CYCLIC POLYMERS

CYCLIC POLYMERS

Edited by

J. A. SEMLYEN Department of Chemistry, University of York, UK

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Softcover reprint of the hardcover 1 st edition 1986

British Library Cataloguing in Publication Data

Cyclic polymers. 1. Polymers and polymerization compounds I. Semlyen, J. A. 547.7 QD381

2. Cyclic

Library of Congress Cataloging in Publication Data

Cyclic polymers.

Bibliography: p. Includes index. I. Polymers and polymerization. 2. Cyclic compounds.

I. Semlyen, J. A. II. Title. QD38l.C93 1986 547'.5 85-16062

ISBN-13: 978-94-010-8354-6

DOl: 10.1007/978-94-010-8354-6

e-ISBN-13: 978-94-010-8354-6

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Preface

Synthetic polymers based on long chain molecules have been investigated intensively for over 50 years. They have found important applications as plastics, fibres, rubbers and other materials. The chain molecules may be simple linear structures or they may be branched or cross-linked.

During the past decade, sharp fractions of the first synthetic cyclic polymer have been prepared. These fractions of cyclic poly(dimethyl­siloxane) consist of ring molecules containing hundreds of skeletal bonds. Some of their properties have been found to be quite different from those of the corresponding linear polymers. Synthetic cyclic polymers, including cyclic polystyrene, have joined the naturally occurring circular DNAs as examples of substantially large ring molecules.

This book aims to review current knowledge of cyclic polymers and biological ring macromolecules. In addition, it discusses theories of cyclic macromolecules and describes cyclization processes involving long chain molecules. Since 1865, when Kekule proposed a simple ring structure for benzene, larger and larger ring molecules have been synthesized in the laboratory and discovered in nature. Many more examples are to be expected in the future. In time, large ring molecules should take their proper place alongside long chain molecules as one of the two possible constituent structural units of polymers.

J. A. SEMLYEN

v

Contents

Preface v

List of Contributors ix

I. Introduction J. A. SEMLYEN

2. Theory of Cyclic Macromolecules 43 WALTHER BURCHARD

3. Preparation of Cyclic Polysiloxanes 85 P. V. WRIGHT and MARTIN S. BEEVERS

4. Comparison of Properties of Cyclic and Linear Poly(dimethyl-siloxanes) 135

CHRISTOPHER J. C. EDWARDS and ROBERT F. T. STEPTO

5. Neutron Scattering from Cyclic Polymers 167 KEITH DODGSON and JULIA S. HIGGINS

6. Organic Cyclic Oligomers and Polymers 197 HARTWIG HOCKER

7. Circular DNA 225 J. C. WANG

vii

viii

8. Cyclic Peptides ALAN E. TONELLI

CONTENTS

261

9. Spectroscopic Studies ofCyclization Dynamics and Equilibria 285 MITCHELL A. WINNIK

10. Cyclization, Gelation and Network Formation 349 S. B. Ross-MuRPHY and ROBERT F. T. STEPTO

Index 381

List of Contributors

MARTIN S. BEEVERS

Department of Chemistry, University of Aston in Birmingham, Gosta Green, Birmingham B4 7ET, UK

WALTHER BURCHARD

Institute of Macromolecular Chemistry, University of Freiburg, Stefan-Meier-Strasse 31, 7800 Freiburg im Breisgau, Federal Republic of Germany

KEITH DODGSON

Department of Chemistry, Sheffield City Polytechnic, Pond Street, Sheffield SI 1 WB, UK

CHRISTOPHER J. C. EDWARDS

Department of Polymer Science and Technology, University of Man­chester Institute of Science and Technology, PO Box 88, Manchester M60IQD, UK (Present address: Unilever Research, Port Sunlight Laboratory, Quarry Road East, Bebington, Wirral, Merseyside L633JW, UK)

JULIA S. HIGGINS

Department of Chemical Engineering, Imperial College of Science and Technology, Prince Consort Road, ILondon SW7 2BY, UK

ix

x LIST OF CONTRIBUTORS

HARTWIG HOCKER

Institute for Macromolecular Chemistry, University of Bayreuth, D-8500 Bayreuth, Federal Republic of Germany

SIMON B. Ross-MURPHY

Unilever Research, Colworth Laboratory, Colworth House, Sharnbrook, Bedford MK44 lLQ, UK

J. A. SEMLYEN

Department of Chemistry, University of York, Heslington, York, Y015DD, UK

ROBERT F. T. STEPTO

Department of Polymer Science and Technology, University of Man­chester Institute of Science and Technology, PO Box 88, Manchester M60 1QD, UK

ALAN E. TONELLI

AT & T Bell Laboratories, Murray Hill, New Jersey 07974, USA

J. C. WANG

Department of Biochemistry and Molecular Biology, Harvard University, 7 Divinity Avenue, Cambridge, Massachusetts 02138, USA

MITCHELL A. WINNIK

Lash Miller Laboratories, Department of Chemistry and Erindale College, University of Toronto, Toronto, Ontario, Canada M5S 1A1

PETER V. WRIGHT

Department of Ceramics, Glasses and Polymers, University of Sheffield, Elmfield, Northumberland Road, Sheffield S10 2TZ, UK

CHAPTER 1

Introduction

J. A.SEMLYEN

Department of Chemistry, University of York, UK

LINEAR POLYMERS AND CYCLIC POLYMERS

In the 1930s, Staudinger's macromolecular hypothesis was generally accepted and his long chain formulae for polystyrene, polyoxymethylene and other polymers became fully established.! As described by Flory, 2 ring structures had been assigned to some polymers earlier in the century but they were later shown to be erroneous. It soon became accepted that synthetic polymers were based on long chain molecules that could be linear, branched or cross-linked to form networks. The linear polymers could have mean molar masses of millions, corresponding to tens of thousands of skeletal bonds. 2

In this book, macromolecules based on large ring molecules rather than long chain molecules are described. Cyclic polymers are compared with linear polymers. Although well-characterized branched and network cyclic polymers have yet to be prepared, it is noted that cyclic polymers could have cyclic or linear branches and networks of rings could be catenated or have no free ends. Furthermore, there is a wide range of possibilities for types of polymer built from long chains and large rings. It will surely be a long time before a substantial number of such structures are synthesized and characterized.

It might be asked, 'How many skeletal atoms (on average) must there be in the ring molecules before we have a cyclic polymer?' There are indications that cyclic poly(dimethyl siloxane) with about lOO skeletal atoms shows the properties expected of a polymer, whereas ring fractions containing substantially fewer skeletal atoms do not. The term macrocyclic

2 J. A. SEMLYEN

(Greek, macros = long) is being used in the literature to describe rings with relatively few skeletal atoms, such as 15 or 20. These 'macrocyclics' do not show macromolecular behaviour and the term is a misnomer. They could be called medium rings or the term mesocyclic (Greek, mesos = middle or intermediate) might be used to describe them. The term macrocyclic could then be reserved for the cyclic polymers and ring macromolecules of the kind described in this book.

Some cyclic macromolecules, including circular deoxyribonucleic acids (DNA) have been found to occur in nature. In Chapters 7 and 8, circular DNA and cyclic peptides are described. Some cyclic oligo saccharides have been discovered, including cycloamyloses (cyclodextrins) and a cyclic oligosaccharide composed of four, five and six trisaccharide repeat units. 3

Much larger cyclic polysaccharides may be found or synthesized in the future. Cyclic biological macromolecules have attracted considerable interest and are expected to become increasingly important in the years ahead.

SOME DIFFERENCES BETWEEN THE PROPERTIES OF CYCLIC AND LINEAR POLYMERS

Ring and chain macromolecules are topologically distinct, so there are many differences in their properties and behaviour. Some examples have been chosen to illustrate these differences in this chapter. Other examples are given later in the book (for example, the stabilization of supercoiling in large DNA rings in Chapter 7)..

The Presence or Absence of End-groups A variety of chemical groups may terminate polymer chain molecules. The nature of these end-groups can be important. For example, they can react with other suitable molecules or they can be used to make analytical determinations as in end-group analysis.

Ring polymers have no ends and no end-groups. The chemistry of end­groups developed by many research workers (see, for example, Ref. 4) is obviously not applicable to large ring molecules. In this connection, it is noted that the name cyclic poly(dimethyl siloxane) is a precise name for the ring polymer. Linear poly(dimethyl siloxane) carries HO-, (CH 3hSi- or other groups at the ends of the chain molecules. No reference is made to these groups in its name.

INTRODUCTION 3

Consequences of Simple Bond Cleavage When a single bond is broken in a cyclic polymer, the product is quite different from those formed by random cleavage of a bond in a linear polymer. Large rings form chains of similar molar mass thus:

By contrast, long chains form two smaller chain fragments, which mayor may not have similar molar masses:

~+ These differences were discussed by Kelen et al. 5 for the hydrolysis of cyclic and linear 1,3-dioxolanes, catalysed by acids.

Reactions Linking Molecules Together When two long chain molecules bearing reactive groups (as in telechelic polymers) link together by an intermolecular condensation reaction, then there is the possibility of different isomers being formed. For example, by the following process:

+

4 J. A. SEMLYEN

By contrast, the corresponding large ring molecules can only form a unique dimer, as shown:

o + D Bond Interchange Reactions The consequences of skeletal bond interchange between two large ring molecules are quite different from those between long chain molecules. Thus, two rings of equal size interchange bonds to form a ring twice the size of each:

D + D By contrast, two long chain molecules undergoing a bond interchange reaction form two other chains, which mayor may not have similar lengths to those of the parent molecules:

+ +

Bond interchange reactions have been widely discussed in chemistry, including the areas of condensed phosphates6 and other inorganic polymers. 7

Catenation Large ring molecules have the ability to form catenanes (Latin, catena = a chain). Obviously, catenane formation is not possible for linear molecules.

Wasserman 8 prepared a catenane by carrying out an acyloin condensation of a diester chain molecule, with a large excess of a partly deuterated C34 cycloalkane being present. The catenane was obtained in c. I % yield and it may be considered to be a topological isomer of the two ring molecules involved.

INTRODUCTION 5

i\...--(CH2)32,\

D'II"~~IIC\'//~O OH

SchiW has reviewed the chemistry and topology of catenanes, including the higher catenanes. An example of a [3]-catenane involving three ring molecules is as follows: 10

Ac

(CII,),,___. (CII,)"-~-(CII,),, /(CII,)" "\

CO AcO f , ~ / II \\ CO \ - N.Ac Ac.N AcO 'I '\ ;;AC

\ AC~ :x-(CH 2)12 '-........(CH) -N-(CH) __ OAc

2 12 I 2 12 (CH 2)12

Ac

Catenated DNA rings are found naturally and are discussed in Chapter 7.

Rotaxanes Rotaxanes (Latin: rota = wheel, axis = axle) can be prepared from ring molecules of suitable size. Obviously, there are no corresponding compounds which can be prepared from linear molecules alone.

The preparation of a rotaxane may be illustrated diagrammatically as follows:

o Heat or catalysts >-0--<

Preparative routes to rotaxanes have been explored by Harrison 11 •12 and by Schill and his co-workers. 9, 13

In a detailed study ofrotaxane formation, Harrison 12 heated a mixture of cyclic paraffins [CH2 ]n with I, 13-di(tris-4-t-butylphenylmethoxy)tri­decane in the presence of a catalyst. After equilibration had been reached, the reaction was quenched and the rotaxanes separated from the mixture.

6 J. A. SEMLYEN

The cyclic molecules were released by treatment with acid and analysed to give the yield of rotaxane for each size of ring. The values obtained showed a steadily increasing yield from 0·0013 % for rings with n = 24 to 1·6 % for rings with n = 33. No rotaxanes were produced when n < 24 or n> 34, as expected from the examination of molecular models.

Harrison 12 also prepared one of the rotaxanes in 1 % yield from the single cyclic [CH2132' by using the procedure just described. Spectroscopic studies together with chemical transformations gave compelling, but not fully conclusive, evidence of the identity of the rotaxane.

Knots The possibility of ring molecules having knots in their structures has been discussed by Wasserman,8 Schill,9 Brochard and De Gennes14 and Roovers and Toporowski. 15 No permanent physical knots can be formed in normal chain molecules. It has been estimated, from the examination of molecular models, that about 50 methylene units are required to make it possible for a knot to be present in an alkane: ring [CH 2 1n. 8 ,9

Network Structures In principle, special network structures can be obtained from cyclic polymers that have no linear analogues. For example, large ring molecules could be linked together to produce networks with no free ends. In the future, it may even prove possible to make 'chain mail' networks of catenated rings.

Using cyclic poly(dimethyl siloxanes) and chains, which can be end­linked into networks, it has been demonstrated recently that ring molecules can be incorporated into network structures without being chemically bonded to the other molecules, thus: 16

-0 When two chains are threaded through the same ring, cross-links of the

following type may be produced:

~~C( ~Y-Y"

INTRODUCTION 7

Now that synthetic cyclic polymers are available, more examples oflarge rings forming all or part of network structures are to be expected. 17

METHODS FOR PREPARING MEDIUM RINGS

Before considering the preparation of large ring molecules by ring--chain equilibration reactions, some synthetic methods which yield medium rings (or mesocyclics) will be described. The rings produced contain up to 40 skeletal bonds and the preparative methods are well known in organic synthesis. The ring forming reactions may be classified into three groups: reactions involving dilution, reactions involving the interchange of skeletal bonds and surface reactions.

The Dilution Method The dilution of a reaction system so as to favour ring formation was discussed and applied by RugglPS in 1912. Linear molecules with reactive ends form large rings under high dilution conditions, because intra­molecular cyclization is favoured relative to intermolecular condensation. Two reactions using the dilution method are as follows:

r----C=NH Thorpe-Ziegler nitrile reaction I

NC-[CH2]n_ l~N ---C--C--~~ [CH2]n- 2 LiN( ,H,)( ,H,) I

Dieckmann reaction ..

,-. -----CH.CN

I r----C=O

[CH2 ]n_2

IL-----CH.COOR

The Thorpe-Ziegler reaction gives good yields of medium rings with n= 14-35 skeletal bonds (see Fig. 1 for some typical results).19-22

Sisid023 has discussed the main features of the experimental results shown in Fig. 1. These include the ease with which the smallest rings are formed, the low yields of rings in the range n = 9-13 skeletal bonds, the general increase in yields of rings in the range n = 13-18 and the alternation of yield in the range n = 15-20. Following the theoretical approaches of Saunders24 and Smith,25 Sisid023 interpreted all these observations by carrying out calculations of intramolecular cyclization using a diamond

8

~ 60 ~ 40 :;:

20

J. A. SEMLYEN

5 10 15 20 25 Number of skeletal bonds in the ring, n

FIG. 1. Typical yields of cyciics produced from NC-[CH21n_ 1-CN in the Thorpe-Ziegler reaction. 19 - 22

lattice model and then Abe, Flory and Jernigan's26.z 7 rotational isomeric state model. Full account was taken of the steric interactions between non­bonded atoms of the X-[CH z],,_ z-Y chains involved and conformations with carbon-carbon distances less than 2· 77 A were rejected. A value in the region 2· 5 A was assumed for the distance within which termini react to form a ring.

The Interchange Method Carothers and his co-workersz8 found that the thermal depolymerization of certain condensation polymers in the presence of catalysts can give good yields of medium rings. For example, in one series of experiments,28.z9 polymeric esters with the general formula +OC(CHz)zCO. O(CHZ)mO-h were prepared from succinic acid. These polymers were heated at 543 K and 1 mm Hg, using 1-3 % SnClz .2HzO as catalyst. The yields of monomeric and dime ric rings obtained are shown in Fig. 2. It was found that cyclic dimers are strongly preferred to cyclic monomers when m = 2 and 3, but when m> 3 monomeric rings are formed in preference to the dimers.

Carothers and Hill z8 .30 have pointed out that depolymerization is involved in the familiar Ruzicka thorium salt method 31 .3z for the synthesis of medium rings with carbon backbones. The classical Ruzicka synthesis31 .3z may be represented as follows:

HOOC-[CHZ]n_1-COOH Thorium salt

)

I c=o ,

The reaction gives no rings with n = 9 and n = 10 skeletal bonds and only

INTRODUCTION 9

80

* 60 "0 OJ >=

40

20

8 9 10 11 12 13 14 15

Skeletal bonds in the monomeric rings (.) FIG. 2. Yields of cyclic monomers (.) and cyclic dimers (.) obtained by heating

polyesters +OC(CH2)2CO. O(CH2)mO-k 28.29

small amounts of rings with n> 10 (see Fig. 3 for some typical yields). Carothers and Hil1 28 .30 viewed the reaction as the depolymerization of 'polyketones' and it gives very low yields of medium rings because the processes involved are not strictly reversible and there are many side reactions. Furthermore, the Ruzicka reaction cannot be carried out under high dilution conditions.

Ring Formation on a Surface The acyloin condensation is believed to be an example of a cyclization reaction which takes place on a surface. 21 ,33 The method is frequently employed and it gives high yields of medium rings. The reaction may be represented:

Acyloin condensation ROOC-[CHZln_ z-COOR -----.. Na

r-I---C=O

[CH,j. , I <-1---4CH.OH

10 1. A. SEMLYEN

100 ~-------=-----:~ ---. 80

* 60 ~ 40 ~

20

6 8 10 12 14 16 18 20

Number of skeletal bonds in the ring, n FIG. 3. Typical yields of cyclics produced from ROOC-tCH21n _ 2-COOR by the acyloin condensation (.)21.33 are compared with those obtained from

HOOC-[CH21n _ 1-COOH by the Ruzicka synthesis (.6.).21.31.32

It has been suggested that the two ends of the reactant molecules become attached to nearby sites on the surface of the metallic sodium. 21 In Fig. 3, typical yields for the acy10in condensation reaction are compared with those obtained by the Ruzicka method described above.

RING-CHAIN EQUILIBRATION REACTIONS

Ring and Chain Molecules Formed in Polymeric Equilibrates Ring-chain equilibration reactions are of particular interest because they produce cyclic populations, which may include a full range of small, medium and large ring molecules. Furthermore, in principle, it is possible to calculate the concentrations of all the individual species produced. In general, there is a most probable distribution of chain lengths in the acyclic portion of the polymer produced in a ring-chain equilibration reaction and the concentrations of the individual chains may be calculated from the number average molar mass of the acyclics, using Flory's re1ationships.2 Similarly, the concentrations of individual cyclics may be calculated using the Jacobson and Stockmayer34 theory, although the theory requires considerable modification before it can be applied to calculate the concentrations of small and medium rings (see below).

In the remainder of this chapter, attention will be confined to reactions where there is a thermodynamic equilibrium between ring and chain molecules. Other cyclization reactions, induding those that are kinetically controlled, will not be considered here (see Ref. 22 and Chapters 9 and 10).

INTRODUCTION 11

Measurement of Equilibrium Ring Concentrations The concentrations of cyclic oligomers and polymers produced in ring-{;hain equilibration reactions can generally be accurately determined by using one or more chromatographic methods. The most widely applicable are gas-liquid chromatography (GLC), gel permeation chromatography (GPC), paper chromatography and high-performance liquid chromatography (HPLC).

x=4 x=5

x=3

A ~ A x=6 IL

t ~ I

445K 545K

FIG. 4. Gas-liquid chromatogram (GLC) of cyclics [CH2 OCH2CH2 0lx obtained by distillation from the oligomeric extract of a ring-chain polymerization of 1,3-dioxolane. The instrument was a Pye series 104 with a flame ionization detector and

temperature programming was employed as indicated. 39

The technique of GLC was introduced in 1952 by Martin and James35 (see also Ref. 36) and it is widely used for the analysis of volatile materials. It has proved a most useful method for analysing cyclics formed in ring-{;hain equilibration reactions. 37 For example, individual cyclic dimethyl siloxanes [(CH3hSiO]x may be separated and analysed using a katharometer detector for values of x up to and including x = 50 (i.e. rings with 100 skeletal bonds).38 A typical GLC of cyclic oligomers from a polyether ring-{;hain equilibration reaction is shown in Fig. 4. 39 The use of thermally stable stationary phases (such as OVI and OVl7 from Phase Separations Ltd, Wales, UK) allows temperature programming up to 673 K and beyond. 38,39

The technique of analytical G PC was introduced by Moore40 in 1964 and 1 year later Maley41 described the first commercial G PC instrument. G PC is a powerful method for analysing and characterizing synthetic polymers. When an instrument has been calibrated, mean molar masses and molar mass distributions can be conveniently and reliably measured. A typical G PC separation of oligomeric rings extracted from a poly( ethylene terephthalate) ring-{;hain equilibrate is shown in Fig. 5. Styragel (from Waters Associates Ltd, England, UK) is still used for many separations, but Dawkins and Yeadon42 have reviewed newer column materials for high performance GPC. These give higher speed and better resolution

12 J. A. SEMLYEN

3

4

55 50 45 40 35 Elution volume (counts)

FIG. 5. Gel permeation chromatogram (GPC) of cyclics [CO.C6 H4'CO.O. CH2 .CH2 .01x with X= 3-9 obtained with columns containing SX-I Bio-beads (from Bio-Rad Laboratories, England, UK) (D. R. Cooper, University of York,

1972).

separations of both oligomers and polymers than the conventional column materials.

Techniques used to analyse polar oligomers and polymers include paper chromatography and gel electrophoresis.43 For example, paper chromatog­raphy has been used to analyse oligomeric metaphosphates [NaP031x from quenched sodium phosphate ring-chain equilibrates at 1000 K.44,45 The quenched melt is known as Graham's sa1t46 and it is a glass with the empirical formula NaP03, consisting of long, linear polyphosphate chains terminated by hydroxyl groups together with - 10% w jw cyclics [NaP031x' The concentrations of these individual cyclics have been determined by paper chromatography for x = 3-6 by van Wazer and his co­workers44 and for x = 3-7 by Thilo and Schulke.45

Until recently, sulphur rings Sx (with x #- 8) were examples of thermally unstable rings that could not be analysed using a chromatographic

INTRODUCTION 13

technique. However, in 1981, Steudel, Miiusle and their co-workers47 - 49 developed a reversed-phase chromatographic method that can be used to analyse all the sulphur rings from S6 to S26. In earlier investigations, sulphur rings were found to decompose to S8 on silica and alumina surfaces, although some success was achieved with column chromatog­raphy using silica gel at - 40°C and carbon disulphide as eluent. The successful HPLC method employs a stationary phase, which Steudel48 has described as consisting of C18H 37 radicals fixed to the surface of an Si02 support via carbon-oxygen bonds. The particle size is only 5-10 11m, giving a large specific surface and approximately 5000 theoretical plates for a 10 cm column. The components are detected by a characteristic ultraviolet absorption. There is excellent resolution for all the rings up to S26' but rings with x> 26 could not be separated because of their low solubility in the polar eluent (methanol for the rings with x ::; 12 and a 69: 31 mixture of methanol and cyclohexane for the larger rings).

Methods for Calculating Equilibrium Cyclic Concentrations The chromatographic methods just outlined make it possible to measure individual cyclic concentrations for a range of ring-chain equilibrates. Methods for calculating these equilibrium cyclic concentrations will now be described.

Consider a ring-chain equilibrate, with an equilibrium between x-meric cyclics Mx and linear chains -My- and -My-x- thus:

(1)

Attention will be confined to unstrained rings, where the standard state enthalpy change for process (1) is zero.

A most probable distribution of chain lengths has been found for many ring-chain equilibrates, so for these systems the molar cyclization equilibrium constant Kx for process (1) above is given by37

(2)

The factor p can be expressed in terms of the number average molar mass Mn and the molar mass of a monomeric unit by Flory's relationship:2

Mo p=l--=­

Mn (3)

F or typical ring-chain equilibrates, values of p are close to unity and so Kx

14 J. A. SEMLYEN

values calculated by equations (2) and (3) are approximately equal to the concentrations [Mxl.

The Jacobson-Stockmayer theory49,50 provided the first theoretical expression for the molar cyclization equilibrium constant Kx' The theory has been discussed and extended by Flory27,51 and, with his notation, the Jacobson-Stockmayer expression for the molar cyclization equilibrium constants is as follows:

(4)

Wx(O) represents the density of end-to-end vectors in the region corresponding to the close approach of chain ends (i.e. where the end-to­end vector r;;;;; 0), N A is the Avagadro constant and O'Rx is the symmetry number of an x-meric ring (values of x or 2x are assigned to all the rings discussed here).

Three methods of calculating experimental cyclic concentrations in ring-chain equilibrates are to be described. the first is the original Jacobson-Stockmayer method and it applies to rings formed from chains of sufficient length and 'flexibility' to obey Gaussian statistics. The second is termed the direct computational method. 52 - 55 Here, no assumption is made about the statistical properties of chains forming rings. Instead, the distances between the terminal atoms of the chains are calculated for all the possible conformations defined by a rotational isomeric state model and those conformations that have their termini close enough for ring formation are determined. The relative orientations of the chain termini can also be investigated. Although the amount of computational time may be considerable, the method has the advantage that it can take account of any special features of the e:nd-to-end distributions in the region corresponding to intramolecular cyclization of the chains. The third method is due to Flory, Suter and Mutter. 56 - 60 These authors modified the Jacobson-Stockmayer theory so as to take account of the directional requirements of chain termini involved in intramolecular cyclization. They used Monte Carlo methods to obtain the density distributions and angular correlation factors required for the calculations.

In the calculations of ring concentrations in polymeric ring-chain equilibrates to be described using these methods, it is assumed that linear chains in the melt, in the amorphous state and in concentrated solution adopt random-coil conformations with similar molecular dimensions to those of the chains in dilute solution at the O-point. This is in agreement with Flory's view that this is the case. 2,2 7 In 1972, Flory61 reviewed

INTRODUCTION 15

convincing experimental evidence to support his theoretical predic­tions. 62 ,63 This includes measurements of stress-temperature coefficients of rubbers, studies of the thermodynamics of concentrated polymer solutions and the direct measurement of chain dimensions in the bulk by low angle X-ray and neutron scattering. As Flory2,2 7 has emphasized, long­range interactions between non-bonded atoms and groups of chains in the bulk state should be neglected. Hence, in the calculations to follow rotational isomeric state models will be used, which only take account of short-range interactions. 2 7 Long-range interactions will be considered only in the case of ring-{;hain equilibria in liquid sulphur, where polymer chains are not present in substantial concentrations.

The Method of Jacobson and Stockmayer49 - 51

Molar cyclization equilibrium constants are calculated assuming that x­meric chains forming rings obey Gaussian statistics, so that64

( 3 )3/2 Wx(O) = 2rc<r;> (5)

Where <r;> represents the mean-square end-to-end distance of the chains and the units of Wx(O) are molecules dm- 3. Combining eqns (4) and (5) gives the following relationship:

Kx = (2rc:r;> yl2 (NAIuRJ (6)

The units of Kx are mol dm - 3. Comparison of experimental Kx values for cyclics formed in ring-{;hain

equilibrates with values predicted by eqn (6) provides a powerful method for investigating the statistical conformations of chain molecules in a variety of environments, as was emphasized in Refs 51 and 65. If the ring-{;hain equilibration reaction is carried out in the melt or in concentrated solution, then the mean-square end-to-end distances can be identified with their unperturbed values <r;>o (as discussed above). The latter can be calculated by the matrix algebraic methods of Flory and Jernigan,66,67 using rotational isomeric state models based on detailed molecular structural information. 2 7 Hence, a direct comparison can be made between experimental K~ values and those predicted by Flory's theories.27 Such a comparison has already been made for a number of polymeric systems and will now be discussed.

The first system to allow a comprehensive test of the Jacobson­Stockmayer theory was poly(dimethyl siloxane) (PDMS). Following

16

'::tC.x C7I o

....J

, ,

J. A. SEMLYEN

\

~:~ \ \

-3 ......... -, " ~

" , " -4

-5

Log X

, , , , ,

FIG. 6. Molar cyclization equilibrium constants Kx (in mol dm - 3) for cyclics [(CH3)2Si0lx in a ring-<:hain equilibrate in toluene solution38.72 (unbroken line)

are compared with values calculated as described in the text (broken line).

earlier determinations of the concentrations of cyclics [(CH3)2Si0lx containing up to 50 skeletal bonds in ring-chain equilibrates of PD MS, 68 - 71 Brown and Slusarczuk 72 measured Kx values for cyclics with x = 4-200 in an equilibrate in toluene solution at 383 K. In Fig. 6, the experimental Kx values are compared with those predicted by the Jacobson-Stockmayer theory, using the rotational isomeric state model of PDMS set up by Flory, Crescenzi and Mark. 73 -75 For the calculation of K x ' eqn (6) can be recast as follows: 27.51

(3/n)3/2 Kx = l6PC3f2x5/2 N (7)

x A

and the characteristic ratios of finite x-me ric chains Cx = <r~>o/2xI2 computed by the methods of Flory and Jernigan. 66 .67 There is good agreement between experiment and theory for rings with more than c. 30

o

x ~ -1 Cl o

...J

-2

-3

INTRODUCTION

Monomer units, x

2 3 4 5 6 7 B

\ \ \\

\ \ \\ \\

\ \ \ \

\

\~--::~::~::

Log X

17

FIG. 7. Experimental molar cyclization equilibrium constants Kx (in mol dm - 3) for cyclics [CHzOCHzCHzOlx in undiluted (0) and solution (e) equilibrates of poly( I ,3-dioxolane) at 333 K compared with values calculated ( x) by the Jacobson

and Stock mayer theory.

skeletal bonds, particularly in the range 15 < x < 40 (see Chapter 3 for more detailed discussions of the PDMS system). Other polysiloxane systems, whose ring-chain equilibria have been investigated, include polysiloxanes with different substituent groups, 76 paraffin-siloxanes 77 and block copolymers of polystyrene and PDMS. 78 Again, good agreement was obtained between experiment and theory.76-78

Apart from polysiloxane systems, there is also good agreement between experimental and theoretical Kx values for the larger rings formed in ring-chain equilibrates of two organic polymers, poly(1,3-dioxolane) and poly(decamethylene adipate). Ring-chain equilibration reactions of the polyether were carried out in the bulk and in solution, starting with monomeric 1,3-dioxolane and using boron trifluoride diethyl etherate as the catalyst at 333 K.79 The equilibrates were quenched in diethylamine and the oligomers extracted and analysed by GLC (see Fig. 4). The Kx values for the cyclics [CH2 OCH2CH2 O]x in an undiluted equilibrate (containing 81 % chain polymer) and a solution equilibrate (containing 14 % chain polymer) are shown in Fig. 7. These experimental values

18 J. A. SEMLYEN

Number of skeletal bonds I n

-1

-3

Log n

FIG. 8. Experimental molar cyclization equilibrium constants Kx (in mol dm - 3) for cyclics [O(CH2)10OCO(CH2)4COlx in poly(decamethylene adipate) melts at 423 K (0) are compared with values calculated (.) by the Jacobson and

Stock mayer theory.

(believed to be correct to within ± 10 %) are also compared with theoretical Kx values calculated by eqn (6) with O"Rx = 2. Values of <r~> required for eqn (6) were computed using a rotational isomeric state model based on the models for other polyethers by Flory and Mark80 ,81 and on the measurements of the characteristic ratio of poly(1 ,3-dioxolane) by Gorin and Monnerie. 82 The close agreement between experimental and theoretical Kx values for the larger cyclics is evidence that 1,3-dioxolane chains with 25-40 skeletal bonds obey the Gaussian relationship for the probability of intramolecular cyclization.

As with the poly(1,3-dioxolane) system, experiment and theory are in close agreement for the larger cyclics [O(CH2)100CO(CH2)4CO]x in the poly(decamethylene adipate) system, where tetraisopropyititanate is used as the catalyst and where the analysis is by G Pc. 83 For this system,

INTRODUCTION 19

Number of skeletal bonds, n

10

o

-1

-2

-3

1-4 1-6 1-8 Log n

FIG. 9. Experimental molar cyclization equilibrium constants Kx (in mol dm - 3) for cyclics [O(CHzhOCO(CH2hCOlx in poly(trimethylene isuccinate) melts at 423 K (0) are compared with values calculated (e) by the Jacobson and

Stockmayer theory.

O"Rx = 2x and values of <r;>o required by eqn (6) were computed by the methods of Flory and Jernigan66 •67 using the rotational isomeric state model for aliphatic polyesters set up by Flory and Williams. 84 Agreement is within experimental error for Kx values corresponding to cyclics with 54, 72 and 90 skeletal bonds, as shown in Fig. 8.

Thus, for a number of ring-chain equilibrates, the Jacobson and Stockmayer theory predicts Kx values for the largest rings analysed that are within experimental error of those determined experimentally. However, there are three polymeric systems where close agreement is not obtained. All three are polar organic systems. Two (Nylon 6 and poly(ethylene terephthalate» will be discussed below, as alternative theoretical approaches to the simple Jacobson and Stockmayer theory are described. The third is poly(trimethylene succinate). Experimental Kx values for

20 J. A. SEMLYEN

cyclics [O(CH2hOCO(CH2hCO]x with x = 4-7 (36-63 skeletal bonds) in a ring--chain equilibrate of molten poly(trimethylene succinate) catalysed by tetraisopropyltitanate are only about half the corresponding theoretical values, as shown in Fig. 9 (see Ref. 84).

The Direct Computational Method The simple Jacobson and Stockmayer theory assumes that chains forming rings obey Gaussian statistics. An alternative approach, which may be termed the direct computational method, makes no such assump­tion. 52 - 55,85 - 88 The distances between the terminal atoms of chains forming rings are calculated for all the discrete conformations defined by a rotational isomeric state model. Cyclization is assumed to take place when chain termini are in close proximity for intramolecular cyclization to occur. Any correlations between the directions of terminal bonds involved in the cyclization process can also be investigated and their effect on calculated Kx values assessed,87,88

The direct computational method avoids any averaging or sampling of the total distribution of end-to-end vectors. Hence, it should be sensitive to any extraordinary features of the chain distribution (for example, there may be a few ring-forming conformations of exceptionally high statistical weight, which might be missed in a sampling method). Although the amount of computational time involved may be considerable, end-to-end distances need only be calculated for the highly coiled conformations and this can result in a considerable saving in time. Other savings result from the equivalence of different chain conformations and conformations given zero statistical weight by the rotational isomeric state model.

Calculation of molar cyclization equilibrium constants Kx by the direct computational method proceeds as follows. Let Z represent the total sum of the statistical weights of all the individual conformations defined by a rotational isomeric state model for an x-meric chain. Then, if z is the sum of the statistical weights for those conformations that have the centres of terminal atoms separated by less than r, the probability densities Wx of the chains are given by54,89

(8)

and from eqn (4):

(9)

INTRODUCTION 21

The radius r of the reaction volume within which chain termini must meet to form a bond has been widely discussed. zz ,z3,54-58,85-88 It may be assumed to be about a bond length or considerably longer depending on the chemical reaction involved. If the x-meric chains forming rings are quite long, then values of Wx may be approximately independent of r in the range of small r. 85 - 88

The direct computational method is a method which has a wide application. It can take into account favourable or unfavourable correlations between the directions of terminal bonds forming rings, as well as any excluded volume effects arising from long-range intramolecular interactions between non-bonded atoms and groups. 85 - 88 The method has been applied to calculate cyclic concentrations in a number of ring-chain equilibrates, including the four polymeric systems to be described here.

(i) Dihydrogen Siloxanes85 ,89

Following earlier investigations of small dihydrogen siloxane rings, 90 - 9Z

Seyferth and his co-workers93 recently described the preparation and characterization of cyclics [HzSiO]x, with x = 4-23. The rings were obtained as a volatile fraction from the hydrolysis of dichlorosilane HzSiClz in dichloromethane solution, by the addition of water under con­trolled conditions. The cyclic tetramer was found to undergo rearrangement in sealed Pyrex glass tubes at room temperature to give a mixture of cyclics [HzSiO]x with x = 4-17. Analysis by GLC showed relatively larger amounts of [HzSiO]8 and [HzSiO]12' The preparation of linear dihydrogen si10xanes (CH3hSiO[HzSiO]ySi(CH3h was also reported. 93

Poly(dihydrogen siloxane) (PDHS) is a polymer of particular interest because all the available experimental data suggest that it should be remarkably flexible, with virtually no restrictions to rotations about its skeletal bonds. A rotational isomeric state model for the linear polymer has been set up, with structural parameters based on the electron diffraction data for disiloxane [HzSi]z094 and the cyclic tetramer [HzSiO]4.92 Thus the bond length dSi - H was taken to be 0·148 nm, dSi - O to be 0'164nm and the bond angle supplements at silicon and oxygen atoms to be ()' = 70° and ()" = 37°, respectively.

There are low energy barriers restricting internal rotation about silicon-oxygen bonds in PDMS. 94 Even lower energy barriers are to be expected for PDHS. Nonetheless, following the rotational isomeric state approach, the continuum of rotational states about skeletal bonds is represented by three rotational isomeric states at ¢ = 0 ° (trans), ¢ = 120 ° (gauche +) and ¢ = 240° (gauche -). For PDHS, virtually free rotation is

22 J. A. SEML YEN

expected, because the hydrogen atoms attached to the chain backbone are so small and because even electrostatic interactions between silicon and oxygen atoms can be neglected. In this approximation, all chain conformations are assumed to be equally probable and the characteristic ratio of PDHS is given by the familiar expression: 2

<r2 >0 (1 + cos 8')(1 + cos 8") -----;;[2 (1 - cos 8' cos 8")

(10)

With the bond angles given above, <r2 >0/nP = 3·3 at all temperatures. The three-state rotational isomeric state model has been used to calculate

the Kx values for cyclic dihydrogen siloxanes in ring-chain equilib­rates,85.89 although the experimental values have yet to be measured. The distances between the centres of terminal atoms of [H2SiO]x chains with x = 4-8 were calculated for all the 32x - 3 conformations defined by the model. Long-range intramolecular interactions were assumed to be absent. Theoretical KCK8 values were calculated for values of r < 0·5 nm. The Kx values were found to be in accord with those expected by comparison with the corresponding values for hydrogen methyl siloxanes and dimethyl siloxanes (see Fig. 10). Furthermore, as x becomes larger, the Kx values approach those calculated for dihydrogen siloxanes by the methods of Flory and Jernigan66.67 assuming that the chains obey the Gaussian relationship (eqn (5)).

If experimental Kx values for dihydrogen siloxanes become available, refinements of the calculations could follow, including estimates of intramolecular steric and Coulombic interaction energies and con­sideration of correlations between the directions of terminal bonds in the intramolecular cyclization processes.

(ii) Dimethyl Siloxanes85 ,87 ,89,95

More experimental information on molar cyclization equilibrium con­stants is available for dimethyl siloxanes than for any other ring system. Precise experimental Kx values have been measured for values of x in the range 3 < x < 200 in undilut(:d 38,65 and solution equilibrates. 38,72,96 Minima and maxima in the log Kx versus log x plots are located at x = 12 and x = 16, respectively (see Fig. 6). In consequence, the system is of special interest in studies of ring concentrations in polymeric systems.

For calculations of equilibrium cyclic concentrations, the Flory, Crescenzi and Mark 75 rotational isomeric state model is available to describe the statistical conformations of dimethyl siloxane chain molecules. The model has been used to calculate the unperturbed

x :::.::: en o

-1

-.J - 2

-3

INTRODUCTION 23

Monomer units, x

0·6 0·8 Log x

FIG. 10. Theoretical molar cyclization equilibrium constants Kx (in mol dm - 3) for cyclic dihydrogen siloxanes (.) calculated by the direct computational method assuming r = O' 3 nm. The unbroken line is for values calculated by assuming the corresponding chain molecules obey Gaussian statistics. 85 ,89 They are compared with the experimental Kx values for the cyclics [H(CH3)SiO]x at 273 K (D) and

[(CH3)2Si0lx at 383 K (0).

dimensions of PDMS,74 the effect of temperature on the unperturbed dimensions 7 3 and the dipole moments of dimethyl siloxane chains. 97 As for PDHS, the structural parameters are as follows: DSi - O = 0·l64nm, ()' = 70 0 and ()" = 37 0 • The statistical weight matrix, which takes account of the mutual interdependence of adjacent pairs of skeletal bonds centred on silicon atoms, is as follows:

(11)

24 1. A. SEMLYEN

and the corresponding matrix for pairs of skeletal bonds centred on oxygen atoms is:

t g+ g

t

[: (J

:J U"=g+ (J (12)

g (j

The statistical weight parameters are 0·327 and 0'082, respectively, at 383 K.

The Kx values of cyclic dimethyl siloxanes have been calculated by the direct computational methodss.s7.s9.9s using Flory, Crescenzi and Mark's 7S rotational isomeric state model. Using this model, K4 , Ks and K6 were found to be far lower than the experimental values (by a factor of more than 103 for K4)' These large differences arise because the rotational isomeric state model does not take account of the mutual interdependence of sequences of bond rotational states. Thus, for example, the acyclic ---f(CH3)2Si0-h.- can adopt the sequence of states tg+ g- g+ g- with no severe steric conflicts between non-bonded atoms or groups, yet the Flory, Crescenzi and Mark model gives this sequence a statistical weight of zero. To demonstrate the effect of using more realistic statistical weights, pairs of g+ g- and g- g+ states centred on silicon atoms were given statistical weights of 20 (not zero), when they were in the centre of the sequences g + g - g + g - and g - g + g - g + . Thus, a sequence t g + g - g + g - t in a chain was accorded a statistical weight of 0·044 rather than zero. The calculated K4 , Ks and K6 values were now close to the experimental values (see Fig. 11).

The rotational isomeric state model does not require modification for the calculation of Kx values for the larger dimethyl siloxane rings and Ks' K9 K10 and Kll were calculated accordingly. A well-defined minimum in the theoretical values was found at x ~ 9, not at x = 12 as observed experimentally. Furthermore, the theoretical Kll value is 10 times that found experimentally and this is believed to be associated with the special geometry of the PDMS chain with its unequal bond angles. The planar all­trans conformation, which is of low energy, corresponds to a closed loop at x = 11. Contributions from this and other conformations rich in trans sequences are believed to be the cause of the maximum in the log Kx versus log x plot for PDMS. Individual experimental Kx values are not reproduced closely for K9 and K ll , as they are for Ks and K10 (see Fig. 11). The discrepancy between experiment and theory at K 11 is believed to arise

x ~

01 o ...J

-1

-2

-3

-4

INTRODUCTION

Number of skeletal bonds I n

0..

10 20 30 40

, , ~

~. \

q \

Log n

25

FIG. 11. Theoretical molar cyclization equilibrium constants Kx (in mol dm - 3) for cyclic dimethyl siloxanes at 383 K, calculated using the modified (.) and unmodified (.) Flory, Crescenzi and Mark rotational isomeric state model, with r = 0·3 nm and r = 0·2nm, respectively. They are compared with the experimental

values (0) also at 383K. 85 ,87.89,95

from the discrete nature of the Flory, Crescenzi and Mark model, with its fixed bond angles and with only three states about each skeletal bond. The experimental Kx values provide a most searching test of such a model and the latter may be modified when more molecular structural data relating to the conformations of dimethyl siloxane molecules become available.

(iii) Sodium Metaphosphates86 ,98

As described above, the concentrations of cyclic oligomers [NaP031x with x = 3~ 7 have been measured in sodium phosphate melts using paper chromatography.44,45 Ring--{;hain equilibria are established in the melt at

26 J. A. SEMLYEN

Monomer units I x

0..

-1

x ~

en 0

-.l

-2

Log X

FIG. 12. Experimental molar cyclization equilibrium constants Kx (in mol dm - 3) for cyclics [NaP03 1x in sodium phosphate melts at 1000 K (0) are compared with values calculated by the direct computational method with r = O· 3 nm (. )86.98 and

by the Jacobson-Stockmayer theory (unbroken line).

c. 1000 K and the Kx values measured by van Wazer and his co-workers44

and by Thilo and Schulke45 are in good agreement for x = 3-5. Thilo and Schulke's values will be assumed for x = 6 and 7.

A rotational isomeric state model for the polyphosphate chain was first set Up99 to calculate the experimental characteristic ratio of the linear polymer under O-point conditions at 298 K, where <r2)0/nP ~ 7.100.101

The model was modified86.98 and used to calculate Kx values for cyclics [NaP03]x in sodium phosphate melts at 1000 K by the direct com­putational method (eqns (8) and (9». The calculated values for K 7-KIO are shown in Fig. 12; Kx values for the smaller cyclics are omitted as the rotational isomeric state model requires further modification to calculate them, as discussed above for dimethyl siloxane rings. The unbroken line in

INTRODUCTION 27

Fig. 12 shows Kx values calculated assuming Gaussian statistics (eqns (5) and (6» and these are in good accord with the experimental data.

The K7 value calculated by the direct computational method is in good agreement with the experimental value. Furthermore, the calculated K8 , K9 and K 10 values are in accord with Thilo and Schulke's conclusion that there are only c. 1·2 % by weight of cyclics [NaP03]x with x ~ 8 in samples of Graham's salt. 45

More detailed theoretical studies of cyclic concentrations in sodium phosphate melts would be expected to follow the publication of further experimental data on equilibrium ring concentrations. The calculations described here support the underlying assumption that chains in sodium phosphate melts adopt random-coil conformations unperturbed by long­range intramolecular interactions.

(iv) Cyclics in Nylon 639 ,88,102,103

. Nylon 6 or polY-e-aminocaproamide has the repeat unit -tNH(CH2)5CO+, It may be prepared by the polymerization of e-caprolactam in the presence of water at temperatures of c. 800 K. A ring--chain equilibrium is set up in the polymeric melt and on quenching approximately 12 % by weight of cyclic oligomers can be extracted from the polymer chip, using hot water or boiling aqueous methanol. The cyclic oligomers may be analysed by G PC (when, for example, Sephadex columns from Pharmacia Ltd, England, UK, are used with glacial acetic acid/water as the solvent,39 the term gel filtration is often used).

Following the investigations of Spoor and Zahn, 104 Andrews and his co­workers39,102 measured Kx values for cyclics [NH(CH2)5CO]x with x = 1-6. Their results for x = 1-5 are believed to be correct to within ± 10 %. Somewhat more uncertainty is associated with the K6 value. 88

Detailed calculations of the Kx values for rings in Nylon 6 ring--chain equilibrates have been carried out using the direct computational method. 88 ,103 The rotational isomeric state model for the linear polymer set up by Flory and Williamsz7 ,105 was employed. The model predicts a characteristic ratio for Nylon 6 of 6·2 at 298 K, which is in good agreement with the experimental values of 6.0 105 and 5,8. 106

The calculations are based on the assumption that chains in molten Nylon 6 adopt similar conformations to those in 8-s01vents and Kx values were calculated for x = 1-4 using eqns (8) and (9). Apart from x = 1, the calculated Kx values are not sensitive to the precise value chosen for the reaction distance r. Thus, the tetrameric aminocaproic acid (tetra(ACA»

NHzCCHz)5CONH(CHz)5CONH(CH2)5CONH(CH2)5COOH

28

o

x ~ -1 01 o -l

-3

J. A. SEMLYEN

0-0 0-2 0-4 Log X

FIG. 13. Experimental molar cyclization equilibrium constants Kx (in mol dm - 3) for cyclics [NH(CH2)sCOlx at 525 K (0) are compared with values calculated by the direct computational method (e) and by the Jacobson~Stockmayer theory

(unbroken line)88.103 (see text)_

has 16991550 individual conformations with the distances between chain termini separated by less than 0·1 nm. When r lies in the range 0·02 < r < 0·1 nm, the calculated log Kx value is always -1·8. Similarly, when 0·1 < r < 0·3 nm for tris(ACA), then -1·5 < log Kx < -1·4. For bis(ACA), when 0·3 < r < 0·5 nm, then -1·3 < log Kx < -1·2. By contrast, for ACA when O· 3 < r < 0·5 nm, then log Kx values lie between - 1·0 and 0·1.

In Fig. 13, Kx values calculated by the direct computational method with r = O· 3 nm (and r = 0·1 nm for tetra(ACA)) are compared with the experimental values and with values calculated using eqns (5) and (6) by assuming Gaussian statistics. Considering the uncertainties involved, there is satisfactory agreement between the experimental values and those calculated by the direct computational method for K2 , K3 and K4 .

The possibility of correlations between the directions of terminal bonds

INTRODUCTION 29

favouring ordisfavouringcyclization has also been investigated. 88.103 Very marked angular correlations were found for the oligomers with x = 1 and 2 in their highly coiled conformations, but little correlation was found for x = 3. For x = 4, the distribution of terminal bonds within the volume of radius r = 0·1 nm was found to be effectively random.

The Method of Flory, Suter and Mutter56 - 60

Flory, Suter and Mutter have developed another method for calculating cyclic concentrations in ring--chain equilibrates and they have applied it to several polymeric systems. In their treatment, the Jacobson and Stock mayer cyclization theory is elaborated so as to take account of correlations between the directions of terminal bonds in the highly coiled chain conformations. In place of the Jacobson-Stockmayer expression (eqn (6», Flory, Suter and Mutter's treatment leads to the following equation:

(13)

where WxCO) is the probability density of end-to-end vectors in the region r = 0 and O"Rx is the symmetry number of the ring.

The factor r 0(')') is the probability distribution, when r = 0, of')' =: cos !1(). The angle !1() is defined as being between a hypothetical bond n + 1 of a chain (with n bonds) and the first bond. Then, r 0(1) is the probability that !1() = 0 for chains with r = O. Methods for evaluating the angular correlation factor 2r 0(1) are described in Refs 56-60.

The probability densities Wx(O), required for eqn (13), were obtained by a number of methods, using rotational isomeric state models to describe the statistical conformations of the chain molecules.

In one approach, Wx(O) was expressed in terms of a scalar Hermite expansion:

(14)

Truncation of the series at g4 was found to give the best agreement with other approximations, in general, where

(15)

30

-1

-2

J. A. SEMLYEN

Monomer units. x 4 6

"1 ',-2 .... ... , -, .... o .............. .:: .....

3.---- '~~, .... ' o .... ,

-- " " ... --4--.!;.J ' .. ,

.... .q' ..... .... ,

................... , 0','"

0·4 0·6

Log x FIG. 14. Experimental molar cyclization equilibrium constants Kx (in mol dm - 3) for cyclics [NH(CH2)sCOlx in the melt at 525 K (0).39.88.102.103 They are compared with values calculated by Mutter, Suter and Florys8 as described in the

text.

The values of <r~) and <r~) were obtained from samples of 30000 chain conformations generated by Monte Carlo methods.

Another approach to calculating Wx(O) was by a Monte Carlo variation of the direct computational method. Instead of generating all confor­mations described by the rotational isomeric state model, just 30000 were selected by Monte Carlo methods and the parameter , in eqn (8) was assigned the value 0·3<,2)1/2 or 0·5<,2)1/2.

Some representative results obtained by Flory, Suter and Mutter, when they applied their methods to real polymeric systems, will now be outlined.

In Fig. 14, molar cyclization equilibrium constants Kx for the formation of cyclics [NH(CH2)sCO]x in ring-chain equilibrates of Nylon 6 at 525 K are compared with the corresponding values calculated by Mutter, Suter and FloryS8 using the Flory and Williams84 rotational isomeric state model. Curve 1 was calculated by the Jacobson and Stockmayer theory assuming Gaussian statistics. 107 Curve 2 was obtained by truncating the

-2

-3

0·2

INTRODUCTION

Monomer units. x 3456789

o

0·4

o

o

0·6 Log x

o o

o

0·8

o

31

1·0

FIG. IS. Experimental molar cyclization equilibrium constants Kx (in mol dm - 3) for cYclics [CO.C6 H4'CO. O.CH2 .CH2. Ot in the melt at 543 K (0).86.108 They

are compared with theoretical values calculated as described in the text.

series for Wx(O) (eqn (14)) at g4 and calculating g4 from eqn (15) with the even moments computed by matrix methods. Curve 3 was obtained by Monte Carlo calculations using a value of r = 0·3<r2 )1 /2 and taking samples of 30000 conformations. Curve 4 was calculated by eqn (13) including the angle correlation factor 2r 0(1).

Application of the Flory, Suter and Mutter methods to the calculation of Kx values of cyclics in PDMS ring--<:hain equilibrates at 383 K resulted in satisfactory agreement for x-meric rings with x> 15.57 However, the minimum at x = 12 and the maximum at x ~ 15 in the log Kx versus log x plot (see Figs 6 and 11) were not reproduced. 57

In Fig. 15, Kx values of cyclics in ring--<:hain equilibrates of poly( ethylene terephthalate) at 543 K86,108 are compared with values calculated by the Jacobson and Stockmayer theory (Curve 1),108 by the direct computational method for K3 and K4 (.)108,109 and by the methods of Flory, Suter and

32 J. A. SEMLYEN

Mutter, excluding (Curve 2) and including (Curve 3) the angular correlation factor. 6o All the theoretical Kx values are substantially lower than those found experimentally in both undiluted and solution equilibrates. This major discrepancy has been discussed by Suter and Mutter60 but no explanation has been established as yet.

Mutter59 has carried out a detailed study of the ease of cyclization of peptide sequences forming rings containing 6-20 amino acid residues, using the methods of Flory, Suter and Mutter. Required statistical weights were obtained by conformational energy calculations using semi-empirical potential functions as described by Brant and Flory. 11 0 Mutter's results for six peptide sequences were in good agreement with observation. For example, ring formation is predicted to take place easily for chains containing 'flexible' Gly residues, whereas sequences of all L-Ala are predicted to have considerable difficulty in cyclizing (see Chapter 8).

Following the theoretical studies of cyclization of phage lambda DNA by Wang and Davison/ 11 Olson 112 has discussed cyclic and loop formation of DNA segments using the Flory, Suter and Mutter method. Cyclization probabilities were estimated as a function of the number of virtual bonds in a 'flexible' DNA helix (see Chapter 7).

Ring-Chain Equilibration Reactions and the Preparation of Cyclic Polymers In general, the techniques of electron microscopy and X-ray crystallog­raphy, that are used to establish the cyclic nature of some biological macromolecules (see Chapters 7 and 8), cannot be applied to the characterization of synthetic polymers. In order to facilitate such characterization, ring-<:hain equilibration reactions can be used. This is because, in favourable cases such as the PDMS system, the concentrations of all the individual ring and chain molecules formed in the reaction can be calculated and there are no other products formed. This means that fractions of cyclic polymers containing molecules with hundreds of skeletal bonds can be prepared, from which it is known a priori that linear molecules are almost completely excluded.

The preparation of cyclic fractions of PD MS prepared by ring-<:hain equilibration reactions is described in Chapter 3. The fractions are being used to investigate the properties of synthetic cyclic polymers (as described in Chapters 4 and 5), as well as to test the theoretical predictions for cyclic polymers discussed in Chapter 2. In Chapter 6, the preparation and properties of cyclic polystyrene are described. Other kinds of cyclization process are reviewed in Chapters 9 and 10.

INTRODUCTION 33

RINGS IN LIQUID SULPHUR

The Molecular Constitution of Liquid Sulphur It is now well established that liquid sulphur consists of mixtures of ring and chain molecules. A thermodynamic equilibrium may be set up in the liquid element at all temperatures, as has been discussed by Gee 113 - 116 and Tobolsky and Eisenberg.117.118

With new experimental evidence, including infrared and Raman spectroscopic investigations and the HPLC analysis of extracts from quenched sulphur melts (see, for example, Refs 47 and 119-125), Steudel48 has concluded recently that the molecular constitution of liquid sulphur has been elucidated. Traditionally, liquid sulphur has been considered to consist of SA (cyclooctasulphur, present at all temperatures), SI' (linear chains, only present at elevated temperatures) and Sn (molecular species believed to be responsible for the well-known freezing-point depression of liquid sulphur).126-129 Since Aten proposed a molecular formula of S4 for Sn' 130 there have been several suggestions as to its identity, including small sulphur rings, 131 -133 large sulphur rings,134.135 a mixture of small and large sulphur rings 136,137 and octaatomic diradical chains. 138 - 140

When liquid sulphur is first melted, its freezing point is identical to its melting point; but, when the melt is maintained at 120 0 e for at least 12 h, equilibrium is reached and the freezing point is now some 5°e lower.48 Analysis of the equilibrated liquid48 gives the following results: 95·1 % S8 (by weight), 2·8 % S7' 0·6 % S6' 1·5 % Sx with x > 8 and negligible amounts of SI" These data give the fraction of Sn to be 4·9 % in agreement with the 5 % estimated from the freezing-point depression. Steudel48 has reported the equilibrium composition of the molten element. Liquid sulphur was quenched from temperatures in the range 115-350 oe and then analysed spectroscopically after rapid quenching to -196 °e. The concentration of cyclooctasulphur decreases sharply from c. 95 % to c. 50 % over the temperature range 150-200°C. Over the same range, the concentration of SI' increases from close to zero to c. 50 %. The concentrations of SA and SI' were both found to be effectively constant over the range 250-350 oe. The concentrations of S6 and S7 rings are c. 2 % and c. 4 %, respectively, over the temperature range 200-350 °e; their concentrations below 160 0 e fall somewhat with decreasing temperature. The concentrations of rings Sx with x > 8 show a steady decrease from their value of 1·5 % at 115 °e to a value about 0·5 % lower at 350 oe. Steudel48 points out that most of the special properties of liquid sulphur are explained by his experimental data,

34 J. A. SEMLYEN

but he states that these data do not support the polymerization theory of Tobolsky and Eisenberg. 117,118

Following the calculations of cyclic concentrations in polymeric ring-chain equilibrates described above, the concentrations of ring molecules in liquid sulphur will now be considered. A rotational isomeric state model is used to describe the statistical conformations of the corresponding open sulphur chains and the direct computational method is applied. The small cyclics S6 and S7 will be omitted from consideration, as the method cannot be meaningfully applied to them. Attention will be directed to those sulphur cyclics Sx with x = 9-26, analysed by HPLC by Steudel and his co-workers. 47 ,48 It is possible that some molecular species in liquid sulphur convert to other forms during the quenching and extraction processes and during HPLC analysis. Further experimental investigations could establish to what extent this occurs.

Rotational Isomeric State Model of Sulphur Chains In 1967, a rotational isomeric state model for polymeric sulphur was set up and used to calculate the dipole moments of some n-alkyl polysulphides, as well as to discuss the conformations of catenated sulphur chains and sulphur rings in the crystalline state. 141 The model was based on Pauling's proposal that mutual repulsions of adjacent lone pairs of Pn-electrons give rise to two-fold intrinsic torsional rotational potentials for sulphur­sulphur bonds in sulphur chains. 142,143 Minima in the rotational potentials were assumed to be located at <p ~ ± 90 0 ,! where <p = 0 0 is the trans state. Experimental estimates of the barrier heights restricting internal rotation about sulphur-sulphur bonds range from 10--70 kJ mol- 1 and each bond in a sulphur chain is considered to be in one of the two discrete rotational isomeric states located at <p = ± 90 0 and designated + and -. Literature data were used to assign the length of a sulphur-sulphur bond in a chain (l = 0·206 nm) and the bond angle supplement (8 = 74°).

It was argued 141 that there should be net forces of attraction between non-bonded sulphur atoms separated by four skeletal bonds in the + -, - +, + + and - - rotational states. The statistical weight matrix U which takes into account the interdependence of bond rotational states is as follows:

+<P -<p

U= ~~ [~ n (16)

INTRODUCTION 35

where states of bond i-I are indexed on the rows and those of bond i on the columns. The statistical weight parameter is given by

(J = exp ( - dE/RT) (17)

where R is the gas constant, T is the temperature and dE is the energy difference between the states + - (or - +) and + + (or - -). Estimates of the van der Waals' forces between non-bonded sulphur atoms as a function of the distance between them led to an estimate of dE = - 1· 2 kJ mol- 1, corresponding to a value of (J = 1·4.

The conformations of some sulphur rings with x > 8, which have been determined by X-ray crystallography,48 are as follows:

S10 - + - - + - + - - + S12 + + - - + + - - + + - -

0(-S18 + - - + + - + + + - + + - - + - -­f3-S 18 + + - + - - + + + - - + - + + - - -

S20 + + + + - + + + + - + + + + - + + + + -Torsion angles in the rings have values centred around cP = ±90°.

Calculation of Equilibrium Ring Concentrations in Liquid Sulphur It is now known that both cyclohexasulphur S6 and cycloheptasulphur S7 are present in liquid sulphur. 48 Both these rings are strained 144,145 and their equilibrium concentrations will be omitted from consideration. Instead, attention will be directed to larger sulphur cyc\ics in liquid sulphur with x> 8.

In a previous calculation, 134,135 the concentrations of sulphur rings with x> 8 were calculated using the Jacobson and Stockmayer theory49 together with the rotational isomeric state model for sulphur chains.141 Excluded volume effects were neglected, so that attractions and repulsions between non-bonded atoms and groups were not considered and the statistical weight parameter (J was set equal to 1. These assumptions would be expected to be valid if substantial amounts oflong polymeric chains were present, and this approach is termed method (A).

F or liquid sulphur below about 150°C, Steudel48 has found negligible amounts of linear polymer. Under these circumstances, excluded volume effects cannot be neglected and full account must be taken of steric interactions between non-bonded atoms. This approach is termed method (B) and it was applied by Sisido23 in his calculation of the intramolecular cyclization of alkane chains.

36 J. A. SEML YEN

In the calculations to be described here, both method (A) and method (B) were used. 146 The distances r between the centres of the terminal atom of sulphur chains Sx were calculated for all the individual conformations defined by the rotational isomeric state model described above. Ring formation was assumed when r < O· 3 nm. In method (A), excluded volume effects were neglected and (J was set equal to 1. In method (B), the distances between all the non-bonded atoms of the chains were calculated for all the individual conformations. When the distances between the centres of non­bonded sulphur atoms (with the exception of the terminal pair) were less than O· 3 nm, the conformation was not included in the calculation. The distance O· 3 nm was estimated from semi-empirical equations of non­bonded energies of interaction between sulphur atoms. 54 The relatively small attractive forces between the non-bonded atoms of the sulphur chains were neglected, so that again (J = 1.

In Table 1, the results of using method (A) and method (B) are shown. 146

TABLE 1

Model (A) Model (B)

Value Number of Total number ofx conformations of conformations in Sx corresponding to ring

formation z Z z Z

9 6 64 0 48 10 4 128 0 88 11 2 256 0 162 12 24 512 4 298 13 30 1024 0 536 14 88 2048 0 980 15 114 4096 0 1790 16 154 8192 0 3270 17 330 16384 4 5974 18 532 32768 16 10896 19 1698 65536 10 19832 20 2362 131072 68 36144 21 4008 262144 44 65724 22 8132 524288 74 119644 23 14382 1048576 236 217716 24 32546 2097152 248 395672 25 55198 4194304 298 26 110 336 8388608 634

INTRODUCTION 37

For the calculation of cyclic concentrations by the direct computational method (eqns (8) and (9», the number of conformations corresponding to ring formation z and the total number of conformations Z are listed.

Method (A), which was used previously,134,135 predicts ring con­centrations for x > 8 to be far higher than found by Steude1.48 With neglect of excluded volume effects, the chain statistics approximate to the Gaussian. 134 As shown in Table 1, in general, calculated ring con­centrations increase with ring size (although Sl1 and S12 fall out of line). The assumption that excluded volume effects can be neglected may be valid when long, linear polymer is present in large concentrations. However, it would not be expected to apply below T.p, where linear polymeric sulphur is known to be present in negligible amounts.

The experimental data of Steudel and his co-workers48 are much closer to the results given by method (B) (see Table 1). The calculations predict zero concentrations of the rings S9' S10' Sl1' S13' S14' S15 and S16 in liquid sulphur. 146 Steudel and his co-workers47 state that although all the rings with x = 6-17 have been detected as components ofliquid sulphur, some of them are only present as traces, especially Sl1 and S13' The calculation by method (B) also predicts S12' S18 and S20 to be present in higher concentrations than the other rings with x> 8. These three cyclics are the only rings with x > 8 to have been prepared in the pure state from the quenched melts. Method (B) predicts considerably lower concentrations of large rings than method (A). Thus, for example, the concentration of S20 is predicted to be lower by a factor of about 10. Total ring concentrations for x > 8, calculated by method (B), with full account taken of excluded volume effects, are in broad agreement with Steudel's estimates.

When the precise concentrations of individual sulphur rings have been determined, it should be possible to refine the calculations presented here. Sulphur (and incidentally selenium) rings and chains are unique in their structural simplicity. More precise experimental information of cyclic concentrations in the liquid elements would provide a most valuable opportunity of relating equilibrium ring concentrations to the statistical conformations of the corresponding open chain molecules.

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38 J. A. SEMLYEN

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INTRODUCTION 39

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(1950) 1607. 51. Flory, P. J. and Semlyen, J. A., J. Amer. Chem. Soc., 88 (1966) 3209. 52. Fluendy,M. A. D., Trans. Faraday Soc., 59 (1963) 1681. 53. Carmichael, J. B. and Kinsinger, J. B., Canad. J. Chem., 42 (1964) 1996. 54. Semlyen, J. A., Trans. Faraday Soc., 63 (1967) 2342. 55. Winnik, M. A., Trueman, R. E., Jackowski, G., Saunders, D. S. and

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Polymer Chemistry, 141st Meeting of the American Chemical Society, Washington, DC, March 1962.

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72. Brown, J. F. and Slusarczuk, G. M. J., J. Amer. Chem. Soc., 87 (1965) 931.

40 J. A. SEML YEN

73. Mark, J. E. and Flory, P. J., J. Amer. Chem. Soc., 86 (1964) 138. 74. Crescenzi, V. and Flory, P. J., J. Amer. Chem. Soc., 86 (1964) 141. 75. Flory, P. J., Crescenzi, V. and Mark, J. E., J. Amer. Chem. Soc., 86 (1964)

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2163. 94. Almenningen, A., Bastiansen, 0., Ewing, V., Hedberg, K. and Traettenberg,

M., Acta. Chem. Scand., 17 (1963) 2455. 95. Scales, L. E. and Semlyen, J. A., Polymer, 17 (1976) 601. 96. Wright, P. V., J. Polym. Sci., Polym. Physics Edition, 11 (1973) 51. 97. Sutton, C. and Mark, J. E., J. Amer. Chem. Soc., 54 (1971) 5011. 98. Cooper, D. R. and Semlyen, J. A., Polymer, 13 (1972) 414. 99. Semlyen, J. A. and Flory, P. J., Trans. Faraday Soc., 62 (1966) 2622.

100. Strauss, U. P. and Ander, P., J. Phys. Chem., 66 (1962) 2235. 101. Strauss, U. P. amd Wineman, P. L., J. Amer. Chem. Soc., 80 (1958) 2366. 102. Andrews, J. M., Jones, F. R. and Semlyen, J. A., Polymer, 15 (1974) 420. 103. Davison, K. and Semi yen, J. A., unpublished calculations. 104. Spoor, H. and Zahn, H., Z. Analyt. Chem., 168 (1959) 190. 105. Saunders, P. R., J. Polym. Sci., A2 (1966) 3765. 106. Bohdanecky, M. and Tuzar, Z., Coll. Czech. Chem. Comm., 34 (1969) 2589. 107. Semlyen, J. A. and Walker, G. R., Polymer, 10 (1969) 597. 108. Cooper, D. R. and Semlyen, J. A., Polymer, 14 (1973) 185. 109. Walker, G. R. and Semlyen, J. A., Polymer, 11 (1970) 472. 110. Brant, D. A. and Flory, P. J., J. Amer. Chem. Soc., 87 (1965) 2791. ll1. Wang, J. C. and Davison, N., J. Mol. BioI., 15 (1966) ll1. 112. Olson, W. K., In: Nucleic Acid Geometry and Dynamics (ed. R. H. Sarma),

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INTRODUCTION 41

116. Gee, G., In: Inorganic Polymers, Chemical Society publication, London, 1961, p. 67.

117. Tobo1osky, A. V. and MacKnight, W. J., Polymeric Sulphur and Related Polymers, Interscience, New York, 1955.

118. Tobo1sky, A. V. and Eisenberg, A., J. Amer. Chern. Soc., 81 (1959) 780. 119. Steude1, R. and Miius1e, H.-J., Angew. Chern. (int. Ed.), 16 (1977) 112. 120. Steudel, R. and Miiusle, H.-J., Angew. Chern. (int. Ed.), 17 (1978) 56. 121. Steudel, R. and Miiusle, H.-J., Angew. Chern. (int. Ed.), 18 (1979) 152. 122. Steudel, R. and Miiusle, H.-J., Chemie in unserer Zeit, 3 (1980) 73. 123. Steudel, R., Z. anorg. Chern., 478 (1981) 139. 124. Steudel, R. and Miiusle, H.-J., Z. anorg. Chern., 478 (1981) 156. 125. Miiusle, H.-J. and Steudel, R., Z. anorg. Chern., 478 (1981) 177. 126. Meyer, B., Chern. Rev., 76 (1976) 367. 127. Schmidt, M., Angew. Chern., 85 (1973) 474. 128. Schmidt, M., Siebert, W. and Bagnall, K. W., The Chemistry of Sulphur,

Selenium, Tellurium and Polonium, Pergamon Press, New York, Oxford, 1975, p. 808.

129. Meyer, B., Sulphur, Energy and Environment, Elsevier, Amsterdam, Oxford, New York, 1977.

130. Aten, A. W. H., Z. physik. Chern., 88 (1914) 321. 131. Krebs, H., Z. Naturforsch., 12 (1957) 785. 132. Schenk, J., Physica, 23 (1957) 325. 133. Schenk, J., Physica, 23 (1957) 546. 134. Semlyen, J. A., Trans. Faraday Soc., 64 (1968) 1396. 135. Semlyen, J. A., Polymer, 12 (1971) 383. 136. Krebs, K. and Beine, H., Z. anorg. Chern., 355 (1967) 113. 137. Harris, R. E., J. Phys. Chern., 74 (1970) 3102. 138. Schenk, P. W. and Thiimmler, U., Z. Electrochem., 63 (1959) 1002. 139. Schenk, P. W. and Thiimmler, U., Z. anorg. Chern., 315 (1962) 271. 140. Wiewiorowski, T. K. and Touro, F. J., J. Phys. Chern., 70 (1966) 3528. 141. Sem1yen, J. A., Trans. Faraday Soc., 63 (1967) 743. 142. Pauling, L., Proc. Nat. A cad. Sci., 35 (1949) 495. 143. Pauling, L., The Nature of the Chemical Bond, Cornell University Press,

Ithaca, New York, 1960. 144. Luft, N. W., Monatsh. Chern., 86 (1955) 474. 145. Berkowitz, J., In: Elemental Sulphur (Ed. B. Meyer), Interscience Publishers,

New York, 1965, p. 152. 146. Dodgson, K. and Semlyen, J. A., unpublished calculations.

CHAPTER 2

Theory of Cyclic Macromolecules

W AL THER BURCHARD

Institute of Macromolecular Chemistry, University of Freiburg, Freiburg im Breisgau, Federal Republic of Germany

INTRODUCTION

The study of cyclic or ring macromolecules may appear at first sight to be a subject of pure mathematics and of little practical interest, though of great beauty. Often the value of this research and its relation to applied polymer science has not been recognized. It is therefore not surprising that the theory of ring macromolecules is incomplete and fragmentary. Several gaps will become evident in this chapter, and only a few of them could be filled when writing it. In other cases, the unsolved problems will be mentioned and it is hoped that they may be considered by theoreticians as a challenge and a spur.

With regard to applications, two main fields can be identified. These are mentioned in other chapters of this book. They will be discussed here briefly for reasons that will become apparent later in this chapter.

A spectacular, though historically not the first, field of application is found in molecular biology. For about 20 years! it has been known that some DNA molecules occur in nature as large closed rings. Furthermore, it was discovered that these rings can be twisted once, twice or even more (see Fig. 1).2-4

Other examples of multiple rings are observed with double-stranded polynucleic acids in the transition region of denaturation. Here a sequence of small and larger loops is formed which is separated by still-intact double helical sections (as shown in Fig. 2).

The other main field of application is concerned with polyfunctional polycondensation and with polymerization of monomers containing a

43

44 WALTHER BURCHARD

FIG. 1. Sketch of a circular double helix of DNA and a multiply twisted form of it. (From Crawford. 3)

small fraction of divinyl compounds. Consider, for instance, the radical polymerization of a mixture of monovinyl and divinyl monomers in a dilute solution. Then excessive cyclization during chain growth will occur and there will be only a little crosslinking by reaction with pendant double bonds from other chains. This is illustrated in Fig. 3. Clearly the dimensions of such a chain will differ significantly from those of a corresponding chain that contains no loops. The problem to be solved here is obviously more complex than for the multiple DNA rings, because in the latter example the rings touch each other only at one point, whereas in Fig. 3 the loops may have fairly large sections in common. In addition to this, the rings will be separated by longer linear sections, and the chains will usually end in open chain segments.

FIG. 2. Schematical graph of an imperfectly reconstituted DNA double helix. Because of imperfect matching between complementary base pairs loops of various

sizes remain.

THEORY OF CYCLIC MACROMOLECULES 45

Cyclization and ring formation is most important in branching and crosslinking reactions. Here, every intramolecular reaction reduces the number of functional groups available for crosslinking by one, and these wasted functional groups cause a shift of the point of gelation towards higher extents of reaction. The investigation of ring-<:hain equilibria is probably one of the most interesting and challenging problems in the statistics of branching. An elegant solution was found for linear chains rather early in the history of polymer science by Jacobson and Stock mayer .5 Fifteen years later the theory was improved further by Flory and Semlyen6 to make it applicable to more realistic chains, taking the directional bond correlations into account.

FIG. 3. Structure of a coiled macromolecule which is formed by copolymerization of monovinyl with divynyl monomers in very dilute solution. Cycles of different size

are created by intramolecular reaction of the pendant double bonds.

These theories demonstrate a close relationship between the probabilities of end-to-end chain closure and the conformational properties of the chains. For this reason the conformational properties of individual flexible rings are treated in this chapter in some detail and both the hydrodynamic and dynamic chain behaviour are considered. The extension to branched structures is not discussed, since this problem is treated in detail in Chapter lO. A discussion of the structure of rigid rings is added. Most interesting and relevant in real applications is, of course, the behaviour of semiflexible chains. Here, however, theory is completely lacking, and only a few remarks can be made. Finally, the close relation of the excluded volume problem to the ring closure probability is outlined in some detail.

46 WALTHER BURCHARD

EQUILIBRIUM PROPERTIES OF FLEXIBLE RINGS

Some General Remarks The basis for all derivations of equilibrium quantities of macromolecular chains is the knowledge of the configurational distribution function W(rsl )

for a pair of chain elements or segments sand t in a macromolecule that may have a structure as shown in Fig. 2 or 3. Once this distribution is known, the mean square radius of gyration (82 ), the -1 th moment <R -1)

and the particle scattering factor P(q) can be calculated from the following equations:

(1)

S<I

N N

<I/R) = N- 2L L<l/rsl ) (2)

s<t

N N

P(q) = N- 2L L <exp (iq.rsl» (3)

where

q = (4:n:/.Ie) sin 8/2 (4)

qis the value of the scattering vector with .Ie the wavelength of the light in the medium and 8 the scattering angle in static light scattering. The angular brackets denote the equilibrium average. It is instructive to write this average explicitly for the case of the pair scattering function occurring in eqn (3):

<exp(iq.rsl» = I'" W(rSI)exp(iq.rSI)drSI (5)

which reveals that this average is actually the Fourier transform of the pair distance distribution. Thus, the particle scattering factor, which describes the angular dependence of the scattered light, contains in principle, though in a coded form, all information on the distance distribution in a chain.

The quantity <l/R) appears in Kirkwood's general equation for the translational diffusion coefficient D7 ,8

D = kT/(Ov) + (kT/6nt/o)< 1/ R) (6)

THEORY OF CYCLIC MACROMOLECULES 47

where' is the friction coefficient of an individual chain segment and rJo is the solvent viscosity.

A Direct Procedure for Deriving Configurational Distributions In the following, Gaussian statistics will be assumed, if not otherwise indicated, for any subchain which contains no ring. Ring closure and ring twisting introduce constraints which change the Gaussian character. In the case of a simple end-to-end closed chain the resulting configurational

N

FIG. 4. Graph of a cyclic chain. If the subchains in a ring obey Gaussian statistics then the distance distribution W(r,,) is the convolution of the two Gaussian chains originating in segment s and ending in segment t, where the one subchain consists of

t - s segments the other of N - (t - s).

distribution can be found by a rather simple consideration. The vector rSI in Fig. 4can evidently be expressed by the vector sum over the (t - s) bonds rs,

rs+ I··· rt or by the sum over the N - (t - s) bonds r t , r t + 1··· rN , r 1 ... rs. The corresponding distributions of these two subchains are

(7)

and

(8)

and the joint distribution in the ring is the convolution of these two distributions, i.e.

W(rst)ring = WI (rs)* W2(rst )

( 3 )3/2 = 2nN(nj N)(l _ nj N)b2 exp ( - 3r;tj(2Nb2(nj N)(l - nj N»)

(9) where n = (t - s) for abbreviation.

48 W AL THER BURCHARD

The Wang-Uhlenbeck Procedure The direct procedure becomes awkward and no longer easily applied to the more complex structures of multiple rings or chains of the type shown in Fig. 3. However, quite general relationships can be derived ifuse is made of the Wang-Uhlenbeck theorem9 for multivariant Gaussian distributions in the generalized form given by Fixman. 1o

5

FIG. 5. Three vectors in a chain shown to illustrate the Wang-Uhlenbeck theorem. Each of the vectors can (according to eqn (10)) be expressed by a linear combination. For the vector V 51 = Ln i/J 51.nr nail i/J 51,n = 0 for n < sand n > t, and for the remaining t - s segments the coefficients are 1. For the vector VI one has non-zero elements for the segments n = i to n =j. The number of segments which

both subchains have in common is then c 51,1 = Ln i/J 51,ni/J l,n = j - s (see eqn (12)).

Let us consider s vectors VI' V 2 ... V s' where each of them is a linear combination of the N bond vectors in a chain, e.g.

(10)

where the coefficients If; pk can have values of 1 or O. For instance for the vector V st drawn in Fig. 5, all coefficients If;st.k = 0 for k < sand k> t and only those of the bonds connecting segments t and s are If; sl,k = 1 for which s < k < t. The joint distribution of the s vectors V p is then given by the equation

Ps(Vo, VI' .. V.) = (3/21lN)3,./2IQ - 3/2exp [ - (3/2b2ICsI)LLcijVYj]

(11)

THEORY OF CYCLIC MACROMOLECULES 49

where ICsl is the determinant of a matrix with elements cij which are given by

N

cij = L !/Jik!/Jjk k = I

(12)

and cij are the corresponding cofactors of this matrix. Note that the indices i andjrefer to vectors Vi and Vj in a chain of N elements. Therefore cijis the sum of unit elements which corresponds to the number of bonds that both vectors have in common. This leads to a procedure that was first given by Casassa: I I

(1) Each diagonal element Cii is the number of segments in a closed loop formed by the vector Vi with the chain.

(2) The off-diagonal elements cij represent just the number of segments which the two loops, spanned by the vectors Vi and Vj' have in common.

Application to Simple and Multiply Twisted Rings Figure 4 exemplifies the situation for the two vectors V 0 = r IN and V I = r st ;

the corresponding elements of the matrix C lr and its determinant are given then by

( N (t - S») C lr = (t - s)(t - s) (13)

IClrl = (t - s)[N - (t - s)] (14)

With (t - s) = nand eqns (13) and (14) inserted into eqn (11), one recovers eqn (9).

The configurational distribution for a subchain (t, s) in a multiple ring can now be written down. First, let us consider a once twisted ring where:

(i) elements 1 and N are connected forming a closed chain; (ii) elements i and j are connected forming the twisting point.

Then the distribution for the distance r st is obtained by integration over all coordinates of the vectors V 0 = r IN and V I = r ij where these vectors are represented as Dirac delta functions, i.e.

p 2r(rst ) = Sp 3(OIN' Oij' V 3 = rst ) dV I dV 21SP 3(OIN' Oij' V 3) dV I dV 2 dV 3

(15)

50 W AL THER BURCHARD

where, for abbreviation, VI = bIN =OIN and V2 = bij=Oij' With the properties of the Dirac delta function, we find immediately from eqn (II) that

In general, one has for the s-tuple ring

and consequently

<r;,) = b21 esl/css

<rs~ I) = (6/nb 2 ) 1/2 (cSS/1 esl) 1/2

< exp (iq. rs'» = exp [ - (b2q2/6)(lesl/cSS )r;,]

(18)

(19)

(20)

The mean square radius of gyration <S2), the hydrodynamic radius Rh or, more generally, the translational diffusion coefficient D, and the particle scattering factor are obtained by summing the segment pair functions of eqns (18)-(20) over all segment pair combinations in the rings. This requires knowledge of the determinant lesl and its cofactors CSS • Following the recipe given by Casassa II one finds

N N-rl N-r l -r2 rs t-s

N- r l N- r l N-r l -r2 rs CIs N-r l -r2 N-r l -r2 N-r l -r2 rs C2s

e= s (21)

rs rs rs rs Cs - I,s t- s CIs c2s Cs - I.s t- s

CSS = r lr2 ··· rs (22)

and for the determinant after some rearrangement

r l 0 0 0 (t-s)-c ls

0 r2 0 0 CIs - c2s

0 0 r3 0 c2s - c3s lesl = (23)

0 0 0 rs Cs - I.s (t-s)-c ls CIs - c2s c2s - c3s Cs - I,s (t - s)

THEORY OF CYCLIC MACROMOLECULES 51

with

(24)

In these equations, rj denotes the number of segments in the jth ring, t - s

the number of segments between the two units t and s on the multiple ring and Cjs is the number of segments which are common in the two loops formed by the vector rSI with the chain and that formed by thejth twisting point and the multiple ring. The situation is illustrated in Fig. 6 for C 25; the common path is marked there by the heavy line.

FIG. 6. Graph of a pentuple ring. The heavy line represents the part of the ring which the loops formed by the vector V st have in common with the ring. The corresponding matrix element in eqn (22) is C25 • Note, coo is the matrix element representing the number of segments which the loop formed by the vector V I N has in common with the total ring; similarly, CII represents that section of the ring which the loop formed by the first twisting point has in common with the total ring, etc.

