The Liquid-crystalline Blue Phases

Rep. Prog. Phys. 53 (1990) 659-705. Printed in the UK

The liquid-crystalline blue phases

T Seideman Department of Chemical Physics, The Weizmaiin Institute, Rehovot 76100, Israel

Abstract

An introductory review of the cholesteric blue phases is presented. The emphasis is on the basic concepts of the theoretical framework and the recent achievements of theory and experiment. We discuss a number of controversial issues indicating where future research is expected to shed light on the remaining unresolved problems.

This review was received in September 1989.

0034-4885/90/060659+47%14.00 0 1990 IOP Publishing Ltd 659

660 T Seideman

Contents

1. Introduction 2. Liquid crystals: an overview

2.1. Thermotropic liquid crystals 2.2. Nematics 2.3. 2.4. The order parameter

3.1. Calorimetric measurements 3.2. Density and refractive-index measurements 3.3. Light reflection 3.4. Polarising microscopy 3.5. Optical rotatory dispersion 3.6. Nuclear magnetic resonance measurements 3.7. Morphological studies 3.8. Electric-field effects 3.9. Crystal growth rate 3.10. Viscoelastic properties 3.1 1. Circular dichroism

4. Theory 4.1. The Landau theory of cholesterics 4.2. The Landau theory of the blue phases

5. Theory versus experiment: higher harmonics 6. Recent progress

The chiral nematic (cholesteric) mesophase

3. Experimental study of the blue phases

6.1. The problem of BPI11 6.2. External field effects 6.3. Multiwave scattering

Acknowledgments Appendix 1. Crystallographic notations and space-group symmetry Appendix 2. Dyadic operator formulation References

7. Discussion

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1 , 1 , 1 , 1 1 I I I I I I I I

1. Introduction

I T I

66 1

Blue phases ( BPS), liquid-crystalline phases appearing mostly in cholesteric derivatives just below the clearing point, were first observed as early as 1888 by Reinitzer, who reported the discovery in a well known letter: ‘On cooling of the molten compound a bright blue-violet colour phenomenon appears which disappears rapidly followed by a non-uniform turbidity. . .’ (Reinitzer 1888). This historical detection of selective reflection in the blue phase of cholesteryl benzoate was in fact the first discovery of mesomorphism. The position of the three blue phases, intermediate in temperature between the cholesteric phase and the isotropic liquid, is shown schematically in figure 1.

I CHOLESTERIC I BPI I B P U I BP m I ISOTROPIC I

Reinvestigation of this modification began in 1956 with the paper of Gray, who termed it ‘blue phase’ (Gray 1956). Preliminary experimental measurements, performed by Barrall et a1 (1967) and later by Arnold and Roediger (1968), did not yet identify the observed phase transition with a transition to the BP. The differential scanning calorimetry (DSC) of Armitage and Price (1975) was the first to provide clear evidence for the existence of a stable phase between the isotropic liquid and the cholesteric phase. Recent experimental work has concentrated on the development of both sensitive detection methods, such as electric-field studies (Porsch et a1 1984, Cladis er a1 1984, 1986, Pieranski et a1 1985, 1986, Pieranski and Cladis 1987a, b, Heppke er al l983,1985a, b, 1987,1989, Yang and Crooker 1988,1990), and improved crystal growth techniques (Blumel et a1 1983, Blumel and Stegemeyer 1984a, b, 1985). A comprehensive review of the experimental aspects was presented by Stegemeyer er al (1986). Very recently the current status of experiment was reviewed by Crooker (1989), who emphasised the question of BPIII, the highest-temperature blue phase (figure 1).

662 T Seideman

From the theoretical point of view, the issue was first addressed by Saupe (1969), who proposed a chiral cubic BP structure without, however, offering theoretical proof. Several other structural models have been formulated in recent years, including the randomly oriented globules theory proposed by Gray and Winsor (1974) and by Demus and Richter (1978), the modified helix structure of Bergmann et al (1979) and the emulsion model developed by Finn and Cladis (1982). Of more impact was the disclination model suggested by Meiboom et a1 (1981, 1983a, b) (see also Berreman 1984). Based on a modification of Frank’s theory, these authors predicted the existence of a three-dimensional ( 3 ~ ) lattice of linear defects with essentially isotropic cores, the director field adopting a double twist configuration. The locally energetically favoured double twisting (i.e. local twisting in two perpendicular directions, as opposed to the single twist of the cholesteric) is nevertheless topologically globally incompatible with the requirement of continuity. This results in the formation of disclinations, i.e. defects in the director field (see figure 2). A similar structure was shown to arise on the basis of the theory of elasticity of liquid crystals.

Figure 2. Configuration of disclination lines in cubic unit cells: ( a ) sc 0‘ symmetry; ( b ) BCC o5 symmetry; (c), ( d ) BCC o8 symmetries. (From Berreman (1984).)

The most successful theoretical study to date is the so-called Landau theory of the cholesteric BP. Inspired by de Gennes’ adaptation of Landau’s theory to cholesterics (de Gennes 1971), this approach was first considered by Brazovskii and coworkers (Brazovskii and Dmitriev 1975, Brazovskii and Filev 1978), who formulated a hexagonal structure model. Later extensive mean-field computations, initiated by Hornreich and Shtrikman (1979), have proved this structure unstable with respect to the 3~ cubic, thus providing the structural ansatz of Saupe with a profound theoretical basis.

