation function, Ii = ki - ka is the vectors of scattering, ki and fa are the wave vectors of the incident and scattered light, q = 4>:n sin!, n is the refractive index, A is the length of the light wave, 0 is the angle of a scattering.

The intensity of a double scattering has the form (Kuz'min et al., 1994)

(6)

(7)

IJ;(2) = f drl f dr21

... e-U~ 12 m~[I- m~ - (mn?lG(qi)G(qi). JVl JV2 Tl - T2 (8)

Here VI and V2 are the illuminated volume and the volume, from which the scattered light enters in the detecting device, m = rl -r2 is the unit vector rl-r2

along the intermediate scattering, qi = km - ki is the wave vector of the first scattering, qi = k( n - m) is the wave vector of the second scattering,

n = ~ is the direction io an observation point, I is the difference between distances, travelled by the single and double scattered light.

The main difference between the single scattering and the scatterings of higher orders implies that the expression for intensity of the single scatter­ing is the algebraical expression from which we can easily find the necessary parameters. In scattering of higher orders the same parameters enter as the

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very complex integral relation, which dependents on the actual of exper­iment geometry. Therefore it is most convenient to measure parameters from the single scattering, excluding previously the contribution of multi­ple scatterings.

The higher order scatterings are complicate for calculations as it is seen from the general expression for double light scattering, Eqs. (6)-(8). There­fore it is necessary to use such geometry of an experiment, as much as possible to simplify the problem of calculations and exception of the mul­tiple scattering.

3. Study of double light scattering

There are some types of geometries of experiment for the study of multi­ple light scattering. Two of these geometries are shown in Fig. 1. In the geometry of Fig. 1a (Kuz'min et at., 1994; Adzhemyan et at., 1980) the recording device, consisting of the system of diaphragms, is focused at a point removed to the distance h from the incident beam instead of at the illuminated volume itself. In this geometry at h i= 0 the scattered light may enter the detector after at least two scatterings. Other geometry is illustrat­ed by Fig. 1b (Bray and Chang, 1975), here the scattered light is collected from the slit of height h, the latter is significantly exceeding the diameter of an incident beam.· In this geometry the detector collects the single- and the multiple light scattering integrated over height h. We discuss the re­sults obtained in the geometry shown in Fig. 1a. Binary solutions in the vicinity of the critical point of stratification were investigated. In this case the constant of a scattering of light has the form

R _ rr2 (8c) 2 kBT (oc) sc -).4 OC p,T P OJ1 p,T'

(9)

where c is the concentration, p is the pressure, T is the temperature, kB is the Boltzmann constant, p is the density, J1 is the chemical potential of the mixture. The correlation function of fluctuations of concentration G(q) is written as

(10)

where T c = TOT-II is the correlation length, T = IT;:;cI is the reduced

temperature, (gc) '" T-"(, 'Y and v are the critical parameters of suscep-/-L p,T

tibility and correlation length respectively. In such systems the correlation length and intensity 'of scattered light grows at T -+ O. The parameters of the correlation length TO and v and parameter 'Yare determinated from the

358

i

~------~~----------~ --r~::C

E

-:L. 1<n

Figure 1. Geometry of experiment of separatinl!; of multiple scattering' a) geometry-with crossed thin cylinders, b) geometry with a narrow slit.

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light scattering experiments. Three solutions with the different optical in­homogeneity were studied: 1. - transparent system, in which the distorted factor is a double scattering only, 2. - mixture, in which there is contribu­tion of higher orders along with double light scattering, 3. - mixture, in which all orders of scattering are important in the whole interval of temper­atures. In critical mixtures the scalar order parameter is the concentration. The depolarized component of single scattering is the very small, and it's magnitude does not dependent on the reduced temperature T. The depo­larized component is caused by the contribution of the multiple scattering of light and has the strong dependence on temperature. The measurements were carried out in the quartz cylindrical cell by 2L = 60 mm diameter. The angle of a scattering (J is varied from 15° up to 140°. The collimating device consists of the system of diaphragms. It forms the visible volume as the cylinder with the effective radius R2 = 1.5 mm. Radius of the laser beam in the cell RI was equal to 0.15 mm.

The first object was a solution ,8,8'-dichlorethil ether (chlorex) - isooc­tane (Adzhemyan et al., 1980). For this mixture it is possible to use an approximation of the double scattering within the measured temperature range since the difference of refractive indices is small. We measured I~ and II; components of the scattered light at h = 0, 4, 6, 8, 10 mm within the range of reduced temperatures T "" 2 10-3 - 10-4 with an angle step 5°. The typical dependence on the height of the both components is plotted in Fig. 2.

To determine the intensities of the double scattering I~(2) at h = 0 we used two expressions

v I~(2)(h = 0) v IV(2)(h = 0) = Ij(h) IJ(e:1Jp) (h), J = V,H (11)

The expressions for components I~ (h) and II; ( h) were obtained for the geometry Fig. 1 from Eqs. (6)-(8). We use the approximation that the volumes VI and V2 are the cylinders of L length and Rl and R2 radii. The dependence of I~ (h) and I;; ( h) component on the height h is plotted in Fig. 2. The curves show the good agreement between the the calculated and measured values.

It is seen a very weak dependence of the depolarized component on h. The polarized component has the very sharp dependence on h. Really at h -+ 0 the intensity I~(2) has the logarithmic divergence. Far from the

critical point it is proportional to In 't. It is follows from the calculations at h = 0, that there is the replacement h by the largest of radii Ri i = 1,2.

360

1(2), relative units J.. +

z

'F-__ ~~~ __

o 8h, mm Figure 2. Components of light scattered at an angle of 90° for system f3f3' - dichlorethyl - isooctane as a function of height for krc = 0.85: 1. I~(2)(h), 2.

Ii;(2)(h). The dots are the experimental data, the solid lines show the calculation, and

+ is the value I~(2) calculated at h = O.

For calculations it is necessary to know the extinction coefficient. In the critical region it has the form (Kuz'min et al., 1994)

(T = Bi{[2 + (krc)-2 + ~(kre)-4]ln[1 + 4(kre)2] - 2 - (kre)-2}, (12)

where B = (t:)2' To the zeroth approximation the values of r e were retrieved from the

angular dependence of I~ (h = 0). Really the I~ intensity is the sum ofthe single and double scattering. The parameters of the correlation length have been found as following ro = 3.2A, v = 0.58.

The constant of scattering Rse was determined from a ratio

R _ Ikexp VG(q) se - IV IV '

Vexp H(2) (13)

,·.'itich follows from the intensities relation of the single and double light scattering, Eqs. (5), (6).

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The value of a double scattering 1~(2)(h = 0) was determined by two ways with the help of Eq. (11). In one way the experimental data of the depolarized component IJ;(e:z:p)(h) were used and the values I~(2)(h = 0)

and IJ;(2)(h) were calculated.

In the second way the data of polarized components l~(e:z:p)(h) and

results of theoretical calculations for a component I~(2)(h = 0), I~(2)(h) were used. If we can extrapolate the polarized component to h = 0, just as the depolarized component, we do not need to carry out the theoretical calculations. Unfortunately, (see Fig. 2) such extrapolation is impossible.

After excluding the contribution of double scattering 1~(2)(h = 0) from the total intensity the corrected value of the correlation length was calculated. The parameters TO and v have became equal to the values TO = (2.36 ± 0.16)A and v = 0.625±0.007. Also from the single scattering intensity the critical exponent 1 = i.21 ± 0.04 was determined. The cor­rected values of critical exponents are in agreement with the prediction of the theory of the critical phenomena.

The input of the double scattering for the given mixture did not ex­ceed 15% from the total intensity.

The nitrobenzene-hexane mixture was the next object of our investi­gations. For T - Tc ::; 0.05 there is an appreciable contribution of the scatterings of order above the second in this mixture. It was detected as more rapid increasing of the depolarization ratio in comparison with the first mixture.

We measured the dependence of light scattering intensity at an angle () and height h, in the range of the reduced temperatures T '" 3 10-3 - 5 10-5 •

As the system approaches to the critical point the value (I~)-1 deviates from the linear dependence on sin 2 ~ at h = 0 as it should follow from Eqs. (5), (10). Therefore in zeroth approximation TO and v parameters were retrieved from experimental data for temperature range, where (1~)-1 is the linear function of sin2 ~ approximately. The values TO = 3.25 A and v = 0.59 were obtained. From measurements of the intensity of the trans­mitted light the extinction coefficient (12) was determined and constant of a scattering Rse = B(kTc)2 is found. The value B has appeared equal to B = 0.32.

Using TO, v and B parameters, we calculated the contribution of a double light scattering at all temperatures and h =I O. Further from the experimental data l~e:z:p(h =I 0) the calculated intensity of the double scat­tering was extracted. The difference represents the contribution of multiple scattering, l(p), with orders above the second. The curve l(p)(h) was extrap­olated to h = O. This procedure is possible by since the higher orders of scattering contain an additional integration over the volume cell. This inte-

362

gration removes the logarithmic divergence of intensity I~ (h) components at h ~ o.

The intensity of the single scattering was determined from the following expression

I~(l)(h = 0) = I~exp(h = 0) - I~(2)(h = 0) - I~(p)(h = 0). (14)

The values I~(2)(h = 0) were obtained by the calculations, and I~(p) - by extrapolation to h = O. The new corrected set of critical parameters was calculated using the values I~(l). It turned out that the iterative procedure quickly converges and has the high stability. The obtained values of critical parameters have appeared to be TO = (2.7 ± 0.2) A, v = 0.62 ± 0.02, 1 = 1.21 ± 0.02, B = 0.29 ± 0.02.

The third object was the solution of the nematic liquid crystal BMOAB and isooctane (Adzhemyan et al., 1982) where the difference between the refractive indices of the components is very large, .6.n = 0.25. In this solu­tion there is the appreciable input of the multiply scattering even at low asymmetry of the single scattering indicatrix. Therefore it is impossible to determine the values of parameters in zeroth approximation.

In this mixture the parameters of the zeroth approximation were ob­tained from temperature dependence of the extinction coefficient. The in­tensity of transmitted light I tr was measured in an interval of temperatures .6.T rv 0.15 - 3,5 K where Itr varies at about in 100 times. The parameters TO, v and B were found from Eq. (12) by the method of the least squares. They were equal to TO = (3.4 ± 0.3) A, v = 0.60 ± 0.03, B = 1.15 ± 0.3. The large dispersion is due to determination of three parameters from the smooth dependence of extinction (J' on temperature .6.T. This set of pa­rameters was used for processing of the light scattering data similar to the previous solution, nitrobenzene and hexane. The obtained values of param­eters are equal to TO = (3.9 ± 0.1) A and v = 0.6 ± 0.04. In this mixture the total contribution of the multiple scattering reached up to 60% in the investigated range of temperatures. Therefore there is a large error in de­termination of the single scattering indicatrix.

To improve an accuracy in determination of parameters in so strong opalescing system we take into account that the data on the extinction and the light scattering contain the mutua.lly complementary information. Therefore we have carried out joint processing of data on the intensity of transmitted light and the scattered light. The iterative procedure in such processing converges quickly and 2-3 iterative steps is needed only. The joint processing has resulted to the sharp reduction of an error in deter­mination of parameters. They have appeared equal to TO = (3.2 ± 0.2) A, v = 0.63 ± 0.01, B = 0.95 ± 0.04. The reason of so high accuracy it is easily to understand from Fig. 3 which shows the admissible values of parameters

363

o ro,A

4,4

2) 8 ~_-'-_---..I"'---___ --=-_

0:56 ~ 60 0,6'4 " Figure 3. Confidence regions of values of parameters Tc and 11 for the system BMOAB-isoocta.ne. 1) extinction experiment, 2) light sca.ttering experiment.

TO and 11 for each of two types of experiments and for joint processing (Adzhemyan et al., 1982). It is seen that these ranges represent the strongly extended ellipses. The small area of the crossing corresponds the values of parameters in the best way describing both experiments simultaneously.

4. Spectral intensity and temporal correlation function of the double scattering of light

Till now we considered the integral intensities of the scattering of light various orders. Here we analyze the spectral structure and the temporal correlation function of the multiple scattering. As before we consider the double scattering by the critical mixtures. The fluctuations of concentration decay due to diffusion law. In this case the spectral intensity I~(l)(w) and

temporal correlation function of fields C~(1)( t) for a single scattering have the form

(15)

(16)

364

The intensity c- intensity correlation function rather than temporal corre­lation function of fields C~(l)(t), is experimenta.Ily studied. In Gaussian approximation it is proportional to

< I~(l)(O)It"(1)(t) "" e-2wct •

Here We = q2 D( q) is the characteristic frequency, D( q) is the diffusion coefficient.

In the critical region the diffusion coefficient depends on wave number q and temperature.

For a double scattering the spectral intensity of the polarized compo­nents can be written as (Romanov and Salikhov, 1985)

here

(18)

Eqs. (17), (18) a.Ilow to calculate the spectral structure of double scattering light similarly to that as it is done for integral intensity. The temporal correlation fu~c~ion is possible to obtain, if in expressi~n for §( iiI, i/2, w) a frequent multiplier to replace on temporary one, e-[we(Qd+we(Q2)]t.

Qualitatively the spectrum of double of scattered light is determined by two types of the contributions. Double scattering of light is essential for the large-scale inhomogeneities with the light scattering indicatrix strongly extended ahead. Therefore the beams with sma.Il angle of scattering con­tribute the main input to the total intensity. On the other hand, in such systems an extinction is essential as a result the beams which travel a sma.Il­er distance in media are important. These two conditions allow to allocate region of angles, which give the basic contribution to the scattering.

The quantitative estimations rather simply turn out, when the light scattering indicatrix is close to circular. In particular, for temporal corre­lation function of a double scattering we have

t 1 9 at <t:: ::;:-JOY cos 2' 2we

t 1 9 at :> ~ cos 2' 2wc

where (J is the angle of a single scattering, w~O) is the characteristic frequen­cy, corresponding to the scattering angle (J = j. It is seen, that the tem­poral correlation function of the polarized component at sma.Il times fa.Ils

I

0.9

0.7

0.5

0.3

O.I

\I Ie v (0) C V(2) v(2J

0.5 I I.5 t 1:

365

Figure 4. Temporal correlation function of polarized component for nitrobenzene and hexane critical mixture (6 = 90°): 1. krc = 0.1, 2. krc = 1, 3. krc = 2,4. e- I / T •

down faster, than for a single scattering, and at large times it falls down slower. The same result turns out for the depolarized component. Within the framework of the geometry shown in Figure la the frequent and the temporal correlation functions of double scattering of light depending on h and krc were calculated. For system nitrobenzene-hexane the comparison with the experiment on measurement of the half-width of a spectrum of scattering was spent at h = 1 mm in a wide range of temperatures T.

The results of our calculations have coincided with the experiment (Bey­sens and Zalczer, 1977). Fig. 4 shows the temporal correlation function of a double scattering at h = 0 for various krc. The simple exponential function is shown for comparison. It is seen, that the temporal correlation function of a double scattering differs greatly from exponential curve.

References

Adzhemyan, L.V., Adzhemyan, L.Ts., Zubkov, L.A., and Romanov, V.P. (1980) Al­lowance for double scattering of light in the determination of critical exponents,

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Zh. Eksp. Teor. Fiz. 78, 1051 [Sov. Phys. JETP 51, 530]. Adzhemya.n, L.V., Adzhemyan, L.Ts., Zubkov, L.A., a.nd Roma.nov, V.P. (1982) Study

of the extinction and scattering of light near the critical point, Zh. Eksp. Teor. Fiz. 83, 539 [Sov. Phys. JETP 56, 295].

Anisimov, M.A. (1987) Critical Phenomena in Liquids and Liquid Crystals, Nauka, Moscow (in Russian).

Beysens, D., a.nd Zalczer, G. (1977) Low frequency spectrum of light multiply scattered by a critical mixture, Phys. Rev. A15, 765-772.

Bray, A.J., and Chang, R.F. (1975) Double scattering in the interpretation of Rayleigh scattering data near the critical point of a binary liquid, Phys. Rev. A12, 2094.

Drewel, M., Ahrens, J., and Podschus, U. (1990) Decorrelation of multiple scattering for a.n arbitrary scattering a.ngle, J. Opt. Soc. Am. A7, 206-210.

Fabelinskii, I.L. (1968) Molecular Scattering of Light, Plenum, New York. Kawase, S., Manuyama, K., Tamaki, S., and Okazaki, H. (1994) The critical exponent of

binary liquid mixtures by Rayleigh and Brillouin scattering measurements, J. Phys.: Condens. Matter 6, 10237-10246.

Kuz'min, V.L., Roma.nov, V.P., a.nd Zubkov, L.A. (1994) Propagation a.nd scattering of light in fluctuating media, Physics Report 248, 71-368.

Roma.nov, V.P., a.nd Salikhov, T.Kh. (1985) Spectrum of double light scattering by con­centration fluctuations near the critical point of separation, Opt. Spektr. 58, 1091-1096 [Opt. Spectr. USSR 58, 666].

Schoeter, J.P., Kim, D.M., and Kobyashi, R. (1983) Multiple scattering of light near critical metha.nol-cyclohexa.n, Phys. Rev. A27, 1134-1145.

Stanley, H.E. (1970) Introduction to Phase Transitions and Critical Phenomena, Claren­don Press, Oxford.

OBSERVATION OF SHEAR-INDUCED GELATION USING

LIGHT SCATTERING IMAGING

Abstract.

DAVID PINE Departments of Chemical Engineering and Materials, University of California, Santa Bar'bam, CA 93 J 06-5080, USA

By direct imaging of scattered light, we observe shear-induced gelation in extremely dilute solutions of wormlike micelles. This gelation is followed by a fracture of the gel which produces extremely elastic gel bands with recoverable strains of up to .5000%. The gelation and fracture account for the unusual shear-thickening and elastic properties of these solutions.

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E. R. Pike and J. B. Abbiss (eds.), Light Scattering and Plwton Correilltion Spectroscopy, 367. @ 1997 Kluwer ACfldemic Publishers.

SINGLE PARTICLE MOTION OF HARD-SPHERE-LIKE

POLYMER MICRONETWORK COLLOIDS UP TO THE

COLLOID GLASS TRANSITION

Abstract.

E. BARTSCH, S. KIRSCH, F. RENTH AND H. SILLESCU

Institut fur Physikalische Chemie der Universitiit Mainz, .fakob- Welder Weg 15, D-55099, Mainz, Germany.

Polymer micronetwork spheres swollen in a good solvent can be re­garded as colloids which require no special stabilisation to avoid aggrega­tion. Their interactions can be tuned by changing the degree of internal crosslinking. The phase behaviour and the static structure factor demon­strate that crosslink density of 1:10 (inverse number of monomer units between crosslinks) is sufficient to achieve hard sphere behaviour. We de­signed a host-tracer system consisting of core-shell micronetwork spheres (core: polystyrene; shell: poly-t-butylacrylate) in a host of refractive-index­matched poly-t-hutylacrylate micronetwork colloids. Employing a crosslink density of 1: 10 and tuning the polydispersity such that crystallisation is just avoided we are able to monitor the self-diffusion of a system with hard sphere interactions up to volume fractions close to the colloid glass tran­sition by dynamic light scattering. In addition the long-time self-diffusion was measured with the forced Rayleigh scattering technique.

We compare the measured long-time self-diffusion coefficients with the prf'(lictions of theoretical concepts which differ in their treatment of hy­drodynamic interactions. We find that mode coupling theory, neglecting hydrodynamic interactions~ is able to describe our data only very close to 4>g . The theory of Medina-Noyola, where hydrodynamic interactions are as­sumed to directly affect only the short-time dynamics, seenis to work only up to moderately high volume fractions. In contrast, a new approach by Tokuya.ma and Oppenheim, where hydrodynamic interactions are explic­itly treated, is fully consistent with our results over the full volume fraction range.

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E. R. Pike andJ. B. Abbiss (eds.J, Light Scattering and Photon Correlation Spectroscopy. 369. © 1997 Kluwer Academic Publishers.

MEASUREMENT OF VISCOELASTICITY OF COMPLEX

FLUIDS WITH DIFFUSING-WAVE SPECTROSCOPY

Abstract.

DAVID A. WEITZ Department of Physics and Astronomy, University of Pennsylvania, 209 S 33m Street, Philadelphia, Pennsylvania 19104-6396, USA

Dynamic light scattering can he used to measure the viscosity of a fluid by determining the diffusion coefficient of a particle of.a. k/lOwn diameter: In addition, it has 1)('en used to measure the fluctuations of a gel network, which can provide some information about the elastic modulus of the gel. Here, we extend these applications to measure the full frequency depen­dent viscoelastic moduli of several different complex fluids. The samples all scatter light very strongly, making it imperative to use diffusing-wave spec­troscopy. This also allows matNials with a IIlllch higher elastic contribution to be studied. This talk will review the general method, and will include a more detailed discllssion of the application of this method to the study of the viscoelasticity of hard splH'rcs near the colloidal glass transition.

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E. R. Pike and J. B. Abbiss (eds.), Light Scattering and Photon Correlation Spectroscopy, 371. © 1997 Kluwer Academic Publishers.