Evaluation of the double summation for the quantities in eqns (18)-(20) is tedious. The hydrodynamic radii have been calculated by Fukatsu and Kurata. 12 For rings of equal size they found

(25)

with the dimensional parameter

s-1 s-k

Ks = TC/S 1/2 + (4/S3/2) L L [2sin- 1 (j-3/2) - (j + 1)1/2sin- 1 (j-l)]

k=1 j=1 (26)

No results for the radii of gyration were given by these authors. For the single, non-twisted ring <S2) has been known since 1946y,13,14 Recently Yang and YU 15 derived an analytic expression for <S2)" by application of graph theory and obtained

(27)

52 WALTHER BURCHARD

Similarly, the particle scattering factor has been calculated for the non­twisted ring only.ll,16,17 The result is

P(q) = (2/3)D(v/2)

where D(X) is the Dawson integral defined by

D(X) = (exp ( - X2» J: exp (t 2) dt

with

(28)

(29)

(30)

The Dawson integral is tabulated in the book by Abramovitz and Stegum. 18

First Cumulant of the Time Correlation Function in Dynamic Light Scattering Before discussing the static properties, an equation for the first cumulant r of the time correlation function g 1 (t) of the electric field in dynamic light scattering may be given. The first cumulant is defined as the initial slope of Ing 1(t) as a function of the delay time t,19,20 where

Seq, t) <IE*(O)E(I)I> g 1 (t) = Seq) = <IE*(O)E(O)I>

(31 )

with Seq, t) and Seq) the dynamic and static structure factors and E(t) the electric field of the scattered wave

1'1

E(t) = L exp (iq. rit ))

j= 1

(32)

The angular brackets denote the average with respect to the space-time distribution. Thus gl (t) is a dynamic quantity which will be discussed in a separate section.

The first cumulant, however, can be calculated as a limiting function at t = 0 with the aid of the equilibrium or static distribution function and is given according to Akcasu and Gurol 21 by

r = q22:2: «l. Dts.l) exp (iqrst) >/[2:2: < exp (iqrst) >] (33)

where I is the unit tensor and Dts is the' diffusion tensor element for a segment pair t and s in the Kirkwood theory 7,8 and is given by

Dst = kT(()st/01 + (l - (jst)/(8n'1o rst)(1 + rslsir;t) (34)

THEORY OF CYCLIC MACROMOLECULES 53

With the distribution function of eqn (17) for multiple rings one obtains

«I. Dst.l)exp(iqrst» = 1>stkTj( + (1 -1>st)

x kT/(61/2n3/21Job)(cSS/1 Csl) 1/24/3 [ - X 2 + (2X- 1 + X- 3)D(X) 1 (35)

with

(36)

and (the friction coefficient of an individual segment. The Dawson integral D(X) is defined in eqn (25).

Again, the first cumulant has been calculated only for the non-twisted ring. 16.17 Applying an approximation for the Dawson integral (which is correct up to terms of order q4) one has

q2 {kT 2kT [rl/2exp( -v2(1 - ~)O r = P(q) N( + 61i2n3i21JobNli2 Jo [(1 _ ~)~P/2 d~

+ (v2/5) fi 2 [(I - O~F/2 exp ( - v2a2 (1 - ~)~) d~]} (37)

where the integrals have to be solved numerically. The first term in the square brackets represents the hydrodynamic pre-average approximation where it is assumed that

«I. Dst.l) exp (iqrst» ;:;;; O. D st .1>< exp (iqrst» (38)

The second term in the rectangular brackets of eqn (37) gives a correction to the error introduced by this assumption where a numerical value of a 2 = 0·72 was found to describe the first cumulant satisfactorily. In the limit of q-+ 0 the dynamic light scattering theory proves r/q2 = Dwhich together with eqns (38) and (34) yields Kirkwood's well-known diffusion equation (eqn (6».

Properties of the Rings Radius of Gyration, Hydrodynamic Radius and Related Quantities The change in the hydrodynamic radius if an open chain is closed to form a ring is of particular interest to biophysicists. This change can be observed by sedimentation measurements which for polynucleic acids can be measured accurately even at very low polymer concentrations. The sedimentation coefficient is given by

(39)

54 WAL THER BURCHARD

where the friction coefficient is

(6')

For long chains the first, free draining, term can be neglected in this equation, and one has

(40)

Fukatsu and Kurata 12 have calculated the effect for twisted rings of equal size, i.e. r 1 = r 2 = rs and the same contour length N. They also examined, for the once-twisted ring, the change of the hr-factor when. the fraction P2 = r 2/(r 1 + r 2) of the individual loop sizes is varied (see Fig. 7).

N

FIG. 7. A once-twisted ring, where the fraction of segments belonging to the

second ring is pz = r z/ N.

0.9 r------,

'" ~ 0.8

0.7 "':-'---'--.L......L.-l

0.0 0.5 P2

FIG. 8. Change of the hydrodynamic shrinking factor h,z when the fraction

Pz of the second ring is varied.

The results are plotted in Figs 8 and 9. Clearly the hydrodynamic radius decreases with the number of rings; the hr factor is 0·849 for the simple ring and approaches a limiting value of 0·5 for a chain with an infinite number of rings. The different sizes of the various rings seem to have only little influence on the hydrodynamic radius; it is most pronounced for rings of equal size.

The g-factor is defined by the corresponding mean square radii of gyration

g, == <S2>'ing/<S2)lin

With eqn (27) and <S2 )lin = b2 N/6, one obtains 15

gsr = (1 + l/s)/4

(41)

(42)

THEORY OF CYCLIC MACROMOLECULES 55

I 1.5·-

~,.-.-.-.-.-:.P--.-1.01\

a. e "-

-.

e, h e-e_e _____ _ e_

0.5

number of rings FIG. 9. Change of the shrinking factors g and h with increasing number of rings

and the corresponding change of the parameter p = <S2>1/2jRh .

In fact, in the limit of infinite twisting, the molecule resembles a double­stranded molecule with an apparent contour length that is half of the total chain length. Thus, if a change in rigidity is neglected, one has g oor = 1/4 and hr = 1/2.

Comparing gsr with hsr one notices that the hydrodynamic radius is less affected by the ring structure than the radius of gyration. The difference between these two shrinking factors is apparently largest for the single ring and becomes gradually smaller with increasing number of rings (Table 1). This fact is also clearly seen in the p-parameter defined as

(43)

For the single ring, Plr = 1·2533 and is considerably lower than Plio = 1·504 for the linear chain, but eventually for infinite twisting this value of 1·504 is again approached (see Fig. 8). It should be mentioned that the theory of multiple rings neglects changes in flexibility when twisting the rings. Thus, the theory is only meaningful as long as the number of rings is small compared with the number of chain segments, because then the assumed ideal flexibility is scarcely affected. With increasing number of rings the

56 W AL THER BURCHARD

TABLE 1 List of the Geometric and Hydrodynamic Shrinking Factors g,,=<S2)sr/<S2)lin and hsr=Rhsr/Rhlin and of the para-

_ _ 1/2 " _ 2 1/2 meter Psr - Rgr/Rhr - (gsr /hsr)Plin, where Rg - <S) and Plin = 1·504. The Index s Denotes the Number of Rings of

Equal Size and s - I is the Number of Twisting Points

S gsr hsr Psr

1 0·500 0,849 1·253 2 0,375 0,829 1,111 3 0·333 0·714 1·216 4 0,312 0,686 1·226 5 0,300 0,666 1·237

10 0·275 0·618 1·276 CIJ 0,250 0·500 1·504

constraints at the tWlstmg points, which reduce the number of configurations considerably, will become noticeable and will make the chain more rigid. Therefore, a lower effect on hr and gr is expected for real multiple rings than is predicted by the Gaussian approximation,

Particle Scattering Factor The change in the particle scattering factor when passing from an open chain to a closed ring is clearly seen in the Kratky plot of Fig, 10, where the particle scattering factor multiplied by u2( = <S2 )q2) is plotted against u. 16 Both particle scattering factors approach asymptotically plateau values with heights 2 and 1 for the open and closed chain, respectively. The plateau is characteristic of Gaussian chains, (Note, chain stiffness as a result of directional bond correlation is not included in the Gaussian chain, The effect of rigidity on the particle scattering factor is discussed in the section 'Rigid Rings',) Of course, the different values of the asymptotes reflect differences in the mean square radii of gyration. The ring shows a striking maximum at u = 2 with a height of about 1·28. A similar maximum occurs for star-branched molecules22 - 24 and for molecules of spherical symmetry. Indeed, the particle scattering factors of a ring and a five-arm star are almost identical. The constraint imposed by fixing the two ends of a chain causes an increase of the segment density around the centre of mass, Evidently, the effect is of about the same magnitude for the five-arm star where only one end is fixed together with the four others to the star centre. This constraint makes these molecules more sphere-like and is responsible for the maximum.

2

1

I

THEORY OF CYCLIC MACROMOLECULES

/ I

/ I

/,

/',,,, .... --------------------.... -

l---'----'-----L...--~, --'----"----'----' o 2 4 6 8 10

u=Rgq

57

FIG. 10. Kratky plots for the particle scattering factors for flexible rings (--) and flexible open chains (- - --) obeying Gaussian statistics. RG = (S2) 1/2,

q = (4nl.le) sin 812.

The First Cumulant Figure 10 shows the corresponding plot of the reduced first cumulant r / q2 D (normalized by the translational diffusion coefficient D) as a function of u2 • Up to u2 = 3, this reduced first cumulant can be written in a power series for u2 , which is exact up to terms of u4 , thus:

(44)

where C is a dimensionless quantity characteristic of chain architecture. Table 2 gives a list of C values for various structures. 24,25 In a region of intermediately large q (i.e. u2 » 1 but bq < 1), the reduced first cumulant approaches a function that is independent of the chain length and its architecture. In this region one has, according to theory,16,25

(45a)

or

(45b)

Since in the plot of Fig. 10 the first cumulant is normalized with regard to D and plotted against u, for the ring the asymptotic slope is by a factor 1·18/1-414 = 1·2 smaller than for the open chain. This explains the behaviour shown in Fig. 11.

In quasi-elastic neutron scattering a region of bq > I is covered, and for

58 WALTHER BURCHARD

TABLE 2 C and p Parameters for Various Polymer Structures24

Structure

Linear chains Monodisperse (Mw/Mn = 1) Polydisperse (Mw/Mn = 2)

Cyclic chain (s = 1)

Star-branched chains Monodisperse, f= 4 Monodisperse, f= 12 Monodisperse, f - 00

Randomly branched chains

Hard sphere

2

5 10 u 2

C

0·1733 0·2000

0·1333

0·1482 0·1187 0·0979

0·2000

0·0000

15

P

1·504 1·732

1·253

1·334 1·172 1·079

1·732

0·775

20

FIG. 11. Reduced first cumulant r/Dq2 as function of u2 = <S2)q2 for flexible rings (--) and flexible open chains (----).

THEORY OF CYCLIC MACROMOLECULES 59

this region of very large q-values the Gaussian chain approximation leads to incorrect results because of the inherent chain stiffness and the rigidity of fixed bond lengths and bond angles. These constraints require a modification of the calculations, which is to be published in the near future by Akcasu. 26

Excluded Volume Effects All considerations made so far are valid for chains in the unperturbed state which can be described by Gaussian statistics. Very little is known of the effect of chain stiffness, though some general remarks can be made without detailed calculations. These questions will be discussed later in connection with the properties of rigid rings.

Chain stiffness is caused by short-range interaction between neighbour­ing units and it can be treated by Markov chain approximations. 27 ,28 The long-range interactions on the other hand, i.e. the interaction between segments sand t, separated by many other segments, are non-Markovian in nature. 29 The excluded volume effect for linear chains has been studied for about 40 years, and satisfactory relationships have been derived. 29,3o

However, little is known about ring molecules. Of special interest is the first segment contact approximation, since this can be treated exactly by perturbation theory. 11,29

The interaction potential between two segments is of very short range in space and can safely be approximated by a delta function

(46)

i.e. only if two chain elements come into close contact will the interaction fJ become effective and otherwise it is zero. Thus, to derive an equation for the effect of excluded volume, the probability P o(rjj = 0) == P(Oij) has to be known; this is the probability that a subchain of j - i units forms a loop. Moreover, since we wish to know the effect of this interaction on < r;,), we need (in addition) the probability P o(rs,' 0;). The change in the distribution P(rsr ) can be expressed by a cluster expansion, which for the present purpose may be truncated after the single contact (for more details see Yamakawa29), thus:

N N

P(rsr ) = P o(rs<) - L L Qo(rSl' OJ) (47)

with (48)

60 WALTHER BURCHARD

where the subscript ° refers to the unperturbed state. Qo(rs,' 0ij) represents the contribution to the distance distribution of the interaction of one special contact. Then, the overall contribution is the sum of all interacting pairs in the chain.

Po(rs,) is the normalized distribution for the unperturbed chain, and P O(Oi) is the probability of finding the two segments i and j at the same position. Similarly, P o(rs" 0ij) is a non-normalized conditional probability distribution, which describes the probability of finding a special distance r SI

in a chain if the two segments i and j are in contact. Thus, the intramolecular excluded volume effect is governed by the probability of ring formation for chain sections of length (j - i).

For closed chain molecules one has to start with the distribution P a (rst, 0lN)' The corresponding perturbed distribution has, in the single contact approximation, the form

N N

P(rsl' GiN) = P o(rst, 0ij) - L L Qo(rs" Oij' DiN) (49)

where now

Configurational averages are obtained by standard techniques. For instance, the mean square radius of gyration is given by

N N

= <S2>0 - N- 2I IfJ r;,Qo(rst,Oi)drs, (51)

s<1

and there is a similar equation for the ring molecule, where Qo(rst' Oij' 0lN) has to be used. Thus the derivations of equations for <S2> reduce to the determination of the probabilities of P O(Oi) and P a (rst, 0ij) for the linear chain and P O(Oij, 0lN)' P o(rst , Oij' 0lN) and P oCrs" 0lN) for the ring molecule. These functions can be derived by applying the Wang-Uhlenbeck procedure. To give an example, start with P oCrs" rij' r iN)' introduce r ij = b(i - j) and r iN = b(l - N) and integrate the distribution over all

THEORY OF CYCLIC MACROMOLECULES

Qh s~h t (Z)h t (Sjh 00 ----- i ---- j is ------ i ------ s ---~- ----- t t sot

I N I N I N IN IN IN

61

FIG. 12. Single contact cluster diagrams for the open chain. The dashed lines represents the contact between segments i and), while the heavy lines represent the distance vectors V" between the two segments sand t. The marked chain sections

represent the subchain which both loops have in common.

coordinates of segments i and) and segments 1 and N; the result is p o(rsp Oij' 0IN). The other probabilities and distributions in eqns (47)-(51) can be found in a similar manner. The elements of the matrices Cs are found by inspection of the cluster graphs; these are given in Figs 12 and 13 for the open and closed chain.

Following the procedure outlined above, one obtains for the open chain

and for the closed chain

CZ=( ; t-s

d

d

h

(52)

t-S) h;

t-s

(53)

where d =) - i is the ring size formed by the contact of segments i and), and h is the number of segments which this loop has in common with the loop formed by the vector rst with the chain. Its value changes with the position of rst ' as is demonstrated by the cluster diagrams of Figs 12 and 13. The rest

FIG. 13. Single contact cluster diagrams for the circular chain. Here three loops have to be considered, i.e. the total ring and the two loops formed by the contact

pair (i.}) and the vector VSI •

62 WALTHER BURCHARD

of the work is straightforward, though there is a tedious integration. The final results for the expansion factors a2 = <S2>/<S2>0 are as follows

a~n = I + (134/105)z - ... (54)

and

a2 = I + (n/2)z _ ... flog (55)

with

z = (3/2nb 2)3/2 N 1/2 f3 (56)

Thus the expansion due to thermodynamic interaction is (with n/2 = 1·571) larger for the closed ring than for the open linear chain, where 134/105 = 1·276. This higher effect results from the constraint of the fixed two chain ends in the ring by which the chain segments are kept confined to a smaller space, which necessarily increases the probability of a contact.

No calculations have been carried out for the higher terms in the cluster expansion for a cyclic chain., i.e. for intermediately large values of z, and no calculations have been performed to derive the asymptotic behaviour at large z. Often similar behaviour is assumed for the cyclic and open chains, and a modified Flory relationship is used: 11.29.31

(57)

with coefficients of K1in = 1·276 and Kring = 1·571 for the open and closed chain, respectively.

The averages for the hydrodynamic radii are obtained in the same manner, which yields

for the ring (58)

compared to

<R; 1> = <R; 1 >0(1 - 0·609z) for the open chain (59)

or

hring = hring.o(1 - 0·609z)/(1 - 0·630z) ---+ hring.O x 0·976 (60)

which holds for the single contact approximationY·29 In the next best approximation by Fixman 32 one has instead of egn (60)

hring = hring.o((l + 1·827z)/(1 + 1·890z»1/2 ---+hring.O X 0·983 (61)

in other words, the h-parameter does not change much when passing from a poor to a good solvent.

THEORY OF CYCLIC MACROMOLECULES

For the p-parameter one obtains

Pring = Pring,O( 1 + O'1554z)

Plin = Plin,o(1 + O·029z)

63

(62) (63)

i.e. this ratio of the geometric to the hydrodynamic radii is more sensitive to the solvent quality for the cyclic chain than it is for the open chain.

The second virial coefficient has again been calculated for the non-twisted chain only by Casassa II within the double contact approximation. For two interacting ring molecules A 2 is given by

A2 = (NA/2M2)L L [1 -1m L LPO(Ohh,Oiti"OINl' OIN)] (64)

it i2 it h

where PO(Ohh' 0itil' 0INl' 0IN) is the probability that a contact is formed between the segment JI of one ring and J2 of another ring and simul­taneously that a second contact is formed between the 'two segments i l

and i2 of the two chains. The situation is demonstrated schematically by the cluster diagram of Fig. 14. Again, the number of contacts is increased

FIG. 14. Double contact cluster diagram for two interacting rings. The dashed lines represent the contact pairs (il' i2) and UI,j2)'

in the two-ring molecules compared to the open chain because of the higher segment density. Thus the coefficient C in

(65)

is with. Cring = 4·457 appreciably larger than C1in = 2·865. Again no calculations were carried out for larger z values, and as an approximation a similar dependence for the cyclic chain is assumed as for the open chain. Only the coefficients Clin and K1in are replaced by the corresponding coefficients of the ring. For instance, if the Casassa-Markovitz 33

relationship is used one has

(66)

64 W AL THER BURCHARD

with

h(i) =(1--exp( -2Ci»/(2Ci) (67)

where i = z/a?, and rx follows a relationship of eqn (57)_ Often it is more convenient to express A2 in terms of the interpenetration function t/J(z), i_e_

(68)

where

t/J'(i) = ih(i) (69)

Since A 2, <S2) and M are measurable quantities the interpenetration function can be obtained from experiment without any assumptions_ The interpenetration function of eqn (67) develops an asymptotic plateau

t/J(i) -+ 1/2C (70)

This plateau is lower by the factor Clin/Cring = 0-642 for the cyclic than for the open chain_ It is interesting to observe that the ratio t/JringNlin = 0-642 is close to the ratio of hydrodynamic volumes which for the unperturbed chains is

Vh_ring = 0-612 X Vh,lin (71)

The reason for this behaviour has recently been discussed in detail by Huber et al,34

Rigid Rings Conformational Relationships The results of the preceding sections are restricted to the assumption of flexible, closed linear chains, which obey Gaussian statistics for the intersegmental distance r st ' Now we shall report some properties of rigid rings, Strictly, rigid structures will not occur in real systems, but their properties are still of interest as the asymptotic limits for semiflexible rings_ Such structures are observed in particular with DNA molecules, but they will also occur for every oligomeric ring because of the existent bond direction correlation, Again, comparison with the rigid open structure (i,e, the rigid rod) will be of interest.

In a similar way to the derivation of the conformational properties of rigid rods, we consider the ring molecule as the limiting structure of a planar polygonal polymer when the number of bonds N goes to infinity_ The static properties are easily derived from simple geometric considera­tions because there are no fluctuations in the dimensions. Hence, the

THEORY OF CYCLIC MACROMOLECULES 65

mean square radius of gyration and the particle scattering factor are (with R the radius of the ring)

<S2)N,ring = R2 = b2j(4sin2 (njN» ~ b2 N 2j4n2 (72) N

P(q)ring = N- 1 I sin (qrn)j(qrn)

N

= ~ \' sin (qb) sin (nn/ N)/sin (n/ N) (73)

N ~ qb sin (nnj N)jsin (nj N) n;l

or N

P(q)ring = N- 1 I sin [(2u) sin (nnj N»)j[(2u) sin (nnj N») (74)

n;l

where

u = q<S2)1/2 = bj[2 sin (nj N»)

The corresponding equations for the rod are

(75)

N-1

<S2)rod=(bjN)2 I (N-n)n2 =b2j12 (76)

n;l

and N-1

P(q)rod=N- 1(1 + (2jN) I (N-n)[Sin(qbn)jqbn)) (77)

(qL P(q)rod --+ (ljqL) Jo sin x/x dx + cos (qL)j(qL)2 (78)

As for flexible rings, the relationship for the translational diffusion coefficient may be given here, although it is a dynamic quantity. However, Kirkwood's general equation for the diffusion coefficient (eqn (6» is expressed in terms of the equilibrium (-l)th moment of the intersegmental distance r", i.e.

N-1

DK=(kTj(N>[1 + «(j6nY/o)N- 2 L (N-n)jrn] (79)

n;l

66 WALTHER BURCHARD

This leads for the ring to

N-l

DK,ring = (kT!(N{ 1+ «(,/6nl1ob) I sin (n/N)/sin (n}/N)] (80)

j= 1

or for large N

DK,ring ~ (kT!(N)[1 + (2(/6nl1ob)(ln N - 0,45)] (81)

The corresponding relationship for the rigid rod is 35

DK,rod ~ (kT!(N)[l + (2(/6nl1ob)(ln N - 1)] (82)

Equation (81) has played an important role in the hydrodynamic theory of diffusion, For a long time Kirkwood's general diffusion equation was considered as being strictly correct, but later it was detected as being an approximation, though a very good one,36-38 Paul and Maz0 39 ,40 have been able to derive an exact solution for the translational and rotational diffusion coefficient, by taking into account the anisotropy of the motion, The corresponding relationships for the translational diffusion coefficient are as follows:

DII ,ring = 1O/9(kT/(N) + (kT/3nl1oNb)(ln N - 0,28) (83)

D .L,ring = kT/(N + (kT/4nl1oNb )(In N - 0-45) (84)

Dring = (29/27)(kT/(N) + (kT/3nl1oNb)«(1l/l2) In N - 0,30) (85)

and for the rotational diffusion coefficient

D II ,rol = (kT!()(2/ NR2) + (kT/2nl1oNbR2)(ln N - 2-45) (86)

D .L,rol = (kTf()(2/ NR2) + (kT/2nl1oNbR2)(ln N - 3,90) (87)

Drol = (kT/0<2/ NR2) + (kTj2nl1oNbR2)(ln N - 2,93) (88)

Comparison of eqn (85) with eqn (81) shows that for large N the Kirkwood approximation overestimates the diffusion coefficient by a factor of39

(DK - D)DK,ring ~ (1/12)(1 - 0·27/ln N) (89)

This demonstrates the high quality of Kirkwood's approximation.

Properties As in the case of flexible rings it is instructive to compare the rigid ring

THEORY OF CYCLIC MACROMOLECULES 67

properties with those of rigid rods of the same contour length. For the two shrinking factors g = <S2>'ing/S;od and h = DroJDring, one has

grigid = 12/4n2 = 0.34;

hrigid = 1 +W3nIJob)(lnN -1) . 1 + W3nIJob)(ln N - 0·45)'

gflexible = 0.50

hflexible = 0.849

(90)

(91)

For comparison, the g and h factors of the flexible rings are given as well. The effect of ring closure on the geometric dimensions is about 40 % larger for the rigid ring than for the flexible ring; it remains constant for all ring sizes. The effect on the hydrodynamic radii is more difficult to interpret. Even at large N the logarithmic terms remain fairly low, so that the free draining term cannot be neglected as was possible for the flexible rings. If as a reasonable estimate we assume (/3nIJob = 1, then we find for the smaliest possible ring of N = 3 a value of h3 = 0'667, and for N = 10, 100, 103 and 104 we have 0'708, 0'894, 0·926 and 0·944, respectively. Thus only for the smallest ring sizes is the h-factor lower than for a flexible ring; with increasing size hrigid increases and gradually approaches unity, while hflexible = 0·849 remains constant if N» 1.

The parameter P == <S2)1/2(6nIJoDlkT) is more important than h, because it can be obtained from a combined static and dynamic light scattering experiment without referring to rods of the same contour length. From eqns (90) and (91) one finds for the ring

Pring = (3IJo bl' - O'45In) + (lin) In N (92)

and for the rod

Prod = (6nIJobl(jU. - 21jU.) + (21jU.) In N (93)

Thus the p-parameters have significantly different molecular weight dependences. Figure 15 shows a plot of the p-parameters versus In N, where 3nIJobl( = 1 has been assumed again. In spite of the different molecular weight dependence, it will not be possible to decide unambiguously from one measurement only whether a rigid rod or a rigid ring is present.

A distinctly different behaviour is obtained, however, from the particle scattering factor at large values of u = <S2)1/2q. Figure 16 shows Kratky plots for a rigid rod and a rigid ring. Above u = 2 the curve for the rod quickly approaches the typical asymptote of a straight line with slope nlji2. The particle scattering factor of a ring, on the other hand, approaches an asymptote which corresponds to a sinusoidal ondulation

a.

FIG. 15.

N

6 2 3 10 10 2 103 104

5

4 ROD

3

2

~ 0 0 2 3 4 5 6 7 8 9 10

In N Chain length dependence of the p( = <S2) 1/2/ Rh ) parameter for rigid rods

and rigid rings.

11.------------------r------~

10

9

8

7

_ 6 g

.e- 5 :J

4

3

2

2 4 6 8 10 12 14 16 18 u

FIG. 16. Kratky plot of the particle scattering factor for rigid rings and rigid rods (u= <S2) 1/2q).

THEORY OF CYCLIC MACROMOLECULES

1.0

0.9

0.8

0.7

0.6

![ O.S ::J

0.3

0.2

0.1

0.00

ROD

2 4 6 8 10 12 14 16 18 U

69

FIG. 17. Plot of the scattering function uP(q) versus u for rigid rings and rigid rods.

around a straight line of slope 1/2. Even more instructive is a plot of uP(q) versus u = <S2>1/2q which is shown in Fig. 17. The difference between the constant asymptote of the rod and the periodic and weakly damped asymptote of the ring is very striking. The particle scattering factor of the rigid ring can be approximated by the empirical relationship

1 sin [2u - 0'7] uP(q) ';;;.-2 + ~

y'12u (94)

Often in experiments the normalized scattering intensity R(q) = r2i(q)/ I 0

is used and divided by the concentration c and a contrast factor K, and this quantity is plotted against q2. In the limit of small q one has the familiar relationship

(95)

which holds for all types of structures. Here i(q) is the scattering intensity, 10 the primary beam intensity and r the distance of the detector from the scattering volume. At large q-values one obtains the asymptotes

qR(q)/ Kc -+ n(M/L) for the rod (96)

70 W AL THER BURCHARD

and

N

qR(q)/ Kc-+ M/(2<S2)1/2)N--1 L sin [(2u) sin (nn/ N)/sin(nn/ N)]

n;1

= ll(M/L) sin (g(u)) for the ring (97)

where

sin (g(u» = N- 1 Lsin [(2u) sin (nn/N)]jsin (nn/N) (98)

Thus in both rigid structures the linear mass density ML = M/L is obtained, where the asymptote for the ring is superimposed by a weakly damped sinoidal modulation.

Semiflexible Rings The different behaviour of rigid and flexible rings gives rise to speculations about what curve may be expected for q2 P( q) with semiflexible rings. These have not been treated theoretically up to the present time. Semiflexible chains are characterized by the number Nk and the length Ik of Kuhn segments, and two limiting cases can be discussed without performing detailed calculations. These two cases are (i) rings with many Kuhn segments, i.e. Nk» 10, and (ii) almost rigid rings where Nk < 10.

(i) Nk » 10 In the low q-region, which corresponds to large dimensions, Gaussian behaviour is observed. For the open chain one has

and since <S2)ring = <S2)lin/2, for the ring we have

<S2)ring = (LIJfl2

where

(99)

(100)

(101)

is the contour length expressed in terms of Kuhn segments of length Ik or chemical bonds of length I. The Kuhn length is twice the persistence length and it can also be expressed in terms of the characteristic ratio CQ'J =6<S2)/NP,41 which with e:qn (99) gives

(102)

THEORY OF CYCLIC MACROMOLECULES 71

for long chains. This relationship is also assumed to hold for rings. Thus for u > 5 but Ikq < lone has

(103)

and

(104)

For large q (i.e. qlk» I) there is rod-like behaviour for both structures and

P(q)-+n/qL

The two sections of the asymptotes intersect at42 - 44

qtn = 12/nlk = 3·82/1k q':ing = 6/nlk = 1·91/lk

(105)

(106) (107)

Thus, in a Kratky plot the transition to the rod-like behaviour occurs at much lower q-values for the flexible ring than for the corresponding open chain. Figure 18 gives a sketch for rings of 50,20 and 10 Kuhn segments, i.e. u* = 3·90, 2·47 and 1·74, respectively. (From eqns (100), (101) and (107) one has u* = 1·91J(Nk/12).)

0-

a.. N ::J

3~-----------r-------r--------~

2

2 4 6 8 10 U

FIG. 18. Kratky plots for semifiexible rings containing N Kuhn segments. The dotted curve represents the function for semifiexible open chains.

72 W AL THER BURCHARD

(ii) Nk < 10 In this case the persistence kink has moved into a region before the maximum, and the kink may not be detectable by experiment. Moreover, for such short rings the Kuhn segments can no longer be approximated by a straight rod. It must possess a certain persistence in curvature. This curvature is not a fixed quantity but it shows fluctuations around a mean value. The curvature will be responsible for an ondulation similar to that in a rigid ring, but the fluctuations will cause a much stronger attenuation of the amplitude when q is increased. In a way, the situation resembles the radial distribution function and the corresponding structure factor of a liquid.

DYNAMICS OF RING MACROMOLECULES IN SOLUTION

Some General Remarks As we have known for a long time, thermodynamic equilibrium means not a static balance of forces (which, for instance, would fix a certain distance rst

between two segments) but short time fluctuations around a mean due to a random force F( t). Experimentally this mean is obtained if measurements are performed over a time interval that is large compared with the characteristic times of fluctuations. For polymeric solutions, these characteristic times lie between 10 - 8 and 10 - 3 s. In many cases, the time for recording a signal is about 1 s and is thus long enough for determining the mean which is characteristic for the equilibrium property.

If shorter time intervals are chosen for the observation of a signal, the measured quantity depends on time. In polymer science, there are two main dynamic quantities which are of interest, i.e. the dynamic intrinsic viscosity [IJL and the dynamic structure factor Seq, t), which can be measured by dynamic light scattering or dynamic (quasi-elastic) neutron scattering.

Application of elementary hydrodynamics establishes the following for the intrinsic viscosity:29

N

[IJ] = - (NA/MIJog) L <FjxY) (108)

j; 1

where

[IJ] = lim (IJ --IJo)/lJoC (109) c-o

THEORY OF CYCLIC MACROMOLECULES 73

In these equations, 1'/ is the viscosity of the solution at concentration c, 1'/0 is the solvent viscosity, M is the polymer molecular weight and g is the shear rate. Fjx is the x-component of the friction force Fj exerted by the sheared fluid on the jth segment of the molecule and Y j is the y-component of the radius vector Rj of thejth segment. Here the average <FjxY) corresponds to the non-equilibrium, space-time distribution.

The shear rate applied in an experiment is often a harmonic function of time

g = go exp (iwt) (110)

and the viscosity is then a complex quantity for which we write

(111)

where 1'/' means the viscosity which is in phase with the shear rate g in eqn (110) and 1'/" is the corresponding out-of-phase quantity. This complex viscosity is phenomenologically related to a complex shear modulus by the equation

Thus we have

where

and finally

[I'/L = [1'/']- i[I'/"]

[GL= [G' ] +i[G"]

[GL = lim (G - iwl'/o[l'/]w)/c c~o

[G]w = iWl'/o[I'/L

[G'L = Wl'/o[I'/"]

[G"L = Wl'/O[I'/']

(112)

(1l3a)

(113b)

(114)

(1l5a)

(1l5b)

(1l5c)

The intrinsic storage modulus [G' ] and the intrinsic loss modulus [G"] are obtained from common viscoelasticity measurements. 44 Because of the relationships of eqn (115), it is sufficient to derive equations for the complex viscosity. The main problem here consists in the derivation of the space-time average <FjxYj).

74 W AL THER BURCHARD

The dynamic structure factor can be derived from elementary scattering theory21,45 and it is given by the equation

N N

S(q, t) = <p*(O)p(t» = L L <exp(iq(r.(t) - rt(O»» (116)

where

N

p(t) = LeXP(iqrs(t» (117)

and the asterisk denotes the conjugate complex quantity. Again, the average has to be taken over the space~time distribution.

fhe Space -Time Correlation Function This distribution function can be derived from Langevin's equation for the motion of Brownian particles46 ,47

(u - v) = F =, - V(kTin tjJ + U) (118)

which represents a balance between the frictional force (left-hand side) and the sum of the internal force between bonds - V U and the random fluctuating force - VkTln tjJ = - V(T M), which is assumed to be entropic in nature. In this equation u is the velocity of the Brownian particle, v that of the solvent and U a potential that describes the segment connectivity in the particle. The vectors u, v and F are sums over the velocities and forces of the individual segments, e.g.

N

F= LFj j= 1

(119)

The velocity field of the solvent near a polymer is perturbed by the presence of the macromolecule. For instance, for the jth segment one has48

N

vs=vos+v"=vos+ L Tslj

j "'s (120)

i.e. the velocity of the solvent at the position ofthejth segment is influenced by the hydrodynamic interaction of the N - 1 other segments. The Oseen

THEORY OF CYCLIC MACROMOLECULES 75

tensor Tsl' which in a first approximation describes this interaction, is given by

(121)

An equation for I/I(t) is obtained by first eliminating the force F from eqn (120) and the left hand side of eqn (lIS) and second from the condition of continuity

N

81/1/8t + I (Vs·V,) =0 (122)

s= 1

which yields

81/1/8t + ~=<Vs· voJ= L L VS· Dst · [Vt + (kT) -ll/lVtU] (123)

with the diffusion tensor

(124)

Solutions for various chain architectures are possible under special assumptions for the potential U( {rs}). The simplest assumption is a harmonic potential between neighbouring segments. For a Gaussian chain this is equivalent to an entropic spring force of(3kT/b2 ). For a single ring one thus finds

Vp =(3kT/b2)(-rN +2r1 -r2 )

=(3kT/b2)(-rs _ 1 +2rs-rs + 1)

=(3kT/b2)(-rN _ 1 +2rN -r1)

for s = 1

for any s

for s = N

(125)

The first and third relationships follow from the fact that the left neighbour of the first segment is the Nth segment and the right neighbour of the Nth element is the first chain element.

Further discussion is greatly simplified when passing to a hypervector/ matrix notation where, for instance, F is a column vector whose com­ponents are the forces of the individual segments. The corresponding row vector is

Ft = (F l' F 2 •.• F N)

and the differential operator

Vt =(V 1 , V2 ··· VN)

Thus, instead of eqn (124) one has

(126)

(127)

81/1/8t + Vt • (voI/I) = (kT/()[Vt. H. (VI/I + 3/b21/1Ar)] (12S)

76 W AL THER BURCHARD

with H the hydrodynamic interaction tensor whose components H st are given by eqn (124) and A the coefficient matrix in the set of equations (125). When written explicitly for the simple ring, this matrix reads

2 -1 0 0 0 -1

-1 2 -1 0 0 0

0 -1 2 -1 0 0

A= 0 0 -1 2 0 0 (129)

0 0 0 0 2 -1

-1 0 0 0 -1 2

To solve eqn (128) the hydrodynamic tensor is often approximated by its pre-averaged quantity; then the components of the hypertensor, which themselves are tensors, reduce tOi scalars, i.e.

A solution becomes possible after transformation of the real coordinates r into normal coordinates (X, Y, Z)

where Q is a matrix which diagonalizes the matrix HA, i.e.

(131a) (13lb)

(132)

The diagonal elements of A are the eigenvalues As which for the cyclic polymer are 12 ,16,48,49

where

As = (2nsIN)2[l +j2hn(-I)sJo(sn)] = (2nsIN)2[1 +A~/n2s2]

(133a) (133b)

(134)

is the so-called draining parameter and Jo(x) is the Bessel function of zero order

Jo(sn) = (2In) I (1 - X 2 )-1/2COS (nsx)dx (135)

THEORY OF CYCLIC MACROMOLECULES 77

The transformation matrix has the property of diagonalizing any type of cyclic matrix. Thus, it also diagonalizes matrix A in eqn (129) where the eigenvalues are

Ils = (2ns/ N)2 (136)

They are those of eqn (132) with H = I, i.e. h = 0 in eqn (133). The elements of the transformation matrix Qst are for the cyclic

polymer 16•47

Qst = N- 1/2 expi(2nst/N) (137)

With eqns (131)-(137), the differential equation for the space-time distribution can be written in a set of decoupled equations

N

t/I = n t/ls (138b)

s= 1

where the relaxation times rs have been introduced. They are related to the eigenvalues As

(139)

As will be shown below, 1/1's can be related either to the intrinsic viscosity at w = 0 or to the zero time diffusion coefficient Do.

Intrinsic Viscosity and Related Properties In eqn (138) a solvent flow in the x-direction was assumed, i.e. for the segment s

VOs = (gys, 0, 0) (140)

Equation (138) can be solved explicitly and a solution for Vo will be given later, when the dynamic structure factor is discussed. For the deter­mination of the frequency-dependent intrinsic viscosity, however, the average <F}xY) can be evaluated analytically without explicit knowledge of t/I. The final result which is derived in great detail in the book by Yamakawa 29 is 48.50,51

N

['11w=(RT/M'10) L -r)(1 +iwr) (141)

j=1

78 W AL THER BURCHARD

and with eqn (115), one finds

[G'Jw = (RT! M) L w 2rJ /(1 + w 2 rJ) [G"Jw = (RT/ M) L wrj(1 + w2rJ)

(142)

(143)

which are quite general equations and hold for all architectures, which satisfy Gaussian statistics for the subchains. The zero frequency (steady state) intrinsic viscosity is

(144)

The first relaxation time can be expressed in terms of A~ if eqns (133b), (139) and (144) are used

N

r1r = Mf/o[~rJo(Rn'l L (1/rj)) j= 1

Numerical integration by Bloomfield and Zimm48 gave

r 1 ,ring = Mrro[f/Jo,ring/5·1 09RT

while for the open linear chain one has

r 1 ,lin = M1]o[f/]O,lin/2·367RT

(145)

(146)

(147)

Alternatively, r 1 ,ring can be expressed in terms of the mean square radius of gyration and the translational diffusion coefficient. From eqn (133a) it can be shown that 1 7

A1,ring = (4n 2/ N 2 )(1 + j2h)

which with eqns (139) and (6') gives

Dr1,ring = <S2>'ing/2n2

(148a)

(148b)

A similar simple relationship cannot be derived for the linear chain. Fukatsu and Kurata 12 have calculated numerically the intrinsic viscosity

and found for the non-draining ring

[f/Jring = <Pring<S2);i;;g/M (149)

with <Pring = 7·69 X 1024, while for the open chain one has<Plin =

4·22 x 1024. The g' factor is defined by the ratio of the intrinsic viscosities at the same

contour length L = Nb, and for this ratio the authors find

(150)

THEORY OF CYCLIC MACROMOLECULES 79

This ratio remains almost unchanged if Fixman's procedure of avoiding pre-averaging is applied. 52

The viscoelastic properties have not yet been calculated and compared with the relationships for [G'] and [G"] for linear chains, which requires some numerical summations, and the usual considerations29 ,44 when high modes of motion become observable.

The influence of excluded volume has been calculated by Fukatsu and Kurata 12 on the basis of the perturbation theory. This was outlined in the section 'Excluded Volume Effects' above. The result is

[11]ring = [11]o.rinP + (n12 + O'630)z) (151)

where the sUbscript 0 denotes the unperturbed state. A qualitatively similar result was obtained by Bloomfield and Zimm on the basis of the·Peterlin and Ptitsyn-Eizner scheme. 53 ,54 The final result is

[11]ring = 63/2(2n 3)1/22 -v/2 NA(6 + 5v + V2)1/2 L<AjI24) (152)

where v is the exponent in <S2) = KM2v. The last two authors also considered the flow birefringence which, however, will not be described here. 48

The dynamics of multiple rings and their viscoelastic behaviour have not been considered in detail so far, although some new results will soon become available by the application of graph-theoretical techniques. 55 - 57

Dynamic Structure Factor To derive an equation for the dynamic structure factor Seq, t), one has to start with eqn (116) and apply three operations which will be briefly outlined. These steps consist of (i) separation of the motion of the centre of mass, (ii) transformation of the equations into normal coordinates and (iii) averaging over the space-time correlation function.

(i) The coordinates of the centre of mass are given by the vector Ro. Thus, when these coordinates are separated from the remaining internal coordinates one obtains

N N

Seq, t) = <exp [iq.(Ro(t) - Ro(O))]) II <exp [iq.(r,(t) - rl(O))])

s I (153)

where rs(t) = R,(t) - Ro(t) now denotes the internal coordinates of the segments. The first factor in eqn (153) can immediately be calculated, so that

Seq, t) = exp (-Dq 2t) LL <exp [iq.(r.(t) - rlO))]) (154)

80 W AL THER BURCHARD

(ii) With eqn (131), transformation into normal coordinates yields

Seq, t) = exp ( - Dq2t) II.! exp (iq . [L Qsj~it) - Qrj~/O) J) j

(155)

(iii) To evaluate the integrals the distribution function 1/1 has to be known. To this end, eqn (138a) has to be solved, where the second term, which describes a drifting velocity, is set to zero, since there is no external driving force. The solution of the equation

(138c)

was given by Wang and Uhlenbeck 9 and may be written as

(156)

where 1/10 = TI I/Ioj is the equilibrium distribution and

I/Io j = (3/2n<e;(0))3/2 exp [- 3e;(0)/(2<e;(0))] (157)

(see eqn (118)) with the variance

<~;(O) = Nb 2(jjN)(1 -jIN) (158)

The time-dependent distribution is again a product of Gaussian distributions, i.e. j{t) = TI.t;<t) and

fit) = (3/2n<~f(t))3/2 exp [-3(~/t) - ~/O) exp( -t/r))2/2<~f(t))] (159)

where

< ~;(t) = < ~;(O)( 1 - exp ( - 2t/r)) (160)

The integration in eqn (155) can now be carried out and this was done by Pecora 58 with the final result

Seq, t) = exp (-Dq2t)L Lexp { - (q 2b2/6)

r

THEORY OF CYCLIC MACROMOLECULES 81

and this equation may alternatively be written as

(162)

with

Pn(q, t) = N- 2L L{exp [ - (q2b2/6) (LJ.lj-l(Q;j + Q~))T s t j

x (1ln !{(q2 b2/6) LJ.lj-l(QSjQ;j + Q~Qt)exp (-tlr)]"} (163)

j

Eventually with the matrix elements Qst of eqn (137) one obtains for Pn(q, t) the general expression 17

P.(q, t) = exp [ _(2U)2 N- 2 LAk- 1 ] N- 2 L L (1 In !)

k s t

x [(2U)2 N- 2 L AI: 1 cos (2nk(t - s)1 N) exp ( - tlrk )]" (164)

k

The terms Po' PI' P 2 and P 3 have been calculated explicitly and are plotted against u = «S2)q2)1/2 in Fig. 19. This series expansion is useful for rings smaller in size than u ~ 3, though even then higher relaxation modes may contribute significantly to the dynamic structure factor.

At values much larger than u = 3, the shape of the time correlation function becomes independent of chain length and molecular polydisper­sity. The reason for this behaviour follows from the fact that at larger q­

values only distances of the order r '" llq are seen, and if the distances are smaller than <S2)1/2 then essentially internal structures are observed. Furthermore, if

qb « 2(lnbtJo (165)

the chain dynamics are governed by the non-drained Zimm hydrodynamics of strong hydrodynamic interaction. An estimation of ( = 6ntJOrh' where rh

is the hydrodynamic radius of one segment, shows that 2(lnbtJo c:::: 3. In light scattering q'" 10 - 3 A-I and b '" 2A, and thus the non-draining limit is

82 WALTHER BURCHARD

U FIG. 19. Dependence of the amplitudes P o(q), PI (q) and R(q) at delay time t = 0 on u = <S2> 1/2/ Rb for ring molecules. The amplitudes are normalized with regard to the static particle scattering factor (Pl(q) = l]n(q». R(q) represents the influence of

the higher modes of motion.

valid in the whole q range aCCl~ssible to light scattering. Under these conditions, eqn (162) has the following asymptotic shape for large u

gl (q, t) -> f) exp [- y - I(y, rt)] dy (166)

with

I(y, rt) = (2In) Leo x- 2 cos (xy)[1 - exp ( - rt(xI2»] dx (167)

THEORY OF CYCLIC MACROMOLECULES 83

where

(168)

Equations (166)-(168) were first derived by Dubois-Violette and de Gennes. 59 As anticipated the asymptotic 'shape function' g 1 (q, t) depends on rt only, and r is a function of q3 alone and no longer depends on the chain length. The dependence of the shape function on rt is unfortunately very complicated and is determined by the relaxation spectrum of Gaussian subchains. A graph of this function is not reproduced here, as it can be found in Ref. 17, but the author emphasizes that a simpler (though only approximate) shape function will be more helpful in practice.

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Press, London, 1978, p. 381. 24. Burchard, W., Adv. Polymer Sci., 48 (1983) 1. 25. Burchard, W., Macromolecules, 13 (1980) 1256. 26. Akcasu, A. Z., Macromolecules, in press. 27. Daniels, H. E., Proc. Roy. Soc. Edinb., 63 (1952) 290.

84 W AL THER BURCHARD

28. Koyama, R. J., Phys. Soc. Japan, 34 (1973) 1029. 29. Yamakawa, H., Modern Theory of Polymer Solutions, Harper and Row, New

York, 1972. 30. Tanaka, G., J. Polymer Sci., Physics Ed., 17 (1979) 305. 31. Flory, P. J., Principles of Polymer Chemistry, Cornell University Press, Ithaca,

New York, 1953. 32. Fixman, M., In: Modern Theory of Polymer Solutions (ed. H. Yamakawa),

Harper and Row, New York, 1972, p. 18. 33. Casassa, E. F. and Markovitz, H., J. Chem. Phys., 29 (1958) 493. 34. Huber, K., Burchard, W. and Akcasu, A. Z., Macromolecules, 18 (1985) 1461. 35. Riseman, J. and Kirkwood, J. G., J. Chem. Phys., 18 (1950) 512. 36. Ikeda, Y., Bull. Koboyashi Inst. Phys. Res., 6 (1956) 44. 37. Erpenbeck, J. J. and Kirkwood, J. G., J. Chem. Phys., 38 (1963) 1023. 38. Zwanzig, R. W., J. Chem. Phys., 45 (1966) 1858. 39. Paul, E. and Mazo, R. M., J. Chem. Phys., 48 (1968) 2378. 40. Paul, E. and Mazo, R. M., J. Chem. Phys., 51 (1969) 1102. 41. Kratky, o. and Porod, G., Trav. Chim., 68 (1949) 1106. 42. Porod, G., Monatsh. Chem., 80 (1949) 251. 43. Heine, S., Kratky, o. and Porod, G., Makromol. Chem., 40 (1961) 682. 44. Ferry, J. D., Viscoelastic Properties of Polymers, 3rd edri, Wiley, New York,

1982. 45. Berne, B. and Pecora, R., Dynamic Light Scattering, Wiley, New York, 1976. 46. Langevin, P., Comptes Rendus, 146 (1908) I. 47. Chandraskhar, S., Rev. Mod. Phys., 15 (1943) I. 48. Bloomfield, V. and Zimm, B. H., J. Chem. Phys., 44 (1966) 315. 49. Bixon, M., J. Chem. Phys., 58 (1973) 1459. 50. Zimm, B. H., J. Chem. Phys., 24 (1956) 269. 51. Zimm, B. H. and Kilb, R. W., J. Polymer Sci., 37 (1959) 19. 52. Fixman, M., J. Chem. Phys., 42 (1965) 3831. 53. Peterlin, A., J. Chem. Phys., 23 (1955) 2464. 54. Ptitsyn, O. B. and Eizner, Yu. E., Zh. Fiz. Khim., 32 (1958) 2464. 55. Eichinger, B. E., Macromolecules, 13 (1980) 1. 56. Forsman, W. C., J. Chem. Phys., 65 (1976) 4111. 57. Yang, Y. L. and Yu, T. Y., Makromol. Chem., 186 (1985) 513, 609. 58.· Pecora, R. J., J. Chem. Phys., 43 (1965) 1562. 59. Dubois-Violette, E. and de Gennes, P.-G., Physics, 3 (1967) 181.

CHAPTER 3

Preparation of Cyclic Polysiloxanes

P. V. WRIGHT

Department of Ceramics, Glasses and Polymers, University of Sheffield, UK

and

MARTIN S. BEEVERS

Department of Chemistry, University of Aston, Birmingham, UK

INTRODUCTION

In studies of ring -chain competition, the polysiloxanes have probably been treated in greater detail than any other synthetic polymer system. They have also proved to be the most convenient source of cyclic material for investigations of the properties of cyclic polymers. Several factors combine to make the polysiloxanes eminently suitable for such purposes. High proportions of cyclic material are generated together with linear components of high molecular weight, so facilitating the separation of cyclic from linear material. While they are generally highly reactive in the presence of certain polymerization or equilibration catalysts, they have excellent thermal stability and adequate chemical stability to allow accurate and reliable analysis and experimentation. In this chapter, we describe the essential procedures which have been followed in preparing and analysing the products of cyclization reactions and in separating fractions of cyclic siloxane material.

85

86 P. V. WRIGHT AND MARTIN S. BEEVERS

THE FORMATION OF SILOXANE RINGS IN EQUILIBRATES

The polysiloxanes are readily prepared by hydrolysis of chlorosilanes to give both linear and cyclic oligomers.

RR'SiCI2 + 2H20 -+ [RR'Si(OH)21 + 2HCl

a[RR'Si(OH)21 -+ HO(SiRR'-O)a-H + (a - I)H20

HO(SiRR'-O)aH~HO(-Si(RR')-O)a_b-H + LfSi(RR')-O=+6!

The cyclics, which are predominantly the cyclic tetramers, pentamers and hexamers, may be recovered by fractional distillation. The pure cyclic tetramers are the starting materials commonly employed in the preparation of high molecular weight polysiloxanes. The polymerizations are conveniently initiated using traces of strong base or strong acid and are normally allowed to proceed to equilibrium whereupon a distribution of cyclics as well as linears is formed as a consequence of backbiting reactions.

HO-(RR')SiO)Y-l-Si(RR')O-K +I~

HO-«RR')SiO)y-x_l--Si(RR')-08K 6l + Lf(RR')SiO=iJ

X= 3;-+ CIJ

The use of oligomeric silanolates K + 0 - (Si(RR')O)y _ 1 Si(RR')-O - K +

as initiators is convenient for the production of equilibrates having linear fractions of high molecular weight. These initiators are readily prepared by reflux of KOH and siloxane in solution. The water formed from silanol condensations is removed as an azeotrope with a suitable solvent such as toluene. l Base catalysed equilibrations may be promoted by alkali metal ion complexants such as polyethers2 RO(CH2--CH2-O)"R (n = 2 -4) or the corresponding cyclic ('crown') ethers. 3 Other catalysts for the production of equilibrates having high molecular weight linear fractions include potassium naphthalene and sodium or potassium mirrors. Morton and Bostik4 proposed that in the latter case ring cleavage proceeds by the following process:

4=Si(RR')O=t.i + 2K -+ K + Si - (RR')-O+Si(RR')O) 2-Si(RR')-O - K +

The extraction and fractionation of such equilibrates (in particular of dimethylsiloxane equilibrates) has been the main source of cyclic material which has been used over the last decade by Semlyen and co-workers in their extensive investigations into the properties of cyclic polymers reviewed in this volume.

PREPARATION OF CYCLIC POLYSILOXANES 87

The hydrogenmethylsiloxane system is highly reactive towards basic catalysts, but in this case extensive cross-linking may occur by removal of the hydrogen substituent if sodium or potassium hydroxide catalysts are used. Lees has discussed the chemistry of this reaction. However, ring--{;hain equilibrates of hydrogenmethylsiloxanes are readily prepared using n-butyllithium as catalyst with 2 % tetrahydrofuran as promoter at a reaction temperature of 273 K. 6

THE FORMATION OF SILOXANE RINGS IN IRREVERSIBLE PROCESSES

The, strained cyclic dimethylsiloxane trimer If{Mez)Si-O=H, which is intrinsically more reactive towards cationic catalysts than its higher cyclic homologues, may be polymerized in the presence of certain catalysts in non-ionizing media by essentially irreversible processes. In contrast to the analogous polymerization with weakly basic catalysts from which only linear species having narrow molecular weight distributions are obtained,7 the cationic polymerizations also include cyclization processes.

Chojnowski and co-workers 8 have studied the hexamethylcyclotri­siloxane-trifluoromethanesulphonic acid system in hydrocarbon solvents. The initiation process is represented by

Using the well-known abbreviation for the dimethylsiloxane unit 0== -Si(Mez)-O- and that for the catalyst AB, the product of the initiation process is AD3B and the kinetically controlled polymerization and cyclization processes may be written

LD~ 3 , AD3(m+ I)B

1 km+l

~+AB

If the rIngs co;;;r are strainless and are sufficiently large, their

88 P. V. WRIGHT AND MARTIN S. BEEVERS

concentration is proportional to the probability of coincidence of their termini according to the Gaussian function

P3m=(2n<:~m>J3/2 v.w

where <r~m>o is the unperturbed mean-square end-to-end distance of the 3m-meric chain, v is the volumt: element within which the - A and - B termini must be located and w is the probability that the mutual

E M c

c,-3Q o

...J

-4·0

x (=3m)

1·0 1·5 Log 3m

FIG. 1. Kinetically controlled (e) and equilibrium (0) distributions of dimethylsiloxane cyclics r:.::"f>? ( == L~D"i,;!) in CIJ? polymerizates at 40 weight % in solution: e, In heptane ·at 303 K with CF3S03H (7 x 1O-4 molkg- 1) at 30 % ~ conversion; 0, in CH 2Ci2 with CF 3S03H (equilibrium CI5.? concen­trations are multiplied by 0·3). (Reproduced from Ref. 8 by courtesy of the

publishers, Hiithig and Wepf Verlag, Basle.)

PREPARATION OF CYCLIC POLYSILOXANES 89

orientations of - A and - B will allow cyclization. Hence, if propagation is much faster than cyclization (kp[D31 » km )

[~l _ <r~(m+ 1)0 _ (m + 1 )3/2 [LD~L 1)1 - <r~m)o - m

Thus, for solely kinetic control, at any stage in the conversion of D 3m,

log [LD"3Jl vs. log 3m has a gradient of - 3/2. This gradient distinguishes the kinetically controlled cyclization reaction from the equilibrium process for which the gradient of the analogous plot is - 5/2 as predicted by theory and observed by experiment (see Chapter 1).

..... E (')·2·5 .s en o -I

-3,5

6 9

x(=3ml

12 15 18 24 30

1·0 1·5 Log 3m

FIG. 2. Distributions of dimethylsiloxane cyclics CD? (=~ in LD;=J polymerizates at 303 K and 30 % ~conversion: ., In benzene with CF 3S03H (7 x 1O- 4 molkg- 1); 0, in CH 2CI 2 with SnCI4 (5·2 x 1O- 3molkg- 1) + H 20 (2·1 X 10- 2 mol kg-I); .A., in CH 2CI2 with CF 3S03H (7·0 x 1O- 4 molkg- 1): 0, in CH2CI2 with Ph3C+SbF6"' (1·0 x 1O- 4 molkg- 1)+HCI (l x 1O- 1 molkg- 1).

(Adapted from Ref. 8 by courtesy of the publishers, Hiithig and Wepf Verlag, Basle.)

90 P. V. WRIGHT AND MARTIN S. BEEVERS

The results of gas chromatographic analyses of \ dimethylsiloxane systems by Chojnowski and co-workers show that in the system ~ (40 % solution in heptane at 30°C) with CF 3S03H (7 x 10 - 4 mol kg - 1) at 30 % ~ conversion, cyclization proceeds with mainly kinetic control (see Fig. 1). The gradient of log [LDT,;D vs. log 3m is -1·7 for 3m ~ 18 in good agreement with the scheme outlined above. The pronounced minimum in the cyclic distribution at 3m = 12 is common to both kinetic and equilibrium distributions. The steeper decline of the log-log plot in the equilibrium case is clearly traced to the possibility of the cyclic reopening at any of the siloxane bonds. Thus, according to the results of Chojnowski and co-workers for 30 % monomer conversions, significantly greater concentrations of rings may be produced by kinetically controlled processes, given sufficient ~ conversion, than occur in the equilibrated systems. This should be particularly true for the larger cyclics as is evident from the divergence oflog [LD3,,!1 vs.log 3m plots for each type of process.

Other cationic catalyst systems investigated by Chojnowski and co­workers 8 such as CF 3S03H in dichloromethane solvent, SnCl4 jH2 0j CH 2Cl 2 or Ph3C+ SbFi jHCljCH2Cl2 give gradients of -2·8 in the log [LD3dl vs. log 3m plots for 3m ~ 18 (see Fig. 2). For these systems the steeper gradients are attributed to the intervention of ring opening reactions which give rise to the appearance of cyc1ics LD3J+ 1 and Ln;J+ 2. However, non-equilibrium values are reported for the cyclics 3m < 15 and the minimum at 3m = 12 has been virtually eliminated when these catalyst systems are used. These variations in the smaller cyclic concentrations are attributed by Chojnowski and co-workers to differences in the location and orientation correlation factors for the short chain termini (v and w) between different catalyst systems. For larger chains AD3mB these differences disappear as the chains become approximately Gaussian in their conformational behaviour.

THE CRITICAL CONCENTRATION IN POL YSILOXANE EQUILIBRATES

Although kinetic processes may generate greater concentrations of siloxane cyclics, the equilibrates are more fully understood. They give rise to linear distributions which conform sufficiently closely to the most probable (Flory) distribution for high extents of reaction so allowing a precise estimate to be made of relative amounts of cyclic and linear species within a given region of the total distribution. The proportion of ring

PREPARATION OF CYCLIC POLYSILOXANES 91

species in the equilibrate is maximized if the equilibration is carried out at a concentration as close as possible to the critical concentration at which, according to theory,9 the siloxane material consists entirely of rings. At the same time, the molecular weight of the chains should be as high as possible so minimizing the region of overlap in ring and chain distributions.

According to the theory of Jacobson and Stockmayer,9 which has been discussed in Chapter 1, the molar cyclization equilibrium constant Kx for an x-meric ring CR? is given by

Kx = [CR;J]/px

Wx(O)

2NAx

(I)

(2)

where p is the extent of reaction of functional groups in the linear fraction of the equilibrate, Wx(O) is the probability of end-to-end closure for an x­meric chain and N A is Avagadro's constant. The weight fraction of rings in the total siloxane, Wr' is given by '"

oc

Wr = L>CR;J]·X.Mo/C (3)

x=4

where Mo is the molar mass per siloxane repeat unit (kg mol-I) and C is the siloxane concentration in kg dm - 3. In eqn (3) the small temperature­dependent concentration of the trimeric ring is ignored.

Assuming a Gaussian form for Wx(r) (where r is the end-to-end vector of the x-meric chain) then

Wx(O) = Cn.c~.2xI2 y/2 (4)

C x( = < r~ > 0/2xf2) is the so-called characteristic ratio of the x-meric siloxane chain, <r; >0 is its unperturbed mean-square end-to-end distance and I is the length of a siloxane bond (0·164 nm).

For high extents of reaction in the linear fraction of the equilibrate (p = 1), eqns (l}-(4) give for the weight fraction of rings

(5)

Theoretical values ofwr are readily calculated if the dependence of Cx on

92 P. V. WRIGHT AND MARTIN S. BEEVERS

X IS known. For poly(dimethylsiloxane) the required values may be obtained by computation using the Flory, Crescenzi and Mark I 0

rotational isomeric state model for this chain. For x> c. 18, Cx may be assumed to have acquired its asymptotic value of 6·7 so that

and x x>c.18 K OCX- 5/2 }

wr =constant.Ix- 3 / 2 -

Since

00 I X- 3 /2 = 2·612

x=1

(see Ref. 9) values for Wr (x = 4 --+ (0) may be evaluated. The analytical methods described below allow a comparison between the

theoretical proportions of rings calculated from eqn (5) and estimates based on extrapolations of lower cyclic concentrations measured by gas-liquid chromatography (GLC)6 or total estimates from gel permeation chromatography (G PC). 11

For undiluted poly(dimethylsiloxane) c = 0·898 kg dm - 3 and the calculated value of Wr is 0·16. This may be compared with the extrapolated experimental value of 0·18 given in Table 1. The latter is the sum of experimental results for x = 4 --+ 17 and the calculated value for x = 18 --+ 00. The calculation was performed by assuming that Cx = CIS is constant for x = 18 --+ 00, CIS being obtained from the experimental Kis .

The extrapolated experimental value is in excellent agreement with the total estimate from G PC data of 0·180.

TABLE 1 Weight Fractions of Cydics in Undiluted Polysiloxane Equilibrates Having High

Molecular Weight Chains (p = 1)

Substituent R in Temperature x=3-6 x= 7-18 x = 19-00· Total ~R( CH 3)Si0--H (K)

H 273 0·055 0·024 0·046 0·125 CH3 383 0·116 0·020 0·047 0·183 CH 3CH l 383 0·198 0·031 0·039 0·258 CH 3CHl CHl 383 0·299 CF3CHl CH2 383 0·776 0·024 0·027 0·827

• Computed by extrapolation of experimental K18 values assuming Kx oc x- l · S (see text).

PREPARATION OF CYCLIC POLYSILOXANES 93

The critical concentration at which, in an inert solvent, the siloxane consists entirely of ring species is then c. 0·898wr kg dm - 3 for dimethylsiloxane. The critical concentrations predicted from theoretical and experimental Wr values are thus 0·144 and 0·162 kg dm - 3, respectively.

However, despite the apparently acceptable accord between theory and experiment for poly(dimethylsiloxane), the discussions in the following sections demonstrate that eqn (5) is totally unsatisfactory as a basis for the prediction of critical concentrations in polysiloxane systems. The theory on which eqn (5) is based has proved of considerable value in understanding the conformational behaviour of long polysiloxane chains as reflected in the equilibrium concentrations of the large :cyclics. However, the latter constitute only a minor proportion of the total weight of the cyclic populations. Gross deviations from the theoretical concentrations of the small cyclics are caused by the effects of alkyl substituents on the silicon atoms and by thermodynamic interactions in good solvents. Thus observed critical concentrations depart considerably from those predicted by the use of eqn (5).

The Influence of Substituents on Cyclic Formation The apparently acceptable discrepancy between theory and experiment for the polydimethylsiloxane system is seen to be fortuitous when the trend of experimental wr values for undiluted equilibrates of several methylsi­loxanes, given in Table 1, is considered. The molar cyclization equilibrium constants for these systems, determined by Wright and Semlyen,6 are given in Fig. 3. With increase in size of substituent, there is a decrease in K 18' This reflects a corresponding increase in C 18 for the more conformationally restricted chains so that wr values calculated according to eqn (5) will decrease with increase in substituent size. The characteristic ratios C 18

obtained from the KI8 values (eqns (2) and (4» for the series (R(CH3)SiO) I 8

are 5·4 for R = H at 273 K, 6·8 for R = CH3 at 383 K and 8·6 for R = CH3CH2 at 383 K. These values are in good agreement with other experimental data and theoretical expectations (see discussion in Chapter 1). However, the value of 13·6 for C18 for R = CF 3CH2CH2 should be too high since the Gaussian region in which eqns (2) and (4) may be applied clearly lies beyond the experimental values in this case. The cyclization equilibria give a value of 12-4 for C20 and Buch et al. 12 obtained a value of 10·6 from intrinsic viscosity measurements on poly(trifluoropropyl­methylsiloxane) at 298 K.

However, the trend in wr calculated from eqn (5) is contrary to the experimental one. Clearly the small cyclic (x = 4-6) concentrations

94

>< liI:: en o oJ

-2-5

-3·5

P. V. WRIGHT AND MARTIN S. BEEVERS

0-6 Log x

FIG. 3. Molar cyclization equilibrium constants for siloxanes [R(CH 3)SiOlx in undiluted equilibrates at 273 K for R = Hand 373 K for the other systems: D, For R = H; 0, for R=CH 3 ; ., for R =CH 3CH2; ~,for R =CH 3CH2CH2; f::", for CF 3CH2CH2' (Reproduced from Ref. 6 by permission of the publishers,

Butterworth & Co. Ltd, Guildford, UK.)

dominate in determining the observed trend in total cyclic concentrations. Thus the data in Table I predict that the methylhydrogensiloxane system should be diluted to approximately 90 volume % of inert solvent in order to attain the critical dilution condition whereas only c. 15 % of solvent is required to bring about the conversion of the methyltrifluoropropylsi­loxane system entirely to rings. These trends in the observed critical

PREPARATION OF CYCLIC POLYSILOXANES 95

concentrations for substituted siloxanes are illustrated in Fig. 4. (The equilibration temperature of 273 K for the hydrogenmethylsiloxane system in the Table 1 and Figs 3 and 4 is 110 K below that for the other systems in the series. The lower reaction temperature is necessary to minimize the cross-linking reaction in this system referred to earlier. However, the lower reaction temperature should not invalidate a meaningful comparison with the other siloxanes. The equilibrium

100 60 20 Volume % siloxane

FIG. 4. Observed trends in the weight fraction of cyclics, Wr' vs. volume percent siloxane for equilibrates of alkylmethylsiloxanes in toluene at 383 K: I,

R=CF 3CH2CH2 ; 2, R=C6 H5 ; 3, R=CH 3CH 2 ; 4, R=CH 3 .

concentrations of rings x ~ 4 should be particularly independent of temperature for siloxanes having small substituents on silicon in view of the small energy differences between accessible conformations. The con­centrations of cyclics in the dimethylsiloxanes are independent of temperature within the error of the experimental techniques 13 and the same should be true of the methylhydrogensiloxanes.)

The marked enhancement of the equilibrium concentrations of small cyclics in siloxane systems having large substituents on silicon, or, expressed conversely, the reduced polymerizability of these systems, can be explained if the conformations of the small rings on the one hand and the

96 P. V. WRIGHT AND MARTIN S. BEEVERS

conformations accessible to the long chains on the other are considered. The conformational behaviour of the dimethylsiloxanes has been discussed in some detail in Chapter 1. In that discussion it was pointed out that the conformations of small siloxane rings include sequences of gauche rotations of opposite sign in profusion. For example, the 'crown form' of the cyclic tetramer is given by (g + g -)4 rotational states about the siloxane bonds. However, g+ g- states centred on silicon atoms are forbidden in the FCM model 10 for long chains of polydimethylsiloxane and are given only low statistical weights when centred on oxygen atoms. These con­siderations must also apply to long chains of polysiloxanes having substituents larger than methyl groups. Furthermore, the larger molecular dimensions for these chains, which are apparent in intrinsic viscosity measurements 12 as well as in the molar cyclization equilibrium constants for large rings (x> c. 18), are consistent with reduced probabilities of all­trans sequences with increase in size of the substituents. However, such conformational restrictions do not hinder the formation of the cyclic tetramer to the same degree. As Fig. 5 shows, quite large alkyl substituents

FIG. 5. The tetrameric dialkylsiloxane cyclic (RR'SiO)4 in a 'crown' (g+g-g+g-g+g-) conformation: e, Silicon atoms; ®, oxygen atoms; 0, alkyl

. substituents.

PREPARATION OF CYCLIC POLYSILOXANES 97

can be accommodated at the 'corners' of the molecule. Accommodation of the substituents in this way is clearly a feature of g + g - states centred on oxygens and ensures maximum separation of substituents on neighbouring silicons placing them on the outside of the ring. The net reduction in the severity of side group interactions relative to those in open chains should also occur in the cyclic pentamer and hexamer. These cyclics also include g + g - sequences in their conformations which place the substituents on the outside of the rings.

Thermodynamic Influences on the Position of Equilibrium The differences in the fundamental conformational features of the small cyclics and the higher molecular weight material are reflected in corresponding differences in physical properties such as density and refractive index. The densities of dimethylsiloxane cyclic tetramer, pentamer and high polymer are 0·853, 0·866 and 0·898 g cm - 3,

respectively, at 383 K. For the cyclics the asymptotic value is virtually attained at x - 9. Elementary thermodynamic considerations therefore suggest that the application of pressure to the equilibrating system should shift the position of equilibrium towards the more dense high molecular weight product. Indeed, equilibrations carried out in a pressure 'bomb,14 show that application of c. 3500 atmospheres pressure brings about a reduction of approximately 25 % in the weight fraction of small cyclics in dimethylsiloxane systems. This corresponds to a reduction of approxi­mately 15 % in the molar concentration of the cyclics at 3500 atmospheres.

For separation of fractions of high molecular weight cyclic material the equilibrate is ideally prepared in solution at a concentration only slightly in excess of the critical concentration. If the molecular weight of the chain fraction is as high as possible the region of overlap in the distributions of chains and cyclics is reduced to a minimum. Thus the use of pure components and trace amounts of siloxanolate initiator in the presence of small quantities of a promoter such as diethylene glycol dimethyl ether (diglyme) should ensure high molecular weight chains. However, the differences in the thermodynamic character of the small cyclics on the one hand and the high polymer on the other indicate that the critical concentration itself should depend on the thermodynamic character of the solvent.

Toluene and xylene have commonly been used as solvents for dimethylsiloxane equilibrations. These are thermodynamically 'good' solvents for poly(dimethylsiloxane). For investigations of equilibration

98 P. V. WRIGHT AND MARTIN S. BEEVERS

under 'theta' ((}) conditions, diglyme was chosen by one of the authors 11 as a 'poor' solvent for the system in view of its auxiliary role as a reaction promoter. However, the small cyclics function as 'good' solvent components of a solvent mixture which includes the inert solvent. At concentrations close to the critical concentration, the high polymer (chains and large rings) is in dilute solution in the mixed solvent at a concentration amounting to only several per cent by weight even though the tota"! siloxane is more than 20 % by weight of the mixture. Theta conditions were therefore established using a solven t mixture consisting of the inert solvent and the small cyclics (x = 4-6) in their approximate equilibrate concentrations. The usual method of observing the temperature of phase separation as a function of high polymer concentration and molecular weight was then employed. Using diglyme as inert solvent in the mixture the (}-temperature was estimated to be 60°C.

TABLE 2 Experimental Molar Cyclization Equilibrium Constants Kx for Dimethylsiloxane

Cyclics c:=r>? in Undiluted and Diluted Equilibrates·

x

4 5 6 7

Undiluted 383 K 898g dm- 3

0·19 0·09 0·03 0·0092

• Accuracy estimated to be ± 10%.

Diglyme solution 333 K; 212gdm- 3

0·21 0·13 0·036 0·0085

Toluene solution 383 K; 224g dm- 3

0·30 0·15 0·044 0·010

The data presented in Table 2 and Fig. 6 illustrate the dependence of the position of equilibrium in dimethylsiloxane systems on the solvent. Following detailed analyses 11 of equilibrates in the bulk, in diglyme solution and in solution in toluene, it is concluded that in the undiluted system (898 g dm - 3 at 383 K) there are 736 g dm - 3 of chains and 162 g dm - 3 of rings; in the 8-s01vent at 218 g dm - 3 of total siloxane there are 55 g dm - 3 of chains and 163 g dm - 3 of rings and in toluene at 224 g dm - 3 there are 8 g dm - 3 of chains and 216 g dm - 3 of rings. The estimated critical concentrations in the bulk (162 g dm - 3) and the 8-s01vent (163 g dm - 3) are apparently in excellent agreement as the Jacobson and Stockmayer theory would predict. Although the close agreement between these two results conceals a mutual cancellation of slightly higher concentrations of small

-0'5

-1,5

)( ~

01 o ....I

-2'5

PREPARATION OF CYCLIC POLYSILOXANES

4

, , , , , , , ,

No of units, X 6 8 10 12 14 16

0'8

, , , , ,

Log X

, , ,

1·0

, , , , , , , "

1·2

99

FIG. 6. Molar cyc1ization equilibrium constants for dimethylsiloxanes: e, in toluene 224gdm- 3 , 383K;14 A, in diglyme 218gdm- 3 , 333K; 0, undiluted 383 K. Dashed line denotes calculated values according to eqns (2) and (4). (Adapted from Ref. II by courtesy of the publishers, John Wiley & Sons, Inc., New

York.)

cyclics, and slightly lower concentrations oflarge ones in diglyme, the total cyclic concentrations in the bulk and the poor solvent are significantly lower than those in the good solvent (critical concentration at 216 g dm - 3). The larger figure in the latter case is due entirely to an enhancement in the concentrations of small cyclics. As Fig. 6 shows, the solvent effect in toluene appears to persist down to x = 12 although it is small for x > 7. The large cyclics x> c. 12 are present in lower concentrations than in the bulk or in diglyme.

The enhancement of small dimethylsiloxane cyclic concentrations in

100 P. V. WRIGHT AND MARTIN S. BEEVERS

equilibrates in xylene solutions have been observed by Carmichael and co­workers. 15 If a crude distinction is made between the accumulated concentrations of small cyclic 'monomer' and the remainder of the siloxane as 'polymer', the monomer ~ polymer equilibrium may be expressed in terms of their respective volume fractions, Vm and vp in the aromatic hydrocarbon (good) solvents by

vm '" 0·21 - O·lvp

and in diglyme at 333 K by

vm '" 0·15 - 0'03vp

The solvent effects may be interpreted 14 semi-quantitatively in an approach similar to that of Ivin and Leonard 16 by using plausible values of the semi-empirical parameters which appear in the Flory-Huggins theory for polymer solutions.

Similar enhancements of the small ring equilibrium concentrations have been observed in the hydrogen methyl- and ethylmethyl-siloxane systems. 14

The Distribution and Configurational Isomers of Cyclic Phenylmethylsiloxanes Beevers and Semlyen 1 7 and Clarson and Semlyen 18 have analysed equilibrates ofphenylmethylsiloxanes using gas-liquid and gel permeation chromatography. The results of both investigations indicate that the influence of the phenyl group on the position of equilibrium in both undiluted and diluted methylsiloxanes is approximately the same as that of an ethyl group. The distribution of cyclic ethylmethylsiloxanes (see Fig. 3) over the range of x = 4-c. 12 is, within the error of the experiments, the same as the distribution for the phenylmethylsiloxanes determined by Beevers and Semlyen (Fig. 7). The Kx values for the larger rings determined by the latter workers prescribe a value of 10·4 for the characteristic ratio of poly(phenylmethylsiloxane) chains in solution in toluene. From GPC data (see below) Clarson and Semi yen estimated a characteristic ratio of 8·8 in agreement with the value obtained by Buch, Klimisch and lohannson 19

from intrinsic viscosity measurements in diisobutylamine at 303 K. These results are close to that obtained from cyclization equilibria for the ethylmethylsiloxane system (C 18 = 8,6). The degree of enhancement of small cyclic concentrations in solution equilibrates is also similar in the two systems and the behaviour of the phenyl substituent in this respect is shown in Fig. 4.

However, each species of cyclic and linear siloxanes (R(Me)SiO)x (R =I Me) of a given· degree of polymerization are present as a mixture of

PREPARATION OF CYCLIC POLYSILOXANES 101

o o

2

LogKx

3

4

0.5 LogX 1.5

FIG. 7. Log Kx vs.logx for phenylmethylsiloxanes at 383 K: ., undiluted; 0, in toluene solution close to the critical concentration (estimated to be 52 volume %

siloxane).

configurational isomers. As with vinyl polymers the substituents R and Me may be placed along the chain in isotactic, syndiotactic or atactic configurations. The separation and identification of these stereoisomers has received relatively little attention although several groups have studied the isomers of the lower cyclics in the phenylmethylsiloxane system using both gas-liquid chromatography17.20.21 and nuclear magnetic resonance spectroscopy. 2 2

Beevers and Semi yen 17 have resolved the isomers of the cyclic trimer, tetramer and pentamer in undiluted phenylmethylsiloxane equilibrates (see Fig. 8) and made assignments to the chromatographic peaks (Fig. 9). The relative amounts of the various isomers of a given degree of polymerization

102 P. V. WRIGHT AND MARTIN S. BEEVERS

FIG. 8. Structural representations of the stereoisomers of the cyclic trimer, tetramer and pentamer of the phenylmethylsiloxanes. Small filled circles are Si atoms. Small open circles are oxygen atoms. Large open and filled circles are phenyl

groups above and below the plane of the ring, respectively.

PREPARATION OF CYCLIC POLYSILOXANES

A 2

min

3 4 5 6 hr

c

7 I~/,I

12 ~-1~4--------~------1~6~--~h~r--Elution Time/min

103

FIG. 9. Gas-liquid chromatogram of stereoisomers of the cyclic trimer, tetramer and pentamer of the phenylmethylsiloxanes with peaks assigned to the isomers

identified in Fig. 8.

are given in Table 3 together with the expected weight fractions for random (atactic) placements of groups. Also included in Table 3 are experimental data for a solution equilibrate (47 % in toluene) and for non-equilibrium chlorosilane hydrolysis and pyrolysis preparations. The configurational isomers are clearly present in proportions close to those predicted from random cyclization of an atactic polysiloxane in each preparation.

Beevers and Semlyen also measured Kx values for the various

104 P. V. WRIGHT AND MARTIN S. BEEVERS

TABLE 3 Weight Fractions of Configurational Isomers of Cyclic Phenylmethylsiloxanes

Stereoisomers Weight fraction of cyclic present as a particular defined in isomer in different mixtures·

Fig. 8 A B C D E

Trimer I 0·25 0·22 0·24 0·26 0·28 Trimer 2 0·75 0·78 0·76 0·74 0·72 Tetramer 3 0·125 0·08 0·10 0·16 0·16 Tetramer 4 0·500 0·49 0·49 0·49 0·51 Tetramer 5 0·250 0·25 0·23 0·20 0·24 Tetramer 6 0·125 0·18 0·18 0·16 0·10 Pentamer 7 0·0625 0·06 0·05 0·10 0·09 Pentamer 8b 0·3125 0·30 Pentamer 9b 0·3125 0·30 0·62 0·60 0·62 Pentamer 10 0·3125 0·34 0·33 0·30 0·29

Reproduced from Ref. 17 by courtesy of the publishers, Butterworth & Co. Ltd, Guildford, UK. • A, This is the isomer ratio that would result from the random cyclization of an atactic polymer; B, undiluted equilibrate at 383 K; C, solution equilibrate at 383 K containing 47 % by volume toluene. Linear polymer is present at concentrations of less than 5 % by weight; D, cyclics prepared by hydrolysing phenylmethyldichloro­silane; E, cyclics prepared by the pyrolysis of poly(phenylmethylsiloxane). High molecular weight polymer was chain stopped bytrimethylchlorosilane. The polymer suffered a loss in weight of only o· 3 % after 12 h at 513 K under 0·05 mmHg. It decomposed at 668 K under 3 x 10 - 3 mmHg to give a distillate consisting of 37 % trimer, 56% tetramer and 7 % pentamer. The residue represented 28·6 % by weight of the starting material. b Temperature programming was used for determining some of the isomer ratios of the cyclic pentamer. Under these conditions isomers 8 and 9 were not completely resolved.

stereoisomers over a range of temperature 383-478 K. They determined the enthalpy of formation I1B; for each isomer according to

Thus the stereoisomers I and 2 of the cyclic trimer have I1B~ values of 27 and 22 kJ mol-l, respectively. These results are largely attributed to Si-O- skeletal angle deformations in closing the ring although some steric interaction between phenyl groups appears to contribute, particu­larly in isomer 1. The cyclic tetramer 3 has I1B:: = 8 kJ mol- 1 which is also attributed to crowding of phenyl groups. However, the remaining isomers were all found to be strain free.

PREPARATION OF CYCLIC POLYSILOXANES 105

PREPARATION OF SILOXANE CYCLIC RESIDUES

A gel permeation chromatogram of an unfractionated dimethylsiloxane equilibrate in toluene (224gdm- 3 , 383 K) is shown in Fig. 10. Fractiona­tion of an equilibrate is preceded by quenching of the chemical reactions. This is readily achieved in base catalysed solution equilibrates by addition of trace quantities of acetic acid. While hydrolysing the silanolate termini the weak acid is inactive towards the siloxane bonds. Undiluted

Cyclics

1.0 0.8 0.6 0.4 Kd

Linear polymer

'"

0.2 0.0

FIG. 10. Gel permeation chromatogram of an unfractionated equilibrate in toluene solution; 224 g dm - 3, 383 K. (Adapted from Ref. II by courtesy of the

publishers, John Wiley & Sons, Inc., New York.)

equilibrates maybe cooled to ambient temperature and dissolved in the presence of acetic acid. In the absence of an equilibration promoter, the rate of reaction at ambient temperature as deduced from the temperature dependence of the rate constant 23 is sufficiently low to ensure that negligible reorganizaton of the dissolved siloxane species occurs before the reaction is terminated.

If unfractionated equilibrate solutions are injected into a gas~liquid chromatograph, the cyclic distribution may be monitored well into the macrocyclic range (x < c. 50) as shown in Fig. 11. However, the G LC instrument has a reduced response to the cyclics in the presence of high

106

40 I

P. V. WRIGHT AND MARTIN S. BEEVERS

30 I

20 I

- elution time

10 I

FIG. II. Gas-liquid chromatogram (a section) of an unfractionated cyclic extract from an undiluted polydimethylsiloxane equilibrate. (100 mesh Embacel solid support coated with 8 % OV-17 stationary phase. Temperature programming at

4K min-I.)