Further studies within the Landau theory framework were carried out by Kleinert (1981), Alexander (1981), Grebe1 et a1 (1983a, 1984) and others. A comprehensive review, mainly of the theoretical aspects, was given by Belyakov and Dmitrienko

The liquid-crystalline blue phases 663

THEORY

(1985). The symmetry properties of the BP order-parameter field were analysed and the underlying connection with the measurement was clearly made (Belyakov and Dmitrienko 1985). Different aspects of the blue-phase theory were recently reviewed by Hornreich and Shtrikman (1988~) and by Wright and Mermin (1989). Extensive theoretical work carried out within the last couple of years had as its objective primarily the understanding of B P I I I (the fog phase) (Hornreich et a1 1982, Hornreich and Shtrikman 1986a, b, 1987a, b, Rokhsar and Sethna 1986, Filev 1986, 1987) and the analysis of external field effects (Hornreich et al 1985a, b, Lubin and Hornreich 1987, Hornreich and Shtrikman 1988a, 1989, 1990). Alternative points of view to the origin and consequences of the double twist configuration characterising the RP have recently been advanced by Sethna (1983a, b), Pansu et a1 (l987), Hornreich and Shtrikman (1988b) and Dubois Violette and Pansu (1989). In figure 3 we have collected some of the important landmarks in the history of research into the properties of the blue phases.

Today, after what has been termed 'a hundred years of puzzle', and after over a decade of intensive study, the subject of the BPS seems to have reached a concluding point. Although, as discussed below, a number of properties remain obscure and a few aspects are still controversial, most underlying features are presently well understood. Since various specific aspects of the blue phases have been extensively reviewed by well known experts in the field, we would like to use the present opporunity to

EXPERIMENT

Reinitzer (18881: 'a bright blue-violet colour phenomenon'

I I Sauoe 119691cubic s t r u c t u A

Gray (1956) the'blue phase' k r - 4 Brozovskii and Dmitriev (1975) Landau theory, hexagonal structure?

Hornreich and Shtrikman (1979) Landau theory, cubic structure

Theoretical anal sis of B P I I I - 1 1 Fluctuation effects

I representation J

Hornreich and Shtrikman 11985-89). field-induced new blue phases

osc f i r s t clear

Somuiski and Luz (1980) N M R

Collings (1984) I O R O s tabi l i ty of BPIII

Figure 3. Landmarks in the history of the blue-phase research.

664 T Seideman

provide a more general framework, on an introductory level, attempting to address a larger audience of less familiar readers. It will be our goal to offer, as far as possible, a self-contained review of the theory and experiments of the blue phases, emphasising both the historical development and the recent achievements. In particular we shall point out the close interplay between theory and experiment throughout the history of the blue-phase research (figure 3). Space limitations preclude us from considering in more detail a number of related topics where recent progress has been made, including the physics of diffraction by imperfect crystals of the BP and methods for describing it theoretically, and fluctuation effects, manifested in pretransitional phenomena such as specific-heat variations (Thoen 1988) and rotation of the plane of light (Dolganov et al 1980). These aspects were investigated by Belyakov and Dmitrienko (1985). An additional question of general theoretical interest that will remain beyond the scope of this work regards the analogies between the physical properties of the BP and colloidal crystals. These features were discussed by Pieranski (1983).

This review is organised as follows. Section 2 consists of a brief overview of the world of liquid crystals, and a more detailed introduction to the cholesteric phase. In section 3 the experimental status is considered and section 4 is devoted to the theoretical counterpart. Starting with a general introductory discussion and proceeding with the basics of Landau’s theory of the BP, this section is primarily intended for the benefit of the less familiar reader. Section 5 compares and contrasts the results and implications of the theoretical and experimental work and in section 6 we discuss questions of current active interest. The final section (section 7) summarises this work, briefly indicating where further research is likely to bring about new developments. A number of supplementary topics are briefly reviewed in the appendices.

2. Liquid crystals: an overview

The liquid-crystalline phase is a state of matter that is observed in certain materials intermediate between the solid crystal and the isotropic liquid. It is thus a particular case of a mesophase (mesomorphic: of intermediate form). A second, basically distinct mesophase is the disordered crystal (plastic crystal). In the literature, however, the terms ‘liquid-crystalline phase’ and ‘mesophase’ are often interchanged owing to the practical importance of the former. For fluid mesomorphism to occur, the constituting molecules must be highly geometrically anisotropic, often elongated (calamitic) but many times disc-like (discotic). They possess some degree of orientational order and in certain cases partial translational order, although the solid-like three-dimensional periodicity has been lost. It is the combination of liquid-like fluidity and solid-like molecular order that imparts the liquid crystals their uniqueness and gives rise to their many interesting properties.

As shown in figure 4 we distinguish two types of fluid mesophases, the lyotropics and the thermotropics. The former are composed of two or more components. Mostly one of the components is amphiphilic and another is water (figures 4(a) and ( b ) ) . The principal interaction producing long-range order is thus the solute-solvent interaction. Phase transitions may be induced by varying the solute concentration. The two most common lyotropic phases are the lamellar phase, exhibiting one-dimensional periodicity (figure 4( a ) ) , and the hexagonal phase, exhibiting two-dimensional periodicity with rod-like aggregates packed into a hexagonal lattice (figure 4(b)). It is


F \ \ r

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interesting to note that cubic lyotropic phases have also been reported in the literature (Lindblom and Rilfors 1989). Lyotropic liquid crystals are of interest biologically and play an important role in living systems. Below we shall be primarily interested in the second class of fluid mesophases.