APPLYING PHOTON CORRELATION SPECTROSCOPY IN SPACE

Abstract

ANTHONY E. SMART 2857 Europa Drive Costa Mesa, California, 92626-3525, USA

Transparent models of condensed matter systems can be studied optically using photon correlation in the microgravity environment of the Space Shuttle or the proposed Space Station. Design and packaging becomes more critical than for Earth-bound equipment. Also, to generate enthusiasm enough to assure its expensive deployment, the system must have a target performance at least as good or better than that possible on Earth. Additional properties must be robustness, small physical size, low weight and power consumption, retention of alignments and a well-controlled thermal environment for the specimen. Confidence may be built by careful design and extensive testing. Optical design must give versatility without unacceptable compromise. Performance must confirm known physics, and separate new phenomena from artifacts of the instrumentation or measurement process. Effectively applied risk analysis must prevent potential defects from compromising the entire mission, preferably by substituting 'graceful degradation'.

1. Introduction,

The Physics of HArd Spheres Experiment [pHASE], an experimental investigation in condensed matter physics due to fly on the Space Shuttle Columbia launch STS-83 as the MSL-l mission in March of 1997, is chosen as an example of the application of photon correlation spectroscopy in space. This paper comments upon why light scattering should be chosen, the relevance and advantages of photon correlation, and the need to perform experiments in the microgravity environment of low Earth orbit. In these contexts the discussion addresses aspects of equipment design for versatility without the compromise of effectiveness and the quantification of performance limits, both desired and likely to be achieved. It also explores how the performance limits may be best exploited with the resources available while substituting potentially graceful degradation for the detrimental possibility or likelihood of complete failure.

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E. R. Pike and J. B. Abbiss (eds.), Light Scattering and Photon Comlation Spectroscopy, 373--386. @ 1997 Kluwer Academic Publishers.

374

Many complex condensed matter systems can be modeled by small 'hard spheres' suspended at various concentrations in a suitable liquid. The 'hard' implies that their mutual interactions are sharply bounded. Transparency to permit optical probing of evolving structures requires the near match of refractive index between the sub-micron diameter spheres and the interstitial liquid. This is not currently available for system components of identical specific gravity, for example coated polymethyl methacrylate spheres in a mixture of decalin and tetralin. Thus, to prevent settling dominating the structure formation process, or damaging the crystals by the frictional effects of Stokes drag, the effects of gravity must be mostly removed.

2. Application in Space

Many aspects affect the suitability or necessity of conducting a particular experiment in the expensive and challenging circumstances associated with off-Earth operation. In order to have the experiment appear politically attractive at the beginning, it must be perceived as potentially useful, even to those who may not initially understand either the physical principles or their relevance. It must be capable of inspiring enthusiasm in all those necessary to its acceptance and its consequent funding. It must at least seem to have the possibility of some previously defmed ultimate success, and it must also appear to be good value for money in the minds of those providing the money.

At least two scientific criteria must be met. The first is that the experiment must show that it can confIrm existing understanding and repeat known observations. Only then can confidence be placed in observations that might potentially lead to new knowledge or insights. Even these must be shown rigorously not to be artifacts of the present instrument nor of the the way it is currently being used, before any confidence may be generated. The meticulous characterization of the equipment prior to its deployment is necessary, both for scientific reasons and to confIrm constancy and reliability of behavior.

On both technical and financial levels, a reasoned trade-off must be made between innovation and risk. Without some innovation, the necessary enthusiasm may not be sustained, but with too much innovation the associated risks may compromise mission success. To make this trade-off satisfactorily, a clear understanding of the relevant technological innovations and risks is necessary, particularly what problems must be solved. At least one potential solution for each must be found close to the beginning of the program.

Only with the above considerations handled do the logistics of how it can all be made to happen become a topic of concern. These include areas over which the designer has at least some choice in the initial stages, but which rapidly become constrained, even 'over-constrained' as those choices are made and confirmed. Typically such areas start with programmatic aspects, the planning, organization, scheduling and funding, but

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rapidly and significantly proceed to the choice and fixing of the system architecture. The next major constraining area for space deployed equipment is the establishment and defmition of interfaces. These include the mechanical, thermal, electrical and communication interfaces where insufficient attention in the early stages can have disastrous consequences. Establishing communication protocols and data structures always takes much longer than anticipated. Power supplies can be particularly troublesome. The infmite resource-sink of software can be better tested if the hardware and software modules are mapped isomorphically when the architecture is initially established.

3. Evolving Programs

To exploit the available resources efficiently, the experiments that can, should or may be performed in space form part of an evolving technology program, where each new experimental stage builds upon the lessons learned from previous activities, but is not necessarily contingent upon their complete success. Examples of such subjects, whose behavior is significantly altered by the reduction of a gravitational field and that can be studied in the Fluids and Combustion Facility of the Space Station are currently crystallization, with which this paper is primarily concerned, rheology, aqueous foams, nucleate boiling and combustion. This is by no means a complete list.

Laser Light Scattering Instrument Advanced Technology

Development March 1997 Technology

Figure 1 Light Scattering Experiment Synergy and Evolution

Proposed or estimated flight dates are shown in figure I, with that in the past being within a rectangular frame.

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For most of figure 1 "Condensed matter physics concerns the phenomena associated with systems containing many interacting particles at temperatures where the interactions dominate. It is differentiated from areas of physics which want to understand the nature of the elementary forces or the properties of individual particles, atoms or molecules, gases and plasmas. It encompasses all of solid state physics plus healthy parts of statistical mechanics, hydrodynamics, biophysics etc." [1]. Figure 2 indicates simply how scattered light has more sharply defmed patterns when the structure is more ordered, permitting the observations of Bragg scattering to be related to evolving order. The figure does not indicate temporal structure of the scattered light from which particle motion and hence other properties of the 'lattice' may be inferred [2,3,4].

Incident Coherent Probe Beam

0000

Narrow Bragg

~crad~~/

ff 00000

00000 00000 0000

Ordered Crystalline

Broadened Bragg Scattering

o:~·:~ QOOUOO

q,...,OOO vOOOO 0000

Disordered Liquid or Glassy

Figure 2 Bragg Scattering from Hard Spheres

4. Innovations for Light Scattering Experiments

In recent years many new devices have become available with properties that make application feasible in space. These properties include high efficiency, small size, low power consumption, thermal controllability, low weight and mechanical robustness. Some of the devices which have these desirable properties are introduced:

• Available solid-state diode-pumped, frequency-doubled, Nd:YAG laser light source delivers 100 mW at 532 nm in TE~ mode with better than 0.05% rms stability from a line input power of less than 25 W.

• Available silicon avalanche photodiode packages have gains around 50 A/W, greater than 50 % quantum efficiency at green wavelengths, intrinsic noise better than 10-14 WI...JHz, low afterpulsing and correlations and the ability to be operated with a smooth transition over the input range from analog detection to the individual quantum realizations of photon counting.

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• Available single-mode, polarization-maintaining, stress-birefringent optical fibers with a 3 J.l.ID diameter quartz step-index core, a numerical aperture of 0.1, and transmission power density tolerance exceeding 1011 W/m2 permit both light input from, and collection to, remote systems. Their single-mode properties have many special advantages, not only for transmission but for beam control by 'perfect' spatial filtration [5].

• High-speed, multifunction correlators are evolving both in capability and packaging as the various topologies are becoming better understood and implemented. The present single-board correlators, replacing large crates of electronics, will soon be consigned to single chips, making their deployment in arrays feasible in terms of cost, physical size and power consumption.

Including the above, there are essentially five innovative technologies based upon which this complexity of measurement system may be contemplated for applications in space. These are primarily new developments and capabilities in laser miniaturization and stabilization, avalanche photo-diode detectors, fiber optics and component design, correlators and signal processing, and system architecture and packaging.

5. Modeling and Design

In systems where not only are the physics and phenomenology complex, but the implementation involves a large number of interacting disciplines, parameters or conceptual relationships, it is beneficial - some might say essential - to have a model. A suitable and useful model can be purely numerical, or it can include physically verifiable elements for empirical closure. Some of the major purposes of a model are shown in table I.

TABLE 1. Purposes of a model

• Introduce and quantify the physics and phenomenology

• Verify that specifications and constraints are compatible

• Understand interactions between parameters

• Optimize a design configuration

• Predict ideal performance possible witl). that design

• Predict performance deterioration when implementation is not ideal

• Diagnose anomalies found during construction, testing and use

• Build confidence in the technology

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6. Architecture

The establishment of a clean, modular and orthogonal architecture for both the hardware and software of a complex system for applications in space is essential if the system is to be adequately tested and characterized. Any area or interface which is unconstrained in the original plan will haunt the program and the users for the life of the instrument. For the PHlsE instrument chosen as an example here, the modularity is shown in the indented table 2. This is not necessarily the best possible nor only suitable architecture but it served well as a design baseline for understanding later interfaces and relationships.

TABLE 2. Architectural Modularity

Optics Module Optical Systems

Specimen cell optics Bragg and Low Angle scattering

Bragg collimator Scattering screen Relay optics and CCD camera

Dynamic and Static scattering Optical launch and dump/monitor fibers Graded index collection lenses and optical fibers

White light visualization lamps and camera Specimen control

Avionics Module

Calibration and test specimen containment cells Cell movement

Cell exchange carousel Cell rotation for static scattering Rheological and mix/melt oscillation

Laser light source module Laser, power supplies and thermal control Intensity control and monitoring Illumination channel selection and fiber launch

Detectors Correlators Motor drives/controllers

Carousel rotation/specimen exchange Rheology and mix/melt motor Collection fiber angular rotation scanning Attenuator and shutter control White light camera and CCD camera control

System Control Real time data viewing Touch screen interface Computing resources Data buffer and archive storage Telemetry interface

Software

Power Drawer Electrical power and thermal maintenance services

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7. Optical Design

The overall design of the system requires a cohesive combination of optical, mechanical, electronic, thermal and interface properties and integration with computing hardware and software. As in this case, where many elements are intended to preserve versatility by performing more than one function, it is essential to consider the mutual interactions under ali conditions of use, so that operation is not compromised in or by originally unanticipated or innovative combinations of applications. Figure 3 shows a much simplified schematic view of the essential optics by which two types of light scattering measurement can be accomplished, although not simultaneously.

The longer vertical path is for the Bragg and Low Angle scattering whereby the light scattered over a forward direction of ± 60 0 from within a 1 cm high, 1 cm radius sample cell may be observed. The collimated illuminating beam is almost Gaussian about 4 mm diameter to the e·2 intensity contour. The screen between cell and camera, although. incurring a penalty in optical efficiency permits the system to indirectly violate the Lagrange invariant, as is required by the original specification. This necessity for the screen is turned to advantage by anti-reflection coating the concave surface and coating the convex surface with a green varnish filter overlaid by a red fluorescent paint. This not only suppresses reflections and backward scattering from the screen, but also shifts the green illumination to red where the CCD is slightly more sensitive. The optical design is a compromise between aberrations, control. of second order reflection ghosts and manufacturability. The screen also supports the combined beam dump and attenuating monitor expanded in figure 4. This dump accepts the 'P' polarization almost without significant backscatter or reflection and transmits about one part in lOs which then appears within the amplitude dynamic range of the camera. Its 1.07 mm diameter occludes only the central 0.3 0 of the forward view, and the change in intensity efficiency due to penumbral shadowing up to 0.40, depending upon the aperture of the relay lens, ideally between fl2.8 and fl4, can be partially compensated.

Part of the lower portion of figure 3 and all of figure 5 show the Static and Dynamic light scattering system. The paraboloidal skirt allows axial access permitting the observation of a range of scattering angles from 10 0 to 170 0, by a simple rotation of the collection optics support manifold about the cell axis. The collection optics, of which the essential elements each comprise a graded index (GRIN) lens glued to a single-mode optical fiber, is slightly shifted radially to permit out-of-plane observation and thus avoid multiple-reflection flare from the plane face of the combined specimen containment cell and Bragg optics.

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K1400 CCD Array

12-bit CCD Camera

B&LA Scattering Screen Green Underlayer with

Red Fluorescent Overcoat

180mm

Spherical Output Surface

D&S Dynamic and Static

B&LA Bragg and Low Angle

G(RIN) GRaded INdex Lens

Relay Lens

Specimen Chamber

Fiber Input

Figure 3 Dual-purpose Optics for Bragg & Low-angle and Static & Dynamic Light Scattering Measurements

B&LABeam

Parabolic Skirt

B&LA Collimator

Incident Polarization 'PO in Plane of Diagram

Focused Incident Beam

Absorptive Cement Loaded with Ground Black GlaM

Optial Density about NO = 5

Brewster Angle Face (55.6°)

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Uniformly Diffusing Attenuated Output Beam

Figure 4 Bragg Beam Dump and Attenuator Internal Optics

Total Internal Reflection

Glass Cell

100 J.lm diameter Gaussian waist

GRIN Lens

IIl~minating

Light Input Optical Fiber

Cell Axis

GRIN Lens

Scattered Light Collection Optical

Fibers (Three)

Slice of Paraboloid

Transmitted Beam Dump and Monitor

Optical Fiber

Figure 5 Optics of Cell for Static and Dynamic Light Scattering

The illuminating beam is projected by the GRIN lens on the end of the single-mode polarization-maintaining fiber and is totally internally reflected from the optical contour of the paraboloidal skirt to form a Gaussian waist about 100 Ilm in diameter to the e-2

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intensity contour in the center of the cell. After crossing the specimen cell with comparatively little aberration the beam is totally internally reflected again and collected by a well anti-reflection coated slightly-inclined GRIN lens glued to a multi­mode fiber with a 60 ~m diameter core. This dump reduces unwanted reflections. After suitable attenuation, the transmitted light is also usable as an independent intensity monitor.

The GRIN lenses and single-mode fibers of the collection optics imaged by the paraboloidal skirt view an image at infmity with the illuminated region in the center of the cell acting as an equivalent aperture stop. Residual aberrations are rendered insignificant by the capacity of the fiber to transmit only a single spatial mode and the 2 m length attenuates much of the detrimental propagation in the cladding. This restriction of observable scattering angle is sufficiently good that a full rotation of the cell yields between 3 x 104 and 7 x 104 independent measurements of static scattering depending upon the scattering angle. This was inferred from the rate of roll-off of a typical correlation function as the cell is rotated.

8. Specifications Desired and Achieved

A group of properties were originally specified as design targets, with the acknowledgment that in such an innovative system and within only a 24 month schedule, some of them might not be achieved. Since this paper also documents lessons, the shortcomings are examined and explained below. The acronyms are B&LA - Bragg & Low Angle scattering system; D&S - Dynamic & Static scattering system; LLM - Laser Light source Module.

TABLE 3. Specifications Met by Design

Cylindrical sample chamber (height & radius) Sample cells (7 specimens + 1 calibration) LLM optical wavelength B&LA scattering angle range (at the screen) B&LA angular resolution (aberration limited) B&LA resolution at CCD (pixels are 6 ~m 0 ) D&S scattering angle range D&S waist diameter at cell center

lcm 8 532nm 0.25 0 to 60 0 t 0.1 0

8~m 10 0 to 170 0

100~m

t As a single example of many fall-back contingencies, if the B&LA illumination were to fail then a Bragg scattering angle range of30 0 to 150 0 remains accessible using D&S light source and the Bragg camera.

TABLE 4. Specifications Validated by Testing

LLM power stability (rms below 0.01 Hz) LLM power stability (rms below 1 MHz) Polarization ratio of fiber-delivered light B&LA collimation B&LA Gaussian beam incident radius D&S independent measurements / cell rotation LLM package weight LLM power consumption

<0.02% <0.05 % > 100:1 30 mradians 4mm > 3 x 104

11.5 kg 24W

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In table 5 below, the first figure, before the colon, is the target specification and the second is that obtained by measurement during testing.

TABLE 5. Specifications Shown Deficient by Testing

LLM power for B&LA (with attenuator)

B&LA forward scattered flare intensity

0.1 - 30 mW : 3 mW

< I x 10-7 : > I x 10-5

The conten~ of table 5 are especially interesting, if disappointing, because they limit the desired performance, even before launch. The most serious power reduction came from the fiber optic connections, which are extremely critical for single-mode polarization-maintaining fibers; a small fraction of a micron misalignment causes significant power loss. Additionally, a small fraction of a degree of differential rotation at the coupling causes cross-polarization leakage and hence the fiber acquires an unexpectedly large sensitivity to mechanical vibration, bending, microphonics and thermal changes, which is almost completely suppressed for a critically well oriented coupling. Such temporal fluctuation can impair the quality of the retrieved correlation function. Power fluctuations are also introduced by feedback into the laser cavity of reflections from either end of the fiber, especially when properly aligned for a high­efficiency launch, and/or from Fabry-Perot effects within the fiber. These were eventually suppressed by fmishing the fiber at an angle; in this system 8 0 seemed a good choice. Such slant-polished ends place additional constraints on alignment of the feed-through connections.

The variable attenuator, which was based upon rotating the frrst of two GLAN prisms, tested as having better than 1:109 ratio in isolation the laboratory and better than 1:107

after installation in the LLM where its performance is more sensitive to angular run-out based upon manufacturing errors in the rotating prism, less than 1.3 minutes of arc for the better of the two prisms. Unfortunately the 'vacuum preparation' of the motor and drive for out-gassing and off-gassing suppression in the Space Shuttle environment caused the unit to seize, rendering it ineffective, fortunately in the maximum

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transmission position. Although this could have been catastrophic, the unanticipated attenuation in the bulkhead feed-throughs introduced a large loss, albeit fixed, which gave a delivered power (3 m W) seemingly acceptable for many of the proposed experiments.

The intensity of forward scattered flare introduced by the collimator is much higher than anticipated; sufficient to impair or perhaps eliminate the capability for very low angle scattering measurements. This arose partially from a mechanism where at each stage of manufacture an originally acceptable design was contingently modified in response to build-up testing, such that each change, while apparently most reasonable at the time, produced an unacceptable final result. The addition of a sampling prism monitor introduced extra surfaces. The adhesive intended to eliminate reflection at the fiber face failed at the high power at which it was tested - marginally higher than was eventually available or even necessary: no acceptable substitute was found in time and the fiber was therefore slant cut with an air gap to prevent reflections from destabilizing the laser. This introduced extra flare and surfaces to the collimator, which even though very well anti-reflection coated compromised the original design concept of a single super-polished output surface. Flare from residual transmission of unstripped cladding modes now seems as if it may have exceeded that from the surface reflection ghosts to which blame was originally ascribed.

9. Aspects of Testing and Verification

Four properties are illuminated by testing: these are initial function, anomalies, deterioration, and contingencies. Initial function confirms the achievement of performance and specifications. The examination of anomalies permits judgment of whether the observed behavior may be considered to be acceptable, and often points to things not suspected earlier: it is well to heed these observations. Deterioration may not be apparent immediately but is often indicated by small changes during environmental, shock and vibration exposures and burn-in. Symptoms may be quite subtle but should be given great credence. The original designed-in contingencies may be verified during testing and new operational protocols explored to circumvent shortcomings arising from discovered defects that cannot be remedied within the available time and resources.

Testing of the PHASE system during build-up and characterization is exemplified by the following notes which dwell primarily upon the features which were NOT as desired, rather that the enormously greater number which proved to be entirely satisfactory .

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The laser light source module was monitored for 1200 hours using data logging of the output intensity with a temperature stabilized silicon detector coupled to a 1 MHz analog to digital converter. Rectifiable defects included instabilities from the subsequently repackaged power supply, its four control loops for the thermal stabilization of the pump diode, Nd:YAG oscillator and frequency doubling crystal, electrical noise in power monitors, destabilizing optical feedback from aligned optical fibers and unstable adhesives.

Specimen containment, cell filling and sealing presented particular problems. Leakage inwards causes bubbles, and leakage outwards changes the concentration ratio upon which experimental data so critically depends. Problems remain in these areas for some of the seven specimen cells.

Stray light suppression using surface coatings, geometrical stops and baffles and careful optical design has proven excellent in the static and dynamic system but less satisfactory in the low angle forward scattering because of design modifications found essential for other reasons during manufacture. An additional flare source arose from inadequately attenuated cladding modes, fiber properties not initially fully understood. This most critical area of flare suppression almost always bounds the ultimately achievable performance.

Testing optical fibers introduced a host of unexpected problems, not all of which are yet understood nor fully rectified. The major problems were with terminations and their alignment, leading to intensity transmission reduction, intensity instability and polarization ratio deterioration and inconstancy.

As always, moving parts are a cause of pain. The mix/melt motor shaft broke during testing and was replaced with a stronger material. Bearings became notchy and were later replaced, the prism rotator seized from inappropriate lubricant too late to replace and the cell positioning carousel locked with temperature changes. Preventive adjustment relaxation led to backlash and hysteresis that has to be tolerated.

Evaluation of other testing by exposure to thermal extremes, shock and vibration, generation of, and sensitivity to, electromagnetic interference showed acceptable performance and no significant deterioration with time or environment. The system also contains many built-in test monitors for continuous health assessment, most of which perform their tasks acceptably.