IS

5

6

4

15 10 5 Elut ion Time min.

FIG. 12. High-performance liquid chromatogram of a phenylmethylsiloxane cyclic extract. (Whatman Partisil PXSIO/25 ODS-2 column with

methanol-tetrahydrofuran solvent mixtures.)

a)

0.8

b)

0.8

PREPARATION OF CYCLIC POLYSILOXANES

X= ,

! /

I i i

/··2 ./

._/""

X=

0.6

100 200

I

0.4

100 I

0.4

0.2

200 I

0.2

107

FIG. 13. Gel permeation chromatograms of dimethylsiloxane macrocyclic residues from (a) an undiluted equilibrate at 383 K and (b) an equilibrate in toluene at 224 g dm - 3, 383 K. Columns supplied by Waters Associates Ltd, Harrow, UK, with gel porosities of 250, 1000, 3000, 3000 and 3000 A. The solid line (I) is the experimental chromatogram; the dotted line (2) is the chromatogram corrected for imperfect resolution; the dashed line {3) is the calculated and estimated chain distribution (see text). (Adapted from Ref. 11 by courtesy of the publishers, John

Wiley & Sons, Inc., New York.)

108 P. V. WRIGHT AND MARTIN S. BEEVERS

polymer. A more precise determination of the overall composition of the equilibrate is obtained by fractionation prior to instrumental analysis.

In the analysis of dimethylsiloxane equilibrates, solvents and cyclics x = 3 to c. 10 are readily removed by fractional distillation at reduced pressures. Each cyclic fraction may then be individually weighed and analysed by gas-liquid chromatography.

High-performance liquid chromatography has been used for the analysis of cyclic and linear phenylmethyllsiloxanes of low molar mass. 18 Figure 12 shows an HPLC tracing of cyclic phenylmethylsiloxanes which was obtained using a stationary phase of silica coated with an octadecyl residue and solvent mixtures of methanol-tetrahydrofuran.

Following separation of the small cyclics, the high molecular weight linear species coalesce into a separate phase from the macrocyclics on cooling of solutions in hot acetone at approximately 10 % concentration. The supernatant contains higher molecular weight material if acetone-butanone solutions are used. After exhaustive extraction of the high molecular weight precipitate the combined supernatant extracts of macrocyclics may be analysed by gel permeation chromatography.

Without further fractionation, a G PC tracing of the macrocyclic residue appears as shown in Fig. 13. However, further fractionation of the macro­cyclic residue is achieved using short-path molecular distillation 6 ,13 or by solvent fractionation from water-acetone mixtures. The gas-liquid chromatographic analysis of fractions obtained by short-path molecular distillation gave Kx values (x:::;; 40) for the dimethylsiloxane system 6 which are in substantial agreement with the more extensive data obtained by gel permeation chromatography. 11

ANALYTICAL GEL PERMEATION CHROMATOGRAPHY

Calibration plots for gel permeation chromatography of cyclic and linear dimethylsiloxanes in toluene eluent are shown in Fig. 14. The cyclic and linear fractions were prepared by fractional distillation, short-path molecular distillation and solvent fractionation and their number-average molar masses were determined 11,14 by gas-liquid chromatography, vapour pressure osmometry and membrane osmometry. Weight average molecular weights of narrow fractions were determined by light scattering. 24 The data are presented as 10g1o (molecular weight) versus the distribution coefficient Kd defined by

PREPARATION OF CYCLIC POLYSILOXANES \09

where Va is the volume of the mobile phase, Vi is the total internal volume of the system (the total volume of the pores of the gel) and Ve is the elution volume for cyclic species of molecular weight Mr and linear species of molar mass MI. For cyclic dimethylsiloxanes Lf(CH 3}zSiO=B the data cover the size range x = 4-250. The slopes of the cyclic and linear plots are apparently identical over this range. However, the plots are displaced such that at constant Kd

(Mr/MI)Kd = 1·24

The calibration plot for ethylmethylsiloxane cyclics is indistinguishable

0.2 0.4 0.6 0.8 Kd

FIG. 14. Calibration plots for gel permeation chromatography of siloxane cyclic and linear molecules: ., dimethylsiloxane cyclics; 0, linear dimethylsiloxane molecules; /:" ethylmethylsiloxane cyclics. Columns supplied by Waters Associates

Ltd, Harrow, UK, with gel porosities of 250, 1000,3000,3000 and 3000A.

from that for dimethylsiloxane cyclics when plotted according to molar mass as in Fig. 14.

It is significant that a similar ratio 1·2 for linear and cyclic polyethers has been reported by Rentsch and Schulz. 2s

It is pertinent to consider this ratio in the light of theories proposed by Casassa26 and others27 - 29 which seek to correlate GPC elution data of linear and branched molecules. Casassa calculated distribution coefficients assuming unperturbed random flight molecules and various geometrical

110 P. V. WRIGHT AND MARTIN S. BEEVERS

forms of the micro pores in the column packing. His theory prescribes a general dependence of Kd on the variable

q = «s2>otla2)1/2g1/6 (6)

where g = <s2>Obr/<s2>O'; <s2>obr and <S2>O' are the unperturbed mean­square radii of gyration of the non-linear and linear species, respectively, of the same molar mass and a is half the cross-sectional dimension of a micropore (e.g. the radius of a sphere or cylinder). Drawing an appropriate analogy between the branched molecules specifically considered in Casassa's theory and the cyclic-linear correlation of distribution coefficients considered in the present context, one obtains for the cyclic molecules of molar mass Mr

(7)

where <S2>01' is the unperturbed mean square radius of gyration for the linear molecule having molar mass M" = Mr. For the linear molecule of molar mass M,

q, = «s2>o,la2)1 /2

At the same Kd , qr = q, and noting that for long unperturbed chains

<S2 >otl M, = constant

one obtains

(8)

(9)

(10)

Theoretical predictions by Zimm and Stockmayer30 and SOlc31 indicate that for unperturbed random flight cyclics and linear species of any polymerization degree

(11)

Recently, Higgins, Dodgson and Semlyen 32 used neutron scattering to measure the z-average radii of gyration of cyclic and linear dimethylsi­loxanes in solution in the thermodynamically 'good' solvent benzene-d6 .

The molecular size range of c. 150-500 skeletal bonds was covered in the experiments and the experimental result gr = 0·53 ± 0·05 confirms the theoretical predictions (see Chapter 5).

Substituting eqn (11) into eqn (10) gives

(12)

This results in excellent agreement with the observed ratio although

PREPARATION OF CYCLIC POLYSILOXANES III

agreement over the full range of molecular size is perhaps surprising. The theory embodied in eqns (6)-(12) assumes/long polymer .chains in a ()­solvent and is not expected to apply rigorously to the smaller cyclic and linear siloxanes. Furthermore, the G PC calibration in Fig. 14 is for toluene eluent, a 'good' solvent for polydimethylsiloxanes, so that for sufficiently high molar mass any effects of excluded volume on the elution behaviour should become apparent. Cyclization equilibria 11 and intrinsic viscosity data 33 show that excluded volume effects become detectable for polysiloxane rings and chains in toluene for x> c. 50. Any significant difference in the relative degree of molecular expansion for rings and linear molecules of the same degree of polymerization should bring about a change in the ratio (Mrl Ml)Kd'

Benoit and co-workers27 •28 and Dawkins and Hemming29 have proposed that molecules of the same hydrodynamic volume given by

(13)

should have the same distribution coefficient. In eqn (13) [17] is the intrinsic viscosity and K and a are Mark-Houwink

parameters for a polymer-solvent-temperature system. Dodgson and Semlyen 33 have determined K" a r and Kl and al (Mark-Houwink parameters for cyclics and linears, respectively) for poly(dimethyl­siloxanes) having up to 500 skeletal bonds in a poor solvent (butanone at 293 K) and in toluene at 298 K.

In butanone at 293K, Kr =I'92xI0- 4, K1=2'8xI0- 4 and a r = a l = 0·58. Equating hydrodynamic volumes for rings and linear species one obtains

This ratio is also in excellent agreement with the corresponding GPC ratio. However, the close accord with Casassa's prediction for unperturbed chains can be appreciated if Zimm and Kilb's 34 approximate expression for the intrinsic viscosity of branched molecules in a ()-solvent is recast for cyclic molecules thus:

(15)

where <I> is the Flory constant. In toluene at 298 K however, Dodgson and Semlyen observed a change

in gradient of log [17] vs. log M at x = c. 25 for both dimethylsiloxane rings and open chains. From the higher molecular weight, steeper region of the plot they found that Kr =K1 =4·21 x 10- 3 and ar =O·77 and a/=0·83.

112 P. V. WRIGHT AND MARTIN S. BEEVERS

There is therefore no unique ratio (Mrl M/)[~]M for x> c. 25 in toluene. In general

M~'77 = Mt'S3 [I]]M constant (toluene 298 K) (16)

Letting Mr = 10000 (x = 135), (MrIM')[~IM = 1·35, and for Mr = 20000 (x = 270), (Mrl M/)[~]M = 1· 38.

Thus the theory of the dependence of G PC elution volume on hydrodynamic volume in conjunction with Dodgson and Semi yen's intrinsic viscosity data predicts a progressive divergence of G PC calibration plots for cyclics and linear species in toluene with increase in molecular weight. The GPC calibration data of Fig. 14 are not in conflict with this predicted trend although the magnitude of the ratio (MrIM/)[~IM is larger than the observed (MrIM,)Kd at least for molar mass < c. 15000 (x = c. 200) over which range the latter apparently adopts the unperturbed value of 1·24. Further GPC elution data are required to clarify the factors which determine the distribution coefficients. However, the observation of Edwards et al. 3 5 that the ratio of diffusion coefficients of cyclic and linear dimethylsiloxanes of high molar mass is insensitive to changes in solvent and to the effects of chain expansion may have some bearing on GPC elution behaviour.

From the calibration data in Fig. 14 it is clear that the unperturbed (Mrl M/)Kd ratio can be applied accurately at the lowest molar masses of the GPC calibration. The statistical principles embodied in eqns (6)-(16) are generally considered inapplicable to small molecules. Nevertheless, although the ratio <S2>o'! M, (eqn (9)) is constant only in the limit of high chain lengths, over the range M, to Ml' ( = Mr) it changes by no more than a few per cent, even at x = 8. Furthermore, for small siloxane molecules, the substituent groups on the silicon atoms should make significant contributions to the overall dimensions. This should be particularly true for the small siloxane cyclics (x = 4--c. 9), the substituents of which are essentially confined to the outside of the siloxane backbone (see Fig. 5). Computations and molecular models indicate that the requirement for the mean-square of the radius of the cyclic dimethylsiloxane tetra mer to be approximately half that of the Icorresponding linear molecule (eqn (11)) is plausibly valid.

For cyclic siloxane tetramers Lf(RR')SiO=+J having short n-alkyl substituents, inspection of models of the puckered conformations such as that in Fig. 5 suggests that the radii should increase approximately linearly with increase in molecular weight. The elution data for the cyclic tetramers shown in Fig. 15 suggest that this is essentially true for Rand R' = H, CH3 ,

PREPARATION OF CYCLIC POLYSILOXANES

" 2.7

Log M

2.5

6

•

• 5

4 •

3

2

•

2.3~------~------~~------~~

0.80 0.85 0.90 Kd

113

FIG. 15. Log10 M vs. distribution coefficient Kd for siloxane tetrameric rings [RR'Si014 on Waters Styragel columns with nominal porosities 250, 1000, 3000, 3000 and 3000A. Numbers refer to substituent groups Rand R' as follows: I, -H, -CH3 ; 2, -CH 3 , -CH3 ; 3, -CH3 , -CH2CH 3 ; 4, -CH2CH 3 ,

-CH2CH 3 ; 5, -CH3 , -CH2CH2CH3 ; 6, -CHzCHzCH 3 , -CHzCHzCH 3 .

Solid line denotes calibration plot for dimethylsiloxane. (Reproduced from Ref. 75 by courtesy of the publishers, Ellis Horwood Ltd, Chichester, UK.)

CzHs, C3 H 7 • The combined data of Figs 14 and 15 suggest a simple procedure for universal calibration of G PC eluent volumes for substituted polysiloxanes of both linear and cyclic species having degrees of polymerization Sc. 250 (sc. 500 skeletal bonds). In general the log M versus Kd calibration plots will lie parallel to those of the dimethylsiloxane system and the plots for the linear and cyclic species will be displaced such that (Mr/M1h = 1·24. In particular, Ilog M vs. Kd for n-alkyl substituents should be dcoincident with the corresponding plots for the dimethylsiloxanes.

When plotted as log x versus K d , calibration plots for all substituents appear as a series of parallel lines. Such plots suggest a simple approximate procedure for correlation of unperturbed dimensions of the various polysiloxanes. According to eqns (7) or (8) molecules of the same species (cyclic or linear) having the same mean-square radius will elute with the same distribution coefficient. Thus substituted siloxane chains

114 P. V. WRIGHT AND MARTIN S. BEEVERS

+(RR')SiO+ of degree of polymerization i elute at the same Kd as dimethylsiloxane chains +(Me2)SiO+of degree of polymerizationj. The characteristic ratio for substituted polysiloxanes of high molar mass is defined

Ci(RR') == lim [6<S2)o;/2il2] (17) i-+ 00

At the same Kd

(18)

where <S2)Oi is the mean-square radius of the dimethylsiloxane chains. Hence

(19)

If the G PC calibration plots for linears and cyclics extrapolate to the lowest degrees of polymerization with constant displacement of their respective calibration plots, the ratio U/i]Kd is also given by elution data for the small cyclics, such as the tetramer, which are probably the most readily obtained of characterized siloxane materials. Since the n-alkyl substituents elute according to the same log M vs. Kd plots, in these cases the ratio U/i]Kd is simply the inverse ratio of the molar masses of the corresponding repeating units. This, of course, reduces the correlations to the seemingly trivial observation that the unperturbed molecular dimensions of polysiloxanes increase with the size of their substituents as cyclization equilibria indicate (Fig. 3). However, the quantitative agreement between the dimensions estimated by G PC data and those from other techniques is good. For example, GPC estimates for the characteristic ratios of phenylmethylsiloxanes18 and di(n-propyl)siloxanes are 8·8 and 11,9, respectively. Corresponding intrinsic viscosity results are 8·8 for poly(phenylmethylsiloxane)19 and 11·9 and 13-9 for poly(di(n­propyl)siloxane) in 2-pentanone at 349 K and toluene at 282 K, respectively. 3 6

Cyclic and Linear Distributions Within Macrocyclic Residues Gel permeation chromatograms of macrocyclic residues from an undiluted dimethylsiloxane equilibrate and from an equilibrate in toluene close to the critical concentration are shown in Fig. 13(a) and (b). The equilibration temperature of both equilibrates was 383 K and their overall compositions were given earlier when the influence of the thermodynamic character of the medium on the position of equilibrium was considered. The equilibrate in

PREPARA TION OF CYCLIC POL YSILOXANES 115

diglyme solution at 333 K (O-solvent), the overall composition of which was also discussed earlier, gave a macrocyclic residue which was also analysed by GPc. However, the GPC tracing of this residue 14 was essentially indistinguishable from that of the undiluted equilibrate (Fig. 13(a» and is not reproduced here.

As described previously the macrocyclic residues were prepared by exhaustive extraction from their mixtures with linear material. The first extracts were obtained using a solvent mixture (90 % acetone-IO % butanone) which extracted material of molecular weight at least at the upper limit of the GPC analysis. Subsequent extracts were obtained with a solvent mixture slightly richer in the thermodynamically more compatible component (20 % butanone). Each extract was then analysed by G PC and a minimum elution volume established above which the relative height of the GPC tracing remains constant. Thus by taking into account the GLC analyses of the lower cyclic distillate_s, it was concluded that for the tracings in Fig. 13(a) and (b) all the material eluting at 0·22:::; Kd :::; 0·63 was present in the macrocyclic residue in its equilibrium concentration.

Prior to analysis of the G PC chromatograms the tracings were corrected for imperfect chromatographic resolution by the method of Pierce and Armonas. 37

It is now well established that the weight fraction Wy of y-meric linear species in dimethylsiloxane equilibrates is given by the Flory distribution of molecular size,38 i.e.

(20)

where p is the extent of reaction. This distribution has been confirmed for acid-catalysed polysiloxanes by Scott39 and by Carmichael and Heffel. 40

However, Mazurek and co-workers41 report that in base-catalysed systems they have observed considerable enhancement of low molecular weight linear species, particularly for y = 4 and 5. In certain equilibrates the concentration of the linear tetra mer was found to be greater than that predicted by the Flory distribution by a factor of approximately 102 • They attributed such enhancement to aggregation of polar siloxane termini in cyclized pairs of the linear tetra mer :

1~

116 P. V. WRIGHT AND MARTIN S. BEEVERS

However, in equilibrates used for the preparation of macrocyclic fractions the molecular weight of the linear fraction is typically sufficiently high that low molecular weight chains cannot be detected by GLC. Thus deviations from eqn (20) should be negligible for the high molecular weight chains. Furthermore, the aggregates proposed by Mazurek and co-workers should be present in negligible quantities in the presence of reaction promoters such as diglyme.

The extent of reaction, p, for the linear component of the equilibrate was determined by measurement of the weight average molecular weight Mw( =(1 + p)/(l - p)) of the separated (major) portion of the chains. For poly(dimethylsiloxane) Mw may be obtained from intrinsic viscosity measurements using the Mark-Houwink relationship of Haug and Meyerhoff,42 viz.

(toluene 298 K)

For the undiluted, toluene and diglyme equilibrates under consideration, the extents of reaction were 0·99985, 0·9989 and 0·9989, respectively.

The linear distribution within the macrocyclic residue was then constructed on the GPC chromatogram (Fig. 13) for Kd ~ 0·22 by a procedure involving a successive approximation.

The height of the corrected chromatogram at a given Kd is the sum of the x-meric cyclic and y-meric linear contributions, i.e.

(21)

The weight concentration in the equilibrate of a y-meric chain, c y' is given by

(22)

where C is the concentration of the macrocyclic residue in the equilibrate and dlny/dKd is obtained from the gradient of the GPC calibration plot. The concentrations of the y-meric chains, cy , were calculated to a first approximation using eqn (20) and the weight of the separated linear fraction. Using eqns (21) and (22) a first approximation of hXK was then calculated for Kd ~ 0·22. Extrapolation of hYK and hXK into the region Kd < 0·22 allowed the relative proportions of cyclic and linear material which elutes in this region to be estimated to sufficient approximation for the linear distribution to be temporarily constructed on the chromatogram. Thus from this first approximation of the calculated I and estimated linear distribution an estimate of the fraction of the total linear material in the equilibrate which is present in the macrocyclic residue was obtained. For

PRE PARA nON OF CYCLIC POL YSILOXANES 117

the undiluted equilibrate (Fig. 13(a)) this amounted to c. 0·2 % and for the toluene and diglyme equilibrates the proportion was approximately c. 5 % of the total. The second approximation of the linear distribution for Kd ;;::: 0·22 was then calculated using the revised weight of linear species. Thus errors in the estimation of the amounts of chains in the macrocyclic residues imparted small errors to the calculated chain distribution for Kd ;;::: 0·22 and negligible errors to the calculated cyclic distributions over this range.

The weight concentrations of the cyclics in the equilibrate were obtained from the hXK using the equation for x-meric cyclics analogous to eqn (22). After converting weight concentrations to molar concentrations, the molar cyclization equilibrium constants Kx were then readily calculated using eqn (1).

Log Kx vs. log x for the three dimethylsiloxane equilibrat~s is plotted over the range 16::::;; x::::;; 202 in Fig. 16. The agreement with the GLC

>< ::.:: Cl o

...J

20 No of units, x

50 100 200

2·4 Log x

FIG. 16. Molar cyclization equilibrium constants for dimethylsiloxanes: e, in toluene 224 g dm -3,383 K; £, indiglyme 218 g dm -3,333 K; 0, undiluted, 383 K. Dashed line denotes calculated values according to eqns (2) and (4). (Reproduced from Ref. II by courtesy of the publishers, John Wiley & Sons, Inc., New York.)

118 P. V. WRIGHT AND MARTIN S. BEEVERS

analysis over the range accessible to this technique is good. K 16 is within 2 % of the GLC result for the undiluted system.

The gradients of the plots for the undiluted and diglyme equilibrates are -2·46 and -2·48. These gradients are in very good agreement with the gradient of - 2·5 predicted by the Jacobson and Stockmayer theory (eqns (2) and (4» or the mean gradient of -2·55 indicated by Flory and Semlyen's43 theoretical plot which takes into account the dependence of C x

on x. One of the stated conditions of eqn (4) is that the polysiloxane chains are unperturbed by excluded volume effects. These results thus represent an early confirmation of the prediction 38 that a polymer chain has the same mean dimensions in the melt as in a 8-s01vent.

Values of characteristic ratios Cx are also obtained from the plots by application of eqns (2) and (4). For the undiluted system C 100 = 7·1 and for the diglyme system C 100 = 7 ·6. The former result is only a little larger than the value of C 18 = 6·8 quoted earlier so that the characteristic ratios obtained from the G PC analysis are close to the asymptotic values. This observation is thus consistent with the form of Cx vs. x as calculated by Flory and Semlyen43 although their calculations prescribed a value of 6-4 for CCfJ as observed by intrinsic viscosity measurements in butanone at 293 K.10 However, from intrinsic viscosity measurements in a CsF s: CCl4 F 2 solvent mixture at 295·5 K (Ref. 10) a value of Coo = 7·6 was obtained. The marked difference in these two results was ascribed to the lower cohesive energy and low dielectric constant of the halocarbon medium. Thus such sensitivity of the poly(dimethylsiloxane) chain to the medium may also account for the higher characteristic ratios observed from cyclization equilibria.

The log Kx vs.log x plot for the toluene equilibrate in Fig. 16 is in broad agreement with the result of the same classic experiment of Brown and Slusarczuk44 (see Chapter 1, Fig. 6) who were the first to analyse large cyclic distributions by gel permeation chromatography. The Kx results are extended to x = 267 (Kd = 0·20) for the toluene equilibrate as the concentration of chains is lower in this case. The gradient over the range 16 < x < c. 40 is -2·50 as found by Brown and Slusarczuk. However, for c. 40 < x < 267 the mean gradien tis - 2 ·69. Brown and Slusarczuk reported a mean gradient of - 2·86 over this range in x. This change in gradient in log Kx vs. log x at x = c. 40 reflects the change in gradient of log [11] versus log M plots for both poly(dimethylsiloxane) cyclics and linear species in toluene at 298 K at approximately the same degree of polymerization as observed by Dodgson and Semlyen. 33 The transitions observed in each experiment denote the onset of detectable expansion of the coil dimensions by excluded

PREPARATION OF CYCLIC POLYSILOXANES 119

volume effects in the good solvent medium. Equations (2) and (4) are recast to take account of excluded volume effects giving for Kx

Kx = (3/2n<r;) )3 /2/2N AX

and the expansion factor rx = «r;)/<r;)0)1 /2.

Thus the ratio

(23)

(24)

For X = 100, rx = 1·08; for x = 202, rx = 1·13. These values of rx are smaller than the rx[~l values interpolated from Dodgson and Semlyen's data. For example, rx[~l = c. 1·2 at x = c. 200. However, agreement with molecular dimensions determined by neutron scattering32 for molecules of this size is closer. The values of <S2)6/2 obtained from cyclization equilibria in toluene and by neutron scattering in benzene-d6 are 50·7A and 49·4A, respectively, for chains of 559 bonds.

The gradient of log Kx vs. log x for the toluene equilibrate ( - 2·69) is more compatible than the gradient (- 2·86) reported by Brown and Slusarczuk with theories of the excluded volume effect45 -48 which predict that rx should increase as XO- 1 in the limit of high chain length. These theories would prescribe a maximum gradient of - 2·80 at high values of x and a somewhat less steep gradient at the comparatively low chain lengths which are being considered in the present context.

PREPARATIVE GEL PERMEATION CHROMATOGRAPHY

A block diagram of the preparative G PC instrument made by Sympson49

for the preparation of macrocyclic fractions is shown in Fig. 17. The column was 120cm in length and 6cm in internal diameter. It was packed and supplied by Waters Associates Ltd, Harrow, UK. Two fifths of the column were packed with Styragel having nominal porosity of 100 nm and the remainder with Styragel of porosity 300 nm. The resolution of the column was 4040 theoretical plates. Toluene was the eluent and was delivered to the pressure pump from an automatic still. The pressure pump was constructed from four stainless steel bellows each 2cm in diameter giving solvent flow rates between 10 and 180cm3 min -1 at pressures up to 7 X 105 Pa (c. 100 psi). The sample eluent stream was directed through a six port injection valve having a sample loop of capacity 1Ocm3 .

The preparative G PC was calibrated using procedures similar to those described and discussed for the analytical instrument.

120 P. V. WRIGHT AND MARTIN S. BEEVERS

K

D

FIG. 17. Schematic block diagram of the preparative GPC instrument constructed by Sympson: 49 A, solvent tank; B, still pump, C, constant level device; D, still; E, column; F, pressure pump; G, pulsation damper; H, pressure control valve; I, filter; J, sample injection valve; K, differential refractometer (and chart recorder); L, solvent and eluent flow sight glasses; M, solvent counter and fraction collector. (Adapted from Ref. 49 by courtesy of the publishers, Butterworth & Co.

Ltd, Guildford, Surrey.)

For each fractionation of poly(dimethylsiloxane) the flow rate was set at approximately 20cm min -1. A maximum of2 g ofpoly(dimethylsiloxane) in c. 12 cm3 of toluene was injected onto the column. The volume of solvent eluted from the column was measured in counts of eluent volume 49·1 cm3

at each discharge of the fraction cutter. The collection of discrete fractions of the sample, each fraction corresponding to a single count, was started when the chromatogram began Ito deflect from the baseline.

The fractions of polymer solution were dried over anhydrous MgSO 4 for c. 12 h, filtered, and the siloxane recovered by removal of toluene using a rotary evaporator. Oxidation products of toluene such as benzaldehyde and benzoic acid were removed by washing the fractions in dry methanol. The fractions were finally dried at 313 K under vacuum.

PREPARATION OF CYCLIC POLYSILOXANES 121

Figure 18(a) and (b) shows preparative GPC and analytical GPC tracings for the fractionation of a typical broad initial fraction of dimethylsiloxane macrocyclics. The initial broad fraction was obtained by solvent extraction of a macrocyclic residue using acetone-water mixtures. Eighteen sharp fractions were collected and their individual analytical GPC tracings obtained (Fig. 18(b)). After correction for imperfect resolution the weight fraction and number-average numbers of skeletal

a

i 40 35 30 25 b

~ c 0 ~ V L.

L.

B v v v

0

65 60 55 50 Elution volume (coums)

FIG. 18. Fractionation of 1·59 g of a broad cyclic poly(dimethylsiloxane) fraction (with Mn=9510 and MwlMn= 1·59) by preparative OPC: (a) Preparative OPC tracing of the broad cyclic sample showing the fractions collected; (b) analytical OPC tracings of the individual fractions. (Adapted from Ref. 49 by courtesy of the

publishers, Butterworth & Co. Ltd, Ouildford, UK.)

122 P. V. WRIGHT AND MARTIN S. BEEVERS

TABLE 4 Number-average Numbers of Bonds fin and Weights of the Cyclic Poly(dimethylsiloxane) Fractions Obtained by Preparative GPC as shown in Fig. 18

Preparative GPC elution volumes of fractions collected

(count numbers)

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

Number-average numbers of skeletal bonds in the cyclic

fractions, fin"

1272 1098

904 759 662 527 468 397 339 277 217 170 132 \07 78 61 47 40

Weights (g) of the cyclic fractions

computed from the tracings shown in Fig. 18

0·005 0·007 0·019 0·038 0·096 0·166 0·219 0·245 0·238 0·185 0·121 0·088 0·049 0·029 0·026 0·022 0·018 0·017

Reproduced from Ref. 49 by courtesy of the publishers, Butterworth & Co. Ltd, Guildford, UK. a The heterogeneity indices of all the cyclic fractions were in the range 1·03 < MwlMn< 1·06.

bonds of each cyclic fraction were determined from the tracings (see Table 4). The heterogeneity indices of all the cyclic fractions were found to be in the range 1·03 < if wi if n < 1·06. Similar results were obtained for the preparative GPC fractionation of broad linear poly(dimethylsiloxane) fractions.

Injections of more than 2g of polymer into the preparative GPC instrument resulted in a broadening of the fractions collected. However, larger amounts of a particular sharp cyclic fraction were readily obtained by combination and refractionation of several fractions of similar molar mass as shown in Fig. 19.

The concentrations of linear species in the cyclic fractions were determined as described previously for the distributions within macrocyclic

a

i 60 55 50 b v II> C 0 ~ II> V L...

L... 0 .... U v .... v

0 35 30 C

60 55 50

E lut ion volume (counts) FIG. 19. (a) Analytical G PC tracing of a combined cyclic poly(dimethylsiloxane) fraction with Mwl Mn = 1·10; (b) preparative GPC tracing of the combined cyclic fraction showing the part retained as a single fraction (shaded area); (c) analytical G PC tracing of the new cyclic fraction retained in (b). It has M wi M n = 1·05 and a number-average number of skeletal bonds nn = 195. (Adapted from Ref. 49 by

courtesy of the publishers, Butterworth & Co. Ltd, Guildford, UK.)

124 P. V. WRIGHT AND MARTIN S. BEEVERS

residues. Thus the relative concentrations of cyclics and linears within the fractions which eluted simultaneously were assumed to be the same as the corresponding relative concentrations within the initial equilibrate. The linear concentrations were then determined by measurement of the concentrations and molar mass of the chains separated from the initial equilibrate followed by the use of eqn (20). The calculated concentrations of chains within three cyclic fractions are shown in Fig. 20. These are

i a b c

~ c 0 Q.

'" Q) ... ... 0 ... u Q) ... Q)

0

60 55 55 50 Elution volume (COumSl

FIG. 20. Analytical GPC tracings (corrected for imperfect resolution) of three cyclic poly(dimethylsiloxane) fractions with number-average numbers of skeletal bonds: (a) nn = 107; (b) nn = 227; (c) nn = 468. The shaded areas show the calculated concentrations of linear poly(dimethylsiloxanes) in the cyclic fractions. (Adapted from Ref. 49 by courtesy of the publishers, Butterworth & Co. Ltd, Guildford,

UK.)

believed to represent upper limits of the linear concentrations since fractionation using acetone-water mixtures apparently precipitates linears in preference to cyclics of similar molecular weight. This conclusion was prompted by the persistence of the ratio [1]]rO/[1]]/O = 0·67 for p<;>ly(dimethylsiloxanes) in butanone at 293 K having weight average numbers of skeletal bonds as high as 500. 33 This ratio is identical within experimental error with that (0,662) predicted by Bloomfield and Zimm,50 Fukatsu and Kurata 51 and Yu and Fujita52 for unperturbed cyclic and linear molecules. If appreciable amounts of linears had been present in the higher cyclic fractions an experimental ratio larger than 0·67 would have been expected.

PREPARATION OF CYCLIC POLYSILOXANES 125

RING-CHAIN EQUILIBRATION OF SILOXANE COPOLYMERS

Synthetic procedures are now available that permit a variety of copolymers to be made in which blocks of one type of polymer structure (-(B)r-) are linked together by the siloxane unit -Si-O-Si-. In principle, a ring-chain equilibration reaction may be achieved for any copolymer that

I I possesses two or more -Si-O-Si- units in its skeletal backbone. It

I I follows that it should be possible to prepare cyclic polymers of the type

I I -t-{B)r-Si-O-Si-h

I I

An example of such a polymer is provided by the paraffin-siloxanes which are comprised of blocks of methylene units -(CHZ)r- linked

I I together by the siloxane moiety -Si-O-Si- to form a regular repeating

I I structure. The resultant polymer may be regarded either as an alternating

I I block copolymer of -(CHZ)r- and -Si-O-Si- or as a homopolymer

I I I I

possessing the repeat unit -(CHZ)r-Si-O-Si-. By addition of small I I

cyclic siloxanes LfRR'SiO=H to the 'live' equilibrates it is possible to alter the average length of the siloxane segment. However, the analysis of equilibrates of these copolymer systems is significantly more difficult than for copolymers in which the siloxane segment is too short to permit the formation of cyclic homosiloxanes. Polymers possessing different sequence lengths of methylene groups may be prepared by the simultaneous polymerization and ring-chain equilibration of suitable cyclic monomers. The latter can be synthesized using the procedures published by Piccoli et at.,53 Kumada and Habuchi 54 and Sommer and Ansul. 55 Block copolymers of siloxanes and polystyrene have also been prepared. 56 - 63 Ring-chain equilibria have been established for the poly(paraffin­siloxanes)64 and for the polysiloxane-polystyrene block copolymers. 6z These two copolymer systems will now be discussed in some detail.

126 P. V. WRIGHT AND MARTIN S. BEEVERS

Ring-chain Equilibration of Poly(2,2,7, 7-tetramethyl-l-oxa-2, 7-disilacycloheptane)64 The cyclic monomer 2,2,7, 7-tetramethyl-l-oxa-2, 7-disilacycloheptane

was synthesized from 1,4-dibromobutane and trimethylchlorosilane using the method of Sommer and Ansul. 55 A summary of the principal steps involved in this synthesis is shown in Fig. 21. A ring-chain equilibration reaction was achieved by the addition of a small quantity (c. 0·01 % by weight) of concentrated sulphuric acid to the pure cyclic monomer (1). The initiation and ensuing polymerization reactions were observed to be quite rapid for temperatures in the range 383-423 K; a highly viscous gum being produced within minutes of adding the initiator. After about 60 h the reaction was quenched by rapid cooling followed by the addition of aqueous ammonia. An analysis of the reaction mixture by G PC showed that the equilibrate consisted of substantial amounts of high molecular weight chains together with a broad distribution of cyclic material. Repeated extraction of the equilibrate, using mixtures of diethyl ether (solvent) and methanol (poor solvent), enabled a cyclic-rich fraction to be obtained. The precipitated residue consisted mainly of high molecular weight linear polymer and was soluble in diethyl ether and toluene.

In order to monitor the efficiency of the extraction procedures and to facilitate the calculation of the concentrations of the cyclic species, known quantities of the internal standards n-hexadecane and 2,6,10,15,19,23-hexamethyltetracosane (squalane) were added to the ethereal solutions of the equilibrate before commencing fractionation procedures.

Proton nuclear magnetic resonance spectroscopy and infrared spectros­copy were used to confirm that the cyclic oligomers and the linear polymers possessed the repeat unit -«CH3)2Si--(CH2)4--(CH3)2SiO}-. The concentrations of individual cyclic oligomers in the equilibrate were determined using gas-liquid chromatography by comparing the peak areas of the eluting species with those found for pure cyclic compounds, due allowance being made for differences in their response factors. The concentrations of individual cyclics were observed to decrease monotoni­cally with increase in ring size, thereby demonstrating behaviour consistent with the predictions of the Jacobson and Stockmayer theory of

PREPARATION OF CYCLIC POLYSILOXANES

! Mg turnings + THF

BrMg-(CH2)4-M gBr

! (CH,),5iCl

(CH3)3Si-(CH2)4-Si(CH3)3 II. H,50,

",,2. H20

CH2-CHz

/ " CHZ CH2 \ /

(CH3}zSi" /Si(CH3}z o

127

FIG. 21. Outline of the method used to prepare 2,2,7,7-tetramethyl-l-oxa-2,7-disilacycioheptane. 5 5

macrocyclization (Fig. 22). The monomer concentration lies below the calculated value but experiments carried out over a range of temperature indicate that this cyclic is strained by approximately 4 J mol- 1 (Table 5). The proportion of polymer in the form of cyclics in the undiluted equilibrate (c. 6 %) would increase at the expense of chains, if the equilibration reaction was performed in solution with a suitable solvent. Assuming that the molar cyclization equilibration constants are unaffected

128 P. V. WRIGHT AND MARTIN S. BEEVERS

x

0.0

-1.0

logKx

-2.0

0.0 0.4 0.8 logX

FIG. 22. Experimental equilibrium molar cyclization constants, Kx (mol dm - 3), for cyclic paraffin-siloxanes [(CH3)zSi---(CHz)4---(CH3)zSi0lx in an undiluted equilibrate at 298 K (0). The solid circles (.) are theoretical values of Kx calculated using the Jacobson and Stockmayer theory of macrocyclization assuming Gaussian behaviour of the conformations of the equivalent open chain species. (Adapted from Ref. 64 by courtesy of the publishers, Butterworth & Co.

Ltd, Guildford, UK.)

TABLE 5 Equilibrium Molar Cyclization Constants Kx for the Paraffin-siloxanes [(CH3)zSi---(CHz)c--(CH3)zSi0lx in Undiluted Equilibrates of Poly(2,2,7,7-

tetramethyl-I-oxa-2, 7 -disilaheptane)

x Equilibrium molar cyclization constants Kx (mol dm - 3)

298K 333K 383K 423K

I 0·128 0·139 0·196 0·179 2 0·0419 0·0396 0·0356 0·0423 3 0·0240 0·0231 0·0210 0·023 9 4 0·010 I 0·0104 0·0097 0·0106

PREPARATION OF CYCLIC POLYSILOXANES 129

by the solvent it is estimated that the critical dilution point will be attained when the volume fraction of the solvent is approximately 0·94. However, as in the case of the pure siloxanes, enhancement of the concentrations of the lower cyclics is to be expected in good solvents.

Ring-Chain Equilibration of Copolymers Containing Polysiloxane and Polystyrene Segments Using vacuum-line techniques, Jones has prepared ABA type alternating block copolymers of poly( dimethylsiloxane) and polystyrene.62 Polystyryl dianions ('living polymer') were prepared by the addition of styrene monomer to a solution of potassium naphthalene in tetrahydrofuran at 285 -288 K. In order to achieve a maximum extent of reaction between the styryl anion and hexamethylcyclotrisiloxane the potassium ions were exchanged for lithium ions by reaction with a slight stoichiometric excess of lithium tetraphenylboride. On addition of hexamethylcyclotrisiloxane, a block copolymer of poly(dimethylsiloxane) and polystyrene was produced. A constitution of this type was consistent with the proton nuclear magnetic resonance spectroscopy of the product. The copolymers could be equilibrated using potassium silanolate catalysts at 383 K. After quenching with glacial acetic acid the equilibrates were fractionated by repeated precipitation. Analysis of the fractions, using gel permeation chromatog­raphy and gas-liquid chromatography, indicated the presence of a distribution of cyclic dimethylsiloxanes similar to that obtained for a bulk equilibration ofpoly(dimethylsiloxanes). However, more importantly, GPC also provided evidence of a more complicated distribution of cyclic species involving blocks of polystyrene and poly( dimethylsiloxane). The molecular weight of one fraction of the copolymer was determined to be 1·15 x 104

using GPC and 1·04 x 104 using vapour pressure osmometry. It was concluded that cyclics of the copolymer, possessing an average structure represented by

LfSt25Si(CH 3M OSi(CH3)ZktJ (x = 3-4) had been formed by a ring-chain equilibration reaction. This work represents the first ring -chain equilibration involving a vinyl polymer.

In addition to the siloxane polymers previously described there are many other copolymer systems which, through reversible interchanges of their skeletal bonds, either in the pure state or in solution, would be expected to produce distributions of both cyclic and linear species. In principle it should be possible to establish ring-chain equilibria for block copolymers of the type AB and ABA, where A denotes the siloxane segments. However,

130 P. V. WRIGHT AND MARTIN S. BEEVERS

the former type of copolymer will, on equilibration, yield cyclics containing only siloxane skeletal bonds. Furthermore, under certain circumstances it is possible that copolymers of the type AB or BAB would produce biphasic mixtures during a bulk equilibration process.

To date, a large number of siloxane copolymers have been based on the poly( dimethylsiloxanes). Some of the earliest work in this area has already been mentioned in the previous section concerned with the preparation and characterization of the paraffin-siloxanes. 1 - 3 Nemetkin and co-workers65 have prepared the six-membered ring 2,2,4,4,6,6-hexamethyl-I,3-dioxa-2,4,6-trisilacyclohexane and have used this material to obtain a variety of cyclic siloxane-methylenes. An interesting development, concerned with the formation of copolymers of siloxanes and methylene sequences, has been the synthesis, by Busfield and Cowie,66 of block copolymers containing segments of polyethylene and polysiloxanes by coupling silane­terminated dimethylsiloxane oligomers and OH-terminated polyethylene in the presence of Sn(lI) octanoate.

Various methods for the preparation of block copolymers containing siloxane and polystyrene sequences have been reported. Early work in this area was carried out by a number of research groups. 56 - 60 Greber et al. 5 7 reacted 'living' oc-w-di-carbanionic oc-methylstyrene oligomers with tri­methylchlorosilyl terminated poly(dimethylsiloxane). Similar approaches to that of Jones62 involving living polystyrene polymer have been pursued by Saam, Gordon and Lindsey. 60 Chaumont and co-workers63 have used a different route to prepare block copolymers ofpoly(dimethylsiloxane) and polystyrene involving a hydrosilation reaction between oc,w-dihydrogeno poly(dimethylsiloxane) and oc,w-di(vinylsilane) polystyrene.

Block copolymers have been formed from the polycondensation of suitable derivatives of p-bis (dimethylhydroxysilyl) benzene and oligomers of dimethylsiloxane. 67 - 69 Poly(arylene siloxanes) have been synthesized by Pittman, Patterson and McManus. 70 Rosenberg et al. 71 have prepared a variety of siloxane-modified polycarbonates and Matzner and co­workers 72 have carried out polycondensation reactions between oc,w­dihydroxy polycarbonate chains and poly(dimethylsiloxane) chains terminated with dimethylaminosilane groups. Alternating block copoly­mers of poly(dimethylsiloxane) and Bisphenol-A carbonate have also been obtained73 and Webster et al. 74 have prepared and characterized block copolymers containing Bisphenol-A isophthalate-terephthalate segments and dimethylsiloxane segments. The application of siloxane redistribution reactions to such systems presents the possibility of preparing some novel cyclic species.

PREPARATION OF CYCLIC POLYSILOXANES l31

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2. Cooper, G. D., J. Polym. Sci. AI, 4 (1966) 603. 3. Okinojima, H., Japanese Patent 7,798,798 (Chem. Abstr., 88 (1978) 74734f);

Evans, E. R., US Patent 4,157,337 (Chem. Abstr., 91 (1979) 92269z). 4. Morton, M. and Bostik, E. E., J. Polym. Sci. A2, 2 (1964) 523. 5. Lee, C.' L., J. Organometal. Chem., 6 (1966) 620. 6. Wright, P. V. and Semlyen, J. A., Polymer, 11 (1970) 462. 7. Lee, C. L. and Johannson, O. K., J. Polym. Sci., Polym. Chem., 14(1976) 729;

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8. Chojnowski, J., Scibiorek, M. and Kowalski, J., Makromol. Chem., 178(1977) 1351.

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563. 13. Semlyen, J. A. and Wright, P. V., Polymer, 10 (1969) 543. 14. Wright, P. V., PhD Thesis, University of York, UK, 1970. 15. Carmichael, J. B., Gordon, D. J. and Isackson, F. J., J. Phys. Chem., 71 (1967)

2011. 16. Ivin, K. J. and Leonard, J., Europ. Polym. J., 6 (1970) 331. 17. Beevers, M. S. and Semlyen, J. A., Polymer, 12 (1971) 373. 18. Clarson, S. J., D Phil Thesis, University of York, UK. 19. Buch, R. R., Klimisch, H. M. and Johannson, O. K., J. Polym. Sci., A2, 8

(1970) 541. 20. Hickton, H. J., Holt, A., Homer, J. and Jarvie, A. W., J. Chern. Soc. (C), (1966)

149. 21. Moore, C. B. and Dewhurst, H. A., J. Org. Chem., 27 (1962) 693. 22. Harris, R. K., Kimber, B. J., Wood, M. D. and Holt, A., J. Organometallic

Chern., 116 (1976) 291. 23. Grubb, W. T. and Osthoff, R. C., J. Amer. Chem. Soc., 77 (1955) 1405. 24. Edwards, C. J. c., Stepto, R. F. T. and Semlyen, J. A., Polymer, 23(1982) 869. 25. Rentsch, R. and Schulz, R. c., Makromol. Chem., 178 (1977) 2535. 26. Casassa, E. F., Macromolecules, 9 (1976) 182. 27. Benoit, H., Grubisic, Z., Rempp, P., Decker, D. and Zilloix, J. G., J. Chim.

Phys., 63 (1966) 1507. 28. Grubisic, Z., Rempp, P. and Benoit, H., J. Polym. Sci. (B), 5 (1967) 753. 29. Dawkins, J. V. and Hemming, M., Polymer, 16 (1975) 554. 30. Zimm, B. H. and Stockmayer, W. H., J. Chem. Phys., 17 (1949) 1301. 31. Sole, K., Macromolecules, 6 (1973) 378. 32. Higgins, J. S., Dodgson, K. and Semlyen, 1. A., Polymer, 20 (1979) 553. 33. Dodgson, K. and Semlyen, J. A., Polymer, 18 (1977) 1265. 34. Zimm, B. H. and Kilb, R. W., J. Polym. Sci., 37 (1959) 19. 35. Edwards, C. J. c., Stepto, R. F. T. and Semlyen, J. A., Polymer, 23(1982) 856.

132 P. V. WRIGHT AND MARTIN S. BEEVERS

36. Lee, C. L. and Emerson, F. A., J. Polym. Sci. A2, 5 (1967) 829. 37. Pierce, P. E. and Armonas, 1., In: Analytical Gel Permeation Chromatography

(J. Polym. Sci. C, 21 (1968) 23) (eds J. F. Johnson and R. S. Porter), Interscience, New York.

38. Flory, P. J., Principles of Polymer Chemistry, Cornell University Press, Ithaca, New York, 1953, p. 318.

39. Scott, D. W., J. Amer. Chem. Soc., 68 (1946) 2294. 40. Carmichael, J. B. and Heffel, J., J. Phys. Chem., 69 (1965) 2213. 41. Mazurek, M., Scibiorek, M., Chojnowski, J., Zavin, B. G. and Zhdanov, A. A.,

Europ. Polym. J., 16 (1980) 57. 42. Haug, V. A. and Meyerhoff, G., Makromol. Chem., 53 (1962) 91. 43. Flory, P. J. and Semlyen, J. A., J. Amer. Chem. Soc., 88 (1966) 3209. 44. Brown, J. F. and Slusarczuk, G. M. J., J. Amer. Chem. Soc., 87 (1965) 931. 45. Flory, P. J., Principles of Polymer Chemistry, Cornell University Press, Ithaca,

New York, 1953, p. 622. 46. Edwards, S. F., Proc. Phys. Soc. (London), 85 (1965) 613. 47. Flory, P. J. and Fisk, S., J. Chem. Phys., 44 (1966) 2243. 48. Alexandrowicz, Z., J. Chem. Phys., 46 (1967) 3789. 49. Dodgson, K., Sympson, D. and Semlyen, J. A., Polymer, 19 (1978) 1285. 50. Bloomfield, V. and Zimm, B. H., J. Chem. Phys., 44 (1966) 315. 51. Fukatsu, M. and Kurata, M., J. Chem. Phys., 44 (1966) 4539. 52. Yu, H. and Fujita, H., work cited by Yamakawa, H., In Modern Theory of

Polymer Solutions, Harper and Row, New York, London, 1971, p. 323. 53. Piccoli, W. A., Haberland, G. G. and Merker, R. L., J. Amer. Chem. Soc., 82

(1960) 1883. 54. Kumada, M. and Habuchi, A., J. Inst. Poly tech. Osaka City Univ., 3 (1952) 65. 55. Sommer, L. H. and Ansul, G. R., J. Amer. Chem. Soc., 77 (1955) 2482. 56. Greber, G. and Metzinger, L., Makromol. Chem., 39 (1960) 167. 57. Greber, G., Reese, E. and Balciunas, A., Farbe u Lecke, 4 (1964) 249. 58. Morton, M., Rembaum,A. A. and Bostick, E. E., J. Appl. Polym. Sci., 8(1964)

2707. 59. Mitoh, M., Tabuse, A. and Minoura, Y., Kogyo Kagaku Zasshi, 70 (1967)

1969. 60. Saam, J. c., Gordon, D. J. and Lindsey, S., Macromolecules, 3 (1970) I. 61. Noshay, A., Matzner, M., Karoly, G. and Stampa, G. B., J. Appl. Polym. Sci.,

17 (1973) 619. 62. Jones, F. R., Eur. Polym. J., 10 (1974) 249. 63. Chaumont, P., Beinert, G., Herz, J. and Rempp, P., Polymer, 22 (1981) 663. 64. Beevers, M. S. and Semlyen, J. A., Polymer, 13 (1972) 523. 65. Nemetkin, N. S., Islamov, T. Kh., Gusel'nikov, L. E. and Vdovin, V. M.,

Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, 6 (1968) 1329. 66. Busfield, W. K. and Cowie, J. M. G., Polym. Bull. (Berlin), 2 (1980) 619. 67. Merker, R. L., Scott, M. J. and Haberland, G. G., J. Polym. Sci. A2, 2 (1964)

31. 68. Yu, N., Masubuchi, T., Ikeda, K. and Sekine, Y., Polymer, 22 (1981) 1607. 69. Yu, N., Ikeda, K. and Sekine, Y., Polymer, 23 (1982) 1646. 70. Pittman, C. U., Patterson, W. J. and McManus, S. P., J. Polym. Sci., Polym.

Chem. Edn., 14 (1976) 1715.

PREPARATION OF CYCLIC POLYSILOXANES 133

71. Rosenberg, H., Tsai, Tsu-tzu., Nahlovsky, B. D. and Kovacs, C. A., Amer. Chem. Soc. Symp. Ser., 121 (1980) 457.

72. Matzner, M., Noshay, A., Robeson, L. M., Merrian, C. N., Barclay Jr., R. and McGraph, J. E., Polym. Sci. (D), Appl. Polym. Symp., 22 (1973) 143.

73. Kambour, R. P., Polym. Lett., 7 (1969) 573. 74. Webster, D. C., Andolino, P. J., Riffle, J. S., Keohan, F. L. and McGraph, J.

E., Polym. Prepr. (Am. Chem. Soc. Div. Polym. Chem.), 24 (1983) 161. 75. Semlyen, J. A. and Wright, P. V., In: Chromatography of Synthetic and

Biological Polymers, Vol. 1, (ed. R. Epton), Ellis Horwood Ltd, Chichester, 1978, Chapter 23.

CHAPTER 4

Comparison of Properties of Cyclic and Linear Poly( dime thy Is ilo xanes)

CHRISTOPHER J. C. EDWARDS* and ROBERT F. T. STEPTO

Department of Polymer Science and Technology, University of Manchester Institute of Science and Technology,

Manchester, UK

INTRODUCTION

Cyclic poly(dimethylsiloxanes) (PDMS) (CH 3SiO)x have been prepared 1.2

in significant quantities up to molar masses of about 3 x 104 g mol- 1. Both cyclic and linear PDMS with molar masses up to this value have freezing points in the range - 36 to - 86°C. 3 Hence, the physical properties which they exhibit at ambient temperatures are those characteristic of polymer melts. In addition, since the molar mass above which transient molecular entanglements become significant is around 2·4 x 104 g mol- 1 for PDMS melts,4 the physical properties of both cyclic and linear PO MS are determined largely by the configurational statistics of single molecules.

Discussion of the properties of cyclic PO MS necessitates a comparison with the linear species of the same molar mass to provide a reasonably well­defined basis for understanding the effects of chain structure and chain length on physical properties. The configurational statistics of a cyclic molecule are dominated by the closure condition, such that segments 1 and

* Present address: Unilever Research, Port Sunlight Laboratory, Quarry Road East, Bebington, Wirral, Merseyside L63 3JW, UK.

135

136 CHRISTOPHER J. C. EDWARDS AND ROBERT F. T. STEPTO

x are identical in, for example, the relation w w

2 I\,\, 2 <s ) = W2 ~ ~ <rij)

i=l j=l

(1)

where <S2) denotes the mean-square radius of gyration and rij the separation of segments i and J in the molecule. For linear chains the maximum value of <rij) occurs for Ii - JI = x, whereas for a cyclic molecule the maximum value occurs for Ii - JI = x/2. Theoretical approaches based on this fact have been shown to predict accurately relative values of such properties as <S2), 5 -7 the intrinsic viscosity, [11],8 - 10 and the translational diffusion coefficient, D,8,9,l1 at high molar mass. However, small angle neutron scattering (SANS) measurements 12 of the scattering function P(Q) at high values of Q have revealed significant deviations from P(Q) calculated using the Gaussian subchain model13 of a ring molecule.

The closure condition also affects the equilibrium shapes of cyclic molecules. Linear chains are known to be asymmetric 14,15 with the degree of asymmetry increasing with increasing stiffness of the polymer chain. 16 - 18 Monte Carlo calculations have shown that cyclic mol­ecules 17 - 19 are somewhat less asymmetric than their linear counterparts although they are still far from spherically symmetric. In addition, <S2) for a cyclic molecule has been calculated to be half that of the equivalent linear chain using the Monte Carlo method,20 in good agreement with analytical calculations5 -7 for the Gaussian subchain model and with experimental SANS data. 21

Molecular shape may be expected to affect bulk properties and indeed the densities of low molar mass cyclic POMS molecules are greater than those of the equivalent linear species. 22,23 In addition, cyclic POMS molecules with x = 11 repeat units exhibit anomalously high bulk densities22 ,23 which correlate well with the peculiar 'disc-like' shapes of these molecules. 20 This feature, which has its origin in the unequal bond angles of the PO MS chain, is discussed in more detail later in this chapter. Other bulk properties considered are dielectric constants,23 bulk vis­cosity,24 and NMR 29Si chemical shifts.2s

With regard to dilute solution properties, chain expansion has been detected experimentally1,21,26-3o for both cyclic and linear POMS with greater than about 100 skeletal bonds. However, the effect of excluded volume on ratios of properties for cyclic and linear species is generally small. The dilute solution configurational properties of cyclic PO MS which have been measured include <S2) derived from SANS measurements21 (see

PROPERTIES OF CYCLIC AND LINEAR POLY(DIMETHYLSILOXANES) 137

Chapter 5) as well as translational diffusion coefficients, D, 28,29,31 - 33 intrinsic viscosities, [" V and second virial coefficients, A 2. 30 Where appropriate, comparisons have been made with Monte Carlo calculations of these quantities. The application of Monte Carlo techniques 19,20 to cyclic PDMS is discussed in the next section. Their use is essential for the calculation of configurational properties, as matrix methods,34 such as may be used for linear chains, cannot be applied to ring molecules. Detailed correlations of experimental data using Monte Carlo calculations have been particularly appropriate for D 3S •36 (this chapter) and particle scattering functions, P(Q) 12 (Chapter 5).

MOLECULAR SIZE AND SHAPE

The shape of a macromolecule is best described as an equivalent ellipsoid of constant segment density. 14,1 5 The three semi-axes of the ellipsoid are generally unequal and may be denoted a, band c with a> b> c. For a given configuration, the values of a, band c depend on the distribution of segments about the centre of mass and they may be evaluated through standard mechanics procedures using inertia tensors and principal radii of gyration. 19 One result is that the conventional mean-square radius of gyration

(2)

whereas the mean-square radius of the equivalent sphere of constant segment density is simply

(3)

Independent of whether the chain is linear or cyclic, <r;) gives a better representation than <S2) of the size of a molecule averaged over all orientations in external space. When <a2) = <b2) = <c2) the molecules have an average spherical shape with circular cross-sections. The degree of eccentricity of a linear PD MS chain is illustrated in Fig. 1. The difference introduced by the use of <S2)1/2 instead of <r;)1/2 is a factor of about two in the effective spherical volume occupied by a molecule.

Information about the shapes of molecules in terms of the semi-axes of equivalent ellipsoids cannot generally be obtained from measurements of equilibrium properties. However, Monte Carlo calculations are ideally suited to this problem and to the calculation of spherically averaged radii of gyration of cyclic molecules.

138 CHRISTOPHER 1. Co EDWARDS AND ROBERT F. T. STEPTO

,.----- ......

OX

....

/

"

FIG. 1. View along OYprincipal axis ofa PDMS chain of 100 skeletal atoms with section of the average equivalent ellipsoid of constant segment density and spheres corresponding to the spherically-averaged radii: ~-, Ellipse of semi-axes < a2 ) 1 /2

and <C2)1/2 along OX and OZ, respectively, ----, circle of radius <r;)1!2 corresponding to sphere of constant segment density; --, circle of radius <S2) 1/2.

Analytical calculations have predicted 5 - 7 that as x ~ 00 the ratio <sf)/<s;) = 2 in the absence of excluded volume effects. The subscripts I and r denote linear and cyclic molecules, respectively. This result has been confirmed experimentally in dilute solution using SANS. 21 To date, no SANS measurements have been reported on bulk cyclic PDMS. The analytical calculations and the SANS measurements are discussed in detail in Chapters 2 and 5 of this book.

Previous numerical calculations 17.18 of the shapes of linear and cyclic molecules have considered fre1ely jointed chains 18 and cubic-lattice chains. 17 However, any attempt to correlate differences between the properties of cyclic and linear PDMS with differences in their respective shapes requires the use of a model which reflects the chemical structure of the chain. Flory, Crescenzi and Mark 37 (FCM) developed the rotational­isomeric-state model (RISM) of the PDMS chain in order to calculate the unperturbed dimensions of linear PDMS. The FCM RISM has been used throughout for the Monte Carlo calculations of properties of cyclic PD MS discussed in this chapter.

In the FCM RISM the Si-O bond length is taken to be 0·164 nm and the

PROPERTIES OF CYCLIC AND LINEAR POLY(DIMETHYLSILOXANES) 139

bond angle supplements at silicon and oxygen are assigned the values 70 0

and 37 0 , respectively. Rotational states (t, g + , g -) are situated at 0 0 , 120 0

and -120 0 and the statistical weight matrix for pairs of adjacent bonds centred on silicon atoms takes the form:

t g+ g-

[:

(J

~] U'=g+ (J (4)

g 0

The corresponding matrix centred on oxygen atoms is:

g+ g-

[:

(J

:] U"=g+ (J (5)

g J

with (J = 0·238 and J = 0·0402 at 298 K. For linear PDMS, <s~> can be evaluated using the FCM RISM and the

generator-matrix approach developed by Flory.34 The generator-matrix method cannot be applied to cyclic molecules because it does not allow selection of the individual configurations accessible to a ring molecule. However, in the absence of ring strain, the same statistical weight matrices can be used. Edwards et al. have described calculations of <S~>20 and the shapes19 of cyclic and linear PDMS molecules using the FCM RISM and a combination of complete enumeration and Monte Carlo techniques.

The complete enumeration method, which was used for x < 12, involves generating systematically all the RISM configurations of a linear PDMS chain of x repeat units. Cyclic configurations are identified using a suitable criterion for ring formation and the properties of interest are evaluated as an average over all possible configurations of the cyclic molecules. The criterion employed for ring formation was that the end separation of the chain should be less than 0·2 nm. Although the configurations generated in this way do not correspond to perfect rings, the error introduced was shown to be small, to be relatively insensitive to the separation used, and to decrease with increasing ring size.

The Monte Carlo technique which was used for the larger ring molecules (12 < x < 50) employed Metropolis sampling. 38 An initial configuration was generated as described above and its configurational energy evaluated using the FCM RISM. A new configuration was then generated by

140 CHRISTOPHER J. C. EDWARDS AND ROBERT F. T. STEPTO

changing the rotational states of a small number of contiguous bonds in the existing configuration and this process repeated until a new cyclic configuration was obtained. The new configuration was accepted according to the Metropolis criterion. 38 For sufficiently large sample sizes the process leads to a Boltzmann distribution of configurational energies.

Figure 2 shows a plot of (s;)/(s;) versus l/w (where w represents the number of skeletal bonds ( = 2x» for linear and cyclic PD MS. In the region 30;:5 w;:5 100 the calculated ratio is in good agreement with both Gaussian subchain theory5 - 7 and SANS measurements. 21 However, for 20 ;:5 w ;:5 30 the ratio is less than 2 and for w;:5 20 the ratio is markedly greater than 2. The origin of these variations in the ratio (st)/(s;) lies in the different, non-Gaussian statistics of linear and cydic PDMS.

Skeletal atoms Cw) 100 5D 40 30 20 18 16 14 12

I I I

2-5- e ~ , , X

~2.0r--.e_--T-•• ---·------_0.._-----

¢ " ~ 1St-

I 0.00 002

I 004 I/w

I I 0.06 008

FIG. 2. .The ratio of the mean-square radii of gyration «s~>/<s~» for linear and cyclic PDMS at 298 K, calculated by the complete enumeration (x) and Monte

Carlo methods (e).

Interesting differences in shapes between the cyclic and linear mol­ecules were found. Figure 3 shows three mutually perpendicular sections through the equivalent ellipsoids for cyclic and linear PDMS with w = 100 normalized to unit <Sf)1/2 .19 These sections show clearly that the major change in shape on ring formation is a shortening of the longest axis with the two smaller axes band c little affected. The shortening of the longest axis is clearly seen in Table 1 where the limiting values of the ratios

142 CHRISTOPHER J. C. EDWARDS AND ROBERT F. T. STEPTO

TABLE I Ratios of Shape Parameters for some Cyclic and Linear Polymers

Linear molecule:cubic lattice! 7

Linear molecule: freely jointed chain!8 Linear molecule: polymethylene!6 Linear molecule: PDMS!9 Cyclic molecule:cubic lattice! 7

Cyclic molecule: freely jointed chain!8 Cyclic molecule: PD MS!9

11'8:2'7:1 12·2:2·7: I 12·9:3·6: I 11·9:2·6: I 6·5:2·4: I 6·5:2·7: I 5·9 :2·6: I

<a2 ):<b2 ):<C2 ) at infinite chain length are compared for various models oflinear and cyclic molecules. The table shows that the comparative shapes of high molar mass linear and cyclic species are relatively insensitive to the details of the chain model employed.

Figures 4,5 and 6 show the variation of the ratios <b2 )/<a2 ), <c2 )/<a2 )

and <c 2 )/<b2 ) with l/w for cyclic and linear PDMS at 298 K. For linear

Skeletal atoms (w)

0·02 0·04

Yw 0·06 0·08

FIG. 4. Ratio of second largest to largest mean-square (ms) semi-axes of average equivalent ellipsoid «b2 > / < a2 > ) against reciprocal number of skeletal atoms (I /w):

., Cyclic PDMS; 0, linear PDMS.

PROPERTIES OF CYCLIC AND LINEAR POLY(D1METHYLSILOXANES) 141

Cyclic Linear

Cyclic Linear

(~t~~r-------~·-------+--~

Linear

'''---~-Cyclic (b12------~----+---~------

FIG. 3. Sections of average equivalent ellipsoids for linear and cyclic PDMS of 100 skeletal bonds normalized to unit <S?>1/2.

PROPERTIES OF CYCLIC AND LINEAR POLY(DIMETHYLSILOXANES) 143

0·2

Skeletal atoms (w)

0·02 0·04 I/w

20 12

0·06 0·08

FIG. 5. Ratio of smallest to largest ms semi-axes of average equivalent ellipsoids «c2 )/<a2» against reciprocal number of skeletal atoms (l/w): ., Cyclic PDMS;

0, linear PDMS.

Skeletal atoms (w)

50 20

0·02 0·04 I/w

0·06

12

0·08

FIG. 6. Ratio of smallest to second largest ms semi-axes of average equivalent ellipsoid «c2 )/<b2» against reciprocal number of skeletal atoms (l/w): ., Cyclic

PDMS; 0, linear PDMS.

144 CHRISTOPHER J. C. EDWARDS AND ROBERT F. T. STEPTO

PDMS these ratios vary smoothly with chain length and tend to the limiting values given in Table 1 at l/w = O. However, for cyclic PDMS quite different behaviour is observed with a maximum in <b2 )/<a2 ) and minima in <c 2 )/<a2 ) and <c 2 )/<b2 ) centred around w = 20. These features suggest that the molecules are tending to adopt average shapes which resemble planar discs in this region of w. Such behaviour is not unexpected because, as a consequence of the unequal bond angles at the silicon and oxygen atoms, the low-energy all-trans configuration of the PDMS chain with w = 22 forms a planar ring. PDMS rings with slightly more than 22 skeletal atoms might be expected to adopt average shapes which resemble puckered planar rings and which account for the low values of the ratio <sf)/<s;) in the region 20 ;$ w;$ 30 (Fig. 2). Rings ofless than 22 skeletal atoms become increasingly configurationally restricted and may be compelled to adopt relatively high energy configurations for steric reasons. This increase in configurational restriction is accompanied by the average shape of the molecule (see Figs 4,5 and 6) returning to the ellipsoidal form characteristic of high molar mass rings and a general rise, albeit with an unexplained minimum, in the ratio <s;)/<s;) as w--+O.

CORRELATION OF BULK PROPERTIES WITH MOLECULAR SIZE AND SHAPE

Density (p) and Refractive Index (n D)

Planar disc-shaped molecules would be expected to pack together more efficiently than ellipsoids and thus produce anomalous behaviour in the bulk properties of cyclic PDMS in the region of w = 22. Figure 7 shows plots of bulk density22 and refractive index22 versus l/w for cyclic and linear PDMS at 298 K. The plots do indeed exhibit maxima in the region of w = 22 which can only be explained in terms of the enhanced packing efficiency of ring-shaped molecules.

Bulk Viscosity (1'/) The bulk viscosity of cyclic PDMS also exhibits a distinct variation with chain length compared with linear PDMS. Figure 8 shows a plot of log 1'/ versus 10gMw for cyclic and linear PDMS at 298K.24 No shear rate dependence of 1'/ was observed for any of the samples, consistent with 'entanglement-free' behaviour4 over the range of chain lengths studied. In the region of higher molar mass (1·8 x 104<Mw <2·4x 104 gmol-l), I'/r/l'/I ~ 0·5 in agreement with the theoretical predictions of Bueche. 39

PROPERTIES OF CYCLIC AND LINEAR POLY(DIMETHYLSILOXANES) 145

0·972

0·970

0·968

0·966

0·964

0·962

1·406 1·405

1·404

1·403

1·402

1·401

1·400

Sk eletal bonds w 20 15

Refractive index J no

0·02 0·04 0·06 0·08

'/w FIG. 7. Densities (p) and refractive indices (nD ) at 298 K plotted against reciprocal

number of skeletal atoms (l/w) for cyclic PDMS.

Rather unexpectedly, the plots of log '1 versus log M w cross at nw ~ 100 so that the cyclic species have higher viscosities at low molar mass.

A detailed interpretation of the curves has yet to be attempted. However, the plots show distinct transitions from short-chain to long-chain behaviour. For linear PDMS, '1 is approximately proportional to Mw with minor variations from this proportionality as chain length increases. For the smaller rings, the enhanced viscosities again reflect configurational restrictions and resultant losses of flexibility. The activation energies for

146 CHRISTOPHER J. C. EDWARDS AND ROBERT F. T. STEPTO

Skeletal bonds nw

30r-----~20~------,50~--~IOO~----------~5rOO~__,

2·5

2·0

15

1·0

0·5

0~------~~~0~-------3~5~------~4~0~------~4.5 -I

Log (Mw/g.mol )

FIG. 8. Plots of the logarithms of the bulk viscosities '1 of cyclic (.) and linear (0) poly(dimethylsiloxanes) at 298 K against the logarithm of weight-average molar

mass Mw and weight-average number of skeletal bonds nw.

viscous flow are also higher for cyclic PD MS at all chain lengths. 24 At w ~ 30 the change in configurational statistics noted previously with reference to <s6) seems to become important and it takes from then until w ~ 400 for the cyclic molecules to attain Gaussian 'equivalent chain' behaviour with a constant ratio 11rl11 ,.

Dipole Moment Ratio «p.2)lnm2) Beevers et al. have derived dipole moments for cyclic and linear PDMS in the bulk from measurements of their static dielectric permittivities. 23

Figure 9 shows a plot of the dipole moment ratio <p.2)/nm2 when p. is the molecular dipole moment, n is the number of skeletal bonds and m is the bond dipole moment. Also inc:1uded in Fig. 9 are previous data for the cyclic tetramer40 (D 4) and a variety of fractions of linear PD MS. 40 - 42

Differences between the various sets of data can mainly be attributed to the variety of experimental techniques employed.

The solid curve in Fig. 9 represents values of <p.2)/nm2 calculated for linear PDMS using the FCM RISM. 37 There is good agreement between the calculated curve and the experimental data of Beevers et al. for linear

PROPERTIES OF CYCLIC AND LINEAR POLY(DlMETHYLSILOXANES) 147

'il 'il 'il

0·3

• • o • •

• •

0·1

I

10 20 30 40

FIG. 9. Dipole moment ratio <J.l2>/nm2 for cyclic PDMS (e, Beevers et al.;23 ., Dasgupta et al. 40) and linear PDMS (0, Ref. 23; 0, Ref. 40; l:::,., Dasgupta and Smyth;41 \7, Sutton and Mark42) in the undiluted state. The unbroken line was

calculated for linear PDMS using the FCM RISM at 298 K.

PDMS at all chain lengths. In addition, for w > 20 the dipole moment ratios for cyclic and linearPDMS are essentially the same. However, for w < 20 the dipole moment ratio for cyclic PD MS is substantially lower than that found for linear PDMS. This difference is another reflection of the configurational restriction of the smaller PDMS rings, although it is interesting to note that there is no apparent difference between the behaviour of cyclic and linear PDMS in the region of w ::::= 22.

29Si NMR Chemical Shift (<5sJ The discrete nature of the average configurations of small PDMS rings is further illustrated by consideration of 29Si NMR chemical shift data. Burton et al. have measured 29Si NMR spectra for several fractions oflow molar mass cyclic PDMS. 25 Figure 10 shows a sample spectrum illustrating that separate reson.ances can be distinguished up to w = 30 (DIs) with all silicon atoms in a given ring having the same chemical shift. The variation in chemical shift with chain length is shown in Fig. 11 where

148 CHRISTOPHER J. C. EDWARDS AND ROBERT F. T. STEPTO

-V

FIG. 10. Typical 19·87MHz 29Si_{lH} NMR spectrum of a low molar-mass fraction of cyclic PDMS. 25

- 19 ~--------------------------------~

-20

Os; /ppm

-21

- 22

- 23

4

•

6 8 10 12 14 Number of monomer units x

FIG. II. Typical plot of observed29Si chemical shifts (<<5) for cyclics ((CH3)2SiO)x versus ring size, x.

PROPERTIES OF CYCLIC AND LINEAR POLY(DIMETHYLSILOXANES) 149

chemical shifts relative to that for (CH3)4Si are plotted against the number of monomer units in the ring (x). The minimum value of JSi occurs for x = 8 and thereafter the chemical shifts tend to a limiting value of - 21· 79 ppm, the same as high molar mass linear PDMS. The observed variations in JSi

can be equated with increased shielding in the region 3 < x < 8 and then a progressive deshielding for x> 8. (A marked deshielding occurs for x = 3 and probably arises from the strained nature of the cyclic trimer (D3) which requires considerable bond angle distortion.) Variations in JSi with chain length for the larger rings would be expected to correlate with chain configuration. However, calculations of JSi using the FCM RISM do not properly reproduce the variations in chemical shift, possibly due to the assumptions of discrete, fixed, rotational states and bond angles inherent in the model.

More generally it is noteworthy that discrete 29Si resonances can be resolved only up to w = 30. In concurrence with this finding, the Monte Carlo calculations discussed earlier 19.20 have shown that marked changes in shape occur up to w = 30, but that for w> 30 essentially limiting behaviour has been reached.

135

125

o 2 3 4 M~l/ 103 mol.g-I

5

FIG. 12. Glass transition temperatures T of cyclic dimethyl siloxanes (e) and linear dimethyl siloxanes (0) versus reciprocal number-average molar mass. 3

150 CHRISTOPHER J. C. EDWARDS AND ROBERT F. T. STEPTO

Glass Transition (Tg) Recent measurements 3 of Tg as a function of molar mass show a marked distinction between linear and cyclic PDMS as illustrated in Fig. 12. Linear PDMS displays the conventional decrease in Tg as M;: 1 increases, which may be attributed to the incre:ase in the concentration of chain ends and hence free volume with decrease in molar mass. In contrast, Tg for cyclic PD MS, in which chain ends are absent, decreases as chain length increases tending to the same value at infinite chain length as linear PDMS. This contrasting behaviour for cyclic PDMS has not been predicted theoretically. However, the decrease to the same limiting value as the linear species at infinite chain length indicates that the same local conformational statistics apply to both linear and cyclic molecules. This conclusion is supported by the previously discussed trends in the relative bulk viscosities, dipole moment ratios and 29Si NMR chemical shifts as chain length increases.

DILUTE SOLUTION PROPERTIES

Intrinsic Viscosity [1]] Dodgson and Semi yen 1 have measured [1]] for linear and cyclic PD MS in a a-solvent (2-butanone/293 K) and in two good solvents (toluene/298 K and cyclohexane/298 K). Figures 13 and 14 show the conventional plots of log [1]] versus log Mw for linear and cyclic PDMS, respectively. For both types of molecule the data an: independent of solvent up to c. 50 skeletal bonds, indicating that excluded volume effects are not detectable below this chain length from intrinsic viscosities.

In 2-butanone at 293 K the data for both linear and cyclic PDMS fall on two parallel lines with slopes of 0·58, which are slightly greater than the expected values of 0·5. The ratio [1]]r/[1]], = 0·67 in 2-butanone at 293 K, which can be compared with the theoretical prediction of 0·662 for cyclic and linear 'equivalent freely-jointed' chains in a a-solvent and in the absence of free-draining. 8 -10 In addition, the data for linear PDMS are in good agreement with the earlier data of Flory et al. 43 for higher molar mass samples under the same solv(:nt conditions.

In both good solvents the cyclic and linear molecules show evidence of chain expansion with the slopes of log [1]] versus log M w plots being the larger for linear PDMS. The ratio [1]]r/[1]], equals 0·60 in toluene and 0·58 in cyclohexane for the highest molar mass samples. There is some disagreement in the literature regarding the theoretical treatment of the

-1·0

-1·5

-2·0

Ske

leta

l bo

nds

nw

50

10

0

log

([l1

.j/d

I.g

-1 )

er

3·0

M/

//ll

3-5

4·0

Log(~g.mo I-

I)

50

0

4-5

FIG

. 13

. P

lots

of l

og [,

.,] v

ersu

s lo

g M

w fo

r se

vera

l fra

ctio

ns

of

line

ar P

DM

S:i

6,.,

2-bu

tano

ne a

t 29

3 K

; 0

, to

luen

e at

29

8 K

; 0

, cy

cloh

exan

e at

298

K.

25

S

kele

tal

bond

s nw

5

0

10

0

50

0

log

( M

j/d

l.g

-I)

-1·0

-1·5

-2·0

.//

/1

t .

... 3.0

3-5

4·0

L

og

(Mw

/g·m

ol-

l )

4-5

FIG

. 14

. Pl

ots

of l

og [,

.,] v

ersu

s lo

g M

w f

or s

ever

al f

ract

ions

o

f cyc

lic P

DM

S:i ., 2

-but

anon

e at

293

K; ., to

luen

e at

29

8 K

;.,

cyc

lohe

xane

at

298

K.

." ~ ;:g ~ ..., m

'" ~ n -< n t"' es :> z o t"' Z

~ ~ ." o t"

' -< B ~ ..., :t

-< t"' '" § >< :> ~ ~

..-

VI

152 CHRISTOPHER J. C. EDWARDS AND ROBERT F. T. STEPTO

effects of excluded volume on this ratio. Readers are referred to a discussion in Ref. 44 from which it appears that the balance of opinion favours a decrease in ['11r/['111 with increasing excluded volume in concurrence with the experimental data. 3 This ratio is governed by the differing effects of chain expansion on equilibrium radii and hydrodynamic radii for cyclic and linear species.

Translational Diffusion Coefficient (D) Values of the ratio DtJDr for linear and cyclic PDMS in dilute solution were first obtained by Edwards et al. 28,29 using the classical boundary-spreading technique. In the impermeable limit the value of 0·84 ± 0·01 found for this ratio concurs within experimental error with that predicted by theory, 8,9,11

namely DtJDr = 8/3n (=0·85). Subsequent investigations using quasi­elastic light scattering (QELS), by Edwards et al., 31 quasi-elastic neutron scattering by Higgins et al.32 and diffusion into PDMS networks by Garrido et al. 33 have confirmed this value. The most detailed analyses of DI and Dr have concerned the boundary-spreading data and subsequent discussion is focused on these.

Figure 15 shows plots of logDI and 10gDr in toluene at 298 K versus lognn (where nn denotes the number-average number of bonds). At

30

2·5

2·0

05 10 1·5 20 25 3-0 Log (nn)

FIG. 15. Loglo D for linear (0) and cyclic (.) poly(dimethylsiloxanes) in toluene at 298 K plotted against loglo nn' The dashed straight lines were constructed using a linear least-squares procedure over the range 15 < nn < 100. nn denotes the

number-average number of skeletal bonds.

PROPERTIES OF CYCLIC AND LINEAR POLY(DIMETHYLSILOXANES) 153

intermediate chain lengths the slopes of the plots are just less than - O· 5 illustrating the approximate proportionality of hydrodynamic and equilibrium radii. The upturns at low molar mass are due to the assumption that nn is proportional to hydrodynamic radius and the downturn above nn :::= 100 reflects chain expansion in a good solvent. The ratio DdDr = 0·84 ± 0·01 over the complete range of chain lengths showing that at high molar mass the effects of excluded volume on this ratio are small, as predicted theoretically.8.9.11

2·5

2·

1·5

0·5 1·0 1-5 2·0 2·0 Log (x)

FIG. 16. Log lO D plotted against loglO x (where x is the number-average number of -fCH3hSiO- friction centres per molecule) for linear and cyclic PDMS in bromocyclohexane at 288 K(linear (0) and cyclic (.» and at 301 K(linear(O) and

cyclic (.».

Figure 16 shows similar plots for cyclic and linear PDMS in bromocyclohexane which is a poor solvent. 29 However, in this case the abscissa is taken as log (x) where x denotes the number-average number of -fSi(CH3h0t- units per molecule. It can be seen that the ratio DdDr tends to unity as x decreases. The data from Fig. 15 also show this behaviour when plotted versus log (x), reflecting the fact that for a cyclic molecule nn = 2x, whereas for a linear molecule nn = 2x - 2.

F or free-draining flow the total molecular friction coefficient f = k T/ D

154 CHRISTOPHER J. C. EDWARDS AND ROBERT F. T. STEPTO

(where k is the Boltzmann constant and T is the absolute temperature) is given by:

J=x.( (6)

where (is the segmental friction coefficient. SinceJin eqn (6) is independent of the spatial distribution of friction centres in the molecule, the ratio DIIDr = !rIft = 1. Thus the observed tendency of the ratio DIIDr to unity as x decreases reflects the increasing importance of free draining for short chain molecules. This feature is discussed in more detail later in the last section of this chapter.

F or x> 50, DII Dr = 0·83, in sensible agreement with the value in toluene. The slight downward curvature of the plots of log D versus log x at high molar mass is characteristic of chain expansion in good solvents. However, bromocyclohexane has previously been shown to be a 8-solvent for PDMS at 301 K45 and so the downward curvature in Fig. 16 cannot simply be accounted for in this way.

The data in Fig. 16 for linear PD MS at 301 K are in good agreement with those of Haug and Meyerhoff45 showing that a slope of less than -t persists at high molar mass. 29 On the other hand, second virial coefficients of linear PDMS in bromocyclohexane have been found to be zero30 and [rill has been found45 to be proportional to M;P and to be characterized by a value of <s6)/nf2 which is close to the value found in other O-solvents.46

This anomalous diffusion behaviour at high molar mass appears to occur for both linear and cyclic PDMS in bromocyclohexane. It may be attributable to a weak specific solvation effect, 4 7 which is not detectable in viscous flow. More measurements are required before a satisfactory explanation can be provided for these observations.48

Concentration Dependence of D Concentration dependences of D for cyclic and linear PDMS in a good solvent (toluene at 298 K) have been measured by Edwards et al. using QELS 31 and boundary-spreading. 28 Figure 17 shows plots of D versus C

for several fractions of linear and cyclic PDMS with narrow molar mass distributions. Also shown in Fig. 17 are interpolated points from previous boundary-spreading data which correspond to the molar masses used in the QELS work. The agreement between the two techniques is within ± 5 % which is acceptable in view of the interpolation procedure used to obtain the boundary-spreading points.

The concentration dependences of D for cyclic and linear PDMS can be well represented by the usual relationship:

D(c) = Do(1 + kDc) (7)

PROPERTIES OF CYCLIC AND LINEAR POL Y(DIMETHYLSILOXANES) 155

12·6.-------------------,

12·4

12·2

8·2

2·2 3

2·0

+ ",",-_C---

1·6

+

1·4 L_=-___ ---e>----~;:::::::::::==i 1·2

100 20·0 C or c/gr1

FIG. 17. D in toluene at 298 K from boundary-spreading (D, ., +) and QELS (e, 0) measurements plotted against c and c, respectively. I: ., cyclic PDMS, Mw =296gmol- l . 2: D, linear PDMS, Mw =627gmol- l . 3: e, cyclic PDMS, Mw =11500gmol- l . 4: 0, linear PDMS, Mw =12400gmol- l . 5: e, cyclic PDMS, Mw =23200gmol- l . 6: 0, linear PDMS, Mw =23300gmol- l . +: Points interpolated with respect to molar mass (M) from boundary-spreading data and plotted at c = 15 g litre -I . The interpolated values of D correspond to those for

the values of M for the present systems 3 to 6.