2.1. Thermotropic liquid crystals

As implied by the title, phase transitions among the thermotropics are most naturally effected by thermal processes (see figure 4). When heating the mesogen, one first reaches its melting point ( T,) where the solid crystal changes into the liquid-crystalline phase. Upon further heating, a second transition point is reached (TJ where the liquid crystal becomes isotropic. Often the liquid-crystalline phase is rather turbid, becoming optically clear at the phase transition to the isotropic. (This feature characterises the nematic mesophase to be discussed below). The second transition temperature is therefore referred to as the clearing point, T,. The melting point T’ and the clearing point T, define the temperature range in which the mesophase is thermodynamically stable. Both phase transitions are first-order as indicated by the occurrence of a latent heat and a discontinuous change in density. Many compounds (termed polymorphic) exhibit more than a single mesophase. It should perhaps be pointed out that phase transitions from one mesophase to another are not necessarily first-order. Our interest in the thermotropic mesophases is twofold. From the basic research standpoint, thermotropics provide a convenient object for experimental study of critical phenomena and a unique proving ground for testing theoretical phase transition models. Their commercial interest, in particular for display systems (Meier et a1 1975, Schadt 1989) stems from the sensitivity of these mesophases to external fields, from their diversity and from their relatively low price.

Following Friedel (1922), thermotropic liquid crystals are classified on the basis of symmetry into two major classes: smectics and nematics. The smectic mesophases possess one-dimensional positional order as well as a certain degree of orientational order. The name ‘smectic’ is derived from the Greek (ukqypci =soap) and reflects their soap-like mechanical properties. All smectic modifications have in common a stratified structure, but a variety of molecular arrangements is possible within each stratification. In smectic A the molecules are on average upright in each layer with their centres irregularly spaced in a random fashion (figure 4 ( c ) ) . Smectic B differs from A in that the molecular centres in each layer are hexagonal close-packed. Smectic C is similar to smectic A but the molecules are inclined with respect to the layers (figure 4 ( d ) ) . In the B, form there is an ordered arrangement within each layer in addition to the tilt. A number of other distinct smectic modifications have been identified (see Gray and Goodby (1984) for a complete discussion).

In what follows we confine attention to the second class of thermotropics-the nematic liquid crystals.

2.2. Nematics

The nematic mesophase has a high degree of long-range orientational order of the molecules but lacks long-range translational order. Consequently there are no Bragg peaks in the x-ray diffraction pattern and the viscosity is low compared to that of the smectic. Nematics differ from the isotropic liquid in that the molecules are spon- taneously oriented with their long axes approximately parallel. The preferred direction


is arbitrary in space, but a homogeneously aligned specimen is birefringent, with uniaxial symmetry. The dual aspects of the nematic are exhibited most spectacularly in its nuclear magnetic resonance ( N M R ) spectrum. The high fluidity leads to narrow liquid-like lineshapes while the anisotropicity results in typical line splittings, absent in the disordered isotropic phase.

The local direction of alignment characterising the nematic phase may be con- veniently described in terms of the director n, a unit vector parallel to the axis of symmetry. Since the axis has no macroscopic polarity, n and -n are equivalent; the sign of the director has no physical significance. In an actual sample the orientation of n is imposed by the boundary conditions and possibly also by external fields. The word ‘nematic’ originates from the Greek ( V ? ~ L I Y =thread). The thread-like defects that are commonly observed in these textures arise from discontinuities in the director pattern, which otherwise varies continuously over distances much larger than the molecular dimension.

Research into the structure of the nematic phase dates back to the pioneering work of Born (1916), who was the first to realise the signifiance of an anisotropic component in the intermolecular potential. A large number of molecular theories of the nematic mesophase were formulated in the years that followed, one of the most successful of which has been the Maier-Saupe mean-field model (Maier and Saupe 1958, 1959, 1960). In spite of its simplicity, the theory developed by Maier and Saupe provided a clear explanation for the alignment tendency of nematics and described fairly well the nematic-isotropic phase transition. Refined versions have been suggested by Gelbart and Gelbart (1977), by Luckhurst and Zannoni (1977) and by others. Con- tinuum theoreis of the nematic state have also been extensively discussed in the literature (see for instance Chandrasekhar (1980, ch 3)).

Within the nematic mesophase we distinguish between the proper nematics and the chiral nematics, commonly known as cholesterics. In the former the constituting molecules are optically inactive (or else the mixture is racemic), while in the latter case the building elements are chiral entities. Historically, cholesteric liquid crystals derived their name from the fact that the first materials that were observed to exhibit the characteristic helical structure were esters of cholesterol. It should perhaps be pointed out that optical activity is found also in various non-steroidal mesophases.

The question of whether the cholesteric (also termed ‘helical’) mesophase should be classified as a subgroup of the nematic mesophase or, in fact, the proper nematic forms a particular case of the more general cholesteric phase (i.e. that corresponding to an infinite pitch) is somewhat philosophical in nature. Below we shall conform to the common use of regarding the proper nematics and the chiral nematics as two distinct classes, referring to the former as the nematic and to the latter as the cholesteric mesophase. It should nevertheless be borne in mind that on the molecular level the two subgroups are indistinguishable.

We shall now confine attention to the proper (i.e. finite pitch) cholesteric subgroup, which will be of prime interest for the discussion of the BP below.

2.3. The chiral nematic (cholesteric) mesophase

Owing to the characteristic left-right asymmetry of the constituting molecules, the chiral nematic mesophase possesses a screw axis superimposed normal to the preferred molecular direction. This configuration is depicted schematically in figure 4( f). On a large scale (as compared to molecular dimensions) the predominant orientation

668 T Seideman

direction varies along an arbitrary spatial axis (i) such that a helix structure is described:

n, = cos( qcz + 4)

ny = sin( qcz + 4) (2.1)

n, = 0.