Software is never fully testable. The best that can ever be achieved is the wide-ranging exercise and bug-fixing until the frequency or seriousness of observed defects reaches some pre-agreed level of tolerability.

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10. Lessons

In conclusion, some of the lessons that can be illustrated by experiences with the PHisE program so far include:

• Have a passion for the science and the subject • Obtain and retain funding by sustained image (cf Apollo Program) • Confirm old knowledge and discover new • Defme 'success' ahead of time: design for 'graceful degradation' • If you insist on versatility, you must accept compromise • Optimize architecture of instrument, program, and experiment • Avoid catastrophe from single-point failures • Demand isomorphic mapping of software and hardware modules • Incorporate system health and diagnostic monitoring • Test everything; heed subtle observations and mysteries • Beware four areas causing anguish: power supplies, software, schedule, cost

11. References

I. Chaikin, P. M., (1996) Princeton University [Private Communication].

2. Rogers, R. B., Meyer, W. V., Turner, W., Zhu, J., Chaikin, P. M., Russel, W. B. Compact Laser Light Scattering Instrument for Microgravity Research. OSA Topical Meeting, August 21-24, 1996, Capri, Italy (To be published in Applied Optics).

3. Lant, C. T., Smart, A. E., Cannell, D. S., Meyer, W. V., Doherty, M. P. The Physics of Hard Spheres Experiment - A General Purpose Light Scattering Instrument. OSA Topical Meeting, August 21-24, 1996, Capri, Italy (To be published in Applied Optics).

4. Meyer, W. V., Tscharnuter, W. W., MacGregor, A. D., Dautet, H., Deschamps, P., Boucher, F., Zhu, J., Tin, P., Rogers, R. B., Ansari, R. R. Laser light scatteringfrom an advanced technology program to experiment in a reduced gravity environment. International Symposium on Space Optics, April 18-22, 1994, Garmisch-Partenkirchen, Federal Republic of Germany, EOSISPIE Joint Venture, Paper # 2210-21.

5. Brown, R. G. W. (1987) Dynamic light scattering using monomode optical fibers. Applied Optics 26, 4846-4851.

HARD SPHERES IN SPACE: LIGHT SCATTERING FROM

COLLOIDAL CRYSTALS IN MICRO GRAVITY

Abstract.

JIXIANG ZHU AND P CHAIKIN Department of Physics, Princeton University Jadwin Hall, Post Office Box 708, Princeton NJ 08544-0708, USA

MIN LI AND W B RUSSEL Department of Chemical Engineering, Princeton University, Princeton, NJ 08544-0708, USA

R ROGERS AND W MEYER NASA Lewis Research Center, Cleveland, OH, USA

R H OTTEWILL School oj Chemistry, University of Bristol, Bristol, U](

AND

STS-73 SPACE SHUTTLE CREW Johnson Space Flight Center, Houston, TX, USA

We report some of the results of our CDOT (Colloidal Disorder-Order Transition) experiments on the space shuttle Columbia, STS-73. The sam­ples (0.518m PMMA spheres suspended in an index matching mixture of decalin and tetralin) ranged in concentration from 0.49 to 0.62 volume frac­tion. Light scattering was used to probe the static structure and the particle dynamics. Digital and 35mm photos provided information on the morphol­ogy of the crystals. The surprises that were encountered in micro-g include the preponderance of RHCP (Random Hexagonal Close Packed) structures and the complete absence of Face Centered Cubic (FCC) structure (until the samples returned to one g), existence of large dendritic crystals floating in the samples where liquid and solid phases coexist and the rapid crystal­lization of samples which exist only in glass phase under the influence of one g. We discuss the possible growth dynamics with and without gravity.

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E. R. Pike and J. B. Abbiss (eds.), Light Scattering and Photon Correlation Spectroscopy, 387. © 1997 Kluwer Academic Publishers.

ZENO: CRITICAL FLUID LIGHT SCATTERING

IN MICRO GRAVITY

Abstract.

ROBERT W. GAMMON, J.N. SHAUMEYER, MATTHEW E. BRIGGS, HACENE DOUKARI AND DAVID A. GENT

Institute for Physical Science and Technology Univer'sity of Maryland, College Pal'k, MD 20742, USA

The Zeno instrument was designed and developed for flight on the U.S. Space Shuttle to measure the density fluctuation decay times of a liquid-vapor critical fluid using photon correlation spectroscopy. The instrument was flown and operated successfully on two flights: March 4, 1994 on STS-62 as part of the USMP-2 payload and Feb. 22, 1996 on STS-75 as part of the USMP-3 payload. This paper will give an overview of the experiment principles, discuss the major design challenges, and show some performance data. Of particular note was the temperature control of better than 3 microK (rms) for hours, which was essential for the effort to make measurements closer than I mK to Tc. Both flights taught us lessons about the difficulty of controlling the local density (at the laser focus) in microgravity. CorreIograms were recorded with a custom interfaced ALV 5000 Correlator. In the second flight, correlograms were recorded from 500 mK down to 2 mK at 24 temperatures, 383 correlograms in all. The fitted decay rates generally gave the desired I % precision. The phase boundary was located with unprecedented precision of ± 20 J.l.K. More details of the experiment Science Requirements, the personnel, apparatus, and results are displayed on the Zeno homepage at http://www.zeno.umd.edu.

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E. R. Pike and J. B. Abbiss (eds.), Light Scattering and Photon Correlation Spectroscopy, 389--400. © 1997 Kluwer Academic Publishers. Printed in the Netherlands.

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ZENO: CRITICAL FLUID LIGHT SCATTERING

IN MICRO GRAVITY

1. Introduction

All fluids exhibit universal, anomalous behavior in their thermodynamic and transport properties in the region of their liquid-vapor critical point. This is most dramatic when one looks at a critical fluid in an optical c~1I and sees the sample become an opaque with opalescence. This cloudiness is the consequence of the enormous thermodynamic fluctuations occurring in the fluid. The renormalization group calculations about critical thermodynamics have shown how to calculate the equilibrium divergences and how to relate different critical systems into universality classes. The theoretical calculation of transport processes such as thermal conductivity and viscosity in critical systems is much less developed.·

The goal of the Zeno experiment is to measure the decay rate of critical fluctuations in xenon close enough to the critical point to determine the limiting behavior. (Critical properties of xenon are: Tc=17.57 °C, Pc=58.4 atm, critical density = 1.110 gm/ec.) The experiment uses photon-correlation light-scattering spectroscopy to measure the characteristics of the density fluctuations as the critical point is approached. The primary measurements are photon correlation functions, digitally processed in real time from the detected scattered-photon pulse stream. We also monitor the sample transmission which provides data about the local density and the sample turbidity when we get close to the critical point.

In the second flight we followed a timeline with only cooling steps of decreasing size and cooling rate in order to avoid problems found in the first flight with the locally perturbed density ncar the sample cell windows. We also took enough data far from Tc to characterize the sample cell reflectivity and thus provide a way to extract an accurate correlation range from the measured intensities. This data should also allow the analysis of the first flight data despite its density uncertainties.

In the following report we will give an overview of the apparatus, review the prohlems found in the first flight, describe the plan for the second flight, and look at the second flight performance. This performance is summarized by the timeline plots of temperature and turbidity data. We show the improved control of density errors, the correlogram data, and our efforts so far to extract the correlation range. The surprise of a large density error from laser heating is discussed. Finally we show plots resulting from the slow temperature scan to locate the phase boundary (phase transition temperature). The reader is urged to look at the USMP-2 One Year Repore to see a discussion of the first Zeno flight and a bibliography of the field.

391

M1 LASER

M2

o PO 1

PMT1

Figure 1. Zeno optical layout. Components: He-Ne Laser, Mirrors M 1-4, Beam Splitter B 1, Shutters S 1-2, Filters F 1-2, Photodiodes PD 1-2, Lenses L 1-4, Apertures A 1-4, Photomultipliers PMT 1-2, and Thermostat TH. The sample cell is in the center of the three shell thermostat. The scattering angles are approximately 12 o and 168 0 for either beam path chosen by shutter.

2. The Apparatus

The optical layout is shown in Figure 1. The sample cell was of unique design. It featured stepped ("top-hat") windows

which allowed a large sample volume (0.7 cm) while allowing the beams down the axis to pass through only a thin, 1 00 ~ sheet of the fluid sample. The thin section limits the multiple scattering from the sample to less than 1 % in our experiments. In the cell flown the windows were an epoxied stack of a larger sapphire widow and a smaller fused quartz window with AR coatings on the outer surfaces and no coatings on the inner surfaces next to the sample. All surfaces were superpolished. The fluid seals (to 70 bar) were crushed copper knife edges machined on the copper flanges. The remaining body parts were copper with some indium used to assure good thermal mating of the parts. The cell was filled to within 0.1 % of the critical density, estimated by the symmetry of the sample volume and the centering of the meniscus when 2 mK below Tc. The filling and sealing valve is built into the cell wall.

The temperature control and measurement is based on using AC bridges which compare a thermistor and stable resistor (1 ppm/K) ratio with the ratio from an computer controlled inductive voltage divider. There are three nested, controlled aluminum cylinders providing the temperature control and setting a final gradient across the sample cell of < 1J.LK1cm. A fourth transformer is switched between five different half bridges, on command, to monitor the temperatures of the sample, the three controlled shells and a Pt calibration thermometer on the sample cell. The least

392

count (digitization) resolution of the measurements is 1 f.1K and the measured RMS is 1.5 J.1K, just larger than the least count of the AID circuit.

3. Plan For The Second Flight

We made our plans for the second flight on the basis of testing for equilibration when 4 K from Tc ("mixing"), making sure that we did not cross into the two-phase region until the end of the experiment, a slow scan search for the phase boundary, and to limit the rate of temperature change so that we kept the density errors from lagging window surface temperatures to < 0.1 %. This required using controlled ramps between temperature points with ever lower slopes. In the course of pre-flight testing we found that we could use much slower scanning rates during Tc searches and we eventually settled on rates of -100 J.1K/hr as a compromise rate for the search.

4. Second Flight Timeline

During the second flight we operated with the flight computer in a paused state and controlled the sequence with a combination of uploaded sequence scripts and ground commanding. We actually carried out the planned timeline. The only changes from the ground baseline were that we waited until the drifts in the transmission (turbidity) signal corresponded to < 0.1 %Ihr before taking data or moving to next temperature. We equilibrated and took measurements at: 4K, 1.4K, 750 mK, 300 mK, 200 mK, 100 mK, 56 mK, 30 mK, 18 mK, 10 mK, 5.6 mK, 3 mK, 1.8 mK, 1.0 mK, 560 J.1K, 300 J.1K, and 180 J.1K. These temperatures are temperature distance above the nominal value of Tc. During the mission we used the data in hand to refine our estimate of the location of Tc on our temperature scale.

The timeline and experiment history is well illustrated by the flight data in Figure 2

and Figure 3. Figure 2 shows the sequence of temperatures seen by the sample over the course of the mission. In this log plot, the temperature steps and slopes all look the same. but they are logarithmically decreasing. Figure 3 shows the that after the laser power was lowered from 17 J.1W to 1.7 J.1W, the turbidity jumped up and then began a strong, uncontrollable drift downward. We have concluded that this drift was real (though never seen on the ground at a power change!) and due to a low density region that had developed from laser heating of the inner window surfaces. When the laser power was changed, the density began moving back towards the average (critical) density. By that time the sample temperature was so close to Tc that this was going to take weeks to relax away. Unfortunately there were only four days of the mission left so we plunged on, taking the planned data. We were also making needed adjustments in our estimate of the Tc location based on the data in hand (turbidity, intensity and decay rates).

393

0.1 · . ••••••••• ".- - - - - -. -' - - '0" - _. _ •• _ •••• _. _ ••• _ •• 0. _ ••• - -. - -. - _. -. -, _. _. _. _.-

-~ -I-() 0.01 •••• 0000 __ ; __ ••• _ ••••• _; __ • __ ....... ~_ •••• _. _ ••• _0 00 _0 •••••••••• _ ••• a:. ___ ..... . · " . · '. . · .. .

I · .. . · .. . I- • " I · .. ,

0.001

0.0001

• • • I • - - _ •• - _ ••• o. •••• _. _. _ •• 1. _. ___ ••••• ~ __ • _. _. ___ ••• _ •• _ •• __ ._

! ! ! ! ! : r\ .......... : ........... : ........... ; ........... : ........... : ........... : ' .. \

: : : : : : I • I • • • • , . . . . . · . . . . . .. ...

0.00001 I • • • • • ........... ~ .. _ ... _. _. "0'···· _. _ .. -"" ... _. _. _. _. 't- 0- -_ ... _ .. ,. ...... _. -'.0- _. _. - _ •• -· . . . . . · . . . . . · . . . . . · . . . . . · . , . . ,

0 2 4 6 8 10 12 14

MET time (days)

Figure 2. The temperature timeline for the second Zeno flight. Notice that the sample did not cross into the two phase region until the last day of the mission.

5. Density Monitoring

We made a quantitative study of the density transients observed on some temperature changes during the mission. An example is shown in Figure 4. Here the thermal expansion contribution bas been subtracted and the residual phase interpreted as due to density changes. Thus we can plot op/Pc during and following a temperature change, in this case from 500 mK to 300 mK. This change was done with a sequence of 10 steps, 6 minumtes apart. Notice that each step can be distinctly seen in the response of the fluid fluctuations. This is a striking demonstration of the effect of the surface film density changes occuring during the temperature changes; the laser beam crosses essentially only fluid which is in such a film. We were delighted to see the good agreement between the calculated density dynamics, modeled with numerical

394

0.25 .-.....,....-r--r--r--.----,r--.----,--.----,--.----,--.-__..

0.20

....... g 3 0.15

~ '0 :e F 0.10

METtime (dayS)

Figure 3. Turbidity measurements over the mission. This is acrually the turbidity-path-Iength product calculated from the narurallog of the transmission Tx. A change of 0.01 in "rurbidity" corresponds to a 1% change in Tx. Notice the dramatic response of this signal at the time of the path change/laser power change at MET day 10, hour 7:00 and the indication of phase separation on MET day 15.

solutions of the adiabatic effect equations for a critical fluid.3 The input to the model are the dimensions of the sample volume, the properties of critical xenon, and the measured temperature of the cell wall during the ramp. The comparison is absolute, without scaling. Attempts to model this effect for terrestial measurements have so far failed because of convective flows which quickly smear the wall induced density changes in the adiabatic expansion layers.

6. Measured Correlation Functions

Precision measurements of the fluctuation decay rates, r, for two supplementary scattering angles were made on both flights. These were the first photon correlation measurements performed in space. A total of 383 valid correlograms were recorded at 24 temperatures from 500 mK down to 2 mK during the second flight. They were recorded in groups of 15 or more at each temperature point. They were recorded in the dual mode allowing both the forward and backward correlograms to be recorded

395

O.CXXXl

-0.(()()5

<..> -0.0010 a..

........ a.. c..o

-0.0015

-0.0020

-0.0025 L..L..-___ --'-___ -L..-___ --'-___ -L..-___ -J

o 2

time (hours)

Figure 4. Comparison of calculated and measured density perturbations due to a temperature ramp applied to sample cell wall. The ramp was from 500 mK to 300 mK from Tc. occurring at MET day 3. hour II :22. The ramp consisted of 10 steps of 20 mK each over an hour interval. The comparison is absolute. without any scaling. The smooth curve is calculated based on the adiabatic equations for a hypercompressible fluid . The noisy curve gives the measurements.

simultaneously. An example of the correlograms produced by our flight adapted, AL V -digital correlator is shown in Figure 5. The figure shows both forward and backward scattering correlograms which were computed and recorded simultaneously using 1.7 JlW of laser power at a temperature of "2 mK" (see below) from Tc. Notice the clear presence of forward scattering dynamics in the backscattering correlogram. This is due to window reflections and is accounted for carefully in the fitting analysis. There is also a small amount of backscattering in the forward scattering correlogram which is also trea~ed in the fitting functions . The signal-to-noise is good, eVen with this low laser power, du~ to the very strong scattering close to Tc.

After least-squares fitting, the correlogram sets recorded at each temperature are used to produce an average fluid critical flucutation decay rate and its statistical error bar. The average decay rates are shown in Figure 6 and Figure 7 for forward and backward scattering angles (11.465 °and 169.546 ° angles, in the fluid) as plots versus

396

2

100 lime (sec)

Figure 5. Correlogram computed and recorded by the AL V correlator in the Zeno instrument during the second flight. Laser power 1.7 mW. Temperature "2 mK" above Tc. The round point data symbols are from forward scattering and the + symbols are from back scattering. The forward scattering has the slower decay rate so drops down 10 the background correlation at longer times. The "faster" fluctuations are seen in backscattering . Notice the clear presence of forward scattering dynamics in the backscattering correlogram from the window reflection.

temperature-distance from Tc. These are all from the second flight. The ground (1 g) points are from immediate post-flight testing at the Kennedy Space Center. The error bars are usually smaller than the plotting symbols here, except for the low power, backscattering correlograms.

7. Locating The Phase Boundary Near Tc

It is a common experience for critical point experiments that have been carried on the Shuttle to experience a shift in temperature calibration from the launch stresses. Zeno has also seen this. Even though the drift rate of the repeated measurement of Tc over 3 years has shown the drift to be < 1 mK/year, the USMP-3 launch caused a -2.2 mK shift in the location of Tc on the instrument temperature scale. Thus we needed to

100 ................... : ..................... ; ........... .

o : 0 0 , ---=---: ' , , , ...... riw ... _ ...... _ ........ MM __ ··"·· .. ___ •

, ,

0.0001 0.001

T - TC (I<)

397

, , ................................

0.01 0.1

Figure 6. Log-log plot of fluctuation decay rates for forward scattering (11.465 0 angle. in the fluid) versus temperature-distance to Tc. Symbols: squares. 17 11 W scattering. flight 2; circles. 1.7 11 W scattering. flight 2; triangles pointed up. 1711W scattering. post-flight 2; triangles pointed down, 1.711W scattering. post-flight 2. The solid curve is a prediction based on equations of Burstyn. Sengers. Ferrell and Bhattacharjee.

adjust our estimate of the location of Tc during the data collection. Then based on our last estimate we set up the final scanned search for the phase boundary on the last day of the flight. The forward scattering intensity seen by the photomultiplier PMT -1 is plotted versus temperature in Figure 8. This data was fitted as two straight lines with the intersection at To. The temperature To we take as the phase separation temperature at the local density probed. The fit gave a best fit value for To with error estimate of± 10 J.l.K. The backscattering data was fitted in the same way. The two fit parameters for To agreed to within ± 20 J.l.K. For comparison, the post-flight,search done in 1 g with all instrument settings the same as flight.

One can see that the maximum intensity reached is 3x lower on earth and the peak is broader with no break in slope at the maximum. The precision found in locating the phase boundary (about 0.6 mK below the Tc) in low gravity is almost a decade better than the best terrestrial measurements. The transition is much: ..:.sharper in microgravity, probably due to the lack of convection and sedimentation, which smear

398

o o 8

Figure 7. Log-log plot of fluctuation decay rates for backscattering (169.546 0 angle, in the fluid) versus temperature-to Tc. Symbols: squares, 17 I1W scattering, flight 2; circles, 1.711W scattering, flight 2; triangles pointed up, 1711W scattering, post-flight 2; triangles pointed down, 1.7 I1W scattering, post-flight 2. The solid curve is a prediction based on equations of Burstyn, Sengers, Ferrell and Bhattacharjee.

the transition. We believe this to be the sharpest critical transition ever seen except for the helium lambda transition.

8. Conclusions

The second flight of the Zeno experiment provided a surprise in the discovery of detectable laser local heating even though the laser power was only 171lW. We took time to characterize the "window interferometer" during the flight. Thus it was possible to use the interferometer to insure good local density stability until late in the mission when the laser power was changed.

399

t\ • • • \ .. .. ..." t .. at ,.,

• ,.'!;:t' , ;., ,...,.,,~ oJ. "~~". :' , • • • • .'t! ... ~,'l':~·''' ..... ·A''· •.• ':. ·t~;'~··::~'L' '1 J

e .. "~' ••• , .~t.*t ~".I':." I t'.. , .. e, ~.. .:" II .. ,:. 'S f .... L. \\.". ..... • .. ~ .. ij . t.· t ,;"" 4' .'..). :L ., , ;. .. , ... ~.,~!

!I~ ~~: f; ) .... " .. .. . it

Figure 8. Microgravity slow-scan search for the liquid-vapor phase-coexistence phase-boundary. Scan rate was -100 I1K1hr. Data are marked with the + symbol while the straight lines were least-squares fitted to the data with the intersection a fitting parameter, To. The transition observed is much sharper than observed on earth. The lower data set is from a Post-flight (l g), slow-scan search for the phase-boundary. Scan rate was -100 I1K1hr, as in the microgravity scan. Data are marked with a dot symbol.