156 CHRISTOPHER J. C. EDWARDS AND ROBERT F. T. STEPTO

TABLE 2 Diffusion Coefficients Do for Three Pairs of Linear and Cyclic PDMS Molecules in Toluene at 298 K, Second Virial Coefficients A 2' and the Coefficients in the Concentration Dependence of D(e) andf(e), Respectively (keo was calculated from

eqn (10))

Mw Do A2 kD kf kfo (gmol- 1 ) (J.lm2s- l ) (10- 4 em 3 g-2 mol) (em 3 g- l ) (em 3 g- l )

Linear, 23300 111·5 8·74 8·18 31·55 6·54 Cyclic, 23 200 130·5 6·55 2·33 27·12 8·96 Linear, 12400 173·5 9·69 3·36 19·66 8·17 Cyclic, II 500 217·5 7-89 -2·00 19·21 14·64 Linear, 627 820·5 15·76 -2·08 2·509 7·93 Cydic,296 1254 17·80 -1·503 1·505 6·65

Diffusion coefficients at zero concentration, Do, obtained by extrapolation of D(c) to zero concentration are listed in Table 2 together with the coefficients ko. The values of ko are positive at high molar mass and become increasingly negative with decreasing molar mass in agreement with previous data in a good solvent, 49 with the change in sign in ko occurring at a higher molar mass for cyclic PDMS.

The diffusion coefficient at finite concentration contains both thermo­dynamic and hydrodynamic contributions. Hence, ko is related to both A z and the coefficient in the concentration depenoence of the friction coefficient ke by the expression

ko =2A zM w - v -k f

where v is the partial specific volume of the polymer and

J(c) = Jo(1 + krc)

(8)

(9)

Values of A z have been measured by Edwards et at.30 using static light scattering for both cyclic and linear PDMS and values interpolated from these measurements and the data of Huglin and Sokro so for very low molar mass PDMS samples are listed in Table 2. Values of krcalculated using eqn (8) are similar for each cy<;lic-linear pair and decrease with decreasing molar mass as expected. Several theories of the concentration coef­ficient,51-S3 k f , agree that

(10)

when N A is the Avogadro constant and Vh is the hydrodynamic volume of

PROPERTIES OF CYCLIC AND LINEAR POLY(DIMETHYLSILOXANES) 157

the molecule. Thus the expression in parentheses represents the effective hydrodynamic volume of the molecule per unit mass and kro is a constant which characterizes the extent of coil interpenetration. Pyun and Fixman 52

have calculated values of k ro using an equivalent sphere, uniform segment­density model. They predict that in a 8-so1vent (maximum inter­penetration) kro = 2·27 and that in a good solvent (zero interpenetration) k ro = 7 ·18. The values of k ro found experimentally are listed in Table 2 and, with the exception of the cyclic sample with M w = 11 500 g mol- 1, the agreement with the good solvent limit of 7·18 predicted by Pyun and Fixman 52 is reasonable.

Second Virial Coefficient (A 2 )

The variation of A 2 with chain length which was determined by Edwards et al. 30 for cyclic and linear PDMS in toluene at 298 K from static light scattering measurements is shown in Fig. 18. Also included in Fig. 18 are data for low molar mass linear PDMS obtained by Huglin and Sokro 50 and

2·6

3·0

3·2

o 3·4

,0.

o

2.0 3·0 4·0 5·0 6·0

Log (Mwl

FIG. 18. LogA z plotted versus logMw for (0) linear and (e) cyclic PDMS in toluene at 298 K: 30 0, data of Huglin and Sokro50 for oligomeric linear PDMS; 0, data of Kuwahara et al. ;55 [::" Price and Bianchi;54 0, Kubata et al. ;56 --,

least-square lines through data; ---, calculated values of A 2r using eqn (11).

158 CHRISTOPHER J. C. EDWARDS AND ROBERT F. T. STEPTO

some previous data 54 - 56 for high molar mass linear PO MS which show no regular dependence on chain length. The data of Edwards et al. 30 are in good agreement with the low molar mass data of Huglin and Sokro 50 and would extrapolate to high molar mass to give approximate agreement with those of Price and Bianchi 54 and Kubata et al. 56

An expression for the ratio A 2.1 A 2/ has been derived by Yamakawa 44 in terms of the modified Flory, Krigbaum, Orofino theory (FKOm):

5·73i/ln(1 + 8·914i,) A 2,IA 2/ = 8.914i,ln (1 + 5.73z/) (11)

where the excluded volume parameter i = zlIY.; with z defined in the usual way and IY.; = <s2>I<s~>. Values of zhave been derived from experimental values of IY./ 1,28 using the modified Flory equation: 44

IY.; - IY.; = KZ (12)

Where for a linear molecule K = 1.276 44 and for a cyclic molecule K = n12. 7

The dashed curve in Fig. 18 shows the values of A 2,' calculated using eqns (11) and (12), relative to the experimental values of A 2/' The agreement with experiment is surprisingly good in view of the probable magnitude of the errors in the derived quantities and the approximate nature of eqn (11). In addition, the ratio A 2.1 A 2/ as defined in eqn (11) is predicted to tend to unity as IY.s -+1 (i.e. for Mw;S4 x 103 gmol- 1), in good agreement with experiment.

DETAILED KIRKWOOD-RISEMAN ANALYSIS OF DIFFUSION COEFFICIENTS

Analyses of the experimental diffusion data28 ,29 for linear35 and cyclic36

PDMS have been made in terms of the now-classical equation due to Kirkwood 57 ,58

kT kT [R- 1 ] D=-+--­

x, 6nl]0 x 2 (13)

The segmental friction coefficient ,= 6nl]oa, where a is the radius of a segment, 1]0 is the solvent viscosity and

[K- 1] = I <r~/> i,j= 1 i~j

(14)

PROPERTIES OF CYCLIC AND LINEAR POLY(DIMETHYLSILOXANES) 159

<riJ 1> is the mean reciprocal separation of segments i and}. Equation (13) is based on several assumptions which are still the subject of discussion in the literature. 59 - 63 However, the Kirkwood equation has provided a rationale of diffusion coefficients of unperturbed polymethylene,64.65 poly(ethyleneoxide)65,66 and poly(hexamethyleneoxide)65 chains in a variety of solvents in terms of the relative contributions of free-draining and impermeable flows to the translational diffusion coefficient. For this purpose eqn (13) can be recast35 ,65 in a form which is independent of solvent viscosity to give:

(15)

when rD is the Stokes-Einstein diffusion radius (=kT/6nY/oD), rF = x. a is the free-draining radius and rE = x 2/[R- 1 ] is the effective impermeable diffusion radius. For the Gaussian subchain model.of a linear chain of in­finite length, Kirkwood and Riseman 58 showed that rE = (3n 1/2/8) x <s~> 1/2. The universality of the numerical constant is now being questioned49,62,63,67 but this has little effect on the relative behaviour of linear and cyclic PDMS. For an infinite Gaussian ring rE = (n/2)1/2<s~>1/2, consistent36 with the value of Dl/Dr = 8/3n discussed previously.

Multiplication of each term in eqn (13) by XI/2 yields

XI/2 XI/2 XI/2 -=-+-r D r F r E

(16)

so that as x~ 00, xl/2/rF~0 and the other two terms tend to a constant value. Thus, eqn (16) enables the relative free-draining and impermeable contributions to D to be illustrated as functions of chain length. 35 ,36 For polymethylene in a variety of solvents it has clearly been shown65 that the effective segment radius a varies with solvent and with chain length in proportion to the effective bond length b = «r~>/n)I/2. The variation with solvent is thought to be due to the assumption of a solvent continuum and the variation with chain length probably arises from the neglect of chain dynamics in the Kirkwood equation.

Figure 19 shows plots of xl/2/rD' xl/2/rF and xl/2/rE versus x for linear PDMS in toluene28 and bromocyclohexane. 29 The data are those obtained by Edwards et al. which were discussed relative to equivalent data for cyclic PDMS earlier in this chapter. Curve 1 (x l /2/r F ) has been calculated assuming that the radius of a segment a = 0·25 nm. Curve 2 was calculated using the FCM RISM of PDMS and the Monte Carlo and exact enumeration techniques described earlier. A segment radius of 0·25 nm is a

160 CHRISTOPHER J. C. EDWARDS AND ROBERT F. T. STEPTO

-'-

.,

'-'-. -'-. -'-'-

- - -- - -"3

- - --2

'-'-'-°O~--------O~--------20~--------30~--------40~--------~

x FIG. 19. Reciprocal diffusion radii versus chain length (x) for «) linear PDMSItoluene/25 °C and (~) PD MS/bromocyclohexane/15 °C. Curves: (I) Free­draining radius, xl/2IrF' a =0'25nm; (2) impermeable radius, x l /2IrE; (3) Kirkwood-Riseman diffusion radius, Xl /2 Ir D; (4) and (5) calculated diffusion radius

with a = (cb)b, using (4) (cb ) = 0·509 and (5) (cb ) = 0·405.

reasonable estimate of the size of an -1Si(CH3h0t- group and gives good agreement between the calculated reciprocal Stokes-Einstein radius (curve 3) and the experimental data for x ~ 10.

The upturn in the experimental data at low x cannot be accounted for with a constant value of the segmental friction coefficient ,. As for polymethylene, the change in effective segment radius can be written

(17)

with a = cb • b and with cb approximately independent of chain length, but dependent on polymer, solvent and temperature. Values of Cb required to reproduce experimental values of D are given in Table 3. The average values of cb (denoted <cb» for toluene and bromocyclohexane are 0·509 and 0,405, respectively. As for polymethylene, <cb> generally decreases as solvent viscosity (110) and molar volume (Vo) increase leading to its interpretation65 as a 'slippage' parameter. Use of the quoted values of <cb> to define a and hence 'Fgives rise to curves 4and 5 in Fig. 19, showing better agreement with the data over all chain lengths.

PROPERTIES OF CYCLIC AND LINEAR POLY(DIMETHYLSILOXANES) 161

TABLE 3 Calculated Values of cb on the Basis of -iSiMe20r-- Segments for Linear PDMS in

Toluene and Bromocyclohexane at 298 K

Toluene" Bromocyclohexaneb

nn c xd blnmd c d b

n C n x d blnm d

6 4 0·447 0·419 6 4 0·477 8 5 0·478 0-433 8 5 0-478

14 8 0·520 0·511 16·5 9·25 0·528 26 14 0·543 0·560 28 15 0·544 36·5 19·25 0·546 0·663 55 28·5 0·555 94 48 0·560 0·468

" Toluene: <cb> = 0·509; 110 = 0·522cP; Vo = 106·4cm3 mol-I. b Bromocyclohexane: <cb> =0·405; 110 =2·44cP; Vo = 122·5cm3 mol- l .

C nn = Number-average number of bonds in chain. d x, b, Cb' 110' Vo: As defined in text.

c d b

0·379 0·425 0·429 0·396 0·397

8r---------------------------------------------------~

• _______ - - - - -- _ _ _ __ _ _ _ 4 • • ....- - ~ _. - •. -- . ___ . -- . -- . -- . ~- . .::. 5

6 •• / ...- • 3 1/ /' + +

I .

nm-I I / I .

'/ 4 I.

{/

2

- - ---'--- - _I -2 OLL ________ L-________ L-________ L-________ L-________ ~

10 20 30 40 50 x

FIG. 20. Reciprocal diffusion radii versus number of friction centres (x) for cyclic PDMS in (e) toluene and (.) bromocyclohexane solution at 298K. Curves: (1) and (2) free-draining radii x l/2lrF with r F = X . a and a = 0·25 nm and a = 0·4 nm, respectively; (3) impermeable diffusion radius, xl/2lrE with rE = x2 I[R- 1]; (4) and (5) Kirkwood diffusion radii, xl/2lrD (sums of curves (1) and (3) and curves (2) and

(3), respectively): x, complete enumeration; +, Monte Carlo calculations.

162 CHRISTOPHER J. C. EDWARDS AND ROBERT F. T. STEPTO

Figure 20 shows a similar analysis 36 of experimental diffusion data 28,29 for cyclic PDMS. A smooth curve (curve 3) has been drawn through the calculated points which show some scatter due to uncertainties in the Monte Carlo calculations. It is apparent from Fig. 20 that a segment radius of 0·25 nm (curve 4) as found for linear (PDMS)35 overestimates the experimental diffusion coefficient for x> 7. Agreement between experi­ment and theory (curve 5) requires that a is increased to c. 0·4 nm when it is again noticeable that the effective segment radius decreases with decreasing chain length for x;S 10. As for linear PDMS, a better fit to the data at all chain lengths can be obtained by allowing, to vary in proportion to an effective bond length. 36 The difference between the values of c. 0-4 nm and c. 0·25 nm for the segment radii of cyclic compared with linear PDMS is consistent with less free-draining flow in the cyclic molecules as would be expected from their higher segment densities.

CONCLUSIONS

The investigations described represent the first detailed studies of a homologous series of cyclic polymers. The basis for comparison has been the relatively well-understood properties of linear PDMS: the natural starting point in the absence of other well-fractionated series of flexible cyclic polymers.

In the high molar mass limit and in the absence of excluded volume effects it is noteworthy that the limiting behaviour predicted by Gaussian, equivalent subchain models for various molecular properties has gener­ally been confirmed. Thus, <s;)/<sf) =0·5,5-7 [17]r/[17]/=0·67 8 - 10 and Dti Dr< = /,/11) = 8/3n. 4,8,11 In addition, the ratio of bulk viscosities39 17r/17/ tends to the predicted limit equal to the ratio of mean-square radii of gyration at high molar mass. Furthermore, refractive index,22 density, 22 29Si NMR chemical shift,26 dipole moment rati0 23 and Tg3 have equal limiting values for both cyclic and linear PDMS at high molar mass.

The critical chain lengths above which Gaussian limiting ratios of properties are attained vary greatly. For radii of gyration it is at w ~ 30, for diffusion coefficients it is w ~ 20 and for bulk viscosities w ~ 400.

The ways in which molecular and bulk properties change with chain length highlight interesting and distinct differences between linear and cyclic PDMS which relate to the detailed chain structures. Thus, at low molar mass (w < 30), density, refractive indices, radii of gyration and bulk viscosities for rings reflect changes in molecular shape and configurational

PROPERTIES OF CYCLIC AND LINEAR POLY(DIMETHYLSILOXANES) 163

restriction which are probably characteristic of the particular geometry of the PDMS chain. 19

Detailed evaluations of <s~> and D for cyclic PDMS have been carried out using Monte Carlo and complete enumeration techniques. 19•20,35,36

They show the importance of taking proper account of chain structure when correlating experimental data. In particular, correlations of D show that the Gaussian subchain model should be used with caution for relatively low molar mass species.

The ratio of intrinsic viscosities [11]r/[I1]1 shows a systematic decrease under the effects of excluded volume. As expected the ratio Dl/Dr is less sensitive than [11]'/[11]1 8 ,9,11 to excluded volume and the ratio A2r/A21 < I and decreases with molar mass in the general manner predicted by theory.44

Although this review has reported numerous investigations, some of these require further work before a complete understanding of the properties of cyclic PDMS emerges. Further properties of cyclic PDMS could usefully be investigated and compared with those of linear PDMS. Such measurements would include bulk and surface thermodynamic properties and the phase behaviour of cyclic PD MS solutions and mixtures of cyclic and linear PDMS. Also spectroscopic investigations may reveal differences in the chain dynamics of linear and cyclic species.

It is hoped that the interesting, and often distinct, behaviour of cyclic compared with linear PDMS discussed in this review will give impetus to the preparation and characterization of different homologous series of cyclic polymers. A general understanding of the relative behaviour of cyclic and linear molecules is not possible on the basis of PDMS alone.

REFERENCES

1. Dodgson, K. and Semlyen, J. A., Polymer, 18 (1977) 1265. 2. Dodgson, K., Sympson, D. and Semlyen, J. A., Polymer, 19 (1978) 1987. 3. Clarson, S. J., Dodgson, K. and Semlyen, J. A., Polymer, 26 (1985) 925. 4. Graessley, W. W., Adv. Polym. Sci., 16 (1974) 1. 5. Kramers, H. A., J. Chern. Phys., 14 (1941) 415. 6. Zimm, B. H. and Stockmayer, W. H., J. Chern. Phys., 17 (1949) 1301. 7. Casassa, E. F., J. Polym. Sci. (A), 3 (1965) 605. 8. Bloomfield, V. and Zimm, B. H., J. Chern. Phys., 44 (1966) 315. 9. Fukatsu, M. and Kurata, M., J. Chern. Phys., 44 (1966) 4539.

10. Yu, H. and Fujita, H., work cited by Yamakama, H., In: Modern Theory of Polymer Solutions, Harper and Row, New York, 1971.

II. Kumbar, M., J. Chern. Phys., 55 (1971) 5046; 59 (1973) 5620.

164 CHRISTOPHER J. C. EDWARDS AND ROBERT F. T. STEPTO

12. Edwards, C. J. C., Richards, R. W., Stepto, R. F. T., Dodgson, K., Higgins, J. S. and Semlyen, J. A., Polymer, 25 (1984) 365.

13. Burchard, W. and Schmidt, M., Polymer, 21 (1980) 745. 14. Sole, K. and Stockmayer, W. H., J. Chern. Phys., 54 (1971) 2756. 15. Sole, K. and Stockmayer, W. H., J. Chern. Phys., 55 (1971) 355. 16. Lal, M. and Spencer, D., In: Applications of Computer Techniques in Chemical

Research (ed. P. Hepple), Elsevier Applied Science Publishers Ltd, London, 1972.

17. Sole, K., Macromolecules, 6 (1978) 378. 18. Mattice, W. L., Macromolecules, 13 (1980) 506. 19. Edwards, C. J. c., Rigby, D., Stepto, R. F. T. and Semlyen, J. A., Polymer, 24

(1983) 395. 20. Edwards, C. J. c., Rigby, D., Stepto, R. F. T., Dodgson, K. and Semlyen, J. A.,

Polymer, 24 (1983) 391. 21. Higgins, J. S., Dodgson, K. and Semlyen, J. A., Polymer, 20 (1979) 553. 22. Bannister, D. J. and Semlyen, J. A., Polymer, 22 (1981) 377. 23. Beevers, M. S., Mumby, S. J., Clarson, S. J. and Semlyen, J. A., Polymer, 24

(1983) 1565. 24. Dodgson, K., Bannister, D. J. and Semlyen, J. A., Polymer, 21 (1980) 663. 25. Burton, B. J., Harris, R. K., Dodgson, K., Pellow, C. J. and Semlyen, J. A.,

Polymer Communications, 24 (1983) 278. 26. Wright, P. V., J. Polym. Sci. (Polym. Phys. Edn), 11 (1973) 51. 27. Semlyen, J. A., Adv. Polym. Sci., 21 (1976) 41. 28. Edwards, C. J. c., Stepto, R. F. T. and Semlyen, J. A., Polymer, 21 (1980) 781. 29. Edwards, C. J. c., Stepto, R. F. T. and Semlyen, J. A., Polymer, 23 (1982) 865. 30. Edwards, C. J. c., Stepto, R. F. T. and Semlyen, J. A., Polymer, 23 (1982) 869. 31. Edwards, C. J. C., Bantle, S., Burchard, W., Stepto, R. F. T. and Semlyen,

J. A., Polymer, 23 (1982) 873. 32. Higgins, J. S., Ma, K., Nicholson, L. K., Hayter, J. B., Dodgson, K. and

Semlyen, J. A., Polymer, 24 (1983) 793. 33. Garrido, L., Mark, J. E., Clarson, S. J. and Sem1yen, J. A., Polymer, 25 (1984)

218. 34. Flory, P. J., Statistical Mechanics of Chain Molecules, Wiley Interscience, New

York, 1969. 35. Edwards, C. J. c., Rigby, D. and Stepto, R. F. T., Macromolecules, 14 (1981)

1808. 36. Dodgson, K., Edwards, C. J. C. and Stepto, R. F. T., British Polymer Journal,

17(1985) 14. 37. Flory, P. J., Crescenzi, V. and Mark, J. E., J. Am. Chern. Soc., 86 (1964) 146. 38. Wood, W. W., In: Physics of Simple Liquids (eds H. N. V. Temperley, J. S.

Rowlinson and G. S. Rushbrooke), North Holland Publishing Company, Amsterdam, 1968.

39. Bueche, F., J. Chern. Phys., 40 (1964) 84. 40. Dasgupta, S., Garg, S. K. and Smyth, C. P., J. Am. Chern. Soc., 89 (1967) 2243. 41. Dasgupta, S. and Smyth, C. p", J. Chern. Soc., 47 (1967) 2911. 42. Sutton, C. and Mark, J. E., J. Chern. Phys., 54 (1971) 5011. 43. Flory, P. J., Mande1kern, L., Kinsinger, J. B. and Schultz, W. B., J. Am. Chern.

Soc., 74 (1952) 3364.

PROPERTIES OF CYCLIC AND LINEAR POL Y(DIMETHYLSILOXANES) 165

44. Yamakawa, H., Modern Theory of Polymer Solutions, Harper and Row, New York,1971.

45. Haug, A. and Meyerhoff, G., Makromol. Chem., 53 (1962) 91. 46. Crescenzi, V. and Flory, P. J., l. Am. Chem. Soc., 86 (1964) 141. 47. Munch, J. P., Herz, J., Boileau, S. and Candau, S. J., Macromolecules, 14

(1981) 1370. 48. Bohdanecky, M., Edwards, C. J. c., Horska, J., Stepto, R. F. T. and Semlyen,

J. A., work in progress. 49. ter Meer, H-U., Burchard, W. and Wunderlich, W., Colloid Polymer Sci., 258

(1980) 675. 50. Huglin, M. B. and Sokro, M. B., Polymer, 21 (1980) 651. 51. Yamakawa, H., l. Chem. Phys., 36 (1962) 2995. 52. Pyun, C. and Fixman, M., l. Chem. Phys., 41 (1964) 937. 53. Imai, S. J., l. Chem. Phys., 50 (1969) 2116. 54. Price, F. P. and Bianchi, J. P., l. Polym. Sci., 15 (1955) 355. 55. Kuwahara, N., Okazawa, T. and Kaneko, M., l. Polym. Sci. C, 23 (1968) 543. 56. Kubata, K., Kubo, K. and Ogina, K., Bull. Chem. Soc. lpn, 49 (1976) 2410. 57. Kirkwood, J. G., l. Polymer Sci., 12 (1954) 1. 58. Kirkwood, J. G. and Riseman, J., l. Chem. Phys., 16 (1948) 565. 59. Zimm, B. H., Macromolecules, 13 (1980) 592. 60. Garcia de la Torre, J., Jimenez, A. and Freire, J. J., Macromolecules, 15 (1982)

148. 61. Garcia de la Torre, J. and Freire, J. J., Macromolecules, 15 (1982) 155. 62. Fixman, M., Macromolecules, 14 (1981) 1710. 63. Edwards, C. J. c., Kaye, A. and Stepto, R. F. T., Macromolecules, 17 (1984)

773. 64. Hill, J. L. and Stepto,R. F. T., Proc. IUPAC Int. Symp. Macromol. Helsinki, 3

(1977) 325. 65. Mokrys, I. J., Rigby, D. and Stepto, R. F. T., Ber. Bunsenges. Phys. Chern., 83

(1979) 446. 66. Couper, A. and Stepto, R. F. T., Trans. Faraday Soc., 65 (1969) 2486. 67. Burchard, W. and Schmidt, M., Macromolecules, 14 (1981) 210.

CHAPTER 5

Neutron Scattering from Cyclic Polymers

KEITH DODGSON

Department of Chemistry, Sheffield City Polytechnic, Sheffield, UK

and

JULIA S. HIGGINS

Department of Chemical Engineering, Imperia] College, London, UK

INTRODUCTION

The particular advantage of neutrons for investigating polymer systems arises from the difference in scattering properties of hydrogen and deuterium, which allows manipulation of the contrast between various parts of the scattering system. This is of most use for studying individual molecules in concentrated solution and bulk samples, but it can be important even for studying dilute solutions in situations where the natural contrast for light or for X-rays is too weak. The wavelength of the thermal neutrons used (1-10 A) allows exploration of spatial ranges comparable with X-ray scattering, but the low energies of the relatively heavy neutrons mean that spectroscopic experiments match more closely with infrared or photon correlation techniques.

For information about neutron scattering in general1.2 and scattering from polymeric systems,2 - 4 the reader is referred to the extensive literature some few examples of which have been cited. From the point of view of information relevant to cyclic polymer molecules, two techniques have

167

168 KEITH DODGSON AND JULIA S. HIGGINS

been important: (i) small angle neutron scattering (SANS) for the study of polymer conformations and dimensions; and (ii) high resolution quasi­elastic scattering which gives information on both dimensions, via measurement of diffusion coefficients, and internal motion. This chapter will discuss the principles of these two techniques and explore the relevance of the experimental results obtained for cyclic polymers.

NEUTRON SCATTERING PRINCIPLES

Neutron energies in these experiments are very much lower than the nuclear binding energy so the scattering from an isolated nucleus is isotropic and the interaction characterized by a single parameter: the scattering length, or amplitude, b. The scattering cross-section, a, is the ratio of scattered neutrons to the incident flux, and for an isolated stationary nucleus

(1)

In a scattering sample containing nuclei with spin, or for an isotopically impure sample, the scattering amplitude which varies with the spin state of nuclei and from isotope to isotope thus varies from site to site.

In a scattering event from stationary nuclei there can be no energy transfer, but the wave vector k of the neutrons changes (k is a vector in the direction of travel of magnitude Ikl = 2n/ A = mv/Ii). The momentum transfer IlQ is defined in terms of the change in wave vector on scattering

(2)

where subscripts i and frefer to the initial and scattered beams, respectively. However, since there is no energy transfer Ikd = Ikrl and simple trigonometry shows

Q = IQI = (4n/A) sin el2 (3)

where e is the angle of scatter. The differential cross-section per atom with respect to solid angle

contains information about the spatial arrangements of the scattering nuclei. In a sample containing N atoms

(4)

n

NEUTRON SCATTERING FROM CYCLIC POLYMERS 169

where Rn is the position vector of the nth nucleus. This expression can be manipulated to give

(5)

where < ) implies averaging over all sites. Only the last term in eqn (5) contains spatial information: the coherent

scattering. The coherent cross-section is given by the mean-square scattering amplitude, (jcoh = 4n<b)2. The first term is constant in a static experiment and forms the incoherent background. The term (jinc = 4n«b2) - <b)2) can also be written as 4n«b - <b»))2, i.e. the mean­square deviation of the scattering lengths from their average value. For an isotopically pure sample of nuclei such as 12C or 16 0 which have zero spin, (jinc itself is zero. For hydrogen it is very large. Table 1 lists values of b, (jinc and (jcoh' together with the absorption cross-section (jabs for nuclei commonly found in synthetic polymers and their solvents.

TABLE 1 Scattering Lengths and Cross-sections for some Common Isotopes

Isotope b x J012 (leoh X 1024 (line X 1024 (labs X 1024

(em) (em 2 ) (em 2 ) (em 2 ) (at 1-08 A)

'H -0-374 1·76 80 0·19 2D 0·667 5·59 2 0-0005 12C 0·665 5-56 0 0·003 14N 0·94 11·1 0-3 I-I 160 0-58 4·23 0 0-0001 19F 0·56 3·94 0·06 0·006 ave_ 2s-06Si 0-42 2·22 0 0·06 32S 0·28 0·99 0 0·28 ave. 3s -sCl 0·96 11·58 3·5 19·5

In general, if the scattering nuclei are in motion on the time scale of the experiment, Ikd ¥-Ikrl. It is then the double differential cross-section d 2(j/dQdEwhich is measured in the experiment and the incoherent term is no longer a flat background, since it carries spatially uncorrelated information about motion of the scattering nuclei. Although for convenience the coherent and incoherent terms are expressed and discussed separately, experimentally they may be very difficult to separate. The

170 KEITH DODGSON AND JULIA S. HIGGINS

scattering cross-sections are related to two space-time correlation functions introduced by Van Hove: 5 G(R, t) and Gs(R, t). G(R, t) ,expresses the probability that if there is a nucleus at position Rj at time t = 0, there will be another nucleus at position Rj at time t. Gs(R, t) is a self-correlation function, expressing correlations between the same nucleus at time t = ° and t.

d 2(j k <b)2 If

d dCOh =~-""'- dRdtexp{i(Q.R-wt)}G(R,t) n E f k j 2nn (6)

:~d~f =:: «b2 );:'Ii<b)2) I I dRdtexp {i(Q. R - wt)}Gs(R, t) (7)

where

"Ii 2 2 2 "liw = I1E= Ef - E. =- (k f -k.) 1 2m 1

(8)

The integrals in eqns (6) and (7) contain all the information about the scattering system and are usually called the scattering laws, with

Scoh(Q, w) = 2ln I fexp {i(Q. R - wt)}G(R, t)dRdt (9)

and an analogous definition for S.(Q, w) in terms of Gs(R, t). If only the spatial Fourier transforms (in eqn (9) and its incoherent analogue) are performed, the quantities obtained are the so-called intermediate scattering functions Scoh(Q, t) and Ss(Q, t).

SMALL ANGLE NEUTRON SCATTERING

Principles We consider the development of the elastic scattering in eqn (4). Assuming the incoherent scattering has been removed and the system is incom­pressible, the second term may be written for a mixture of components

(10)

We assume small scattering angles such that spatial correlations are explored over distances, Rij, large compared with interatomic spacing and Oi' OJ are average scattering lengths per unit volume. Sij(Q) is the scattering

NEUTRON SCATTERING FROM CYCLIC POLYMERS 171

law arising from all pair correlations i-j within the systems; Sij(Q) =

exp(iQ. Rij)' For an incompressible system

(11)

and thus allows eqn (10) to be reduced for a two-component system to

(12)

It is usual when dealing with polymer solutions to split S22(Q) into two terms, one referring to intramolecular correlations, p2(Q), and a second referring to intermolecular interactions, Q22(Q).

Equation (12) then becomes

(dO") - - 2 2 2 dO = (b 2 - b l )[N2Z 2P2(Q) + N 2Z 2Q22(Q)] coh

(13)

where Z 2 is the degree of polymerization and N 2 the number of molecules per unit volume.

If we substitute N2 = cNA/Mw and Z2 = Mw/m where Mw is the molar mass, c the concentration in g cm - 3, m the monomer molar mass and NA Avagadro's number, eqn (13) becomes

(dO") (52-51) 2 dO COh= m 2 N A[cMwP2(Q)+cNAQ22(Q)] (14)

The intermolecular function Q22(Q) can be written in terms of the Flory-Huggins interactions parameter, X12' and the osmotic pressure second vi rial coefficient A 2 (Q):

For c ~ I, eqn (14) can be inverted to the familiar Zimm form

Rc

I(Q)

(15)

(16)

where R = «NA(51 - 52)2)/m)D and D is a normalization constant for the spectrometer used. I(Q) is now the observed scattering intensity as a

172 KEITH DODGSON AND JULIA S. HIGGINS

function of Q so D takes account of incident beam intensity, detector geometry, etc. For a Gaussian chain of N steps oflength I, P(Q) will be the familiar Debye function

(17)

where /l = N1 2Q2/6. For such a chain «NI2)/6)1/2 = (S2)1/2 = R g, the radius of gyration of the polymer molecule. When /l ~ 1, eqns (16) and (17) reduce to

(18)

«S2)1/2 will be a z-average value for a polydisperse sample) whereas at high Q, P(Q)-+21/l and

Rc Q2(S2) Q(S2)1/2 ~ 1 + 2A (Q) (19)

I(Q) = 2Mw 2 c

Thus the ratio of the slopes at high and low values of Q of a Zimm plot (1II(Q) against Q2) will be 3/2 at infinite dilution. For cyclic polymer molecules it has been shown that 6

for (/l ~ 1)

for (/l ~ I)

P-l(Q) = (1 + /l2/6)

P-l(Q)::= t(/l- 2)

(20)

(21)

Thus the slope of the Zimm plot at low Q is predicted to be half of that for a Gaussian linear polymer of the same molecular weight. The effective radius of gyration of a cyclic polymer molecule is predicted to be «NI2)/12)1/2 which is l/fi that for the equivalent Gaussian linear polymer chain.

The slope of a Zimm plot at high Q values, governed by local conformation, is predicted to be the same for cyclic and linear polymers (i.e. /l). Thus for a cyclic polymer, the ratio of the slopes of a plot of III(Q) against Q2 in the high and low Q limits should be 3, and the function is therefore expected to appear much more curved than that for the corresponding linear polymer.

Further discussion of the scattering functions of cyclic polymers is presented in the sections devoted to experimental results.

Experimental Considerations Apparatus Figure I shows a typical small angle neutron scattering apparatus. 7 The main features are: (i) the velocity selector, which allows a choice of

NEUTRON SCATTERING FROM CYCLIC POLYMERS

FROM REACTO R

~ __ selector

r 1;: :::::::" ~J~ sample

I I

I I

I I

I I

I i'---

/

multidetector

173

FIG. 1. D178 small angle neutron scattering spectrometer at the Institut Laue­Langevin, Grenoble.

wavelength typically within the range 2 < A. < 15 A available from the Maxwellian distribution of a thermal reactor; and (ii) the large area detector which increases the counting rate by allowing simultaneous measurements over a range of values of e. The range of e available is extended by changing the sample to detector distance. Spectrometers in which the whole detector can be positioned ofT-axis (as seen in the diagram) have been constructed for cases where measurements at very high values of e are required. By these means the Q-range available is 10 - 3 < Q < I A - 1,

the lower limit being determined by the maximum sample to detector distance.

Samples As described, the contrast in an SANS experiment is usually obtained by

174 KEITH DODGSON AND JULIA S. HIGGINS

exploiting the difference between the scattering lengths of hydrogen and deuterium. Thus, in solution experiments the solvent is usually a perdeutero analogue. The beam area is usually about 1 cm2 and the sample path length, calculated for 50 % transmission, is between 2 and 5 mm. Liquid samples are contained in quartz cells or cells with single crystal aluminium windows: materials with low incoherent cross-sections and little structure in the relevant scale of dimensions. Normalization for the instrument factors included in the term D in eqn (16) is made using the observed incoherent scattered intensity from a sample of water in the same experimental geometry. 9

Experimental Results Determination of the Radii of Gyration of Cyclic and Linear Polymers A variety of neutron scattering measurements have been made on cyclic and linear poly(dimethyl siloxanes) (PDMS), poly(phenylmethyl siloxanes) (PPMS) and polystyrenes (PSTy), using spectrometers at the ILL, Grenoble, France, and at AERE, Harwell, UK.

The cyclic poly(dimethyl siloxanes) [(CH3)2SiO]x studied were fractions with z-average values of x in the range 7 <.xz < 275. The linear poly(dimethyl siloxanes) (CH3)3SiO[(CH3)2SiO]ySi(CH3)3 studied for the purposes of comparison were fractions with z-average values of y in the range 7 < Jlz < 280. The heterogeneity indices of all of the cyclic and linear PDMS fractions were in the range 1·01 < Mw/Mn < 1,1. 10,11

The scattering observed at 292 K from solutions of the PDMS fractions in benzene-d6 was found to be isotropic in all experiments. Therefore, the measured neutron counts were integrated over equal radial distances from the centre of the detector to give radial distributions of scattered neutron intensities for the solutions. Corrections were applied for background scattering and then the distributions were each normalized using the procedures described in the previous section to give the scattering intensity functions I(Q).

The z-average radii of gyration (S2);/2 for molecules in each of the cyclic and linear PDMS fractions were obtained by plotting c/I(Q) against Q2 (see Fig. 2 for some examples) and using the relationship derived from eqn (18)

(S2)1!2 = {3 .x gradient}1!2 z mtercept

(22)

Benzene-d6 is expected to be a 'good' solvent for PDMS at 292 K since

NEUTRON SCATTERING FROM CYCLIC POLYMERS 175

10 a

c= 00349 j 8

6 • 1 c=0·00B7

4

d

--u 10 ""' b 0

c=0-0362 8

6

4

2

0 2 4 6 8 10 12 14

104 Q2(A-2)

FIG. 2. Examples of the effect of concentration on the plots of the inverse neutron scattering intensities cj I(Q) against the scattering vectors Q2 for (a) cyclic PDMS fraction R12, and (b) linear PDMS fraction LlO, in benzene-d6 at 292 K (concentrations c are in gcm -3). The arrows indicate the upper limits of the Guinier

ranges where Q<S2>:/2 = 1.

benzene is a 'good' solvent for linear PDMS at this temperature. 12 Hence the experimental (S2) z values were expected to be concentration dependent as a result of excluded volume effects. Therefore, measurements were made at several concentrations (typically in the range 0·5 < c < 5 wt %) for each of the cyclic and linear PDMS fractions studied. Plots of 1/(s2)z against concentration c were used to obtain values for the mean square z-average

176 KEITH DODGSON AND JULIA S. HIGGINS

radii of gyration at infinite dilution <S2)z.c=0 by making use of the relationship 1 3

1 1 --2 - = < 2) {I + 2A 2M wRc } <s >z s Z,c=O

(23)

The molar masses and radii of gyration <S2);!c2=0 for the cyclic and linear PDMS fractions are given in Table 2.

In Fig. 2, cj I(Q) values are plotted against Q2 values for the cyclic and linear PDMS fractions Rl2 and LlO (high molar mass fractions). The plots relate to the highest and lowest concentrations that were studied. The

TABLE 2 Molar Masses and Radii of Gyration <S2>~!c2=O of the Cyclic and Linear PDMS

Fractions at 292 K Measured by SANS

Fractions M. ii. Mw/Mn <S2>~!c2=O (gmo/- 1) (A)

Rl 530 14 1·01 4·9 R2 600 16 1·02 5·0 R3 830 22 1·04 5·8 R4 940 25 1·10 5·9 R5 1110 30 1·01 7·1 R6 1420 38 1·01 8·0 R7 2290 62 1·04 10·1 R8 2880 78 1·03 11·0 R9 4950 133 1·05 13·7 RIO 7540 203 1·05 18·5 Rll 11640 314 1·06 23·7 R12 20210 545 1·04 34·1

LI 720 17 1·03 7·2 L2 860 21 1·01 8·1 L3 1 100 28 1·01 9·5 L4 1260 32 1·03 9·6 L5 2290 59 1-01 1306 L6 3720 98 1·10 16·8 L7 4990 132 1·05 18·6 L8 8670 231 1·05 25·2 L9 12890 345 1·05 33·8 LlO 20880 561 1·05 49-4

Mn , Mw and M. denote number-average, weight-average and z-average molar masses, respectively. iiz denotes z-average number of bonds.

NEUTRON SCATTERING FROM CYCLIC POLYMERS 177

arrows shown in Fig. 2 indicate the upper limits of the Guinier ranges (where Q<S2)1/2 = 1) calculated using the measured values of <S2);/2 for each of the solutions. Although the experimental data exceed the Guinier limit of Q<S2) 1/2 = 1 in some cases, the linearity of the data indicates that eqn (18) can be considered to apply for both cyclic and linear PDMS fractions over the range of scattering vectors used.

Experiments using the cyclic and linear PDMS fractions Rl2 and LlO were also carried out using a wider range of scattering vectors than are shown in Fig. 2. The plots of cj I(Q) against Q2 obtained are shown in Fig. 3. In this figure, the parameter Q<S2)1/2 is in the range 0·7 < Q<S2)1/2 < 3·2 for fraction R12, and 1·0 < Q<S2//2 < 4·6 for fraction LlO. Over a wide range of Q2 values the plot of cj I( Q) against Q2 for a monodisperse solution of a linear polymer chain would be expected to show upward curvature as the scattering vector is increased from within the Guinier range described in eqn (18), to the high Q range where eqn (19) is applied.

Slight upward curvature is observed in the plot of cj I(Q) against Q2 for the linear PDMS fraction LlO (see Fig. 3(a)). The limiting gradient of this plot in the range where Q2 > 6 x 10- 3 A - 2 is higher than the limiting gradient in the Guinier Q range by a factor of approximately 1·2.

The plot of cj I(Q) against Q2 for the cyclic PDMS fraction Rl2 (see Fig. 3(b)) shows pronounced upward curvature in agreement with Casassa's prediction.6 Furthermore, the limiting gradient of this plot in the range where Q2 > 6 x 10- 3 A - 2 is higher than the limiting gradient in the Guinier Q range by a factor of2·6, which is close to the value of3 predicted for cyclic polymers. Further experimental and theoretical data concerning the scattering functions of cyclic and linear PDMS are presented in the next section.

In Fig. 4, values of log <S2 )z,c= 0 are plotted against log iiz for all of the cyclic and linear PDMS fractions listed in Table 2. The values of <S2>z,c=0 determined for the higher molecular weight linear fractions L 7-LlO have been found to be in good agreement with a range of calculated values deduced from data obtained by other experimental approaches. They include molar cyclization equilibrium constants,10 diffusion measure­ments 14 and theoretical calculations. 14,15 The reader is referred to Chapter 4 and to the literature cited for more detailed discussions.

Excluded volume effects have been shown to be present for linear PDMS with iiz ~ 100 in toluene. 16 There is also experimental evidence to suggest that chain expansion due to excluded volume effects is also significant for cyclic PDMS containing over 100 bonds in toluene and in other good solvents such as benzene-d6 used for the experiments described here (for

178 KEITH DODGSON AND JULIA S. HIGGINS

o ..... 25--. u

m o

20 -

o

b

I

5

• • •

103 Q2(A-2)

• • •

•

I

10

FIG. 3. Plots of the inverse neutron scattering intensities c/I(Q) against the scattering vectors Q2 for (a) linear PDMS fraction LIO, and (b) cyclic PDMS fraction R12, in benzene-d6 at 292 K. The broken lines ----- were obtained by

extrapolation, using the points in the linear Guinier regions of the plots.

NEUTRON SCATTERING FROM CYCLIC POLYMERS 179

3·0

0

" N-

/\. N

VI V

C>

01 2·0 0

....J slope =1

1·0 1-5 2·0 2·5 3·0

Log10 ii z

FIG. 4. Logarithmic plots of the z-average mean square radii of gyration <S2)z.<=0 at zero concentration of the linear (0) and cyclic (e) PDMS fractions against their

z-average numbers of bonds, in benzene-d6 at 292 K.

examples, see Chapter 4). The upward curvature of the plots shown in Fig. 4 for the samples with the higher molar masses (i.e. samples L7-LlO and R9-R12) is believed to be due to the onset of excluded volume effects.

The ratio <sf>l<s;> (l and r refer to linear and ring molecules, respectively) is predicted theoretically to be 2·0 for discrete linear and ring polymer molecules containing the same number of skeletal bonds and unperturbed by excluded volume effects. 6,15,17 -21 Least squares analysis of the data for samples L 7-LlO and R9-R12 gives a value for the ratio <sf>z,c=ol<s;>z,<=o=1·9±0·2 in the region iiz=500 (see Ref. 10). However, the error limits associated with the data do not allow a definite conclusion to be drawn as to whether or not the ratio <sf>zl<s;>z at infinite dilution is greater or less than 2·0 in good solvent media (see Refs 6 and 22-28 for conflicting predictions). The data plotted in Fig. 4 for samples Ll-L6 and RI-R8 also give a value for the ratio <sf>z,c=ol<s;>z,c=o = 2·0 ± 0·2 in the region iiz = 25. These experimental ratios are close to those

180 KEITH DODGSON AND JULIA S. HIGGINS

which have been predicted theoretically for cyclic and linear polymers and provide support for the validity of general treatments of the confor­mational statistics of 'flexible' large ring molecules.

A limited amount of small angle neutron scattering data has been obtained recently for cyclic and linear poly(phenylmethyl siloxanes).29,30 The z-average mean-square radii of gyration <S2) z of three cyclic and three linear PPMS fractions with iiz in the range 53 < iiz < 307 are plotted in Fig. 5. The data were obtained for PPMS in benzene-d6 solution at 293 K and are in good agreement with the theoretical prediction that <sf)/<s;) = 2·0 for unperturbed linear and cyclic polymers. In Fig. 5, the line drawn through the data for the linear PPMS fractions was fitted assuming a gradient of 1. The line drawn through the data for the cyclic fractions was constructed parallel to the line drawn through the data for the linear

N /'0.

10

NV) 2·5 '-/

0'1 o -l

2·0

1-5

3·5 4·0

Log 10 Mz

4·5

FIG. 5. Logarithmic plots of the z-average mean-square radii of gyration <S2)z of linear (0) and cyclic (.) PPMS fractions against their z-average molar masses, in

benzene-d6 at 293 K.

NEUTRON SCATTERING FROM CYCLIC POLYMERS 181

fractions and assuming the ratio (st)/(s;) = 2·0. It is noted that the radii of gyration measured for the cyclic and linear PPMS fractions are similar to those obtained for cyclic and linear PDMS fractions containing corresponding numbers of skeletal bonds.

Finally, the radii of gyration of five cyclic and five linear polystyrene samples with molar masses M in the range 1·25 x 104 < M < 2·2 X

104 g mol- 1 have been obtained by small angle neutron scattering. The data are presented in Chapter 6 of this book.

Theoretical and Experimental Particle Scattering Functions of Cyclic and Linear Silicone Polymers In this section, calculations of the particle scattering functions P(Q) for cyclic PDMS are presented. They are compared with Monte Carlo calculations of P(Q) for linear PDMS and with experimental scattering functions obtained using small angle neutron scattering. 31

The particle scattering function, P(Q), is related to the intensity of scattered radiation in the following way2,32,33

P(Q) = /(Q)I/(O) (24)

where /(Q) is the scattered intensity at scattering vector Q and /(0) is the corresponding intensity when Q = O.

The exact isotropic expression for P(Q) is34

(25)

where rij represents the separation of segments i andj and gi and gj are their respective contrast factors. The appropriate value of g for neutron scattering is given by

(26)

where p( = L blv) and Pm are the scattering length densities of a segment and of the background medium, respectively. L b is the total scattering length of a segment and v is its volume. The latter may be obtained from experimental densities or van der Waals' radii. Values of b for commonly occurring nuclei are given in Table 1.

Previously mentioned approaches to the calculation of P(Q) for ring molecules have used analytical methods. Casassa 6 and Burchard and Schmidt35 assumed that, for sufficiently large rings, the segmental pair distribution function differs from that of a Gaussian chain only in that the

182 KEITH DODGSON AND JULIA S. HIGGINS

first and last segments are joined together. Hence, the pair distribution function can be represented by the convolution of two Gaussian distributions for the subchains connecting any pair of segments. This method gives rise to the following expression6 ,35

(27)

where u = Q<S2)1/2 and 9fi(x) denotes the Dawson integral, which is tabulated in Ref. 36. The corresponding expression for a Gaussian chain, eqn (17), was first described by Debye. 37 Equation (27) predicts a maximum in the Kratky plot (u2 . P(u) against u) at u = 2'15, as shown in Fig. 6. As u -+ 00, the normalized Kratky plots for cyclic and linear species according to eqns (27) and (17) tend to asymptotic limits of 1·0 and 2·0, in accord with <sf)/<s;) = 2·0.

Calculations of P(Q) have been carried out for cyclic PDMS molecules

3.0

SOL 100 L

5.0 10.0

u = Q <S2)1/2

FIG. 6. Normalized Kratky plots calculated for PDMS molecules at 298 K (u = Q<S2)1/2). The ratio of contrast factors gJg. = 0·135. Cyclic PDMS is denoted R, linear PD MS is denoted L, with the number of skeletal atoms indicated. - . - . - is the Gaussian approximation to P(u) for cyclic PDMS using eqn (27) and ----- is

the Gaussian approximation to P(u) for linear PDMS using eqn (17).

NEUTRON SCATTERING FROM CYCLIC POLYMERS 183

containing up to 100 skeletal bonds. The calculations used eqn (25) and a Monte Carlo method which employs Metropolis sampling38 •39 as described in Ref. 15. The calculations used the rotational isomeric state model (RISM) developed by Flory, Crescenzi and Mark40 to describe the statistical conformations of a PDMS chain. The ring molecules were the fraction of the chain conformations with end separations less than 2·0 A. Scattering centres were taken as being situated on atoms in the molecular backbone and appropriate contrast factors gi and gj were used. The effective centre of the -Si(CH3)2- scattering unit in PDMS molecules is slightly offset from the skeletal atom, but it has been shown41 that this produces a relatively small effect at high values of Q and it is not taken into account in the calculations.

The calculated, normalized, Kratky plots for cyclic PDMS molecules at 298 K containing 40-100 skeletal bonds are shown in Fig. 6 together with the plots predicted by eqns (17) and (27). The latter are independent of chain length. The ratio of contrast factors used corresponds to PDMS in benzene-d6 , the solvent used in Ref. 10.

There is an essentially common maximum in the calculated Kratky plots for cyclic PDMS at u ~ 2·0. This is in reasonable agreement with Casassa6

and Burchard and Schmidt's35 prediction of a maximum at u = 2·15 for Gaussian rings. However, the maximum is less pronounced than that predicted by the analytical method and the plots do not attain the limiting value of 1·0 as u --+ 00. In addition, the plots for linear PDMS do not attain the expected asymptotic limit of 2·0. This reflects deviations from the Gaussian statistics assumed in the derivation of eqns (17) and (27). Although the pair distribution function W(r i) may be Gaussian for large separations Ii - il for both linear and cyclic molecules (i.e. at low Q), this approximation fails for small values of Ii - ii, resulting in the observed discrepancies at high Q in Fig. 6.

In addition, as shown in Fig. 6, cyclic PDMS with 40 skeletal bonds displays a second maximum in the Kratky plot in the region u = 5·0. This is not shown by rings with 50 or more skeletal atoms. Calculations of P(Q) were extended to higher values of Q for PDMS rings with 12-26 skeletal atoms (see Ref. 31 for details). The results of these calculations demonstrate that second maxima again occur with these smaller rings in the region u ~5·0.

To explore this feature further, calculations of P(Q) were carried out for short polymethylene (PM) rings. The results demonstrate that maxima corresponding to those observed for cyclic PDMS occur for cyclic PM at the same values of u. However, the second maxima for cyclic PM are less

184 KEITH DODGSON AND JULIA S. HIGGINS

pronounced. These results show that small cyclic PDMS and PM rings have a special annular character. As ring size increases this character becomes less pronounced.

In the limit of infinite chain length, the only distinctions between a cyclic molecule and the equivalent linear molecule are the markedly increased segment density and the slightly more spherically symmetrical segment distribution of the cyclic species. 42

Comparison with experimental data. In order to obtain experimental data at values of Q<S2);/2 higher than those shown in Fig. 3, neutron scattering measurements were carried out using wave vectors in the range 0·024 < Q < O' 2 A-I. The samples investigated were cyclic and linear PDMS fractions with number-average molar masses of2·14 x 104 gmol- 1

and 2 ·13 x 104 g mol-l, respectively (corresponding to ;:::: 550 skeletal bonds), and with heterogeneity indices Mw/Mn = 1·09.

-::J

N

::J

6.0

4.0

O~------~------~------~------~-------J 2 4 6 8

u = Q <52) 1/2

FIG. 7. Normalized Kratky plots of u2 ](u) against u for cyclic and linear PDMS. Experimental data for the cyclic (Mn =2·l4x 104 gmol-l) and linear (M n = 2 ·14 X 104 g mol- 1) fractions are denoted x and +, respectively. ---, calculated for linear PDMS using eqn (17) and normalized to the same asymptotic limit as the experimental data. -. -. -, calculated for cyclic PDMS using eqn (27) and assuming <s?>I<s;> = 2·0. -----, calculated using eqn (25) and the Monte

Carlo method for a 100-bond PDMS ring and reproduced from Fig. 6.

NEUTRON SCATTERING FROM CYCLIC POLYMERS 185

In Fig. 7, the experimental normalized Kratky plots are shown for the cyclic and linear PDMS fractions at 296 K, together with plots calculated using eqns (17) and (27). Also shown in Fig. 7 is the curve for cyclic PDMS with 100 skeletal bonds, calculated using eqn (25) and the Monte Carlo method. The calculated curves were evaluated in terms of u2 P(u) and converted to values of u21(u) by multiplying by a constant factor of 2·6 which is equal to the ratio of the experimental value of u2 leu) at high u for linear PDMS to the Gaussian asymptotic value of 2·0. In addition, values of the z-average radii of gyration (S2);/2 were chosen to yield the best agreement between the experimental data and the calculated plots at low u. The values of (s;);/2 and (s?);/2 obtained in this way are 25 ±4A and 36 ± 4 A for the cyclic and linear fractions, respectively, giving a ratio (s?);/2/(s;);/2 = 2·1 ± 0·3. These values can be compared with those interpolated from the experimental data obtained at low Q shown in Fig. 4, viz. (s;);/2 =29A and (s?);/2 =40A.

There is good agreement between the experimental data for linear PDMS and the plot obtained using eqn (17) over the complete range of u.

The agreement between the experimental data for the cyclic PDMS and the curve given by eqn (27), which is based on Gaussian statistics, is not so good. However, the experimental data for the cyclic PDMS show a slight maximum and agree well up to u = 2 with the curve calculated for a 100 bond ring using the Monte Carlo method. The calculated curve then shows a broad shallow minimum which is not followed by the experimental data. This may be due to the combined effects of chain length polydispersity of the samples, since the Monte Carlo calculations indicate that P(u) is sensitive to ring size in this region of u.

Figure 8 shows Kratky plots of the neutron scattering data obtained for a cyclic PPMS fraction (iiz =291 and (s;);/2 = 19·oA) and for a linear PPMS fraction (iiz = 307 and (s?);/2 = 26·7 A)29,30 in benzene-d6 at 293 K. In contrast to the experimental Kratky plots shown in Fig. 7 for the high chain length cyclic and linear PDMS samples, these plots show pronounced upward curvature at high u values. They are similar in form to the scattering functions calculated for the relatively small cyclic and linear PDMS molecules shown in Fig. 6, although the predicted maximum at u ~ 2 for ring polymers is not clearly resolved. It is hoped that the use of the rotational isomeric state model for PPMS43 •44 together with Monte Carlo computational techniques will prove to be useful in interpreting these PPMS data.

The particle scattering functions of polymeric molecules are sensitive to chain length, chain structure and chain flexibility (further discussions of

186

...... N

::J

KEITH DODGSON AND JULIA S. HIGGINS

6

2

O~~------~--·------~----------L-----~ o 2 4 6

u = Q (S2)1/2

FIG. 8. Kratky plots ofu 2 J(u) against u for cyclic PPMS fraction with n = 291 (.) and linear PPMS fraction with n = 307 (0), in benzene-d6 at 293 K.

recently modelled systems can be found in Chapter 2). The experimental data presented here represent the first attempts to characterize the differences between the spatial arrangements of atoms in cyclic and linear polymer molecules using neutron scattering techniques, and to compare the results with theoretical predictions.

QUASI-ELASTIC NEUTRON SCATTERING

Principles The type of motion observed in a scattering experiment from polymer molecules in dilute solution depends on the frequency and distance scales explored. The frequency scale is limited by the experimental energy resolution, while the distance scale is defined by Q-1. For reasonably large molecules, centre of mass diffusion has been observed by light scattering45

(e.g. photon correlation spectroscopy) in the range Q(S2) 1/2 < l. Neutron scattering explores rather short distances and is sensitive to local conformational changes (Q - 1 < 30 A). Motion of such short chain lengths is governed by bond lengths and rotational potentials and is therefore sensitive to chemical structure, while the long-range diffusive motion

NEUTRON seA TTERING FROM CYCLIC POLYMERS 187

depends upon the molecular dimensions. Between these two extremes the connectivity of the chain ends leads to a universal behaviour independent of both molecular structure and dimensions where the characteristic frequency is proportional to (kB T/Yfs)Q3, where Yfs is the solvent viscosity. The range of this universal behaviour is limited at low Q by Q<S2)1/2 ~ 1 and at high Q by Qa ~ 1, where a is an imprecisely defined length below which chemical structure becomes important. It is conveniently taken to be of the order of the statistical segment length. The scattering behaviour in this Q3 range has been dealt with in some detail by Pecora46 and de Gennes47 .48 using concepts introduced by Rouse49 for a spring-bead model (where a then becomes the spring length). Zimm50 subsequently modified the calculation to incorporate the solvent hydrodynamic effects. The Q3 behaviour of the characteristic frequency is, however, a general result and arises directly from the connectivity of the molecular structure. The range Q<S2)1/2 > 1 has been explored using light scattering experiments for very large molecules 5 1.52 (Mw> 106 , <S2)1/2 > 103 A) while the highest resolution neutron experiments53 can now achieve Qa ~ 1. Both light and neutron data are well fitted by the calculations of Akcasu et af.54 which extend the calculations of Dubois-Violette and de Gennes beyond the two inequalities <S2) -1/2 < Q < a- 1.

The resolution required to observe the long-range internal and centre of mass motion is at the limit of quasi-elastic neutron scattering techniques. The highest resolution is obtained using the neutron spin-echo spectrometer described below and which measures the coherent in­termediate scattering function Scoh(Q, t) introduced in the section 'Neutron Scattering Principles'.

The analytical form of Scoh(Q, t) has been calculated by Dubois-Violette and de Gennes48 for spring-bead models in the so-called Zimm limit in the intermediate Q region <S2) -1/2 < Q < a- 1. In addition, Akcasu et af.54

have further developed the calculations to predict Scoh(Q, t) throughout the entire Q regime. The intermediate scattering law is expressed in terms of an inverse correlation time, which is defined as the first cumulant of the intermediate coherent scattering law, i.e.

Q = -lim din (Scoh(Q, t)) I~O dt

(28)

In the intermediate Q region, for 8-solvent conditions

(29)

188 KEITH DODGSON AND JULIA S. HIGGINS

The factor I/6n becomes 0·0625 if the Oseen tensor is not pre-averaged. Scoh(Q, t) has not been calculated for good solvent conditions, but shows the same Q3 dependence with the numerical prefactor increasing from 0'053, i.e. (1/6) to 0·071 (Ref. 54) (or 0·079 for non pre-averaging).

At low Q (Q<S2)1/2 ~ 1) and low frequencies (w;S 1/'1) where '1 is the correlation time of the first Rouse mode, Brownian diffusion of the whole molecule dominates and Scoh(Q, t) is a simple exponential of the form exp ( - nt)Q, which is now proportional to Q2, gives the temperature­dependent diffusion coefficient, D(T) =n/Q2 in the Q-+O limit. 45 ,52

1000

N

0« > QJ

:::1.. 100 N

CJ

~

10

0·001 10

Q ($,-1)

FIG. 9. Variation of Q/Q2 with Q, calculated for benzene-d6 at 303 K for three different values of B (see Ref. 58).

Figure 9 shows the variation of n/Q2 with Q calculated by Akcasu for a chain of 104 segments of unit length (0' = 1) in deuterated benzene at 30°C and for three values of the parameter B. Apart from a factor 21/2, B is identical to the reduced draining parameter h* = h/ N. h was introduced by Kirkwood 55 as a measure of the strength of the hydrodynamic interaction. As can be seen from Fig. 9, B only affects the values of n for QO' ::::: 1.

The resolution of the spin-echo technique limits observation to values of Q > 0·02 A -1. For most polymers studied, 53 values of 0' are apparently of the

NEUTRON SCATTERING FROM CYCLIC POLYMERS 189

order of 20 A or larger so that most neutron data lie within the range Q ~ (J-1 where deviation away from Q3 towards Q2 behaviour may be observed. For PDMS, however, (J is quite short (15 A) and a reasonable range of Q3 behaviour is observed at low Q. However, if <S2)1/2 is made very small the lower limit appears within the neutron scattering Q range, and for small PDMS molecules, therefore, a region of Q2 behaviour associated with centre of mass diffusion should appear at low Q below the Q3 behaviour.

Experimental Considerations A detailed discussion of the neutron spin-echo technique is not presented here, but the interested reader is referred to the extensive literature. 56,57 Currently the only spectrometer in existence is the INII spin-echo spectrometer situated at the high flux reactor at the Institut Laue­Langevin. 56 This instrument allows measurement of very small changes in energy of a beam of polarized neutrons scattered by a sample by monitoring the neutron beam polarization non-parallel to the magnetic guide field. The resultant polarization, when normalized against a purely elastic scatterer, is directly proportional to the cosine Fourier transform of the coherent scattering law, ScohCQ, w), i.e. ScohCQ, t) is measured directly and energy changes down to 3 ne V may be measured. The wavelength of incident neutrons in the experiments described here was 8·3 A, with a spread of /).Aj). of 10%. The Q values were in the range 0·026 < Q < 0.106A -1.

Interpretation of neutron scattering data involves extracting n from the measured scattering functions. As discussed in Ref. 53, the approach taken in the intermediate Q region is to fit the appropriate de Gennes correlation function 48 to the data and obtain the corresponding inverse correlation time r. It can be shown 53 ,54 that r is then simply related to n, the initial slope of ScohCQ, t) via the expression n = fir. For the low Q regime, where simple diffusion is observed, and the scattering law is a single exponential, n may be obtained directly from the slope of a logarithmic plot of ScohCQ, t) against time. It will become clear in the next section that, for most of the samples investigated, the available ranges of Q and molecular weight lead to observations in the region of crossover from simple diffusion to internal segmental motion. Under these circumstances, the form of SCQ, t) must be studied closely52 - 54 and the appropriate method of obtaining n chosen at each Q value. Having obtained n as a function of Q, it is possible to compare the experimental variation of n/Q2 against Q with that predicted theoretically.

190

~ > OJ ::i.

KEITH DODGSON AND JULIA S. HIGGINS

800 R ! 1100 L I 50----+:;1+-7

2700 R I 2700 L I

50 ------~-h--H- .... /\I ~ I , 7 j--t--t-f--t-

/ i N 6300R i 6400 L ~ 50 i

~---tT!/~t/ 50

·01

15400 R

0·1 ·01

15100 L

0-1 Q (.li.-1 )

FIG. 10. Experimentally determined behaviour of Q/Q 2 for cyclic and linear PDMS fractions. Cyclic PDMS is denoted R, linear PDMS is denoted L, with the number average numbers of skeletal bonds indicated. The broken lines show the experimental values of D(c) and the solid lines show the Q behaviour predicted by

eqn (29).

NEUTRON SCATTERING FROM CYCLIC POLYMERS 191

Diffusion of Cyclic Molecules The values of fl/Q 2 plotted against Q for a series of low molar mass cyclic and linear PO MS fractions at concentrations of 2·5 to 3 % in benzene-d6 at 303 K are shown in Fig. 10. 54 The numbers shown are the number average molar masses Un of the samples (R and L denote ring and linear POMS, respectively). The crossover from Q2 to Q3 behaviour for each of the cyclic and linear PO MS fractions studied is shown clearly. In Fig. 10 the solid line indicates the Q3 behaviour predicted by de Gennes (eqn (2)), whereas the broken line shows the Q2 behaviour in the limit Q < <S2> 1/2. It can be seen that the critical values of Q, i.e. Q* (defining the crossover point Q<S2>1/2 ~ 1), for the linear silicones are in general lower than the Q* values observed for the corresponding cyclic silicones with similar molar masses. This arises because, for linear and cyclic POMS containing the same number of skeletal bonds, <sf> 1/2 > <s) 1/2.

The diffusion coefficients may be obtained from the horizontal parts of the curves in Fig. 10. Because benzene is a good solvent for linear POMS, a concentration-dependent diffusion coefficient D(c) is expected of the form

D(c) = D(O)[l + c/J(N)c] (30)

where c/J(N) is both molar mass and solvent dependent. However, Fig. 11 shows, within experimental precision, that no effect of concentration can be detected (this implies c/J(N) ::;0·12 x 1Q6 cm 5 s-1 g-l), in agreement with

6 -

., 5 -VI

N

E u

'" o x

Cl 3 t-

2t-

1

,

: : ¢ D· 1-9----t-:- ring -r-i-T- ¢ ---t--l-Olinear

I . I 1 1

2 3 4 5

concentration (wt/vot%)

FIG. 11. Concentration dependences of the diffusion coefficients of cyclic and linear PDMS fractions (each with fin = 54) in benzene-d6 at 298 K.

192 KEITH DODGSON AND JULIA S. HIGGINS

results obtained by other workers. 14 The values of D obtained are therefore shown in Fig. 12 without any attempt to correct to infinite dilution.

For the samples with lower molar masses, where the Q2 region is well defined experimentally, the diffusion coefficient may be found directly. For the cyclic and linear silicones with molar masses of 15000gmol- 1 , D(c) may be obtained from the limiting value of Q/Q 2 found by fitting Akcasu's curve to the experimental data. (Due to difficulties in obtaining a good fit, 58

these values are inherently less accurate than those obtained directly.) Figure 12 compares the values of the diffusion coefficients obtained for

cyclic and linear PDMS obtained using quasi-elastic neutron scattering with those obtained using classical boundary spreading techniques 14 and photon correlation spectroscopic methods. 59 The lines drawn in the figure are taken from Ref. 14. The values of D(c) from neutron scattering measurements were obtained using solutions of concentration between 2 and 3 % and the data have been normalized (according to

3-D

, III

N

E 'Il :1.

~ Ci 2·5 <> 'lope~-~ en

0 ....J

2·0

1-5 2·0 2·5

Log 10 fin

FIG. 12. Comparison of the values of the diffusion coefficients of cyclic and linear PDMS obtained using quasie1astic neutron scattering (~cyclic and D linear) and other methods. All data are normalized to solvent viscosity and temperature (toluene at 298 K): • cyclic and 0 linear PDMS, data obtained using boundary spreading techniques (Ref. 14); () cyclic and [;iii linear PDMS, data obtained using

quasi-elastic light scattering (Ref. 59).

NEUTRON SCATTERING FROM CYCLIC POLYMERS 193

D = Dmeasured(~slT) (298 K/~toluene)) to account for measurements made in different solvents and at different temperatures. This allows direct comparison between the sets of data. The absolute precision of the neutron scattering data is no better than 3 neV, which typically corresponds to an error of ± 10 % on a value of D(c) of the order of 3 x 10- 6 cm2 s -1, whilst data obtained using light scattering give values of D with an experimental error of the order of ± 5 %, and the precision of the data obtained by boundary spreading is estimated to be about ± O' 5 %. 14 The ratio Dlinear/Dcyclic is found to be 0·84 ± 0'016, which compares favourably with the predicted value of8/3n = 0·85 for cyclic and linear polymers of the same molecular weight and in the absence of free draining and excluded volume effects. 19 •23 Knowing the diffusion coefficients for all of the samples, it is possible to calculate the hydrodynamic radius and compare this with data from other sources.

(a) The hydrodynamic radius, RH , may be calculated directly from D via the Stokes-Einstein equation 14.19

kBT D = (31)

6n~sRH

where all symbols carry their usual meanings. (b) For both cyclic and linear polymers, R*, which defines overall

molecular dimensions, and is thus comparable with <S2)1/2, is defined by Q*, the crossover from Q2 tOQ3 behaviour. For 8-solvent conditions, Akcasu's curve shows this crossover to be defined by Q* R*:~ 1. Thus a crude experimental value of R* may be obtained directly from the data plotted in Fig. 9 by taking R* = I/Q*. These values are quoted in Table 3 and are of the same order of magnitude as the molecular dimensions meoasured using the methods described in the section 'Small Angle Neutron Scattering' .

Table 3 includes the values of <S2);/2 obtained by small angle scattering, so that the ratio <S2);/2/RH may be compared with theoretical predictions. For the linear samples, it may be shown that in the Gaussian limit23 the radius of gyration is expected to be related to the hydrodynamic radius via

<S2)1/2 = 3n81/2 RH = 1·505RH (32)

However, recent experimental determinations of the relationship60

<S2);/2/RH suggest a somewhat lower value than this of 1·27 ± 0·06. The experimental values presented here have an average of 1·21. For the cyclic samples the theoretical ratio of <S2);/2/RH' again in the Gaussian limit, is

194 KEITH DODGSON AND JULIA S. HIGGINS

TABLE 3 Dimensions of Cyclic and Linear Poly(dimethylsiloxanes)

(All dimensions are in A)

Un nn Rg R* Rg RiRH RH (SANS) ( =1IQ*) (QENS) (boundary

spreading or QELS)

(a) Linear samples (T = 293 K) 1100 24 7 6 6-4 1·1 6·7 2100 54 10·2 7·6 1·34 2700 74 12-4 12 12·1 1·02 6400 172 21·2 18 19·6 1·1

15100 408 40·0 26·4 1·5 26·4

(b) Cyclic samples (T = 293 K) 800 20 5 5·1

2000 54 7·8 6·8 1·15 2700 73 9·5 II 10·7 0·89 9·9 6300 170 14·3 16 14·7 0·97

15400 415 29 22 18·9 1·5

Rg = <S2)1/2.

predicted to be (2In) -1/2 = 1.2533. 14•35 The values presented in section (b) of Table 3 suggest again that the experimentally determined value appears to be somewhat lower than that predicted, with an average of 1·15.

REFERENCES

1. Willis, B. T. M. (Ed.), Chemical Applications of Thermal Neutron Scattering, Oxford University Press, Oxford, 1973.

2. Kostorz, G. (Ed.), Treatise on Materials Science and Technology, Vol. 15, Neutron Scattering, Academic Press, London, 1979.

3. Maconnachie, A. and Richards, R. W., Polymer, 19 (1978) 739. 4. Higgins, J. S., In: Developments in Polymer Characterisation-4 (ed. J. V.

Dawkins), Elsevier Applied Science Publishers Ltd, London, 1983, pp. 131-76. 5. Van Hove, L., Phys. Rev., 95 (1954) 249. 6. Casassa, E. F., J. Polym. Sci., A, 3 (1965) 605. 7. Ibel, K. J., J. Appl. Cryst., 9 (1976) 296. 8. Neutron Beam Facilities Available to Users, Scientific Secretariat, Institut

Laue-Langevin, 156X Centre de Tri, 38042, Grenoble Cedex, France. 9. Jacrot, B., Rep. Prog. Physics, 39 (1976) 911.

10. Higgins, J. S., Dodgson, K. and Semlyen, J. A., Polymer, 20 (1979) 553.

NEUTRON SCATTERING FROM CYCLIC POLYMERS 195

11. Dodgson, K., Higgins, J. S. and Semlyen, J. A., in preparation. 12. Kuwahara, N., Miyake, Y., Kaneko, M. and Furuichi, J., Rep. Prog. Polym.

Phys. Jpn, 5 (1962) 1. 13. Richards, R. W., Maconnachie, A. and Allen, G., Polymer, 19 (1978) 266. 14. Edwards, C. J. c., Stepto, R. F. T. and Semlyen, J. A., Polymer, 21 (1980) 781. 15. Edwards, C. J. c., Rigby, D., Stepto, R. F. T., Dodgson, K. and Semlyen, J. A.,

Polymer, 24 (1983) 391. 16. Wright, P. V., J. Polym. Sci. (Polym. Phys. Edn.), 11 (1973) 51; Semlyen, J. A.,

Adv. Polym. Sci., 21 (1976) 41. 17. Zimm, B. H. and Stockmayer, W. H., J. Chern. Phys., 17 (1949) 1301. 18. Kramers, H. A., J. Chern. Phys., 14 (1946) 415. 19. Yamakawa, H., In: Modern Theory of Polymer Solutions, Harper and Row,

New York, 1971. 20. Boots, H. and Deutch, J. M., Macromolecules, 10 (1977) 1163. 21. Prentis, J. J., J. Chem.·Phys., 76 (1982) 1574. 22. Bloomfield, V. and Zimm, B. H., J. Chern. Phys., 44 (1966) 315. 23. Fukatsu, M. and Kurata, M., J. Chern. Phys., 44 (1966) 4539. 24. Kumbar, M. and Windwer, S., J. Chern. Phys., 50 (1969) 5257. 25. Yu, H. and Fujita, H., unpublished work cited in: Yamakawa, H., Modern

Theory of Polymer Solutions, Harper and Row, New York, 1971, pp. 321-3. 26. Kumbar, M., J. Chern. Phys., 55 (1971) 5046. 27. Lax, M. and Windwer, S., J. Chern. Phys., 55 (1971) 4167. 28. Naghizadeh, J. and Sotobayashi, H., J. Chern. Phys., 60 (1974) 3104. 29. Clarson, S. J., PhD Thesis, University of York, 1985. 30. Dodgson, K., Clarson, S. J. and Semlyen, J. A., in preparation. 31. Edwards, C. J. c., Richards, R. W., Stepto, R. F. T., Dodgson, K., Higgins,

J. S. and Semlyen, J. A., Polymer, 25 (1984) 365. 32. Allen, G., In: Structural Studies of Macromolecules by Spectroscopic Methods

(ed. K. J. Ivin), Wiley, New York, 1976. 33. Maconnachie, A. and Richards, R. W., Polymer, 21 (1980) 745. 34. Debye, P., Ann. Physik., 46 (1915) 809. 35. Burchard, W. and Schmidt, M., Polymer, 21 (1980) 745. 36. Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions,

Dover Publishers, New York, 1970. 37. Debye, P., J. Phys. Chern., 51 (1947) 18. 38. Rigby, D., PhD Thesis, University of Manchester, 1981. 39. Lal, M. and Stepto, R. F. T., J. Polym. Sci., Polymer Symp., 61 (1977) 401. 40. Flory, P. J., Statistical Mechanics of Chain Molecules, Interscience, New York,

1969. 41. Edwards, C. J. c., Richards, R. W. and Stepto, R. F. T., Macromolecules,

submitted for publication. 42. Edwards, C. J. c., Rigby, D., Stepto, R. F. T. and Semlyen, J. A., Polymer, 24

(1983) 395. 43. Mark, J. E. and Ko, J. H., J. Polym. Sci., Polym. Phys. Edn., 13 (1975) 2221. 44. Freire, J. J. and Rubio, A. M., J. Chern. Phys., 81 (1984) 2112. 45. Berne, J. and Pecora, R., Dynamic Light Scattering with Applications to

Chemistry, Biology and Physics, Wiley, New York, 1976. 46. Pecora, R., J. Chern. Phys., 49 (1968) 1032.

196 KEITH DODGSON AND JULIA S. HIGGINS

47. de Gennes, P. G., Physics, 3 (1967) 37. 48. Dubois-Violette, E. and de Gennes, P. G., Physics, 3 (1967) 181. 49. Rouse Jr., P. E., J. Chem. Phys., 24 (1956) 269. 50. Zimm, B. H., J. Chem. Phys .. ,i 24 (1956) 269. 51. Adam, M. amd Delsanti, M., Macromolecules, 10 (1977) 1229. 52. Han, C. C. and Akcasu, A. Z., Macromolecules, 14 (1981) 1080. 53. Nicholson, L. K., Higgins, J. S. and Hayter, J. B., Macromolecules, 14 (1981)

1836. 54. Akcasu, A. Z., Benmouna, M. and Han, C. C., Polymer, 21 (1981) 866. 55. Kirkwood, J. G., J. Polym. Sci., 12 (1954) 1. 56. Dagleish, P., Hayter, J. B. and Mezei, F., Neutron Spin Echo, Springer Verlag,

Berlin, 1980. 57. Nicholson, L. K., Con temp. Phys., 22(4) (1981) 45. 58. Higgins, J. S., Ma, K., Nicholson, L. K., Hayter, J. B., Dodgson, K. and

Semlyen, J. A., Polymer, 24 (1983) 793. 59. Edwards, C. J. c., Bantle, S., Burchard, W., Stepto, R. F. T. and Semlyen,

J. A., Polymer,' 23 (1982) 873. 60. Schmidt, M. and Burchard, W., Macromolecules, 14 (1981) 210.

CHAPTER 6

Organic Cyclic Oligomers and Polymers

HARTWIG HOCKER

Institute of Macromolecular Chemistry, University of Bayreuth, Federal Republic of Germany

INTRODUCTION

It is not only because of their 'beauty' but also because of their special properties that cyclic molecules have been attracting the special interest of chemists for a long time. The most 'beautiful' molecules incorporating cyclic structures might be the 'platonic hydrocarbons' such as the cubane 1, the tetrahedrane 2 and the dodecahedrane 3 system.!

: I / -"";--.1<."­

j //: '··· .. r··

.... t

1 2 3

They are characterized by a high degree of symmetry. On the other hand, there are a large number of cyclic molecules with pronounced functionality such as bicyclic polyaminoethers, the so-called cryptands 4 which are known to form cryptates with cations of suitable size. 2 Also the monocyclic crown ethers ('coronands') 5 exhibit selective complexing ability for mainly alkali and alkaline earth metal ions forming the 'coronates'.

Moreover, the antibiotic activity of some macrocyclic natural com­pounds 4 is, most certainly, a consequence of their complexing ability. As

197

198 HARTWIG HOCKER

examples, the cyclodepsipeptide valinomycin 6 and the tetrahydrofuran units containing macrolide nonactine 7 should be mentioned. The increasing interest in macrocycliccompounds with physiological activity has motivated organic chemists to develop a variety of elegant synthetic methods for cyclic systems. 5

7

Synthetic cyclic hydrocarbons, ethers, amines and sulfides, as well as cyclic esters, amides and urethanes, may be regarded as model compounds.

Beside these low molecular weight compounds, nature also provides high molecular weight cyclic or circular material, such as cyclic polypeptides and DNAs which will be covered by the following chapters. It should be noted here, however, that it was the discovery of the circular character of some DNAs which motivated theoreticians to study the dilute solution properties of ring polymers long before their synthesis by anionic methods as suggested by Casassa 6 was achieved.

SYNTHESIS AND PROPERTIES OF CYCLIC OLIGOMERS

General Aspects Cyclic molecules generally are prepared by high dilution principle

ORGANIC CYCLIC OLIGOMERS AND POLYMERS 199

techniques. 7 The starting materials are bifunctional compounds for which, under the chosen dilution conditions, the probability of the intramolecular reaction is significantly larger than that of the intermolecular reaction.

To achieve high yields of a particular cyclic compound, the template effect, the rigid group principle, the gauche effect, the caesium effect and other effects may be considered (for references, see Ref. 7).

These cyclization procedures shall not be treated in the present chapter, however. Rather, it will be concerned with cyclic oligomers and polymers which result from typical polyreactions. In polyreactions the distribution of cyclic oligomers in the thermodynamic equilibrium is controlled by the probability that the two 'ends' approach each other to make a reaction possible, and this for each step of the reaction. Three points are noted in this connection: (i) not only chain ends may react with each other {'end­biting')-the reaction may also occur between an active end and a reactive group within the same chain (back-biting reaction); (ii) the distribution of cyclics under kinetic control generally differs from that under thermo­dynamic control, i.e. the formation of cyclics may be enhanced or depressed in the early stages of the polymerization; (iii) there are only very exceptional cases for which a certain cyclic is formed with high preference, e.g. in more than 90 % yield. 8

Polymerization by both step reactions and chain reactions may generate an equilibrium of cyclic oligomers and acyclic polymer (ring--chain equilibrium). Typical examples for step reactions are the poly­condensation reaction of diacids and diols (or of hydroxyacids) and of diacid dichlorides and diamines. As pointed outlabove, not only the ends of the molecule may react with each other. In a similar way, the end group of a molecule may react with an internal ester/amide group or two internal groups may react with each other following a transesterification or a transamidation reaction. In all cases, intramolecular reactions result in cyclic molecules.

On the other hand, ring opening polymerization reactions can establish ring--chain equilibria as well. Two prominent examples are the cationic ring opening polymerization of heterocycles and the metathesis reaction of cycloolefins. Here the active end (i.e. cationic species or transition metal car bene) may react with each 'functional group' (i.e. heteroatom in the case of cationic polymerization or C=C double bond in the case of the metathesis reaction) along the chain (back-biting reaction) or with the functional group at the other end of the same molecule (end-biting reaction) to form cyclic molecules.

200 HARTWIG HOCKER

Ring-Chain Equilibria Systems which result in ring--{;hain equilibria have been reviewed in detail by Semlyen. 9 The equilibrium concentration of the cyclic oligomers yields valuable experimental information on the conformations of chain molecules, in particular of low molecular weight chain molecules. For Gaussian chains, the lacobson-Stockmayer cyclization theory10 can be applied to systems of this kind .. According to this theory, the equilibrium constant Kx of a macrocycle with degree of polymerization x is proportional to X- 5/2 .

It should be noted that Kx (which to a first approximation is equal to the inverse cyclic concentration) is independent of the initial monomer concentration. Consequently, for each system, a maximum initial monomer concentration can be found up to which only cyclics are formed. On the other hand, ring formation can be kinetically enhanced (or depressed), as was discussed in detail for the cationic ring opening polymerization by Matyjazewski et al. 11 A kinematic enhancement of macrocyclic formation, in the early stage of the polymerization, becomes evident if the heteroatom carrying the initiating group is much more reactive towards the growing centre than the other heteroatoms along the growing chain; and if, simultaneously, the rate constant of the intramolecular addition reaction of the heteroatom carrying the initiating group to the growing end is larger than that of the addition reaction of the monomer to the growing end.

Kinetic depression will occur if the rate of the addition reaction of the monomer to the growing end is much faster than that of any back-biting reaction. In this case, the formation of cyclics does not start before the monomer is completely consumed. Examples are the metathetical polymerization of strained cycloolefins with low activity catalysts 12 and the anionic polymerization of s-caprolactone. 13.14

Cyclic Hydrocarbons An interesting method for preparing macrocyclic hydrocarbons starts from o:,w-diacetylenes which are cyclized by oxidative coupling with oxygen, cuprous chloride and ammonium chloride in aqueous ethanol and benzene (Glaser conditions). A review concerning the preparation of higher annulenones using this method was published by Sargent and Cresp.15

Synthesis of Cyclic Hydrocarbons via the Metathesis Reaction The metathesis reaction of cycloolefins is a polymerization reaction which allows the preparation of cyclic oligomers. A suitable catalyst is WCl6 in

ORGANIC CYCLIC OLIGOMERS AND POLYMERS 201

conjunction with EtAIC12 or (CH3)4Sn. The active species is assumed to be a tungsten carbene complex 816

8 9 10

which forms a metallacyclobutane intermediate 9 with the olefin and eventually a new carbene complex 10. The new car bene complex can be regarded formally as the insertion product of a cycloalkene into the metal--carbene bond. The cyclic oligomers are then formed by the back­biting reaction of a long chain car bene complex, i.e. by the backward direction of eqn (l).

Figure 1 shows a gel permeation chromatogram of the reaction products of cyclododecene. As is seen from the figure, no oligomer is particularly favoured although the tetra mer of cyclododecene was chosen as the

30 40 50 60 vE in counts

FIG. I. G PC of a homologous series of oligomers and a polymer (at the exclusion limit) as generated from the tetra mer of cyclododecene, C4sHSS; the broken lines

show distributions before equilibrium is achieved.