The spiral structure can be represented by an axial vector (pseudovector) along the helix axis. Thus, the structure is periodic with its spatial period ( L ) equal to half the pitch length ( p )

J - ' = p / 2 = ../lqcl.

The sign of qc distinguishes between left and right helices while its magnitude determines the period.

As pointed out above, the presence of chiral molecules is a prerequisite for the appearance of the cholesteric mesophase. Following de Gennes (1974) we consider a general twisted structure n, = cos[ e( z ) ] , ny = sin[ e( z ) ] , and express the free-energy density F as a function of the twist q = a e / a z . In materials where right and left are indistinguishable, F ( q ) must be invariant with respect to sign, F ( -4) = F ( q ) . In that case the free energy is minimised either at q = 0 or symmetrically about zero at q = * q l (figure 5(a)) . The second solution requires the interaction range to extend up to second neighbours. In nematics, where short-range forces dominate, the first solution is observed. In the case of optically active molecules, corresponding to the cholesteric, F ( q ) is not symmetric and the optimum twist qc is in general non-zero (figure 5(b)).

Figure 5. Variation of the free energy with pitch: ( a ) systems that do not distinguish right from left (curve A corresponds to the nematic liquid crystal); ( b ) systems where right and left are distinguishable, such as the cholesteric liquid crystal. (From de Gennes (19741.)

The requirement of optically active building elements formed the basis of both the molecular-statistical theory developed by Goossens (1971) and the phenomenological model suggested by de Gennes (1974, chs 3 and 6). The former relies upon an extension of the Maier-Saupe idea. The latter utilises the concept of long-range distortions induced by chiral molecules in a nematic matrix. Following the work of Goossens, several molecular-statistical models have been formulated in an attempt to explain the spontaneous cholesteric twist (Priest and Lubensky 1974, van der Meer and Vertogen 1976, Lin-Liu et a1 1977). The Landau theory approach to the cholesteric structure (de Gennes 1971) is reviewed in section 4.1.

From the point of view of its optical characteristics, the cholesteric mesophase was investigated in detail by de Vries (1951), by Berreman and Scheffer (1970) and by Dreher and Meier (1973). A complete analysis of either the origin or the consequences of the unique helical structure is beyond the scope of this introduction. We note,


however, that the spiral arrangement is responsible for a number of remarkable features of the cholesterics, which are in turn manifested in their various applications. These features include an extraordinarily strong rotatory power and Bragg reflections at optical wavelengths (i.e. selective reflection of circularly polarised light.) Since the cholesteric pitch and hence the wavelength of the Bragg reflected light are sensitive to external agents (such as temperature and pressure), the colour of the mesogen can change drastically in a small temperature or pressure interval. This property leads to a number of applications: detection of hot points in microcircuits, localisation of fractures and tumours, conversion of infrared images to visible, etc. Cholesteric devices are particularly useful in applications where large-area temperature or pressure profiles must be determined. For a more complete discussion of applications of liquid crystals in general and cholesterics in particular, see Schadt (1989) and references therein.

2.4. The order parameter

The self-organising tendency of fluid mesophases may be cast in a quantitative form by defining an order parameter such that it vanishes in the highly symmetric isotropic liquid and remains non-zero in the ordered phases. Regarded from the microscopic point of view, physical insight may be gained by considering first a simplified model of rigid-rod-like nematic molecules (i.e. both the phase and the constituent molecules are assumed uniaxial). The cylindrical symmetry and the tendency to align about the director may be expressed via an angular distribution function f( 4, 8, I)) satisfying

f(4, 4 I)) = f ( O ) ( 2 . 2 ~ )

(see figure 4 ( e ) ) and

f ( e ) = Y ( ~ - e) =f(cos e) (2 .2b )

The simplest way of defining a measure of alignment is by the parameter S, first (recall that n and -n are equivalent).

introduced by Tsvetkov ( l 9 4 2 ) ,

s = (3(C0S2 e)-1)/2. (2.3)

Equation (2 .3) may be regarded as the second term (the quadrupole moment) of an expansion in Legendre polynomials, where we omitted the zero-order (constant) term and noted that the first term (the dipole moment) vanishes identically: (cos e )= j f (e) cos 8 dR = 0 (see equation (2.26)). Clearly S = 1 for the completely ordered phase while S = 0 for the disordered isotropic.

In describing a macroscopic order parameter we keep in mind the shape anisotropy of the constituent molecules, due to which all molecular response functions are anisotropic. It is a consequence of the long-range order that the same property characterises the response of the bulk material. We are thus led to describe the degree of orientational order in terms of the anisotropic part of a macroscopic tensor property. This notion will be made more precise in section 4 below.

3. Experimental study of the blue phases

Early experimental identification and characterisation of the BP has been carried out along a number of major lines.

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3.1. Calorimetric measurements (figure 6)

In their pioneering differential scanning calorimetry ( DSC) studies, Armitage and Price (1975) measured a finite but very small density change at the transition to the BP, showing the stable phase to exist in a very narrow temperature range (-1 K) and indicating a discontinuous transition (first-order) between the cholesteric and the coexisting BP. Later, better resolved DSC thermograms of cholesteryl nonanoate (CN)

(Bergmann and Stegemeyer 1979a, b) detected an additional peak, thereby providing evidence for the existence of two different polymorphic forms of BP. Of particular interest are investigations of so-called induced cholesterics, formed by mixing optically active cholesterics with a nematic phase, thus permitting an effective continuous change of the helical pitches. The phase diagram of such mixtures was determined calorimetri- cally by Bergman (1980), who showed the BP to appear only in systems characterised by a short pitch. Further specific-heat measurements of C N were conducted by Kleiman et a1 (1984), who detected the third BP in this substance. Mapping out the phase diagram of CN, Kleiman and coworkers have shown the cholesteric-, BPI, BPI -+ BPII

and BPII + BPIII transitions to be first-order. Regarding the nature of the B P I I I -, isotropic phase transition, no firm conclusion could be reached. The thermodynamic properties of CN were recently reinvestigated via adiabatic scanning calorimetry by Thoen (1988) (see figure 6). This modified mode of operation permitted a proper differentiation between pretransitional effects and true latent heats, which could not be attained in previous studies. All phase transitions, including the BPI I I + isotropic, were conclusively shown to be first-order, the reported transition temperatures being consistent in all cases with the data of Kleiman et a2 (1984).