This experiment has reached the limits of the particular sample flown. The only way to lower the amount of local heating is to use a cell with all sapphire windows (14x higher conduction makes the local temperature rise 14x smaller). We have produced a flight qualified version of such a cell but it was made too late to be integrated into the Zeno experiment. One could lower the laser power, but only if detectors with much higher quantum efficiency were used: the laser powers used in the Zeno flights were at the shot-noise limits for recording precise correlograms in the time available. Finally, it would be good to know why the window surface absorption was 24x higher than the expected losses from superpolished surfaces (1 ppm). It seems that the greatest challenge turns out to be, not the approach to eqUilibrium, but rather the tiny temperature non-uniformities in the sample fluid, which make themselves strongly felt only in microgravity.

Finally, we have found the best, detailed confirmation of the adiabatic effects occurring in critical fluids in response to wall temperature changes. The successful modeling of the temperature-step-induced density changes during the second mission was only possible because of the data from the low gravity measurements in the unique Zeno sample cell.

400

Acknowledgements

This work was sponsored by the Microgravity Applications and Science Division of NASA through NASA-Lewis under contract NAG3-25370. We wish to acknowledge the tireless efforts of our project managers, John Borden at the University of Maryland, Richard Lauver at Lewis Research Center, NASA, and Richard Reinker at Ball Aerospace, who kept all the parts of this project moving to the end. Robert Stack of Ball Aerospace worked energetically to get the instrument checked out and tested for the reflight, and prepared himself to be an expert at commanding. Robert Berg and Michael Moldover were always sympathetic and helpful as they helped us and worked to get their related experiment on the viscosity of xenon, CVX, designed and built. Richard Ferrell and Jan Sengers have offered much stimulation and guidance over many years on the importance and subtle issues of critical transport coefficients.

References

1. R.W. Gammon and J.N. Shaumeyer, "Science Requirements Document for Zeno," (NASA, Microgravity Science and Applications Division (MSAD), Washington, D.C.,1988).

2. R.W. Gammon, J.N. Shaumeyer, M.E. Briggs, H. Boukari, D.A. Gent, R.A. Wilkinson, "Highlights of the Zeno Results from the USMP-2 Mission," NASA Technical Memorandum 4737, (NASA, Marshall Space Flight Center, Huntsville, AL, 1996), p.5-135.

3. H. Boukari, J. N. Shaumeyer, M. E. Briggs, and R. W. Gammon, "Critical Speeding Up in Pure Fluids", Phys. Rev. A 41,2260-2263 (1990); H. Boukari, J. N. Shaumeyer, M. E. Briggs, and R. W. Gammon, "Critical Speeding Up Observed," Phys. Rev. Letters 65, 2654-2657 (1990); and references therein.

STATIC AND DYNAMIC LIGHT SCATTERING IN PHASE­

SEPARATING SYSTEMS

Abstract.

S. V. KAZAKOV AND N.I. CHERNOVA Faculty of Physics, Lomonosov State University, Moscow 119899, Russia

The diffusion coefficients, susceptibility, surface tension and viscosity are measured by quazielastic light scattering spectroscopy for several phase­separating mixtures in the single-phase region along lines of constant con­centrations and in the two-phase region along the coexistence curve. An analytical form of the universal functions both for static and for dynamic properties is derived using the concept of pseudo-spinodal and quasichem­ical approach to the symmetrization of the binary coexistence curves. The behaviour of the mobility of components in the vicinity of the critical point is explained.

401

E. R. Pike andJ. B. Abbiss (eds.J, Light Scattering and Photon Co"e/Jltion Spectroscopy, 401-422. © 1997 Kluwer Academic Publishers.

402

STATIC AND DYNAMIC LIGHT SCATTERING IN PHASE-SEPARATING SYSTEMS

1. Introduction

The light scattering (light beating or photon correlation) spectroscopy is being used to an increasing extent in connection with its high informativity. Quazielastic light scattering spectroscopy (LSS) is a noninvasive and nondestructive probe of diffusion in complex fluids. LSS has been used successfully to study solutions of different nature, including molecular mixtures of organic liquids and solutions of macromolecules, proteins, polysaccharides, synthetic polymers, colloidal particles and aggregates, micelles, and microemulsions. Specific interest has been aroused by investigations of bulk or surface static and dynamic scattering in the neighborhood of the consolute critical point (CCP) as well as by a study of common dependences of transport and equilibrium thermodynamic properties in the region of phase transition.

In this paper we consider, as an example, liquid-liquid phase transitions in binary mixtures, taking into account that the recent theory of critical phenomena [1-4] attributes all liquids, their mixtures, magnetics and 3D-Ising model to the same class of universality. Since the study of critical phenomena concerns with the behavior of a system whose correlation length is very large compared to interatomic spacings, it is natural to suppose that many of the microscopic details will be unimportant for the critical behavior [4].

The concept of the universality of critical phenomena generalizes the law of corresponding states and assumes that the anomalous thermodynamic property Y(x,T), whose asymptotic behavior near the critical point is determined by the critical exponent cp,

lim Y(x, T) = Yo 't'P, 't ..... 0

is described in the hydrodynamic approximation by the following expression:

Y(x,T)/(y 0 AxcplP Zo CP) = fy(z),

(1)

(2)

403

where fy(z) is the universal function of the scaling argument z = 't I axll13, 't = IT-Tell Tc is the reduced temperature, Tc' the critical temperature; ax = IX-XciI Xc is the reduced concentration; Xc, the critical concentration; 13, the critical exponent of the coexistence curve, and Yo is the critical amplitude. The curve z = - Zo is the coexistence curve (CC). The universality of critical exponents and ratios of critical amplitudes are followed from here.

According to the theory of dynamic critical phenomena [5] the order parameter relaxation rate has a homogeneous scaling form

rD = 00 qA Q(Y,z), (3)

where Q(Y,z) is the homogeneous function of the scale arguments y = qrc and z; rc' the correlation length; Qo is the parameter introducing the time scale; A, the dynamic critical exponents; q, the module of wave vector. In the hydrodynamic approximation, qrc« 1, this law is transformed to the form

(4)

here v is the critical exponents for the mutual diffusion D; Do, the nonuniversal amplitude.

It is essential that under the conditions of the hydrodynamic regime, which were experimentally maintained in this study, the concept of universality both for the static and dynamic properties has the similar representation. The theory of static critical phenomena gives an analytical dependence of the universal function (e.g., linear model [6]) on thermodynamic states, and in the frame of the theory of dynamic critical phenomena the dependence of the universal function on the thermodynamic scale argument z is hidden within the spatial scale argument y (or correlation length).

Recent development of the concept of the pseudo-spinodal [7] gives a new convenient way to characterize data and allows to obtain a unified analytical form for the static and dynamic universal functions.

The aims of this study are (i) to measure the temperature and concentration dependences of mutual diffusion coefficient D, inverse scattering intensity 1-1, shear viscosity l'\, and interfacial tension cr, which can be obtained by the bulk and surface light scattering spectroscopy, in homogeneous one-fluid phase region of phase diagram approaching the miscibility gap, as well as in the two-fluid coexisting phases along the liquid-liquid CC; (ii) to analyze and elucidate in detail the concept of universality of critical phenomena for the static and dynamic properties of binary mixtures at the hydrodynamic approximation; (iii) to investigate the shape of the CC; (iv) to explain the behavior of the mobility of components in the vicinity of the CCP.

For the purposes we use bulk and surface light scattering spectrometers with simultaneous measurements of integral intensity.

404

2. Molecular Motion and Light Scattering (Bulk and Surface)

2.1. BULK SCATIERING IN THE BINARY LIQUID MIXTURES

The existence of light scattering in condensed matters results from thermal collective and single-particle motion. Kinetics of the thermal motion is revealed in an unshifted spectral line of the scattering light. Spatial and energy distribution of scattering radiation is connected with spatial and temporal molecule distribution in the matter [8]

10 V 000 4

l(R.oo) = ---- (5)

00

where S(q,oo) = J J exp(ioot - qr) G(r,t) dt dr, the generalized structure factor which is V-oo

connected with the Van-Hove correlation function G(r,t) = <~x(r,t) ~x(r,O» = <~x2> exp(-Dq2t), i.e. with the statistical characteristic of liquid medium; R, the radius-vector from scatter to the point of observation; c, the velocity of light; <1>0' the angle between R and vector of electric field E; q = qo - q' = 2Qo sin(812), Qo and q' are the initial and final wave-vectors of scattering light, respectively; 8 is the scattering angle, V is the volume of scattering. The value of <~x2> characterizes the fluctuations in order parameter. The strength of these fluctuations is determined by the "generalized susceptibility" [9]. For binary mixtures the susceptibility refers to (8xjl0J.li)T,p where Xi is the concentration and J.li is the chemical potential of either component.

The generalized structure factor S(q,oo) can be obtained in hydrodynamic approximation, herein the interpretation of the scattering light spectrum is found on the basis of Onsager hypothesis. According to this hypothesis, spontaneously increased excitations, the measure of which is fluctuation, vary with time like the excitations, caused by the external actions. Hereby their behavior is described with usual macroscopic equations of hydrodynamics.

The spectral distribution of the intensity of light, scattered on the concentration fluctuations:

rDht l(R.oo) = 100 <~x2> -----

(OO-(Oo)2+rD2 (6)

This equation describes central line of light scattering spectrum, having Lorentzian form with r D half-width, connected with diffusivity and integral intensity proportional to the strength of fluctuations, i.e. to the differential of the chemical potential J.li with respect to concentration Xi.

405

2.2. SCATTERING ON THE SURFACE WAVES IN THE BINARY MIXTURES

A liquid surface is subjected to continuous disturbances on the molecular level. These disturbances appear in the form of capillary gravitational surface waves (ripplons) which in turn are present due to random pressure fluctuation in the liquid bulk. The surface tension plays the role of a restoring force for short-length capillary waves (Ar « 1) which are damped by the bulk viscosity, and gravitational forces are restoring forces for long-length gravitational waves (Ar» 1), both of them tend to return liquid surface into the equilibrium state.

The molecular interfacial excitations can be considered as superposition of small amplitude (-10 A 0) capillary waves. Detail consideration of the liquid surface waves has been carried out in [10] and for the liquid-liquid interfaces has been developed in [11]. More general cases - the capillary wave propagation on a monolayer-covered liquid surface, possessing visco-elastic properties - has been considered in [12].

The dispersion equation for capillary waves' propagation on the liquid-liquid interface has the form

[ll 1 (q+kI)+1l2(q+k2)Hll IkI/q(q+kI)+1l2k2/q(q+k2)+crq2/ro]= = [llI(q-kI)-1l2(q-k2)]2 (7)

where ki2 = q2 + roPi/% i = 1 or 2 for the upper or lower phases, respectively, q = 2rrJAr, ro = -iroo - rD is the complex frequency, roo is the temporal frequency of the waves; Pi, lli and O"i are the density, the shear viscosity and the surface tension, respectively.

For the capillary waves on the liquid-gas interface the dispersion equation is simplified

(8)

where ro0 2 = crq3/p and rD = (1l/P)q2. The solution of the dispersion equations with respect to surface tension and shear

viscosity was computed by a Newton-Raphson method which was found to be insensitive to the initial approximation and convergence was rapid.

The general theory of light scattering on the surface of liquid or the interface between two liquids gives following expression for the intensity of diffusion scattering

I(R,ro) = (9)

where IR is the intensity of reflected light, So is the area of scattering surface, 9i is the angle of falling. Time variations of molecular disturbances on the interface according to the Doppler effect cause the modulation of scattered light, i.e. the line of monochromatic radiation is shifted and broadened.

406

Using the connection of the generalized structure factor S(q,ro) with the autocorrelation function of surface fluctuations, one can obtain the spectral distribution of the intensity of light, scattered on the surface waves:

+ ], (10)

where a strength of surface fluctuations is expressed by the value <12> = kbT/( crq2 + .1pg); g, the gravity constant. Wave vectors % = roJc and q are connected by following manner q = qolsinSi - sinS I = qo I.1Sil cosSi' We can neglect .1pg compared with crq2. It follows from here that the intensity of surface scattering rapidly increased with decreasing of scattering angle. The surface scattering is dominated under .1Si < 10 or q < 105 m-I .

We can establish that as for the bulk scattering (artificial heterodyning) and surface capillary waves (natural heterodyning) the information about the statistical characteristics of molecular motion are localized within the values of roo and r D of fixed q. High resolution of heterodyne light beating spectroscopy [13] allows to detect small signals and low frequency shifts.

2.3. THE EXPERIMENTAL TECHNIQUES

2.3.1. Light-Scattering Setup for Bulk Scattering A schematic drawing of the optical heterodyne spectrometer is shown in Figure 1. A laser beam from an He-Ne laser at 632.8 nm wavelength and 30-mW power is splitted onto a reference (h) beam and a beam (s) which excited scattering radiation in the bulk of a sample. The latter is modulated by piesoceramic vibrator with the frequency of 60 kHz. The beam (s) is focused by lens L3 to the cell with studied mixture. Lens Ll focuses the reference beam to the point where both beams are crossed under scattering angle, S = 10 -150 .

Thus, the adjustment of scattering radiation collecting system is carried out by reference beam, which selects needed spatial components, and the ideal photomixing conditions are fulfilled. Radiation collecting system (lens L2, pinholes PI and P2) yields the image of the scattering volume on the surface of photomultiplier where the optimal conditions of coherent photo mixing are provided [14].

The intensity of scattered and reference beams are measured by the system with working stroboscopic modulator M1.

2.3.2. Light-scattering Setup for Surface Scattering A schematic drawing of the optical heterodyne spectrometer for the surface scattering is shown in Figure 2. The laser beam is splitted onto the signal (s) and reference (h) beams by means of the diffraction grating (G). Optimal intensity of reference beam is reached by the nutral filter F. The beams (s) and (h) are formed by lenses Ll and L2 and have diameter d = 1 mm on the scattering surface. An image of the diffraction

407

M1 F L1 L2 ... 1 1 PMT

.,:~. - .... --------- -------.

1

P1 P2

5

Figure J. Schematic oflaser ligbt-scattering apparatus (bulk 8CI!ttering): MI, sIrobosoopic modulator; M2, piesoceramic modulator; F, filter; LI, L2, L3 are the lenses; PI and Pl, pinholes; PMT, photomultiplier; (s),

signal beam; (h), reference beam; I, sample cen in thermostat; 2, system oflaser intensity oontrol; 3, generator; 4, system of temperature oontrol; S, system ofmeasurement.

M1 1

Computer

Figure 2. Schematic oflaser ligbt-scattering apparatus (surface scattering): G, difliadion grating; ADC, fast analog-to-digital convertor. For notation of another terms see Figure J.

grating on the liquid-gas or liquid-liquid interfaces is produced by the lens L2. The wave vector of capillary waves selected by the beam (b). In our measurements q = (3 + 6).104 m-l. The system of radiation collecting includes lens L3 and pinhole P3. The scattering angle, or wave number q of surface waves, is determined from the image of reference and signal beams passing through the sample cell [15].

408

2.3.3. Registration System of the Power Spectrum and Autocorrelation Function The system of photocurrent registration and analysis is the same for both bulk and surface scattering, because of the heterodyne scheme is applied. The photocurrent treatment and analysis can be carried out as in spectral and in time domain. In the first case a signal after wide-band amplifier is directed to the spectrum analyzer, then digitized and subsequently treated by means of computer. Experimentally obtained power spectrum is approximated by shifted Lorentzian

A P(co)=----- +B. (11)

If the photocurrent analysis is performed in the time domain the signal after wide­band amplifier is put forward on the analog-to-digital converter. The data subsequently are processed by mM PC. The autocorrelation function G(t) is approximately an exponentially damped cosine function

G(t) = A exp(-rOt) cos(coot) + B (12)

and we chose this functional form to fit our experimentally measured autocorrelation functions. Values of parameters rO, coo' A, B are found from the minimum of a functional F = A-2l:(Fn - Fa>, where Fo is P(co) or G(t), and Fn is Pn(dco·n) or G(dt·n), i.e. the measured samples of the power spectrum or autocorrelation function.

The instrumental spectrum broadening and its sources have been investigated in [16]. The corrected value ofro has to be calculated from ro = ro(1 -ra2tr02) [17], where r a the width of the instrumental function.

2.3.4. Intensity Measurements The bulk thermodynamic properties of binary mixtures characterized by the differential of the chemical potential Ili with respect to concentration can be described by the quantity

(13)

10 and I are the intensities of the reference beam before and after passing through the sample cell, and KI is a coefficient, which allows for the difference between the fluctuational and macroscopic values of differential of the dielectric constant with respect to concentration (OsIOx>P,T; Isc' the measured scattering intensity.

2.3.5. Conditions of Measurements Bulk scattering was carried out in the single-phase region along the lines of constant concentrations (10-4 < 't < 0.2 and 10-4 < Ax < 0.25), as well as in the two-phase region along the CC of three binary systems: heptanelnitrobensene (7 samples), heptanelmethanol (3 samples), and decanelnitrobensene (7 samples). The conditions of hydrodynamic approximation are fulfilled (qrc < 0.3). The values of rO and I were measured simultaneously at each temperature and on the same sample.

409

The temperature and concentration dependences measurements of surface scattering were carried out on the liquid-gas interface along the lines of constant concentrations and in two-phase region (10-4 < 't < 0.1 and IAxl < 0.6). The surface tension coefficients were measured on the liquid-liquid interface along the CC of the systems: pentanelacidic anhydride (7 samples), heptane/methanol (1 sample), and heptane/perfluorodecaline (I sample). The conditions of hydrodynamic approximation are fulfilled (qrc < 0.03).

In both cases temperature was controlled better then SmK for several hours.

3. Results

3.1. MUTUAL DIFFUSION COEFFICIENT AND RECIPROCAL INTENSITY ALONG THE LINES OF CONSTANT CONCENTRATIONS

The families of experimental curves of the diffusion coefficients and the thermodynamic properties for seven C7HI6/C6HSN02 mixtures of different compositions are shown in Figure 3. Here t = (T - T p-)IT c, T p- is the visually determined temperature of phase separation. The slopes of the lines for three samples with near critical compositions, 0.S303, 0.S312, and 0.S289 mole fractions of heptane, have the same value giving critical exponents: for diffusion coefficient, v = 0.63 ± 0.02, which coincides with the critical exponent of the correlation length rc' and for r I, 'Y = 1.26 f O.OS. For the mixtures of non-critical compositions under closing up to the separation temperature diffusion coefficients and 1-1 tend to finite values. The analogous have been obtained for other mentioned systems critical parameters for which are presented in TABLE I.

3.2. D AND rl ALONG THE CC (T < Tc)

The temperature dependences of D and r 1 within two coexisting phases are illustrated in Figure 4. As for the phenol/water mixture in [18,19), we can see evident asymmetry of the diffusion coefficient and the thermodynamic properties with respect to the critical state, i.e. their asymptotic behavior along the left and right branches of CC is described by the power law with different critical exponents and amplitudes: DoL = (32 ± 4).10-10 m2/s, VL = 0.60 ± O.OS; DoU = (68 ± 3).10-10 m2/s, Vu = 0.68 ± O.OS; 10L -I = (25 ± 3).107 a.u., 'YL = 1.23 ± O.OS; Iou-I = (SI±5) .107a.u., 'YU = 1.30 ± 0.06.

We introduced an asymmetry factor accounted for asymmetry of the physical property around the critical concentration. The factor may be a complicated, but nonsingular, function of x and/or T [20). We have noticed empirically that it can be approximated by (xlxC)0.5 [21]. Using the asymmetry factor we obtained that D('t)(xlXc)O.5 and rl('t)(xlxC)0.5 are described by the same manner in lower and upper phases (see Figures 4 b and d): Do' = (44±4).1O-1O m2/s, v = 0.63±O.03; 10' -I = (36±4).107 a.u., 'Y = 1.26±O.OS.

410

10g(D) -8

a C7H,6/C6H5N02

0 ++

-10 0.8308

'fir 0.OS12

0 o.oaee 0 o.ana o O.eon 'V OA477

+ o.neo

-12 -.. -3 -2 -1 0

log(t)

log{ 111) 8

b C7H,lC6H5N02

0 o 0 00 ~-+O~

o *+ e + + ~-l+ + + ++++

0 O.OIOS

'fir 0.08'2 .. 0 0.1288 00 0 0.1100

0 0.80.3

+ 0.0860

1I -.. -3 -2 -1 0

log{t)

Figure 3. Temperature dependences of the diftbsion coefficient <a) and inverse light scattering intensity (b) along the lines of constant oooceotrationa for the mixtures. Solid lines are calculated using simple scaling with critical

amplitudes shown in TABLE 1.

TABLE 1. Critical parameters of diftbsion coefficient and generalized susceptibility for three systems studied

SystemAIB 0 0.101°, 0 0 •• 1010, llIo·10·7, llIo··10·7, 00'100 IJIo' p B

.......................................... ~!.~ ................ '!!.~~~ ............. ~~: .............. ~~: ..................................................... Jp..~:~~~>. .. C7HI6/CJI,N~ 22.0 20.0 8.1 7.1 2.0 4.0 0.63 0.818 CloHnlCJI,N~ 14.3 7.3 2.0 0.63 0.870 C7HI6I'CH30H 48.2 16.S 8.7 1.3 2.2 4.4 0.613 1.047

411

log(Dl log[D(x/xc) 0.6 J

e b

-11.15 -8.8

-10 -10

-3 -2 -3 -2

log(lIl) log!l!1 (xlxcl06 J

-3 -2

Figure 4. Di1fusion coefficient (a), symmetric diffusion coefficient (b), inverse scattering intensity (c), and syhunetric scattering ~ (d) VB reduced temperature below the critical point: ., upper phase, o,lower phase.