202 HARTWIG HOCKER

starting olefin. Because of the relatively large difference in molecular weight (166 units) between the single oligomers, a preparative separation by gel permeation chromatography can be achieved.

As published in detail elsewhere, 1 7 the following characteristics are observed:

(i) The oligomers exhibit cyclic structures, as proven by means of spectroscopic methods, in particular by mass spectroscopy.

(ii) At low initial monomer concentration, up to a critical concentration, the so-called cut-off point, only cyclic oligomers are observed and practically no polymer is detected.

(iii) The cut-off point depends on the chemical structure of the monomer. For norbornene the critical initial monomer concentration was found to be 0·125 mol litre -1, and for cyclooctene 0·2 mol litre -1 (Fig. 2).

(iv) At initial monomer concentrations higher than those corresponding to the cut-off point, the overall oligomer concentration in the solution of the reaction product remains constant; the excess monomer is converted into high polymer.

(v) In the kinetically controlled regime, the cyclic oligomer distribution is given by [M xlt = At __ ,,- 3/2ax with x being the degree of polymerization, a = I-IIPn and At = constant, Pn being the number-average degree of polymerization (Fig. 3(a»; in the thermodynamically controlled regime the distribution is given by the lacobson-Stockmayer theory (Fig. 3(b»:

(vi) The distribution of the (open chain) polymer may be represented by a most probable distribution function (Fig. 4).

Properties of Cycloolefins and Cycloparaffins The cyclic oligomers of cycloalkenes, obtained by the metathesis reaction, 18 show an increasing portion of trans configurated double bonds with increasing degree of oligomerization. Thus, the dimer of cyclooctene exhibits a trans content of 20 %, the decamer of 45 %. The boiling points increase from 415 K for cis-cyclooctene to 760 K for the pentamer (isomeric mixture, 35 % trans) of cyclooctene. The index of refraction, n = 1·46 for cyclooctene, quickly approaches the limiting value of 1·49. The proton resonances exhibit an upfield shift going from the monomer to the dimer and then remain constant.

The mass spectra of the oligomers of cyclooctene and cyclododecene have been studied in detail. 19 They show high intensity molecular ion

ORGANIC CYCLIC OLIGOMERS AND POLYMERS

-0 E

0.3-

~ 0.2-o .0 -. ~ .. !5 0.1-.2' o U

b

O~ 0.2

203

I O,2-r------------------------------------------------, o E QJ VI o .0 -. ~

-

~ 0,1-.2' '0

U

b o o 0

o 0

a

I/° o ~----._---.I-----r----'I-----r----.-I----.----1

o 0,1 0,2 0,3 0,4

[NBEloi mol 1-1

FIG. 2. 'Cut-off' point found in the metathesis reaction of cycIooctene (COE) and norbornene (NBE).

peaks. The most prominent fragments have the general formula CnH 2(n-p)-1 with O::;;p::;; x, where x is the degree of polymerization of the respective oligomer (or the number of double bonds in it) and p represents the number of double bonds in the fragment.

After GPC separation the individual oligomers can be readily hydrogenated to form the respective cycloalkanes. In Fig. 5, the melting points of the cycloalkanes are shown as a function of ring size and compared with those of n-alkanes. 2o Formally, the melting points of the

x Ol

~

lJ)

-1,5

+

!..... -0 E

;:::" ::l'

Ol

~

-2,0

-2,5

","'""

'-", "" '- "',

° 0' 0

"

"'"

"

°'0

""

""'0

00

"

'0

'",

""0

":to'"

~o '0

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\,

'0

0, 0'0

'0 "0

'0 "0

" 0'0

.0

o 2

4 6

8 10

12

14

(a)

x

"L

-0 E

..... )(

:.::

Ol

~

-2,0

-2,5

-3,0

2

q2

4 6

q6

x 8

10

1,0

log

X

10-2

10-3

!- -0 E

)(

:.::

FIG

. 3.

D

istr

ibut

ion

of

CO

E o

ligo

mer

s (a

) in

the

kin

etic

ally

con

trol

led

regi

me:

0,

6·5

%; 0

,21

·0 %

; D,.,

35·

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ORGANIC CYCLIC OLIGOMERS AND POLYMERS

~ ""'0 t -6,5 - "'--0 Ol o

-1.0 -

-1,5 -

-8,0 -

I 1,0

"'" 0"",

I 1,5

O~

O~

I 2,0

O~

I 2,5

O~

I 3,0

O~

3,5 10-3. P

205

a.. ......

t

FIG. 4. Most probable distribution of the polymer population generated by the metathesis reaction of norbornene.

fr ..... E

100

80

60

40

20

10 n

FIG. 5. Melting points of cycioalkanes (., data of compounds from different sources; ., data of compounds from metathesis reaction of cyciododecene and hydrogenation) as compared with those of n-alkanes (0) as a function of the

number of C-atoms, n.

206 HARTWIG HOCKER

cycloalkanes can be represented by an empirical equation proposed by Broadhurst 21 for n-alkanes

Tm = T*(n + a)/(n + b) with T* = 414·3 K

a = -1·4 (-1'5)

b = 5·75 (5·0)

where the numbers in parentheses refer to n-alkanes.

(2)

Synthesis of Cyclophanes via Polycombination Reaction Induced by Electron Transfer to Divinylidene Compounds An interesting class of cyclic hydrocarbons has been reviewed recently in great detail, i.e. the calixarenes, which are poly-aryl m-methylene-bridged macrocyclic compounds and can be prepared by arene-aldehyde condensation reactions as well as by stepwise synthesis. 22 These molecules are [1 n] metacyclophanes with basket-like structures and they have the ability to form inclusion complexes. They generally possess intra annular hydroxyl groups which playa particular role in this host-guest chemistry.

Boekelheide has recently reviewed the synthesis and properties of [2"] cyclophanes. 23 In the following paragraph a rather unusual way of preparing cyclophanes shall be summarized.

Since the work of Szwarc et al.,24 it has been known that 1,1-diphenylethylene (D PE) upon electron transfer forms the monomer radical anion which immediately and exclusively forms the dimeric dianion. The homopolymerization of DPE, on the other hand, does not occur, very probably because of steric reasons. Molecules which are bifunctional with respect to the characteristic group of D PE react in a corresponding way upon electron transfer:

e H2C==C(C6Hs)-C6H4--C(C6Hs)-CH2

I

e - 2e-+

H2C==C(C6Hs)-C6H4-~(C6Hs)-CH2 (11a) e

(3)

They, however, possess another vinylidene group which is still eligible for an electron transfer reaction. Whether this electron transfer reaction occurs with a similar rate as the first one or at a much lower rate depends on the substitution mode of - C6 H4-. If it is a p-phenylene (or 4,4' -biphenylene)

ORGANIC CYCLIC OLIGOMERS AND POLYMERS 207

group, the negative charge formally placed at the neighbouring carbon atom is de localized over the whole molecule, the colour of the solution (in THF) is deep violet, and the electron affinity of the residual double bond is so low that, with lithium as an electron transfer reagent, no further reaction occurs at all; the dimeric (J(,OJ-diene (lla) can be obtained in nearly 100 % yield. With sodium as an electron transfer reagent, the dimer is formed as well; but in the further course of the reaction, further electron transfer occurs by which new radical species are formed which combine. Thus, the even members of the homologous series of the oligomers are obtained.

In contrast, if the middle group is an m-phenylene group (or -p--C6HC-(CH2)n-p--C6H4- or -p--C6H4--CH(C6H5HCH2)n­CH(C6H5)-p--C6H4-) the two double bonds react individually, i.e. the electron affinity of a double bond is independent of whether the twin double bond has been converted to a carbanionic species already or not. Consequently, intramolecular reactions (combination reactions/radical addition reactions) become possible by which cyclic oligomers, cyclo­phanes, are formed. The colour of the THF solution is red.

All molecules may react intermolecularly to build up an oligomer with a certain degree of polymerization. But once they have reacted intra­molecularly they do not react further; the cyclics are formed irreversibly. 25 It has to be mentioned that the divinylidene compound containing the m­phenylene unit is an exceptional case since the dimer highly favours the intramolecular reaction. Thus the cyclic dimer (lIb in its protonated form) can be isolated in more than 90 % yield. 26

Molecules with a similar geometrical structure behave in a similar way, i.e. divinylidene compounds which contain the I ,5-naphthylene group (12) or the 1,4-pyridylene group (13).27

208 HARTWIG HOCKER

@c~J§J " II CH 2 CH 2

12 13

The other divinylidene compounds mentioned below, i.e. 14 and 15, form a homologous series of oligomers without favouring

@-Yi-@-<CH 2)n-@-Yi-@ CH 2 CH 2

14

Q Q Q Q H2C=C--< )-CH-CH 2-CH 2-CH-Q-l=CH 2

15

a certain oligomer. The formation of a cyclic monomer can be observed for n 2 3 and reaches a maximum for n = 6. The individual oligomers (degree of polymerization (DP) = x) contain 2x centres of asymmetry, which for the cyclic monomers and dimers is clearly reflected in the NMR spectra. The larger the rings, the more the chiral centres are decoupled and the less is the effect on the nuclear resonances, i.e. the less is the number of lines. 27

Cyclic Ethers, Sulphides and Amines Cyclic ethers, including acetals., sulphides and amines, are generally obtained by cationic ring opening polymerization of the respective heterocyclic monomers.28

ORGANIC CYCLIC OLIGOMERS AND POLYMERS 209

0+ 0+ ···X······CH ······X······CH

1 1 2 1 1 2 (4)

(CH 2)n-Y (CH 2)n-Y

C X ······ CH ...... 3c 10+ 1 z ~

(5)

(CHzt-Y

X: heteroatom 0, N, S Y: ° or CH 2 if X = 0; otherwise CH2

While polymers are built up by the addition of a monomer molecule to the active end of a growing chain (eqn (4», the intramolecular reaction of the active end (eqn (5» in most cases has to be envisaged as well. The back­biting reaction eventually leads to the formation of cyclic molecules, while the residual part retains the active end; this reaction is to be understood as an intramolecular chain transfer reaction (termination reaction) followed by reinitiation. The systems approach ring--chain equilibria.

In this sense, the situation closely resembles that of the. metathesis reaction of cycloolefins described above.

Cyclic Oligomers of Oxiranes While ethylene oxide and its derivatives with small substituents result in a series of cyclic oligomers starting from the dimer, ethylene oxides with bulky substituents yield the corresponding cyclic tetramers with high preference. 29

Cyclic Oligomers of Tetrahydrofuran and 2,3-Dihydrofuran When THF is polymerized cationically (in CH3N02 as the solvent), the formation of cyclics is very much dependent on the initiator. While with CF 3S03CH3 as the initiator, beside the acyclic oligomers, only small amounts of cyclic oligomers are formed, CF 3S03H gives rise to the exclusive formation of the cyclic oligomers (from trimer to octamer as demonstrated by GC).30

+/CH 2) H+O--(CH2)4+nO,,--

16 CH 2

The oxygen in the hydroxy group of 16 is more nucleophilic than the ether oxygens. Thus the hydroxyl oxygen will react with the chain end much faster than any other oxygen. Further, the macrocyclic oxonium ion

210 HARTWIG HOCKER

formed will transfer its proton to the more nucleophilic THF monomer and remain as cyclic oligomer. In the case of 17, there are three equivalent sites of reaction (methylene groups adjacent to the oxonium): reaction with the monomer or a terminating agent with two of these sites will lead to acyclic molecules; reaction with the exocyclic methylene group, however, will yield a cyclic oligomer.

The ring opening polymerization of 2,3-dihydrofuran cannot be achieved by cationic initiators because the cationic polymerization of this monomer, being an enolether, yields a polymer with pertained five-ring structure (18):

18

The same result is observed when (CO)s W=C(OCH3)C6Hs is used as a catalyst. If, however, (CO)sCr==C(C6HS}z and (CO)sCr=C(OCH3)C6Hs are used as initiators a metathetical ring opening polymerization is observed. 31

(6)

Beside the polymer, homologous cyclic oligomers are formed (Fig. 6) which, after hydrogenation, are identical to the cyclic oligomers based on THF.

Cyclic Oligomers of Cycloacetals Schulz et al. 32 have reported on the cationic polymerization of cyclic acetals such as I ,3-dioxolane (19), I ,3,6-trioxocane (20), 1,3,6,9-tetraoxa­cycloundecane (21) and 1,3,6,9,12-pentaoxacyclotetradecane (22).

20 21 22 23

ORGANIC CYCLIC OLIGOMERS AND POLYMERS

" . , \1\ 7: ' '. u' -, -,-,-,-,'----------

35 30 25

- VE/m(

211

FIG. 6. 2,3.Dihydrofuran products obtained via metathesis reaction with (CO)sCr=C(C6H s)2 as the catalyst: - - ---, UV detector; ---, differential

refractometer as detector of gel permeation chromatography.

In particular in the case of 21, in the early stages of the reaction (in CHzClz with CF 3S03H as initiator) exclusively cyclic oligomers are observed. This behaviour is characterstic of a kinetic enhancement of ring formation as described above.

According to Penczek et al.,33 during 1,3-dioxolane polymeri-

zation both the cyclic living species H-6:::J and the linear living ones +

H-· .. --CHz-O::: coexist in the solution and the molar ratio of both is dependent on the degree of polymerization.

212 HARTWIG HOCKER

Yamashita et al. 34 studied in detail the ring-chain equilibria of 20 to 23 (l,3,6,9,12,15-hexaoxacycloheptadecane) in methylene chloride as the solvent and with BF 3. O(C2H 5h as the catalyst.

Not only saturated but also unsaturated oxacyclics have been subjected to ring opening polymerization.

Schulz et al. described the cationic ring opening polymerization of 4H, 7 H-l ,3-dioxepin 24 35 and of 3,5-dioxabicyclo[5.4.0]undec-9-ene 25. 36

n O~

24 25

In the former case the cyclic oligomers up to the hexamer could be separated by G PC, and the cyclic dimer was isolated. In the latter case, the cyclic structure of the oligomers was proven up to the pentamer; a prominent formation of the cyclic dimer was observed.

Cyclic Oligosulphides In the cationic polymerization of propylene sulphide the formation of a tetrameric species, a 12-membered ring (26) is favoured by an intra­molecular chain transfer mechanism. 37

26

Cyclic Oligoamines The cationic polymerization of N-substituted aziridines generally leads to the formation of low molecular weight products because of the occurrence of a termination reaction between the active species and an amino function of the polymer. The resulting quarternary ammonium salts, either branched molecules or cyclics with pendant chains, are not effective in re­initiation and hence do not form cyclics. With increasing bulkiness of the

ORGANIC CYCLIC OLIGOMERS AND POLYMERS 213

substituent at the nitrogen or at the (X-C atoms, termination reactions can be suppressed and open chain 'living' species may be obtained. 38

Cyclic Oligoesters, Oligoamides and Oligo urethanes As already reviewed by Semlyen,9 the formation of cyclic oligomers from poly(decamethylene adipate) and from poly(trimethylene succinate) upon equilibration reactions has been described. 39 After extraction from the equilibrates, the cyclics were separated by means of G PC and identified by comparison with authentic samples.

Individual cyclic oligo(ethylene terephthalate)s up to the hexamer were synthesized 40 and characterized by various methods. 41 Poly(ethylene terephthalate) was, moreover, equilibrated in the presence of antimony trioxide as catalyst at 543 K in the melt and in I-methyl naphthalene solution with zinc acetate as catalyst. 42 Oligomers from the trimer to the nonamer were identified by means of G Pc.

Cyclic oligomers of e-caprolactam were extracted from the melt equilibrate of the polymer and separated and identified following methods described in a wide range of former works (cited in Ref. 9). (See Chapter I for a description of some of the results for cyclic oligo esters and oligoamides.)

A series of papers is devoted to the preparation and isolation of cyclic oligoamides of the nylon 6,43 nylon 6, 10 44 and nylon 6,6 type,45 as well as of other kinds. 46

The preparation of cyclic oligourethanes has also been described at length. As an example, cyclic oligomers obtained from diethylene glycol (50) and hexamethylene diisocyanate (h) up to the heptamer shall be

TABLE I Melting Points T m and Long Periods LP of Cyclic Oligourethanes from Diethylene Glycol (50) and

Hexamethylene Diisocyanate (h), i.e. (50 h)n

n Tm(K) LP(nm)

I 412 104 2 443 1·5 3 405 (2-8) 5-4 4 424 (408) 3-8 5 411 4·8 6 419 (407) 6·0 7 407 6·5

Numbers in parentheses: values before annealing.

214 HARTWIG HOCKER

mentioned. The results of an investigation in the solid state (melting points and long periods) are given in Table 1.47

CYCLIC POLYMERS

As was pointed out in the foregoing paragraphs, cyclic oligomers are formed during the polymerization of cycloolefins and certain heterocycles, as well as during polycondensation and polyaddition reactions (polyesters, polyamides, polyurethanes). In these systems the cyclics are present within a ring-{;hain equilibrium. The concentration of high molecular weight cyclics is low. The isolation, except in the oligomer region, is difficult or impossible. The preparation of high molecular weight cyclics was first achieved by Brown and Slusarczuk48 and later by Bannister and Semlyen. 49 They obtained high molecular weight cyclic poly(dimethyl­siloxane)s as described in detail in Chapter 3.

Cyclic Polystyrenes and Poly( styrene-b-dimethylsiloxanes) More recently, organic ring polymers were prepared by anionic polymerization and coupling techniques. The advantage of the method is, in particular, the narrow molecular weight distribution of the cyclics obtained. The basic strategy followed by different groups is the bi­functional anionic polymerization of a suitable monomer such as styrene to obtain a bifunctionally 'living' polymer with a certain degree of polymeri­zation which is then reacted, according to high dilution principle tech­niques, with a bifunctional electrophile to form the cyclic polymer (eqn (7)).

Hild et al. 50 used potassium naphthalene as an initiator for the anionic polymerization of styrene in benzene/THF as the solvent and rJ.,rl­dibromo-p-xylene as the bifunctional electrophile. Vollmert and Huang 51

applied sodium naphthalene as the initiator, THF as the solvent and the same monomer and electrophile as Hild et al. to obtain the cyclic polymer.

Geiser and Hocker 52 applied sodium naphthalene as the initiator, tetrahydropyran as the solvent and rJ.,rJ.' -dichloro-p-xylene as the electro­phile. The deep red solution of the bifunctional living polymer and the colourless solution of the bifunctional electrophile were slowly added to a certain volume (e.g. 2litres) of the solvent following the principle of mutual titration to guarantee an equimolar ratio of the reactants. At the end of the cyclization reaction, an excess of electrophile was provided to end-cap the acyclic molecules with X-groups. Then the polymer was isolated, dissolved and reacted with high molecular weight living polystyrene. By this reaction,

ORGANIC CYCLIC OLIGOMERS AND POLYMERS 215

Mt: Na, K X: Cl, Br

the acyclic material was fixed to the high molecular weight polystyrene and the cyclic material was isolated by fractionation.

In this work, the cyclic polymers obtained had molecular weights up to 25000. For a number of reasons cyclic polymers with a higher molecular weight are desirable. They were prepared by Roovers and Toporowski 53

using sodium naphthalene as the initiator for the anionic polymerization of styrene in THF and dimethyldichlorosilane as the electrophile closing the ring; the living polystyrene solution was diluted with cyclohexane. In this way cyclic polystyrenes with molecular weights from 5000 to 450000 were obtained as measured by light scattering in cyclohexane at 308 K. Further, the authors showed that ring and linear polymers can be fractionally precipitated using a methanol-benzene mixture.

An interesting way of preparing cyclic polystyrene has been described by Jones. 54 He polymerized styrene in THF with potassium naphthalene as

216 HARTWIG HOCKER

initiator, exchanged the counterion by adding lithium tetraphenylborate, and polymerized hexamethylcyclotrisiloxane to build up the triblock copolymer poly-( dimethylsiloxane-b-styrene-b-dimethylsiloxane) with methyl end groups, achieved by killing the living system with methyl iodide. The method is designed to make use of the labile dimethyl siloxane linkage to effect a ring--chain equilibration reaction.

The purified block-copolymer was equilibrated at 383 K in toluene solution by adding 0·01 %w/v of catalyst (50% KOH in diglyme suspension). The reaction was carried out up to the point at which the concentration of cyclic low molecular weight siloxanes had become constant (as monitored by GC). Then the solution was cooled and quenched by adding acetic acid.

After equilibration, the toluene solution was washed and dried, and the polymer was precipitated from methanol. The remaining solution was concentrated, diluted with toluene and again poured into methanol to obtain a second fraction. The same procedure was performed to obtain a third fraction. The polymer remaining in solution was taken as the fourth fraction. The different fractions were analysed by G PC and NMR. The results showed that the first two fractions consisted of acyclic alternating block-copolymers while the third fraction was composed of cyclic block copolymers with the average structure

Dilute Solution Properties of Macrocyclic Polystyrene When preparing macrocyclic polymers it immediately becomes evident that the G PC elution volume V E of cyclics is larger than that of acyclic molecules. The reason for this behaviour is to be found in the smaller molecular dimension of cyclics as compared with acyclic molecules. In the case of polystyrene the so-called calibration curve, that is the log M vs. V E

plot of cyclic molecules, is, within the limits of experimental error, parallel to that of acyclic molecules but shifted to higher elution volumes (Fig. 7). The ratio of molecular weights of cyclic and acyclic polymers eluted at a certain elution volume is close to 1·4 in THF as eluting solvent.

Since GPC is sensitive to the hydrodynamic volume of the respective molecules, according to Benoit,S 5 a 'universal' calibration curve is obtained if log M . [,,] is plotted instead of log M, [,,] being the intrinsic viscosity.

[,,] reflects most obviously the hydrodynamic volume of polymer molecules in dilute solution (Fig. 8). Measured in cyclohexane at 307 K the ratio of the intrinsic viscosity of cyclic and acyclic polystyrene was found to

ORGANIC CYCLIC OLIGOMERS AND POLYMERS 217

(al

103~~~ __ ~~ ____ ~~ __ ~~ ____ ~ ____ ~~ __ ~~ ____ ~~ 115 120 125 130 135 140 145 150

vE' cts

(bl

104~ ______ ~ ________ 4-______ ~~ ______ -L ________ L-____ __

120 125 130 135 140 145 Ve:' cts

FIG. 7. GPC calibration (a) and universal calibration (b) curvefor cyclic (e) and acyclic (.) polystyrenes.

218 HARTWIG HOCKER

be 0·66. This value is in close agreement with the theoretically expected value. 56

The most direct evidence for the hydrodynamic volume of both cyclic and acyclic molecules is the radius of gyration which was obtained from small angle neutron scattering in toluene-ds.57 Although toluene is a thermodynamically good solvent, the theoretically expected ratio of the mean square radii of gyration of cyclic and acyclic molecules, i.e. a value of 0·5 (as expected for 8-conditions), was found. The dependence of the radius of gyration on the molecular weight is shown in Fig. 9.

...., ~

'" o -'

1·0

0·5

3·5 4·0 4·5 Log M

FIG. 8. log [11] vs. log M curves for cyclic <e) and acyclic <.,.6.) polystyrene.

Other Cyclic Macromolecules Horbach et at. 5S reported on macrocyclic polycarbonates, prepared by an interphase polycondensation reaction of the bischloroformic acid esters of 4,4' -isopropylidenediphenol (Bisphenol A) in water- and alcohol-free methylene chloride as solvent in conjunction with an aqueous solution of triethylamine. These polycarbonates were found to suffer from a 95 % deficit of end groups with respect to their molecular weight and to exhibit a lower intrinsic viscosity than acyclic polycarbonates of comparable molecular weight (as determined by light scattering). The authors proved the predominantly cyclic character of the polymers further by controlled saponification with piperidine as a base. The molecular weights ranged up to 70000.

5.0

4.0

3.0

E c

-L. N '­_ 0

- u N'"

a::: v

2.0 1.0

ORGANIC CYCLIC OLIGOMERS AND POLYMERS 219

t t

t t t

t tt t

f , , , I I , , I , ,

1.5 2.0

Mw /(10 4 g.mol -I)

FIG. 9. Radius of gyration as a function of molecular weight for cyclic (.) and acyclic (0) polystyrene as obtained from low angle neutron scattering in

toluene-dB'

Macrocyclic Associates Finally, a particular effect of macro cyclic formation by association is to be mentioned. As Szwarc 59 has pointed out, 'living lithium polystyrenes with two growing ends should undergo intramolecular association'. Although these polymers have not yet been obtained, the anionic polymerization of methyl methacrylate in THF with Na + as counter ion and using a bifunctional initiator such as the tetrameric dianion of ex-methyl styrene yields intramolecular associates 26 which are stable up to a considerably

220 HARTWIG HOCKER

O-CH3

I O=C

I Na+-C-CH2

I CH3

26

high degree of polymerization and which add the monomer with a rate that is lower by almost one order of magnitude as compared with that of the non-associated species. 60 The formation of very large rings has been observed as well in systems PS - -(Cs +)2 and PS - -Ba + +.61.62

CONCLUDING REMARKS

In the early days of polymer chemistry the virtual absence of end-groups in natural polymers such as natural rubber, cellulose, etc., was attributed to a cyclic structure rather than to high molecular weight. Later, Staudinger 63 in his first publication Uber Polymerisation presented acyclic polymer molecules with radical ends, and eventually Kern and Kiimmerer 64 proved the presence of end-groups, i.e. the acyclic character of the respective macromolecules.

In the meantime, the existence and the importance of natural cyclic polymers has become evident. Synthetic cyclic oligomers are so far of special interest only. The properties of synthetic cyclic polymers still have to be investigated in detail. Preliminary results show, e.g., that the diffusion of cyclic polystyrene molecules in a high molecular weight acyclic polystyrene matrix under certain conditions of preparation is strongly hindered. 65

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ORGANIC CYCLIC OLIGOMERS AND POLYMERS 221

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222 HARTWIG HOCKER

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and Trent, L. C., J. Polymer Sci. B, 5 (1967) 1057; Mersakentis, E. and Zahn, H., Chem. Ber., 103 (1970) 3034; Zahn, H. and Repin, J. F., Chem. Ber., 103 (1970) 3041; Repin, J. H. and Papanikolau, E., J. Polymer Sci. AI, 7 (1969) 3126.

41. Seidel, B., Z. Elektrochem., 62 (1958) 214; Grime, D. and Ward, I. M., Trans. Faraday Soc., 54(1958) 959; Ward, I. M., Chem. Ind., (1956) 905, (1957) 1102; Binns, G. L., Frost, J. S., Smith, F. S. and Yeadon, E. c., Polymer, 7 (1966) 583; Ito, E. and Okajima, S., J. Polymer Sci., B7 (1969) 483 and Polymer, 12 (1971) 650; Hashimoto, S. and Sakai, J., Kobunshi Kagaku, 23 (1966) 422; Hashimoto, S. and Jinnai, S., ibid., 24 (1967) 36.

42. Cooper, D. R. and Semlyen, J. A., Polymer, 14 (1973) 185. 43. Rothe, M. and Kunitz, F. W., Liebig Ann. Chem., 609 (1957) 88. 44. Zahn, H. and Gleitsmann, G. B., Makromol. Chem., 60 (1963) 45. 45. Zahn H. and Kusch, P., Chem. Ber., 98 (1965) 2588. 46. Hahn, G. E., Kusch, P., Rossbach, V. and Zahn, H., Makromol. Chem., 186

(1985) 297. 47. Heitz, W., Hocker, H., Kern, W. and Ullner, H., Makromol. Chem., 150(1971)

73. 48. Brown, J. F. and Sluzarczuk, G. M. 1., J. Amer. Chem. Soc., 87 (1965) 931. 49. Bannister, D. J. and Semlyen, 1. A., Polymer, 22 (1981) 381. 50. Hild, G., Kohler, A. and Rempp, P., Europ. Polym. J., 16 (1980) 525. 51. Vollmert, B. and Huang, J. X., Makromol. Chem., Rapid Commun., 1 (1980)

333; 2 (1981) 467. 52. Geiser, D. and Hocker, H., Polym. Bull., 2 (1980) 591; Macromolecules, 13

(1980) 653. 53. Roovers,]' and Toporowski, P. M., Macromolecules, 16 (1983) 843. 54. Jones, F. R., European Polym. J., 10 (1974) 249. 55. Benoit, H., Grubisic, Z., Rempp, P., Decker, D. and Zilliox, B. J., J. Chem.

Phys. Phys.-Chem. Bioi., 63 (1966) 1507; Grubisic, Z., Rempp, P. and Benoit, H., J. Polym. Sci., Part B, 5 (1967) 735.

56. Bloomfield, V. and Zimm, B. H., J. Chem. Phys., 44 (1965) 315; Fukatsu, M. and Kurata, M., J. Chem. Phys., 44 (1966) 4539.

57. Ragnetti, M., Oberthur, R. c., Geiser, D. and Hocker, H., Makromol. Chem., 186 (1985) in press.

58. Horbach, A., Vernaleken, H. and Weirauch, K., Makromol. Chem., 181 (1980) Ill.

59. Szwarc, M., Carbanions, Living Polymers and Electron Transfer Processes, Interscience Publishers, New York, 1968, p. 498.

60. Warzelhan, V. and Schulz, G. V., Makromol. Chem., 177 (1976) 2185; Warzelhan, V., Hocker, H. and Schulz, G. V., Makromol. Chem., 181 (1980) 149.

61. Bhattacharyya, D. N., Lee, C. L., Smid, J. and Szwarc, M., 1. Phys. Chem., 69 (1965) 612; Bhattacharyya, D. N., Smid, J. and Szwarc, M., J. Amer. Chem. Soc., 86 (1964) 5024.

ORGANIC CYCLIC OLIGOMERS AND POLYMERS 223

62. Mathis, c., Christmann-Lamande, L., Francois, B. and Hebd, C. R., Seances Acad. Sci.,Ser. C, 280(1975) 941; Mathis, C. and Francois, B., J. PolyrnerSci., Part A-J, 16 (1978) 1297.

63. Staudinger, H., Ber. Dtsch. Chern. Ges., 53 (1970) 1073. 64. Kern, W. and Kammerer, H., J. prakt. Chern. N.F., 161 (1942) 81; 161 (1943)

289. 65. Antonietti, M., Diploma thesis, University of Mainz, 1983.

CHAPTER 7

Circular DNA

1. C. WANG

Department of Biochemistry and Molecular Biology, Harvard University, Cambridge, Massachusetts, USA

INTRODUCTION

Discovery DNA is unique among polymers in its being the genetic material of living organisms. The first hint of the existence of circular DNA came not from physical measurements but from genetic analysis. When the relative positions of a larger number of genes of the bacterium Escherichia coli were determined by genetic crosses, the genes were found to form a single circular linkage map.!

Physical evidence for the existence of cyclic DNA came shortly from several parallel lines of pursuit. Fiers and Sinsheimer found in 1962 that the single-stranded DNA of the bacterial virus (phage) ¢X174 is resistant to degradation by an exonuclease, which progressively degrades linear DNA strands from their ends. 2 This resistance suggests that ¢X174 DNA molecules might be circular, and are thus devoid of ends. Support for this notion was obtained by the use of an endonuclease from the pancreas that cleaves DNA strands internally. It is found that a single cleavage of the ¢X 174 DNA chain converts it to a distinct form that sediments more slowly than the intact molecule; furthermore, this endonucleolytic cleavage product is no longer resistant to the exonuclease. 3 These findings are entirely consistent with intact ¢X174 DNA being in the form of a ring that can be converted to the linear form by a single endonucleolytic hit. Finally, electron microscopy reveals that ¢X174 DNA is ring-shaped. 4 The rapid recent progress in DNA biochemistry is illustrated by the determination of

225

226 J. C. WANG

the order of arrangement (sequence) of all 5386 nucleotides of ¢X174 DNA in 1977. 5 •6 The sequencing work confirms fully the cyclic nature of the sugar-phosphate chain.

Around the time the circular nature of the single-stranded ¢X174 DNA was being elucidated, physical evidence indicating the circular nature of the much larger double-stranded bacterial DNA was also accumulating. In the early 1960s, Cairns developed an auto radiographic method in which bacterial cells containing 3H-Iabelled DNA are lysed and deproteinized very gently, and the DNA is allowed to adsorb on a dialysis membrane. The membrane is overlaid with photographic emulsion and the decay of 3H in the DNA triggers the formation of silver grains along its length; these grains can be visualized under an electron microscope after photographic manipulations. The resulting autoradiogram reveals that the DNA of the bacterium Escherichia coli is a ring-shaped molecule over I mm in contour length. 7.8 Early electron microscopic examination of the DNA of the bacterium Micrococcus IUleus (then termed Micrococcus lysodeikticus) also shows the molecule as a very long thread with no visible ends. 9

Because of the double-helix structure of DNA in which the two strands of complementary nucleotide sequences revolve around each other, a double­stranded cyclic DNA that contains no interruptions in its strands can be viewed as two multiply-intertwined single-stranded rings. The giant duplex DNA rings of bacteria, with molecular weights in the billions (where billion = 109) are rather fragile and are easily broken by hydrodynamic shearing forces. It is therefore difficult to establish whether the strands in such molecules are intact (see, however, Refs 10-12). The first double­stranded DNA shown to have two intact circular strands is that of the animal virus polyoma. In 1963, Wei I and Vinograd showed, by a combination of sedimentation analysis and digestion of the DNA with nucleases, that this small duplex DNA ring (molecular weight around 3 x 106) is made of two topologically intertwined single-stranded ringsY Double-stranded rings of this class are termed covalently closed double­stranded rings or closed-duplex rings and will be discussed in detail later.

Unique Aspects of Cyclic DNA Compared with Other Cyclic Polymers Cyclic DNA is in a distinct class by itself among cyclic polymers. First, being the genetic material, an intricate biochemical machinery has evolved in nature to ensure the faithful duplication of a particular DNA. Thus monodispersity and homogeneity are usually the rule rather than the exception. Secondly, there exist a larger number of enzymes for the manipulation of DNA.14 Deoxyribonucleases with a wide spectrum of

CIRCULAR DNA 227

specificity are known; of particular importance are the hundreds of restriction endonucleases that cleave DNA at specific nucleotide sequences. DNA ligases join juxtaposed pairs of 5'phosphoryl and 3'hydroxyl groups to form phosphodiester bonds. DNA topoisomerases catalyse the interconversion among different topological isomers such as simple rings, catenanes and knots. DNA polymerases and their associated factors catalyse DNA synthesis from deoxyribonucleoside triphosphates. These and many other enzymes provide powerful tools that are not available for other polymers. Third, the development in recombinant DNA method­ology has made it possible to construct DNA molecules containing desired sequences, to replicate these molecules in cells, to express the genetic message encoded by the cloned sequences, or to integrate the sequences into the chromosomes of organisms. Fourth, the double-helix structure of DNA is rare among polymers. As will be discussed later, this double-helical structure influences the cyclization of DNAs shorter than 1000 base pairs (bp) in length and also imposes interesting topological properties on the DNA when it is in the form of a closed-duplex ring.

Abundance Cyclic DNA is widely present in nature. 1S All bacteria possess ring-shaped DNA. Many DNA viruses that infect bacteria or animals have circular genomes. Plasmids, which are omnipresent self-replicating entities in cells, are usually circular: so are the DNAs of mitochondrion and chloroplast. Although the chromosomes of eukaryotic organisms are linear, the extremely long DNA threads (a typical human chromosome consists of a single DNA duplex 1 m in contour length) are organized in loops with properties resembling those of cyclic DNAs.

DNA CYCLIZATION

It was observed in the early 1960s that the DNA of phage A possesses cohesive ends that allow the molecules to join intermolecularly to form aggregates or intramolecularly to form circles. 16 The entire nucleotide sequence of ADNA is now known (the strain sequenced contains 48 502 bp) and the cohesive ends of each molecule are a pair of short pro­truding single-strands. 17 The sequence of the left protruding end, 5'GGGCGGCGACCT, is complementary to that of the right protruding end, 5' AGGTCGCCGCCc. The cyclization (or intermolecular aggre­gation) of the DNA involves the pairing of the single-stranded ends to form a 12 bp helix.

228 J. C. WANG

Phage ADNA provided a convenient source of material for earlier studies of DNA cyclization. 18 •19 At concentrations of the order of 10 jig ml- 1 or below, linear and cyclic monomers are the major species. The process is reversible: low temperature favours base pairing and cyclization, and high temperature favours unpairing and linearization. The midpoint of inter­conversion is 51 DC in aqueous solution containing 0·13 M Na +, and 64 DC when the Na + concentration is increased to 2 M. The cyclization process has an activation energy of about 100 kJ mol- 1 and therefore reaction mixtures can be quenched for analysis of the relative amounts of linear and cyclic forms. For a DNA of the size of A, fragmentation of the linear monomers in to half-size mo lecules can be achieved by con tro lled h ydrod ynamic shear: by passage of the DNA solution through a capillary tube, for example. The half molecules so obtained can be conveniently used to study the intermolecular joining of the cohesive ends. The experimentally determined parameters for cyclization and for the intermolecular joining can then be compared to deduce the unique features of cyclization.

or

The equilibrium constant for the joining of the halves can be written as

[JH] [LH][RH] = K

[JH] = K[RH] [LH]

(la)

(1 b)

where [JH], [LH] and [RH] are the concentrations of the joined halves, the left-half and the right-half molecules, respectively, at equilibrium. The equilibrium constant for cyclization can be expressed as

[C] [Lf= Kc =Kj (2)

where [C] and [L] are the equilibrium concentrations of the cyclic and linear monomers, Kc is the equilibrium constant of cyclization, and K is the same constant as in eqn (1), the equilibrium constant of joining. The cyclization equilibrium is characterized by the parameter Kc or the parameter j = Kj K. The physical meaning of j can be seen by comparing eqns (1b) and (2). Equation (1b) says that the ratio of left-halves in the joined form to free left-halves is a constant K multiplied by the concentration of free right-halves. Therefore, j is the effective con­centration of one end (the right end, for example) in the vicinity of the other end (the left end, for example). This parameter has been referred to as the

CIRCULAR DNA 229

Jacobson-Stockmayer factor of cyclization or the ring-closure prob­ability.20 The parameter} similarly characterizes the kinetics of DNA cyclization. If the forward rate constant is krfor the bimolecular joining of the halves, the forward rate constant is }kr for the cyclization reaction. 1S .19

For .leDNA, the molecule is long and the random coil model is expected to apply. It is shown 20 that for a Gaussian chain:

}=(2:blYI2 (3)

where I is the contour length of the DNA and b is the Kuhn statistical segment length. The value of) determined from thermodynamic and kinetic measurements is about 3 x 1011 molecules ml- 1 or 5 x 10- 10 M for a strain of phage .le with a contour length of 13 11m; 19 the value calculated from eqn (3), taking b = 0·1 11m, is 3·7 x 10- 10 M.

Although for a Gaussian chain the ring-closure probability increases monotonically with decreasing chain length (eqn (3)), for a polymer like DNA) is expected to drop precipitously when the molecule is shortened to the extent that the ends cannot be brought together without seriously distorting the molecule. Theoretical treatment indicates that for a stiff polymer} approaches a maximum when the chain length is around one statistical segment length (0·1 11m or about 300 bp for DNA); further shortening of the molecule is accompanied by a rapid drop in j.

With the discovery of restriction endonucleases that cleave at specific DNA sequences, the ring-closure probability of DNA molecules of a broad size range can be studied. The restriction enzyme EcoRl, for example, cleaves the sequence

! -GAATTC---CTTAAG-

i

at the points indicated by the arrows, yielding fragments with self­complementary single-stranded ends that are four nucleotides long. Molecules joined by such short ends are stable at O°C but not at room temperature. By adsorbing the molecules at low temperature to a supporting film, it has been observed by electron microscopy thal ring­formation probability indeed increases with decreasing DNA chain length until the size falls below several hundred base pairs. 21

A more extensive study of the length dependence of ring closure probability was undertaken more recently. 22 - 24 DNA ligase was used to

230 J. C. WANG

join permanently a pair of ends that became affixed by base-pairing. From the rates of forming DNA ligase joined rings and intermolecular adducts, the ring-closure probabilities of DNA molecules from several thousand to a few hundred bp in length were evaluated. It is observed thatj approaches a maximum around a DNA length of several hundred bp (Fig. I).

More strikingly, for short DNA fragments j is an oscillatory function, with a periodicity of about 10· 5 bp (Fig. 2). Qualitatively, this periodicity can be attributed to the orientation dependence of the ring-closure probability. Earlier in this section, j is interpreted as the 'effective' concentration of one end in the neighbourhood of the other end. It is apparent from eqns (I b) and (2) that 'effective' is implicitly a thermo­dynamic description: the joining of the ends is taken to have the same intrinsic equilibrium constant K, and differences in the free energies of the intra- and intermolecular processes are grouped in the parameter j. (In the

,. ... I ,

10- 7 , 1, I , r It !'

I , .. ,

'i I ! \ I \!

~ , ,

.2 , \ ! , !i , : I

, 10-8 \ ,

1 \ , , , ,

\ , , , , , , , 10-9

Id 103 104

DNA length (base-pairs)

FIG. I. The ring-closure probability j as a function of DNA length. The dashed curve is that calculated from eqn (62) of Ref. 92. (Reproduced with permission from

Ref. 23.)

f­

f-

I

235

I

J

CIRCULAR DNA

I I

1"

f \ I \ I \ r,

I \ I \ • • \

\

• I I I

I I I \

I I •

• I

I

240

I \ \ I • I \ . ~ I

I ,.I I

245 I

250

DNA LENGTH (bose pairs)

\

\ \ \ \ •

255

231

-

-------

-

FIG. 2. The ring-closure probability j as a function of DNA length for a series of DNA fragments between 230 and 260 bp. (Reproduced with permission from

Ref. 23.)

kinetic formulation,} is present in the cyc1ization rate; the reverse step, the disjoining of the cohered ends, is assumed to be identical to the disjoining of cohered ends of intermolecular adducts.)

Because the cohesion of the ends involves the formation of a short helical segment that becomes imbedded in the helical stretch encompassing the rest of the molecule, a pair of ends to be joined must be oriented properly with respect to each other and with respect to the rest of the segments linked

a -7

-,

go -8

-9

0.37 0.38

,,- ........ , , , , , , / '

oj

0.40 0.41

-6~------~------~-------.------~

-,

'" o

-8

(J' 0.5

_9L-~LU~~-------L-------L----__ ~ 2.0 2.5 3.0 3.5 4.0

log nbp

FIG. 3. (a) Double-logarithmic plot of the ring-closure probability j (in M) as a function of DNA length in number of base pairs (n bp). The filled circles are experimental data from Ref. 23. The full curve represents best-fit theoretical values with a statistical segment length of 940 A, a Poisson ratio of - 0·4 and a DNA helical periodicity of 10·46 bp per turn. The broken curve represents those with 950A, -0·2 and 1O·46bp per turn for the corresponding parameters; the largest three observed values are ignored in this fit. (b) Double-logarithmic plot of the ring­closure probability j (in M) against the number of base pairs per DNA molecule nbp .

The filled circles are data from Ref. 23. The full curves represent the best-fit theoretical values of the upper and lower bounds with a statistical segment length of 950 A and a Poisson ratio of 0; the broken curves are those with a Poisson ratio of o· 5. The theoretical values of the angle-independent j factor according to Ref. 25 (denoted 10 in the figure) and Ref. 92 (denoted 16s) are also shown for the same value of statistical segment length. (Reproduced with permission from Ref. 25.)

CIRCULAR DNA 233

to the ends. When the ends are on a short DNA, their orientation is strongly affected by the helical geometry of the chain connecting them. 22,23

Shimada and Yamakawa 25 have carried out statistical mechanical and mechanical calculations for the ring-closure probability of a helical wormlike chain. The best-fit theoretical curves for small DNA and the experimental data of Shore and Baldwin 23 are shown as a double­logarithmic plot in Fig.3(a). The predicted ), in agreement with the experimental points, oscillates through upper and lower bounds with a periodicity of 10· 5 bp, which is the helical periodicity of B-form DNA in solution. 24,26 - 32 Their calculated results for larger DNA are shown in Fig. 3(b): the pair of solid lines are the upper- and lower-bound values for a helical chain with a statistical segment length b of950 A and a Poisson ratio of 0, and the pair of dashed curves are the upper and lower bound values for the same b value and a Poisson ratio of 0·5. (The Poisson ratio is the quotient of the bending and torsional forces constants minus one.) The curve labeled 1(0) in Fig. 3(b) represents the orientation independent) factor calculated by the same authors for b = 950 A.

COVALENTLY CLOSED DOUBLE-STRANDED DNA RINGS AND DNA SUPERCOILING

The Linking Number, Linking Difference and Specific Linking Difference As mentioned earlier, a double-stranded DNA ring in which both chains are intact can be viewed as two topologically linked single-stranded rings. The linking number (which is also termed the topological winding number in the earlier literature) is a parameter that characterizes the order oflinkage. If the duplex ring is laid in a plane, the linking number ('j, is simply the net number of right-handed turns the two strands revolve around each other. So long as the continuity of both strands is maintained, ('j, is a topological invariant. The linking number is an integer for individual molecule, although the population average is usually not.

If the continuity of one of the strands of a covalently-closed duplex ring is disrupted by the use of an endonuclease to break a phosphodiester bond in the sugarophosphate chain, then the topological constraint on ('j, is removed. Resealing of the break by the use of a DNA ligase is expected to give a molecule with a linking number ('j, 0, which is determined by the most stable structure of DNA under the conditions at which chain continuity is restored. Thus the cycle of nicking and resealing frees a closed duplex ring from its topological constraint and allows it to 'relax'. Instead of the

234 J. C. WANG

sequential use of an endonuclease and a ligase, an appropriate DNA topoisomerase can also be used to relax a closed duplex ring.

The well-known Watson-Crick B-structure of DNA predicts that strands revolve around each other in a right-handed way once every 10 bp; thus 0(0, the linking number of a relaxed DNA, is expected to be around N/ 1 0, where N is the number of bp per molecule. It will be described later that several types of measurements show that 0(0 = N/IO·5 in a dilute aqueous buffer near room temperature.

The relaxed duplex ring serves as a useful reference state for topological isomers (topoisomers) of the same DNA that differ only in their linking numbers. The difference between the linking number 0( of a DNA and 0( ° of the same DNA is termed the linking difference. The linking difference divided by 0(0, (0( - 0(0)/0(0, is termed the specific linking difference. The linking difference and specific linking difference correspond to the less precisely defined terms 'number of super helical turns' and 'superhelical density' in the earlier literature. 33 . 34

Preparation of Topoisomers with Different Linking Numbers A common procedure involves the relaxation of a closed-duplex DNA ring in the presence of an agent that unwinds or winds the helical structure. A drug ethidium, for example, binds to DNA by intercalating in between two adjacent base pairs 35 and unwinds the double-helix by about 26 0.36 Thus by sealing a nicked DNA ring with a DNA ligase or relaxing a covalently closed DNA ring with a DNA topoisomerase, in the presence of varying amounts of ethidium, DNA topoisomers with linking numbers lower than the value of 0( ° in the absence of ethidium can be obtained. 37.38 The bound ethidium, which is positively charged, can be readily removed by solvent extraction or passage through a cation exchange resin. If v ethidium is bound to each DNA nucleotide during ligation or relaxation of the DNA, the change in 0( ° due to the binding of ethidium is LlO( ° = 2Nv¢/360, where N is the number of base pairs per DNA molecule and ¢ = - 26 ° is the change in helix rotation angle of the DNA per ethidium bound (the minus sign signifies an unwinding of the angle; in the li terature before 1974, ¢ was taken to be - 12 ° and corrections are thus needed in using the earlier data). Topoisomers with linking numbers higher than 0(0 of pure DNA can be prepared similarly by the use of a drug netropsin. 39 Changes in temperature and ionic medium also result in small changes in 0( 0.40 - 42

DNA topoisomerases that utilize A TP hydrolysis to actively drive linking number changes are known. Bacterial DNA gyrase, which couples A TP hydrolysis with linking number reduction, has been studied

CIRCULAR DNA 235

extensively. 43 -45 A 'reverse gyrase' from Sulfolobus acidocaldarius, an acidothermophile cultured from hot springs, has been reported to increase the linking number in an ATP-dependent way.96

Supercoiling When the linking number a. of a closed duplex DNA ring differs appreciably from that of the same DNA in the relaxed state, a. 0 ,

deformation of the molecule occurs in a way analogous to that of a torsionally unbalanced multi-stranded rope or a twisted rubber tubing. 46 A DNA with a linking number a. that differs appreciably from a. 0 is termed a supercoiled or superhelical DNA. If the linking difference (a. - a. 0) > 0, the

FIG. 4. Electron micrograph showing a relaxed (left) and a supercoiled (right) DNA molecule. (Reproduced with permission from Ref. 33.)

236 J. C. WANG

DNA is said to be positively supercoiled; if(o: - 0:0) < 0, the DNA is said to be negatively supercoiled. Figure 4 shows two electron micrographs of a typical DNA in the relaxed and supercoiled forms.

The Twist Number Tw and the Writhing Number Wr Mathematical studies of the properties ofa twisted ribbon in space have led to a theorem in differential geometry47 that is related closely to the properties of a supercoiled DNA. 48 - 50 If the two ends of a twisted ribbon are joined so that each edge of the ribbon joins only with itself, the two edges become linked. The linking number 0: between the two lines marking the edges, which for ease of comparison with a double-stranded DNA with anti parallel chains can be taken as two lines of opposite polarity, has been shown to be equal to the sum of the twist number Tw and the writhing number Wr:

0:= Tw+ Wr (4)

The twist number is a well-known quantity in classical mechanics. At any point A along the central axis curve of a very narrow thin ribbon, if Tis the tangent vector and u is a vector perpendicular to T and lies on the ribbon, then the twist number of the ribbon between point A and another point B on the axis curve is the total number of turns u revolves around T as it moves from A to B along the axis. For convenience, right-handed turns will be taken as positive and left-handed turns as negative.

The writhing number of a space curve can be obtained from the Gauss integral 48 - 51

(5)

where e=[x(s2)-x(st))/lx(s2)-x(sl)1 is the normalized connecting vector between two points X(SI) and X(S2) along a space curve xes) parameterized by the arc length s, and x and. represent the cross and scalar product, respectively. The writhing number of a ribbon is determined solely by the spatial shape of its axis curve.

The ribbon model provides several important concepts for under­standing the properties of covalently closed DNA rings. For example, if Wr is zero, 0: is equal to Tw from eqn (4). It can be shown that Wr of a closed curve is zero if it lies in a plane or on the surface of a sphere; 48 - 50 thus for a DNA ring lying flat in a plane its linking number can be counted as the number of revolutions the strands make around each other, as mentioned earlier. Equation 4 also shows that, for a topoisomer with a fixed 0:, a change

CIRCULAR DNA 237

in the angle the base pairs twist around the helix axis must be accompanied by a change in the spatial writhe of the helix axis to keep the sum of Twand Wr constant. Because the hydrodynamic properties of a DNA are expected to be sensitive to its three-dimensional shape and thus the writhe, small changes in the helical geometry of the base pairs in a closed-circular DNA may change the hydrodynamic properties of the DNA significantly. (If the helix rotation angle between adjacent base pairs changes by 0.1 a,

for a 10000 bp closed-duplex ring Tw and thus Wr are changed by 10 000 x 0·1/360 or about three turns.) It will be shown in later sections that for a DNA of this size a change of Wr by a fraction of a turn is readily detectable, and thus closed-duplex rings can be used to study minute changes in DNA helical structure.

The Energetics of DNA Supercoiling A supercoiled DNA is in a higher energy state compared with a relaxed DNA. 52 The dependence of the free energy on a can be determined by examining the Boltzmann population of topoisomers. Figure 5 illustrates an experiment in which a nicked DNA about 10000bp in size is treated with a DNA ligase and the product is analysed by electrophoresis in an agarose gel. The four samples loaded in lanes from left to right were treated with DNA ligase at 37, 29, 21 and 14 ac, respectively. The topmost (slowest migrating) band in each case is the nicked DNA that was not sealed by the enzyme. Under this nicked DNA band in each case is a ladder of bands. This is particularly evident for the samples run in the three lanes on the right. It has been shown that the ladder of bands represents covalently closed topoisomers that differ only in their linking numbers, and that the linking numbers of two adjacent bands, of the ladder shown in the third lane from the left for example, differ by 1. 38 . 53 Within a certain range of the specific linking difference from 0 to approximately ± 0·05, the gel electrophoretic mobility of a topoisomer oflinking number a increases with the magnitude of (a - an where a~is the linking number of the DNA when relaxed under gel electrophoresis conditions. One way to deduce whether the topoisomers are positively supercoiled [(a - a~) > 0] or negatively supercoiled [(a - an < 0] during electrophoresis is by adding a small amount of an intercalating agent, such as ethidium or chloroquine, to the electrophoresis buffer. This decreases a~. If (a - a~) > 0, the linking difference becomes greater when a~ is reduced; thus the bands would migrate faster. If (a - a~) < 0, the opposite is true. For the samples shown in the figure, the topoisomers can be shown to be positively supercoiled.

For each of the samples the ladder represents the Boltzmann population

238 J. C. WANG

FIG. 5. Gel electrophoretic patterns of a 10 000 bp DNA covalently closed by DNA ligase at, from left to right, 37, 29, 21 and 14°C. (Reproduced with permission

from Ref. 41.)

of topoisomers that are formed by the sealing of the nick in the original DNA. Prior to the sealing of the: nick, the DNA is relaxed and the number of revolutions the strands go around each other in the double helix is not restricted topologically; after sealing the nick, the linking number becomes a topological invariant. Thus the sealing of the nick by ligase effectively 'freezes' the Boltzmann population under the conditions of ligation.41 ,97

Figure 6 depicts densitometer tracings of the electrophoretic patterns shown in the second and fourth lanes. In each case the closed topoisomer bands are enveloped by a Gaussian curve. For a DNA of this size, it can be shown that the centre of the Gaussian envelope corresponds to a value IX~,

CIRCULAR DNA 239

(a' Clos ... Temperature 14-C

1----------------.-----

(b) Closure Temperature Zg-'C

- DISTANCE MIGRATED

FIG. 6. Densitometric tracings of the second and fourth lanes of the negative of the photograph shown in Fig. 5. The dotted line indicates the position of a small amount of linear DNA in this preparation; the linear DNA band is the fastest migrating band resolved from other species in the left-most lane. (Reproduced with

permission from Ref. 41.)

the population average linking number of the relaxed DNA under ligation conditions. The Gaussian shape of the envelope results from the excess free energy of a topoisomer of a linking number rx, over that of a hypothetical relaxed topoisomer rx~, which is proportional to (rx - rxD2:

!J.Gr == G(rx) - G(rxD = K(rx - rxD2 (6)

240 J. C. WANG

In general, if rx 0 represents the linking number of a hypothetical topoisomer of minimal free energy under a given set of conditions (rx 0 may not be an integer whereas the linking number of any real topoisomer must be an integer), then the excess free energy of a topoisomer with linking number rx under the same conditions is

(7)

The magnitude of K decreases with increasing size of a DNA. Figure 7 shows some of the measured values of K of DNAs of different lengths. 32

For DNAs larger than 2000bp in length the product NK, where N is the length of a DNA in bp, is a constant around 1100 RT. Calorimetric measurements with a DNA of a specific linking difference of about - 0·07 shows that the positive free energy of DNA supercoiling is due to a positive enthalpic term. 54

The samples shown in Fig. 5 also illustrate the point raised earlier, that the coupling between Tw and Wr in a closed duplex ring can be used to study minute changes in the DNA helical structure. The average linking

o o o

~

5.---------------------------------------------.

o 1000 5000

Lenglh (bpi

FIG. 7. Dependence of the product NK as a function of DNA length. NK is in units of bp' RT. (Reproduced with permission from Ref. 32. For additional data

which differ significantly from data shown in the figure, see Refs 23 and 24.)

CIRCULAR DNA 241

number of the Gaussian population of topoisomers increases progressively as the temperature ofiigation is decreased. Since Wr for a relaxed DNA is close to zero, the temperature dependence of (X~ reflects the temperature dependence of Tw, the twist of the relaxed DNA. The tracings shown in Fig. 6 show that from 14 to 29 °C, (X~ decreased by about five turns, corresponding to an unwinding of about 0·01 degree per bp per °C increment in temperature.

EtIects of DNA Supercoiling on its Structure and its Interactions with Other Molecules When the linking number (X of a closed duplex DNA ring deviates significantly from (X 0, torsional and flexural stresses are induced, which cause deformation of the molecule from its shape in the relaxed state. Figure 8 illustrates several idealized modes of distortion by which a negative value of the linking difference ((X - (X 0) can be accommodated. In cases (a)-(d), the deficiency in linking number is accomodated entirely by a reduction in the twist number. Case (a) illustrates a more-or-Iess uniform unwinding of the helical structure so that the average helical rotation between adjacent base pairs is slightly reduced. Case (b) illustrates a reduction in Tw by flipping a short stretch of base pairs from the right­handed B helical-structure to a left-handed helical structure. Case (c) illustrates a reduction in Tw by the disruption of a short helical segment. Case (d) illustrates a special case of (c) when the disrupted sequence is palindromic so that the single-stranded region shown in (c) can form a pair of short hair-pinned structures termed a 'cruciform'. Cases (e) and (f) illustrate two modes of deformation that involve primarily a change in Wr. Case (e) is termed the 'interwound negative supercoil' in which the duplex coils right-handedly upon itself. 52 Case (f) is termed the 'toroidal negative supercoil' in which left-handed toroidal turns are present. 52

The actual mode or combination of modes of deformation depends, of course, on the free-energy changes associated with the various modes of deformation. Thermodynamic and statistical-mechanical treatments of some of these cases involving localized structural changes have been carried OUt.52.55-63.98-100 In one approach,33.55-57 it is assumed that, for a reasonably large DNA, eqn (7) remains valid and that a local structural change is assumed to change only (X0. For the case of cruciform formation illustrated in Fig. 8(d), as an example, the thermodynamic cycle depicted in Fig. 9 can be used to relate I1G 2 , the standard free energy change for the

242 J. C. WANG

a b

c d

e f

FIG. 8. Schematic drawings illustrating several idealized modes of deformation resulting from a linking number e< lower than that of the relaxed DNA, e<0. For simplicity, the intertwined strands are shown as parallel lines: (a) represents a uniform unwinding of the double-helix to absorb the linking difference (e< - e(0); (b) represents the flip of a short segment (hatched region on top) from the right-handB­helical structure to a left-handed helical structure; (c) represents the disruption of a short helical segment; (d) represents cruciform formation (the formation of a pair of hairpinned structures) for a palindromic sequence; (e) represents the 'interwound' supercoiling of the duplex; (f) represents the 'toroidal' supercoiling of the duplex.

process in a supercoiled DNA of linking number IX, to the standard free­energy change llGc when cruciform formation occurs in a relaxed or linear DNA: 57 •63

llG2 = ~Gc - llG 1 - llG2

llG 1 is the standard free energy of supercoiling when the linking number changes from IX 0 to IX:

CIRCULAR DNA 243

B,~= a c, ~= a

FIG. 9. A thermodynamic cycle for deducing the relation between the free energy of cruciform formation in a relaxed and in a supercoiled DNA. For simplicity the two intertwined strands of the double-helical DNA are drawn as parallel lines. The intrinsic energy change I'lGe when a cruciform forms in a linear or nicked circular DNA can be calculated from the sum of I'lGI' I'lG2 and I'lG3 . I'lG j is the free energy of supercoiling corresponding to a change of the linking number from Cl 0 to Cl, for the DNA with the cruciform. - I'lG3 is the free energy of supercoiling corresponding to a change of the linking number Cleo to Cl. Cl 0 and Cleo are the linking numbers of the relaxed DNA without and with the cruciform. The linking number for the I'lG2 step is unchanged. For a particular topoisomer of a linking number Cl, species Band C are present in equal amounts in an equilibrium mixture, and t1G2 is zero at this value of Cl. In the illustration, the t1Ge is shown for the relaxed covalently closed DNA, and the linking number of the DNA is altered in the process. It is conceptually helpful to imagine that a nick is first introduced into A, followed by the formation of the cruciform, and ending with the re-closure of the nick to give D.

(Reproduced with permission from Ref. 63.)

244 J. C. WANG

!:J.G 3 is assumed to obey the same relation as !:J.G 1 :

!:J.G3 = K(rx - rxcO)2

except that rxco refers to the linking number of the DNA with a cruciform but is otherwise relaxed. The length of the palindromic sequence forming the cruciform is taken to be short compared with the size of the DNA, so that K can be assumed to be unchangl~d in the calculation of !:J.G1 and !:J.G 3 • It then follows that

or

!:J.G2 - !:J.Gc == K[(rx - rxcO)2 - (rx - rxO)2]

= 2K[rx - (rxo + rxcO)/2](rxo - rxCO) == 2K(rx - aO)(rxO - rx CO)

(8)

where aO is the average linking number of the two relaxed states and brxo is the difference in linking numbers of the two relaxed states.

The difference !:J.G2-!:J.Gc can also be obtained directly from the change in !:J.G r when the linking number of the reference state changes by quantity brxo. 33 When brxo is small compared with rx-rxo, it is convenient to calculate the change in !:J.G r corresponding to a change in brxo from

MGr=(d!~r)brxo =2K(rx-rxO)brxo

= O·19NK(J brxo

(8a)

(8b)

The relations rxO = N/lO·5, where N is the size of the DNA ring in bp, and (J = (rx - rxO)/rxO are used in obtaining eqn (8b) from eqn (8a). Since N K is approximately constant for large N, b!:J.G r is proportional to (J, the specific linking difference, for a given change in brx 0.

Figure 10 illustrates an experiment 62 demonstrating a negative supercoiling-induced structural change in DNA, in which a short stretch of alternating CG flips from the right-handed B-helical geometry to the left­handed Z-helical geometry. 64 A mixture of topoisomers of different linking numbers was loaded in the upper-left corner of a square agarose gel slab and first-dimension electrophoresis in the direction from top to bottom was carried out in a TRIS-borate-EDT A buffer. The gel was then soaked in the same buffer plus an intercalating agent, chloroquine in this case, and

o , 6

CIRCULAR DNA

I I

o

•

I I

245

FIG. 10. Dependence of the right- to left-handed helix transition of a sequence d(pCpG)'6 . d(pC-G) '6' cloned into a 4400 bp plasmid DNA by recombinant DNA methodology on DNA supercoiling. See text for details. (Reproduced with

permission from Ref. 62.)

second-dimension electrophoresis in the direction from left to right was carried out in the chloroquine-containing buffer. Such a two-dimensional electrophoresis technique 34 .62.65 expands the range in linking number of topoisomers that can be resolved. The group of spots shown in the left panel of Fig. 10 are isomers of a control plasmid pTR 161, with a size of about 4400bp. Spot 0 is nicked pTR161; the other spots are the closed topoisomers. The apex (I) of the arc of spots represents the position of a hypothetical topoisomer (of linking number IX~) which is completely relaxed under the first-dimension electrophoresis conditions: its mobility is the lowest in the first dimension compared with adjacent topoisomers with higher or lower linking numbers , and is closed to that of the nicked ring. Similarly, the tip of the arc near spot 15 (II) represents the position of a to poi somer (of linking number IXn which is completely relaxed under the second-dimension electrophoresis conditions. The difference IX~ - IX~ can be obtained by counting the number of spots along the arc connecting the two positions, since adjacent spots represent topoisomers differing by I in their linking numbers. For the example shown, this difference is about II; IX; is lower than ct~ because of the intercalation of chloroquine into the DNA helix under the second-dimension electrophoresis conditions. It can be easily seen that spots with indices higher than 20 have about the same mobilities in the first dimension, but the reduction of their linking differences during second dimension electrophoresis makes it possible to resolve them on the same gel.

246 1. C. WANG

The group of spots shown on the right panel of Fig. 10 are derived from pTRI61 DNA containing a 32 bp alternating CG insert. The arc of spots 1-16 for the plasmid with the (CG) 16 insert is very similar to the arc of spots 1-16 of the control without the insert. Spot 17 of the right panel, however, shows a dramatic shift in mobility in the first dimension compared with spot 17 of the left panel.

The abrupt change in mobility signifies a highly cooperative super­coiling-induced structural change of the inserted CG sequence. During the first-dimension electrophoresis, the position of the relaxed state is in­between spots 3 and 4 for the (CG) 16-containing plasmid, thus the linking differences of spots 10 and 16, as examples, are -(10 - 3·5) = -6·5 and -(16 - 3·5) = -12·5, respectively. When the linking difference is de­creased further, the free energy of supercoiling favours the flipping of the CG insert from the right-handed B-form to the left-handed Z-form. This transition reduces Tw and hence Wr (0: remains unchanged for any topoisomer). The magnitude of the change in Tw can be deduced if one makes a reasonable assumption that the gel mobility is primarily determined by the average spatial configuration of the axis curve and is not sensitive to the way the base pairs rotate around the axis curve; in other words, the mobility is assumed to be sensitive to Wr and not to Tw. Since spots 17 and 12 of the right panel have approximately the same first­dimension mobility, they can be assumed to have the same Wr. The linking number of spot 17 is, however, lower than that of spot 12 by 5; therefore Tw of to poi somer 17 is lower than that of 12 by 5. Detailed analysis of the data and calculations of the free-energy parameters of the transition are described in the original work,62 and additional studies have been reviewed.64

In addition to influencing structural changes of DNA, supercoiling also affects interactions between DNA and other molecules. Consider the binding of a certain molecule L to a relaxed DNA with an association constant Ka.R and a corresponding free-energy change !}G~ = - RT In Ka.R. If the bound L alters the helical structure of DNA in some way, then the linking numbers of a closed-duplex ring relaxed in the presence and absence of a bound L would be different by an amount 60: o. Thus when L binds to the same DNA in the supercoiled form the expected free-energy change is,33.55-57 when 60:° is small relative to (0: - 0:0),

o 0 d!}G, 0

!}Gs = !}GR + ~ 60:

== !}G; - 0·19 N Krr( 60: 0)

It follows that

or

CIRCULAR DNA

I1G;-I1G;= -RTln Ka,s + RTln Ka,R

= - RT In (Ka,s/ Ka,R) = -0'19NKO'(ba D)

Ka,s/ Ka,R = exp 0·19N KO'(baD)/RT

247

(9)

In eqn (9), Ka,s is the association constant for the binding of L to a supercoiled DNA of a specific linking difference O'. The simple analysis above shows that the binding constant can be strongly affected by supercoiling. For example, if 0'=-0·05 and baD=-I, KaS/KaR is calculated to be 3·4 x 104 for a typical DNA with a ~alu~ of NK= 1100RT. 33 ,56,57

In the thermodynamic approach described above, the experimentally determined free-energy dependence on the linking difference is used and there is no need to separate the contributions of torsional and flexural stresses; values of the torsional and bending moduli of DNA are not needed. By expressing the free energy of supercoiling, I1G" in terms of the experimentally measurable parameter (a - aD), the particular approach also avoids the explicit use of Tw and Wr as variables, which, though conceptually well-defined and useful, are difficult to measure experimen­tally. The approach assumes, however, that eqn (7) holds before and after the structural change. Additional examples and different thermodynamic and statistical-mechanical treatments of the energetics of supercoiling can be found in Refs 52 and 55-60, and references therein.

Shape of Supercoiled DNA As illustrated in Fig. 8, for a given linking difference (0( - a D), different modes of deformation are possible. The average shape of a supercoiled DNA of a given linking difference is determined by the conformational free energies of the various forms, some of which can be calculated, as illustrated above. For a typical DNA under physiological conditions (dilute aqueous buffer containing 0'1-0·2 M Na + or K + and several millimolar Mg2+, room temperature), flipping of the helical hand (Fig. 8(b», disruption of helical regions (Fig. 8(c» or cruciform formation (Fig. 8(d» does not occur when - a < 0·03. The modes of deformation in such a moderately supercoiled DNA are primarily small changes in the average twist, and interwound and toroidal supercoiling (Fig. 8(a), (e) and (f).

Several types of calculations have been carried out to gain insight into the

248 J. C. WANG

general shapes of supercoiled DNAs. For a circular DNA much longer than the persistence length and a linking number close to IX 0 , calculations of the writhing numbers of a large number of computer-generated closed polygonal lines have provided the fluctuation in Wr and the confor­mational entropic term associated with it. 66.67 For small DNA circles with a contour length shorter than the persistence length, elastic 68 and conformational analyses 69 have been carried out to determine defor­mations due to supercoiling.

A number of experimental approaches have also been used. Electron microscopy usually reveals the interwound form (see Fig. 4), in particular for large DNAs with high values of (J. The technique, however, usually requires the adsorption of the DNA to a supporting film, and thus the shape viewed is the one after flattening of the molecule by forces during sampling, adsorption and drying. Under some conditions, spiral-shaped rather than interwound DNA supercoils have been seen (see, for example, Ref. 70). A toroidal type conformation has also been suggested from calculation of the sedimentation coefficients of various geometric shapes, 71

from light scattering 72 and X-ray scattering 73 measurements. First-order elastic treatment, however, disfavours the toroidal shape. 68 The non­monotonic variation of the sedimentation coefficient of a supercoiled DNA with its specific linking difference (J has been interpreted in terms of a toroidal shape at low values of (J and an interwound form at high values of (J.74

CATENANES AND KNOTS 75

DNA catenanes or interlocked rings were first observed in vitro by cyclizing a linear DNA in the presence of a high concentration of a circular DNA 76

and in vivo by electron microscopic examination of mitochondrial DNA isolated from human tissue culture cells. 77 The sedimentation coefficients of a variety of DNA catenanes containing single- and double-stranded rings have been determined. 78 More recently, it has been shown for simian virus 40 that a pair of newly replicated progeny DNA rings are multiply­intertwined catenanes; 79,80 an example is shown in Fig. II.

In vitro, catenation and decatenation of DNA rings are catalysed by DNA topoisomerases. 43 -45 The type I topoisomerases, which break and rejoin one DNA strand at a time, can catenate single-stranded rings or a pair of double-stranded rings provided that at least one of the pair contains a pre-existing nick. The type II topoisomerases, which break and rejoin

CIRCULAR DNA 249

FIG. II . Two dimeric catenanes, each consisting of a pair of newly replicated simian virus 40 DNA molecules. (Electron micrograph courtesy of Professor Alex

Varshavsky.)

250 J. C. WANG

both strands of a duplex DNA ring at the same time, can catenate or decatenate duplex rings with or without nicks. In vivo the resolution of catenanes following a round of replication appears to be carried out by a type II topoisomerase.45 ,79.8o

Although cyclization of a linear DNA is expected to yield some knotted rings,81 the probability of knotting is small and the existence of knots following cyclization of a DNA has not yet been substantiated. Knotted DNA rings were first observed by treatment of a single-stranded cyclic DNA with a bacterial type I topoisomerase. 82 The formation of double­stranded knotted rings has been observed in type I and type II topoisomerase catalysed reactions, and in cyclic phage P2 and P4 DNA extracted from a phage capsids without tails.43-45

Figure 12 illustrates several plausible modes of intramolecular strand passage, catalysed by DNA topoisomerases, that could lead to knot formation. Figure 12(a) shows a simple ring a and the twisting of the ring at one location, dividing the molecule into two loops (b), linking of the two loops by a strand passage eVt:nt can yield the molecule c, which is a trefoil (d). Figure 12(b) illustrates a case in which a segment ofa duplex DNA ring

OOCG{)~& ~ c d

la l

A

B

Ie I

I bl

FIG. 12. Three plausible modes of DNA knot formation catalysed by DNA topoisomerases. See text for explanation. (Parts (a) and (b) reproduced with

permission from Ref. 93 and Ref. 94, respectively.)

CIRCULAR DNA 251

is wrapped around a topoisomerase, which introduces a transient double­stranded break at the point indicated by the arrow, to permit the passage of another segment (AB or CD). If AB crosses, the ring remains a simple one, although its linking number is altered by 2; 49,8 3 - 8 5 if CD crosses, a knot is introduced. Figure 12(c) illustrates a simple double-stranded ring: (i) if a pair of cuts are made across ab and cd, and a is then joined to d and b to c in the manner shown (ii), a knot results. A number of topoisomerases, in particular some of the enzymes that catalyse site-specific recombination, are known to give knotted products if their sites of action (ab and cd in Fig. 12(c)) are appropriately oriented along a cyclic DNA.43 -45

The detailed topology of DNA catenanes and knots can be visualized by the use of special electron microscopic methods. 86 - 88 Figure 13 illustrates two cases. In each one, the DNA is incubated with a protein that forms a unique complex with DNA to give a filament with a diameter much larger than that of DNA itself. The thickening of the diameter makes it easier to distinguish, when the molecule is viewed in the electron microscope after shadowing with evaporated metal, whether one segment is crossing over or going under another. It is readily seen that the molecule shown in Fig. 13(a) is a trefoil knot, the one shown in Fig. 13(b) is a knot with five crossovers and the one shown in Fig. 13( c) is a catenane with the two rings going around each other twice. The topological characteristics of catenanes and knots resulting from a topoisomerase catalysed reaction are intimately related to the way the topoisomerase interacts with the DNA substrate; thus the chemical topology of DNA rings provides a powerful tool in the analysis of the mechanism of catalysis by the topoisomerases (see Refs. 86-89 and references therein).

GENERAL PROPERTIES OF DNA DEDUCED FROM STUDIES OF DNA RINGS

Because of the double-stranded helical structure of DNA, certain properties of cyclic DNA are uniquely coupled to its helical structure. It has been described earlier in this chapter that the ring-closure probability of a short DNA is strongly modulated by its helical periodicity, that the coupling between the twist number Tw and the writhing number Wr of a covalently closed duplex ring makes it possible to detect minute changes in the helical geometry of DNA, and that DNA structure and its interactions with other molecules are strongly influenced by supercoiling. Studies of the topological properties of closed-duplex DNA rings have also established

252 1. C. WANG

(a)

(b)

FIG. 13. Electron micrographs showing the details of two knotted DNAs «a) and (b» and a catenane (c). In (a) and (b), the DNA was first coated with a protein (the uvsX gene product) isolated from phage T4 infected cells (micrographs courtesy of Professor Jack Griffith) ; in (c), the DNA was first coated with a protein (the recA gene product) isolated from the bacterium E. coli (micrograph courtesy of Dr

Andrei Stasiak).

CIRCULAR DNA 253

(c)

FIG. 13-contd

the double-helix nature of DNA independent of the X-ray diffraction results . 53 Several examples are described below.

The Handedness of the DNA Double Helix The invariancy of the linking number of a closed duplex ring has been used to show that the DNA double helix is a right-handed one. 90 The scheme used is sketched in Fig. 14. A single-stranded ring with a sequence of nucleotides complementary to the terminal sequences of a linear DNA is annealed with the linear DNA to form a short helical region. The product is

-FIG. 14. An illustration of the scheme used to determine the handedness of the

DNA double-helix.

254 J. C. WANG

then treated with DNA ligase Ito seal the nick and thus linking topologically the two single-stranded rings. Subsequent disruption of base pairing between the segments of complementary sequences yields a catenane with the two single-stranded rings intertwined. Electron microscopy is then applied to visualize whether a strand is crossing over or under another. The sign of the linking number of the catenane can thus be determined, which in turn gives the handedness of the original short helical segment.

The Gaussian-Centre Method and the Determination of the Helical Periodicity of DNA in Solution 26

It was described earlier that the distribution of a Boltzmann population of topoisomers differing only in their linking numbers is Gaussian centring at a position rx 0 which is usually non-integral and thus does not coincide with any of the real topoisomer positions. In Fig. 15, the separation between the Gaussian centre and the most abundant topoisomer of the equilibrium population is denoted w. If a segment of a particular nucleotide sequence x bp in length is inserted into the original DNA by standard recombinant DNA methods, w remains unchanged if x is an integral multiple of h 0, the helical periodicity of DNA, and is modulated by the non-integral residual of x if it is not. Strictly speaking, here the helical periodicity h 0 refers to the number of base pairs of the inserted sequence that increases rxO by unity. For a DNA segment of a typical sequence, its intrinsic writhe is small and h 0 is very close to the number of bp per helical twist.

I I

---~

-+ """, ---..........

" " /' , " , ,

, " /

/ I

Linking Number

FIG. IS. A densitometric tracing showing a Boltzmann population of DNA topoisomers differing only in their linking numbers. The size of the DNA is about

4400 bp. (Reproduced with permission from Ref. 95.)

CIRCULAR DNA 255

The Band-Shift Method and the Determination of the Helical Periodicity of DNA in Solution A demonstration of the intricate coupling between the twist and the writhe in a covalently closed DNA is the band-shift experiment 27,28 illustrated in Fig, 16(a), The electrophoretic patterns of several DNAs in agarose gel are shown. In each lane, the slowest migrating (topmost) band is the nicked circular form, which is preceded by a group of closed topoisomers. The DNA run in the middle lane is about 4400 bp in length. If the length is in­creased by 26 bp, the pattern shown in the left lane is obtained. The right­most lane shows a mixture of the two DNAs run simultaneously to facilitate a comparison of their mobilities. Clearly, the group of topoisomers of the longer DNA is shifted relative to the group of topoisomers of the shorter one: it appears that each topoisomer of the longer DNA is shifted upward from a corresponding topoisomer of the shorter DNA by about six-tenths

(a) (b)

FIG. 16. The band-shift method for the determination of the helical periodicity of DNA in solution. See text for description. (Reproduced with permission from

Ref. 27.)

256 J. C. WANG

of the interband spacing. That the mobility changes of the topoisomers are intimately related to the topology of the DNA can be seen in two ways. First, the mobilities of the nicked DNAs differ much less than those of the closed topoisomers. Secondly, for two DNAs that differ by 6 rather than 26 bp, the closed topoisomers of the longer DNA are also shifted from the corresponding shorter topoisomers by about six-tenths of the interband spacing (Fig. 16(b)).

The shifts of the topoisomer bands when the length of a DNA changes can be understood in the following way. 33 Let N be the number ofbp of the shorter DNA and ... i-I, i, i + I, . .. be the linking number of the topoisomers produced by ligase treatment. The gel electrophoretic mobility of a topoisomer oflinking number rx can be expressed as a function of the linking difference under electrophoresis conditions, rx - rx~. If h ° is the helical periodicity under the electrophoresis conditions, rx~ = N/ h 0.

Thus for the set of topoisomers with linking numbers ... i-I, i, i + 1, ... , their corresponding linking differences are ... i - I - N/ho, i - N/ho, i + 1 - N /h 0, •••• Similarly, upon the insertion of 6 bp into this DNA (and thus increasing rx~ to (N + 6)/h 0), the linking differences corresponding to topoisomers with linking numbers ... i -1, i, i + 1, ... are ... i - 1-(N+6)/ho, i-(N+6)/ho, i+I-(N+6)/ho, ... , etc. Thus for a topoisomer with a linking difference k in the first set, there is a topoisomer with a linking difference k - 6/h ° in the second set. Because the interband spacing between two adjacent topoisomer bands represents a difference of 1 in the linking number or the linking difference, it follows that all topoisomer bands with N + 6 bp per molecule are shifted from the corresponding topoisomer bands with N bp per molecule by a fraction 6/ h ° of the interband spacing, in the direction of decreasing linking number. I tis readily shown that if x = integer. h ° + y, where y < h ° is the residue, then all topoisomers are shifted by y/h ° times the interband spacing. This dependence on the non-integral residue makes it possible to obtain an accurate value of h ° by using progressively longer inserts. It has been shown 2 7 ~ 29 that DNA of a typical sequence has a value of hOof 10·6 bp; the value is for electrophoresis conditions (dilute TRIS-borate buffer, room tempera ture).

Two additional points are noteworthy in regard to the band-shift method. First, because even for a nicked DNA ring the mobility is dependent on its length, for topoisomers with long inserts corrections for the intrinsic dependence of electrophoretic mobility on length are necessary. One way of correcting the length effect is to carry out the band­shift measurements using topoisomers that are positively supercoiled and

CIRCULAR DNA 257

using topoisomers that are negatively supercoiled under gel electrophoresis conditions. 2 7 .28 When the topoisomers are positively supercoiled, the ones with higher linking numbers migrate faster; when the topoisomers are negatively supercoiled, the ones with the higher linking numbers migrate slower. Because shifts due to changes in the linking differences are in the direction of decreasing linking number, they can be either coincident with or opposite to the direction of migration. The intrinsic length effect always reduces mobility with increasing length, whether the DNA is positively or negatively supercoiled. Thus when to poi somer shifts are measured for the two positively and negatively supercoiled sets, the intrinsic length effect can be readily subtracted out. 27,28 Secondly, in the paragraph above an increase in length by x bp is assumed to increase (X~ by xl h 0, and thus the predicted shift is a fraction of xl hOof the interband spacing (if x < h 0) in the direction of decreasing linking number. But this assumption is correct only if the x bp is assuming a right-handed helical form. If the inserted segment is in a left-handed helical form, then the shift wouid be xlh ° times the inter band spacing in the direction of increasing linking number. Thus the band-shift method can also be used to determine the relative handedness of the helical structure of a particular sequence. This method has been applied to show that alternating CG sequence assumes a left-handed helical structure in a negatively supercoiled DNA,62 in agreement with other studies. 64

The Torsional Rigidity of DNA When the linking number (X of a closed duplex DNA ring deviates from (X 0, torsional and flexural deformations occur. Mechanical stress con­siderations suggest that if the ring is small (a couple of hundred bp in contour), the torsional changes dominate. 68 From the data shown in Fig. 7 and similar data in the literature,23.24 N K increases with decreasing Nand reaches a value 3910 RT at N = 210 bp, the smallest ring for which N K has been measured. If it is assumed that for a duplex DNA ring of this size torsional deformation dominates when (X deviates from (X0, it is readily shown that K is related to the torsional rigidity C of an isotropic rod of length L by

(10)

where N A is the Avogadro number. Substituting N K = 3910 RT and L=3·4Nx 1O- 1o m into eqn (10) gives C=2·9 x 1O- 1g ergcm or 2·9 x 10- 28 J m, suggesting that DNA might be considerably more rigid torsionally than indicated by values of C around 1·5 x 10- 28 J m from

258 J. C. WANG

spectroscopic measurements using probes that are rigidly bound to DNA. 23,24,32,91 A more rigorous calculation gives C = 3·2 x 10 - 28 J m from the measured values of N K. 101

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2. Fiers, W. and Sinsheimer, R. L., J. Mol. Bioi., 5 (1962) 408. 3. Fiers, W. and Sinsheimer, R. L., J. Mol. Bioi., 5 (1962) 4214. 4. Freifelder, D., Kleinschmidt, A. K. and Sinsheimer, R. L., Science, 146(1964)

254. 5. Sanger, F., Air, G. M., Barrell, B. G., Brown, N. L., Coulson,A. R., Fiddes, J.

c., Hutchinson, C. A., III, Siocombe, P. M. and Smith, M., Nature, 265 (1977) 687.