3.2. Density and refractiue-index measurements (figure 7 )

The optical isotropy of BPS, first demonstrated photographically by Saupe (1969), was later confirmed by quantitative refractive-index measurements (Pelzl and Sackmann 1973, Demus et al 1978), which have shown a zero birefringence in BP and BPII, as opposed to the cholesteric. Investigating the temperature dependence of the density of cholesteryl myristate (figure 7 ) , Demus et a1 found evidence for a first-order

100 - 89.5 90.0 90.5 91.0 91 5

T P C ) Figure 6. Calorimetric measurements: the reduced heat capacity per mole (C , /R) for the temperature range covering all BP transitions in cholesteryl nonanoate. (From Thoen (1988).)

The liquid-crystalline blue phases 67 1

09*' 1 Blue phase

0 918 I I I I I . l , l . . I I . I I I , I . . .

-0 10 -0 05 0 0 05 0 10 T-T, i0C1

Figure 7. Density and refractive-index measurements: density change at the sp-isotropic transition of cholesteryl myristate, for decreasing temperature (-) and increasing temperature (----). (From Demus er al (1978).)

isotropic + BP transition, while the BP -+ cholesteric transition was claimed to be second- order. More recently, studies of BPII I (Stegemeyer et a1 1986) have proved this phase to be non-birefringent, as is the case for the two lower-temperature BPS.

3.3. Light reflection (Jigure 8)

Preliminary selective reflection (SR) measurements have shown the BPI and BPII signals to be of comparable intensity to that of the cholesteric, and of the same sense of the reflected circularly polarised light, while well shifted in energy. The angular dependence of SR was found to be analogous to that of cholesterics (Bergmann and Stegemeyer 1979b) in contradiction to early results (Goldberg and Schnur 1970). Subsequent temperature-dependence measurements (Bergmann et a1 1979) have clearly identified two distinct phases below the cholesteric clearing point-BPr and BPII. The equivalent pressure-dependence measurements of selective reflection of BPS were carried out by Pollmann and Scherer (1980), who first constructed the full, four-component phase diagram. Supporting evidence for the role of the helical pitch length was provided by SR measurements of a mixed cholesteric-nematogenic compound by continuously varying the mole composition. The critical pitch (p,) below which the formation of BP is no longer possible was quantitatively measured. Of much significance are the early experiments of Meiboom and Sammon (1980), which were the first to indicate a cubic BP. Clearer evidence for a cubic phase was derived by Johnson et a1 (1980,

1 380 nm 530 nm 1

Figure 8. Light reflection: polarised Bragg scattering measurements of the two BPI1 lines in a C B I ~ / E ~ mixture. The incident and reflected beams are right (left) circularly polarised for the upper (lower) curve. (From Tanimoto and Crooker (1984).)

672 T Seideman

1984), who identified four Bragg reflections for both BPI and B P I I and assigned them to the (1 lo}, (200}, (2 11) and (220) planes of a body-centred structure. The experiments of Marcus and Goodby (1982) were the first to describe the variation of the chirality parameter alone, thereby permitting a direct and unambiguous comparison against theory. Mixtures of the chiral and racemic forms of the same compound were studied as a function of composition and temperature, presenting evidence that the system’s thermodynamic properties do not change as the chirality is altered by addition of the racemic mixture. Observations of B P I I reported a single peak of width greater by a factor of 3.8 than that found for BPI and BPI] (Demikhov et a1 1985). These studies, currently of active interest, may not be considered conclusive. This is primarily due to the experimental resolution, which is, as yet, hampered by the difficulty in obtaining large enough B P I I I crystals (Demikhov et a1 1985). More recently the phase diagrams of compounds that exhibit all three blue phases were measured by Yang and Crooker (1987).

3.4. Polarising microscopy (figure 9)

Mixed nematogenic-cholesteric systems were investigated by means of polarising microscopy by Bergmann and Stegemeyer (1979b) and later by Blumel and Stegemeyer (1984a, b), who estimated the BP temperature range as a function of pitch length. The possibility of a third BP between the isotropic liquid and the B P I I platelet texture was first alluded to by the polarising microscopy work of Stegemeyer and colleagues. The monomorphic (dimorphic) pitch, pm ( p d ) , defined as the pitch length above which only one (two) BP is (are) stable (Stegemeyer et a1 1986), as well as the critical pitch, p c , were measured for a variety of systems.

3.5. Optical rotatory dispersion (figure 10)

Early studies were carried out mostly in Stegemeyer’s laboratory. The spectra measured by Stegemeyer and Bergmann (1980) exhibit the same qualitative features of the SR

experiment, i.e. the same sign of the anomalous optical rotatory dispersion (ORD) curve, however red-shifted as compared to the cholesteric, and the appearance of an additional signal (attributed to B P I I ) at a somewhat higher energy. The ORD measurements of Collings (1984a) first provided convincing evidence for the stability of B P I I I .