The behavior of any physical quantity Y(x, T) along the CC contains the concentration dependence which can be expressed via the parameters of the CC (Ax All ±B'tf3): Y(x,T) = Yo Zo~ ~/f3 and Yo· = Yo Zo~ (see TABLE 1).

3.3. RESULTS OF SURFACE SCATTERING MEASUREMENTS

3.3.1. Surface Tension of the Binary Mixtures on the Liquid·Liquid Interface The temperature dependences of the surface tension coefficient on the liquid-liquid interface for the systems pentane/acidic anhydride and heptane/methanol are analyzed in [22] by means of the scaling power law CJ = CJo 'til. The values of critical exponents Il and amplitudes CJo are gathered in TABLE 2.

TABLE 2. Critical parameters of shear viscosity and correlation length for the systems studied

IS System AlB (JolOl, N/m J1 (Jo. '10', N/m J1* rc ,A v ........ c;H;·.;ic;;;F;; ...... ·_ .......... ·:i:'C .......... _ .... ·'i:2· .... ·_ .............. · .............. _ ............. _ ...................... ..

C,Hn/(CH,COhO 3.3 1.24 4.0 3.9 1.9 0.63 C7HI~CH,OH 1.7 1.23 2.3 3.9 2 .3 0 .63

412

The critical exponents are very close to the values predicted by fluctuation theory of phase transitions and experimentally obtained for the binary systems [23]. Parameters of the concentration dependences of a along the CC are displayed in TABLE 2, too. It is clearly that critical exponents (Il and Il.) and critical amplitudes (ao and a o.) are connected with each other via the shape of the CC (B, P).

3.3.2. Surface Tension of the Binary Mixtures on the Liquid-Gas Interface From the data of temperature dependence of surface tension coefficient on the liquid­gas interface in one- and in two-phase regions for the systems pentane/acidic anhydride, heptane/methanol, and heptane/perfluorodecaline it was found [22] that experimental points for the off-critical composition mixtures form the curves, which are in the one-liquid region and terminate on the boundaries of phase-separation. For the system pentane/acidic anhydride it was indicated that a varies relatively strong when the system transits to the two-phase region and then moves along the CC. This is connected with the richness of the upper phase by pentane having small surface tension. For other systems, in which the difference between surface tensions of the components is too small, the permanent transition from one-liquid region to two-liquid region can be observed. In the critical point a has a finite value, but more detail experimental data are required to obtain the analytical form of the temperature and concentration dependences.

3.3.3. Shear ViSCOSity of the Binary Mixtures near CCP The measured curves of shear viscosity coefficients vs T for the systems pentane/acidic anhydride, heptane/methanol, and heptane/perfluorodecaline [22] have an abnormal increasing in the vicinity of CCP which can be approximated by the expression

(14)

where 11R is the regular part of shear viscosity, + and 110 are the critical exponent and amplitude, respe¢vely. For the system heptane/methanol the values 11R = (0.41 ± 0.02).10-3 Nslm2 and 110 = 0.79 ± 0.03 were obtained at fixed + = 0.04.

4. Discussion

4.1. MOBILITY OF THE COMPONENTS OF BINARY LIQUID MIXTURES

In liquid mixtures the mutual diffusion coefficient D can be expressed in terms of two factors [20, 24]: thermodynamic (the differential with respect to concentration Xi of the chemical potential ~) and kinetic (the macroscopic mobility bi of the ith component; the value, reversed to bi, is referred to the coefficient of friction):

(15)

413

There are essential differences between the macroscopic mobility bi of the ith component and the microscopic mobility directly connected with molecular motion. Let us consider connections between D, Onsager's coefficients and mobility of components.

On the one hand, the diffusion coefficient is defined by Fick's law in the following form

(16)

where Ji is the vector of flux of thermodynamic quantity, p is the density of solution. This law is one of the experimental confirmations on which the theory of linear reaction is based [24]. But, on the other hand, according to the thermodynamics of irreversible processes in the binary mixture a heat Jq and a mass Ji fluxes are written as the linear combinations of the relevant thermodynamic forces

(17)

-Jq = Loo grad(lnT) + Lol gradT<J.ll - J.l2),

where gradT(J.lj) = grad(J.lj) - Si grad(T), Si is the partial entropy, Lij is the matrix of Onsager phenomenological coefficients. And, in particular, the flux Ji is proportional not to the gradient of concentration, but to the gradient of chemical potential.

"In the irreversible thermodynamics the heat of transfer Qi is defined [25] as the heat which is transferred by a unit of mass of ith component under a constant temperature

(18)

From here the Onsager coefficients can be represented via the usual transport coefficients by the following way

a) LOO = A. T, b) Lal = p D x2 Qi (mqI8J.1.i)T,p' (19) c) LIO = pDT T Xl x2, d) Lll = p D x2 (mCjI8J.lj)T,p,

where DT is the thermodiffusion coefficient and A. is the coefficient of heat conductivity. So, on description of diffusion processes in nonideal multi-component liquid mixtures it is necessary to distinguish molecular mobility of particles and mobility of' components. In experiments, where we can watch for the motion of separated molecules, the microscopic molecular mobility is revealed (NMR methods, cell with diaphragm, method of labelled atom). In the cases, when we observe macroscopic variations of concentration, the macroscopic mobility of component is revealed.

By combining data of the mutual diffusion coefficient D and the scattered intensity I - (8xjl8J.lj)T p we not only determined the critical exponent", of the macroscopic mobility bi' 'but also carried out systematic investigations of the behavior of the mobility in a wide region around the critical mixing point including the two-phase region.

414

For compositions close to the critical value the dependence of the mobility upon temperature is fairly well approximated by the power relation

b· = (ax:lou.:.)T -nIx· = b 't-'" 1 1 q ,p- 1 0 . (20)

The critical parameters bo and bo', and '" and ",' for three systems studied are calculated by least mean square fitting and given in. the TABLE 3. Here, primed letters referred to the two-phase region. The values of the critical exponents '" are similar for all the systems within experimental errors and correspond to the critical exponent v which determines the divergence of the correlation length rc.

TABLE 3. Critical parameters of the mobility of components for the systems studied

b •. 10.4, a.u. ' Ii' . I

System AlB '" b.·10 ,a.u. '" b.,lb.

C,H.6I'~,N~ 0.273 0.627 0.123 0.642 2.2

C,H.6I'CH,OH 0.487 0.631 0.244 0.63 2.0

C.ofhzl~,N~ 0.182 0.638

In the phenomenological theory of diffusion processes the mobility of the components is related to the Onsager kinetic coefficients Lij by the expression

(21)

In analyzing the data obtained for coefficient L 11 the critical exponent '" should be comparable with the exponent v of the correlation length. Experiment shows that the coefficients Loo [26], LIO [27, 28], and Lol [29] change regularly in the neighborhood of the critical point. It follows that only one Onsager coefficient, L 11, diverges in the CCP. Direct measurements of the heat of transfer in the system isobutyric acid/water [29] gave a value of the critical exponent of Ql close to 2/3 which agrees reasonably with the obtained value of the critical exponent",.

4.2. UNIVERSALITY OF TRANSPORT AND THERMODYNAMIC PROPERTIES OF LIQUID MIXTURES IN THE CRITICAL REGION

Our experimental data on D and (Oxilo~>T permit us to study an analytical form of the universal functions fy[(z + Zo>/ZoJ [21]'¥or the static and dynamic properties. On the basis of a scaled equation of state (linear model) [6J the generalized susceptibility can be represented by following way:

~1-(2) - (1~e2)(8-1)

ey/P(bl-l) (22)

415

Here z = (l-])2e2)/(k®)lIP, a is the critical exponent, ])2 = (y-2P)/y(1-2P), kllP= ~­l)/Zo. But there are no predictions of the dynamic universal function analitical form.

Another way of analyzing the results is connected with the concept of the pseudo­spinodal. Recently, it has been shown that the pseudo-spinodal acts not only as a convenient way to characterize data but also appears to be a generalization of the mean field concept of a spinodal [7, 22, 30]. The properties Y(x,T) of binary mixtures near their stability limit depend on how far the given state is from the spinodal curve. As temperature is varied along an off-critical line of constant concentration a given parameter Y(x, T) will display divergent behavior identical to that observed along the line of critical composition except that the singular temperature is T sp(x) rather than the critical temperature Tc as described in

Y(x,T) = Yo K-<i> , (23)

here K= (T-Tsp>lTc, Tsp(x) represents the pseudo-spinodal curve. We propose to divide new reduced temperature on to two terms K= (T -T C>IT c + (Tc-T sp>1T c = 't+ 'tSl>' one of them, 't, gives the relative distance of the system from CCP in one-liquid regIon, and the other, 'tsp' represents the relative distance of the system from CCP along the spinodal. The second term can be expressed via the equation of pseudo-spinodal which can be derived [7, 22, 30] from the binodal curve with the help of the universal parameter p, whose value is close to 2/3:

~ = s~gn(L\x) pB 'tP(l + Bl~ + B2-r2L\ + ... ) + A'tl-a + Al't + A2't2P + ... =

= sign(~) pB 'tP + ~y~ . We separated out the main term and obtained the i

argument K= 't + (pBrllp[~ - sign(~) ~CfA]lIP, which yields the generalized form i

of the universal function

fy(z) = {z/Zo + p-lIP[l - sign(~) ~ci-rd111P}<p . (24) i

The usage of this function is restricted by necessity of introducing correction terms ~Ci'A which are mostly due to the clear asymmetIy of the spinodal (or binodal). To simplify the analitycal form of the universal function Eqn.(24) we studied the skewness of the CC of binary mixtures, we determined the factors which are responsible for the asymmetry, and investigated possible ways of symmetrizing the curves.

4.3. SYMMETRIZATION OF THE COEXISTENCE CURVES

Compositional asymmetry of CC shows itself as a deviation of critical mole fractions from 0.5 and as an anomalous properties of the rectilinear diameter of CC. Experimental studies on liquid-liquid equilibria show that CC symmetIy depends on the structural differences between component molecules and their energy ability to form mobile liquid local structures, whose nature is not the same in various systems.

416

According to our idea [31, 32] active exchange between coexisting phases occurs by means of dynamically stable and energetically preferred molecular units. Components of solution are asymmetric relative to the composition of molecular aggregates: component A is "a solvent" which forms matrix represented by multimer molecule [rnA], and component B, associating with molecules of A component, yields the compound [nA·kB] dissolved in the matrix.

Formation of the indicated structure units can be represented by the quazichemical equation:

x A + (I-x) B ~ Xs [rnA] + (l-Xs> [nA·kB], (25)

x is the mol.fr. of component A, Xs is the normalized mol.fr. of aggregates [rnA]. The transformation mole ratios X = x/(l-x) of initial components to symmetric

coordinates is expressed by the formula

Xs = Xg'(I-XsF (X - Xo> IS. (26)

where S = Xc - Xo = m I k, Xo = n I k, Xc = xd(I-Xc> = (m+n) I k. Parameter Xo characterizes some relative composition of component A, limiting the region at which isotropic solution of dispersed aggregates [nA·kB] exists.

TABLE 4. ConoenIratiOll parameters of studied coexistence curves.

Type of Xc (oils.) Xc (calc.) [Al-k'BkJ [mA] SystemA/B CCP mol.&. mol.&. Xo=n S=m

k m

SiOz/LizO upper 0.89 8/9 2 6 113 2

Cyclohexane/Polystyrene upper 0.9999'9 0.9999'99 672' 182'2 0.0001' 2.7 (M=200000)

1-Propoxy-propane-201l upper 0.077 0.083 0.001 0.089 0.9991 0.089 Water lower 0.077

C7H"I/~,NOz upper 0.'304 0.'32 0.062 1.08 0.842 1.14

CloHnlC~,NOz upper 0.4268 0.422 0.062 0.67 0.942 0.'2

C7HI&'CH]OH upper 0.3721 0.3833 0.07 0." 0.93S 1.22

To test this idea, we took parameters S and Xo obtained from the X-ray diffraction data for the oxide system Si02ILi20 [33] . The results of symmetrization are shown in Figure 5a. For the systems studied, including very asymmetric cyclohexanelpolystyrene mixture [34] and l-propoxypropane-20Vwater system with closed-loop CC [35], parameters S and Xo are calculated independently by the least mean square fitting, using the properties of symmetric CC (see Figures 5 b,e and TABLE 4).

TI1b

0.. a ................. ~~ ... Y.6r-_ ................. .

o.e ,/, •• ···2 I ./ !

0.7 ! ! 0.' f !

! ! o.a! i

! OAL-----~------~~--~------~----~

o 0.. 0.4 0.. 0 ..

mol'r. 0' 810.(1) or rn8Io. (2)

moL'r. 0' mC,HI • (2) 1~.~TI1b~ _____ 0~~_. _______ o.eo.-______ O',7r· ______ -,I~

4eOrT~~~:----=~===+====~--------~ (1······· .... 400 ;

! I

-I t

2 c

eooL·~~~·:·~:··:~=·~~·· __ ==:::::jt:::::==~~ ____ ~ o 0.. G.4 U 0 ..

moLtr. 0' AU) or mA(2) OOIIIPO'*It

417

Figure 5. Primary (1) and symmetrized (2) coexistence curves for the systems a) Si021Li20, b) cyclohexanel polystyrene, and c) l-propoxypropaue-2ollwater. Points are the experimental data represented in terms of mole fraction of component A and aggregates [mA] using Eqo.(26) and parameters from TABLE 4. The solid lines

were calculated by simple acaling Eqo.(27) and critical amplitudes B from TABLE I.

418

4.4. UNIVERSAL FUNCTIONS FOR STATIC AND DYNAMIC PROPERTIES

Symmetrized binodal (or spinodal) is described in tenns of the simple scaling,

Ax = "s - 0.5 = sign(Ax) B't~

and we obtain the following generalized form of an universal function

which can be extended both to static and dynamic properties.

- 9

- 11

-1 3

- 8

- 10

0.11303 _ (1) 'R 0 .11112

o 0.112811 ~~;$ ,r o 0.1I043~ " 0 .4477

2

o 0.27011 ..ere1S ",;0---+ 0.081111 n~jj£'e (2) _. 0.4271

t;. T. ~- + 0 .41147

o 0.4074

• 0.3122 _ (3) 'il 0.5245

+ 0 .2818 - :.:.,~ 0 0.3845

o 0.4311 ~- t;. 0 .2521

/::,. T. Tc *- 0.8a811

-12L-~--~--~~L-~--~--~~L-~ __ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0

log( ')( )

-9

- 1 1

-13

(27)

(28)

Figure 6. Diffusion coefficient D VI the argument Ie: for three liquid mixtures of different concentrations: beptane/nitrobensene, (1), decane/oitrobensene, (2), and heptane/methanol, (3). Points are the experimental data.

Solid linee are culculated by the simple scaling with parameters &om TABLE 1 ..

Moreover, one can see from Eqn. (23) that each of experimental functions,

(x/Xc> 0.5D(x, T) = DoXY;

(x/Xc> 0.511(x, T) = 1101C'I'; o(x, T) = 0 0 1("", (29)

behaves on double logarithmic plots as the single line common for every given system (see Figures 6, 7, and 8). Herein an critical amplitudes are the only nonuniversal parameters. The critical exponents and amplitudes of the dependences Eqns.(29) for the mixtures studied represented in TABLES 1 and 3.

419

7 . 0 .6303

'Ie 0 .6312 8 0 0 .112811

5 0 0 .8043

0 0 .2788 e 3

C::. T c To . 0.4271

+ 0.4641 4

6 0 0 .4014

4

\l O.SUS 2 0 0.3846

0 0.4371 C::. 0 .2621

C::. T c To 'Ie 0 .8S86

2 -5 -4 -3 -2 -1 0

log( ')()

Figure 7. Inverse scattering intensity rl va the argument IC for three liquid mixtures of different concentrations. For notation sec Figure 6.

In addition to the critical exponents the certain ratios and combinations of the critical amplitudes of different properties carry important information about the universality classes. According to the linear model of the equation of state [4, 6] the relations of critical amplitudes for the thermodynamic quantities are the functions of only critical exponents,

1'0-11Io -I = (y/~) [y(I-2~) / 2b(y-I)]y-1 , (30)

where 1'0-1 is the critical amplitude of the susceptibility temperature dependence alon~ the CC (T<T C>. In the frame of renormalization group approach up to the terms of &

(& = 4-d, d is the space dimension), the authors [36, 37] have obtained

(31)

From our consideration we can state, that

(32)

, where D 0 is the critical amplitude of the mutual diffusion coefficient temperature dependence along the CC (T<T C>.

420

IOg[(x/xcjO.5 ytlYkJ log(6)

* 8 -4

* * 4 * -5

* * -3 -2 - 1 10g(K)

Figure 8. Coefficients of shear viscosity 'IJ. (1), and surface tension G. (2). VB the argument 11: for the liquid mixture heptanelmethanol.

These equations can in turn give the information about p the calculated values of which (see TABLE 1) are in good agreement with the value p = 0.613 obtained from Eqns.(32) for the 3D-Ising model critical exponents, a fact that confirms universality of this parameter.

There are two amplitude numbers which relate the critical amplitudes of the correlation length, ro ' the surface tension, 00 ' the diffusion coefficient, Do ' and the viscosity, Tlo and TlR The amplitude number Rar= 00 rolkbTc was calculated for two mixtures on the basic of data on the temperature dependence of surface tension on the liquid-liquid interface and correlation length: Rcrr = 0.20 ± 0.04 (heptane/methanol); Rar = 0.26 ± 0.04 (pentane/acidic anhydride). This values agree with theoretical calculations for 3D-Ising model and confirm their universality. Obtained value ~Dr= 61tTloTlRDrofkbTc = 1.12 ± 0.15 agrees well with the results of the dynamic scaling theory.

5. Conclusions

The results of this work are strong proof that information of light scattering spectroscopy can be significantly gained by combining of both bulk and surface scattering at the same experimental conditions. The measurements of diffusion coefficients, susceptibility, surface tension and viscosity are performed for several phase-separating mixtures in the single-phase region along lines of constant concentrations and in the two-phase region along the coexistence curve.

On the basis of the presented results and our earlier findings universal general­purpose functions describing the temperature and concentration dependence of the dynamic and static properties of binary phase-separating mixtures are obtained under

421

the conditions of the hydrodynamic regime. An analytical form of the universal function can be predicted for static properties. Using the concept of pseudo-spinodal, we derived the universal function not only for static but for dynamic properties as well. A new quasichemical approach to the symmetrization of the binary coexistence curves of liquid-liquid equilibria has been applied to simplify an analytical form of universal function.

6. References

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3. Griffiths, R.B. (1967) Thennodynamic functions for fluids and ferromagnets near the critical point, Phys.Rev.

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4. Anisimov, M.A (1981) CrlttcalPhenomena in Liquids and Liquid Crystals, Gordon and Breach, London.

5. Hohenberg. P.C. and Halperin, B.I. (1977) Theory of dynamic critical phenomena, RevMod.Phys. 49, 435-

479, and references contained therein.

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8. Fabelinsky, I.L. (1965) Molecular Light Scattering, Nauka, Moscow, (in Russian).

9. Landau, L.O. and Ufihitz, t.M. (1958)StattsttcaIPhysics, Addison-Wesley, Reading. Mass.

10. Levich, V.G. (1962) Physicochemical HydrodynamiCS, Prentico-Hal~ New York.

11. Papoular, P.M. (1968) Ondes de surface dans un systeme de deux phases fluids superposees,

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12. Langevin, o. (1992) Light Scattering by Liquid Surfaces and Complementary Technique, Marcel Derrer,

New York, and references contained therein. 13. Cununins, H.Z. and Pike, E.R. (1974) Photon Correlatton and Light Beattng Spectroscopy, Plenum Press,

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16. Samokhin, S.P. and Chernova N.I. (1988) Instrumental function of laser heterodyne spectrometer with a

difti"action grating. Opttcs and Spectroscopy (USSR) 64, 460-461.

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surface properties. Theory and experiments, J.AppLPhys. 47, 2433-2442.

18. Pusey, P.N. and Goldburg. W.I. (1971) Ught-scattering measurement of con<:enIration fluctuations in

phenol-water near its critical point, PhysRev. lA, 766-776.

19. Jasnow, 0 and Goldburg, W.I. (1972) Asymmetry in criticallight-scattering from binary mixtures of phenol

and water in two-phase region, PhysRev. 6A, 2492-2498.

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region, Ber .Bunsen. Ge,., Phyl.Chem. 76, 234-249.

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properties ofliquid mixtures in the aiticaI pbase-leparation region, JE1'P Lett. 36, 47-51.

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thennodynamicpropertiesofbinaryphase-separatinmixtures,FluidMechanicsRueorchZl,105-108.

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Amsterdam.

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systems, Sov.J.Phys.Chem. 30, 2424-2427.