6. Sanger, F., Coulson, A. R., Friedmann, T., Air, G. N., Barrell, B. G., Brown, N. L., Fiddes, J. c., Hutchinson, C. A. III, Siocombe, P. M. and Smith, M., J. Mol. Bioi., 125 (1978) 225.

7. Cairns, J., J. Mol. Bioi., 6 (1963) 208. 8. Cairns, J., Cold Spring Harbor Symp. Quan. Bioi., 28 (1963) 43. 9. Kleinschmidt, A., Lang, D. and Zahn, R. K., Z. Naturforsch., 166 (1961) 730.

10. Stonington, O. and Pettijohn, D., Proc. Natl. Acad. Sci. USA, 68 (1971) 6. II. Benyajati, C. and Worcel, A., Cell, 9 (1976) 393. 12. Kavenoff, R. and Zimm, B. H., Chromosoma, 41 (1973) 1. 13. Weil, R. and Vinograd, J., Proc. Natl. Acad. Sci. USA, 50 (1963) 730. 14. Boyer, P. (Ed.), The Enzymes, 3rd edn, Vol. XIV, Nucleic Acids Part A,

Academic Press, New York, 1981. 15. Helinski, D. R. and Clewell, D. B., Ann. Rev. Biochem., 40 (1971) 899. 16. Hershey, A. D., Burgi, E. and Ingraham, L., Proc. Natl. Acad. Sci. USA, 49

(1963) 748. 17. Sanger, F., Coulson, A. R., Hong, G. F., Hill, D. F. and Petersen, G. B., J.

Mol. Bioi., 162 (1982) 719. 18. Wang, J. C. and Davidson, N., J. Mol. Bioi., 15 (1966) Ill. 19. Wang, J. C. and Davidson, N., J. Mol. Bioi., 19 (1966) 469. 20. Jacobson, H. and Stockmayer, W. H., J. Chem. Phys., 18 (1950) 1600. 21. Mertz, J. E. and Davis, R. W., Proc. Natl. Acad. Sci. USA, 69 (1972) 3370. 22. Shore, D., Langowski, J. and Baldwin, R. L., Proc. Natl. Acad. Sci. USA, 78

(1981) 4833. 23. Shore, D. and Baldwin, R. 1L., J. Mol. Bioi., 170 (1983) 957. 24. Shore, D. and Baldwin, R. L., J. Mol. Bioi., 170 (1983) 983. 25. Shimada, J. and Yamakawa, H., Macromolecules, 17 (1984) 689. 26. Wang, J. c., Cold Spring Harbor Symp. Quan. Bioi., 43 (1979) 29. 27. Wang, J. c., Proc. Natl. Acad. Sci. USA, 76 (1979) 200. 28. Peck, L. J. and Wang, J. c., Nature, 292 (1981) 375. 29. Strauss, F., Gaillard, C. and Prunell, A., Eur. J. Biochem., 118 (1981) 215. 30. Rhodes, D. and Klug, A., Nature, 286 (1980) 573. 31. Rhodes, D. and Klug, A., Nature, 292 (1981) 378.

CIRCULAR DNA 259

32. Horowitz, D. and Wang, J. c., J. Mol. Bioi., 173 (1984) 75. 33. Wang, J. C., Trends in Biochem. Sci., 5 (1980) 219. 34. Wang, J. c., Peck, L. J. and Becherer, K., Cold Spring Harbor Symp. Quan.

Bioi., 47 (1983) 85. 35. Lerman, L. S., J. Mol. Bioi., 3 (1961) 18. 36. Wang, J. C., J. Mol. Bioi., 89 (1974)783. 37. Wang, J. c., In: Procedures in Nucleic Acid Research, Vol. 2 (eds G. L.

Cantoni and D. R. Davies), Harper and Row, New York, 1971, p.407. 38. Keller, W., Proc. Natl. Acad. Sci. USA, 72 (1975) 4876. 39. Malcolm, A. D. B. and Snounou, G., Cold Spring Harbor Symp. Quan. Bioi.,

47 (1983) 323. 40. Wang, J. c., J. Mol. Bioi., 43 (1969) 25. 41. Depew, R. E. and Wang, J. c., Proc. Natl. Acad. Sci. USA, 72 (1975) 4275. 42. Anderson, P. and Bauer, W., Biochemistry, 17 (1978) 594. 43. Cozzarelli, N. R., Science, 207 (1980) 953. 44. Gellert, M., Ann. Rev. Biochem., 50 (1981) 879. 45. Wang, J. c., Ann. Rev. Biochem., 54 (1985) 665. 46. Vinograd, J., Lebowitz, J., Radloff, R., Watson, R. and Laipis, P., Proc. Natl.

Acad. Sci. USA, 53 (1965) 1104. 47. White, J. H., Amer. J. Math., 41 (1969) 693. 48. Fuller, F. B., Proc. Natl. Acad. Sci. USA, 68 (1971) 814. 49. Crick, F. H. c., Proc. Natl. A cad. Sci. USA, 73 (1976) 2639. 50. Bauer, W. R., Crick, F. H. C. and White, J. H., Sci. Amer., 243 (1980) 118. 51. Braun, W., J. Mol. Bioi., 163 (1983) 613. 52. Bauer, W. R., Ann. Rev. Biophys. Bioeng., 7 (1978) 287. 53. Crick, F. H. c., Wang, J. C. and Bauer, W. R., J. Mol. Bioi., 129 (1979) 449. 54. Seidl, A. and Hinz, H.-J., Proc. Natl. Acad. Sci. USA, 81 (1984) 1312. 55. Bauer, W. R. and Vinograd, J., J. Mol. Bioi., 47 (1970) 419. 56. Davidson, N., J. Mol. Bioi., 66 (1972) 307. 57. Hsieh, T.-S. and Wang, J. c., Biochemistry, 14 (1975) 527. 58. Anshelevich, V., Vologodskii, A., Lukashin, A. and Frank-Kamenetskii,

M. D., Biopolymers 18 (1979) 2733. 59. Benham, C. J., Cold Spring Harbor Symp. Quan. Bioi., 47 (1983) 219. 60. Frank-Kamenetskii, M. D. and Vologodskii, A~ V., Nature, 307 (1983) 481. 61. Singleton, C. K., Klysik, J. and Wells, R. D., Proc. Natl. Acad. Sci. USA, 80

(1983) 2447. 62. Peck, L. J. and Wang, J. c., Proc. Natl. Acad. Sci. USA, 80 (1983) 6206. 63. Courey, A. J. and Wang, J. c., Cell, 33 (1983) 817. 64. Rich, A., Nordheim, A. and Wang, A. H.-J., Ann. Rev. Biochem., 53 (1984)

791. 65. Lee, C.-H., Mizusawa, H. and Kakefuda, T., Proc. Natl. Acad. Sci. USA, 78

(1981) 2838. 66. Vologodskii, A. V., Anshelevich, V. V., Lukashin, A. V. and Frank-

Kamenetskii, M. D., Nature, 280 (1979) 294. 67. LeBret, M., Biopolymers, 19 (1984) 619. 68. LeBret, M., Biopolymers, 23 (1984) 1853. 69. Olson, W. K., Marky, N. L., Srinivasan, A. R., Do, K. D. and Cicariello, J ..

In: Molecular Basis of Cancer, Part A: Macromolecular Structure,

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Carcinogens, and Oncogenes (ed. R. Rein), Alan R. Liss, Inc., New York, 1985, p.109.

70. Pulleyblank, D. E. and Morgan, A. R., J. Mol. Bioi., 91 (1975) 1. 71. Gray, H. B., Biopolymers, 5 (1967) 1009. 72. Campbell, A. M. and Jolly, O. U., Biochem. J., 133 (1973) 209. 73. Brady, G. W., Fein, D. B., Lambertson, H., Grassian, V., Foos, D. and

Benham, C. J., Proc. Natl. Acad. Sci. USA, 80 (1983) 741. 74. Upholt, W. B., Gray, H. B. and Vinograd, J., J. Mol. Bioi., 61 (1971) 21. 75. Frisch, H. L. and Klempner, D., In: Advances in Macromolecular Chemistry,

Vol. 2 (ed. W. M. Pasika), Academic Press, New York 1970, p. 149. 76. Wang, J. C. and Schwartz, H., Biopolymers, 5 (1967)'953. 77. Hudson, B. and Vinograd, J., Nature, 216 (1967) 647. 78. Wang, J. c., Biopolymers, 9 (1970) 489. 79. Sundin, O. and Varshavsky, A., Cell, 21 (1980) 103. 80. Sundin, O. and Varshavsky, A., Cell, 25 (1981) 659. 81. Frank-Kamenetskii, M. D., Lukashin, A. V. and Vologodskii, A. V., Nature,

258 (1975) 398. 82. Liu, L. F., Depew, R. E. and Wang, J. C., J. Mol. Bioi., 106 (1976) 439. 83. Fuller, F. B., Proc. Natl. Acad. Sci. USA, 75 (1978) 3557. 84. Brown, P. O. and Cozzarelli, N. R., Science, 206 (1979) 1081. 85. Liu, L. F., Liu, c.-C. and Alberts, B. M., Cell, 19 (1980) 697. 86. Stasiak, A., DiCapua, E. and Koller, Th., Cold Spring Harbor Symp. Quant.

BioI., 47 (1983) 811. 87. Krasnow, M. A., Stasiak, A., Spengler, S. J., Dean, F., Koller, T. and

Cozzarelli, N. R., Nature, 304 (1983) 559. 88. Griffith, J. and Nash, H., Proc. Natl. Acad. Sci. USA, 82 (1985) 3124. 89. White, J. H. and Cozzarelli, N. R., Proc. Natl. A cad. Sci. USA, 81 (1984)

3322. 90. Iwamoto, S. and Hsu, M.-T., Nature, 305 (1983) 70. 91. Shimada, J. and Yamakawa, H., Biopolymers, 23 (1984) 853. 92. Yamakawa, H. and Stockmayer, W. H., J. Chem. Phys., 57 (1972) 2843. 93. Wang, J. c., Sci. Amer., 247 (1982) 94. 94. Wang, J. c., In: Nucleases (eds R. Roberts and S. Linn), Cold Spring Harbor

Laboratory, Cold Spring Harbor, New York, 1982, p.41. 95. Wang, J. c., In: Genetic Engineering Techniques (eds P. C. Huang, T. T. Kuo

and R. Wu), Academic Press, New York, 1982, p.41. 96. Kikuchi, A. and Asai, K., Nature, 309 (1984) 677. 97. Pulleyblank, D. E., Shure, M., Tang, D., Vinograd, J. and Vosberg, H. P.,

Proc. Natl. A cad. Sci. USA, 72 (1975) 4280. 98. Lilley, D. M. J., Proc. Natl. Acad. Sci. USA, 77 (1980) 6468. 99. Singleton, C. K. and Wells, R. D., J. Bioi. Chem., 257 (1982) 6292.

100. Lyamichev, V. I., Panyutin, I. G. and Frank-Kamenetskii, M. D., FEBS Letters, 153 (1983) 298.

101. Shimada, J. and Yamakawa, H., J. Mol. Bioi., 184 (1985) 319.

CHAPTER 8

Cyclic Pep tides

ALAN E. TONELLI

AT & T Bell Laboratories, Murray Hill, New Jersey, USA

INTRODUCTION

Cyclic peptides are an interesting class of biological molecules found to occur widely in nature both in animals and plants. They perform a variety of biological roles functioning as hormones, toxins, antibiotics, complex­ing agents or ionophores, and as elements of proteins where they are incorporated through cysteine~ysteine S -S bridges. Interest in cyclic peptide research was born nearly 40 years ago, when the residue sequence was determined 1 for the cyclic decapeptide antibiotic gramicidin-So Since that time, and especially over the last 20 years, research into the structures and functions of cyclic peptides has flourished.

In addition to the many naturally occurring examples discovered to date, a large number of cyclic peptides have been synthesized either to mimic or modify the behaviours exhibited by cyclic peptides found in nature or as models of protein structures and/or functions. Much of the work dealing with cyclic peptides has focused on establishing their conformations both in solution and in the solid state, with the hope of learning the source of their biological activities, establishing the connection between primary (residue sequence) and secondary (conformation) structures, and determin­ing the conformational preferences of peptide residues and sequences so crucial to the problem of protein folding.

Though in this brief review the syntheses and biological functions of several cyclic peptides will be touched upon, emphasis will be placed on establishing the conformations of cyclic peptides. The methods used to study the conformational properties of cyclic peptides will be described,

261

262 ALAN E. TONELLI

and the results of their application to several specific examples will be presented. When possible, it will be attempted to draw connections between their structures, conformations and biological functions.

SYNTHESIS

Polypeptide synthesis involves the coupling of amino or imino acids through formation of peptide bonds. To produce a cyclic peptide

R 0 R' 0 I II I II

H2N-CH-C-OH + H2N-CH-C-OH --

R 0 R' 0 I II I II

H N-CH-C-N-CH-C-OH + H 0 2 I 2

H

the final peptide bond

formed must be intramolecular, connecting the N(-NH2) and

cCLOH) termini of the polypeptide chain. To avoid polycondensation to longer, linear polypeptide chains, syntheses of cyclic peptides are performed under conditions of high dilution. Te:n- to hundred-fold dilution is commonly employed, which most often leads to cyclic peptide yields of much less than 50 %. Cyclic dipeptides (diketopiperazines) are exceptions and are formed in virtually quantitative yield.

Cyclization oflinear peptides may be accomplished either by leaving the terminal amino group unprotected and activating the terminal carboxyl group or, more commonly, by n~moving the amino protecting group while the carboxyl group is activated. A variety of protecting and activating groups have been employed2 - 10 in the solution synthesis of cyclic peptides.

CYCLIC PEPTIDES 263

During the cyclization of certain peptides it is observed 3,4,8,11-14 that a 'doubling reaction' sometimes occurs. For example, cyclization of a linear tripeptide leads to a cyclic hexapeptide rather than to a cyclic tripeptide. Linear, tri- andpentapeptides were found to double upon cyclization to the corresponding cyclic hexa- and decapeptides by Schwyzer and co­workers, 12,14 while linear di-, tetra-, and hexapeptides do not. They suggest association phenomena between the linear peptides and the average distance between the N-terminal and C-terminal groups as important factors in the doubling of linear to cyclic peptides.

Mutter1S has discussed the influence of peptide residue sequence on the ability of a polypeptide to undergo cyclization. Glycine (Gly) residues and regularly alternating D, L sequences favour cyclization due to inherent residue flexibility in the former case and the small persistence length of racemic D, L peptides. Hexapeptides are expected to cyclize more readily than polypeptides of similar residue sequence, but with different numbers of residues, because of the favourable angular correlation * between the N and C terminal residues in hexapeptides. Mutter's expectation is in agreement with the experimental fact that hexapeptides give higher yields in cyclization reactions compared with other chain lengths.

Cyclic peptides have also been synthesized in solution while attached to insoluble high molecular iweight' polymer carriers. This method 16 is schematically illustrated in Fig. I. By spacing the linear peptides attached to the polymer carrier at reasonably 10!lg distances, intramolecular cyclization rather than intermolecular condensation occurs on removal of the amino blocking groups. A several-fold improvement in cyclic peptide yield over that produced by traditional dilute solution cyclization can be achieved by this method.

Peptides, both linear and cyclic, can also be synthesized on solid-phase polymeric supports, which are usually cross-linked polymer gels, by the Merrifield method. 1 7 Again, the growing peptide is attached by its carboxyl end to the polymer, while its amino end remains protected. N-protected amino acids are added stepwise to the polymer-attached peptide chain, whose own amino protecting groups are removed just before the addition of the next amino acid. Finally, the peptide is cleaved from the supporting polymer gel and the amino protecting groups removed. In the synthesis of cyclic peptides, removal of the amino protecting groups after cleavage from the polymer support is performed under high dilution to ensure cyclization.

Many microbes produce a wide variety of cyclic peptides. Bacteria, fungi

* See Chapters I, 2 and 9 of this book.

264 ALAN E. TONELLI

0 I 0 OH C=O I I C=O +

PEPTIDE I C=O I PEPTIDE E~TIDE NH I I NH2 NH

Z

FIG. 1. Schematic illustration of peptide synthesis in solution while attached to insoluble high moll~cular weight polymer carrier.

and streptomycetes synthesize antibiotics, hormones, toxins and iono­phores, which complex with metal ions and transport them across membranes. Many of these compounds are cyclic peptides. It appears 18.19 that the biosynthetic pathway followed by the microbes in the synthesis of these cyclic peptides is different from the ribosome-RNA system of protein synthesis.

Soluble enzyme systems are apparently used to synthesize microbially produced cyclic pep tides by a route termed the protein template mechanism. 20 - 25 This mechanism may be illustrated by describing the biosynthesis of gramicidin-S,18 a cyclic decapeptide antibiotic with the following primary structure: cyclo(-o-Phe-Pro-Val-Om-Leu-)2.

Two protein factors with molecular weights of 280 000 and 100000 are involved in the synthesis of gramicidin-S by Bacillus brevis. The lighter factor racemizes L- Phe to 0-Phe and activates the latter. The remaining four amino acids are activated by the heavier protein. The light factor with o-Phe bound to it initiates the sequential addition of Pro, Val, Om and Leu on the heavier protein. Production of the cyclic decapeptide results from the condensation of two bound linear pentapeptides o-Phe-Pro-Val-Om­Leu, although it is not yet clear 19 whether both pentapeptides are bound to the same or separate heavy protein factors.

The biosynthesis of cyclic peptides in animals differs from the route described above for microorganisms. 19 They are produced 26 from larger proteins by the action of specific proteinases. As an example, oxytocin H­Cys-Tyr-Ile-Gln-Asn-Cys-Pro-L1eu-Gly-NH 2 , a nonapeptide whose first and sixth residues are linked by a disulphide bridge forming a six residue cycle, is a hormone present in mammals, producing uterine contractions

CYCLIC PEPTIDES 265

and milk ejection. Oxytocin is produced,27. 2 8 as are many other peptide hormones 26 in the body, by enzymatic cleavage of a portion of a higher molecular weight protein, resulting in two fragments. The smaller fragment is oxytocin, which manifests its biological properties only after being cleaved from its parent protein. The larger fraction is neurophysin, the protein which binds and carries oxytocin into the blood stream. 28

CONFORMATIONS OF CYCLIC PEPTIDES

Aside from the obvious constraint imposed by cyclization, specification of the conformation of a cyclic, or linear, peptide requires knowledge of the backbone (¢, 1/1, w) and side chain (Xl' X2) rotation angles * in each residue (see Fig. 2). Of the three backbone bonds, w rotation about the peptide bond is usually restricted 30 to the nearly planar trans and cis conformations. This is a result of the partial conjugation of electrons between the C=O and

o II C~N

I H

bonds. Amino acid residues nearly always have trans peptide bonds,31 while the peptide bonds in Pro, Sar and other imino acid residues can adopt either the cis or trans conformations.

The remaining two backbone rotation angles ¢ and 1/1 in each residue are able to adopt a large variety of values depending on the chemical structure of the residue. Allowable conformations are determined by non-bonded steric and electrostatic interactions occurring in a given peptide residue. When these interactions are estimated via semiempirical potential energy functions, a measure of the inherent conformational flexibility of a given peptide residue can be obtained.30.32.33

In Fig. 3 the conformational energy maps for three typical peptide residues are presented:

Gly(R == H), Ala(R == CH 3), and pro(~~-?-l) c",-c/C

* Throughout this discussion the author adopts the 1966 peptide rotation angle convention 29 which assigns ¢ = 1/1 = OJ = 0 0 to the planar zig-zag conformation shown in Fig. 2.

266 ALAN E. TONELLI

(0)

H R H 0 \ I I 4> 1/1 II \ N '-J '"\. C' CQ

~ C'/ /,CCI/ t ~N / ~ II /\ I o / H

o II

R H

(b)

/C'~

(c)

H 0 I 4> 1/1 II

CQ

~C,/ N ~>C\ ~ c' '" II o

I~ H

N/ I H

CQ

FIG. 2. Portions of an all L polypt:ptide chain illustrating (a) the three backbone rotation angles 29 cp, t/I. w, (b) side chain rotations 29 Xl' X2' etc. and (c) the peptide

fragment employed 30.32.33 in conformational energy calculations.

Energies were calculated 30.32 for the peptide fragment illustrated in Fig. 2(c), where all peptide bonds were assumed to be trans. It is apparent that Gly residues are more flexible than Ala or Pro residues and may adopt more conformations. The pyrrolidone ring in Pro fixes ¢ at c. 120 0 and only ljJ = 90-150 0 and 270-10 0 conformations are energetically permitted. Thus Pro is the least flexible of the three residues discussed. When the peptide bond in Pro or other imino acid residues is cis, then the energy maps are modified 34.35 resulting in a significant reduction in the numbers of energetically allowed conformations. These conformational energy

(b)

CYCLIC PEPTIDES

(0)

(551-E "­o u 31-"""'.

(c)

Vp~U I I I I I I

60 120 180240300 0 Ij! (<I> = 122°)

267

FIG. 3. Conformational energy maps30.32 for (a) Gly, (b) L-Ala and (c) L-Pro peptide residues calculated for the peptide fragment shown in Fig. 2(c). For the Gly and L-Ala residues only the 5 kcal mol- 1 energy contours are shown, and x

indicates the lowest energy conformation.

maps are valuable aids in the search for the solution and even the solid-state conformations of cyclic peptides.

CYCLIC PEPTIDE CONFORMATION IN SOLUTION

Of all the spectroscopic techniques applied to the elucidation of cyclic peptide conformation in solution,36 NMR spectroscopy 37 is clearly the method of choice. 1 H NMR resonances may be most commonly assigned to particular amino acid residues in a given cyclic peptide by homonuclear

268 ALAN E. TONELLI

spin-decoupling (double resonance)38 of peptide HN protons from the a­CH protons. Next, the side-chain protons at the p-carbon atoms are spin­decoupled from the a-CH protons, and this procedure is repeated for side­chain protons further removed from the peptide backbone. Heteronuclear spin-decoupling 39.4o of 1 Hand 13C can also be used to assign the proton resonances in the 1 H NMR spectra of cyclic polypeptides.

Recently the advent of two-dimensional (2-D) NMR techniques 4l .42 has begun to revolutionize the assignment of polypeptide NMR spectra and provide conformationally sensitive information. Without going into detail it can be stated that 2D_l H NMR can be utilized to assign the proton resonances of a cyclic peptide to its individual residues. In addition, one variation of this technique also provides a measure of the distances between protons, information which is related directly to the conformation of the cyclic peptide.

The most useful information about polypeptide conformation obtained from 1 H NMR is the vicinal coupling constants (J) for the H-N-C"-H and H-C"-CPH protons. A Karplus43 relation

J = A cos 2 () + B cos () + c (1)

where () is the dihedral angle between the vicinal protons which are spin-spin coupled, seems to account for the coupling constants observed in polypeptides. Consequently, measurement of JNC' and JoP provides a potential means to determine the backbone rotation angle ¢ and the side­chain rotation angle Xl (see Fig. 2).

In Fig. 4 we have plotted a typical Karplus relation 44 for the spin-spin coupling and dihedral angle () between NH and C"H protons in polypeptides:

J NC' = 8·9 cos2 () - 0·9 cos () + 0·9 (2)

Large coupling constants are expected for () ~ 0 and 180 0. Since () = I¢ - 240 °1, it is possible to deduce the conformational state of the N-C" backbone bond (¢) in a peptide residue from the ,J NC' observed for that residue. For an L-amino acid residue, large JNC' implies ¢ = 20-100° or 210-270°, while small JNC' are expected for the remaining conformations about the N-C" backbone bond. This information is valuable to the search for solution conformations of cyclic peptides.

Another means for narrowing the range of conformations considered for cyclic peptides is provided by their conformational energy maps. Each residue in the cyclic peptide is required to adopt a low energy conformation, such as those within the 5 kcal mol- l contours of the energy

N I

tI o z ...,

12

o

-2

o 45

CYCLIC PEPTIDES

90

e

269

135 180

FIG. 4. Vicinal N-H to C'-H spin-spin coupling iNC' calculated according to the Karplus 43 relation (eqn (2))44 and presented as a function of the dihedral angle

e between N-H and C'-H.

maps 30 displayed in Fig. 3. The rationale for this approach is the empirical fact that even in proteins,45 where long-range interactions abound, peptide residues nearly always adopt conformations' (¢, 1/1) within or very close to the low energy contours of their conformational energy maps.

Finally, in the search for solution conformations of cyclic peptides each potential conformation must be tested for ring closure. Several methods 15 ,46.47 have been developed for this purpose, but each requires that the first and last residues meet the spatial restrictions imposed by the formation of a cyclic peptide structure.

Having derived 48,49,87 several cyclic conformations for a given cyclic peptide, each having all residues in low energy conformations which are consistent with the observed vicinal couplings INC" how do we determine which of these structures represents most closely the experimentally observed sol!ltion conformation? Intramolecular hydrogen-bonding of the amide NH protons often stabilizes cyclic peptide conformations in

270 ALAN E. TONELLI

solution. Observation of the temperature dependence of the amide proton chemical shifts can identify which, if any, of the amide protons in a cyclic peptide are involved in intramolecular hydrogen bonds. 48 - 50

In hydrogen bond accepting solvents such as water, ethanol, dimethyl­sulphoxide, etc., those amide protons exposed to these solvents will be bound to them, while those forming intramolecular hydrogen bonds will not. As the temperature is increased an increasing fraction of the solvent to amide proton hydrogen bonds will be broken resulting in an upfield chemical shift for these amide protons. Intramolecularily! hydrogen­bonded NHs should not be sensitive to temperature and their chemical shifts are expected to have a m:gligible or small temperature dependence compared to the amide protons exposed to hydrogen-bonding solvents.

When this method detects intramolecular hydrogen-bonded amide protons their observation can be used to select from the derived structures the ones most appropriate for the solution conformation of the cyclic peptide. Only those derived structures possessing the appropriate intramolecularily hydrogen-bonded NHs need be considered as candidates for the solution structure.

13C NMR has not been used to study polypeptide conformation. Aside from determination of the cis or trans character 51 of the X-Pro peptide bond, 13C NMR has not been used in the conformational analysis of cyclic peptides. Instead, 13C NMR has been extensively used to study the dynamics 52 of polypeptide chains. However, it was recently 53.54 pointed out that the 13C chemical shifts of backbone carbonyl and side-chain f3 carbons in polypeptides provide structural information as well.

In Fig. 5 we present Newman projections along the two flexible bonds in a peptide residue: the N---ca bond (¢-rotation) and the ca_C' bond (ljJ­rotation) (see Fig. 2). It has been demonstrated 55 that when a carbon atom is in close spatial proximity to a non-hydrogen atom three bonds removed (y-substituent) its nucleus is shielded and resonates upfield from a similar carbon atom that is further removed from its y-substituent. As an example, the f3 carbon in Fig.5(b), which is near to its y-substituent N i + l , will resonate upfield relative to the: f3 carbon in part (a), because in (a) it is distant from its y-substituent C; _ l' In other words, the relative 13C chemical shifts of q depend on the conformation (¢, ljJ)i of its constituent residue. Backbone carbonyl carbons have y-substituents whose distances of separation are dependent on backbone rotations ¢ of the residue containing that carbonyl carbon and the subsequent residue and on the side-chain rotation Xl of the constituent residue (see Figs 2 and 5).

The conformational sensitivity of the 13C NMR chemical shifts of

CYCLIC PEPTIDES 271

(a)

( b)

FIG. 5. (a) Newman projection along the N-C· backbone bond in a polypeptide chain illustrating tP-rotation; (b) Newman projection along the C·---C' backbone

bond in a polypeptide illustrating t/I-rotation.

backbone carbonyl and side-chain f3 carbons has been used 53,54 to obtain structural information about polypeptides, including cyclic peptides. This approach has permitted conclusions to be drawn about both the primary (residue sequence) and secondary (conformation) structure of polypeptides.

Instead of drawing from among the many interesting biologically active cyclic peptides to demonstrate the traditional method for establishing their solution structures, we will use the synthetic cyclic hexapeptide 15L-Ala . D-Ala I as an example,56 We select this example to illuminate two major limitations inherent in the approach. It is not always possible by traditional! H NMR experiments 37 to associate the observed amide proton resonances with specific residues in the polypeptide, and it is possible that residues with intermediate iNC- values are interconverting rapidly on the NMR time scale between conformers with large and conformers with small vicinal amide to IX-proton couplings (see eqn (2»,

! H NMR studies 56 of 15L-Ala , D-Ala lIed to the following observations: (i) iNC' = 9,8, 8,0, 7·0, 5·5, 5·0 and 4·5 Hz; (ii) five residues exhibited temperature coefficients for their amide proton chemical shifts (0'0032-0·008 j ppm/ 0C) which were comparable to that measured for the NH proton of the simple amide N-methylacetamide (0,0061 ppm/"C), while a single residue showed a small negative temperature coefficient ( -0,0014 ppm/ 0C) for its amide proton; and (iii) similar rates of deuterium exchange (amide protons rigidly hydrogen-bonded are expected 49,50 to

272 ALAN E. TONELLI

undergo a reduced rate of exchange for deuterium compared to solvent­exposed amide protons) were observed for all six amide protons. The six residues in 15L-Ala. o-Ala Iwere: categorized as follows: (1) the residue with J NC' = 9·8 Hz; (2) the two residues with JNC' = 7·0-8·0 Hz; and (3) the three residues with JNC' = 4·5-5·5 Hz. Because it was not possible to specifically assign JNC' to individual residUl~s, 60 different combinations of these three classes of residues were considered in the search for low energy cyclic conformations.

The residue with JNC' = 9·8 Hz was allowed to adopt ¢L = 60 ° and t/I L = 0, 120,240,270,300 and 330°; qlL = 290° and t/lL = 240 and 270°. The two residues with JNC' = 7·0-8·0 Hz were permitted to have the conformations ¢L = 30 and 90 0, and t/I L = 0, 120, 240, 270, 300 and 330 0; ¢L = 240 ° and t/lL = 240 and 270°. Because JNC' = 4,5-5,5 Hz may result from averaging of conformations corresponding to both low and high JNC', the three residues with these couplings were permitted to adopt all of their low energy conformations, in addition to 41L = 15 and 105 0, and t/I L = 0, 120, 240, 270, 300 and 330°, which correspond to JNC' =4·5-5,5 Hz. In each case (¢, t/I)o = ( - ¢, - t/I)L and each correspond to the lowest energy confor­mations consistent with the measured amide to IX-proton couplings (see eqn (2) and Figs 3 and 4).

In this manner over; 109 conformations were tested for ring closure by calculating the distance between the IX-carbons in L-Ala4 and L-Ala s which terminate the acyclic hexapeptide L-Ala 5-o-Ala-L-Ala j -L-Ala2-L­Ala3-6L-Ala4 . If this distance 30 was between 3· 7 and 3·9 A, then the distances between No-Ala and C~.Ala, and H(N)o.Ala and 0L.Ala, were calculated to ensure ring closure with a trans peptide: bond of the correct length. The two lowest energy cyclic conformations generated in this manner are listed in Table 1. Structure 1, which is 2 kcal mol- j lower in intramolecular conformational energy than structure 2, has no internally hydrogen-bonded amide protons, while in structure 2 (N-H)L.Ala2 is hydrogen bonded to (C=O)L.Ala, (where 1 cal = 4·1841).

TABLE 1 Lowest Energy Cyclic Conformations Generated for 15L-Ala. o-Ala I

Structure

1 2

D-Ala

330, 120 120, 120

15, 120 15, 270

L-Alaz

15, 240 15, 120

30, 120 30, 120

15, 120 105, 120

60,270 60, 0

CYCLIC PEPTJDES 273

The amide proton with the zero to slightly negative temperature coefficient belongs to the alanine residue with 'NCo = 4·5 Hz. This is consistent with structure 2, which is the lowest energy hydrogen-bonded conformation generated, where the N-H of L-Ala2 is hydrogen-bonded and <PL.Ala, = 15 0

, or e = 135 0, corresponds to 'NC';:::; 5 Hz. The deuterium

exchange study, on the other hand, indicates that all amide NHs are equally accessible to the solvent.

A rapid equilibrium between conformations such as the two generated above, one with a single intramolecular amide proton hydrogen bond (structure 2) and the other (structure 1) with all amide protons exposed to solvent, is consistent with the deuterium exchange study, which shows that the amide proton with the nearly temperature-independent chemical shift exchanges at a rate similar to the remaining amide protons. This is also in agreement with the intermediate 'NC' couplings of c. 5 Hz observed for three residues suggesting a conformational averaging of several values of the N-Ca bond rotation angle <P in these residues.

In addition to the flexibility of 15L-Ala . D-Ala I, the slightly negative sign of the temperature coefficient of the chemical shift of the highest field amide proton resonance can also be interpreted. An increase in the internal burial or hydrogen bonding of an amide proton as the temperature is increased would lead to a negative temperature coefficient. The fraction of hydrogen­bonded conformers must be increasing at the expense of the non-hydrogen­bonded conformers as the temperature increases. Hence, it is likely that the hydrogen-bonded conformers have energies slightly higher than the remaining conformers involved in the rapid equilibrium. Indeed the lowest energy hydrogen-bonded conformer generated for 15L-Ala . D-Ala 1 (struc­ture 2) does have an intramolecular energy 2 kcal mol- 1 in excess of the lowest energy, non-hydrogen-bonded conformer (structure 1) found.

Though the lack of residue symmetry and specificity and the absence of residues with bulky side chains lead to complications in determining the solution conformation of 15L-Ala . D-Ala I, even in this case 56 the coupling of 1 H NMR measurements with approximate conformational energy estimates permits several probable conclusions to be drawn regarding the conformations and flexibility of this cyclic hexapeptide. It would be interesting to study the solution conformations of 15L-Ala . D-Ala 1 by 13C NMR to see if the conclusions reached using the 1 H NMR data could be used to rationalize 53 .54 the 13C chemical shifts and their temperature dependences.

The solution conformations of many cyclic peptides both synthetic and naturally occurring have been determined 48 - 50 in the manner described

274 ALAN E. TONELLI

above. The impetus for these studies is the belief that cyclic peptides serve as useful models for aspects of protein conformation and that biologically active cyclic peptides owe their activity, at least in part, to their solution structures. With this in mind, it is interesting to learn if the solution structure of a cyclic peptide is also stable in the solid state.

SOLID STATE CONFORMATIONS OF CYCLIC PEPTIDES

The crystalline conformations of a number of cyclic peptides have been determined by X-ray diffraction analysis. 57 - 71 Both synthetic and naturally occurring cyclic peptides have been studied, and several in both the free and complexed (bound to metal ions) states.

Recently 13C NMR spectroscopic techniques 72 -74 such as cross­polarization, high power proton-decoupling and magic-angle sample spinning have produced 75. 76 high resolution spectra of cyclic peptides in the solid state. This technique circumvents the requirement of growing fairly large single crystals of a cyclic peptide in order to perform X-ray diffraction analysis of its solid··state structure. In addition, comparison of the 13C chemical shifts observed in the solid state and solution spectra of a cyclic peptide provides a means for determining their degree of similarity.

Let us use the cyclic hexapeptide cyclo-(Gly l-L-Pro-GlY2}z [C-(G 1 PG2)2] as an example for elucidation of solid-state cyclic peptide structure. Unlike 15L-Ala. D-Ala I, C-(G1 PG2)2 possesses two-fold residue symmetry and two chemically different residues. Kostansek et af.70 have used X-ray diffraction methods to determine the solid-state structure of C-( G 1 PG 2h. They find C-( G 1 PG 2}z to adopt an asymmetric conformation in the solid state with a single intramolecular hydrogen bond between the N-H of one Gly 1 residue and the c=o of the other Gly 1 residue. This structure is presented in Table 2.

Pease and co-workers 77 using 1 H NMR studies of specifically deuterated C-(G 1PG2)2 have derived a two-fold symmetric structure for the solution conformation of this cyclic hexapeptide. This rigid structure is charac­terized by two intramolecular hydrogen bonds between the N-H and C=O groups of Gly 1 residues and is described in Table 2. A comparison of the solution and solid-state structure indicates similar conformations for half of this cyclic peptide, i.e. the -Gly'l-Pro'-Gly~- portion, while the other half clearly adopts widely different conformations depending on whether or not it is in the solid or solution states.

Using the techniques recently developed to obtain high resolution 13C

CYCLIC PEPTIDES

TABLE 2 Solid-State and Solution Conformations of Cyclo(Gly I-L- Pro-Gly 2h

Residue

G1YI Pro G1Y2 Glil Pro' Gly~

Solid-state 70

<P t/J

38 7 114 144 65 173 30 358

127 306 263 177

Solution 77

<P t/J

0 0 120 300 270 180

0 0 120 300 270 180

275

NMR spectra of solid samples, Gierasch et al. 76 have recorded and compared the high resolution spectra ofC-(G I PG2)2 in both the solid and solution states. Consistent with the X-ray diffraction-derived crystalline conformation, 70 the solid-state l3C NMR spectrum ofC-(G 1 PG2)2 reflects the absence of two-fold symmetry. For example, two resonances are observed for the Pro CH~ carbons in the solid-state spectrum, with the most upfield resonance occurring at the same field strength as observed for the Pro CH~ resonance in solution.

In the crystal 1/1 Pro = 144° and 1/1 Pro' = 306°, while in solution 1/1 Pro = 1/1 Pro' = 300 0. In the crystal (CH~hro has no y-substituent in close proximity, because 1/1 Pro = 144 ° (see Fig. 5(b)), while (CH2)~ro' in the crystal and both Pro and Pro' CH~s in solution are proximal (1/1;;::: 300°) to their carbonyl oxygens. We therefore expect S3 - ss and Gierasch et al. 76 observe that (CH~hro' in the crystal and both CH~s in solution resonate at the same field strength and occur upfield from (CH~hro in the crystal.

Many other cyclic peptides have been subjected to X-ray diffraction analyses and their solid-state structures have been derived. Several of these cyclic peptides have also been analysed by solution NMR techniques leading to determination of their solution structures. We now compare the solution and solid-state structures of several cyclic polypeptides. 78

COMPARISON OF CYCLIC PEPTIDE CONFORMATION IN SOLUTION AND IN THE SOLID STATE

Antamanide is a cyclic decapeptide (see Fig. 6) isolated from the poisonous mushroom Amanita phalloides. 79 It is observed to protect against

276 ALAN E. TONELLI

ANTAMANIDE

8 9 10 1 2 PRO - PHE - PHE - VAL - PRO

I I PRO - PHE - PHE - ALA - PRO

7 6 5 4 3

a-AMANITIN OH I CH OH

2 I 'CH

'CW ... CH3 I 4 5

HN -CH-CO-NH-CH-CO-NH -- CHZ-CO I 3 II I oc H2C):Q NH CH

OH I I I / 3 V-CH 2 O~S N h OH 6 CH-CH

Hf\-~ \ H I \ CH2 CO C2H5

I 1 I 7 I OC - CH - NH - CO -- CH - NH - CO - CH2 - NH

I 8 CH2 I

CONH Z

PHALLOIDIN

OH I

5 6 7 HZ1 H3C-CH -CO -NH -CH - CO - NH - CH - CH2-C -CH3

I / I bH NH H2C CO

I ~ I CO S)lN~ NH P4 H2~; H I N - CO-CH HCI1 CH3

I 2 HN-CO-CH -NH - CO

OH I HO - CH

I CH 3

FIG. 6. Residue sequence and chemical structures of the biologically active, cyclic, mushroom peptides antamanide, IX-amanitin and phalloidin.

CYCLIC PEPTIDES 277

phalloidin, but not against amanitin, the two constituent poisons (see Fig. 6) of Amanita phalloides. The conformations of free antamanide and antamanide complexed with metal ions (Na + and Li +) have been determined both in solution 80 - 82 and in the crystalline 67.68 states.

Cis peptide bonds between the two pairs of Pro residues, i.e. Pro2-Pr03 and Pr0 7 -Pr08, are the feature common to the structures of antamanide, whether free or complexed, in solution or in the crystal. In non-aqueous media antamanide apparently adopts two different conformations depending on the hydrogen bond acceptor strength of the solvent. 82 In weak hydrogen-bond acceptor solvents antamanide assumes a structure (2,7-cis I) characterized by four strong and two weak intramolecular hydrogen-bonded amide protons, while all amide protons are exposed (2,7-cis II) to solvents that are strong hydrogen-bond acceptors (see Table 3).

In the crystal,68 free antamanide adopts a conformation (A-xtal) distinctly different from its solution structures (see Table 3). The crystalline structure is characterized by two intramolecularily hydrogen-bonded amide protons belonging to Phes and Phe lo ' Water molecules are integral to the crystal and are hydrogen-bonded to the remaining amide protons.

TABLE 3 Solution and Crystalline Conformations· of Free and Complexed Antamanide

Conformation 4>.>/1

Vall Pro2 Pro3 Ala4 Phe, Phe. Pro7 Pros Phe9 Phe lO

A_xtal 68 67. 338 116.341 100.160 77. 158 250, 210 102,341 118,340 88. 176 79. 158 236, 228 2,7-cis 180 - 82 60,300 120. 330 120,150 90,270 60,200 60.300 120, 330 120.150 90,270 60.300 2,7-cis 11 80 - 82 30, 300 120,300 120, 330 60,300 90,270 30. 300 120,300 120. 330 60,300 90,270 A_M+_xtaI 67 65.318 115,319 97, 327 113, 166 96,174 57,319 106, 324 111,324 102, 165 92. 187 2,7-c;s-M+ 8O - 82 60, 315 111,315 115,345 120, 135 90, 255 60, 330 111,315 115,345 120, 120 75, 225

.. All antamanide conformations have cis peptide bonds between Pro 2-Pro) and Pro 7-ProS '

When antamanide is bound to metal ions, such as Na + and Li +, it adopts a structure characterized by two intramolecular hydrogen bonds between the amide protons of Vall and Phe6 and the carbonyl oxygens of Pro8 and Pr03, respectively. This feature is maintained in the structure of the antamanide metal complex independent of crystallization (A-M + -xtal) or dissolution (2,7-cis-M+). Unlike the free peptide, when antamanide is bound to metal ions it adopts solution and solid-state conformations which are very similar. 8 2

The conformations of the two Amanita phalloides toxins, amanitin and phalloidin, have also been studied.71.83.84 In Table 4 backbone

278 ALAN E. TONELLI

conformations are presented for IX-amanitin and phalloidin (see Fig. 6). IX­Amanitin adopts very similar conformations in solution 84 and in the crystal,71 while only the solution structure 83 of phalloidin has been determined. Each toxin conformer is stabilized by an intramolecularly hydrogen-bonded Cys amide proton, and in addition the IX-amanitin structure is further stabilized by two internal hydrogen bonds involving the Trp4 and Gly 5 amide protons.

TABLE 4 Conformations" of the Mushroom Toxins Amanitin 84 and Phalloidin 83

ResiduebjToxin

2 3 4 5 6 7 8

lX,f:I-Amanitin 355, 5 120, 135 90, 120 90, 120 300, 60 120, 300 225, 240 60, 120 Phalloidin 100, 330 270, 90 100, 330 122, 125 100, 120 60, 330 110, 0

a All peptide bonds are trans. b See Fig. 6 for residue sequence.

As observed for IX-amanitin, 71.84 the solution and solid-state structures of phalloidin are also expected to be very similar. This expectation is based on our experience with IX-amanitin and the fact that, in addition to the common bicyclic structures of both mushroom toxins, phalloidin has one less residue than IX-amanitin which further favours the adoption of a rigid conformation.

We can summarize the comparison of solution and crystalline conformations for the three cyclic peptides obtained from the poisonous mushroom Amanita phalloides by saying that the bicyclic mushroom toxins IX-amanitin and phalloidin have similar structures in solution and in the solid state, as does the phalloidin antagonist antamanide when complexed to metal ions. Free antamanide, on the other hand, exhibits distinct structures in solution and in the crystal. A survey49.50.78 of other cyclic peptides also leads to the conclusion that generally there is agreement between the solution and solid-state structures of cyclic peptides if they possess one or more conformation stabilizing factors such as bulky side chains, proline residues, complexed metal ions, and a bicyclic structure produced by side-chain coupling.

CYCLIC PEPTIDES 279

BIOLOGICAL FUNCTIONS OF CYCLIC PEPTIDES

The mushroom toxins a-am ani tin and phalloidin function via distinct molecular mechanisms. 79 a-Amanitin inhibits the synthesis of ribonucleic acid (RNA) by binding to and blocking a protein, RNA-polymerase, which catalyses the synthesis of RNA. Phalloidin binds selectively to the actin protein of liver cells membranes, thereby disturbing their integrity and permitting a five-fold excess of blood above the normal level to collect in the liver. Antamanide which functions as an antitoxin against phalloidin is known not to compete with phalloidin for the same actin protein binding site, but instead must strengthen some other liver cell membrane structure against the attack by phalloidin.

Chemical substitution and modification reactions have been used 79 to map out the residues which are essential for the biological activity of each of these mushroom cyclic peptides. However, the molecular details of how each of these cyclic peptides interacts with its receptor site, which for antamanide is not known, and how it modifies the normal behaviour of its target, are not yet available. As we have seen, the toxins a-amanitin and phalloidin are constrained to adopt rigid conformations, while free antamanide is more flexible and becomes rigid only after complexation with metal ions. It would appear that until these cyclic peptides from mushrooms are studied when bound to their target molecules, unravelling the details of their biological functions will be difficult.

Precisely these types of experiments have been performed on the neurohypophyseal hormone oxytocin (see Fig. 7) when bound to its target protein neurophysin (NP). Because oxytocin can be readily synthesized and NP isolated in gram quantities, the peptide hormone-target protein interactions are amenable to study. Blumenstein, Hruby and co­workers 85.86 have specifically enriched oxytocin with 13C at various positions in the hormone. Then they measured the 13C NMR spectrum of the enriched oxytocin when bound to NP and compared it to the spectrum of free oxytocin in solution.

Residues I and 2 were found to be crucial for binding of oxytocin to NP, residue 3 less important, and residues 4-9 only marginally important. They found the backbone conformations of free and bound oxytocin to be similar, while the side-chain conformations of several residues were altered on binding to NP. The side chains of CYS1' Tyr 2 and Leu3, which interact directly with NP, and of Gln4 , which does not interact with NP, change their conformations upon binding to NP. The 13C chemical shifts of the tripeptide tail of oxytocin (-Pro7-Leu8-GlY9-NH2) are identical in the free

280 ALAN E. TONELLI

1 2 3 H-CYS-TYf;: - ILE

1 1 6 5 4

CYS - ASN - GLN 1

7 8 9 PRO - LEU - GLY-NH2

CH3 1

Cs H4 OH CH3 CH 2 1 '\ /

NH2 0 CH CH 1 II 1 2 //0 1

CH 2-CH - C-NH-CH-C-NH-CH 1 1 23 1 5 C =0 1 1 5 0 0 NH 1 6 II 5 ,,41 CH -CH -NH -C -CH-NH-C-CH-(CH 2)2- CONH 2

2 1 1 C = 0 CH2 1 1

CH 2-N 0 CONH2 0

I \ 7 II 8 // 9

/CH - C-NH-?H-C-NH- CH 2-CONH 2

CHz-CHz CH z 1

/C-tt CH3 CH3

FIG. 7. Residue sequence and chemical structure of the neurohypophyseal hormone oxytocin.

and bound states, and reflect an absence of interaction between this portion of the hormone and its receptor protein.

Though a step in the right direction, without similar detailed structural knowledge of the free and bound forms of the oxytocin target protein N P, it remains to establish the molecular details of the hormone-protein interaction which is apparently at the heart of the biological activity of oxytocin.

CONCLUDING REMARKS

This brief review of cyclic peptides has attempted to describe several properties of this interesting class of biologically significant molecules. The syntheses of cyclic peptides both in nature and in the laboratory were

CYCLIC PEPTIDES 281

touched upon. Mention was made of several of the interesting biological functions performed by many cyclic peptides. Emphasis, however, was placed on establishing the conformations of cyclic peptides and upon the methods used to determine their structures in solution and in the solid state.

With the advent of 2D-NMR techniques,41.42 future conformational studies of cyclic peptides will no longer find it difficult to assign NMR resonances to specific residues and even to distinguish between two or more residues of the same amino or imino acid. Also the ability 72 -74 to record high resolution 13C NMR spectra on solid cyclic peptide samples 75.76 means we can determine whether or not the crystalline and solution conformations of a cyclic peptide are similar without investing the effort required to obtain an X-ray diffraction-derived structure on the crystalline sample. In addition, recent awareness 55 - 57 of the sensitivity of the 13C NMR chemical shifts observed in polypeptides to their conformations, both in solution and in the solid state, will hopefully rekindle interest in this powerful spectroscopic technique, which has until recently been relegated 51.52 to studies of the cis or trans character of the imino peptide bonds and the dynamics of polypeptide chains.

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2. Kenner, G. W. and Turner, J. M., Chem. Ind. (Lond.), (1955) 620. 3. Schwyzer, R. and Sieber, P., Chimia (Aaran), 10 (1956) 265; Helv. Chim. Acta,

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Otd. Khim Nauk, (1959) 690. 10. Brockmann, H. and Graef, Y., Naturwissenschajten, 49 (1962) 540. 11. Schwyzer, R. and Aung, T. K., Helv. Chim. Acta, 45 (1962) 859. 12. Schwyzer, R., Carrion, J. D., Gorup, B., Notting, H. and Tun-Kyi, A., Helv.

Chim. Acta, 47 (1964) 441. 13. Schwyzer, R., Kappeler, H., Iselin, B., Rittel, W. and Zuber, H., Helv. Chim.

Acta, 42 (1959) 1702. 14. Schwyzer, R., Sieber, P. and Gorup, 8., Chimia (Aaran), 12 (1958) 90. 15. Mutter, M., J. Amer. Chem. Soc., 99 (1977) 8307.

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16. Fridkin, M., Patchornik,A. and Katcha1ski, E., J. Amer. Chem. Soc., 87(1965) 4645.

17. Merrifield, R. B., J. Amer. Chem. Soc., 85 (1963) 2149. 18. Perlman, D. and Bodanszky, M., Ann. Rev. Biochem., 40 (1971) 449. 19. Kurahashi, K., Ann. Rev. Biochem., 43 (1974) 445. 20. Kurahashi, K. et al., Cold Spring Harbor Symp. Quant. Bioi., 34 (1969) 815. 21. Saito, Y., Otani, S. and Otani, S., Advan. Enzymol., 33 (1970) 337. 22. Lipmann, F., Gevers, W., Kleinkauf, H. and Roskoski, R., Jr., Advan.

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(1969) 805. 25. Laland, S. G. and Zimmer, T.-L., Essays Biochem., 9 (1973) 37. 26. Krieger, D. T., Science, 222 (1983) 975. 27. Brownstein, M. J., Russell, J. T. and Gainer, H., Science, 207 (1980) 373. 28. Land, H., Schutz, G., Schmale, H. and Richter, D., Nature, 295 (1982) 299. 29. Edsall, J. T., Flory, P. J., Kendrew, J. c., Liquori, A. M., Nemethy, G.,

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50. Sarkar, B., In: Progress in Macrocyclic Chemistry, Vol. 2 (eds R. M. Izatt and J. J. Christensen), Wiley-Interscience, New York, 1980, Chapter 5.

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CHAPTER 9

Spectroscopic Studies of Cyclization Dynamics and Equilibria

MITCHELL A. WINNIK

Department of Chemistry, University of Toronto, Ontario, Canada

INTRODUCTION

This chapter reviews cyclization of polymers and oligomers measured in real time by spectroscopic techniques. The requirement for such an experiment is that the macromolecules must have groups attached to the chain ends (e.g. D - - - - A) such that proximity of A and D, or reaction between A and D, gives rise to an observable event.

D~A - CDi» - ob,,,,,bl, mnt (1)

There are clearly many possibilities for such experiments, only a fraction of which have already been exploited. Many examples are described in a recent review of hydrocarbon chain cyclization. 1

The single most important advantage of using spectroscopic methods to study the cyclization of labelled chains is that of sensitivity. Linear polymers have only a small fraction of conformations with their chain ends in proximity. Traditional methods for studying ring closure reactions have relied upon product isolation and characterization. These methods become increasingly more difficult at very low concentrations where one uses dilution to suppress competing bimolecular processes. Spectroscopic methods also suffer this limitation. The range of accessible experimental

285

286 MITCHELL A. WINNIK

sensitivities, however, often exceeds that of product isolation by several orders of magnitude.

Spectroscopic methods also introduce new dimensions into the study of cyclization. Not only can one carry out experiments at extreme dilution in simple solutions, but one can, in addition, study very concentrated polymer solutions, or even polymer melts, containing only traces of the labelled chains. The experiment, by its very nature, is blind to the presence of even a vast excess of unlabelled material. One can study the entire range of dilute to concentrated polymer solutions, examining the behaviour of individual chains as they are affected by their environment.

Spectroscopic methods permit one to study systems at equilibrium or to study reaction kinetics. One approach to cyclization equilibria involves measuring donor -acceptor complex formation between a donor group 0 and an acceptor group A attached to the ends of a polymer chain. Such complexes are frequently characterized by a new absorption band to the red of that of 0 or A, and the fraction of cyclized chains is enhanced by the free energy of DA complex formation. 2 If the molar extinction coefficient B, of DA is large, even very low concentrations of cyclized chains can be measured. Since the system is studied at equilibrium, one learns about cyclization probability.

Ring closure reactions can be studied to determine the rate of product formation. Alternatively one can study a catalytic process, involving cyclized intermediates, where a group at one chain end catalyses a chemical transformation of a substituent at the other end of the chain. Kinetic studies on such systems give information about the ease or rate of cyclization. Particularly attractive choices of end groups are those that give a product or by-product that can be detected with great sensitivity. 4-Nitrophenylesters in aqueous base, for example, give the 4-nitrophenolate anion as a hydrolysis by-product. 3 Because of its large B, value, this species is easily detected by its visible absorption. Fluorescence is an even more sensitive technique. 4 With a non-emissive group on one end of a polymer chain that could react to form a highly fluorescent product, one could study its cyclization by monitoring the total fluorescence intensity as the reaction proceeds. 5a•h

Q F~Q ~ C.(FQ)0 ~ new emission 0' quenching

A wide variety of spectroscopic processes, referred to collectively as

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 287

luminescence quenching reactions, are very useful for studying the probability and dynamics of polymer cyclization. Luminescence que.nch­ing 6 is defined as a process arising from the interaction of two groups (here F* and Q) which decreases the intensity of emission from F* and shortens its excited state decay time. Fluorescence quenching refers to deactivation of the excited singlet state of F (F* 1), whereas phosphorescence quenching refers to triplet state (F* 3) deactivation. Normally one studies groups whose interaction in the ground state is sufficiently weak that it can be ignored.

Quenching is defined phenomenologically. Many quenching mech­anisms exist. 6 Some involve formation of excited state complexes (excimers and exciplexes) which emit light. Others involve energy transfer from F* to Q, electron transfer or chemical product formation. In these experiments one examines the kinetics of interaction between F* and Q. The photochemist studies these reactions in order to learn about the detailed mechanism of interaction between F* and Q.7 The polymer chemist studies polymers containing an F* and a Q group on the chain ends in order to obtain the rate constant key for end-to-end cyclization of the chain. 1

The interpretation of the experiment depends sensitively on the efficiency of the interaction between the end groups. If these groups interact on every encounter, the kinetics experiment measures cyclization dynamics. If, on the other hand, the reaction is inefficient, so that fewer than I % of the encounters lead to quenching, a pre-equilibrium precedes the rate-limiting step. These values of key provide a measure of cyclization equilibria.

In a luminescence quenching experiment, one of the important features is that the excited state lifetimes limit the time scale of the measurement. One can think of F* as an internal stopwatch in the molecule. In one lifetime r, lje of the excited state molecules decay. After several lifetimes, very few excited molecules remain. If cyclization is slow compan;d to r, the experiment will be unable to detect it. By fluorescence quenching, cyclization rates of 104 s -1_109 S - 1 have been measured. 8 Triplet state processes have permitted key values as low as 103 s - 1 to be determined. 9

Thus these techniques have a dynamic range of 106 .

There is in principle no lower limit to cyclization rates that can be studied spectroscopically. Examples can be imagined involving the chemical reactions of photogenerated transients with lifetimes on the order of seconds or longer.

On the other hand there are two serious problems with these methods. First, one has to synthesize the labelled polymers. Secondly, one has to

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292 MITCHELL A. WINNIK

establish that the label groups themselves do not perturb the process one wishes to investigate. These problems are common to all labelling experiments, in membrane biochemistry,l°a.b.c in immunology, 10b,c and in other areas of polymer science. 11

Examples of reactions and processes that lend themselves to the study of cydization phenomena are indicated in Table 1. Those cited are meant to be suggestive rather than exhaustive. Most of the examples involve determination of reaction rates. One measures the rate of product formation or excited state quenching spectroscopically. In example 2, product formation is followed by means of a bromide-sensitive electrode. While not a 'spectroscopic' technique per se, this electrochemical method has provided very useful measures of cyclization rates with results pertinent to other examples presented here.

Some of the examples in Table 1 represent true equilibrium processes. These include donor-acceptor complex formation (example 10) and acid-base equilibria (example 11). Kinetic studies are more difficult to interpret. Obtaining rate constants from data requires recourse to a model. One has to establish whether the data fit the rate laws predicted by the assumed kinetic scheme. In addition, one has to determine whether the reaction is diffusion controlled, in which case information on cyclization dynamics can be obtained, or whether the rate-limiting step is preceded by a pre-equilibrium. Under these conditions inferences about cyclization equilibria can be drawn.

Examples 6 and 11 have been applied only to the study of bimolecular processes, with the 0 and A groups attached to the ends of different chains. Studies of cyclization requiring polymers A - - - - 0 with different end groups place stringent demands on the synthetic chemist. Where these demands can be met, rather elegant experiments are possible. One of the advantages of examples 4, 8 and 12 is that cyclization processes can be studied in polymer systems containing the same group on both ends of the polymer.

Example 13 is purely hypothetical. It was chosen to indicate that release of an end substituent allows it to rotate faster. ESR line widths and steady­state fluorescence depolarization experiments are sensitive to this motion, in principle allowing ring closure kinetics to be studied.

The major motivation behind all spectroscopic studies of cyclization is the recognition that one can in principle examine molecules of unique chain length. One avoids many of the problems inherent in competitive cyclization-polymerization experiments, where polydispersity effects increase the separation between the data and their interpretation. One

SPECTROSCOPIC STUDIES OF CYCLIZA TION DYNAMICS AND EQUILIBRIA 293

normally tries to examine polymer systems of very narrow polydispersities (i.e. M wi M n < 1·1). Hence synthetic strategies frequently use anionic polymerization or other 'living polymer' methods in order to obtain end­functionalized polymers with narrow molecular weight distributions (MWD).

Some experiments, particularly those relying on fluorescence measure­ments, can be carried out on 50-200 j1g polymer samples. Under these conditions it becomes practical to use GPC methods to fractionate mg quantities of polymers. From one synthesis, one can obtain several fractions of different molecular weights, each characterized by low values of MwiMn.

CONCEPTS, MODELS AND THEORY

This chapter is concerned with the quantitative determination of cyclization rates and equilibria, and with comparison of theory with experiment. Most theoretical models used to describe polymer cyclization invoke assumptions that may not always be applicable to those particular molecules employed for spectroscopic studies. For example, even in 8-solvents the end-to-end distance distribution function W(h) is Gaussian only for chains of sufficient length. 22 Furthermore, these models frequently ignore the influence of the terminal substituents on W(h) or on the cyc1ization probability, WOo In some circumstances, these features are important; whereas at other times, they seem to make little difference in influencing the comparison of theory and experiment.

Experimental Considerations To be detected, cyclization of a polymer must lead to an observable event. The phenomena underlying the observable event (such as energy transfer, electron transfer, bond formation) establish criteria which must be incorporated into a theoretical model. The experimental data must then be treated in a way which permits comparison between theory and experiment.

All models of cyclization designate a region of space around one chain end such that the presence of the other end within that space can lead to reaction. Most generally one can define a sink function S to describe the distance (and orientation) requirements for reaction. Its mean value <S) describes the shape of the reactive volume, which for convenience is frequently taken to be spherically symmetric. In the simplest model of a chemical reaction, <S) is a sphere with a capture radius R. Approach of the

294 MITCHELL A. WINNIK

chain ends to a distance h < R leads to reaction with an intrinsic rate constant k.

To compare experimental results to theory one makes use of the concept of effective concentration Co. 23 Co is defined as the ratio of a first-order to a second-order rate or equilibrium constant

or (2)

and presumes that for any intramolecular process (k~~~ or K~W there exists a corresponding bimolecular reaction (k~~ or K~~D for which the encounter pair has the same intrinsic reactivity k. 1 ,2,13,23

k~~lf k --A + --D~ -'·AD -- __ observable event

k~~:r (3)

In bimolecular reactions (eqn (3» -A and - D must diffuse together [k~~lrl before they can react. When designing an experiment to study cyclization, the most difficult part is choosing the appropriate end groups for the polymer. This choice must be based upon knowledge of the interaction process between A and D. One important question is whether their reaction is diffusion controlled. In addition, the bimolecular reaction (3) serves as a model for the intramolecular reaction. Proper data interpretation normally requires one to determine both the first-order and second-order rate or equilibrium constants, and to compare them according to eqn (2).

Co has units of moles per litre. It is related to Wo by the expression

C _lOOOWo 0- NA<S)

where NA is Avogadro's number.

-D A*----D < k? > ~)0 ~

kD -1

(4)

observable event (5)

In a cyclization reaction such as (5), the experimental first-order rate constant kl (or k~~D determines the fraction cp of [A * - - D) remaining at time t:

~dcp=kCP dt 1

(6)

This reaction is said to be diffusion controlled (k 1 = kf) if every encounter

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 295

between A * and D leads to a reaction. 24a,b,c If each encounter leads to reaction with low probability (small k), a pre-equilibrium is established. The reaction is said to operate under conformational control. Since Wo = k?/kr: l' one can write

I I 1 -=-+--k1 k? kWo

(7)

In the limit k-+oo, k1 =k? When k is very small, k1 =kWo, the equilibrium rate constant. Both k? and Wo decrease with increasing chain length. Whether a reaction is diffusion controlled or not will depend upon the relative magnitudes of k? and k WOo When these values are comparable, the reaction is said to be partially diffusion controlled. 25a,b

One needs experimental criteria for establishing whether a reaction is diffusion or conformation controlled. Since k? '" T/rJo where rJo is the solvent viscosity, diffusion-controlled reaction rates should vary as rJ;; 1.

The rates of reactions involving pre-equilibria should not depend upon rJo' Partially diffusion-controlled reactions should show some modest (and non-linear) sensitivity to rJ;; 1. One has to be aware here that a change of solvent can affect more than just rJo' A difference in solvent quality can also affect cyclization rates through changes in the end-to-end separation, and in other factors to be described in later parts of this chapter.

Cyclization Equilibria The detailed theory of cyclization equilibria has been treated in Chapter 1 of this book. Certain salient features are described here, either as a basis for comparison with experiment or to define notation.

One class of models is those used to develop analytical theories of chain statistics. 2 2 These models normally require chains of sufficient length N that the mean-squared end-to-end length <h 2 ) and mean-squared radius of !gyration Rb are small with respect to the contour length. C:~lculation of the cyclization probability Wo from these models involves the additional assumption that the capture radius R about the chain end is much smaller than <h 2 )1/2. Many of the experiments reported in this chapter involve oligomers sufficiently short that these assumptions are not satisfied.

Discrete models avoid this problem by simulating the behaviour of 'realistic' chains. These are generated in terms of the rotational isomeric state [RIS] model. 22 These can include not only the structure of the polymer backbone, but also the space-occupying characteristics of the end groups.15,16 Such models in principle permit one to incorporate more or

296 MITCHELL A. WINNIK

less realistic details of the end-group interaction which one detects spectroscopically: the structure of the transition state for the rate-limiting step of ring closure reaction studied kinetically, or the distance and orientation dependence of an electron transfer reaction. Chain properties are calculated by matrix multiplication, exact enumeration or Monte Carlo methods. 22.26.27a.b Simulations in such detail are obviously more tractable for static properties of polymer chain than for dynamic properties. Dynamics opens the system to many more degrees of freedom.

One of the theoretical predictions of analytic models which has been the target of many investigations 28a,b,29 is the ring closure exponent rJ. in the expressIOn Wo - Na. For Gaussian chains, since <h 2 ) - N,

(8)

rJ. = - 3/2. One of the curious features to emerge from the study of polymers in 8-solvents (see below) is that the rJ. = - 3/2 behaviour is frequently observed even in oligomers where W(h) is not Gaussian.

Some authors have applied scaling arguments to cyclization in good solvents,30,31 assuming that, as in eqn (8), Wo continues to depend upon a single length scale represented by RG. In good solvents, RG - N 3v with v ~ 3/5;32a,b,c therefore one predicts that Wo - N- 9 / 5 . Calculations based upon discrete models 33 have shown an even steeper dependence upon chain length, with rJ. = - 1·92.

It has recently been shown that in the presence of excluded volume Wo depends as well upon a second length scale (and a second exponent e) which describes the strength of the repulsive interaction between the chain ends at small r. 32a,b,c,34 Unlike Gaussian chains for which Wo is a maximum at h = 0, excluded volume chains have a minimum at h = 0; and in the vicinity of h = 0, depressed values of W(h), whose magnitude depends upon the strength of the excluded volume interaction. 34 This depression in W(h) for small h is referred to as a correlation hole in the end-distance distribution function. Using the de Gennes and des Cloizeaux32 value of e one predicts Wo - N- 1·9 .

One way of assessing the importance of the second scaling length in the magnitude of W(h) would be through careful evaluation of the ring closure exponent, rJ. = -1·8 or -1,9. Alternatively, one could examine solvent effects on WOo Equation (1) predicts that changes in Wo will reflect only changes in <h 2 ). The correlation hole concept suggests that between good­and 8-solvents, Wo will change by much more than <h 2 ) -312.

SPECTROSCOPIC STUDIES OF CYCLIZA nON DYNAMICS AND EQUILIBRIA 297

Cyclization Dynamics The theory of cyclization dynamics was originally treated by Wilemski and Fixman (WF).35 Aspects have been examined in greater detail by Doi and co-workers in Japan,36a.b by Evans in the USA,37a,b by Perico and co­workers in Italy, 38a,b,c and by Cuniberti and Perico (CP). 39 A recent review by CP provides a formal solution of the WF model and a much more detailed treatment of the theory than presented here. 39 For consistency, much of the notation used by CP in their review will be followed here.

One begins the theory by writing down a generalized diffusion equation (1) in the configurational space of n particles interacting via a potential U(R I , R z, ... , Rn), which is a function of the particle coordinates. This equation describes essentially a balance between averaged Brownian forces, interaction forces and friction forces. Inertial forces in solution are considered unimportant.

dZRj B A m dtZ = 0 = F j - VjU(R I , R z, ... , Rn) + F j (9)

Ft describes the friction force acting on particle j (with a friction coefficient (). Ff are the averaged Brownian forces, given by

Ff = - V}n t/I (10)

where t/I is the distribution function for the configuration {R;} at time t. One treats the solvent velocity as a sum of two contributions: the sum of the unperturbed fluid velocity plus a perturbed velocity due to the presence of friction forces from the particles (the Oseen interaction) acting on the solvent.

The system in the absence of reaction is conservative

f t/I(RW .. ,Rn)dRI,· .. ,dRn=1 (11)

Equation (3) states that the distribution of particles may change in time but the number of particles does not. In a reactive system the number of particles will change in time. Equation (11) must be replaced by a non­conservative distribution function.

f t/ld {R} = ¢(t) (12)

Here ¢(t) is defined as the probability of finding the n particles unreacted at time t.

298 MITCHELL A. WINNIK

Non-conservative distribution functions can be obtained either by introducing appropriate boundary conditions into the diffusion equation, or by introducing a 'sink term' proportional to the intrinsic rate constant k between a pair of adjacent reactant species. For species at the ends of a polymer molecule, one boundary condition is that they can never be further apart than the length b of the fully extended chain. Reaction can be denoted by a Smoluchowski boundary condition at R: 20a,b

t/J(R) = ° (13)

A somewhat more general boundary condition allows the particles eventually to react at R 24a,b,c

flux (R, t) = kt/J(R, t)

In the limit k-- 00, eqn(6) reverts to eqn(5).

at/J -+Gt/J=-kSt/J at

(14)

(15)

An alternative method is to introduce a sink term - kSt/J into the diffusion equation (e.g. eqn (15»). 28a,b In St/J, Sis an operator which delimits the region of configurational space in which the reaction can occur. The probability Wo of finding reactive groups within that volume at t = ° depends upon the fraction of t/J within S.

Wo ,= f t/JeqSd{RJ (16)

In eqn (15) G is the generalized diffusion operator. It contains the Brownian and friction forces as well as the details of the interaction potential U between the particles. If one sets U = 0, eqn (15) describes diffusion-controlled reactions between unconnected particles. For linear polymer chains, the particles are connected and one has to treat a real many-particle system. The boundary condition methods are unwieldy. The sink term method of WF provides an appropriate solution to the cyclization process.

A first approach to intramolecular reactions in polymers might be two free particles (U = 0), having a Gaussian end-distance distribution t/J at t = 0, trapped inside a sphere of diameter b. 31 Somewhat more realistic is the harmonic spring (HS) model. 35,36a,b Here the polymer chain is replaced by a harmonic spring (U = harmonic spring potential) connecting the two particles at the chain ends. These models can be evaluated with single

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 299

particle Smoluchowski equations. A more detailed model adequate for large amplitude motions of polymer chains is the bead and spring model of Rouse and Zimm.40a,b,41

The Cyclization Reaction WF35 factored the time dependent distribution function into an equilibrium part ljJeq and a time-dependent part p( {RJ, t). In a photochemical reaction, the sink does not become operative until the light is turned on at t = O. One considers the time evolution of the system starting from the equilibrium distribution ljJeqPO' The sink operator S causes the establishment of an exponentially decaying state at large time. As time increases, the distribution function IjJ becomes spatially distorted due to the depletion of configurations for which the reactive groups have entered the reaction volume.

In order to solve the appropriate equations to obtain the long time rate constant k?, certain approximations about S need to be made. WF assumed a closure approximation, namely that the distorted part of the exact distribution function within S is proportional to IjJ eq' This choice was initially viewed with some scepticism, since there was concern that the mathematical convenience in the assumption might be made at the expense of an unrealistic depiction of the reaction process. The validity of this assumption was examined by various authors in the context of the HS model where alternative analyses are possible. 36a,b,38a,b,c,39 These in­vestigations suggest that the closure approximation does indeed take into account the essential features of the distorted distribution function in the calculation of the cyclization rate constant, even in the case of very efficient reaction between the end groups, k ---+ 00.

The dynamic component of the reaction rate depends upon the fluctuations in S. The time correlation function <S(t)S(t + r» describes the time-decaying probability that a group present in S at t will also be present in S at some later time (t + r). The diffusion-controlled rate constant k? depends upon the autocorrelation function of C5S(t) == S(t) - <S), the instantaneous fluctuation from the mean, which CP30 refer to as [D(t) - W61 * In terms of the WF model, one finds

(17)

* Note that CP39 use (dimensionless) Co to mean the same as Wo here. Our Co has units of moles litre - '.

300 MITCHELL A. WINNIK

Rouse-Zimm Dynamics In the Rouse-Zimm (RZ) model (Fig. 1), one replaces the polymer chain with n beads of Min mass and friction coefficient C connected by (n - 1) massless harmonic springs of mean-squared length a2 • This model exhibits a spectrum of relaxation times characteristic of large-scale motions delocalized over the entire chain, as well as short-scale diffusion over distances comparable to the length of one or two springs. The distribution function", is Gaussian for bead-and-spring chains, with the mean-squared end-to-end length <h 2 ) = a2 (n - 1).

FIG. 1. Representation of the Rouse-Zimm bead-and-spring model for a polymer chain. Here cyc\ization is detected by the presence of the other chain end with the

sphere of radius R cf:ntred on one end of the chain.

To calculate values for k 1 and k?, expressions are needed for D(t) and pet), with U equal to the bead-spring potential. Evaluating D(t) requires that certain choices be made in the form of S, ideally to accommodate realistic descriptions of the reaction geometry. For computational convenience, one normally chooses spherically symmetric forms for S, the simplest being a capture sphere of radius R.

WF evaluated k? in the limit of infinitely long chains. 35 They considered both the free-draining Rouse model40a (no hydrodynamic coupling between the particles) as well as the non-draining Zimm model40b (all solvent molecules trapped within the polymer coil). Real polymers are finite in length and partially draining. In order to make the theory accessible to experimentalists, CP rewrote WF theory in terms of exact eigenvalues and eigenvectors of the Rouse-Zimm model for finite chains. 39.42 They provide expressions which permit realistic values of k? to be calculated in terms of

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 301

two' parameters characteristic of each individual polymer-solvent com­bination. These are the spring length a and (r' a measure of the strength of the hydrodynamic interaction between particles.

In a careful and thorough examination of the problem CP indicated that under most circumstances it suffices to use the Rouse eigenvectors Qil' simply evaluated from

Q~ = t.o cos- i +-(2 - b )1/2 nl( 1) n n 2

(18)

plus a combination of both the Rouse eigenvalues Ar and the exact eigenvalues At to obtain reasonable values of pet):

(19)

(20)

(21)

The diffusion-controlled cyclization rate constant k? can be calculated from

where

with

[ rOO ( K(t») J-1 k?= Jo K(oo) -1 exp(k 1t)dt

K(t) = erf [z(t)] - (2/n 1/2)z(t) exp ( - Z2(t»

K(oo) =erf(y) _(2/n1/2)y exp( _1'2)

(22)

(23)

(24)

(25)

(26)

Note that the dimensions for the capture radius R enter the calculation of k? through the term y.

In the limit of very long chains, I' - 0, but k? / W 0 for RZ chains converges

302 MITCHELL A. WINNIK

to a non-zero value. This asymptotic value is related quite simply to the bead-and-spring model expression for the translational diffusion coef­ficient of the chain: for non-draining chains

k?(ND) ~ 1·98DND/(h 2> and for free draining chains

k?(FD) ~ 13·6DFD/(h 2>

(27)

(28)

The former is more realistic for polymers in dilute solution and predicts that k?(ND) - N- 3 / 2 , sinceD _ N-- 1/ 2 and (h 2> - N. For finite chains the ring closure exponent will differ somewhat from the value of -1·50 because of the influence of the RI(h 2> term in eqn (26).

Oligomer Cyclization WF theory, as presented above, is inapplicable to short chains. Rotation over internal barriers, which is a dominant feature of their motion leading to cyclization, is not considered in the bead-spring model. Alternative approaches have been develope:d which examine the motion of individual molecules in terms of the RIS model.

Sisido and Shimada 43 were the first to consider cyciization dynamics of alkanes. They carried out careful RIS model calculations of the end-to-end distance distribution function W(h) for N-(CH2)n-N, where N is the I-naphthyl substituent, and h is the distance between centres of the N groups. Dynamics were introduced simply as a mean lifetime for cyclized conformations, taken to be the same as that for the corresponding bimolecular reaction.

Nairn and Braun 37b developed a dynamic model in which polymethylene chains were embedded in a diamond lattice. Each chain was subjected to a series of random-number-determined bond motions, and each motion was assigned a time (c. 100 ps) depending upon its nature and location within the chain. These times were assigned empirically on the basis of 13C NMR relaxation measurements on comparable molecules. Eventually the two chain ends come into contact and that chain is deleted from the sample. Repetition of the process allows a histogram of surviving chains ¢(t) to be generated, and the rate constant k? is obtained from a least squares fit to a plot of log ¢(t) vs. time.

James and Evans 37a carried out Brownian dynamics simulations for 4-, 7-, 8-, 10- and 15-bond chains with rigidly fixed bond lengths and bond angles, and a torsional potential with three-fold rotational barrier of 3 kcal mol- 1 . Self-correlation functions were determined for the end vector

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 303

h, for h2 and for the sink function S. Correlation times were calculated analytically. The appropriate correlation time for the diffusion-controlled chemical reaction was found to be related to rK , which measures the rotational isomer interconversion time in n-butane (r; 1 = k(rot)).

The cyc1ization rate constant k? was found to consist of a static and a dynamic part, with the former containing the chain length dependence. Thus if one mimics the rate by the average sink function, as Sisido did,43 one finds an N- 3/2 dependence attributable to <S).

The dynamic component shows a near N independence for long chains and approaches a value that is a multiple of the butane rotational rate.

James and Evans37a are thus able to write

(29)

where kBis the Boltzmann constant, b the C-C bond length, and (Rjb) the relative capture radius. Q is the rotational barrier in butane.

Both James and Evans 37a and Nairn and Braun 37b applied their models to calculate the rates of intramolecular electron exchange in the molecules naphthalene-{CH2)n-naphthalene studied experimentally by Shimada and Szwarc. 21a,b Both sets of calculations gave k? values an order of magnitude larger than found in the experiment. Neither pair of authors recognized that these intramolecular electron exchange rates are less than diffusion controlled (vide infra), perhaps by an order of magnitude. The model calculations are very good indeed.

EXPERIMENTS IN DILUTE SOLUTION

Cyc1ization experiments have been carried out on a variety of polymers, some involving end-to-end cyc1ization and others involving reaction between groups interior to the chain. While the organization of this section is by type of polymer, polymers presented first are also those for which experiments yield information primarily on cyc1ization probability. The methods for studying cyc1ization dynamics are described in the context of polystyrene, where they have been most intensively applied. A discussion of pyrene excimer formation to study cyc1ization dynamics of poly(ethylene oxide), poly(dimethylsiloxane) and poly(vinyl acetate) completes this section.

304 MITCHELL A. WINNIK

Polyacrylamide The most difficult aspect of studying end-to-end cyclization of long polymers is the synthesis of appropriate end-labelled macromolecules. An alternative approach is to synthesize polymers containing a distribution of reactants along the chain. Such polymers are easy to prepare and they are a source of rich information on cyclization between elements interior to the chain, provided one can interpret the data.

Such experiments were first carried out on long polymers by Goodman and Morawetz. 44a ,b They prepared acrylamide copolymers containing low concentrations of 4-nitrophenylester side chains and 4-alkylpyridine groups. Because of the method of synthesis, these groups are statistically distributed along the polymer. These papers are particularly important because they provide the foundation and fundamental concepts for treating the kinetics of intramolecular reactions in polymers. 23 ,44a,b

I +

Scheme 1

Since the raw data could not be fit to a simple rate expression, the authors assumed a model predicting a distribution of first-order rate constants, n(k), corresponding to a distribution of contour separations of reactive and catalytic groups on the polymer. The fraction of unreacted ester groups <p(t) is a function of this distribution:

<p(t) = In(k) exp( -kt)dk (30)

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 305

and the initial apparent first-order rate constant was shown to be equal to the average value of k, <k):

kin it = _ d4>(t) = S kn(k)dkIS n(k)dk = <k) dt

(31)

As the authors point out, the interpretation of their experiments is made difficult by the need to relate n(k) to W 0 for groups within the chain and its length N dependence. By assuming a relationship in which Wo '" N a , the authors 44a.b were able to parameterize their model and determine a value for the ring closure exponent of r:J. ~ 2.

Polymethylene Chains The cyclization properties of hydrocarbon chains have been reviewed in detail. 1 Thus, a few examples here will suffice. These have been chosen somewhat arbitrarily to illustrate important features of hydrocarbon chain cyclization.

From a historical point of view , the seminal cyclization experiments were the ring-chain polymerization studies of w-hydroxycarboxylic acids [HO(CH2)n_2C02Hj by Stoll and Rouve 45a•b from which cyclization equilibrium constants could be calculated. More recently Mandolini and Illuminati 13 have investigated the ring closure kinetics of the reaction

° II C""

OH')o-, + S,- (32)

which was followed by determining the rate of Br - release. These results are compared in Fig. 2. Since the geometry of the transition state for lactone formation by an SN2 mechanism is not the same as the geometry of the lactone itself, it would not be surprising if the two plots showed differences. Nonetheless they are surprisingly similar. It appears that both the transition state and the product are similarly sensitive to the constants of cyclization.

*. Me 2N--(CH l)n-N Mel

306 MITCHELL A. WINNIK

I 0 k~1 ) 0 II II

Id 0 Br (CH2)n-2CO(-) - (C'O) 0 K(ll

" --...h. 10 • HO (C~n-2COH ...,,------ (C~)n-2

I u Q) (f)

~ 10° 0

0 LD

:::. ..., ..:.::

.... 0 Id --0' 0 co

:::::~ --y

104

o 5 10 15 20 25 RI NG SIZE

FIG. 2. Molar equilibrium constants K(1) forcyclization ofHO(CH 2)n_ 2C02H at 80°C (.) determined from ring-<:hain polymerization studied by Stoll and Rouve.45a •b First-order rate constants k(l) for cyclization of Br(CH2)n-2C02 (-) at 50°C (0) studied by Illuminati et al. 13 (Reproduced with permission from Ref. I by

courtesy of the American Chemical Society, Washington, DC.)