In a temperature range of a few hundredths of a kelvin (==0.05 K) below the clearing point, he observed ORD spectra with a rather different pattern from the typical dispersion curve found in BPI and BPII , indicating a different structure. BPIII was detected only in systems with very short pitches, its temperature span decreasing rapidly with increasing pitch. Experimental phase diagrams containing data of all three BPS were subsequently obtained by the ORD studies of Collings (1986).

3.6. Nuclear magnetic resonance measurements (figure 1 1 )

Early fluorine-19 and proton nuclear magnetic resonance ( NMR) investigations of the BP were carried out by Collings et a1 (1976). These, however, did not yield quantitative information, owing to the relatively low field and the weak anisotropic interactions of the nuclei employed. Conclusive evidence for a cubic structure was given by the deuterium NMR measurements of Samulski and Luz (1980), thus verifying the Meiboom and Sammon suggestion and ruling out the hexagonal symmetry assumption. The


Figure 9. Polarising microscopy: ( 0 ) three-dimensional RI' single crystals with faces of the form ( 1 10) and ( 2 1 1 ) ; ( h ) a construction of the complete hahit observed, involving the forms ( 1 1 0). ( 2 1 1 ) and ( 1 0 0). ( From Rlumel and Stegemeyer (1984a).)

spectral lineshape has been shown to be sensitive to the dimension of the elementary structural units, the average order parameter and the diffusion rates (see figure 11). Interpreting the spectra in terms of a simple theoretical model, Samulski and Luz ( 1980) related the correlation time for the director reorientation to the average quadrupolar splitting and extracted the former from the experimental data. Subsequent, confirming N M R studies were carried out by Yaniv er a1 (1983).

3.7. Morphological studies ( jgure 12)

The determination of the space group that describes the UP lattice symmetry may also be achieved by morphological studies of single crystals. Blumel er a1 (1983), Blumel and Stegemeyer (1984a, 1985), Onusseit and Stegemeyer (1981, 1983, 1984) and Onusseit ( 1983) reported the observation of such crystals, characterised by well defined

674 T Seideman

Figure 10. Optical rotatory dispersion spectra: the BP of cholesteryl nonanoate. ( a ) BPI ,

( b ) BPII , (c ) BPII I . (From Collings (1984b).)

T c - 0 . 5 K T c - 1 . 5 K

Figure 11. Nuclear magnetic reasonance: deuterium N M R spectra in the BP of ( a ) , ( b ) neat deuterated liquid crystals (slow diffusion) and (c), ( d ) deuterated probe molecules dissolved in normal liquid crystals (fast diffusion). The spectra on the left correspond to a temperature of 0.5 K below the clearing point (TJ. Those on the right were recorded at 1 K lower. (From Samulski and Luz (1980).)

77ie liquid-crystalline blue phases 675

Figure 12. Morphological studies: microphotographs of RP single crystals. ( a ) R P I I crystal, 9290°C; ( h ) the same crystal after 10s; ( c ) the same crystal after the transition to R P I I ,

92.55 "C. (From Stegemeyer el a/ (1986).)

habit. The phase sequence isotropic-, R P I I I -, R P I I -, R P I on decreasing the temperature of short-pitch cholesterics ( p < p d ) was accompanied by a reproducible variation of colour and shape: rapidly growing quadratic crystals of blue selective reflections change into crosshatched green l m at the second transition temperature, with which the growth stops. In monomorphic blue-phase systems ( pm < p < pc) , where BPI single crystals grow directly from the isotropic phase (Blumel er a1 1983, Blumel and Stegemeyer 1984a), a rather different structure was found due to the slower growth rate. Three- dimensional rhombic BPI crystals were developed by Blumel er a1 (1983) and by Stegemeyer er a1 (1986), the former reporting up to 0.3 mm sized single crystals of different colours, including green, orange and violet reflections. The medium-pitch range, pd < p < pm, although less easy to characterise experimentally, was again proved to involve a different habit of the much slower growing BPI] crystals. Attempts to obtain similar crystallites by 'annealing' R P I I I have been unsuccessful.

3.8. Electro-field effects (figure 13)

Early experimental work confined attention to the study of materials with positive dielectric anisotropy ( E : , ) . The possibility of field-inducing a RP + cholesteric phase transition by increasing the pitch to above pc was investigated by Armitage and Cox (1980) and by Finn and Cladis (1982). Porsch er a1 (1984) investigated the thermal R P I I + R P I transition in the presence of a constant external field. On cooling a cholesteric-nematic mixture, a typical striated texture was obtained, the symmetry of which indicated field-induced birefringence. Detailed investigation led to the conclusion that the cubic structure has been distorted by the field, thus allowing optical biaxiality. Complete voltage-temperature phase diagrams were next presented by Stegemeyer er a1 (1986). Electric-field-induced structural transitions among rhe various

676 T Seideman

Figure 13. Electric-lield effects: a microphotograph showing the phase transition from 1)PI

to H P I I (left to right) via a distinct, field-induced RI'. The voltage is 23.8 V and the temperature 33.56 "C with a small gradient from left to right. (From Porsch and Stegemeyer (1987).)