27. Giglio. M. and Veodramini, A (1975) TbemiaI-diftbsion measurements near a consolute critical point,

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nature: symmetrization and scaling description, Rus.J.Phys.Chem. 71, 242-247.

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the critical point, Phyl.lwv. 137,232-239.

SHEAR INDUCED DISPLACEMENT OF THE SPINODAL,

AND SPINODAL DEMIXING KINETICS UNDER SHEAR

Abstract.

JAN K. G. DHONT van't lIoff Labomtory, Utrecht University Padualaan 8, 3584 Gil Utrecht, The Netherlands

An equation of motion for the macroscopic density of a sheared sus­pension is derived from the Smoluchowski equation, with an appropriate closure for the pair-correlation function. This equation of motion is used to (i) predict the shear induced shift of the spinodal (both of its off-critical part and the critical point itself), and (ii) to analyse the anisotropic initial spinodal decomposition kinetics under shear flow.

The off-critical part of the spinodal is predicted to be shifted linearly with the shear rate 'i , for not too large shear rates, while the critical temperature is shifted like i'0.81. The location of the cloud-point curve is argued to be much more sensitive to shear flow than the location of the spinodal. The cloud-point cuurve of a sheared suspension no longer coincides with the spinodal, but is located far below the spinodal, in the unstable part of the phase diagram.

The predicted characteristics of ansotropic spinodal decomposition ki­netics are found to be in accord with experiments on polymer systems and binary fluids. No experiments on colloidal systems exist as yet to quantita­tively test the theoretical precictions developed in this paper.

423

E. R. Pike and J. B. Abbiss (eds.), Light SctJllering and Photon Corre1ation Spectroscopy, 423-440. © 1997 Kluwer Academic Publishers.

424

SHEAR INDUCED DISPLACEMENT OF THE SPINODAL,

AND SPINODAL DEMIXING KINETICS UNDER SHEAR

1. Introd uction

To predict the shear induced shift of the critical point for molecular systems, Onuki et al. [ 1 ,2] analysed renormalization group theoretical expansions in ... - d (with d the spatial dimensionality). They find that the location of the critical temperature varies with the shear rate 1 like 1", with]J ~ 0.52. In this paper, a relatively simple calculation is presented for the shift of the spinodal due to shear flow for colloidal systems. The prediction is that the off-critical part of the spinodal shifts linearly with the shear rate 1, while the critical temperature is shifted", 1th, with '1 the critical exponent for the inverse equilibrium compressibility. Since equilibrium critical exponents for molecular fluids are the same as for colloids, and the critical exponent f for molecular fluids is 1.23, this predicts that the critical temperature for colloids is shifted like 1" with 1J ~ 0.81.

Shear flow has a profound effect on the temporal evolution of the long wavelength density inhomogeneities that exist in the initial stages of spin­odal decomposition. These effects arc already pronounced for shenr nlles so small, that the shift of the location of the spinodal is negligible: effects on demixing kinetics are due to distortion of large scale inhomogeneities, while the shift of the spinodal is related to distortion of small scale corre­lations.

Both the displacement of the spinodal and cloud-point curve, and the effect of shearing motion on the initial stages of spinodal decomposition will be addressed in the present paper, which is organized as follows. First of all, in section 2, an equation of motion for the macroscopic density is derived from "the Liouville equation for colloidal systems", the Smolu­chowski equation, together with an appropriate closure relation for the pair-correlation function. This closure relation may be regarded as the statistical mechanical analoque of thermodynamic local equilibrium. In section 3, this equation of motion will be used to predict the shear induced shift of the spinodal, both critical and off-critical. In addition, the dis­placement of the cloud-point curve is addressed. Disregarding the effects of shear flow on short-ranged correlations, the equation of motion for the

425

macroscopic density is analysed in section 4 to describe shear induced anisotropic spinodal decomposition kinetics.

Throughout this paper, without loss of generality, the externally im­posed shear flow velocity u(r) at position r is taken equal to,

u(r) = r· r,

with the velocity gradient matrix r equal to,

1 o o

(1)

(2)

with '1 the shear rate. This corresponds to a flow velocity in the x-direction, with its gradient in the y-direction. The x-, y- and z-direction are referred to as the flow, gradient and vorticity direction, respectively.

2. Equation of motion for the Macroscopic Density Matrix

The Smoluchowski equation is the equation of motion for the probability density function P = P(rt,· .. rN, t) of the position coordinates {rj} of the N Brownian particles in the system. It is in fact a conservation equation for the number of points in the phase space spanned by the position coordinates, and reads,

a N -P = - '"'v·· {v·P} at ~ J J ,

J=1

(3)

where V j is the gradient operator with respect to r j, and v j is the velocity of the ph Brownian particle. On a time scale much larger than the time required for a Brownian particle's velocity to relax to thermal equilibrium with the heat bath of solvent molecules, due to friction with the solvent, the velocities can be expressed in terms of position coordinates. On this coarsened time scale, the so-called diffusive- or Smoluchowski time· scale, inertia of the Brownian particles may be neglected, which means that, without shear flow, all remaining (non-inertial) forces add up to zero. There

426

are three such forces: (i) the direct force -V'j~, with ~ the potential energy of the assembly of Brownian particles, (ii) the Brownian force -kBTV'j In{P}, which assures that the system ultimately attains a state of minimum Helmholtz free energy, and (iii) the friction force with the solvent -67r'11oa Vj, with '110 the shear viscosity of the solvent and a the radius of the spherical Brownian particles. In writing this expression for the friction force, hydrodynamic interaction between the Brownian particles is neglected. As mentioned above, these forces cancel instantaneously (on the diffusive time scale), so that,

Vj = -Do [V'j In{P} + ,BV'j~] .

Here, Do = kBT /67r'11oa is the Stokes-Einstein diffusion coefficient, and ,B = 1/ k B T. When there is an externally imposed shear flow present, there is an additional contribution to the above "thermalized" velocity of the Brownian particles equal to the locally imposed fluid flow velocity, r . r j (hydrodynamic interaction is again neglected here). Adding this shear flow induced contribution to the velocity we thus find,

(4)

Substitution of this expression into the conservation equation (3) yields the Smoluchowski equation without hydrodynamic interaction,

8 N -8 P = L: V'j . {Do [V'jP + ,BPV'j~] - (r· rjP) } . (5)

t j=l

Clearly, without shear flow, the equilibrium solution of this equation is the Boltzmann exponent P '" exp{ -,B~}. The particular form of the Brownian force given above is in fact dictated by the requirement that the equilibrium solution is indeed the Boltzmann exponent.

The above derivation of the Smoluchowski equation may be extended to include hydrodynamic interaction [3], and derivations (without shear flow) starting from the Liouville equation for the entire system of Brownian particles and solvent molecules can be formulated [4-6].

An equation of motion for the macroscopic density p( r, t I 1') may be obtained from the Smoluchowski equation, noting that,

p(r,tli') = N j dr2 ... j drNP(r,r2,···,rN,t). (6)

427

Integration of eq.(5), assuming a pair-wise additive potential energy <P, yields the following equation of motion for the macroscopic density,

:t P(r,tl'7) = Do{\72 p(r,tli')

+ (3\7. p( r, t Ii') jdr' [\7V( I r- r' I)] p( r', t 1'7) g( r, r', t Ii')}

-\7. (r·rp(r,tli')) , (7)

where V is the pair-interaction potential, and 9 is the pair-correlation func­tion, which is defined as,

p(r, t Ii')p(r', t Ii')g(r, r', t Ii') = N 2jdr3 " .jdrN P(r, r', r3,"', rN, t). (8)

The pair-correlation function measures the correlation between two parti­cles, resulting from both their pair-interaction as well as interactions medi­ated via other colloidal particles. The equation of motion (7) must be solved for smooth density variations, that is, for small wavevectors. The smooth density variations are the unstable modes, as gradients in the density tend to increase the free energy of the system. Larger gradients, corresponding to larger wavevectors, give rise to a larger increase of the free energy as compared to smooth gradients, corresponding to small wavevectors. The small wavevector density variations are the unstable ones, the amplitude of which will increase during the initial stage of demixing, while larger wavevector modes are stable.

The closure relation

To obtain a closed equation of motion, the pair-correlation function in eq.(7) must be expressed in terms of the macroscopic density. This can be done by noting the following features :

• The pair-correlation function in eq.(7) is multiplied in the integrand by the pair-force \7V(1 r - r' I). A closure relation is therefore needed only for distances I r - r' I between two Brownian particles which are less than the range Rv of the pair-interaction potential. Now, relaxation or growth rates of a density variation of wavelength ,.\ is I"V 1/,.\2 . 1 Moreover, as will

IThis expresses the fact that the mean squared displacement varies linearly with time. The time T required for diffusion over a distance A thus varies like", A 2 , leading to the '" 1/ A 2 -dependence of relaxation and dernixing rates.

428

be seen shortly, in the neighbourhood of the spinodal the (absolute value of the) effective diffusion coefficient for small wavevectors is much smaller than for larger wavevectors (this is usually referred to as "critical slowing down"). The dynamics of short wavelength density variations is therefore much faster than for large wavelengths. Relaxation of the stable short wavelength density variations is much faster as compared to relaxation or demixing rates of long wavelength density variations. Correlations over distances less than Rv therefore establish much faster than the rate of relaxation or demixing of the long wavelengths of interest. Therefore, the pair-correlation function, for the small distances of interest in eq.(7), may be replaced by the stationary solution 9stat of its equation of motion (which may, in principle, be obtained also from the Smoluchowski equation (5»,

9(r, r', t 11) = 9stat(r, r', t 11') , jor 1 r - r'l~ Rv . (9)

In case no shear flow is applied, 9stat is the equilibrium pair-correlation function, and the above reasoning may be considered the statistical me­chanical analoque of thermodynamic local equilibrium .

• Our interest here is in the dynamics of smooth density variations, that is, smooth on the length scale of the range Rv of the pair-interaction po­tential. The stationary pair-correlation function may therefore be evaluated for a homogeneous system with a macroscopic density equal to the density inbetween the points r and r' :

9stat(r,r',tl1')=9stat(lr-r'II-Y)I_ r+r' . ,jor Ir-r'I~Rv.(lO) p=p( 2 ,tb)

This way to go about is valid in the initial and intermediate stages of phase separation, but is certainly wrong in the transition and final stages, where in general non-equilibirum interfaces of a thickness of the order Rv exist.

• The extra term due to shear flow in the equation of motion for the pair-correlation function is much of the same form as that in the equation of motion (7) for the macroscopic density, namely: - l' . (r - r') 9. This shear perturbing term is bounded from above by a form that is linear in shear rate, provided that 1 r- r'l~ Rv : 1 1'· (r- r')91~ -yRV9+, where 9+ is the contact value of the pair-correlation function. All the remaining terms in the equation of motion for 9 are of equal order of magnitude. One of these terms is the contribution that stems for the Brownian force, as for the macroscopic density in eq.(7), namely 2Do \19, where the diffusion coefficient is now

429

twice the Stokes-Einstein diffusion coefficient, since the dynamics of 9 pertains to the motion of pairs of particles. A typical order of magnitude of this term, for distances smaller than Rv, is, 2Do I dgeq I drl+ I, where the derivative of the equilibrium pair-correlation function geq , without shear flow, is to be evaluated at contact. The effect of shear flow on pair­correlations for distances smaller than the range Rv of the pair-interaction potential is measured by the ratio of these two contributions. A little rearrangement thus shows that when the so-called bar Peclet number,

P 0 ~ 1R~ (11) e - 2Do '

is smaller than Rv 1 d In {geq} I drl+ I, the effect of shear flow on correlations over distances smaller than Rv is small. Since the derivative at contact of the pair-correlation function is large for the attractive forces at hand, this means that, for not too large bar Peclet numbers, the pair-correlation function may be expanded in a power series of Peo (with R = r - r'),

stat(R I .) 9 I 1-_ (r±rl tl') p-p 2 ,'Y

= geq(R)I __ (!:±..!:. tl,)+F(R)I __ (!:±..!:. tl') Peo p-p 2 ' 'Y p-p 2 ' 'Y

+O((PeO)2) ,forR~Rv. (12)

The function F here describes the distortion of the pair-correlation function for small distances, and can in principle be calculated as the linear response solution of the equation of motion for the pair-correlation function.

The expressions (9-12) specify the closure relation for the pair-correlation function that can be used in the equation of motion (7) for the macroscopic density, once the equilibrium pair-correlation function of a homogeneous system and the linear response function F(R) are known.

After a quench into the unstable part of the phase diagram, when the system just started to phase separate, inhomogeneities of the macroscopic number density of cQlloidal particles are small. The equation of motion can then be linearized both with respect to gradients of the density,. and with respect to the change 8 p of the density,

8p(r, t 11) = p(r, t 11) - p, (13)

where p = N IV is the density of the homogeneous system. Substitution of the closure relation (9-12) into the equation of motion (7), linearization

430

with respectto h p, Fourier transformation, and expanding upto () (( kRv )4), which implements the leading order expansion in gradients of the density, yields,

with kj the ph component of the wavevector, and where the effective diffusion coefficient is equal to,

Deii (k h) = Do [f3 8II~ T) + fo(k 1,0, T)PeO (15)

+k2 (f3 'E(,o, T) + II (k 1,0, T)PeO) + () ((PeO)2, (kRv )4)] , with,

the eqUilibrium osmotice pressure of the unsheared, equilibrium system (the density and temperature dependence of the pair-correlation function is denoted explicitly here), and,

'E(- T)=27r -JoodRR5dV(R) [ eq(RI- T)+~ _dgeQ(RI,o,T)] >0 p, 15 Pio dR 9 p, 8 P d,o ,

(17) is directly proportional to the Cahn-Hilliard square-gradient coefficient. Furthermore,

f (kl- T)=-Q -JdR(koR)2...!.. dV(R) [F(RI- T) ~ - 8F(RI,o,T)] JO p, fJ P R dR p, + 2 P 8,0 ,

(18)

f (k 1- T) = ~ Q -JdR (koR)4...!.. dV(R) [F(R 1- T) ~ - dF(R 1,0, T)] 1 p, 6 fJ P R dR p, + 8 P d,o ,

(19) where k = kj k is the unit wavevectoro These integrals can be evaluated after the function F is determined as the linear response solution of the equation of motion for the pair-correlation function for small distances R =Ir - r'l:::; Rv o

431

Notice that close to the spinodal, where I f3 arr/ap I is small, the effective diffusion coefficient (15) for small wavevectors is much smaller than Do. Together with the factor P rv 1/>.2 in eq.(14) that multiplies the diffusion coefficient, this implies the separation of time scales for long and short wavevelength dynamics referred to earlier in connection with the construction of the closure relation.

The equation of motion (14-19) will be used in the following section to study the shift of the spinodal due to shear flow, and in the subsequent section to describe the anisotropic spinodal demixing kinetics under shear.

3. Displacement of the Spinodal

The spinodal is defined as the set of densities and temperatures where the system becomes absolutely unstable against density fluctuations of infinite wavelength. This happens if, and only if, the effective diffusion coefficient (15) at zero wavevector becomes equal to 0,

D""(k = 0 I t) = Do [,8 8II~ T) + io(klp, T) peo] = o. (20)

At still lower temperatures, the effective diffusion coefficient becomes negative for certain wavevectors, leading to "uphill diffusion", that is, to diffusion of colloidal particles toward regions of higher concentration, which is the initial stage of the phase separation process. Due to the shear flow induced anisotropy of the microstructure, the temperature where a density wave with wavevector k becomes unstable depends on its direction k = k/ k. Mathematically, this anisotropy is described by the function fo in eq.(20). Wavevectors with the particular direction where fo attains its minimum value become unstable first. Let us therefore introduce the minimum value of fo with respect to all directions of the wavevector, for a given density and temperature,

fJ-)(p,T) = mJn fo(klp,T). (21) k

The implicit relation between the temperature and the density that defines the spinodal now follows from eq.(20),

f3 arr~ T) + fJ-)(p, T) Peo = 0 , defines the spinodal T = T(p) . (22)

432

For zero shear rates this reproduces the well-known thermodynamic def­inition of the spinodal, namely dTI(p, T) / dp = O. The above relation is a generalization of that thermodynamic definition which includes effects of shear flow, up to leading order in Pea.

Since fJ -) is related to the short-ranged shear induced distortion of the pair-correlation function, it is well-behaved on the spinodal.

First consider the displacement of the spinodal away from the critical point, where a2TI(p, T~)/apaT~ =J 0.1 Here, the subscript "s" refers to quantities on the spinodal, and the superscript "0" refers to the quiescent, unsheared system. Let STa denote the shift of the spinodal temperature at a given fixed number density p when a shear flow is applied. Writing Ta = T~ + STa, with Ta the spinodal temperature of the sheared system, and expanding eq.(22) up to leading order in STa, yields,

~rp fJ-)(p,T~) P ° O.1a = -{3a2TI(-TO)/a-aTO e. p, a P a

(23)

The shift of the spinodal temperature at a given density is thus seen to vary linearly with the shear rate.

The above arguments fail at the critical point since there the second order derivative a2TI(p,T~)/apaT~ is O. Here, the subscript "e" is used to indicate quantities at the critical point, and, as above, a superscript "0" refers to the quiescent, unsheared equilibrium system. Close to the critical point of the unsheared system, the divergence of the reciprocal equilibrium compressibility is characterized by the critical exponent"

(3 aTI~T) = e(T - T~)'Y . (24)

where e may be regarded independent of density and temperature. Substi­tution of this expression into eq .(22) yields the following prediction for the shift STc of the critical temperature,

(25)

1 A little thought shows that dll/dp changes with temperature (so that 82 ll(p, T.O)/8p 8T.o :/; 0), provided that the slope of the spinodal (plotted as T versus p) is neither infinite or zero. The latter happens at the critical point.

Figure 1: A schematic of the displacement of the spinodal due to shear flow, where a lowering of the spinodal due to shear is assumed. The dotted line represents the cross­over from linear displacement of the off-critical part of the spi­nodal to '" iO.81 displacement ~ of the critical temperature. The e dashed-dotted line is a sketch of ~ the cloud-point curve of the shea- ~ red system, which is located in the unstable part of the phase diagram. The un sheared cloud­point curve coincides with the un sheared spinodal.

433

density

Thus, away from the critical point the displacement of the spinodal is linear in the shear rate i, whereas the displacement of the critical tem­perature is proportional to i P, with p = 1/, = O.S1. These predictions are summarized in fig. 1 , where it is assumed -that shear flow stabilizes the system (as is usually found experimentally). The dotted line depicts the cross-over between these two scenarios.

In order to determine whether the spinodal temperature is increased or decreased by applying a shear flow, the linear reponse solution of the equation of motion for the pair-correlation function for small distances I r - r' I:::; Rv should be calculated. So far, such a calculation has not been done.

The turbidity of a colloidal system measures the total amount of light scattered by the colloidal particles, and is formally equal to an integral of the scattered intensity over all scattering directions. The cloud-point curve is defined as the set of temperatures and densities where the turbidity diverges. This divergence is due to the development of long-ranged structures, that is, due to the development of long-ranged character of the pair-correlation

434

function. These long-ranged correlations give rise to a strong upswing of the scattered intensity at small scattering angles. The displacement of the cloud-point curve due to shear flow is therefore related to the effect that the shearing motion has on the long-ranged structure of the pair-correlation function. As discussed above, this is different for the displacement of the spinodal, which is related to the effect of shearing motion on the short­ranged behaviour of the pair-correlation function. Since the shear rate de­pendence of the long-ranged part of the pair-correlation function is highly non-linear, contrary to its short-ranged part (see eq.(12)), the displacement of the cloud-point curve is expected to be much more pronounced than for the spinodal. Shear flow acts more strongly on large scale structures than on small scale structures. In the quiescent dispersion the cloud-point curve coincides with the spinodal. In a sheared system, however, the two do not coincide, since for the spinodal and cloud-point curve the distortion of the short-ranged and long-ranged part of the pair-correlation function are respectively responsible for their displacement. The cloud-point curve in a sheared system is expected to be located below the spinodal, in the unstable part of the phase diagram, as sketched in fig. 1. The experimental consequence is that a sheared system becomes unstable, without being very turbid, contrary to quiescent dispersions.

There are a number of experiments on systems with an upper criti­cal point where the critical temperature is found to be lowered by shear flow, such as polystyrene/celluloselbenzene [7], polystyrene/polybutadie­ne/dioctylphthalate [8,9], isobuteric acid/water [10,11,12] and polystyre­ne/trans-decaline of low molecular mass [13] (polystyrene solutions in trans-decaline with high molecular mass show a decrease of the critical temperature). For deformable particles, such as high molecular weight polymers, and anisometric particles, there is an additional effect of shear, not included in the present theory, related to changes of interactions on the pair-level as a result of single particle deformation and alignment.

In the above mentioned systems an exponent for the shift of the critical temperature of about 0.50 is reported, except in ref.[7], where a linear displacement of the critical temperature is found. It should be mentioned that in some of these experiments the location of the cloud-point curve might have been probed instead of the critical point. It seems that the displacement of the cloud-point varies with the shear rate as -rP , with

435

P ~ 0.50. The displacement of the cloud-point as reported in ref.[7] can indeed be fitted with such a power law.