A somewhat similar dependence of cyclization rate on chain length is shown in Fig. 3 with data from intramolecular fluorescence self-quenching of aliphatic amines. 46 Electronically excited tertiary alkyl amines fluoresce in the UV. The lifetime of the fluorescence is very sensitive to the presence of other amine groups. The rate constant for bimolecular self-quenching is very large, about a factor of 5 less than that for diffusion-controlled processes. Nonetheless, the intramolecular process does not show the key

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 307

Me2N (CH2)n-2 NMe2 0

109

10- 1

108

'u .... Q.I ... \

10-2 ~ VI \ /. \ / u

.r:!" \ /

• ~

107

163

106

0

FIG. 3. Estimated rate constant kiq (s -I) for intramolecular fluorescence quenching in the molecules Me2N(CH2)n_2NMe2 studied by Halpern et al.46 Co values are calculated from (kiq/k~2)), where k~2) = 3·9 X 109 M- 1 S -I is the bimolecular rate constant for self-quenching in Me2N(CH2)8NMe2. (Reproduced with permission from Ref. I by courtesy of the American Chemical Society,

Washington, DC.)

feature of a diffusion-controlled process, that k jq = k 1 k 2/(k -1 + k 2) be proportional to the reciprocal solvent viscosity '1;; 1. A plot oflog k jq vs. ring size is presented in Fig. 3 and shows the influence of transannular CH2/CH 2 interactions suppressing the cyclization to form medium rings.

The key features that are apparent in Figs 2 and 3 are the ring strain effects for very small rings, the ease of cyclization for 5- and 6-membered rings and the difficulty of forming medium (7- to 16-membered) rings. In polymethylene chains, transannular interactions between CHz groups

308 MITCHELL A. WINNIK

cause torsional and bond angle distortions, and inhibit cyclization. These local effects are not considered in analytic models of polymer cyclization. The ring closure exponent could only be determined for the study of polymethylene chains with n > 20.

Two features reduce the influence of the transannular interactions. Replacing CH2 groups with oxygen atoms or with planar groups such as alkenes, esters or aromatic rings decreases the steric hindrance to cyclization. Alternatively, reactive processes with large capture radii, electron transfer 4 7a,b,c or Forster energy transfer, 48a,b allow 'cyclization' to be detected with the chain ends sufficiently far apart that steric demands on the chain itself are minimal.

In the classic intramolecular electron transfer experiments of Shimada and Szwarc, 21a,b the electron exchange rate pO)(s -1), obtained from ESR measurements, decreases monotonically with n for N---{CH2)n-N (1) (Fig. 4), where N is I-naphthyl. 1 is produced from N---{CH2)n-N by sodium metal reduction. The monotonic dependence of pO) on n is a consequence of the fact that electron transfer can occur over 8-10 A. Curiously, the corresponding dectron exchange rate in PI-CH2-PI -, where RI is the phthalimido group, shows a minimum at n = 6. 21a,b This result suggests that the geometric requirements for electron exchange in PI are much more stringent than in N. The exchange reaction can be represented by the scheme

This reaction would be diffusion controlled if k t » k l' One of the reasons these studies have become classic is that they were erroneously interpreted as diffusion-controlled cyclization reactions. As such they had a powerful influence in stimulating the attention of theoreticians to the subject of cyclization dynamics. Unfortunately, the reaction is not diffusion controlled: kt is limited by the rate of counterion exchange and solvent relaxation. Experimentally one observes the exchange reaction to go faster in hexamethylphosphoramide than in dimethoxyethane even though the former is lO-fold more viscous. Professor Szwarc now concurs with this interpretation.49 It is interesting that dynamic simulations of poly­methylene cyclization rates gives values an order of magnitude larger than P.

310 MITCHELL A. WINNIK

Two other examples of conformationally controlled cyclization of hydrocarbon chains are noteworthy. In these examples the site of cyclization is separated from the chain origin by an aromatic ring. Winnik and his co-workers 50 - 52 have published an extensive series of papers examining the kinetics of intramolecular phosphorescence quench­ing of benzophenone derivatives:

2 Q=-CH 3

-CH=CH 2

-C~H quenching photo-products

Interaction of the chain with the ketone oxygen of the benzophenone triplet causes a decrease in the phosphorescence lifetime from which first-order quenching rate constants k jq = kK can be calculated. When Q = -CH3'

OJ

H2N,

~H2)n

S02NH

•

C~2=CH2\ 5

~SH2)n

6 CO2 14

~ ~32 N~ ~ 0.06 o E

00.04

.~ QO:'--<I--=-_....g. __ ........,-';:--____ ~----___::_'::J~

.:: 0.08

15 20 Chain Length n

FIG. 5. Experimental values for Co for (6) intramolecular phosphorescence quenching in the molecules benzophenone-4-C0z-{CHz)n -CH=CHz in CCl4

at 22 °C;18b.SO for (e) ring closure in the molecules l-ClOzS-naphthalene-5-SOzNH(CHz)n-NH Z· 5

)(

<II

o

~ -1 a... en .2

-2

-3

0·4

i

n 4 8 12 16 20 6c

1-2 logn

FIG. 4. Dependence of the frequency P (s -1) for intramolecular electron exchange divided by the corresponding bimolecular rate constant kex (M- 1 s -1) as a function of chain length for the molecules N-(CH2)n-N°. (N is the I-naphthyl group). Experimental values from Shimada and Szwarc 21 in HMPA solvent (e) and in DME solvent Ca.); 6, 0, 0, calculated values based upon the RIS model using6, 8 and 10 A, respectively, as the reactive distance for electron transfer. Theoretical values from Sisido and Shimada.43 (Reproduced with permission from Ref. 69(c) by

courtesy of the Chemical Society of Japan, Tokyo.)

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 311

only hydrogen abstraction from the chain interior occurs. 50 When Q =

-CH=CH2, or -C==CH, quenching by exciplex predominates and information about end-to-end cyclization can be obtained. 51 ,52 The corresponding bimolecular rate constants k(2) for model reactions range from 3 to 5 orders of magnitude less than diffusion controlled.

Co = k~~)/k(2) values (Fig. 5) give information about cyclization probability. No intramolecular quenching occurs for n < 8: the chain cannot reach. Co goes through a maximum at n = 12 for end-to-end cyclization and decreases thereafter. Fraser and Winnik 52 simulated this process using Monte Carlo calculations of cyclization probability with the RIS model for the chain. They took account of the steric effects of the benzophenone group and chose a capture radius of R = 3 A centred on the ketone oxygen. The salient features of the intramolecular quenching process were reproduced in the calculation.

(36)

non-fluorescent fluorescent

Mita et al. 5 have recently developed an exceptionally clever method for studying cyclization rates in very dilute solution. Chlorosulphonate derivatives of aromatic molecules are non-fluorescent because of rapid internal quenching. Their sulfonamides are very fluorescent. In the reaction of NHZ----{CH2)n-NH 2 with 1 ,5-naphthalenedichlorosulphonate, the first substitution is sufficiently faster than the second step (eqn (36» that kb can be determined. 5b One measures the product formation by the growth in fluorescence intensity. This cyclization reaction is conformationally controlled: values of kb are of the order of 10 - 5 S -1. In Fig. 5, Co = k~l)/k(2)

for 3 are plotted vs. n. The value of k(2) is taken from the reaction of l-n­C4 H9 NH02S-naphthalene-5-S02CI with n-butyl amine. 5b These results are also compared in Fig. 5 to Co values for intramolecular phosphores­cence quenching for 2 obtained by Winnik et al. 51 ,52 The differences in curve shape for chains of these lengths reflect different geometric demands of the individual chemical reactions.

A curious feature of the cyclization reaction is the exceptionally low values of Co. Mita has shown that the corresponding diamines [NH2-(CH2)n-NH21 undergo intramolecular hydrogen bonding. 5b Such

312 MITCHELL A. WINNIK

cyc1ized conformations in 3 might lower the probability of the -NHz group occupying reactive volume about the -SOzCI group, thus suppressing the rate of the intramolecular reaction.

The only experimental studies yet published which seem unambiguously to describe diffusion-controlled cyc1ization of hydrocarbon chains involve (i) magnetic resonance studies (CIDNP) of photochemically generated

o II o. .

/C, (C . CHz C' )' -"+ ) ~ (CHz)n- 3 (CHZ)n_ Z (37)

biradicals, and (ii) excimer formation in polymethylene chains containing pyrene derivatives at both ends. In the photo-CIDNP studies,53a,b the magnetic interaction between the radical centres is distance dependent, and the intensity of the emission signal in the NMR depends upon the biradical lifetime and the strength of the applied field. By combining CIDNP theory with a model of alkane chain dynamics, De Kanter et al. 53a,b found that chain end motion could be described by an effective diffusion coefficient Dl = 5 x 1O-5 cm -2 s-1.

-- (38)

fluoresces blue fluoresces green

Intramolecular excimer formation between pyrene groups at the ends of a polymethylene chain was first reported by Zachariasse. 15 The excimer forms at the diffusion-controlled rate, and, except at high temperatures, ring re-opening is slower than decay of the excimer to ground state. His plot of the excimer to pyrene ('monomer') fluorescence intensity ratio IJ 1M for various n (Fig. 6) has many features in common with other examples of chain cyclization. These data do not permit rate constants to be calculated. Spectral shifts of the excimer demonstrate the consequences of strain in the formation of medium rings. Rate constants are in principle available from fluorescence decay measurements. Zachariasse 77 has made many such measurements. He argues quite forcefully that the values one calculates and the conclusions one draws are sensitive to the kinetic model one assumes. For chains of n < 23, the fluorescence decay data are inconsistent with the

SPECTROSCOPIC STUDIES OF CYCLIZA TION DYNAMICS AND EQUILIBRIA 313

20

I • Py( n) Py

15 excimer to -

monomer ratio

t 10 I'll

5 /\ • V-- './ ~.

\ ,....~_II o o 2 5 10 15 20 22 25

n-FIG. 6. Dependence of the excimer-to-monomer emission intensity ratio (J E/ I M) as a function of chain length for the molecules Py-{CH2)n-Py. (Reproduced with permission from Ref. l5(b) by courtesy of Zeitschrift fur Physikalische Chemie NF,

Munich.)

simplest model (which seems to apply for longer polymers), and more complicated models which fit the data available are not unique.

Nishijima and his co-workers 55 have published studies of excimer formation in I-pyrenemethyl esters of dibasic acids. lEI 1M values have been reported for n = 2-22, and rate constants have been calculated assuming the simple two-state model for the kinetics of excimer formation and decay. The two ester groups mitigate transannular interactions in the cyciization: no minimum is seen in the lEI 1M vs. n plot.

Poly( N-methylglycine) Sisido and hisco-workers56 - 58 in Japan have carried out an important series of experiments on the cyciization of end-substituted poly(N-methylglycine)

314 MITCHELL A. WINNIK

(polysarcosine). This polymer can be prepared anionically using sub­stituted amines as initiators. By proper choice of initiator, one can control the end groups on the polymer. The polymerization mechanism ensures narrow molecular weight distribution (MWD). One interesting feature of polysarcosine is that the amide group can exist in both the cis and trans conformation. (Polypeptides are exclusively trans.)

cis trans

Using conformational calculations based upon the RIS model, Sisido et al. 5 6.5 8 have explored the consequences of this equilibrium on the conformational properties of the chain, and used NMR methods to demonstrate that the equilibrium itself is sensitive to solvent polarity.62 To examine cyclization properties of polysarcosine, two sets of experiments were carried out. In one, the kinetics of ester hydrolysis were studied, in aqueous solution, for a polymer containing a nitrophenyl ester at one end and a pyridine group at the other. In the other, donor-acceptor complex formation was studied as a function of chain length and solvent for polysarcosine samples containing donor groups at one end and acceptor groups at the other.

Catalysed Ester Hydrolysis In the molecules 4 and 5 above, hydrolysis of the nitrophenyl ester produces nitrophenol and the nitrophenolate anion, with a strong UV-visible absorption. The reaction kinetics can be followed conveniently by spectroscopic methods. In addition to spontaneous hydrolysis, there is catalysis by the pyridine group. Thus, the observed first-order rate constant

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 315

is composed of two terms: ko for the uncatalysed process and kl for the pyridine-catalysed step. Since the polymer 15 can undergo only spontaneous hydrolysis, ko can be determined, and values of k 1 calculated.

The mechanism of the catalysed hydrolysis involves formation of a covalent intermediate, cf. Scheme 1, which involves ring closure in the intramolecular reaction of 4. Thus k l' the rate constant for this process, is proportional to the cyclization probability of the chain.

Data for studies in aqueous solution at pH 6·1 are shown in Fig. 7. One

o U

0.5

D/A complex

hydrolysis

theoretical

-- ...... ",.... "

", " W:' " o "

1.0

" " " " " Log (n +2)

, ,

1.5

, , ,

limiting slope

-1.2

-1.3

, ,

~ -2.0

2.0

FIG. 7. Log-log plot of Co vs. N for the end-to-end cyc1ization of polysarco­sine chains: 0, Intramolecularly catalysed ester hydrolysis in the molecules 4-pyridine-CH2NH~[COCH2N(CH3)lnCOCH2C02C6H4-4-N02 in water at pH 6·1; 55 ,0" intramolecular D/A complex formation in the molecules 4-Me2N-C6H4~NH[COCH2N(CH3)]nCOC6H3~3,5--{N02)2;2 --, values of Co calculated from Monte Carlo estimates of Wo, using an RIS model for the

polysarcosine chain. 71 The capture radius R is taken to be 4A.

316 MITCHELL A. WINNIK

notes that Co (and hence k I) decreases with increasing chain length. The conformational calculations also predict that the cyclization probability Wo should decrease with increasing n. There are, however, important differences between the experimental data and the predictions of the theory. These emphasize some of the difficulties in comparing theory with experiment.

Temperature Effects In carrying out conformational calculations one normally assumes that the only relevant energies are those of the rotational state isomers. From the point of view of the model, the only differences one should observe between the activation enthalpy flH'" (or Ea) of the intramolecular reactions and the corresponding bimolecular process should be those due to conformational population differences in the cyclized chains and the total ensemble of chains. These differences are normally small. For small rings, torsional and bond angle strain will make flH",(I) larger than flH",(2). Solvent effects are not taken explicitly into account.

Since hydrolysis is an activated process, k 1 increases with temperature. Arrhenius studies by Sisido and his co-workers 56 indicate that the sensitivity of k 1 to temperature increases with increasing chain length. They find, for example, for n=10, l!!.H"'=-13kcalmol- 1 , /).S"'=

-37K- 1 mol- l , whereas for n=25, flH"'=-17kcalmol- 1 , flS"'=

- 25 cal K - 1 mol-I (where I cal = 4·184 J). The decrease in k 1 with increasing n is due to a change in activation energy, and the activation entropies become less negative for longer chains.

On the other hand, the activation parameters of the corresponding bimolecular reaction are quite normal, with flH",(2) = - 14 kcal mol- I, /).S",(2) = -17 cal K -I mol-I. The /).S",(2) value is typical of bimolecular reactions in which the loss of translational entropy is compensated by a gain in vibrational entropy in the transition state. 71

The peculiar behaviour of flH",(1) and M",(1) with chain length must reflect polymer-solvent interactions which change with temperature and are affected by cyclization. These are neglected in the model. While experiment and theory could, in principle, be compared rigorously at the ()­temperature for unperturbed chains or in an athermal solvent for non­intersecting chains, these experimental conditions are not close to either reference state.

Donor-Acceptor Complex Formation Sisido et al. 2 prepared polysarcosine derivatives 6 containing a powerful

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 317

electron donating group (Me2N-C6 H4-NH-) at one end and a strong electron acceptor (3,5-dinitrobenzamide) at the other. In dilute solution in chloroform, ethanol and N-N-dimethylformamide (DMF), one can observe absorption of visible light (Amax = 455 nm) due to a donor-acceptor complex. Detailed studies on the model compounds 7 and 8 established the extinction coefficient c and the bimolecular equilibrium constant for complex formation:

o CH 0 N02

~_ II 1311~ Me2N---&-NH--(CCH2N)-C-~

D 6 A N02

o 0 0 N02

~II II II'~ Me2N---&-NHCCH3 + Me2NCCH2N--C~ -- --DjA--

o 7 8 A N02 (40)

K(2) values are close to 1·0 M -1 (where M- 1 represents litres per mole) at room temperature in CHCl3 and ethanol. The binding energy I1.H(2) = - 1·4 kcal mol- 1 is weak, but the entropy change is not very unfavourable, M(2) = - 4·9 cal K - 1 mol- 1. K(2) values could also be obtained from the study of the bifunctional polymer 6, since the absorbance of the band at 455 nm varies modestly with concentration over the range 0·2 x 10- 3 to 5 X 10- 3 M. Data for various chain lengths give very similar K(2) values, almost identical to those for the model reaction (7 +8).

The fraction of cyclized chains was obtained by extrapolating the results to zero concentration and by assuming that c for the intramolecular DA complex is the same as that found from the bimolecular experiments. This fraction is substantial, and exceeds 10% at short chain lengths. This reflects the bias imposed upon the system by the free energy of DA complex formation. 2 From this data, the intramolecular equilibrium constants can be calculated. K(1) values decrease with increasing chain length, varying in ethanol solution from a value of 8·6 x 10- 2 for n = 6 to 2·6 X 10- 2 for n = 32. The n refers to the mean chain length calculated from Mn values. Activation parameters were obtained for experiments in chloroform as well as in ethanol. I1.H 1 values decrease with increasing chain length, while, curiously, M 1 values ( - 8 cal K - 1 mol- 1) do not change.

318 MITCHELL A. WINNIK

For short chains, discrete isomers could be obtained by chromato­graphic separation. 57 From the intensity of the DA absorption band, the fraction of cyclized species could be calculated, assuming that B for the intramolecular DA complex was the same as for the intermolecular complex between 7 and 8. These values are shown in Fig. 8, where the open points refer to the individual oligomers and the closed points to chains having a Poissonian MWD. Note that in DMF, hardly any cyclization is detected, a consequence of the fact that K(2) in that solvent is very small.

10

o 5

..... ..... .... .....

..... ................................ ..... ..... ..... .... .... .... .... ....

............. ......~ ..... '"' .......... ~ ~ --

............ ./ ..... ' " .............. "' ....... / .... ,..:;' ;e... - -.......... ..... .... -13y ......... --_ ..... -~210 ----:_--"! ,\30·r - ...

43°

10 15 n

FIG. 8. Comparison of cyc1ization equilibria values of D-polysarcosine-A for discrete oligomers (0) and for oligomeric mixtures (.) with Poissonian molecular weight distributions. (Reproduced with permission from Ref. 57(b) by courtesy of

John Wiley & Sons, Inc., New York.)

Comparing the Data It would be very useful to be able to compare the results of both sets of experiments. Sisido has done this by interpreting the hydrolysis data in terms of eqn (2), choosing R = 4A and calculating hydrolysis Wo values, which he calls the equilibrium constant for cyclization unperturbed by the energy of the chain end interaction. K( 1) was converted to experimental Wo values, multiplying each K(l) by exp ( -!1G d RT) to correct for the free

SPECTROSCOPIC STUDIES OF CYCLIZA TION DYNAMICS AND EQUILIBRIA 319

energy of binding in the complex, with (~Gl = 1·1 kcal mol- 1) taken from the reaction of7 + 8. Sisido et al. compare the experimental and theoretical Wo values.

Here a somewhat different approach is taken, although the general conclusions one draws from the various experiments are similar. This author prefers to consider Co values, obtained as the ratio of first-order to second-order rate or equilibrium constant. Figure 7 presents a log-log plot of Co = k 1/k~2) from the ester hydrolysis against the average degree of polymerization (n + 2). In counting the degree of polymerization one considers each end substituent to contribute one monomer unit to the chain length. This plot is curved at short chains and appears to have a limiting slope of c. -1·3. For comparison, values of Co = K(1)/ K(2) for DAcomplex formation in ethanol are also plotted (top curve). The data fit a straight line, with the possible exception of the shortest chain, and the slope is -1,2.

While these slopes are nearly identical, they differ from that oflog Wo vs. log n' (n' = n + 2) obtained theoretically. The theoretical model considers non-intersecting chains generated from a RIS model of polysarcosine. The model is realistic in its depiction of the chain structure, although the criteria for cyclization (R = 4 A) are necessarily oversimplified. The calculations show a maximum in Wo for n' = 9 suggesting that steric effects inhibit cyclization of short chains. The limiting slope is c. - 2, consistent with the prediction of the more abstract theory for excluded volume chains.

Among the serious concerns about the data in Fig. 7 are the numerical differences in Co values. Not only are those for ester hydrolysis a factor of 4·5 smaller than Co for DA complex formation, but both sets of data are larger than Co values calculated from Wo using R = 4A in eqn (4). These differences can arise from polymer-solvent interactions, which are not considered in the theoretical model. One would like to believe that solvent effects on the intra- and intermolecular reactions would be similar, so that these contributions would cancel in the ratio Co = k(1)/k(2) or K(l)/ K(2). This seems unfortunately not to be the case for polysarcosine in water, ethanol or chloroform. Since ~Ho and ~H# for the cyclization processes vary with chain length, even the chain length dependence of Co will depend upon the temperature of the experiment.

Polypeptides-Energy Transfer Studies Electronic energy transfer by dipole coupling between groups, Forster energy transfer, can occur over substantial distances (i.e. up to 80-100 A). 48a,b The rate and efficiency of this transfer vary as (Ro/r)6, where r is the distance between the centres of the transition moments of the

320 MITCHELL A. WINNIK

donor D* and acceptor A groups, and Ro is a distance characteristic of the DA pair. It depends only upon the spectroscopic properties of 0 and A and their net orientation. It represents the distance at which the rate of energy transfer is equal to the reciprocal lifetime r;; 1 of D. To study energy transfer in a polymer system, one prepares molecules of the form 0 - - - - A and irradiates the sample at a wavelength where 0 is preferentially excited. One obtains cPET' the

kET D*----A __ D----A*

<PET (41)

energy transfer efficiency from measurements of the ratio loj I A of the 0* and A* fluorescence intensities. Values of the rate constant for energy transfer must be obtained from fluorescence decay studies of ID(t) and IA(t)·

Results of these kinds of experiments are difficult to interpret. 58 If the chain ends move a distance comparable to Ro during the lifetime ro of D*, diffusion contributes to energy transfer. In viscous solvents r may not change on this timescale: a non-exponential lo(t) reflects the distribution of D-A separations.

NMez

~ SOz

o II

NHCHC I

(CHz)z I C=O I NH I

(CHzh I

OH

-NHCHCH!§:@ I C=O I NH I

(CHz)z I

OH 9

Katchalski-Katzir, Stein berg and Haas 19a,b have prepared oligopeptides of the structure 9 above which contain the 2-naphthylmethyl group as 0* and l-dimethylaminonaphthalene-5-sulphonamide (dansyl) as A. The Ro for this pair is 22 A assuming random mutual orientations. Initial experiments were carried out in a viscous solvent, glycerol, where it was assumed that the relaxation time for the end distance h was slower than rD.

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 321

The non-exponential decay of Io(l) was measured and fitted to a distribution W(h) of end separations

Io(t) = S W(h) exp [-kET(h)t]dh (42)

A Gaussian form for W(h) would not fit the data; whereas a good fit was obtained with a polynomial form for W(h) suitable for finite chains.

In fluid solvents of similar polarity, eqn (42) was no longer suitable. Chain dynamics contributes to the energy transfer process, requiring a diffusion term to describe the relaxation of the end-distance distribution:

I(t) = S DW(h) exp [-kET(h)t]dh (43)

While D is presumably a function of h, the data could be fit by a single parameter, the effective mean chain end diffusion coefficient whose value depends upon chain length and solvent viscosity.

Polystyrene Polystyrene can be prepared by anionic polymerization with control of end group substituents and with narrow molecular weight distribution. It is natural that it has become the prototype polymer for testing ideas about cyclization dynamics. Two independent approaches have been taken. Winnik and his group8,59a,b,60 in Canada prepared pyrene end-labelled polystyrene 10 and examined the kinetics of intramolecular excimer formation for molecules of M = 2900-100000. Pyrene was chosen not only by virtue of its long fluorescence lifetime (c. 200 ns) but also because excimer formation is diffusion controlled.

Slow cyclization processes require longer-lived probes, based, for example, on the triplet states of aromatic chromophores. Mita and his group9,6la,b in Japan prepared anthracene end-labelled polymers 11 and examined the kinetics of intramolecular triplet -triplet (Tn annihilation for molecules of M = 10000-300000:

° ° II II (CH2)3COCH2CH2-polystyrene-CH2CH20QCH2)3

10

322 MITCHELL A. WINNIK

H2---polystyrene---CH2

11

Kinetics of Excimer Formation Excimer formation is a one-photon process. Cyclization of photo excited 10 forms pyrene excimer (PyPy)* in a diffusion-controlled process with rate constant k 1. One can measure the intensities of the (blue) pyrene fluorescence 1M , the (green) excimer fluorescence IE and their decay profiles in fluorescence decay experiments.

Py1J-PY ':: ~J ~ ,~ A

Py U Py

Scheme 2

Two coupled differential equations describe the excited state populations

d[Py*]/dt = (k 1 + kMHPy*] - k - d(pyPy)*]

d[(PyPy)*]/dt = (kE + L l)[(PyPy)*] - k dPy*]

(44)

(45)

Solution of these equations gives the expressions to which data are fit:

IM(t) = A exp ( - A1 t) + exp ( - A2 t)

IE(t) = B[exp( - A1 t) - exp ( - A2 t)]

(46)

(47)

IM(t) decays as a sum of two exponential terms; IE(t) grows in and decays as a difference of two exponential terms, with the same short and long A values in each experiment. The AS are related to the rate constants in Scheme 2 by the expressions

2A1,}'2 = (k1 + L 1 + kM + kE) ± [{(k 1 + kM) - (L 1 + kE)}2 + 4k1L d1/2 (48)

A = (k 1 + k M ) - A1 "2 - (k1 + kM)

(49)

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 323

To determine k l' one needs to measure IM(t) and IE(t), verify that similar AS are obtained in each experiment and substitute the values of Ai' A2 and A into eqns (48) and (49), with kM known independently from measurements made on a model polymer containing pyrene at one end only. At room temperature and below, the dissociation rate constant k -1 is quite small. When 4k 1 k _ 1 is small compared to the squared term in the brackets of eqn (48), IM(t) becomes exponential, and k 1 can be determined from

(50)

Alternatively, one can use steady-state fluorescence intensities to obtain information about the cyclization process. For example, the ratio of excimer to pyrene fluorescence quantum yields is

cPE cP~kE( kl ) cPM = cP~ kM k E + L 1

(51)

where cP~ and cP~ are the quantum efficiencies of the pure components. Perhaps the most useful application of these measurements is in the study of polymers of different lengths in one solvent at a single temperature where (k E + k _ 1) do not vary. Then the ratio of I E/ 1M values for two chain lengths is equal to the ratio of their cyclization rate constants

(52)

Kinetics of TT Annihilation In order to observe intramolecular TT annihilation both chromophores at the chain ends must be excited to their triplet states.48a Since the pro babiJity of dou ble exci tation is low, the presence of singly excited species must be taken into account.

A*3-polystyrene-A *3

X

2 A-polystyrene-A * 3

Y

delayed fluorescence

klZl

-C;AA)~ Scheme 3

324 MITCHELL A. WINNIK

For the mechanism in Scheme 3, kT describes the decay rate of A * 3, k~2) the rate of bimolecular TT annihilation, and k 1 the cyc1ization rate. Mita 61a,b

has shown that, under these experimental conditions, the reaction rates are adequately described by the equations

d[X] - dt = (k 1 + 2kT) [X] (53)

- d~] = kT[y] + 2k~2)[YF - 2kT[X] (54)

where [X] and [Y] are the concentrations of doubly and singly excited polymers. Note that TT annihilation is neglected as an additional source of A * 3. Experimentally one measures either the decay of the intensity of delayed fluorescence IOF(t) or the decay of the absorption AT(t) oflight due to A*3.

IOF(t) '" k 1 [X] + k~2)[YF AT(t) '" 2[X] + [Y]

(55)

(56)

Solving eqns (53) and (54) and substituting the results into the expressions (55) and (56), one obtains for IOF(t) and AT(t)

IoF(t) '" Zlk1XO exp [-(kl + 2kT)t] + Z~k~2)YJ exp( -2kTt) (57)

AT(t) '" 2Z1XO exp [-(k 1 + 2kT)t] + Z2YO exp ( -kTt) (58)

where the pre-exponential terms also depend upon time

, {k(2) (kT+kl) 1+ :T (l-exp(-klt))

X[Yo+2XOkT~kl (l-eXP(-(kT+k1)t))]}

(59)

Here Xo and Yo are the initial concentrations of X and Yproduced by flash photolysis. One sees that the measured signals are not simply a sum of two

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 325

exponential terms but have a more complicated time dependence. At very long times, the kT dominates the AT(t) measurement, and bimolecular processes dominate the decay of delayed fluorescence. If k I is very much larger than kT and one examines the system in a time domain where Z I and Z2 remain reasonably constant with time, the two rate constants can be evaluated from examination of the short-time and long-time portions of the decay curves. 39 ,6Ia,b

Experimental Results Fluorescence spectra of pyrene-end-capped polystyrene are shown in Fig. 9 for two different chain lengths, Mn = 2900 and Mn = 15500. In both cyclo­hexane at 35-4 °C (Fig. 9(a» and in toluene at 22°C (Fig. 9(b» the shorter chain gives a more intense excimer emission than the longer chain. These results are in accord with the WF theory, which predicts that cyclization rates decrease with increasing chain length. 35 Comparing Fig. 9(a) and 9(b) one also notices that there is significantly more excimer emission in cyclohexane at 34·5 °C, a 8-solvent for polystyrene (PS), than in toluene, a good solvent for PS. One sees that chain length and solvent quality are important parameters which affect polymer cyclization.

Fluorescence decay experiments on these molecules also indicate that cyclization rates decrease strongly with increasing chain length. A similar conclusion can be drawn from TT annihilation experiments on anthracene tagged polystyrene.

WF theory applies rigorously only to 8-solvents. Consequently experiments at the 8-point, cyc10hexane at 34·5 0, for PS provide the most meaningful tests of the WF model.

8-Solvent For PS samples of M < 30000 in cyclohexane at 34'5°C, fluorescence decay and steady-state studies of pyrene excimer formation give essentially identical values of the cyclization rate constant <k I)' The angular brackets < ) emphasize that experimental values of a molecular weight sensitive quantity are averages over the MWD of the sample. For higher molecular weights, Al is nearly equal to kM ; eqn (50) gives <k l ) values with a large uncertainty. Under these circumstances, eqn (52) provides more reliable data. Measurements of lEI 1M become progressively more difficult for <k 1) values less than 2 or 3 x 104 S - I.

F or short chains where the 4k 1 k _ I term in eqn (48) is significant, I M(t) decays as a sum of two exponential terms. Under these circumstances, values of kE and k -I can also be determined. At 34'5°C these values are

326 MITCHELL A. WINNIK

100

(a ) 80

60

en -c: 40 :J

>-~

a ~

20 -.a ~

a

0 ~ 360 (iJ 100 z

( b) w f-Z 80

w u z 60 w u (f) W 0:: 40 0 :::> ~ lL

20

FIG. 9. Fluorescence spectra of Py-polystyrene-Py, 2 x 10- 6 M. (a) In cyclohexane at 34·5°C, Mn 2900 (upper curve) and Mn 15600 (lower curve). The curves are normalized at 378 nm. These spectra were run with 10 nm emission slits, and are uncorrected for the spectral response of the emission monochromator and photomultiplier tube; (b) in toluene at 22 DC, Mn 2900 (upper curve) and Mn 15600 (lower curve). The curves are normalized at 378 nm. These spectra were run with

2 nm emission slits, and are fully corrected.

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 327

3·5 X 106 s- 1 and 1·8 x 107 s- 1, respectively. They are independent of chain length, within experimental error. The former is very sensitive to temperature (Ea = 11 kcal mol-I) whereas kE depends only weakly upon temperature (E. = 1·5 kcal mol-I). From this data, an apparent binding energy of -!J.H 0 = 8 kcal mol- 1 can be calculated for the intramolecular pyrene excimer. This compares to a value of 8 kcal mol- 1 reported by Birks for bimolecular excimer formation from pyrene itself in cyclohexane. 62

The chain length dependence of <k 1) is shown in Fig. 10. Data from excimer formation give a linear log-log plot, with values bracketed by those predicted by eqn (22) as calculated by CP, 39 using the parameters a = 60 A and" = 0'23, and those predicted from eqn (27) using the literature values 63

of D = 1·21 X 10- 4 M- O'49 (cmZ s -1) and <hZ) = 4·9 x 10- 17 M(cmZ) for polystyrene in cyclohexane at 34·5°C. Equation (22) generates a slope of - 1·43 over the moelcular weight range indicated in Fig. 11, whereas the D/<h2) plot has a slope of - 1·49. If a first-order correction is applied to the experimental data to accommodate the fact that the polydispersity of two highest molecular weight samples is somewhat higher than that of other samples, the least squares slope is - 1·68 ± 0·10.

Also shown in Fig. 10 are data obtained from AT(t) measurements on TT annihilation experiments on A-PS-A in cyclohexane. Here, too, <k 1 )

decreases with increasing N, but the magnitude of these values is c. 30 times smaller than that obtained from excimer studies. A line of slope - 3/2 passes reasonably through the data for M < 105 ; including all the data, however, gives a smaller slope. The origin of the discrepancy between the two sets of data is not clear at this time.

o II

C~-POIYstY"ne--CH'CH' OC(CH,) ,

N / "­

CH 3 CH 3

12 DMAP-PS-Py

In order to provide an independent measure of PS cyclization rates and an assessment of the effect of the end group on the cyclization, Winnik, Sinclair and Beinert 16 examined a PS derivative 12 of M n = 11 000 (Mw/ Mn = 1·1). This polymer undergoes intramolecular exciplex for­mation between the Py* and the MezN-C6 H 4- groups. Exciplexes differ

328 MITCHELL A. WINNIK

cyclohexane 34.5°

o Py -polystyrene-Py*1 *3 *3 A A -polystyrene-A

• DMAP-~ystyrene-Py*l

2 Log N

FIG. 10. Log-log plot of (k 1) vs. N for polystyrene in dilute solution at the 8-temperature in cyc1ohexane: 0, Values determined from pyrene excimer formation in Py*l-polystyrene-Py, taken from Ref. 8; 6" values determined from A(t) measurements on A*3-polystyrene-A*3, taken from Ref. 61. The solid line is calculated from the expression k? ~ 1·98D/(h 2 ), the asymptotic limit for long

chains using data for PS in cyc10hexane at 34·S°c.63

330 MITCHELL A. WINNIK

The validity of this correction is substantiated by the finding that Y1o<k l )

values for 12 in benzene, toluene and tetrahydrofuran are within a few per cent of one another. 16 These data are presented in Fig. 11.

The <k I) values obtained from intramolecular excimer formation in toluene are substantially smaller than corresponding values obtained in cyclohexane. 6o The data appear to have a slight downward curvature in the log-log plot. <k I) values of 2 x 10 - 4 S - 1 are at the very limit of meaningful IE/1M values. The <k l ) values for the sample of M=41000 have a significant uncertainty, and the M = 105 sample has been deleted from the plot.

The normalized <k I) values from both transient triplet absorption and delayed fluorescence studies on 11 are comparable in magnitude to those obtained from excimer experiments on 10, and over a small range of M the data coincide. Treated alone, the AT(t) data show a significantly smaller slope.

According to the literature values 64 of D and for PS in toluene (D = 3·92 x 10- 4 M- 0 ' 574 (cm2 s -I); <h 2 ) = 9·36 x 10- 18 M 1' 19 (cm)) k 1 ~

D/<h2) should decrease as N- 1 ·76 . A line of this slope passes nicely through the excimer data and some of the TT annihilation data. This treatment assumes that the only consequence of excluded volume is expansion of the mean dimensions of the chain, so that <k I) is affected only by factors that affect D and <h 2 ). With this assumption, k? values can be calculated from eqn (22), since even in good solvents <h 2 ) = 6<R~) to within a few per cent.

Experimental values of <k 1 > in toluene are significantly smaller than values calculated 39 from k? = 1·83(D/<h2»). The difference is about a factor of 6. This difference suggests that another factor besides the change in chain dimensions is responsible for the decreases in cyclization rate in good solvents.

Solvent Effect The Correlation Hole. According to de Gennes,32a,b the end-to-end

distance distribution function W(h) has a minimum at the origin in the presence of excluded volume. The minimum arises from the tendency of polymer segments at or near the origin to repel other segments from their vicinity, including the other chain end. Such a minimum in a probability distribution function is often referred to as a 'correlation hole', The depth and breadth of the minimum depend upon the strength of the excluded volume interaction (solvent quality). 34 In a 8-solvent it disappears entirely: W(h) is Gaussian and has a maximum at h = O.

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 329

Mn 100000 200000

*1 Py -polystyrene-Py 0

toluene 22°

IcY

100 Chain Length N

FIG. II. Log-log plot of (k 1> vs. Nfor polystyrene in toluene at 22°C; 0, Values determined from pyrene excimer formation in Py*l-polystyrene-Py, taken from Ref. 8; values (benzene, 30°) corrected to toluene at 22°C determined from (6) A(t) and CA.) IDF measurements on A* 3-polystyrene-A* 3 , taken from Ref. 61(a),(b). The solid line is calculated from k?::::; 1·98D/(h 2 >, using data for PS in toluene at

22°C.

from excimers in that they dissociate into ions in polar solvents and kE is very solvent sensitive. The cyclization rate for this polymer (the point. in Fig. 10) is within 20 % of that predicted by interpolation of the <k 1 > values obtained from excimer studies.

Good Solvents Excluded volume effects should contribute to polymer cyclization in good solvents. TT annihilation experiments have been carried out in benzene, at 22 °C for AT(t) measurements and at 30°C for delayed fluorescence measurements. 61 Excimer and exciplex formation has been studied primarily in toluene at 22°C. 16,59,60 These various results can be compared by normalizing them to a common 17o/Tvalue, here that oftoluene at 22°C.

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 331

Two sets of experiments provide evidence for the role of the correlation hole in suppressing cyc1ization in good solvents. Both involve polymers sufficiently small (Mn = 11 000 and 2900) that a change in solvent has only modest effects on <h2 ). Intramolecular exciplex formation in DMAP­PSIIOOO-Py, 12, has been studied in five solvents at room temperature. 16

In THF, benzene and toluene, good solvents for PS, YJo<k 1 ) values are virtually identical (Fig. 12). In cyc10pentane (8 = 23°C) and in cyc10hexane

1.5 ~--r-----r-----,.---"'-----r--r---'--'

CH. 1.3

.CP

lu Q.l I/) 1.1

to 10

)(

a: a: 0.9 0 u

/'0..

~

"-/

0.7 T. :& •

THF 8

FIG. 12. Values of <k1>corr= <kl> (YJo!YJc H ) vs. solvent viscosity YJo for DMAP-PSlIOOO-Py in cyclohexane (CH), 'cyclopentane (CP), benzene (B), toluene (T) and tetrahydrofuran (THF). (Reproduced with permission from Ref. 16

by courtesy of the National Research Council of Canada. Ottawa.)

332

Q)

""

'I o

....

3

5

..lr:' 4

•

a.. °3

{:"" . " CH {:""2

U)

'0

• /CP

MITCHELL A. WINNIK

FIG. 13. Values of the rate constants for cyclization <k 1>' and ring opening (k -1)e, calculated from I E(t) data on Py-PS2900-Py, corrected to the viscosity of cyclopentane (CP), and plotted vs. the Hildebrand solubility parameter "H" Other solvents are: CH, cyclohexane; EA, ethyl acetate; T, toluene; THF, tetra-

hydrofuran; MEK, 2-butanone; A, acetone.

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 333

(8 = 34·5 0q, rJo<k I) values are significantly larger. These differences are too large to be explained only by changes in mean polymer dimensions.

A more compelling experiment involves solvent effects on room temperature cyclization of Py-PS290o-Py. In this case data are available in sufficient detail to permit both <k I) and k _ I to be evaluated. 65 It is unfortunate that the data are not as simple as one would like: IM(t) and IE(t) decays, particularly in poor solvents, give A. I, A.2 values that differ by 20 %, so that the actual <k I) and k _ I values one calculates depend upon which set of data one uses. Nevertheless, one finds from either set that not only is <k I) diminished in good vs. poor solvents, but k _ 1 is enhanced (Fig.l3). Thus their ratio <Key) = <kl)/L I is an order of magnitude larger in cyclopentane and acetone than in benzene, toluene or THF. Such results are consistent only with some special factor, the correlation hole, suppressing cyclization in good solvents.

Poly(vinyl acetate) Partial hydrolysis of poly(vinyl acetate) (PVAc) produces hydroxyl groups available for chemical reaction. Neighbouring group effects in that reaction enhance the probability of adjacent pairs of -OH groups, but at very low conversion their distribution ought to be statistical. These groups can then be reacted with pyrenebutyric acid to produce a polymer containing Py labels internal to the chain. Fluorescence studies of these materials provide information on internal chain dynamics.

13 X~IOO

Such experiments were reported by Cuniberti and Perico in 1980 and 1981 for PVAc 13 labelled with 1 % Py groups.66a,b They observed that

334 MITCHELL A. WINNIK

IE/1M increased with M, reaching an asymptotic level at M = 105. IE/1M increased linearly with '10 1 in methanol--ethylene glycol mixtures, both poor solvents for the polymer, whereas at low viscosities this plot was curved for ethyl acetate-glycerol triacetate mixtures. To the extent that cyclization depends upon coil expansion, as reflected in the intrinsic viscosity ['1] of the polymer, these differences should disappear in a plot of '1o['1]IE/IM vs. ['1] or '10· While this plot vs. '10 shows some scatter, the data are best fit by a line of zero slope.

Poly(ethylene oxide) (PEG) Hydroxy-terminated PEO is available commercially in high purity and narrow MWD. Hence it is a natural choice for studies of cyclization in which both end groups are the same. Shimada and Szwarc 67 pre­pared oligo-PEO derivatives with phthalimido groups on each end; PI-CH2CHiOCH2CH2)n_l-PI. As in the case of the corresponding n-alkane derivatives, they prepared the anion radical by metal reduction and used ESR techniques to follow the rate of electron exchange, viz.

(PI-;- PI3 M+

14 (61)

PI--CH2CHz{OCH2CH2)n-t-PI-;-, M +

These rate constants were obtained as a function of chain length and temperature. While the k(2) values for the model reaction are large (c. 108 M - 1 S - 1), the intramolecular reactions are not diffusion controlled as originally reported. The rates are not decreased by increasing solvent viscosity. It appears that the rate-limiting step is not electron exchange, but rather counter-ion exchange accompanied by solvent reorganization.

0(-)

$ Mandolini and Illuminati 68 have examined the kinetics of ring closure

reactions by SN2 mechanisms of oligo-PEO derivatives (eqn (62». The rates were followed by means of a bromide sensitive electrode which measured

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 335

the rate of Br - release. The corresponding bimolecular reaction has a k(2)

value 15 orders of magnitude slower than the diffusion-controlled reaction. These reactions are governed by chain statistics. Calculated Co values indicate that transannular interactions, which suppress formation of common- to medium-sized polymethylene rings, are absent here.

o N~H2CH2CH20-(CH2CH20)n--CH2CH}O--@--N02

14

Cyclization kinetics under conformational control were also examined by Sisido et al.,69a,b,c who prepared the polymer 14 and studied the kinetics of intramolecularly catalysed ester hydrolysis in water at pH 6·1. Samples were examined with n = 5-38. Co values were calculated and these are compared in Fig. 14 to those of polysarcosine cyclization. Co goes through a maximum at n = 8 and declines thereafter. It is noteworthy that cyclization is less probable in PEO than polysarcosine. Monte Carlo-RIS model calculations were also carried out for non-intersecting PEO chains, with cyclization detected for chain ends within capture radii of R = 6 A and 4 A. Cyclization of PEO was found to be favoured over poly sarcosine for n < 8, but to be significantly more difficult for chains of n = 10-20. Unfortunately it is not possible to extract a ring closure exponent from these data.

o 0 II II

Py(CH2)3COCH2CH2-(OCH2CH2)n_I-OQCH2)3Py

Py-PEO-Py 15

Diffusion-controlled cyclization of PEO was first reported in 1977 by Cuniberti and Perico. 14 They prepared Py-PEO-Py from commercial samples of PEO and I-pyrenebutyric acid using acid-catalysed esterifi­cation. These polymers were studied by steady-state fluorescence spectroscopy in air-saturated solution, and relative <k I> values were obtained from measurements of IE/1M , These were the first experiments to examine diffusion-controlled cyclization oflong polymers. In a more recent paper, they used an absolute (k I> value reported by Cheung, 70 obtained by fluorescence decay measurements on Py-PE09000-Py, to calibrate their IE/1M scale. A log-log plot of their values of (k 1 > vs. chain length is shown in Fig. 15.

Cheung, Redpath and Winnik 70 used a combination of steady-state and

i

N 51

.. ..

N

"c 2 00

PO

E(2

S·)

PS

" -..,

10

a

! N

51 .. ..N

r

........ ''I

t __

__

__

._ 30

n

=m

-2

3r,--------,---------~--------r_----_,

• \ \ \ \ \ -'

~ \

.. \

/1 \

/. ~

\PS

(25

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E

\ 1~ \

(Rin

g-C

ha

in)

\ I.

\

,,~

\,

. .

, ~

.,

b

. ,

~ ' ..

...

D

..... - .-

15

10

5

~

I _ ' _

' ~ '0

l " .i >-

FIG

. 14

. C

ompa

riso

n o

f th

e ca

lcul

ated

(a)

and

the

obs

erve

d (b

) ef

fect

ive

conc

entr

atio

n C

o =

kl

/k2

for

intr

amol

ecul

arly

cata

lyse

d hy

drol

ysis

of N

32

)- P

Oly

mer

-C0

2--@

-N

02

: P

OE

ref

ers

to p

olym

ers

= -

{C

H2C

H20

).-;

PS

refe

rs t

o

poly

mer

= -

{C

OC

H2N

(CH

3))

n-'

In

(a)

the

data

for

n =

4 a

nd 5

are

dec

reas

ed b

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x 10

-2 M

for

cla

rity

. Tak

en f

rom

Ref

. 69

. T

he c

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E (

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ts t

he y

ield

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omer

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tain

ed i

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atio

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f eth

ylen

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epor

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2

\H

\H

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z ~

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 337

+0.5

0> o -.J

o

-0.5

1.0

PEO Molecu lar

Py-POLY(ETHYLENE OXIDE)-Py

Log N

5.0

2.0

1'-0 1.0 -

FIG. 15. A plot of log [(4) ON) - I) vs.log Nfor Py---{polyethylene oxide)--Py in tetrahydrofuran solution, using data from Cuniberti and Perico. 14 These values are proportional to the cyc1ization rate constant k 1. The absolute value for k 1 C&.) shown was obtained from fluorescence decay measurements as reported by

Cheung.7o

fluorescence decay techniques to obtain <k 1 ) values for a single polymer sample, Py-PE09000-Py in a variety of solvents. Their results confirm that for most solvents <k 1) is proportional to 1'/0 1 as predicted for diffusion-controlled processes. For water and methanol, both IJ 1M and apparent <k 1) values were larger than one would expect. Furthermore, in the IE(t) measurements, significant excimer was seen at t = O. They interpreted these observations in terms of partial association of the pyrene groups due to unfavourable interactions with solvent.

Poly(dimethylsiloxane) (PDMS) The most difficult aspect of the study of PDMS cyclization has been the synthesis of appropriately end-labelled chains. Winnik and his co­workers 73 carried out the synthesis of l-pyrene-(CHz)sSiMezCI, which

338 MITCHELL A. WINNIK

NUMBER OF BONDS

100 200 500 1000

2 a

Id

b 5

IU 3

c Q.l

VI 2

"" .x

"" 106

5

3

2

2.0

FIG. 16. Log-log plot of the mean cyclization rate constant (k 1) vs. mean chain length N. Curve a: data from these experiments for Py-PDMS-Py in toluene at 22°C (e, from IE/1M ; ., from fluorescence decay); the line drawn through the points has a slope of - 3/2. Curve b: summary of previous results for Py-PS-Py in cyclohexane at 34· 5°C; the experimental slope is - I· 52. Curve b': data of curve b corrected ('lo/T) to 22 °C and the viscosity of toluene. Curve c: summary of previous results for Py-PS-Py in toluene at 22°C. (Reproduced with permission from Ref.

73 by courtesy of Butterworth Scientific Ltd, Guildford, UK.)

SPECTROSCOPIC STUDIES OF CYCLIZA TION DYNAMICS AND EQUILIBRIA 339

could be attached to the ends of commercial HO(SiMe20)nOH to give pyrene end-labelled polymer, Py-PDMS-Py. Since the commercial polymer has a broad MWD, samples were fractionated on analytical GPC columns to give milligram amounts of fractions, ranging from Mn = 5000 to Mn = 33 000 with Mw/Mn ~ 1·1.

Intramolecular excimer formation was studied in toluene at 22°C, using both steady-state and fluorescence decay techniques. Results were comparable within experimental error. Rate constants for diffusion­controlled cyclization were calculated from the data. A plot oflog <k 1> vs. log N is shown in Fig. 16, and a line of slope - 3/2 is drawn through the data. Although toluene is not a 8-solvent for PDMS (the Mark-Houwink exponent = 0·68), no curvature in the log-log plot is apparent in the data. The experimental slope is consistent with the predictions of WF theory.

TABLE 2 Comparison of <k1 ) Values for Different Polymers in Toluene at 22°C

Py-pol y( dimethylsiloxane)-Py Py-poly(ethylene oxide)-Py Py-polystyrene-Py

25800 9600

33600

N

692 655 650

1-4 X 106

8·3 X 105

9·0 X 104

According to the CP model,39 PDMS chains are predicted to cyclize a factor of two faster than PS chains of comparable length. Such a comparison of experimental results should be made only for data carried out in 8-solvents. Unfortunately <k 1 >e data for PDMS are not yet available. Qualitatively, however, one can compare <k 1> values for PDMS in toluene with corresponding values for PS of similar length in both cyclohexane at 34·5 °C and toluene at 22°C, since the former is a 8-solvent and the latter is a better solvent for PS than for PDMS. Presumably, a solvent of poorer solvency for PS would give <k 1> between these limiting values. After correcting for solvent viscosity and temperature differences, the PDMS cyclization rate is comparable to twice the PS rate in an intermediate solvent. Part ofthis increase is due to the smaller <h 2> value of PDMS than PS. The remainder is due to its greater dynamic flexibility.

In this qualitative spirit, values of <k 1> are compared in Table 2 for various polymers having c. 660 bonds between their chain ends.

340 MITCHELL A. WINNIK

EXPERIMENTS IN CONCENTRATED SOLUTION

One of the great virtues of labelled-chain experiments is that one can study the behaviour of marked chain in the presence of a vast excess of unlabelled chains. In consequence one can study the properties of individual polymer molecules over the entire range of very dilute to very concentrated polymer solutions without interference from intermolecular interactions. Many phenomena are open to study in this way. These include screening of excluded volume effects in experiments sensitive to chain conformation, as well as screening of hydrodynamic interactions and entanglement effects in measurements of chain dynamics.

To date, only three studies have appeared which examine the consequences of macromolecule concentration on the cyclization of labelled polymers. These have involved pyrene excimer formation, end-to­end cyclization for polymers with Py- groups on the chain ends, and internal cyclization for polymers containing Py- groups along the chain backbone. In none of these experiments are the chains long enough, or the solution sufficiently concentrated, for entanglement effects to be important.

Poly(vinyl acetate) Addition of unlabelled PV Ac to dilute solutions of the labelled polymer 13 causes changes in lEI 1M • 30.39 The nature of these changes differs in poor solvents (methanol, 20°C) and good solvents (THF, 20°C). As the lower line in Fig. 17 indicates, there is an increase in lEI I M for 13 in THF for PV Ac concentrations above c ~ 0·02 g ml- 1 which then levels off at higher concentrations. In the poor solvent, one observes first a decrease in lEI I Min this PV Ac concentration range followed by an increase and a subsequent decrease at higher concentrations. The initial changes occur at con­centrations approximately equal to c* = MI N A(2Ro)3 for the unlabelled polymer.

In both poor and good solvents, one expects the screening of hydrodynamic interactions to increase the net friction felt by the chain, causing the mean cyclization rate k to decrease. In good solvents, this retardation is compensated by excluded volume screening, which decreases the mean separation of Py groups and suppresses correlation hole effects. From the net increase in IJ 1M for 13 in THF, it appears that the latter effects predominate. Unfortunately, there seems to be no simple explanation at this time for the more complicated behaviour of 13 + PV Ac in a poor solvent.

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 341

1.5

0.5

0.05 0.10 0.15

( ,g/ml

FIG. 17. The excimer-to-monomer intensity ratio FE/ F M for a sample of poly(vinyl acetate) of M ~ \05, containing 1% pyrene groups, as a function of the concentration of unlabelled PVAc at 20 0e in methanol (0, e) and tetra­hydrofuran (1:';). The unlabelled PVAc samples had (0, 1:';) M = 1·8 X \05 and (e) M = 2·2 X \06. (Reproduced with permission from Ref. 30 by courtesy of John

Wiley & Sons, Inc., New York.)

Polystyrene Cyclization rate constants have been obtained74 for Py-polystyrene-Py samples of Mn = 9200 and 25000 as a function of unlabelled PS concentration for weight fractions of PS up to 0·6. As in the case of PVAc, the concentration dependence of <k 1) depends sensitively on the quality of the solvent (Fig. 18). In cyclohexane just above the 8-temperature (35°C), one observes a monotonic decrease in <k 1) by a factor of c.4 for P:' -PS9200-Py, and a factor of c. 8 for Py-PS25000-Py. These results are consistent with screening of hydrodynamic interactions.

In toluene, a good solvent for PS, <k 1) is substantially slower at infinite dilution than in a 8-s01vent, a fact which was attributed to the effect of the correlation hole on cyclization. At increased PS concentrations, <k 1) decreases only modestly, so that above c::;:::200mgml-t, <k 1 ) values in good and poor solvents are comparable. This demonstrates the compensatory nature of excluded volume and hydrodynamic screening.

342 MITCHELL A. WINNIK

4 a

3

;-<II

II 2 .:.< v

.j) I

Q

o~----~----~----~----~----~~~ o

8

6

II .:.<4 V

III

Q

01

2 Toluene. 22°C

02 03 04 05

b

o~----~----~----~----~----~--~ o 01 02 03 04 05

Weight fraction PS

FIG. 18. A plot of <k I > vs. the weight fraction of PS: (a) Test chain of M n = 9200 in the presence of unlabelled PS of Mn = 17500(0, .), Mn = 860000(0, .), and HO~PS~OH of Mn = 8900 (MwiMn = 1·3) (.6., ... ); (b) test chain of Mn = 25 000 in the presence of PS of M n = 17500 (0, '\7). Open points refer to steady-state measurements; closed points to fluorescence decay measurements. (Reproduced with permission from Ref. 74 by courtesy of Butterworth Scientific Ltd, Guildford,

UK.)

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 343

An interesting feature of these experiments is that <k 1> depends only upon c and not upon the molecular weight of unlabelled PS. This stands in curious contrast to the results for PV Ac in methanol (Fig. 17).

x ~ y ~ 14 15

Because of the difficulties involved in interpreting experiments on polymers with a distribution of pyrene separations along the chain contour, Winnik et al. developed a synthesis of polystyrene containing evenly spaced pyrene groups, 15. 75 This polymer was prepared by condensing HOCH 2CH2-polystyrene-CH2CH20H (Mn = 3100) with I-pyrene­methylmalonic acid. Intramolecular excimer formation in 15 was studied as a function of solvent quality in dilute solution,75 and as a function of unlabelled PS concentration. 76 It is these latter experiments which are of interest here.

Figure 19 presents results of I E/ 1M measurements for a sample of 15 of M n ~ 14000 (n ~ 4) in three solvents for unlabelled PS concentrations up to a weight fraction of 0·7. Even for these low molecular weights (M( PS) = 17500) the samples of highest concentrations are rubbery solids that barely flow on the time scale of weeks. In these experiments 15 is present at a concentration of 10ppm.

One sees in Fig. 19 that, in a 8-solvent, IE/1M for 15 decreases monotonically with increasing c, mimicking the behaviour of end-labelled polymers. In the good solvent toluene, IE/1M at [PS] = 0 is substantially reduced over that in cyclohexane, but is relatively uninfluenced by even very large amounts of added PS. Increases in bulk solution viscosity of several orders of magnitude reduce IE/1M for 15 in a 8-solvent by only a factor of c. 3 and barely affect the internal cyclization rate of 15 in a good solvent.

In 2-butanone, a modest solvent for PS, an intermediate behaviour is observed. There is a pronounced decrease in IE/1M at low c, followed by a plateau region at intermediate c. At high PS concentrations I JIM decreases

344 MITCHELL A. WINNIK

1.5 -(TPS3000-k 22°

cyclopentone Py .

:::!: H o toluene

"" W o 2-butanone H

O~ __ ~~~ __ -L __ ~ __ -L __ ~~~L-~~ __ L-~ o W 00 100

WEIGHT PERCENT PS 17500

FIG. 19. Plot of lEI 1M vs. weight per cent of polystyrene (Mn = 17500) in toluene (0), a good solvent, and 2-butanone (0), a modest solvent, at 22°C. The thin solid line presents the data for cyc\opentane, a 8-solvent, at 22°C. (Reproduced with permission from Ref. 76 by courtesy of the American Chemical Society,

Washington, DC.)

once more, with values superimposing on those of 15 in toluene at similar PS concentrations. One has rather detailed insights into the interplay of excluded volume and hydrodynamic interactions on internal chain dynamics, and the screening effects of added unlabelled polymer.

CONCLUSION

Spectroscopic techniques provide powerful methods for studying cycli­zation processes. One requires polymers of the form A - - - - D, where proximity of A and D (or A * and D) give rise to an observable event. Many such pairs of groups exist; their choice represents one of the most delicate aspects of the experimental approach. One must know the mechanistic details of the AID interaction. Interpretation of the experimental results requires the data from the intramolecular reaction to be compared with

SPECTROSCOPIC STUDIES OF CYCLIZATION DYNAMICS AND EQUILIBRIA 345

corresponding data for the bimolecular reaction of suitable - - A and - - D molecules.

The power of these spectroscopic techniques derives from their sensitivity and versatility. One can study fast processes which are diffusion controlled and slower processes which depend upon a conformation equilibrium or pre-equilibrium. Molecules can be examined at very low concentrations. Perhaps that most important application, and the least explored, is the study of traces of labelled chains in semi-dilute and concentrated polymer solution. These experiments give one the power to examine the effect of the environment on the cyclization of individual chains.

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Japan, 50 (1977) 1807-12. 3. Morawetz, H. and Zimmering, P. E., J. Phys. Chem., 58 (1954) 753-6. 4. Lakowicz, J. R., Principles of Fluorescence Spectroscopy, Plenum Press, New

York, 1983. 5. (a) Okamoto, A., Toyoshima, K. and Mita, I., Eur. Polym. J., 19(1983) 341-6.

(b) Mita, I., Toyoshima, K. and Okamoto, A., Eur. Polym. J., 19 (1983) 657-60.

6. Birks, J. B., Photophysics of Aromatic Molecules, Wiley-Interscience, New York, 1971, Chapter 10.

7. DeSchryver, F. c., Boens, N. and Put, J., Adv. Photochem., 10(1977) 359-465. 8. Redpath, A. E. C. and Winnik, M. A., Ann. NY Acad. Sci., 366 (1981) 75-92. 9. Mita, I., Ann. NY Acad. Sci., 366 (1981) 62-75.

10. (a) Badley, R. A., Modern Fluorescence Spectroscopy, Vol. II (ed. E. L. Wehry), Plenum Press, New York, 1976, Chapter 3. (b) Chen, R. F. and Edelhoch, H., Biochemical Fluorescence, Vois I and II, Marcel Dekker, New York, 1975. (c) Cundall, R. B. and Dale, R. E., Time-Resolved Fluorescence Spectroscopy in Biochemistry and Biology, Plenum Press, New York, 1983.

11. Beavan, S. W., Hargreaves, J. S. and Phillips, D., Adv. Photochem., 11 (1979) 207-303.

12. Sisido, M., Mitamura, T., Imanishi, Y. and Higashimura, T., Macromolecules, 9 (1976) 316--19.

13. Illuminati, G. and Mandolini, L., Accounts Chem. Res., 14 (1981) 95-102. 14. Cuniberti, C. and Perico, A., Eur. Polym. J., 13 (1977) 369-74. 15. (a) Zachariasse, K. and Kuhnle, W., In: The Exciplex (eds M. Gordon and

W. A. Ware), Academic Press, New York, 1974. (b) Zachairasse, K. and Kuhnle, W., Z. Phys. Chem. (Wiesbaden), 101 (1976) 267-73.

16. Winnik, M. A., Sinclair, A. M. and Beinert, G., Can. J. Chem., 63 (1985) 1300-7.

346 MITCHELL A. WINNIK

17. (a) Horie, K. and Mita, I., Polym. J., 9(1977) 201-8. (b) Mita, I., Horie, K. and Takeda, M., Macromolecules, 14 (1981) 1428-33.

18. (a) Winnik, M. A., Accounts Chem. Res., 10 (1977) 173-9. (b) Mar, A., Fraser, S. J. and Winnik, M. A., J. Amer. Chem. Soc., 104 (1981) 4941-3.

19. (a) Haas, E., Wilcheck, M., Katchalski-Katzir, E. and Steinberg, I. Z., Proc. Nat. Acad. Sci., 72 (1975) 1807-11. (b) Haas, E., Katchalski-Katzir, E. and Steinberg, I. Z., Biopolymers, 17 (1978) 11-31.

20. (a) Worsfold, D. J., J. Polym. Sci. Polym. Chem. Ed., 21 (1983) 2271-6. (b) Black, P. E. and Worsfold, D. J., ibid., 19 (1981) 1841-6.

21. (a) Shimada, K., Moshuk, G., Connor, H. D., Caluwe, P. and Szwarc, M., Chem. Phys. Lett., 14 (1972) 396-401. (b) Shimada, K. and Szwarc, M., J. Amer. Chem. Soc., 97 (1975) 3313-19.

22. Flory, P. J., Statistical Mechanics of Chain Molecules, Wiley-Interscience, New York, 1969.

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Prog. Reaction Kinetics, 1 (1961) 197-214. (c) Saitiel, J., Shannon, P. T., Zafiriou, O. C. and Uriarte, A. K., J. Amer. Chem. Soc., 102 (1980) 6799-808.

25. (a) Mozumder,A., J. Chem. Phys., 69(1978) 1384-91. (b)Andre,J. c., Bouchy, M. and Ware, W. R., J. Photochem., 14 (1980) 177-97.

26. Sisido, M., Imanishi, Y. and Higashimura, T., Macromolecules, 9 (1976) 320-4, 389-95.

27. (a) Winnik, M. A., Trueman, R. E., Jackowski, G., Saunders, D. S. and Whittington, S. G., J. Amer. Chem. Soc., 96 (1974) 4843-8. (b) Fraser, S., Saunders, D. and Winnik, M. A., Eur. Polym. J., 19 (1983) 991-6.

28. (a) Kuhn, W., Kolloid Z., 68 (1934) 2-15. (b) Jacobson, H. and Stockmayer, W. H., J. Chem. Phys., 18 (1950) 1600-6.

29. Flory, P. J. and Semlyen, J. A., J. Amer. Chem. Soc., 88 (1966) 3209-12. 30. Cuniberti, c., Musi, L. and Perico, A., J. Poly. Sci. Poly. Lett., 20 (1982)

265-72. 31. Brereton, M. G. and Rusli, A., Polymer, 17 (1976) 395-8. 32. (a) de Gennes, P. G., Scaling Concepts in Polymer Physics, Cornell University

Press, Ithaca, New York, 1979. (b) de Gennes, P. G., Macromolecules, 9(1976) 587-93. (c) des Cloizeaux, J., J. Phys. (Paris), Col/oq., C2 (1978) 135-42.

33. Martin, J. L., Sykes, M. S. and Hioe, F. T., J. Chem. Phys., 46(1967) 3478. 34. Oono, Y. and Freed, K. F., J. Phys. A. Math. Gen., 15 (1982) 1931-50. 35. Wilemski, G. and Fixman, M., J. Chem. Phys., 60 (1974) 866-77, 878-90. 36. (a) Doi, M., Chem. Phys., 9 (1975) 455-66. (b) Sunagawa, S. and Doi, M.,

Polym. J., 7 (1975) 604-12; 8 (1976) 239-46. 37. (a) James, C. and Evans, G. T., J. Chem. Phys., 76 (1982) 2680-6. (b) Nairn,

J. A. and Braun, C. L., J. Chem. Phys., 74 (1981) 2441-9. 38. (a) Perico, A. and Battezzati, M., J. Chem. Phys., 75 (1981) 4430-4.

(b) Battezzati, M. and Perico, A., J. Chem. Phys., 74(1981)4527-32; 75(\981) 886-91. (c) Szabo, A., Schulten, K. and Schulten, Z., J. Chem. Phys., 72 (1980) 4350-7.

39. Cuniberti, C. and Perico, A., Prog. Polym. Sci., 10 (1984) 271-316. 40. (a) Rouse, P. E., J. Chem. Phys., 21 (1953) 1272-80. (b) Zimm, B. H., J. Chem.

Phys., 24 (\956) 269-78.

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41. Yamakawa, H., Modern Theory of Polymer Solutions, Harper and Row, New York, 1971.

42. Perico, A. and Cuniberti, c., J. Polym. Sci. Polym. Phys. Ed., 15 (1977) 1435-50.

43. Sisido, M. and Shimada, K., Macromolecules, 12 (1979) 790-2, 792-5. 44. (a) Goodman, N. and Morawetz, H., J. Polym. Sci. Part C,31(l970) 177-92;

(b) Goodman, N. and Morawetz, H., J. Polym. Sci. Part A-2, 9(1971) 1657--68. (c) Cho, J. R. and Morawetz, H., Macromolecules, 4 (1973) 628-31.

45. (a) Stoll, M. and Rouve, A., Helv. Chim. Acta., 17 (1934) 1283-8; 18 (1935) 1087-25. (b) Morawetz, H. and Goodman, N., Macromolecules, 3 (1970) 699-702.

46. Halpern, A. M., Legenze, M. W. and Ramachandran, B. R., J. Amer. Chern. Soc., 101 (1979) 5736-43.

47. (a) Simon, J. D. and Peters, K. S., J. Amer. Chern. Soc., 103(1981) 6403--6; 104 (1982) 6542-7. (b) Hub, W., Schneider, S., Dorr, F., Oxman, J. D. and Lewis, F. D., J. Amer. Chern. Soc., 106 (1984) 701-8, 708-15. (c) Weller, A., Z. Phys. Chern. (Wiesbaden), 101 (1976) 371-90; 130 (1982) 129-38; 133 (1982) 93-8.

48. (a) Birks, J. B., Photophysics of Aromatic Molecules, Wiley-Interscience, New York, 1971, Chapter II. (b) Berlman, I. B., Energy Transfer Parameters of Aromatic Compounds, Academic Press, New York, 1973.

49. Szwarc, M., personal communication. 50. Winnik, M. A., Basu, S. N., Lee, C. K. and Saunders, D. S., J. Amer. Chern.

Soc., 98 (1976) 2928-35. 51. Mar, A. and Winnik, M. A., Chern. Phys. Lett., 77 (1981) 73-6. 52. Fraser, S. F. and Winnik, M. A., J. Chern. Phys., 75 (1981) 4683-95. 53. (a) De Kanter, F. J. J., Sagdeev, R. Z. and Kaptein, R., Chern. Phys. Lett., 58

(1978) 334-9. (b) De Kanter, F. J. J. and Kaptein, R., Chern. Phys. Lett., 58 (1978) 340-5.

54. (a) De Kanter, F. J. J., der Hollander, J. A. and Kaptein, R., Mol. Phys., 34 (1977) 857-74. (b) De Kanter, F. J. J., Kaptein, R. and Van Santen, R. A., Chern. Phys. Lett., 45 (1977) 575-9. (c) Closs, G. L. and Doubleday, C. E., J. Amer. Chern. Soc., 95 (1973) 2736-7.

55. Kanaya, T., Goshiki, K., Yamamoto, M. and Nishijima, Y., J. Amer. Chern. Soc., 104 (1982) 3580-7.

56. Sisido, M., Takagi, H., Imanishi, Y. and Higashimura, T., Macromolecules, 10 (1977) 125-30.

57. (a) Sisido, M., Kanazawa, Y., Imanishi, Y. and Higashimura, T., EUCHEM Conference 'Ring Closure Reactions and Related Topics', Rome, August 1978, C28. (b) Sisido, M., Kanazawa, Y. and Imanishi, Y., Biopolymers, 20 (1981) 653--63.

58. Sisido, M., Imanishi, Y. and Higashimura, T., Macromolecules, 12 (1979) 975-80.

59. (a) Winnik, M.A.,Redpath,A. E. C. and Richards, D. H., Macromolecules, 13 (1980) 328-35. (b) Redpath, A. E. C. and Winnik, M. A., J. Amer. Chern. Soc., 102 (1980) 6869-71.

60. Winnik, M. A., Redpath, A. E. c., Paton, K. and Danhelka, J., Polymer, 25 (1984) 91-9.

348 MITCHELL A. WINNIK

61. (a) Ushiki, H., Horie, K., Okamoto, A. and Mita, I., Polym. J., 13 (1981) 191-200. (b) Horie, K., Schnabel, W., Mita, I. and Ushiki, H., Macromolecules, 14 (1981) 1422-8.

62. Birks, J. B., Rep. Prog. Phys., 38 (1975) 903-74. 63. Brandrup, J. and Immergut, E. H. (Eds), Polymer Handbook, 2nd edn, John

Wiley, New York, 1975, Section IV. 64. Raczek, J. and Meyerhoff, G., Proceedings of the 27th International

Symposium on Macromolecules, Strasbourg, France, July 1981, pp. 700-3. 65. Winnik, M. A., Li, X. B. and Guillet, J. E., J. Polym. Sci. Polym. Symp., in

press. 66. (a)Cuniberti, C. and Perico, A., Eur. Polym. J., 16 (1980) 887-93. (b) Ann. NY

Acad. Sci., 366 (1981) 35-43. 67. Shimada, K., Shimozato, Y. and Szwarc, M., J. Amer. Chem. Soc., 97 (1975)

5834-41. 68. Illuminati, G., Mandolini, L. and Masci, B., J. Amer. Chem. Soc., 103 (1981)

4142-5. 69. (a) Takagi, H., Sisido, M., Imanishi, Y. and Higashimura, T., Bull. Chem. Soc.

Japan, 50 (1977) 1807-12. (b) Sisido, M., Imanishi, Y. and Higashimura, T., Bull. Chem. Soc. Japan, 51 (1978), 1469-72. (c) Sisido, M., Yoshikawa, E., Imanishi, Y. and Higashimura, T., Bull. Chem. Soc. Japan, 51 (1978) 1464--8.

70. Cheung, S. T., Redpath, A. E. C. and Winnik, M. A., Makromol. Chem., 183 (1982) 1815-24.

71. Page, M. I., Chem. Soc. Rev., 2 (1973) 295-323. 72. Davies, H. W. and Schwarz, M., J. Org. Chem., 30 (1965) 1242. 73. Redpath, A. E. C., Svirskaya, P., Danhe1ka, J. and Winnik, M. A., Polymer, 24

(1983) 319-22. 74. Redpath, A. E. C. and Winnik, M. A., Polymer, 24 (1983) 1286-90. 75. Winnik, M. A., Li, X. B. and Guillet, J. E., Macromolecules, 16 (1983) 992-5. 76. Winnik, M. A., Li, X. B. and Guillet, J. E., Macromolecules, 17 (1984)

699-702. 77. Zachariasse, K. A., Duveneck, G. and Busse, R., J. Amer. Chern. Soc., 106

(1984) 1045-51.

CHAPTER 10

Cyclization, Gelation and Network Formation

S. B. Ross-MURPHY Unilever Research, Colworth Laboratory, Bedford, UK

and

ROBERT F. T. STEPTO

Department of Polymer Science and Technology, University of Manchester Institute of Science and Technology,

Manchester, UK

INTRODUCTION

Earlier chapters in this volume have considered the properties of cyclic polymers, which are formed principally from linear polycondensations or random polymerizations (as defined in Ref. I) under conditions of chemical equilibrium. Such polymerizations have been used to good advantage, particularly by Semlyen and collaborators (see earlier chapters), for the preparation of cyclic polymers and the study of the cyclization probabilities for linear chains. However, most preparative polymerizations are carried out irreversibly so that the population of cyclic species increases in amount and to a lesser extent in average molecular size, for example M, <S2), as a polymerization proceeds. In addition, there is a sharp contrast between cyclization in linear and non-linear random polymerizations. In the former, the number of reactive groups per molecule remains at two throughout a polymerization. In the latter, the average number increases without limit as the gel point is passed and a network is formed, with resulting greatly increased opportunities for cyclization.

349

350 S. B. ROSS-MURPHY AND ROBERT F. T. STEPTO

In terms of cyclization and network formation, one may delimit three types of polymerization process: the random, non-linear polymerizations already mentioned; non-linear addition or sequential polymerizations; and vulcanization or the cross-linking of preformed, high molar-mass polymer chains. All three types are used extensively for the preparation of elastomeric and glassy network polymers. However, only in the case of random polymerizations have reaction systems been devised which allow the clear investigation of the respective opportunities for intramolecular versus intermolecular reaction. They undergo intermolecular reaction essentially in a spatially and chemically random fashion and the gel point occurs well into the polymerization reaction so that it may easily be varied and monitored. Accordingly, most experimental and theoretical investi­gations have used random polymerizations and this chapter is concerned mainly with results from such systems.

GELATION IN THE ABSENCE OF INTRAMOLECULAR REACTION

The gel point, i.e. that of incipient network formation, is a key point in the formation of networks. 2 Because, in non-linear polymerizations, the average number of opportunities for reaction per molecule grows without limit, a point in the polymerization process is reached where species of essentially infinite molar mass, i.e. limited only by the macroscopic size of the polymerization mixture, are formed. Given the random reaction between reactive groups or sites, the more complex molecules grow at the expense of the smaller ones because of their larger number of opportunities for reaction. At some critical conversion (the gel point) the molar-mass distribution, as measured by M wi M n' becomes infinitely broad. From this point onwards, the polymerizing mixture consists of a gel fraction (of weight fraction W g) of species of infinite molar mass and a sol fraction (of weight fraction ws) of species of finite molar mass. The former increases at the expense of the latter, giving an increasing proportion of infinite species or network material. In the idealized case, the gel fraction (Wg) = 1 when all groups have reacted. This situation may be closely approached in practice, provided that the exact stoichiometry of reactive groups is achieved, the polymer is formed above the Tg of the final product, which can increase as a polymerization proceeds, 3 and cyclization amongst species does not lead to a sol fraction consisting of molecules with no reactive ends.

The phenomenon of gelation in the absence of intramolecular reaction is

CYCLIZATION, GELATION AND NETWORK FORMATION 351

well-understood in terms of the Flory-Stockmayer theory 2,4 and the resulting gel point can be used as a reference point for the consideration of the effects of intramolecular reaction, At the gel point, the species of infinite molar mass have a tree-like structure permeating through the whole reaction mixture. The critical conversion occurs when there is a non-zero probability that a randomly chosen chain continues to infinity. 1,2,4 Given the previously mentioned random reaction (or equal reactivity) of like functional groups or sites, the gel point may be quite generally predicted.

F or the self-polymerization of RA f monomers, the equation for Xw is 2

l+p x =----

w 1-(f-1)p (1)

where p is the extent of reaction of A groups, and gelation occurs when p =Pc' with

Pc(f -1) = 1 (2)

In a random RA2 + RB f polymerization, the type which has been most studied experimentally, the expression for the weight-average degree of polymerization (xw) at extents ofreactionp. of A groups andpb ofB groups is 4

(3)

It can be seen that for a stoichiometric mixture, in which P. = Pb = I at complete reaction, the second factor in the denominator becomes equal to zero and Xw --+ 00 before complete reaction, provided!. or!b is greater than 2. The factor is equal to zero when

(4)

where ()(c = (PaPb)c, the critical value of the product of extents of reaction at gelation. For non-stoichiometric mixtures(r = c.O/cbO = pJpc =1= 1) gelation can occur provided eqn (2) is satisfied before the minority group is completely consumed. That is, for

(5)

Equation (3) also holds for a linear random polymerization (f. =!b = 2); however, x --+ 00 only for a stoichiometric mixture and then only at complete reaction.

At that point, Xn is also infinite and xw/xn is finite. For non-linear polymerizations, Xn is finite and Xw becomes infinite when ()( = ()(c. Thus,

352 S. B. ROSS-MURPHY AND ROBERT F. T. STEPTO

eqns (I) and (3) summarize the essential difference between linear and non­linear polymerizations. In the latter, increase in the number of opportunities for reaction per molecule as a !polymerization proceeds leads to an 'infinitely broad' distribution of molar masses.

For the general polymerizing mixture of a variety of reactants bearing A and B groups separately, eqn (4) becomes1,4,5

rJ.c(faw -l)(fbw - I) = 1 (6)

where faw and fbw are, respectively, the so-called weight-average functionalities of the reactants bearing A and B groups.faw is defined by the equation

(7)

withfbw defined similarly. Nai is the number of moles of reactant bearingfai reactive A groups. Notice that NaJai is just the number of moles of A groups belonging to species i. Hence, eqn (7) in effect defines the average functionality weighted according to numbers of reactive groups.

In many random polymerizations, the assumption of the equal reactivity of like groups is not justified. Intrinsic unequal reactivity or induced unequal reactivity (substitution effect) may exist. 6, 7 ,8 Equations for the gel point are then much more complicated and the reader is referred to the literature for the treatment of such systems. Given the present development of understanding of cyclization in non-linear polymerizations, it is difficult in an arbitrarily chosen system to separate unequivocally the effects of unequal reactivity and cyclization in the interpretation of experimental gel points.

The gel point in addition or sequential polymerizations may be treated in a similar way, with the extent of reaction, p, replaced by the probability of chain propagation, (, which relates to the relative rates of propagation, transfer and termination reactions. 9 Unfortunately, the resulting ex­pressions refer only to fixed monomer concentrations. Further, the assumption of the equal reactivity of reactive sites is usually a poor approximation. These limitations result in interpretations of gelation and cyclization data from sequential polymerizations being less well-defined and more difficult to achieve. Theories of cyclization in such systems have been reviewed recently by Dusek. 10 The experimental work of Dusek, Galina and Mikes 11 is particularly instructive; they reacted styrene and ethyldimethacrylate monomer in benzene to high conversions. By

CYCLIZATION, GELATION AND NETWORK FORMATION 353

establishing the critical concentration of the divinylidene monomer for gelation, the fractions of the divinylidene units engaged in intermolecular cross-links were estimated. These fractions were much smaller than expected, i.e. the amount of intramolecular reaction at the gel point was very high. The implication is that many more short range cycles are formed, as if a microclustering of reactant groups occurs during the cross­linking. Under these circumstances, intermolecular reaction is not governed by the usual Gaussian chain statistics, but rather the statistics are similar to those of a chain with pendant reactive groups along the chain which can interact pairwise. Such a model has been discussed by Allen and co-workers,12 and the theory of such reactions in a linear polymerization has also been discussed by Ross-Murphy13 and by Martin and Eichinger. 14

Gelation in the cross-linking of preformed high molar mass linear chains, subject to the random reaction of groups or sites on the chains, may be treated in a manner similar to that for random polymerizations. Considering chains of weight-average degree of polymerization (dp) Xw and one reactive group or site (e.g. side group or main-chain double bond) per structural unit, their average functionality is simply Xw. For chains with pendant B groups being cross-linked by a difunctional monomer, RA2, eqn (6) shows that gelation occurs when

(8)

For the vulcanization of unsaturated monomers, (PaPb)c is replaced by Pc' the critical cross-link density or fraction of sites reacted.2 Importantly, in both cases, the average functionality is large. Hence, the gel point is passed very early in the reaction and is not so amenable to determination. In addition, because of substitution effects, the assumption of random reaction is often not justified.

INTRAMOLECULAR REACTION IN LINEAR AND NON­LINEAR POLYMERIZATIONS

The important distinction between intramolecular reaction in linear and non-linear polymerizations is well-illustrated by the comparison 15 between the data of Stepto and Waywell16 and Stanford and Stepto 17 on, respectively, number fractions of rings (Nr ) in bulk linear and non-linear polyurethane-forming polymerizations (reaction of NCO with OH). Nr is the average number of ring structures (pairs of intramolecularly reacted groups) per molecule and Fig. I shows Nr as a function of extent of reaction (p).

354 S. B. ROSS-MURPHY AND ROBERT F. T. STEPTO

0'4

0·3 non-linear HDf+LG56

0·2

0·1

linear HDI+PEG200

P

FIG. 1. Number fraction of ring structures per molecule (Nr) as a function of extent of reaction (p) for bulk, linear and non-linear polyurethane-forming reactions with approximately equimolar concentrations of reactive groups (r = [NCO]o/[OH]o = 1).16,17 Conditions: 0, linear polymerisation, hexane-l,6-diisocyanate (HDI) + poly(ethylene glycol) at 70°C, [NCOlo = 5·111 mol kg- 1, [OH]o=5'188molkg- 1, number-average Dumber of bonds in chain forming smallest ring structure v = 25·2; e, non-linear polymerization, HDI + poly­oxypropylene (POP) triol at 70°C, [NCO]o =O'9073molkg-t, [OH]o =

O·9173molkg- 1, v= 115.

For the non-linear system, Nr is much larger than for the linear system and is increasing rapidly as the gel point is approached. N r was evaluated by measuring l6 - 18 the number-average molar masses (Mn) of samples withdrawn at various values of p (measured by titration of ~NCO groups) and comparing the values obtained with those expected from classical Flory-Stockmayer theory. 2,4 Changes in Mn record changes in numbers of molecules due to intermolecular reaction, and p records the total disappearance of reactive groups due to inter- and intramolecular reaction. Thus, from Mn and p, values of N r, the number of cyclization (intramolecular) reactions per molecule, may be deduced.