BP.S were first observed by Pieranski et a1 (1985) in a mixture of biphenyls ( c B I ~ / E ~ ) . The simple cubic phase of B P I I , oriented with its fourfold axis parallel to the field, transformed first into a phase of tetragonal symmetry, next to a 3~ hexagonal structure and finally into a ZD hexagonal phase. The phase transitions were recorded by observations of morphological changes of single crystals in contact with the isotropic liquid. Cladis et a1 (1986) employed the technique of Kossel diagrams (single-crystal diffraction patterns formed by a convergent beam) to follow the field-induced transitions in the BPI of a similar biphenyls composition. This experiment permits direct observation of the evolution of the unit cell in reciprocal space as a function of field. It allows, furthermore, differentiation between different cubic crystals, since the diffraction patterns reflect the underlying symmetry. Studying the field-induced variation of reciprocal-space and real-space structural parameters, Cladis et a1 (1986) showed that an increasing field applied parallel to the (1 10) direction of the BCC monocrystal transforms the twofold axis into the fourfold axis of a BC tetragonal crystal. It was concluded that the phase transition is weakly first-order. Subsequent Kossel diagram studies confirmed the detection of tetragonal 14122 (D:') (Pieranski and Cladis 1987a) and 2~ hexagonal structures (Jorand and Pieranski 1987, Jerome and Pieranski 1988, 1989), in addition to the 3~ hexagonal P6222 (D:) (Pieranski et a1 1985). Very recently, a similar technique has been employed for the study of molecules with negative dielectric anistotropy (Heppke er a1 1989). Similar symmetry arguments to those derived for the E,>O case (Jerome and Pieranski 1988, 1989) led to proposing the space group P6222 (Dd) (or P6422 (D;)) for the newly observed 3~ hexagonal structures. The selective reflection of B P I I I samples in the presence of electric fields was measured by Yang and Crooker (1988, 1990). Materials of positive as well as negative dielectric anisotropy were analysed, and a number of theoretical B P I I I models were discussed in light of the experimental data.


" - _

400 I - c .- E E -I 200-

3.9. Crystal growth rate (Jigure 14)

This recently developed technique contributed in a number of cases to the elucidation and understanding of BP crystal structure. The observed linear dependence of the growth rate on the pitch in BPII (figure 14) explains why the largest and best developed single crystals were obtained for the longest alkyl chain derivatives. The remarkably slow growth of BPI crystals developing directly from the isotropic liquid, as compared to those undergoing BPI I+ BPI transition (leading to the much better quality of the former), arises from the differing growth mechanisms characterising the two processes (Stegemeyer et a1 1986).

-

3.1 0. Viscoelastic properties (Jigure 15)

Mechanical measurements of the properties of BPS were first performed by Stegemeyer and Pollmann (1982) and later by Clark et a1 (1984) and Cladis et a1 (1984). These studies were, however, limited to the lowest-temperature phase (BPI ) (see figure 15). Complementary to their thermodynamic experiments, Kleiman et a1 (1984) conducted high-precision shear-modulus and viscosity measurements on cholesteryl nonanoate

L

0

I' /' I ,

- /i// , p , I I I I

p l n m l

Figure 14. Crystal growth rate: the growth rate O f BPI1 crystals as a function ofthe cholesteric pitch. Shown are two different compositions of the mixed system SOBE/CV. (From Stegemeyer et a1 (1986).)

-2 -1 0 1 T-T, P C )

Figure 15. Viscoelastic measurements: kinematic viscosity of cholesteryl nonanoate/cholesteryl chloride mixtures, for a concentration ratio that shows a BP range (-) and a concentration ratio that does not show a B P range (----). (From Stegemeyer and Pollmann (1982).)

678 T Seideman

(CN) for a temperature range covering all phase transitions involving the three BPS of this substance. The shear elasticity was measured as a function of frequency, showing BPI and BPII to exhibit a shear modulus that remains finite as the frequency approaches zero. From the discontinuities in the viscoelastic properties, these authors confirmed their conclusions regarding the nature of the cholesteric+ BPI, the BPI + BPII and the BPII + BPIII phase transitions, and provided supporting evidence to the stability of B P I I I

as a distinct thermodynamic phase. The much enhanced bulk viscosity and viscoelas- ticity of BPS as compared to the isotropic as well as to the cholesteric state supports the view that BPII crystal habit is preserved during the BPII + BPI transition. Con- sequently it has been suggested that the observed change of the reflection colour is due to a change of either the lattice parameter or the BP space group on the BPII + B P I

transition. The well known ability of BPI to be supercooled may be a result of these viscous properties. It seems plausible that mechanical forces within a BPI structure of high stiffness result in the preservation of BPII habit. Interpreting their results for the two lower-temperature BPS in the context of the Meiboom-Sammon model (Meiboom et a1 1981, 1983a, b, Sammon 1982) (figure 2 ) , Kleiman et a1 have shown that the viscoelastic properties of C N are consistent with the identification of BPI as O*‘-’(BCC) and BPII as 02(sc). The detailed structure of B P I I I and the nature of the B P I I I + isotropic transition were left as open questions.

0 04

0 03

0 02

0 01

AD 0

0 0 4

0 03

0 02

0 0’

( 200 400 600

X (nm)

Figure 16. Circular dichroism: spectra in BPII of a cholesteryl nonanoate/cholesteryl chloride misture, at ( a ) normal incidence and ( b ) oblique incidence. Full and broken curves correspond to different directions of incidence on the crystal. (From Kizel’ and Prokhorov (1983).)


3.11. Circular dichroism (jigure 16)

An informative optical technique is the study of circular dichroism. For probing chiral objects this method has a significant advantage over conventional reflection and transmission measurements, in that the signal relates directly to the difference in transmission of left- and right-polarised light. Kizel’ and Prokhorov (1983, 1984) measured circular dichroism spectra for the BPS in a series of cholesteryl nonanoate and cholesteryl chloride mixtures. Their study confirmed the view that BPI and BPII

belong to the Os and O2 symmetry space groups, respectively. It was furthermore asserted that the structures are universal for all investigated mixtures, displaying only a slight dependence on chirality and composition. For the third BP, circular dichroism of the same order of magnitude as in the other phases was found, which decreased sharply with increasing temperature and increased with light frequency (Kizel’ and Prokhorov 1984). It was suggested that the former effects reflect structure or pitch variation with temperature, the latter arising from large-scale inhomogeneities of the chiral structures.