No experiments on colloidal systems that unambiguously probe the location of the spinodal exist as yet to test the above predictions.

How to probe the sheared spinodal

In most light scattering experiments on sheared systems, the location of the cloud-point is probed, instead of the spinodal. A possible experiment to probe the location of the spinodal is as follows. The most important difference between phase separation kinetics after a quench into the meta­stable region and the unstable region of the phase diagram, is that in the former case there is a finite delay time for density inhomogeneities to oc­cur, while there is no delay in the latter case. Above the spinodal, in the meta-stable region of the phase diagram, density fluctuations with some minimum finite amplitude are the only unstable modes. The probability for the occurence of such finite amplitude fluctuations decreases strongly with increasing amplitude. As a result, an increasing delay time for phase separation is observed as the minimum amplitude for a density fluctuation to be unstable increases. On the spinodal, the amplitude of the unstable mode is equal to zero, and phase separation occurs without any time delay. What is expected than is, that when the turbidity is measured as a function of time directly after a quench, its initial increase changes drastically on crossing the spinodal: in the meta-stable region the initial slope of the turbidity versus time is small, due to the finite delay time, while below the spinodal the turbidity increases without any time delay, resulting in a large initial slope. Notice that the increase of the turbidity in a sheared system is due to developing inhomogeneities as a result of ongoing phase sepa­ration, not because of the development of long-ranged correlations. The temperature where the initial slope of the turbidity versus time drastically changes, measures the location of the spinodal. The cross-over from a small to a large initial slope is gradual, however, since on appraoch of the spinodal from the meta-stable side, where the amplitude of the unstable density fluctuations becomes smaller, the probability for the occurence of an unstable density fluctuation gradually increases. In fact, a "spinodal like demixing scenario" already occurs at temperatures in the meta-stable region where the amplitude of unstable modes has become so small that these give rise to an increase of the free energy of only ~ kB T or less: such

436

fluctuations are almost always present, and demixing will occur without significant time delay, as in case of spinodal decomposition. In practice the spinodal must be found from experiments by a convenient interpolation of the time dependent turbidity data. Such experiments have been proven useful to determine the location of the spinodal in an unsheared colloidal system [13], but' have not been used to locate the spinodal of a sheared suspension yet, as far as I know.

4. Initial Spinodal Decomposition Kinetics of Sheared Suspensions

In this section we analyse the effect of shear flow on the temporal evolution of the macroscopic density in the initial stage of spinodal decomposition, where linearization with respect to changes of the density is allowed. The Peclet number P eO measures the distortion effects on structures of linear di­mensions Rv. Since the density develops inhomogeneities on much larger length scales, large effects of shear flow on the decomposition kinetics are expected even for small values of P eO. Anisotropic demixing kinetics is therefore already pronounced at shear rates where the displacement of the spinodal is negligibly small. It is therefore sufficient to consider the dif­fusion equation (14), where in the effective diffusion coefficient in eq.(15) the shear contributions may be neglected. Hence,

with,

For larger shear rates, where P eO is not small compared to the slope of the pair-correlation function at contact Rv I dIn {geq } / drl+ I, demixing kinetics is also affected by distortion of short-ranged correlations, in which case the full equation of motion (14,15) must be analysed. Here, we do not consider these correl~tion distortion contributions, and assume not too large values of Peo.

437

~

IJ, " .. ' - . . , . . ..

Figure 2: The anisotropic growth rates _Deff (k, t 1.:y)I<2, with K = k Rv a dimen­sionless wavevector, for various times.:yt (see eq.(30)). The ratio of the two dimensionless numbers (3dIT/dp and (3'£/ R~ is taken equal to -1/10 here. Negative values for the growth rate, corresponding to stable fluctuations, are not shown. The left column of figures is for I<3 = 0, the right column for I<2 = O. Scales are the same for all the theoretical figures. The two most lower figures are experimental scattering patterns from a binary fluid, which are taken from ref.[15).

438

The solution of the Smoluchowski equation (26) reads,

b'p(k,tl-y) = b'p(k=(kl,k2+i'klt,k3),t=01-y)exp{-neff(k,tl-y)k2t~, <28)

where the time and shear rate dependent effective diffusion coefficient is equal to,

1 [k2Hklt (_ / ) k2+x2+k2 neff(k,tl-y) = i'k1tJk2 dxneff Vk?+x2+k~ 1 k2 3. (29)

Notice that there is a time dependence in the exponential prefactor. Besides the exponential function, also the wavevector dependence of the initial density variation contributes to the evolution of the density. The integral in eq.(29) for the effective diffusion coefficient can be done explicitly, with a little effort, after substitution of the small wavevector expansion (27) for neff (k), to yield,

neff(k,tl-y) = no[/1~~{1+KlK2i't;:K~(i't)2}

+ (/1 EI R~) {(Kl + K;) (1 + K~ + 2KIK~! + ~K~( i't)2) (30)

+ K~ + 2KIKii't + 2Kl K~( ~;2 + K~ K 2 ( i't)3 + ~Kt( i't)4}].

Here, K = k Rv is a dimensionless wavevector. Since the time always appears in eq.(29) as a product with kb it follows that there is no effect of shear flow in directions where kl = 0,

(31)

Density variations in the (y, z )-plane, that is the gradient-vorticity plane where kl = 0, are therefore not affected by shear flow.

Apart from the prefactor in eq .(28), the growth rate of density variations is equal to _neff (k, t 1 i') k2. This is an anisotropic function, that is, a function of the vector k, not just of its length k = 1 k I. Moreover, the anisotropy changes as time proceeds. A plot of the anisotropic growth rates at various values of i't is given in fig.2. The two most lower figures are experimental scattering patterns of a binary fluid [15], and indeed are quite

439

similar to the theoretically predicted patterns. The spherical symmetrical growth rates become ellipsoidal like for small times. In the velocity­gradient plane, where ]{3 = 0, the ellipsoid makes an angle with the ]{l and]{2 axes, while in the velocity-vorticity plane, where]{2 = 0, the long axis of the ellipsoid is along the line where]{l = O. The angle of the major axis of the ellipsoidal distortion in the (I{t, ]{2)-plane with the ]{2-axis is seen to decrease for larger values of:yt, in accord with experiments on binary fluids. For somewhat larger values of -yt, the only unstable wavevectors are those with a relatively small value of the wavevector component kl in the flow direction. As a result, density inhomogeneities develop in an almost two-dimensional fashion, leading to elongated structures along the flow direction. Such two-dimensional growth is indeed observed experimentally in binary fluids [16] and polymer systems [17]. The first to predict such two-dimensional growth theoretically are Imaeda and Kawasaki [18]. The prediction (31) that there is no effect of shear flow in the directions where kl = 0 is also confirmed experimentally [16,19]. A subtle effect is the decrease of the intensity along the major axis of the scattering pattern in the (kt, k2)-plane, indeed observed experimentally [15,16].

There are no data available, as far I know, for spinodal demixing under shear of colloidal systems, allowing for a quantitative test of the above theory.

5. References

[1] Onuki, A and Kawasaki, K (1979) Ann. Phys. (NY) 121,456. [2] Onuki, A, Yamazaki, K, and Kawasaki, K (1981) Ann. Phys. (NY) 131,217. [3] Dhont, J.KG. (1996) An Introduction to Dynamics of Colloids, Elsevier, Amsterdam. [4] Mazo, R.M. (1969) 1. Stat. Phys. 1,89, 101 and 559. [5] Deutch, 1.M., and Oppenheim, I.J. (1971) Chern. Phys. 543547. [6] Murphy, T.J., and Aguirre, J.L. (1972) J. Chern. Phys. 57,2098. [7] Silberberg, A (1952) Interfacial Tension and Phase Separation in two-Polymer-Solvent Systems, Thesis, University of Johannesburg, South Africa, Silberberg, A, and Kuhn, W. (1952) Nature 170, 450, (1954) 1. Polym. Sci. 13,21.

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[8] Hashimoto, T., Takebe, T., and Asakawa, K. (1993) Physica A 194, 338. [9] Fujioka, K., Tabeke, T., and Hashimoto, T. (1993) J. Chern. Phys. 98, 717. [10] Beysens, D., Gbadamassi, M., and Boyer, L. (1979) Phys. Rev. Lett. 43,1253. [11] Beysens, D., Gbadamassi, M., and Moncef-Bouanz, B. (1983) Phys. Rev. A. 28, 2491. [12] Fukuhara, K., Hamano, K., Kuwahara, N., Sengers, J.V., and Krall, A.H. (1993) Phys. Lett. A 176, 344. [13] Verduin H., and Dhont J.K.G. (1995) J. ColI. Int. Sci. 172425. [14] Wolf, B.A. (1984) Macromolecules 17, 615. [15] Baumberger, T., Perrot, E, and Beysens, D. (1991) Physica A 174 31. [16] Perrot, E, Chan, C.K., and Beysens, D. (1989) Europhysics lett. 965. [17] Hashimoto, T., Matsuzaka, K., Moses, E., and Onuki, A. (1995) Phys. Rev. lett. 74126. [18] Imaeda, T., and Kawasaki, K. (1985) Prog. Theor. Phys. 73559. [19] Chan, C.K., Perrot, E, and Beysens, D. (1991) Phys. Rev. A 431826.

SPECTRAL KINETIC AND CORRELATION CHARACTERIS­

TICS OF INHOMOGENEOUS MIXTURES IN THE VICINITY

OF THE CRITICAL POINT OF STRATIFICATION

Abstract.

A. D. ALEKHIN, S. G. OSTAPCHENKO, D. B. SVYDKA AND D. I. MALYARENKO

Kiev Taras Shevchenko University, Physics Department, 6 Glushkova St., Kiev 252127, Ukraine

The height and temperature dependences of the total scattered light intensity and Rayleigh line width for a binary mixture of methanol-hexane were studied near the critical consolution point by means of static and dynamic light scattering. The temperature dependence of the diffusion co­efficient along the critical isochore and coexistence curve, and the field of dependence of the diffusion coefficient as a function of height were stud­ied. The correlation length temperature dependence along the critical iso­chore, the coexistence curve, and the field dependence along the critical isotherm were analyzed using Ferrell's formula derived in the framework of the Kawasaki coupling mode theory.

441

E. R. Pike andJ. B. Abbiss (eds.), Light Scattering and Photon Correlation Spectroscopy, 441-460. @ 1997 Kluwer Academic Publishers.

442

SPECTRAL KINETIC AND CORRELATION CHARACTERIS­

TICS OF INHOMOGENEOUS MIXTURES IN THE VICINITY

OF THE CRITICAL POINT OF STRATIFICATION

At the present time research of the critical phenomena in liquids continues to remain one

of the actual directions of development of condensed matter physics. The significant

experimental material on equilibrium properties on individual substances and binary

solutions in the vicinity of a critical point liquid-vapor [I, 2] has been already

accumulated that allows to study the equation of state in close as well as wide vicinity of

a critical point. There much less experimental investigations on kinetic properties of

binary solution in the vicinity of a stratification critical point. The problem is

complicated by the fact that really equilibrium state is spatially inhomogeneous one due

to unlimited growth of solution susceptibility under gravity [3, 4]. Moreover kinetics of

equilibrium state establishment in these spatially inhomogeneous systems has been

shown [5, 6] to qualitatively differ from kinetics of homogeneous systems. It has

appeared that the maximal time of equilibrium establishment in the vicinity of a

443

stratification critical point in a binary solution under gravity does not correspond to a

critical temperature Tc but temperature AT=T-Tc~10 K [6].

In this connection the investigation of kinetic properties of binary solutions in

the vicinity of a stratification critical temperature is actual enough at the present time.

During experimental investigation of equilibrium and kinetic properties of

system under gravity in the vicinity of a critical temperature of stratification we should

pay the major attention to time of equilibrium establishing in such inhomogeneous

system. One usually states, on the phase transition fluctuation theory grounds [7], that

time of equilibriwn establishment in a system tp -t - D-1 _e-o decreases at increasing

the distance from a critical point ('t -relaxation time, D - R -! -diffusion coefficient, R.: -

T-T correlation radius, e = --' -reduced temperature, v - critical index of fluctuation

T,

theory [7]).

However this conclusion of the phase transition fluctuation theory of spatially

homogeneous system can not simply be applied to inhomogeneous systems under

external gravity. Indeed, optical examinations of gravitation effect have showed [8] that

correlation radius has an ambiguous behavior at different altitudes of inhomogeneous

system in the vicinity of critical point. At altitudes h« eJlS , h = p ,I¢z (pc' P, - critical P,

density and pressure of a substance, llz- altitude measured from a level of critical

density) where critical densities and concentrations of binary solution are realized, the

correlation radius R, -e- I increases under e~o. Under h»eJlS,when density and

444

concentration of substance differs significantly from critical values, correlation radius

behavior described by relation [7, 8]

(1)

From (1) it follows that in the inhomogeneous system at these altitudes h» epa

correlation radius decrease under approach to critical temperature (e~O). So based on

substance correlation properties only any prediction of dependence to(e) for

inhomogeneous system is impossible. In this connection to solve this problem we have

carried out experimental study of establishing time of equilibrium gravitation effect in

binary solution methanol-hexane in the vicinity of critical temperature of stratification.

The experimental method implies that binary solution methanol-hexane at

methanol critical mass concentration x=0.33 was positioned within a flatparallel optical

cell of high pressure [9] placed in a thermostat. A chamber with substance was heated

relatively fast (for thirty minutes) from room temperature T=297 K up to T~Tc=307.1 K

and for an appreciable length oftime was thermostabilized with accuracy 0.01 K. Later

dn behavior of refractive index gradient - of inhomogeneous binary solution was

dz

investigated by refractometry method [10] during thermostabilizing as it transition in

dn equilibrium state [6]. Figure I shows the dependence of - on time at altitude z=0

dz

where critical density and concentration of the solution are realized under different e.

0.12 ri .• -1

D.D8

O.IM

l~ 21 52 t, haurs

Figure 1. The temp~ral dependencies of derivatives ~: at level Z = 0 (cril.ical isochorc) at temperatures: 1)!!J.T = T - T. = 0.39° 1\, 2) flT = 1.76°/\', 3)!!J.T = 3.96°1\, 4) ~T = 14.1°/\

445

We consider that the time of equilibrium establislul1ent is equal to the span ailer

dn which value of gradient - becomes stable. A number of experiments were carried out

dz

to be ceJ1ain that equilibrium state achieved. The essence of they is depicted below.

Initially a system was overheated up to temperatw'e T, exceeded critical one on 10-15

dn K. After prolonged thennostabilizing (up to time when value of gradient - at level z=O

dz

becomes stable) a binary solution was quickly cooled down to T2 which is in the vicinity

dn of critical point. Value of gradient - at level z=O increases as approach to equilibrium

dz

state, up to maximal value at equilibrium state. As we ean see in Figw-e I equilibriwl1

values of substance refractive index under the temperatw-e T2 coincides very closely

within the limits of experimental en-ors under ditferent direction of approach to

equilibrium state: either from room temperature or temperatw-e T,>T2. So we can

consider that equilibriwll state in inhomogeneous state is achieved.

446

On the base of these data we have examined the temperature dependence of the

equilibrium state establishing time in inhomogeneous binary solution methanol-hexane

16 J. Obtained results are presented in Figure 2. As we can see the dependence to(O) is

non-monotone. As moving away from critical temperature the in inhomogeneous system

does not decrease as it follows from [7] but increase up to maximal value 10 ~38 hours at

~T=T-Tc=10 K. During further moving away from critical temperature the equilibriwn

state establishing time decrease as it follows from [7].

40

30

20

10

to. hours

o JO

Figlll'e 2. The dependence or the gravitation time eITect on the proximity to the critical temperature

30 6T, K

To can)' out qualitative analyze the cited dependence 10(9) we should use data

on gravitation effect in the vicinity of critical point. According to these data during

transition of inhomogeneous system into equilibriwn state there are a transfer of solution

components in vertical direction from bottom phase into upper one and vise versa. In

this case we can consider that the equilibrium state establishing time in such

447

inhomogeneous system is significantly detennined by two factors: 1) declaration of

diffusion properties of substance at level z=O where solution critical density and

concentration are realized and where relaxation time • - D -I ~ <Xl under e ~ 0 (Diffusion

properties of other layers of substance placed far from level z=O takes essentially smaller

influence on process of vertical transfer of solution components in vertical direction); 2)

thickness of layer of inhomogeneous substance Az near level z=O where under e ~ 0

relaxation time • - Rc increase according to the law • - e-{ 1- :: - .. .) [7].

For qualitative estimation of behavior of time of equilibrium establishing in

inhomogeneous system it can be presented in form

z,

to "" J.(z)dz "" :r(z = 0XZ 2 - zJ (2) z,

According to [8] layer width tlz = Z2 - Zl increases conforming to the law

tlz_eP8 as increasing ofO. So having regard to [2]:

(3)

Since index in (3) P8--v>O (P~""1.5; v=2/3 [1]) then during increase of 0 to increases too,

and this fact is confirmed by our experimental data.

Value of Az becomes equal to total height of optical cell (Az,.,L=const) during

further moving away from the critical temperature. Then time

(4)

448

will decrease at increase of e as follows from fluctuation theory of phase transition [7].

This result is confirmed by our experimental data too (Figure 2).

~ oM I R' D,U iIi •• -

" . " : II

....

'.-Figure 3. The altitude profile of refract.ive index gradient of an inhmogeneolls binary solution of mel.hanol and hexane at temp­eratures: 1)..:1T=6.90J(. 2)..:1T=l.i°K. 3)..:1T=O.3101(

We have carried out simultaneous investigation of altitude and temperature

dependence of equilibrium properties of scattered light intensity I(z,T) as well as

dn refractive index gradient - of inhomogeneous binary solution of methanol-hexane after

dz

establishing of the equilibrium in a system in the vicinity of a stratification critical

temperature on experimental installations described in [9].

We consider that the temperature is equal to critical one if phase interface

disappears during slow heating with step 0.01 K or appears during slow cooling. At this

temperature scattered light intensity on interface level (z=0) and substance refractive

index gradient hav~ maximal values. We have determined the stratification critical

temperature of binary solution methanol-hexane as Tc=307.1 K by means ofa number of

449

above-mentioned examination. The accW"3CY of such detennined critical temperature is

0.01 K.

'50 /ICd I I I I

1l1li

,.

I, ..

Figure 4. The altitude dependence of scattered light intensity /(z) in an inhomogeneous binary solution at Lemferatures: 1) ~T = 7.3~K, 2) ~T = 2.750K, 3) ~T = 0.36 K

Experimental data on equilibrium values of altitude and temperature

dependence of equilibrium properties of scattered light intensity I(z, T) as well as

refractive index gradient of examined solution are presented in Figures 3, 4. On the basis

of presented data we can draw a conclusion that in the equilibrium state in this system

dn gravitation effect is realized. Presented data on I(z, T) and point that spatial

dz

inhomogeneity of binary solution presents up to temperatures aTRllO-lS K above critical

temperature and aTRlS K below one. This range oftemperatures is nearly 10 times more

width than the range of gravitation effect existing in the vicinity of critical point liquid-

vapor [9].

450

Analysis of obtained experimental data. on the altitude and temperature

dependence of scattered light intensity of inhomogeneous substance under gravity in the

vicinity of critical point liquid-liquid allows to fmd [11] at the ftrst glance unusual

dn temperature dependence of above-mentioned optical characteristics I(z,T) and - under

p gz fixed liclds h=-<-;eO.

p< 0.11

a._

I .•. 1

0.01 a ...

Figl\~'e 5. The temperature dependence of the derivative ~~ at altlt.udes: I): = 0111111, 2) : = h"l1I. 3) : = 2111111, 4) : = 3111111 5) : = 4111111, 6) : = 8ml1l

dz

Analysis of presented data shows that during closing to the critical temperature

of stratification Tc value of scattered light intensity and refractive index gradient of the

solution sharply increase at altitude z=O only where critical concentration is realized.

But they decrease at altitudes ~ 1 sm. under T ~ Te.

To analyze the found dependence regularity let us to examine the temperature

dn dependence of derivative - at different fixed altitudes z corresponded to different field

dz

451

variables h. The pointed dependencies are presented in Figure 5. The symmetrized

di1 ~dn dn] dn values - = -( zO)+ -( z < 0) are shown. Values -( z) were measured at altitudes dz 2dz dz dz

dn symmetrical as to z=O. As we can see from the Figure 5 values of gradient - and

dz

de consequently concentration gradient as well as binary solution susceptibility

dz

* * dn * * ----~ have inmonotonous character at altitudes z;t:O. As closing to Tc ---dll dh dh dll dn

increases initially up to maximal value under Tm>Tc and then decreases. One can see

that value of T m decreases as decreasing of field variable h too. At altitude h=O maximal

value of is observed at T=Tc. as follows from [7]. The results are in qualitative agree

with detail investigation in the vicinity of critical point liquid-vapor carried out early

[11 ].

Analysis of obtained results was perfortIJ.ed on the basis of phase transition

fluctuation theory [7]. According to [7] in the vicinity of critical point a susceptibility of

binary solutions directly connected to

de 2 -=A·Rc •

dll (5)

potential.