The results shown in Fig. I were selected from investigations 16,1 7 of intramolecular reactions which show that N r increases with the dilution of a reaction mixture by solvent and is maximal for equimolar concentrations of reactive groups (r = [NCO]o/[OH]o = 1). The difference between the two curves in Fig. I can be understood in qualitative terms by considering the

CYCLIZATION, GELATION AND NETWORK FORMATION 355

C int Cext

c: B,-,"",-

B----B-------

B~ B------

[a] [b]

FIG. 2. Concentrations of B groups around a reference [AI group about to react. (a) RA2 + RB2 polymerization; (b) RA2 + RB3 polymerization. cint is the concentration of B groups from the same molecule, cext is that from groups on other

molecules.

instantaneous probabilities of intramolecular and intermolecular re­action 1,19 which relate closely to those devised by Jacobson and Stockmayer 20 for ring---<:hain equilibrium. They are illustrated in Fig. 2 for linear (RA2 + RB2) and non-linear (RA2 + RB3) polymerizations. The groups [AI are about to react and competition between intramolecular and intermolecular reaction with B groups is shown. The rate of intramolecular reaction is proportional to Cint (int = internal), the concentration of B groups on the same macromolecule as [AI, and that of intermolecular reaction is proportional to Cext (ext = external), the instantaneous concentration of all other B groups in the reaction mixture. It is obvious that there are many more opportunities for intramolecular reaction in the non-linear case. Approximate analysis 15 in fact shows that the probability for each pair of groups to react intramolecularly is higher for the linear polymerization shown in Fig. 1 than for the non-linear polymerization. The larger number of ring structures per molecule in the non-linear polymerization therefore illustrates the effects of the much larger number of opportunities for intramolecular reaction in that case.

The data in Fig. I refer to bulk polymerizations. Thus, even at the maximum concentration of reactive groups (cexJ, intramolecular reaction cannot be neglected ab initio. The factors which affect cyclization, namely reactant molar mass, functionality, chain structure, ratio and dilution of reactive groups, may possibly reduce it to negligible amounts, but in non­linear polymerizations this has to be established rather than assumed. Cyclization will affect the gel point and also the properties of the network

356 S. B. ROSS-MURPHY AND ROBERT F. T. STEPTO

material finally formed at complete reaction. As will be discussed later, the effect on the gel point may nominally be small in terms of extents of reaction, but the properties of the network material formed at complete reaction are extremely sensitive to cyclization or loop formation.

PROBABILITIES OF INTRAMOLECULAR REACTION AND THEORIES OF GELATION

In this section, intramolecular reaction in random polymerizations is considered. Referring to Fig. 2, only ring structures of certain sizes can form, as defined by the repeating unit of the chain structure. The situation is illustrated in Fig. 3 with reference to RA2 + RB2 and RB2 + RB3 polymerizations. v is the number of bonds in the chain which forms the smallest ring structure, and ring structures of approximately v, 2v, 3v, ... bonds may form, depending on the number of bonds lost in the chemical reaction between A and B groups. For a given ratio of reactants, the probability of intramolecular reaction may be characterized by the ring­forming parameter1.21

A. = Pab/cao (9)

where

(moles functional groups per unit volume) (10)

In eqn (9), CaO is the initial concentration of A groups (one of the types of reactive groups) and defines a scale for cext ' which of course decreases as a polymerization proceeds. Pab is the mutual concentration of reactive groups separated by a chain of v bonds, assuming the chain obeys Gaussian

r -------------: A ~ ~ 8 B I '- -------------..

[a] [b)

FIG. 3. Repeating units of chain structures. (a) RA2 + RB2 polymerization; (b) RA2 + RB3 polymerization. The repeating units have v bonds separating the groups ~ and 00 which may react to form the smallest ring structure. Ring

structures of approximately v, 2v, 3v, ... bonds may form.

CYCLIZATION, GELATION AND NETWORK FORMATION 357

statistics. N is the Avogadro constant, and b is the effective bond length of the chain, such that <r2) = vb2, where <r2) is its mean-square end-to-end distance. Hence, Pab is dependent on the chain structure (or b) and the molar mass (or v) of the reactants.

The expression for Pab follows from Jacobson and Stockmayer's treatment of linear ring-chain equilibria. 20 The justification for assuming Gaussian statistics (at r ~ 0) may be based on the results of experimental studies of such equilibria (see Chapter 1) which indicate that, for reasonably flexible chains v must be greater than about 30 bonds. Satisfactory interpretations1,16,21-23 of total ring fraction data (as in Fig. 1) have also been made on the basis of Gaussian statistics for linear polyurethane-forming reactions.

For a linear polymerization, Cint,i for an A-B chain of i repeat units (see Fig. 2) is

C. . = Pab/i3/2 Int,1 (11)

For a non-linear polymerization, Cint,i is proportional also to the number of opportunities for intramolecular reaction with the given group. This proportionality leads to the key approximation in the various theories of non-linear polymerization which include intramolecular reaction. The way to count exactly the numerous opportunities which exist for the intramolecular reaction of a given group has yet to be devised. In principle, it requires the detailed specification and evaluation of the statistics of the complex, branched structures connected to that group. The theories simplify the structures which need to be considered and thus reduce the number of differential equations to be solved. These describe relative rates of intermolecular and intramolecular reaction. Probabilities of intramolecular reaction do not decrease in a simple fashion with molar mass as in linear polymerizations. Amongst the theories may be mentioned the cascade theory of Gordon, Dusek and collaborators,22,24,25 the rate theory of Stanford, Stepto and collaborators1,21,26 and reaction kinetics approaches. 27 Computer simulations have also been employed. 28,29 A detailed comparison of the various theories and their predictions has yet to be attempted, although a comparison of rate and cascade theories of linear random polymerization has been made. 1 The field is a developing one and beyond the scope of this review at the present time. However, comparison between experiment and theory is essential if useful approximations are to be found. In this respect, it is not sufficient to investigate the gel point alone, a single point in a polymerization, but rather it is necessary to examine how a theory may explain pre-gel intramolecular reaction (such as the Nr versus

358 S. B. ROSS-MURPHY AND ROBERT F. T. STEPTO

f-1 B~BA1_A"-:-BA-ABABA -ABABA-AB- - - --B~ /;~ /\ /\

B B B B B B A A A I I I A A A

f-2 f-2 f-2 FIG. 4. Generalization32 of Kilb's linear sequence of branch units to define the

conditions for gelation in an RA2 + RBf polymerization.

p data in Fig. 1) together with the gel point. Such data are sparse for polymerizations carried out under the more usual non-equilibrium, irreversible conditions compared with those available for ring--chain equilibria in linear polymerizations.

In addition to the theoretical approaches mentioned, there are what may be termed 19 approximate theories of gelation which seek to derive analytical expressions for the gel point. They use a single, average value for cext ' so that the relative probability of intramolecular and intermolecular reaction is constant throughout a polymerization. The earlier expressions of Frisch 30 and Kilb 31 have been superseded by that due to Ahmad and Stepto,32 who showed that Frisch's expression seriously underestimated the probabilities of intramolecular reaction and that Kilb's expression resulted from an incorrect evaluation of the probabilities for gelation, or unlimited chain growth.

Kilb's model of a growing linear sequence of branch units was generalized 32 to that shown in Fig. 4. Only limited structures for the pendant chains are considered, so that (f - 2) opportunities for intramolecular reaction with the growing sequence exist at each branch unit. Consideration of the probabilities of passing by intermolecular reaction from group B1 to A 1 to B2 in Fig. 4 gives

(12)

at gelation. Aab is the ring-forming parameter defined by the equation

cint if-2)Pab. cp(l, 3/2) (13) Aab=---

cint + cext if - 2)Pab. cp(1, 3/2) + cext

cint is the internal concentration of A or B pendant groups around B1 or A 1, respectively. In the expression for C int , the combined factor

Pab.(p(l, 3/2) = Pab L lii- 3 / 2 = Pab. 2.612··· (14)

i= 1

CYCLIZATION, GELATION AND NETWORK FORMATION 359

is proportional to the sum of internal concentrations of A or B pendant groups around B 1 or A 1 assuming only one group from the branch point at generation i(seeeqn (11». The factor (j - 2) is the number of opportunities for intramolecular reaction at each branch point. As stated, cext must be chosen arbitrarily and may be equated with the initial or gel-point concentration of reactive groups (cao + cbO) or (cae + cbe), respectively, as extreme values.

In order to interpret experimental values of (Le' eqn (12) is rearranged to give

(15)

where, from eqn (13),

A~b = Cint (j - 2)Pab. 4>(1,3/2) cext cext

(16)

Thus, plots of A~b versus dilution (c;,,}) should be linear with slopes proportional to (j - 2) and Pab, allowing the increase in intramolecular reaction with dilution and functionality to be understood, and, through Pab (see eqn (10», interpretation of its increase with decrease in reactant molar mass (v) and chain stiffness (b). The basis of this theory is too approximate to allow it to be used for the absolute evaluation of the effective bond length, b, but useful correlations between values of b and the structures of reactants can be obtained, as will be discussed shortly.

EXPERIMENTAL STUDIES OF CYCLIZATION AND GELATION IN NON-LINEAR RANDOM POLYMERIZATIONS

Early experimental determinations 2 of the gel point, (Le' published subsequent to Flory and Stock mayer's elucidation of the phenomenon of gelation 2,4 used polymerization systems for which the assumption of the equal reactivi ty of like functional groups was not justified. For example, the intrinsic reactivities of the primary and secondary OH groups of glycerol in the oft-quoted work of Kienle and Petke 2 on glycerol/adipic acid polymerizations are different. In addition, once one group has reacted the other groups are sufficiently close for induced unequal reactivity (substitution effect) to enter. 33 This effect of induced, unequal reactivities of OH groups should also be considered when interpreting the results of Stockmayer and Weil 34 on pentaerythritol/adipic acid polymerizations.

360 S. B. ROSS-MURPHY AND ROBERT F. T. STEPTO

However, these authors did demonstrate the effects of dilution on the gel point. This idea is now embodied in Cext in the parameter A.~b of eqns (i5) and (16) according to which IXc ~ (j - 1) - 1 or A.~b ~ 0 in the limit C e~tl ~ O. That is, extrapolation of experimental values of IXc to zero dilution of reactive groups should give IXc = 1/3 for the pentaerythritoljadipic acid system. The experimental value ofO' 334 ± 0·006 obtained for this system 34

is in excellent agreement with the expected value. However, the agreement should be viewed with caution as the induced unequal reactivity will reduce the probabilities of intramolecular reaction.

More recently, polymerization systems have been devised in which the effects of unequal reactivity are minimized. Gordon and collabo­rators 35 - 39 have used reactions of benzene-I ,3, 5-triacetic acid and decane­I,IO-diol, and Stepto and collaborators 1,15,40-43 have used polyoxypro­pylene triols or tetrols reacting with diacid chlorides or diisocyanates. In succeeding sections these studies will be discussed in more detail.

Benzene-l,3,5-triacetic AcidjDecane-l,lO-diol (BTAjDMG) The polymerization of the two monomers may be represented schemati­cally as in Fig. 5. Extents of reaction may be measured by titration of COOH groups or from the pressure of steam generated in a closed system. In bulk at r( = [OH]o/[COOH]o) = 1, IXc = 0·518 showing apparently only

BTA DMG

+ • •

FIG. 5. The BTAjDMG system and corresponding 'tree' representations illustrating the condensation to a dimer (and higher species ... ) by elimination of

water.

CYCLIZATION, GELATION AND NETWORK FORMATION 361

0'76

1 lXd'20'75

1 O. 74

0·73

0·72

0·71

0'70

0·69 0~---0~.-=5---~1.L:'0~---1"""5~-----=-2.'"=0-

0-FIG. 6. 11.:/ 2 plotted against dilution, D (D = (polymer volume + diluent volume)jpolymer volume) for the BTAjDMG system. 36 The various symbols refer

to replicate experiments carried out at the same dilution.

small deviations from the value of 0·5 expected in the absence of cyclization. This is reasonable because the system was chosen to reduce the likelihood of such deviations by substituent effects and intramolecular reaction.

A detailed investigation by Ross-Murphy,36 using a series of polymerizations at different initial dilutions of the tridiphenylmethylester of BT A as solvent (chosen so that any effects of solvent on the reaction mechanism would be minimal) showed that 1X~/2 increased approximately linearly with the initial dilution of reaction groups as illustrated in Fig. 6. The value of IXc at zero dilution was found to be 0'486, suggesting a small, positive substitution effect (enhancement of COOH reactivity once one of the groups of a BT A molecule has reacted), producing gelation earlier than expected from random reaction. The kinetics of the polyesterification were analysed using cascade theory, allowing Pab to be evaluated from the ratio of rate constants for intermolecular and intramolecular reaction. From Pab and the value ofv( = 17 for the BTA/DMG system), bwas found to be 0·433 nm, a value which agrees well with that calculated from rotational­isomeric-state calculations and that found from light scattering measure­ments. 38 Other work on this system includes extraction of the sol fraction and determination of its molecular weight,35 and measurements of elastic moduli of the gel. 37.39

CH2 -+OCH2 CH*-: OH I n

I CH3

CH2+OCH2 CH~ OH

I I n CH3

(CH2)3

c.]

o 0 ~C+CH +-: C~

CI/ 2 m 'CI

d.]

CH2 -+ OCH2 CH~ OH I n

I CH3

CH -+ OCH2 CH+':- OH I n

I CH 3

CH2 -+ OCH 2 CHi:: OH I n CH3

b.]

O=C=N-tCH2~ N=C=O

e.]

O=C=N-@-- CH2-@--N = C=O

f.] FIG. 7. Reactants used in studies of gelation using POP triols.l.15.40-43 (a) Niax Triols LHT240 and LHTll2 (Union Carbide); (b) Niax Triol LG56 (Union Carbide); (c) POP tetrol based on pentaerythritol; (d) adipoylchloride (AC) (m = 4) and sebacoyl chloride (SC) (m = 8); (e)(hexane-l ,6-diisocyanate (HDI)(m = 6) and decane-I,IO-diisocyanate (DDI) (m = 10); (f) diphenyl-l,4-diisocyanto methane (MDI). In the polyols, (a), (b) and (c), ii is the number-average of oxypropylene

units per arm. In (a), ii for LHT240 < ii for LHTI12.

CYCLIZA TION, GELATION AND NETWORK FORMATION 363

Polyoxypropylene (POP) Polyol Polymerizations The experimental investigations of cyclization and gelation using POP triol- and POP tetrol-based systems have been extensive. They have used commercial triols based on the chain extensions of glycerol and hexane-1,2,6-triol and specially synthesized tetrols based on chain extensions of pentaerythritol. The pol yo Is have been reacted with adipoyl chloride (AC) and sebacoyl chloride (SC), and hexane-I,6-diisocyanate (HDI), decane-1,10-diisocyanate (DDI) and diphenyl-I,4-diisocyanato methane (MDI). The reactants are summarized in Fig. 7, giving values of v ranging from 29 to 136 for the various pairs of reactants used.

POP triols reactions with HDI in bulk and in benzene have been used to determine total ring fraction data (Nr versus extent of reaction) in the pre­gel regime. 1 7 Some of these data have been reviewed recently1 and show clearly that Nr increases with initial dilution of the reaction mixture and decreases as r moves away from unity. Figure I showed that many more structures are formed in non-linear polymerizations compared with linear polymerizations. The interpretation of the total ring fraction data using the rate theory of non-linear random polymerization is in progress. 26 ,44

POP triols and tetrols have been used in gelation studies 1,6,15,19,32,40 - 43

which have explored the relationships between O(c or A:b, and molar masses, chain structures and function ali ties of the reactants, as well as the initial dilutions of reactive groups, as described by eqns (10), (I5) and (16). By choosing different polyols, v has been varied (see Fig. 3) and, for each reaction system, a series of gel points at different initial dilutions has been determined, from bulk until perhaps 70 % solvent, or until gelation was no longer observed. A:b is then plotted versus c;~}. The plots obtained often show decreases in slope as dilution increases, as illustrated in Fig. 8, for polyurethane-forming polymerizations. 15,40,42 Thus, deviations from the linear behaviour predicted by eqn (16) are apparent. Slightly more curvature occurs when the gel point dilution, rather than initial dilution, of reactive groups is used as the abscissa. The curvature is a manifestation of changes in Cex! and in opportunities for intramolecular reaction at different generations as a polymerization proceeds. In Fig. 4, there are assumed to be always (f - 2) opportunities at each generation.

The points at the lowest dilutions in Fig. 8 represent bulk reaction mixtures, indicating significant amounts of intramolecular reaction at the maximum obtainable concentrations of reactive groups. The slopes of the curves show the general dependences expected from eqns (10) and (16), with f and v being the main factors affecting A:b and with a secondary dependence on chain stiffness (cf. systems I and 3).

364 S. B. ROSS-MURPHY AND ROBERT F. T. STEPTO

4

0·2

0.' 2

3

0·' 0·2 0'3 0'4 0'5 (CIIO+ Cbo)-Ykg mol-'

FIG. S. Analysis according to eqn (J6)with ceX! = caO + cbO of gel point data 1~.40.42 from reactions of HOI and MOl with POP triols LHT240 and LHTll2 and POP tetrols OPPE-NHI and OPPE-NH2 in bulk and in nitrobenzene solution at SO°C. Systems I and 2, HOI + POP triols; system 3, MOl + POP triol; systems 4 and 5, HOI + POP tetrols. (I) HOI + LHT240, v = 33; (2) HOI + LHTI12, v = 61; (3) MOl + LHT240, v = 30; (4) HOI + OPPE-NHl, v = 29; (5) HOI + OPPE-NH2,

v=33.

Although plots of A~b versus initial or gel dilution often show some curvature, the analysis of initial slopes according to eqns (10) and (16) gives values of b which show sensible correlations, as illustrated in Table 1. Within groups of similar systems, entries are tabulated in order of decreasing v2/v, where V2 is the contribution to v from the difunctional reagent, whose chain structure, based on CH2 units, should be stiffer than that of the POP chain. Accordingly, the derived values of b decrease as v2/v decreases. (Compare systems 1 and 2, 4 and 5, and 6 to 11.) The absolute values of b derived using Cex! equal to the gel concentration of reactive groups (cac + cbc) are in better accord with those expected from solution properties. This result is indicative of most of the intramolecular reaction occurring near the gel point, consistent with the increasing complexity of molecular structures as gel is approached.

CYCLIZATION, GELATION AND NETWORK FORMATION 365

TABLE I Values of Effective Bond Length (b) of Chains Forming the Smallest Ring

Structures of v Bonds (Values derived from initial slopes of A.~b versus c;./ plots according to eqns (10) and

(16))

System f v v2 /v b/nm(i)a b/nm(ii)b

I.C HDI/LHT240 2.c HDI/LHTlI2 3.c MDI/LHT240 4.c HDI/OPPE-NHI 5.c HDI/OPPE-NH2 6.d SCjLHT240 7. d ACjLHT240 8.d SCjLHTll2 9.d ACjLHTll2

lO.d SCjLG56 II.d ACjLG56

a (i) cex! = caO + chO •

b (ii) cex! = cac + cbc '

3 3 3 4 4 3 3 3 3 3 3

33 0·303 0·247 61 0·164 0·222 30 0·233 0·307 29 0·345 0·240 33 0·303 0·237 41 0·268 0·318 37 0·189 0·313 70 0·157 0·293 66 0·106 0·270

136 0·081 0·267 132 0·053 0·260

c Systems 1-5: polyurethane-forming reactions in nitrobenzene at 80°C. d Systems 6-11: polyester-forming systems in diglyme at 60°C.

0·400 0·363 0·488 0·356 0·347 0·508 0-480 0-433 0·399 0·390 0·371

v2 /v fractional length of unit from difunctional reactant in chain of bonds.

Table 1 shows that the polyester chains 1 •32 generally have larger values of b than the polyurethane chains. The reason for this needs to be investigated by independent theoretical and experimental chain­configuration studies. The BTA/DMG system discussed earlier and system 3 have stiffer, aromatic-containing chain structures and give relatively large values of b.

The effects of chain stiffness and functionality on the gel point are further exemplified by results of Ahmad, Stepto and Sti1l 43 on polyester-forming systems using (a) SC and mixtures of a POP triol and diol, and (b) SC and three POP triols reacting separately at different temperatures. In (a), the diol and triol were shown by kinetics studies to have OH groups of approximately equal reactivities and the values of v for the six systems used were approximately equal.

Figure 9 shows r:t.; 1 versus initial dilution of reactive groups and the curve through the experimental points for each system has been extrapolated to the value of (fw - 1) at zero dilution, consistent with eqn (6). The similar initial slopes of the curves in Fig. 9 indicate that intramolecular reaction is not sensitive to functionality in these systems.

366 S. B. ROSS-MURPHY AND ROBERT F. T. STEPTO

2·0

I

o

2

3

1·1 6 5

1·0 o:----,o~.2.------::()oJ.,4,----:0~.6~----,;:0 ..... 8,....---1-,1..0:------,.I1.2

([COCI1 0 + [OHlot'lk9 mol-1

FIG. 9. Reciprocal product of extents of reaction (lXc-l) versus initial dilution of reactive groups «[COCllo + [OHlo) -I) for SC/POP diol/POP triol mixtures in diglyme at 60°C. Curves extrapolated to IX;; I = fw - I at zero dilution in accordance

with eqn (6). Systems 1-6: fw = 3'00, 2'81, 2'65, 2'50, 2·35, 2'19, respectively.

CYCLIZATION, GELATION AND NETWORK FORMATION 367

However, analysis on the basis of eqns (10), (15) and (16) shows that this apparent insensitivity may be interpreted as a decrease in b with fw' resulting in relatively higher probabilities of intramolecular reaction per pair of groups which can so react as fw decreases. The decrease in chain stiffness as functionality decreases is a subtlety in behaviour not previously expected and probably reflects the inadequacy of using linear, Gaussian chain statistics for species, independent of the functionality of branch points.

In investigation (b), using SC and POP triols, the variations of A~b with dilution show that b increases with temperature. 43 Values of d In <r~)/dT = din < vb2 )/dT have been derived. They are larger than the values expected for linear chains of similar structures, again indicating that more precise chain statistics should be used in interpretations of results.

Conclusions The general conclusions from gelation studies are that the dilution of reactive groups, the functionality (j) and molar masses (or v) of the reactants are the main factors affecting gel points. In addition, variations in chain stiffness (b) have to be accounted for. The dependence of !Xc (or A~b) on these factors is understood in general terms. However, the quantitative, ab initio prediction of the delay in gel points in all systems is not yet possible. The use of cascade and rate theories in describing N r versus p data and gel points has been mentioned. 26,36,39,44 Such approaches use a value of CeXl which naturally decreases as a polymerization proceeds, thus removing the key assumption that A~b remains constant throughout a polymerization.

CYCLIZATION AND NETWORKS

Prior to the formation of gel, intramolecular reaction may be regarded as a perturbation to the classical, Flory-Stockmayer description of polymeriz­ations. After the gel point, where molecules with unlimited numbers of reactive groups or sites increasingly exist, intramolecular reaction is, even in classical terms, necessary for network formation. Such reaction influences the molecular structure and properties of the network or gel and of course the molecular species in the sol fraction (as before the gel point). For example, for non-glassy networks, cyclization can lead to elastically ineffective chains or loops and a reduction in modulus. However, in the classical approach, all post-gel reaction leads to elastically effective chains.

368 S. B. ROSS-MURPHY AND ROBERT F. T. STEPTO

The prediction of properties of even perfect, rubbery networks from their molecular structure is still a matter of some debate. 45 ,46 The term 'perfect' implies that all chains joining two chemical junction (or branch) points are elastically active. There has been much theoretical work in this area relating to the importance or not of topological effects (entanglements) and to the affine/phantom behaviour of individual chains, which lies beyond the scope of this review. However, such work has largely and unjustifiably neglected the effects of cyclization in producing inelastic loops. In practice, of course, topological effects including inelastic loops may be regarded as network defects, and the ideal behaviour occurs only for 0( :::::: O(c' As cx increases above CXc ' the formation of elastically ineffective loops becomes of increasing significance. The number and effects of such loops on physical properties in the post-gel regime have been investigated theoretically but there are very few (if any) related experimental studies. For this reason the present section concentrates on linked experimental and theoretical studies which illustrate the post-gel effects of intramolecular reaction. The studies carry through from those of the pre-gel regime and, hence, concentrate on random polymerizations, that is those forming end-linked networks. Two regimes have been emphasized in experimental studies to date; that around the gel point, the sol-gel transition; and that of the network at complete reaction.

The Sol--Gel Transition Much of the experimental work on the sol-gel transition has not been designed specifically to investigate the importance of intramolecular reaction. For the BTA/OMG system, measurements of Mw as gel is approached and of M w of the sol fraction after the gel point 35.38 indicate low amounts of cyclization, consistent with the previously discussed gelation results for this system. 36 In addition, the shear modulus (G) of the BTA/OMG polymerizing system just after gel follows the expected relationship

(17)

for no intramolecular reaction. 37.47 Studies 48 of an HOI/POP triol system also gave a similar exponent. The results are only approximate but, as expected, indicate little effect of cyclization on the tree-like gel fraction near the gel point, provided the experimental value of cxc ' rather than cx: ( =(j - 1) - 1), is used as the point of incipient gelation.

The recent interest in the sol-gel transition region has been motivated by developments in the theory of critical phenomena, particularly using percolation models. Such models imply that 'polymerization' occurs

CYCLIZATION, GELATION AND NETWORK FORMATION 369

through reactions on a rigid lattice. 49 The sol-gel transition, for example, is treated in an analogous way to liquid/gas and magnetic phase transitions. The model, which enables the large fluctuations in properties which occur very near to critical points to be treated, predicts universal exponents (c5) in relationships such as

f3 = (1£ - nc>fJ (18)

where f3 is some measured property and 1£ - nc represents displacement from the critical point, nc .. Thus,· percolation models predict the exponent in eqn (17) to be 1'6-1'7.

Note that questions still remain to be answered concerning the applicability of percolation models to polymerizing systems. They bring into question the wealth of evidence for random intermolecular reaction in the pre-gel regime: evidence which spans the history of quantitative polymer science. For example, the most probable distribution for linear random polymerization and the Flory-Stockmayer gel point equations are said from the percolation viewpoint to represent only special types of molecular growth. However, percolation models have relevance very near to critical points, where, in polymerizations, random molecular movement may be suppressed.

It is a matter of present debate how important it is; in general polymerization terms, to understand the fine details of the sol-gel transition. It is an extremely narrow regime (typically (1£ - nc)/nc < 0·01), which has traditionally been adequately treated as a point of discontinuity. In addition, percolation treatments are limited in that they concentrate on proportionalities and exponents rather than absolute magnitudes of properties. As discussed earlier in this chapter, with respect to ring-fraction and gel point data, provided systems with like groups of equal reactivity are chosen, the classical Flory-Stockmayer model of random, intermolecular reaction may be taken as the basis for understanding polymerization processes. The deviations from this model, e.g. the increase in ac over acD

(=(j - 1) -1 from RA2 + RBf polymerization) and the reduction in Mn (which leads to values of N r), are adequately explained by the competition of intramolecular reaction (between spatially related groups) and random intermolecular reaction. To this may be added the wealth of evidence from linear ring-chain equilibria from Semlyen and collaborators (Chapter 1) for which the lacobson-Stockmayer theory provides a near-quantitative explanation in similar terms. In addition, the studies relating to the sol-gel transition just mentioned 35,37,38.47 -49 indicate that experimentally it is very difficult to get close enough to the transition to enter its percolation

370 S. B. ROSS-MURPHY AND ROBERT F. T. STEP TO

'sub-regime'. Burchard and collaborators 50 have also noted this problem in studies of changes of sizes of clusters near the gel point. The sensitivity of the 'universal' critical exponents to the amount of intramolecular reaction is currently being investigated. 51

Effective and Ineffective Loops Close to the gel point, the structure of a network is most nearly tree-like (as discussed in the previous section). By contrast, at high concentrations, and for a so-called 'perfect' network, all chains joining two junction points are elastically effective. In the intermediate regime a number of junction points will close loops which are not attached to an elastically effective chain except at the ring closing junction: this ring is therefore elastically ineffective (Fig. lO(a)). As the cross-linking proceeds further a junction point may be formed between a site on this ring and another elastically effective chain, and the ring becomes part of a new elastically effective chain (it is said to be 'activated') (Fig. lO(b)). There have been a number of

(a)

(b)

Elastically ineffective cycle

FIG. 10. (a) Formation of an elastically ineffective cycle by intramolecular cross­linking of loose material attached to the gel; (b) Reactivation of this elastically ineffective loop by further cross-linking to another elastically active chain. The

arrows indicate that a path from such a point continues to 'infinity'.

CYCLIZATION, GELATION AND NETWORK FORMATION 371

theoretical papers in this area, but experimental studies have so far been limited because of the possible contribution to elastic properties from other network defects (e.g. entanglements), and the difficulty in separating these different contributions in an experimental study.

In 1974, for example, Tonelli and Helfand 52,53 asserted that 'before any properties of a rubber are seriously discussed, the presence of elastically ineffective crosslinks terminating intramolecular loops must be rational­ized and dealt with'. They evaluated the effect of loops for a cis-1,4-polyisoprene chain using a Gaussian probability corrected for the smaller rings. For < 16 bonds they used an RIS model, for 16-100 bonds a fourth moment corrected Gaussian probability and for> 100 the usual Gaussian probability. From their calculations it was found that the fraction of elastically ineffective loops was in the range 10-20 % for bulk systems, rising to around 70 % for networks swollen by a large excess of solvent.

Dusek and co-workers 55,56 have recalculated this fraction using a slightly different model, but recognising the contributions from a number of different sources. For example, cross-links can be formed and/or wasted in elastically ineffective loops (l) during intermolecular cross-linking, (2) by cyclization in the remaining sol fraction (as in the section 'Probabilities of Intramolecular Reaction and Theories of Gelation'), and (3) by cyclization within the gel.

Contribution (3) included both intramolecular reaction within a pre­existing elastically effective chain and intramolecular reaction in the loose material attached to the gel by one bond. Gel-gel reaction steps are assumed to be governed by the law of 'mass action' (as in intermolecular cross-linking), rather than by the probability of ring closure in a sol, intramolecular step. The justification for this latter procedure was asserted many years ago by James and Guth. 54

The approach 6f Refs 55 and 56 suggests that the earlier estimates of Tonelli and Helfand gave rather too high a figure for the fraction of wasted cross-links. Nevertheless, the proportion does increase as expected on passing to swollen systems. However, the most relevant finding was that when results were rescaled to the 'experimental' gel point (i.e. to include the delay due to intramolecular reaction in the pre-gel regime) the contribution from elastically ineffective loops was all but eliminated, even quite far from the gel point.

Networks at Complete Reaction For the use of polymers as solid materials, the main interest in networks, or products of non-linear polymerizations, lies in the materials formed at

372 S. B. ROSS-MURPHY AND ROBERT F. T. STEPTO

'complete' reaction. The interpretation of the mechanical and thermo­dynamic (e.g. swelling) properties of rubbery as distinct from glassy networks materials in molecular terms stems from the statistical theory of rubber elasticity,57 in which the overriding effect on deformation is the change in entropy of the individual chains between (elastically effective) junction points. More recent considerations45.46.58 have focused on the energetic changes also occurring on chain deformation, the topology of the network and the effects of physical interactions (entanglements) on the modulus, and the expected behaviour of a collection of flexible chains connected by chemical function points, leading to affine and phantom behaviour as idealized extremes. It should be emphasized that all these considerations have been made on the basis of the chemically perfect network (as previously defined), and comparisons with experiments 58 have usually assumed such perfection in the materials studied. Mechanical or swelling behaviour is then interpreted in terms of the non-affine behaviour of chains and/or the effects of chain interactions introducing additional (transient) junction points.

Recently the work of Stepto and collaborators15 .42 ,44.59-61 has shown that the absolute value of the modulus of a rubbery, end-linked network at complete reaction is related closely to the amount of intramolecular reaction occurring in a polymerization, and that only a small increase in IJ(c

over IJ(co can be characteristic of a large reduction in the modulus of the network at complete reaction.

Intramolecular reaction is essential for network formation and need not necessarily lead to inelastic chains. However, the smallest ring structures (or loops) so formed will be elastically ineffective. Thus, although the effects of topology and the behaviour\ of larger loops cannot be unequivocally resolved, the effects of the smallest loops can be predicted with more certainty. * From the considerations of probabilities of intramolecular reaction given in previous sections, such loops will be the most numerous and will be formed both in the pre-gel and post-gel regimes.

Figure II illustrates the occurrence of the smallest loop structure in an RA3 polymerization at complete reaction. The modulus of the (dry) network is proportional to the number of elastically active chains per unit volume (vN) between elastically effective function points or inversely

* The introduction of entanglements by the intercatenation of loops may be neglected as its occurrence requires the formation of two loops in a given small volume, the probability of which is an order of magnitude less than for individual loop formation.

CYCLIZATION, GELATION AND NETWORK FORMATION 373

-AA>-AA­- AA

State 4 (a)

I>-AA-State 6

(b)

-AA~AAWAA--!(AA--AA~ I AA--

A'J- ~AA~ I AA"'-...4/AA~ A T AA~

A A

A A-A

A

(e) A A-A

l'J-AAyAA~I ~

(f) AAA A/-- ~A I AA I A A

(9)

FIG. 11. Occurrence of the smallest loop structures in an RA3 polymerization at complete reaction, -< ,elastically active junction point. «a) and (b)) possible states of the RA3 unit assuming only the smallest ring structure forms (denoted state 6 in rate theory 61); (c) junction point in the perfect network; «d) and (e)) occurrence of the smallest ring structure (state 6) in the network: «f) and (g)) occurrence of the smallest ring structure in the sol. Structure (c) gives ~ = 1; structures (d) and (e) show that two elastic junction points are lost for every ring structure formed; (f) and (g)

constitute the sol fraction, reducing the mass of network formed.

proportional to Me' the (number average) molar mass between such points where p/ Me = VN for the dry network. For an J-functional network, VN

equals (f/2) x the number of junction points per unit volume. Using rate theory, Lloyd and Stepto 61 calculated lXe and the proportions

p 4 and P 6 ofRA3 units existing in states 4 and 6 at complete reaction. From structures (d) and (e) one has

1 c; = GO/G = M /Mo =-:--~ eel - 2P6

(19)

where G is the observed modulus and GO that of the perfect dry network. Here, c; is understood to be the proportionate reduction in modulus of the

374 S. B. ROSS-MURPHY AND ROBERT F. T. STEPTO

dry network due to elastically ineffective loops (ElLs). For an RA3 polymerization, P 6 is related directly to A of eqn (9) (remembering there are only A groups present), which also defines the gel point. Accordingly, Fig. 12 shows relationships between calculated values of A at complete reaction and the extent of reaction at gel (Pc)' For an RA3 polymerization, Pc = IXc

(cf. eqn (2) for no cyclization). Experimentally, for a given RA3 reactant, Pc and A would be varied by carrying out polymerizations at different initial dilutions in solvent.

The difference between the curves 1 and 2 in Fig. 12 emphasizes the importance of post-gel intramolecular reaction. Although the calculations

8~----------------------------------------------,

6

4

2

2 ----------1~--~=:~~~-~-----------~----------~--------~ 0·50 0-52 0-54 0-56

~I ______ ~I~ ____________ ~I ____________ ~I ________ ~I Pc

o 0-02 0-06 0-10 0-14 A FIG. 12. Proportional reduction in modulus of the dry network formed from complete reaction of an RA3 polymerization, or proportional increase in effective molar mass between elastic junction points due to elastically ineffective loops (~) versus extent of reaction at gelation (Pc) or ring-forming parameter (A,). Calcu­lations according to rate theory.61 Curve 1: Effect of pre-gel and post-gel intra­molecular reaction on ~. Curve 2: Effect of pre-gel intramolecular reaction on ~.

CYCLIZATION, GELATION AND NETWORK FORMATION 375

consider only the smallest ring structure and make other approximations,61 they show the sensitivity of modulus to intramolecular reaction and the possibility of using excess reaction at gel to characterize and predict modulus. Considerations of the smallest ring structures formed in polymerizations using reactants of higher functionalities if = 4, 5, ... ) show that it is the modulus of trifunctional networks which will be most sensitive to loop formation.

Experimental correlations between ~ at complete reaction and Pr,c' the extent of intramolecular reaction at gelation, have been made15,40,42,59,60 using some of the POP polyol-based polyurethane- and polyester-forming systems employed previously for gelation studies. 1,15,4o-43 Results for

Pr,c

FIG. 13. ~ versus extent of intramolecular reaction at gelation (Pre) for polyurethane networks formed at complete reaction. 15,40.42 The reaction mixtures and gel points are the same as in Fig. 8. ~ evaluated from small-strain measurements

on dry and swollen networks.

376 S. B. ROSS-MURPHY AND ROBERT F. T. STEPTO

polyurethane systems are shown in Fig. 13. The values of ~ have been evaluated assuming affine chain behaviour. The networks were formed using samples from the same reaction mixtures as those used to determine CXc and, hence, Prp which is defined by the equation

P = CX 1/2 _ (j- 1) - 1/2 f,C C (20)

The results clearly show the increase in ~ with cyclization. The reader is referred to Refs 15, 40, 42 and 59 for detailed discussions of the relative behaviour of the different systems relating ~ to v and f In general, a relatively small amount of pre-gel intramolecular reaction is indicative of a large reduction in modulus. Further, the experimental points at the smallest values of Pr.c are for bulk reactions. Thus, the effects of intramolecular reaction on network properties cannot be neglected a priori. System 1 shows that 5 % excess reaction at gel (Pr.c ~ 0·05) can be characteristic of a five-fold reduction in modulus of the dry network (~ ~ 5). It should not be inferred from the results in Fig. 13 that pre-gel intramolecular reaction alone leads to increases in ~. Contributions also result from post-gel intramolecular reaction (cf. Fig. 12). Figure 13 merely shows that extents of reaction at gelation can be used as a basis for correlating the effects of intramolecular reaction on modulus at complete reaction.

The Tg values of networks have also been shown 15.59.62 to decrease as CXc

or ~ increases due to more freedom of chain motion. Thus, measurements on system 3 of Fig. 13 predict that the Tg drops by 12K between the perfect network and the linear polyurethane of the corresponding chain structure.

Conclusions The effects of the systematic variation of intramolecular reaction on network structure and properties have been studied using networks at complete reaction. The indications are that the gel point can be used to characterize such effects, with apparently small amounts of sol and gel intramolecular reaction being characteristic of large reductions in modulus. It is recommended therefore that the preparation of end-linked networks for determinations of absolute values of modulus (and other properties) be accompanied by determinations of the gel points of the reaction systems used. It is not sufficient to assume perfect networks are formed by reactions in bulk.

Theoretical predictions of network imperfections need further develop­ment. They have to account not only for the complexity of the molecular structures which can undergo intramolecular reaction in the pre-gel and

CYCLIZATlON, GELATION AND NETWORK FORMATION 377

post-gel regimes but also for the proportion of post-gel intramolecular reaction which, even for perfect gelling systems, leads to elastically ineffective loops. The extrapolation of points to ~ = 1 at Pr.c = 0 in Fig. 13 rests on the assumption that an ideal gelling system gives a perfect network. This assumption has yet to be verified.

The effects of intramolecular reaction on properties nearer to the gel point are perhaps easier to interpret except in the critical region of the sol-gel transition. The classical tree-like structures formed when Pr.c = 0 can justifiably be used as a basis for interpreting properties. However, well­defined experiments are then more difficult to make. Studies of properties such as sol fractions (in systems with r =F 1) and modulus, over the whole regime between gel point and complete reaction, would be valuable in establishing bases which can be used for interpreting properties.

REFERENCES

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2. Flory, P. J., Principles of Polymer Chemistry, Cornell University Press, Ithaca, New York, 1953, Chapter IX.

3. Gillham, J. K., Brit. Polym. J., 17 (1985) 224 (Networks 84 Issue). 4. Stockmayer, W. H., J. Chern. Phys., 11 (1943) 45; 12 (1944) 125; J. Polym. Sci.

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178. 9. Bamford, C. H., Barb, W. G., Jenkins, A. D. and Onyon, P. F., The Kinetics of

Vinyl Polymerisation by Radical Mechanisms, Butterworths Scientific Publications, London, 1958.

10. Dusek, K., In: Developments in Polymerisation~3 (ed. R. N. Haward), Elsevier Applied Science Publishers Ltd, London, 1982, p. 143.

11. Dusek, K., Galina, H. and Mikes, J., Polym. Bull., 3 (1980) 19. 12. Allen, G., Burgess, J., Edwards, S. F. and Walsh, D. J., Proc. Roy. Soc.

(London), 334 (1973) 453, 465, 477. 13. Ross-Murphy, S. B., Polymer, 19 (1978) 497. 14. Martin, J. E. and Eichinger, B. E., Macromolecules, 16 (1983) 1345, 1350. 15. Stanford, J. L. and Stepto, R. F. T., In: Elastomers and Rubber Elasticity (eds

J. E. Mark and J. Lal), ACS Symposium Series 193, Amer. Chern. Soc., Washington, DC, 1982, Chapter 20.

16. Stepto, R. F. T. and Waywell, D. R., Makromol. Chern., 152 (1972) 263. 17. Stanford, J. L. and Stepto, R. F. T., Brit. Polym. J., 9 (1977) 124. 18. Stepto, R. F. T. and Waywell, D. R., Makromol. Chern., 152 (1972) 247. 19. Stepto, R. F. T., Faraday Disc. Chern. Soc., 57 (1974) 69.

378 S. B. ROSS-MURPHY AND ROBERT F. T. STEPTO

20. Jacobson, H. and Stockmayer, W. H., J. Chem. Phys., 18 (1950) 1600. 21. Stanford, J. L., Stepto, R. F. T. and Waywell, D. R., J. Chem. Soc. Faraday

Trans. I, 71 (1975) 1308. 22. Gordon, M. and Temple, W. B., Makromol. Chem., 160 (1972) 263. 23. Gordon, M. and Temple, W. B., Makromol. Chem., 152 (1972) 277. 24. Gordon, M. and Scantlebury, G. J., J. Chem. Soc. C, (1976) I. 25. Dusek, K., J. Polym. Sci., Polym. Phys. Ed, 12 (1974) 1089. 26. Cawse, J. L., Stanford, J. L. and Stepto, R. F. T., Reprints IUPAC 26th Inti

Symposium on Macromolecules, Mainz, 1979, p.693. 27. Dusek, K., Macromol. Chem., Suppl. 2 (1979) 35. 28. Leung, Yu-Kwan and Eichinger, B. E., In: Characterization of Highly Cross­

linked Polymers (eds S.,S. Labana and R. A. Dickie), ACS Symposium Series 243, Amer. Chern. Soc., Washington, DC, 1984, Chapter 2.

29. Leung, Yu-Kwan and Eichinger, B. E., J. Chem. Phys., 80 (1984) 3877. 30. Frisch, H. L., Paper presented at 128th Meeting Amer. Chem. Soc., Polymer

Division, Minneapolis, 1955. 31. Kilb, R. W., J. Phys. Chem., 62 (1958) 969. 32. Ahmad, Z. and Stepto, R. F. T., Colloid Polym. Sci., 258 (1980) 663. 33. Kodama, Y., Temple, W. B. and Ross-Murphy, S. B., Brit. Polym. J., 9 (1977)

117. 34. Stockmayer, W. H. and Weil, L. L., results cited by Stockmayer, W. H., In:

Advancing Fronts in Chemistry (ed. S. B. Twiss), Reinhold Publishing Corp., New York, 1945, Chapter 6.

35. Peniche-Covas, C. A. L., Dev, S. B., Gordon, M., Judd, M. and Kajiwara, K., Faraday Discuss. Chern. Soc., 57 (1974) 165.

36. Ross-Murphy, S. B., J. Polym. Sci. Symposia, C53 (1975) II. 37. Gordon, M. and Roberts, K. R., Polymer, 20 (1979) 681. 38. Gordon, M., Kajiwara, K., Peniche-Covas, C. A. L. and Ross-Murphy, S. B.,

Makromol. Chem., 176 (1975) 2413. 39. Gordon, M., Ward, T. C. and Whitney,R. S., In: Polymer Networks(edsA. J.

Chompff and S. Newman), Plenum Publishing Corporation, New York, 1971, p. I.

40. Stanford, J. L., Stepto, R. F. T. and Still, R. H., In: Reaction Injection Molding and Fast Polymerisation Reactions (ed. J. E. Kresta), Plenum Publishing Corporation, New York, 1982, p. 31.

41. Ahmad, Z. and Stepto, R. F. T., Polymer J., 14 (1982) 767. 42. Stanford, J. L., Stepto, R. F. T. and Still, R. H., In: Characterization of Highly

Crosslinked Polymers (eds S. S. Labana and R. A. Dickie), ACS Symposium Series 243, Amer. Chern. Soc., Washington, DC, 1984, Chapter I.

43. Ahmad, Z., Stepto, R. F. T. and Still, R. H., Brit. Polymer. J., 17 (\985) 205. 44. Stanford, J. L. and Stepto, R. F. T., in preparation; see Askitopoulos, V., MSc

Thesis, University of Manchester, UK, 1981. 45. Rubber Elasticity Conference 1975, Proc. Roy. Soc. (London), A351 (\976)

295-406. 46. Networks 84 issue, Brit. Polym. J., 17(2) (\985). 47. Gordon, M. and Torkington, J. A., Pure Appl. Chem., 53 (\981) 1461. 48. Adam, M., Delsanti, M., Durand, D., Hild, G. and Munch, J. P., Pure Appl.

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49. Stauffer, D., Coniglio, A. and Adam, M., Adv. Polym. Sci., 44 (1982) 103. 50. Schmidt, M. and Burchard, W., Macromolecules, 14 (1981) 370. 51. Kajiwara, K., Burchard, W., Nerger, D., Dusek, K., Matejka, L. and Tuzar,

Z., In: PMM 1984, 27th Prague Microsymposium, Physical Optics of Dynamic Phemomena and Processes in Macromolecular Systems (ed. B. Sedlacek), Walter de Gruyter and Co., Berlin, 1985, p.47.

52. Tonelli, A. E. and Helfand, E., Macromolecules, 7 (1974) 59. 53. Helfand,E. and Tonelli, A. E., Macromolecules, 7 '(1974) 248. 54. James, H. M. and Guth, E., J. Chem. Phys., 15 (1947) 669. 55. Dusek, K., Gordon, M. and Ross-Murphy, S. B., Macromolecules, 11 (1978)

236. 56. Dusek, K. and Vojta, V., Brit. Polym. J., 9 (1977) 164. 57. Treloar, L. R. G., Physics of Rubber Elasticity, 3rd edn, Clarendon Press,

Oxford, 1970. 58. Mark, J. E., Adv. Polym. Sci., 44 (1982) 1. 59. Stepto, R. F. T., Polymer, 20 (1979) 1324. 60. Fasina, A. B. and Stepto, R. F. T., Makromol. Chem., 182 (1981) 2479. 61. Lloyd, A. C. and Stepto, R. F. T., Brit. Polym. J., 17 (1985) 190. 62. Feger, C. and MacKnight, W. J., Macromolecules, 18 (1985) 280.

Index

To avoid repetition of'Cyclic' and 'Poly' in the Index, compounds are listed under the monomer names (e.g. 'Cyclic peptides' will be found under 'Pep tides', and 'Poly(dimethylsiloxane), under 'Dimethylsiloxane')

Absorption scattering cross-section, values listed, 169

Acetals, cyclic oligomers, 210-12 Acrylamide polymers, cyclization

kinetics, 304--5 Acyloin condensation, 9-10

Amines

Addition polymerization, treatment of gel points, 352

cyclic oligomers, 212-13, 306, 307 cyclization studies, 306-7

Aminocaproamide, ring--chain polymer equilibrates, 27-9, 30

Analytical gel permeation chromatography, siloxanes, 108-19

Adipoyl chloride (AC) molecular structure, 362 POP polyol polymerization with,

363, 365 Affine chain behaviour, 372, 376 Alanine [5L-Ala.D-Ala] hexapeptide,

NMR studies, 271-3 Alkanes, compared with cycloalkanes,

205, 206 Amanita phalloides, cyclic peptides,

275,277, 178 IX-Amanitin

biological functions, 279 chemical formula, 276 conformation data, 278

Amides, cyclic oligomers, 27-9, 30, 208-9,213

Annular character, small cyclic rings, 184

Antamanide, 275 biological functions, 277, 279 chemical formula, 276 conformation data, 277, 278

Anthracene end-groups, 289 Anthracene-labelled polystyrene,

321-2, 325, 327 Antibiotic activity, macrocyclics,

197-8 Approximate theories of gelation, 358 Association, macrocyclic formation

by, 219-20

Back-biting reactions, 86, 199

381

382 INDEX

Band-shift method, DNA helical periodicity determined by, 255-7

Bead-and-spring models, 187, 188, 299,300

Benzene-d6 , 174-5, 177, 178, 188, 191 Benzene-I,3,5-triacetic acidjdecane­

I,IO-diol (BTAjDMO) system, 360-1,368

dilution effects, 361 tree representation of

polymerization, 360 Benzophenone derivatives, cyclization

studies, 289,310-11 Bessel function, 76 Biological functions, cyclic peptides,

279-80 Block copolymers, styrene-siloxane

129-30,214-18,219 Bond cleavage effects, cyclic vs. linear

polymers, 2 Bond interchange reactions, cyclic vs.

linear polymers, 4 Boundary-spreading techniques, 152,

154, 192, 194 Bromocyclohexane, 153, 154, 159-61 Brownian motion, space-time

correlation function for, 74 Bulk properties, dimethylsiloxane

cyclics, 144-50 Bulk viscosity, dimethylsiloxane

polymers, 144-6 2-Butanone, as 8-solvent, III, 150,

151,343,344

Calibration (OPC) curves polysiloxanes, 108, 109, 113 polystyrenes, 216, 217

Calixarenes, 206 e-Caprolactam, cyclic oligomers, 213 Car bene complexes, metathesis

reaction, 20 I Carbonates, cyclic oligomers, 218 Casassa-Markovitz relationship, 63 Casassa theory, 109-10 Cascade theory (gelation theory), 357 Catalysed ester hydrolysis, 288,

314-16

Catenanes, 4-5, 248-50, 253 naturally occurring, 5

Catenation reactions, 4, 248, 250 Chain expansion, excluded volume

effects, 118-19, 177 Chain stiffness, cause of, 59 Characteristic ratios, siloxanes, 91,

144 Chemical shifts (NMR)

dimethylsiloxane cyclics, 147-9 peptides, 270, 271, 272

Chlorosilanes, hydrolysis reactions, 86 Chi oro sulphonate derivatives, 311 Closed-duplex DNA rings, 226, 227,

233-7 Cluster diagrams, 61, 63 Coherent intermediate scattering

function, 187 Coherent scattering cross-section, 169

values listed, 169 Competition, inter-jintramolecular

reactions, 355 Complete enumeration method,

dimethylsiloxane polymers, 139, 140, 159

Complete reaction, networks at, 371-6

Concentrated solutions, cyclization studies, 340-4

Concentration dependence, diffusion coefficients, poly(dimethyl­siloxanes), 154-7, 191

Configurational distribution function, 46

Configurations, dimethylsiloxane cyclics, 136, 144

Conformational energy maps, pep tides, 266-7

Conformation-controlled cyclization reactions, 295

Conformations N-methylglycine polymers, 314 peptides, 265-7

comparison of solid-state and solution, 275-8

solid-state, 274-5 solution, 267-74

rotation angle convention for peptides, 265

INDEX 383

Conformations-contd. siloxanes, 96-7 sulphur rings, 35

Copolymers, siloxanes, 125-30, 214-18,219

Correlation hole, solvent effects, 330-3 Critical concentration, siloxane rings,

90-3 substituent effects, 94-5

Cross-linked polymers, gelation equations for, 353

Crown conformation, dialkylsiloxane tetramer, 96

Crown ethers, 197 Cruciform structure, DNA rings, 241,

242,243 Cryptates, 197, 198 Cubane system, 197 Cuniberti-Perico (CP) model, 297,

339 Cut-off point, metathesis reaction,

202, 203 Cyclization

dynamics definition of, 287 models, 297-303

effect on networks of, 374-5 equilibria, 295-6

definition of, 287 see also Ring--chain equilibration

reactions factors affecting, 355 gel point affected by, 356, 370-1 models, 293-4

experimental considerations, 293-5

reaction rate constants, 299 Cycloalkanes, melting points of, 203,

205, 206 Cycloalkenes, properties of, 202-3 Cyclododecene, oligomers, 201-2 Cyclo(Gly,-L-Pro-GIY2)2,274-5 Cyclohexane, as lI-solvent, 150, 151,

325-9, 341, 342 Cyclooctene

metathesis reaction cut-off point, 202,203

oligomer distribution, 202, 204 oligomer properties, 202

Cycloolefins, properties, 202-3 Cyclopentane, as lI-solvent, 331, 332,

344 Cyclophanes, synthesis of, 206-7

Dansylend-groups,320 Dawson integral, 52, 182 Debye function, 172; see also Particle

scattering factors/functions Decamethylene adipate, ring--chain

polymer equilibrates, 18-19 Decane-I,IO-diisocyanate (DDI),

molecular structure, 362 Decane-I,IO-diol, 360-1, 368 Densities, dimethylsiloxane polymers,

97, 144, 145 Deoxyribonucleic acids (DNAs)

closed-duplex rings, 226, 227, 233-7 free energy considerations,

239-40, 241-4, 246-7 covalently closed double-stranded

rings, 226, 233-7 cyclic form, 43, 44

abundance of, 227 compared with other cyclic

polymers, 226-7 first discovered, 225

cycIization activation energy, 228 cyclization processes, 227-33 fragmentation into half-size

molecules, 228 handedness of double helix, 253-4 helical structure studied by c1osed­

duplex ring changes, 237, 240, 251, 253-8

helix transition, dependence on supercoiling, 244-6

ligases, 227, 229-30, 233, 238 loops in double helix, 43, 44, 227 phage A., 227-8 phage ¢XI74, 225--6 ribbon model, 236-7 ring-closure probability, 229-33 supercoiled, 235-6

deformation modes listed, 242, 248 effects on interaction with other

molecules, 246-7 effects on shape, 247-8

384 INDEX

Deoxyribonucleic acids (DNAs) -contd.

supercoiled-contd. effects on structure, 241-7 energetics, 237-41

topoisomerases, 227, 234-5, 248 torsional rigidity of, 257-8

Dialkylsiloxane tetramer, crown conformation, 96

Diamines, cyclization studies, 311-12 Dieckmann reaction, 7 Diethylene glycol dimethyl ether

(diglyme), 97, 98, 99, 100, 115, 117,118,366

Diffusion coefficients, dimethylsiloxane polymers, 156, 191-3

Kirkwood-Riseman analysis, 159--62

Diffusion-controlled cyclization reactions, 294-5

experimental evidence, 308, 312 poly(ethylene oxide), 335

2,3-Dihydrofuran,'cyclic oligomers, 210, 211

Dihydrogensiloxanes, ring--chain polymer equilibrates, 21-2, 23

Dilute solution(s) cyclization experiments, 303-44 cyclization studies, 303-39 properties

dimethylsiloxane polymers, 150-8 styrene cyclic polymers, 216-18

Dilution effects, gelation, 359, 360 Dilution method,

mesocyclics/oligocyclics preparation, 7-8, 198-9

I-Dimethylaminonaphthalene-5-sulphonamide end-groups, 320

Dimethylsiloxanes abbreviation used for, 87 analytical gel permeation

chromatography calibration plots, 108, 109, 113

block polymers with styrene, 129-30,214-18, 219

catalyst systems for cyclization, 87, 89,90

Dimethylsiloxanes-contd. comparison of cyclic and linear

polymers, 135-63 conformational behaviour of cyclic

and linear forms, 24, 96-7 cyclic polymers

configuration changes, 144, 149 density anomaly 136, 144, 145 density variation with ring size,

144, 145 dimensions listed, 194 disc-like configuration, 136, 144 limiting size, 149 molar masses, 135 NMR chemical shifts, 147-9 polymers, 194 precise naming of, 2 refractive index variation with

ring size, 144, 145 cyclic trimer, 87

reactions 87-90 cyclization studies by spectroscopy,

337-9 dilute polymer solution properties,

150-8 dipole moment ratios, 146-7 glass transition temperatures of

cyclic and linear polymers, 149, 150

macrocyclic residues cyclic/linear distributions in,

114-19 preparation of, 105-8, 119-21

molecular shape parameter ratios, 141-4

polymers bulk viscosity, 144-6 dimensions listed, 194 neutron scattering studies, 174-9,

190-4 radius of gyration, 136, 137,

174-9 ring--chain polymer equilibrates, 11,

15-17 compared with kinetically

controlled distributions, 87-90 direct computational method,

22-5

INDEX 385

Dimethylsiloxanes-contd. ring-chain polymer equilibrates

-contd. F1ory-Suter-Mutter

computational method, 31 substituent effect on, 93, 94, 95

translational diffusion coefficients for dilute polymer solutions, 152-4

concentration dependence, 154-7 Dinaphthyl polymethylene chains,

cyclization studies, 308-9 3,5-Dioxabicyclo[5.4.0]undec-9-ene,

212 4H,7H-I,3-Dioxepin,212 1,3-Dioxolane

cyclic oligomers, 210, 211 cyclic polymers separated by GLe,

II ring-chain polymer equilibrates, II,

17-18 Diphenyl-I,4-diisocyanto methane

(MDI) molecular structure, 362 POP polyol polymerization with,

363, 364, 365 I,I-Diphenylethylene (D PE), 206 Dipole moment ratios,

dimethylsiloxane polymers, 146--7

Di(n-propylsiloxane), characteristic ratio quoted, 114

Dirac delta functions, 49, 50 Direct computational method,

calculation of cyclic concentrations, 14, 20-9

examples, 21-9 Direct procedure, configurational

distribution derivation, 47-52 Disc shapes, dimethylsiloxane cyclics,

136, 144 1,13-Di(tris-4-t-butylphenyl-

methoxy)tridecane, 5 Divinylidene compounds, 206--8 DMAP-labelled polystyrene, 327-9 DNA. See Deoxyribonucleic acids

(DNAs) Dodecahedrane system, 197

Donor-acceptor complex formation, 290, 316--18

Double contact cluster diagram, 63 Double-helix structure (of DNA),

uniqueness of, 227 Doubling reactions, peptide

cyclization, 263 Draining parameter, 76 Dynamic intrinsic viscosity, 72, 73 Dynamic light scattering, 52-3 Dynamic structure factor, 72, 74,

79-81

Effective-concentration factor, 294 Elastically ineffective loops, 370-1,

374 Electrochemical method, cyclization

kinetic studies, 288, 292 Electron spin resonance (ESR)

spectroscopy, ring closure kinetic studies, 291, 292

Electron transfer reactions, 206, 291, 308-9

Ellipsoid shape parameters, dimethylsiloxane polymers, 141-4

End-biting reactions, 199 End-groups, 220

cyclic vs. linear polymers, 2 spectroscopic studies, 286

Enzymes, DNA-cleavage, 226--7 Equilibrium cyclic concentrations. See

Ring-chain equilibration reactions

Escherichia coli, cyclic DNA in, 225, 226

Esters, cyclic oligomers, 213 Ethers, cyclic oligomers, 208-9 Ethyldimethacrylate-styrene system,

gel point studies, 352-3 Ethylene, block copolymers with

siloxanes, 130 Ethylene glycol polymers,

intramolecular reaction studies, 354

Ethylene oxide cyclic oligomers, 209

386 INDEX

Ethylene oxide-contd. polymers

cyclization studies, 334-7 pyrene-Iabelled, 335, 337, 339

Ethylene terephthalate cyclics, 31,213 separation by GPC, 12

Ethylmethylsiloxane cyclics, 92, 93, 94,95

G PC calibration plot, 109 Exact enumeration methods,

dimethylsiloxane polymers, 139, 140, 159

Excimer formation, 288, 312, 313, 322-3,339

Exciplex formation, 289, 311, 327 Excluded volume effects, 59-64, 296

cyclization spectroscopic studies, in, 329

liquid sulphur calculations, 35, 36 screening of, 340, 341 siloxane polymers, 118-19, 177, 179

First cumulant, 52-3 Flexible rings, equilibrium properties

of,46--64 Flory constant, III Flory-Crescenzi-Mark (FCM) model,

16, 22, 24-5 dimethylsiloxane polymers, 92, 96,

138, 159, 183 treatment of opposed gauche

rotations, 24, 96 Flory-Krigbaum-Orofino modified

theory (FKOm), 158 Flory (modified) equation, 158 Flory molecular size distribution, 115 Flory relationship, 10, 13 Flory-Stockmayer model, 369

gelation phenomena, 350-1, 369 Flory-Suter-Mutter method, 14,

29-32 compared with

lacobson-Stockmayer theory, 30, 31

Fluorescence, 286 Fluorescence quenching, 287, 306,

307,310-11

Fluorescence spectra, pyrene-Iabelled polystyrene, 326

Forster energy transfer, 319

Gas-liquid chromatography (GLC), 11

phenylmethylsiloxane oligomer stereoisomers, 10 I, 103

unfractionated dimethylsiloxane equilibrate, 105, 106

Gaussian statistics, 14, 15, 182,183 dihydrogensiloxane cyclics, 23 DNA cyclization, 229 gelatin processes, 357 meta phosphate cyclics, 27 Rouse-Zimm model, 300 siloxane cyclics, 23, 91 see also lacobson-Stockmayer

theory Gaussian-centre method, DNA helical

periodicity determined by, 254 Gel electrophoresis

band-shift technique, 255-7 DNA rings, 237, 238, 244-6, 255-7

Gel filtration, 27 Gel permeation chromatography

(GPC), 11-12, 27 analytical application for siloxanes,

108-19 calibration curves

polysiloxanes, 108, 109, 113 polystyrenes, 216, 217

dodecene cyclic oligomers, 201 preparative application for siloxane

cyclics, 119-24 siloxane polymers

analytical application, 108-19 calibration curves, 108, 109, 113 preparative application, 119-24

unfractionated dimethylsiloxane equilibrate, 105, 107, 108

Gel point definition of, 350 dilution effects on, 360 effects of cyclization on, 356, 370-1 factors affecting, 365, 367 probability equation for, 351

INDEX 387

Gelation absence of intramolecular reaction,

in, 350-3 experimental studies of cyclization

effect on, 359-67 studies

benzene-I,3,5-triacetic acid/decane-I, 10-diol polymerization, 360-1

polyoxypropylene polyol polymerizations, 362-7

theories, 357-9 Generator-matrix method, 139 Glaser conditions, 200 Glass transition temperatures

dimethylsiloxane polymers, 149, 150

networks, 376 Glycerol/adipic acid polymerization

studies, 359 'Good' solvents, 329-30; see also

Tetrahydrofuran; Theta(B)­solvents; Toluene

Graham's salt, 12 Gramicidin-S, 261, 264 Graph method, configurational

distribution derivation, 47-52 Guinier ranges, neutron scattering,

175, 177 Gyration, mean square radius of, 46,

54,60 dimethylsiloxane polymers, 136,

137, 174-9, 194 neutron scattering determination,

174-81, 194 neutron scattering studies, 174-81 ratio to hydrodynamic radius,

193-4

Harmonic spring (HS) model, 298 2,2,4,4,6,6-Hexamethyl-I ,3-dioxa-

2,4,6-trisilacyclohexane, 130 Hexane-l,6-diisocyanate (HDI)

intramolecular reaction studies, 354 molecular structure, 362 POP polyol polymerization with,

363, 364, 365, 368

High-performance liquid chromatography (HPLC), 13

liquid sulphur, 34 unfractionated dimethylsiloxane

equilibrate, 106, 108 Hydrocarbons, cyclics, 200-8 Hydrodynamic properties, DNA

rings, 237 Hydrodynamic radius

calculation from diffusion coefficient, 193

calculation of, 51 change on ring closure, 53, 62 ratio to radius of gyration, 193-4

Hydrodynamic screening, 340, 341 Hydrodynamic volume, G PC

sensitivity, 216 Hydrogenmethylsiloxane

cyclics, 92, 93, 94, 95 ring-chain equilibration reactions,

87 w-Hydroxycarboxylic acids,

cyclization studies, 305, 306

ILL (Institut Laue-Langevin) small-angle neutron scattering

spectrometer, 173, 174 spin--echo spectrometer, 189

Incoherent scattering cross-section, 169

values listed, 169 Ineffective loops, 370

estimation of fraction of, 371 reactivation of, 370 RIS model, 371

Intermediate coherent scattering law, 187

Intermediate scattering functions, neutron scattering, 170

Intermolecular condensation reactions, cyclic vs. linear polymers, 3-4

Interwound supercoil structure, DNA rings, 241, 242, 243, 248

Intramolecular reaction(s) gelation in absence of, 350-3

388 INDEX

Intramolecular reaction( s )-contd. linear and non-linear

polymerizations, 353-6 network formation requirement, 372 probabilities of, 35&-9

Intrinsic loss modulus, 73, 78 Intrinsic storage modulus, 73, 78 Intrinsic viscosity, 77-8, 111,216

dimethylsiloxane polymers, 15G--2

Jacobson-Stockmayer cyclization factor, 228-9

Jacobson-Stockmayer theory, 10, 14, 15-20, 45, 200

dimethylsiloxane ring-<:hain equilibrium, 98

direct computational method, compared with, 20, 23, 26, 28

dodecene cyclic oligomers, 202 Flory-Semlyen refinements, 45 Flory-Suter-Mutter method,

compared with, 30, 31 paraffin-siloxane cyclics, 12&-7, 128 siloxanes, 91

James-Evans dynamic model, 302-3

Karplus relation, polypeptides, 268, 269

Kilb model, 358-9 Kinetically controlled cyclization

reactions, 199,202 Kinetics, siloxane cyclization, 87-90 Kirkwood diffusion equation, 4&-7,

65, 66, 158, 159 Kirkwood-Riseman analysis,

poly( dimethylsiloxane) diffusion coefficients, 159-62

Knots, ring molecules, 6, 25G--I, 252 Kratky plots

dimethylsiloxane polymers, 182, 183, 184, 185

flexible rings, 56, 57 phenylmethylsiloxane polymers,

185, 186 rigid rings/rods, 67, 68 semiflexible rings, 71

Kuhn (segment) length, 70, 229

Labelling, problems encountered, 287, 292

Langevin equation, 74 Linear polymerization, gelation

during, 351, 354, 357 Linear polymers

compared with cyclic polymers, 1-7 size, 1

Linking number, DNA rings, 233-4, 236

free energy effects, 240, 241 Living polymers, 129,214, 219 Long-range interactions, 15, 269 Loops

effect on networks of, 368 effective/ineffective, 37G--I human DNA, 227 multifunctional polymerization,

43-4,45 polynucleic acids, 43, 44 smallest structures in RA3

polymerization, 372-3 Low-angle neutron scattering: see

Small angle neutron scattering Luminescence quenching, 287

Macrocyclics naturally occurring, 2, 197-8

see also Deoxyribonucleic acids; Peptide cyclics

term defined, 1-2 see also entries under individual

compounds Mark-Houwink parameters, III Mark-Houwink relationship, 116 Medium rings. See Mesocyclics;

Oligomeric cyclics Merrifield method, peptide synthesis,

263 Mesocyclics

preparation methods for, 7-10 term defined, 2

Metaphosphates, cyclic polymer concentration, 12, 25-7

INDEX 389

Metathesis reaction, 199 cyclic hydrocarbons synthesised

using, 200-2, 203, 205 Methylene-chain polymers

cyclization dynamics, 302, 305-13 particle scattering function

calculation, 183--4 N-Methylglycine polymers

catalysed ester hydrolysis, 314--16 cyclization studies, 313-19 donor-acceptor complex formation,

316-18 temperature effects, 316

Methylsiloxanes cyclics weight fractions, 92, 95 substituent effects, 93-7 see also Ethylmethylsiloxane

cyclics; Hydrogenmethyl­siloxane; Phenylmethyl­siloxane polymers; Propylmethylsiloxane; Trifluoropropylmethylsiloxane

Metropolis sampling, 139, 140, 183 Molar cyclization constants

dimethylsiloxanes, 117-18 w-hydroxycarboxylic acids, 306 paraffin-siloxane ring-chain

polymer equilibrates, 128 siloxanes, 94, 95

solvent effects, 98-9 substituent effects, 93, 94

Molecular shape, 137-8 dimethylsiloxane polymers, 138-44 correlation with bulk properties,

144--50 Monte Carlo techniques

benzophenone derivatives, 311 dimethylsiloxane polymers, 139,

140, 159, 183, 185 Flory-Su ter - Mutter compu ta tion,

30 Multiply twisted rings, configurational

distribution for, 50-1 Mushroom toxins, 277-9

Nairn-Braun dynamic model, 302, 303

Naphthalene salts, as initiators, 214, 215

Naphthyl end-groups, 308, 309 2-Naphthylmethyl end-groups, 320 Network(s)

complete reaction, 371-6 cyclization effects on structure,

375--6 formation, 367

cyclization effect on, 367-76 glass transition temperatures, 376 modulus, 372, 374 structures, 6-7

Neurophysin (NP), 279-80 Neutron scattering

advantages of, 167 data for various isotopes, 169 principles of, 168-70 wavelength used for, 167

Neutron spin-echo spectrometer, 187, 188,189

Neutron wavelengths, scattering studies, 167, 189

Newman projection, polypeptide chain, 270, 271

Niax triols, 362 Nitrile condensation reactions, 7, 8 Nonactine, 198 Norbornene, metathesis reaction of,

202, 203, 205 Non-linear polymerization, gelation

during, 351-2, 354, 357-8, 359-67

Nuclear magnetic resonance (NMR) spectroscopy

dimethylsiloxane cyclics, 147-9 peptides in solution, 267-74 solid-state peptides, 274--5 two-dimensional technique, 268,

281 Nylons, cyclic oligomers, 27-9, 30,

213

Observable event definition, 285 examples listed, 288-91

Oligomer cyclization dynamics, 302-3

390 INDEX

Oligomeric cyclics, 198-214; see also individual entries under appropriate chemical group name

Once-twisted rings configurational distribution for,

49-50 hydrodynamic radius affected by

rela tive size, 54 Oseen interaction, 297 Oseen tensor, 75, 188 OVI/OVI7 stationary phase (GLC),

11,106 Oxiranes, cyclic oligomers, 209 Oxypropylene polymers

intramolecular reaction studies, 354 polyol polymerizations, 362-7

polyester-forming reactions, 365-7

polyurethane-forming reactions, 364, 365

Oxytocin, 264-5 target protein, 279-80

Pab definition of, 356--7 evaluation of, 361

Paper chromatography, 12,25 Paraffin-siloxane copolymers, 125,

126--9 Particle scattering factors/functions,

46 calculation for dimethylsiloxane

cyclics, 181-3 calculation for polymethylene rings,

183-4 change on ring closure, 56, 57 deviation from Gaussian model, 136 flexible rings, 52 isotropic expression for, 181 neutron scattering data

dimethylsiloxane polymers, 184-5 phenylmethylsiloxane polymers,

185-6 rigid rings/rods, 65, 69 semiflexible rings, 71 sensitivity of, 185-6

Partisil (HPLC) column packing, 107 Pentaerythritol/adipic acid polymeri­

zation studies, 359-60 Pentaerythritol-based POP tetrol, 362 Peptide cyclics, 261-81

biological functions, 279-80 conformations, 265-7

compared (solution and solid state), 275-8

solid state, in, 274-5 solution, in, 267-74

naturally occurring, 261, 263-4, 275-80

synthesis, 262-5 Peptide hexamers, cyclization, 263 Peptide polymers

energy transfer studies, 319-21 see also N-Methylglycine polymers

Percolation models, 368-9 Phalloidin

biological functions, 279 chemical formula, 276 conformation data, 278

Phenylmethylsiloxane polymers characteristic ratio quoted, 114 cyclics

configurational isomers, 101-4 distribution of, 100, 101 weight fraction curve, 95

small-angle neutron scattering data, 180-1, 185-6

Phosphate melts, ring-chain polymer equilibrates, 12, 25-7

Photon correlation spectroscopy, 186, 192; see also Quasi-elastic light scattering

Phthalimido end-groups, 308 'Platonic' hydrocarbons, 197 Poisson ratio, DNA, 233 Poly ... , see main entry under

monomer name Polycombination reactions, 206--8 Poly(dihydrogensiloxane) (PDHS).

See Dihydrogensiloxane Poly(dimethylsiloxane) (PDMS). See

Dimethylsiloxane Poly(ethylene glycol) (PEG). See

Ethylene glycol polymers

INDEX 391

Polymethylene (PM) polymers. See Methylene-chain polymers

Polyoma (animal virus), double­stranded cyclic DNA in, 226

Polyoxypropylene (POP). See Oxypropylene polymers

Poly(phenylmethylsiloxane) (PPMS). See Phenylmethylsiloxane polymers

Preparation methods dilution method, 7-8 interchange method, 8-9 macrocyclics, 10-32 mesocyclics/oligocyclics, 7-10,

198-202, 206-16, 218, 219-20 peptide macrocyclics, 262-5 ring--chain equilibration reactions,

32; see also Ring--chain equilibration reactions

surface cyclization reactions, 9-10 Preparative gel permeation

chromatography, siloxanes, 119-24

Propylmethylsiloxane, cyclics, 92, 94 Protein template mechanism, peptide

synthesis, 264 Puckered configuration, large

dimethylsiloxane rings, 144 Pyrene end-groups, 288, 312-13 Pyrene-Iabelled polymers

poly(dimethylsiloxane), 338, 339 poly(ethylene oxide), 335, 337, 339 polystyrene, 288, 289, 321, 322,

325-9, 331-3, 339 poly(vinyl acetate), 333-4

I-Pyrenemethyl esters, cyclization studies, 313

Quasi-elastic light scattering (QELS), 152, 154

data for dimethylsiloxane polymers, 192, 194

Quasi-elastic neutron scattering (QENS)

experimental considerations, 189 experimental results, 190-4 principles, 186-9

Quenching mechanisms, 287

Random polymerization processes, 349, 350

Rate theory (gelation theory), 357, 373

Reduced first cumulant, 57-8 Refractive index

cyclooctene oligomers, 202 dimethylsiloxane cyclics, 144, 145

Relaxation time, 78 Repeating units, chain structures, 356 Restriction enzymes, DNA cleavage,

227,229 Rho(p )-parameter

definition of, 55 flexible rings, 55 rigid rings/rods, 67, 68 twisted rings, 56 values listed, 58

Rigid rings conformational relationships, 64-6 equilibrium properties of, 64-70 properties, 66-70

Ring--chain equilibration reactions, 10-32

concentration calculation methods, 13-32

concentration measurement methods, 11-13

molecules formed, 10 preparation of cyclic polymers

using, 32 siloxane copolymers, 125-30 siloxanes, 86-7 thermodynamic influences on

position of equilibrium, 97-100 Ring--chain equilibria, 199, 200 Ring strain effects, 8, 9, 10, 307 Ring-closure probability, DNA

cyclization, 229 chain length dependence, 229-33

Ring-opening polymerization reactions, 199, 208-9

Rods, compared with rings, 67-9 Rotational diffusion coefficient, exact

solution, 66

392 INDEX

Rotational isomeric state (RIS) model, 295-6

benzophenone derivatives, 311 cyclization dynamics of alkanes,

302 dimethylsiloxane polymers, 138,

159, 183 phenylmethylsiloxane polymers, 185 sulphur chains, 34-5 see also Flory-Crescenzi-Mark

(FCM) model Rotaxanes, 5

preparation of, 5-6 Rouse eigenvalues, 301 Rouse eigenvectors, 301 Rouse (spring-and-bead) model, 187,

188, 300 Rouse-Zimm (RZ) model, 299, 300-2 Rubbery networks

compared with glassy materials, 372 modulus of, 372

Ruzicka synthesis method, 8-9, 10

Sarcosine polymers, 313-19; see also N-Methylglycine polymers

Scattering cross-section, 168 Scattering function. See Particle

scattering factors/functions Scattering laws, neutron scattering,

170 Scattering lengths, values listed, 169 Sebacoyl chloride (SC)

molecular structure, 362 POP polyol polymerization with,

363, 365-7 Second virial coefficients, 63, 137

dimethylsiloxane polymers, 156, 157-8

Sedimentation coefficient, 53-4 Segmental friction coefficient,

dimethylsiloxane polymers, 158, 160

Self-quenching (of fluorescence), 306, 307

Semiflexible rings, equilibrium properties of, 70-2

Sephadex (gel filtration) column packing, 27

Sequential polymerization, treatment of gel points, 352

Shape function, 82, 83 Shape parameter ratios,

dimethylsiloxane polymers, 141-4

Silanolates, 86 Silicon-oxygen bond length, 138 Siloxane cyclic residues, preparation

of, 105-8 Siloxane polymers, 85-130, 132-63

block copolymers with styrene, 129-30

copolymers with paraffins, 125, 126-9

ring~hain equilibrium, thermodynamic influences, 97-100

ring~hain polymer equilibrates, 86-7

critical concentration, 90-3 see also Dihydrogensiloxane;

Dimethylsiloxane; Ethylmethylsiloxane cyclics; Phenylmethylsiloxane polymers

Siloxane rings formation in irreversible processes,

87-90 formation in ring~hain polymer

equilibrates, 86-7 Siloxane tetrameric rings

crown conformation, 96 OPC elution data, 112-13

Single contact cluster diagrams, 61 Sink term, cyclization dynamics, 298,

299 Small angle neutron scattering (SANS)

apparatus described, 172-3 experimental results, 174-86 principles, 170-2 radii of gyration determination,

174-81,194,218,219 sample details, 173-4

Smoluchowski boundary condition, 298

INDEX 393

Sol-gel transition, 368-70 Solvent effects

cyclization kinetics, 295 cyclization studies, 330-3 dimethylsiloxane ring-chain

equilibrium, 98-100 Sommer-Ansul synthesis method,

126, 127 Space-time correlation functions,

74-7 neutron scattering, 170

Spectroscopic methods, 285-345 advantages of, 285-6 sample size required, 293

Spring-and-bead models, 187, 188, 299,300

Star-shaped molecules, 56, 58 Staudinger macromolecular

hypothesis, I Step reactions, examples quoted, 199 Stereoisomers, phenylmethylsiloxane

oligomers, 102-3 Stokes-Einstein diffusion radius, 159 Stokes-Einstein equation, 193 Styragel (G PC) column packing, II,

119 Styrene--ethyldimethacrylate system,

gel point studies, 352-3 Styrene polymers

block polymers with dimethylsiloxane, 129-30, 214-16

concentrated solution studies, 341-4

cyclization studies, 321-33 dilute solution properties, 216-18 polymers, 214-18, 219 small angle neutron scattering data,

181,218,219 Substituents, effect on siloxane cyclics

formation, 93-7 Sulphamide bond formation, 288, 311 Sulphides, cyclic oligomers, 208-9,

212 Sulphonamides, fluorescence, 311 Sulphur

cyclic octamer, 33

Sulphur-contd. cyclic polymer concentration, 12-13 liquid

freezing point data, 33 molecular constitution of, 33-4 ring-chain polymer equilibrium

calculations, 35-7 rotational isomeric state model

for, 34-5 Supercoiled DNA rings, 235-6

effects on interaction with other molecules, 246-7

effects on shape, 247-8 effects on structure, 241-7 energetics, 237-41

Synthesis. See Preparation methods

Temperature effects, cyclization studies, 316

Tetrahedrane system, 197 Tetrahydrofuran, cyclic oligomers,

209-10 2,2,7,7-Tetramethyl-I-oxa-2, 7-

disilacycloheptane, ring-chain polymer equilibrates, 126-9

Thermodynamic considerations, ring-chain equilibrium, 97-100

Theta( O)-so Ivents siloxane polymers, 98, 115, 150,

154 styrene polymers, 325-9, 343, 344

Theta( O)-tem pera tures cyclohexane, 325, 341 dimethylsiloxane ring-chain

equilibrium, 98 Thorpe-Ziegler reaction, 7 Toluene, as solvent for

dimethylsiloxane polymers, 16,97, 98,99, III, 117, 118, 150, 151, 159-61

styrene polymers, 343, 344 Toluene-dB' 218, 219 Topoisomers, DNA closed-duplex

rings, 234-5 Toroidal supercoil structure, DNA

rings, 241, 242, 243, 248

394 INDEX

Translational diffusion coefficients concentration dependence, 154-7 dimethylsiloxane polymers, 152--4 exact solution, 66 Kirkwood's approximation, 46, 65

Tree structure, polymers near gel point, 351, 360, 368, 370, 377

Trifiuoromethanesulphonic acid, 87, 88,89,90

Trifiuoropropylmethylsiloxane, cyclics, 92, 93, 94, 95

Trimethylene succinate, ring--chain polymer equilibrates, 19, 20

1,3,6-Trioxocane, cyclic oligomers, 210, 211

Triplet quenching, 289, 311 Triplet-triplet (TT) annihilation, 289,

323-5,327 Twist number, DNA rings, 236-7 Twisted rings

configurational distributions for, 49-51

shrinking factors affected by number of twisting points, 55-6

Universal calibration (GPC) curves, polystyrenes, 216, 217

Urethanes cyclic oligomers, 213-14 gelation formation, 365-6 networks, 375-6

Valinomycin, 198

Vinyl acetate polymers concentrated solution studies,

340-1 spectroscopic studies, 333--4

Vinyl monomers, crosslinking in loops, 44, 45

Viscoelastic properties, 79 Viscosities: see Bulk viscosity;

Intrinsic viscosity

Wang-Uhlenbeck procedure, 48-9, 60,80

Watson-Crick structure for DNA, 234

Weight average molecular weights, siloxanes, 116

Weight concentration calculations, siloxane ring--chain equilibrate, 116-17

Weight fraction, siloxane rings, 91, 92 Weight-average functionalities, 352 Wilemski-Fixman (WF) model, 297,

299, 302 experimental results, 325

Writhing number, DNA rings, 236-7

X-ray diffraction analysis, cyclic peptides, 274, 275

Xylene, as solvent in dimethylsiloxane ring--chain equilibrium, 97, 100

Zimm hydrodynamic correction, 187 Zimm hydrodynamics, 81 Zimm (non-draining) model, 300 Zimm plot slope, 172