452

(6)

in the vicinity of critical isothenn (Z2 • « 1)

~2( Z2 .) == ibn ( Z2 .) n (7) ... 0

If we substitute in (5) scaling functions ct>1 and ct>2 by expressions (6) and (7)

and take into accoWlt that Wlder gravity altitude variation of chemical potential obeys to

relation L1J.L(h) == dJ.L. h we can find derivative ~(dJ.L) at different extreme cases: dh dt de

1) in the range of altitudes h are in the vicinity of critical isochore h· e-P6 «1

(8)

2) in the vicinity of critical isotherm Wlder condition h·e-P6 »1

(9)

As we can see the carried out calculations give qualitatively correct description

dn de of experimental results on - == - presented in Figure 5.

dz dJ.L

The function <I{ Zl·) is differentiable in supercritical region so it has and

extremwn in the interval Zl·» 1 and Zl·« 1. Then condition of extremwn existing

d [de) - - == 0 in the range 0 < z· < 00 can be presented in form de dJ.L

453

d d~ dz* -cI(z*)=--=O. (10) d9 dz* d9

It follows that scaling variable ZI * = .1J,19-111 has constant value along line of

extremal values of binary solution susceptibility. Condition ZI * = canst simultaneously

d~ means constancy of scaling fimctions ell( z*), - and the same along this line. This fact

dz

is equivalent to temperature dependence of correlation radius Rc - 9-v of concentration

(.1c*-9P), susceptibility dc _9-r (as on critical isochore or on phase interface) or dj.L

"field" altitude dependence of concentration [:: - h -(&-1)15 ) and heat capacity Cy _ h -a/III

(as on critical isotherm).

To study correlation and kinetic properties of the solution we have applied the

method of photo correlation spectroscopy [12]. The measurements have been carried out

on the experimental installation Malvern System 4700c. Measurement of half-width r of

Raileigh's central line in the scattered light spectrum allows to find diffusion coefficient

D of a solution in the vicinity of a stratification critical tempeJ7ltllTe

2 47t 9 ( )

-2

D=q- r= -;:sin;- r (11)

where q is module of scattered lig}-> wavevector.

These data in hydrodynamics approach Rcq « 1 can be used to define system

correlation radius Rc [1, 8]:

454

kT R

c 61t11D (12)

where k is Bolzman constant, 1') is solution viscosity.

The experiment was carried out as follows. Examined solution under critical

concentration was filled into cylindrical optical cell with diameter d=7 mm up to the

height IS mm placed in liquid thermostat and thermo stabilized with accuracy of 0.1 K.

Initially the solution was heated up to T=323 K from room temperature (AT=12.8 K) in

a hour.

At transition through a critical temperature T c the examined solution becomes

spatially inhomogeneous (phenomenon of the gravitation effect [5, 6]). It was shown

early [5, 6] that this spatial inhomogeneity manifests and in the range AT:S:20-25 K.

Therefore and at T=50 K altitude inhomogeneity continues to be visible. In this

connection examined solution was strongly mixed by intensive shaking of ampoule with

the substance. After shaking the examiped system becomes macroscopic homogeneous.

Intensity of light scattered by the system considerably increase (about 100 times). This

result is evidence of microscopic inhomogeneity of the solution after shaking. The

averaged size R of disperse phase particles from (12) on data of Raileigh central line is

1-2 J.IIIl at temperature AT=12.8 K. With the further solution thermostabilizing at this

temperature intensity of scattered light I and R decrease. This fact shows that this mixed

macroscopic homogeneous solution is non-equilibrium.

At the following stage the solution was cooled down to temperature being in the

vicinity of the critical point (AT:S:1 K) and therinostabilizing in 5-6 hours. In process of

455

thennostabilizing through each 5-10 minutes measurements of the diffusion coefficient

were carried out. The received data of dependence of diffusion coefficient on time OCt)

for three temperatures are shown in Figure 6.

10,0 0xl0-7, ra2/111C

8,0

4,0

"a a 100 200 B, IIin

Figure G. The change of the dirrusion (oefficient with time during the transil.ion of the system 1·0 t.he equilibrium state: 1)t:.T=O.;oh·, 2)AT= 1.2°1\, 3)t:.T=2.So/\

We can see as the system approaches to equilibrium state the diffusion

coefficient increases and achieves equilibrium state in 4-5 hours at the temperatures.

The measured value of OCt) should be noted to oscillate arolUld monotonous curve

described with exponential relation 0 -~ ,tJ There" is time of relaxation of size of

disperse phase R to equilibriwn value, the last equals to correlation radius at the

temperature. We consider that time of establishment of equilibrium state is equal to

span of time after which value of diffusion coefficient becomes stable within

experimental errors.

456

The temperature dependencies of equilibriwn values of diffusion coefficient

D(T} along critical isochore (T>Tc) and phase interface (T<Tc) are presented in the

Figure 7.

10,0

8,0

',0

4,0

2,0

~.8 ~.' 0,0 0.4 0.8 T, K

Fignre 7. The temllcrature dependence or the diffusion coefficient in the vicinil.y or the st.ratilication critical temperature or methanol­hexane solution along: 1) phase boundary, 2) critical isochore

One can see that under T -+ T c diffusion coefficient increases up to maximal

value Dc. Such behavior of D(T) is predicted by Farrel formula deduced in the frame of

Kawasaki theory of connected modes:

kT (2 _2)112 D=-- q +k..

16'1_ (13)

where lletTect is effective viscosity.

As we can see from the equation (13) under T -+ Tc <Rc-+«» diffusion

coefficient

457

kT D~D =--q=const

c 1611ef1'ec:t (14)

According to assumption of scaling theory in the vicinity of T c the temperature

dependence of correlation radius along critical isochore and phase interface can be

presented as

T-T where t = __ c •

T C

Proceeding from (13) and (14) we can write correlation radius in a form

kT (2 2)-U2 R =-- D -D .

C 16 C llef1'ec:t

Then using (15) we obtain

(15)

(16)

(17)

On the basis of obtained experimental data presented in Figure 7 values of

amplitudes Do as well as critical exponents v were calculated along critical isochore

(T>Tc) and phase interface (T<Tc). The temperature behavior of correlation radius along

critical isochore and coexisting curve was investigated. Values of the amplitude and

critical parameters are:

Ro(.5.8 ± 0.5) ·10-8 sm v=0.62±0.02, T>Tc

458

Ro(.3.6 ± 0.3) ·10-lsm v=0.61±0.02, T<rc (18)

The obtained results are in good agreement with prediction of scaling theory

and existing experimental data in the vicinity of a critical point of stratification.

459

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ustanovleniya ravnovesnogo gravitazhionnogo effecta vblizi kriticheskoy

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387-390.

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svoystvakh veschestva v gravitazhionnom pole v blizi kriticheskoy tochki.

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PARTICIPANTS

ABBISS, John, Singular Systems, 19451 Sierra Raton Road, Irvine, California 92715, USA

BARTSCH, Eckha.rd, Universitat Mainz, Institut fUr Physikalische Chemie, Jakob-Welder-Weg 15, D-55099, Mainz, Germany

BERTOLOTTI, Mario, Dipartimento di Energetica, Universita. degli Studi Roma La Sapienza, Via Scarpa 14-16,00161 Rome, Italy

BROWN, Robert, Sharp Laboratories of Europe Ltd, Edmund Halley Road, Oxford Science Park, Oxford OX4 4GA, UK

CASTANIIO, Miguel, Centro de Quimica Fisica Molecular, Complexo LI.S.T., 1096 Lisboa Codex, Portugal

CHAIKIN, Paul, Department of Physics, Princeton University, Princeton, New Jersey, USA

CHRISSOPOULOU, Kyriakh, IESL-FORTH, PO Box 1527, Vassilika Vouton, Heraklion 71110, Crete, Greece

DEGIORGIO, Vittorio, Dipartimento di Elettronica, Universita. di Pavia, Via Ferrata 1, 27100 Pavia, Italy

DHONT Jan, Utrecht University, Vakgroep Fysische en CoIloid Chemie, Paduala.an 8, 3.584 CII, Utrecht, The Netherlands

DIERKER, Stephen, Department of Physics, University of Michigan, 2071 RandaIl Laboratory, 500 E University Avenue, Ann Arbor, Michigan 48109-1120, USA

FRISKEN, Barbara, Department of Physics, Simon Fraser University, Burnaby BC, V.5A IS6, Canada

FYTAS George, IESL-FORTH, PO Box 1527, Vassilika Vouton, Heraklion 71110, Crete, Greece

461

462

GAMMON, Robert, Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA

GOLDBURG, Walter, Department of Physics and Astronomy, University of Pittsburgh, Pennsylvania 15260, USA

GRAN Michael, Department of Physics, King's College London, Strand, London WC2R 2LS, UK

JAKEMAN, Eric, Department of Electronic and Electrical Engineering, University Park, Nottingham, NG7 2RD, UK

JANSZKY, Jozsef, Crystal Physics Research Laboratory, H-l.502 Budapest, PO Box 132, Budapest, Hungary

KAZA KOV, S. V., Division of Low Temperature Physics Faculty of Physics, Lomonosov State University, Vorob'evy Gory, 119899 Moscow, Russia

KIERK, Isabella, Manager, Microgravity Technology Development and Transfer Programs, NASA Jet Propulsion Laboratory M.S. 233-200, 4800 Oak Grove Drive Pasadena, CA 91109-8099, USA

KLEIN, Rudolph, Universitat Konstanz, Fakultat fiir Physik, Universitatstrasse 10, D-78434, Konstanz, Germany

KLYURlN, V.V., Department of Colloid Dispersions, tebedev Rubber Research Institute, Gapsalskaya str.l, 198035 Sankt-Petersburg, Russia

KOSTKO, Andrei P., Physical Department, State Academy of Refrigeration and Food Technologies, Lomonosova str.9, 191002 Sankt-Petersburg, Russia

KOTOYANTS, Karine, Institute of Applied Mathematics, Academy of Sciences of the Republic of Kazakhstan P.O.Box 223, 480000, Almaty, Kazakhstan

MANN, .Jay Adin, Jr, Department of Chemical Engineering, Case Western Reserve University, A. W. Smith Bldg., Rm 129, 10900 Euclid Ave., Cleveland, 011 44106-7217 USA

MEYER, William, NASA / Ohio Aerospace Institute NASA Lewis Resea~ch Center, 21000 Brookpark Road, M.S. 105-1 Cleveland, 011 44135-3191, USA

NIKOLAENKO, G.L., Oil and Gas Research Institute of the Russian Academy of Sciences, Leninsky Prospect 63, 117917 Moscow, Russia

OSTAPCHENKO, S., Faculty of Physics, Taras Shevchenko State University, Glushkova Prospect 6, 252017 Kiev, Ukraine

PIKE, E. Roy, Department of Physics, King's College London, Strand, London WC2R 2LS, UK

PINE, David, Department of Chemical Engineering, University of California at Santa Barbara, CA 93106-5080, USA

PUSEY, Peter, Physics Department, University of Edinburgh, King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK

RARITY, John, DRA, P Building, Malvern, Worcs WR14 3PS, UK

RICKA, Jaroslav, Institut fiir Angewandte Physik, Universitat Bern, Sidlerstr. 5, CH-3012 Bern, Switzerland

ROGER, Philippe, Institut National de la Recherche Agronomique LBTG Rue de la Giraudihre BP 71 62744316 Nantes Cedex 03, France

SMART, Anthony, Titan Spectron, 2857 Europa Drive, Costa Mesa, California 92626-3525, USA

STEPANEK, Petr, Institute of Macromolecular Chemistry, 162 06 Praha 6, Czech Republic

463

TRAPPE, Veroniqlle, Institut fiir Makromolekulare Chemie, Sonnenstr. 5 (Ecke Kleierstr), Universitat Frieburg, Frieburg, Germany

van MEGEN, William, Royal Melbourne Institute of Technology, Departm(>nt of Applied Physics, Melbourne, Victoria 3000, Australia

WEISSMULLER, Max, Institut fiir Makromolekulare Chemie, Sonnenstr . .5 (Ecke Kleierstr), Universitat Frieburg, Friebllrg, Germany

WEITZ, David, Department of Physics and Astronomy, University of Pennsylvania, 209 S 33rd St, Philadelphia PA, USA

YUDIN, Igor, Oil and Gas Research Institute of the Russian Academy of Sciences, Leninsky Prospect 63, 117917 Moscow, Russia

ZII U, Jixiang, Department of Physics, Princeton University, Princeton, USA

ZUBKOV, L.A., Institute of Physics, State University, San kt- Petersburg, Russia

E-mail addresses

abbiss@aol.com bartsch@vpcmza.chemie.uni-mainz.de bertoI88@itcaspur.caspur.it rgwb@sharp.co.uk pcmcastanho@alfa.ist.utl.pt chaikin@pupgg.princeton.edu degiorgio@ipvsp4.unipv .it kiki@iesl.forth.gr dierker@umich.edu frisken@sfu.ca fytas@t.alos.cc.uch.gr rgammon@roissy.umd.edu goldburg@vms.cis.pitt.edu mg@maxwell.ph.kcl.ac.uk janszky@sparc.core.hu kazakov@lt267.phys.msu.su ikierk@pop.jpl.nasa.gov rudolf.klein@uni-konstanz.de klyubin@veren.cyt.ras.spb.ru vlad@loc.spb.su karin@kit.kz jam 12@po.cwru.edu rphbm@bunyip.ph.rmit.edu.au bill@sarah.lerc.nasa.gov nikol@fluid.msk.ru ostap@molphys.ups.kiev.ua erp@maxwell.ph.kcl.ac.uk pine@engineering.ucsb.edu pusey@ed.ac.uk jgr@fakor.dra.hmg.gb rick@fluor.iap.unibe.ch roger@nant.es.inra.fr stepan@imc.cas.cz trappe@ruf.uni-freiburg.de weit.z@dept.physics.upenn.edu yudin@fluid.msk.ru jzhll@phoenix.princeton.edu zu bkov@onti.niif.spb.su

464

INDEX

Advanced Photon Source, 71 Amyloplectin, 225 Amylose, 225 Analytic continuation, 299, 300 Asphaltene aggregation kinetics, 348ff Asymmetry, 409, 415 Avalanche photodiodes, 117, 254 Binary mixtures, 161, 162, 167ff, 404, 405, 408,

411ff, 415, 420 Binary fluid, 161, 162

solutions, 443ff Biovectors, 174 Birefringence, 206 Block copolymers, 133, 190ff, 195ff, 199, 206 Bragg scattering, 213, 215, 218 Branched polymers, 147

systems, 152, 154, 155, 157, 158 Brownian motion, 7, 8

particles, 425ff Bulk and surface scattering, 403, 404, 406 Bunching, 232, 242 Classical nucleation theory, 220 Cloud-point curve, 424 Clusters, 195ff Coated sphere model, 182 Coexistence curves, 403, 409, 411, 412, 415, 416, 419 Coherent X-rays, 66 Colloidal glass transition, 371

particles, 7,8, 9, 14,129,427,429,433 suspensions, 39

Colloids, 24, 25, 211 Compositional heterogeneity, 191, 192 Concentrated suspensions, 2 Condensed matter, 373, 376 Consolute critical point, 402,403,412,414,415

465

466

Cooperative diffusion, 201 Critical behaviour, 162

exponents, 424 isochore, 446, 447 fluid,389 parameters, 357, 362 point, 423, 424 systems, 161,390 temperature, 390, 432, 434, 454, thermodynamics, 390

Cross-correlation, 4.5ff, 51, 54, 57ff, 85, 119, 233, 242ff Crude oil, 347, 348 Crystallisation, 210ff, 220, 221 Data inversion, 313,314 Dead time, 123, 125, 127 Decay rates, 389ff Density fluctuations, 37,390 Depolarized light scattering, 7, 8, 16 Design, 373, 377, 385 Diblock copolymers, 131fT Diffusing-wave spectroscopy, 66, 326, 330, 339, 371 Diffusion, 214,216,218

free-particle, 2 long-time, 3 rate, 3 relative, 4 self, 5,369 short-time, 3 time, 2 zero shear rate, 3

Dipole approximation, 85 Disorder, 161, 162, 169 Disorder-order transition, 387 Disperse particle sizing, 342 Dispersed systems physics, 24 Domain growth, 169

state, 161, 165, 166 Double scattering, 51, 60, 62, 356, 357, 359, 360, 361, 363ff

Drug delivery system, 173, 174 Dynamic structure factor, 37,68 Dynamic scaling, 161 Dynamics of colloidal coagulation, 25, 26 Elastic modulus, 371 Enhanced backscattering, 81, 90 Entanglement, 277, 279, 286, 288, 290ff First-kind Fredholm integral equations, 296, 297 Fluctuation Spectra, 99 Fluid viscosity, 371 Forced Rayleigh scattering, 369 Fractal dimensions, 228 Fraunhofer diffraction, 75 Gel fragments,174, 175, 180 Glass transition, 70 Gold-coated latex beads, 174 Grazing incidence, 91 Growth dynamics, 387 Hard spheres, 10,17,18,19, 373ff, 212, 213,

218,220, 222 liard-sphere behaviour, 369

interaction, 1, 3 Hydrodynamic regime, 403

size measurements, 26, 28 Ill-conditioned problems, 313,316 Ill-posedness, 297 Image scatterers, 88 Interface/s,97ff Interference, 248, 2.51, 260 Intermediate scattering function, 1

structure factor, 4 Internal motions, 142 Inverse problems, 296 Kratky plot, 227 . Laser velocimetry, 264 Light transport, 328 Liquid-liquid equilibria, 415,421

interfaces, 405,407,409,411

467

468

Liquids, 68 Local dynamics, 158 Long-range density fluctuations, 195 Maltodextrins, 174, 175, 180, 182 Mean-square displacement, 5 Micelles, see Wormlike Microgels, 226 Microgravity, 373, 389, 399 Mobility of components, 403, 412ff Molecular vibration states, 277 M uel1er matrices, 84 Multi-path effects, 81, 91 Multiple scattering, 39ff, 45ff, 51ff, 328, 329, 331,333, 334, 354, 355, 357,

3.58, :362, 363 Non-gallssian enhancement factor, 86

polarisation fluctuations, 80, 86 Nonclassica.1, 278,279

vibrational states, 278, 279 Nonlinear materials, 232, 237, 238

processes, 236 waveguide, 232, 233

Opaque fluids, 342ff, 347, 348 materials, 66

Opto-electronic devices, 26:3, 27lff Order-to-disorder transition, 190, 195, 196,203,206 Packaging, :37:3 Pair-correlation function, 427ff Particle scattering models, 80

size, 313, 316, 317 Phase transition, 354 Photon counting, 117ff

diffusion, 326, 329, 331,332, 339 pair, 2.52, 257

PMMA spheres, 387 Poisson statistics, 118, 256, 261 Polarisability tensor, 85 Polarization, 355, :361 Polyelectrolytes, 176

Polymer blends, 131, 135 starlike, 179 solutions and melts, 190, 191, 195, 197

Polymers, 131, 132, 136 Polysaccharides, 225 Probability density of intensity, 83 Q-dependence, 142, 143, 147, 155ff Quantum Cryptography, 232 Random walk, 81 Rayleigh line, 453, 454

scatterers, 8.5 scattering, 369

Regularisation, 300,301 Rotational diffusion, 7, 8, 17, 18 Rough surfaces, 80 Scaling theory, 448

property, 1 Scattering vector, 4 Schrdinger-cat state, 283ff Shear thickening, 367 Shear flow, 424ff, 428, 429, 431ff, 436, 438, 439 Silica gel, 161, 162 Single-mode fibres, 129 Single photons, 248, 249, 259, 260 Singular functions, 297, 300 Singular-value decomposition, 305 Smoluchowski equation, 37, 424ff, 438 Space Shuttle, 389

Columhia Launch, :373, environment, 383 Space Sta.tion, 375 Spatia.l photon correlation, 232 Spatially inhomogeneous systems, 442 Speckle, 71 Spheroid, 85 Spinodal, 423,424,431, 432ff, 439

decomposition, 423, 424, 436 Squeezing, 2;12, 242, 244, 245, 278, 282

469

470

Starch, 225 Static structure factor, 2

and dynamic properties, 402, 403, 404, 411 light scattering, 147, 149, 156

Stokes parameters, 84 Stokes-Einstein, 426,429 Stratification critical point, 443 Structural relaxation, 4 Super-aggregates, 226 Surface chemistry, 97, 112

light scattering spectroscopy, 97, 99, 107ff science, 112 visco-elastic coefficients, 106

Synchrotron , 71 Temporal correlation function, 363ff Temperature control, 389, 391 Turbidity, 44

of colloids, 26, 32 Turbulence, 323 Two-colour technique, 2 Universal functions, 403,415,418 USMP-2, 389, 390

-3,389,396 Vaccine, 173 van Cittert-Zernicke theorem, 71 Vibrational states, 281ff, 285ff, 289, 292 Viscoelastic moduli, 371 Vitreous, human eye, 129 Wormlike micelles, 367 Xenon, 390,400 X-ray photon correlation spectroscopy, 66 XPCS,66 Zeno, 389ff,