4. Theory

As described in section 2 , the distinguishing feature of liquid crystals is the rich variety of phase transitions displayed. The most widely used tool for the theoretical description of this feature has been the Landau theory of phase transitions (Landau and Lifshitz 1958). The popularity of Landau’s theory owes much to its relative simplicity, the small number of free parameters involved and the diversity of its predictions. A number of inherent limitations will become evident below.

In the essence of the Landau theory (Landau and Lifshitz 1958) and later extensions (de Gennes 1971) is the assumption that the liquid-crystalline order can be regarded as a small perturbation imposed upon the isotropic state. Accordingly, the free-energy density is expanded in powers of the order parameter and its spatial derivatives, neglecting higher-order tems (see e.g. equation (4.2) below).

A second ingredient in our theoretical treatment is the use of reciprocal-space Fourier analysis to describe the director field. As alluded to in the introduction, in BPS the order parameter, characterising the correlation in the orientation of the molecules, displays three-dimensional periodicity. In analogy to the standard treatment of periodic structures in the theory of the solid state, we therefore expand the order parameter in Fourier components, carrying out the discussion entirely in terms of the reciprocal-space quantities. The advantage of this representation for the analysis of Bragg scattering data should be evident. Real-space studies of plausible BP structures were presented by Hornreich and Shtrikman ( 1 9 8 1 ~ ) ~ by Meiboom et a1 (1983a, b), by Grebe1 et a1 (1984) and by Berreman (1984). The latter representation has the merit of being physically transparent and readily visualised. The type of real-space ordering envisioned in cubic systems is illustrated schematically in figure 2.

4.1. The Landau theory of cholesterics

De Gennes (1971) was the first to apply Landau’s model to the treatment of the cholesteric phase. His approach, although frequently criticised in later studies (Brazovskii and Dmitriev 1975), seems instructive not just from the historical point of view but also as a clear introduction.

680 T Seideman

As pointed out in section 2, the orientational order of liquid crystals is manifested as an anisotropy in the system’s second-order tensor properties. It is therefore appropriate to describe the degree of macroscopic order in terms of a tensor order parameter. Thus, for example, the dielectric permittivity tensor sd( r ) is locally anisotropic in liquid crystals. Separating the anisotropic traceless part E ( r ) from Ed(r) , one defines the degree of order in terms of the rank-2 tensor

Ey(r) = E:(d -Tr(EdVV/3 (4.1)

(Grebe1 et a1 1983a, b, Belyakov and Dmitrienko 1985). It should be emphasised that any other tensor property, such as the magnetic susceptibility (de Gennes 1971), could have been selected.

Following the standard principles of Landau’s theory, we proceed by expanding the average free-energy density in powers of sV and its gradient as

F = V-’ d r [ ( a ~ , ~ , , / 2 - P E ~ E , ~ E I , + Y E ~ E , I E ~ ~ E , , )

+ (cia/&,ar~,t + ~2dr&,d /~0) /2 - de,,r~,,a,~,t,l. (4.2)

The second line of (4 .2) allows for spatial variation in the order parameter. The last term, linear in the gradient of E , is a consequence of the absence of inversion symmetry of the cholesteric liquid crystal. It is a pseudoscalar and therefore its coefficient necessarily vanishes in nematics. Finally d r = a/dxr and e,, signifies the Levi-Cevita tensor, satisfying

I

i j l = xyz, yzx, zxy

e,,/ = -1 ijl = yxz, zyx, xzy (4.3) [ : otherwise

(Priestley et a1 1974). Aside from minor notation changes, introduced so as to conform with more recent

literature, we alter de Gennes’ expression merely by including the single fourth-order invariant (the last term in the first parentheses). Note that the omission of second derivatives from the Landau expression is in fact practically rigorous. This follows since the volume integral of V(r) can be converted into a surface integral (by Gauss’ theorem) and is thus negligible, surface contributions being assumed small. The volume integral of e ( r ) V ( r ) is equivalent to that of [V(r)]’.

The various coefficients in the above have the following physical significance: a, as in the original Landau theory, is the single temperature-dependent parameter, proportional to T - T,, where T, denotes the isotropic+ nematic (I-N) phase transition temperature (rigorously T, should be replaced by T,*, where T,* is the computed temperature at which the phase transition would have occurred had it been of the second order); c1 and c2 are associated with the elasticity of the liquid-crystalline phase and p is related to the non-equivalence of states differing by sign change of ~ ( r ) ; y ensures stability with respect to increase in E, while d is associated with the left-right asymmetry.

The order parameter describing a general transition from the isotropic to an orientationally ordered phase may be expanded in terms of the spherical harmonics Y J , , (Imi S j ) , being further characterised by a set of basic wavevectors (4,). Since Yo,o belongs to the unit representation, while the j = 1 harmonics do not satisfy the requirement of invariance under a 7r rotation about the nematic axis, we may express the


order parameter as a linear combination of the five Y2,m. For a system with no inversion symmetry,

(4.4a)

Equivalently, the tensors E,, can each be expanded in terms of five basis tensors M,(a) as

~ i , = C ~ - ‘ ” ~ m ( u ) { Y 2 , m ( 6 u , 4u) exp[iq(hx+ky+zz)+i$m(a)I}. m u

= N-’”E,((T){[M,((T)] exp[iq(hx+ky+lz)+i$,(c+)]} (4.4b) mo

(see appendix 2). The latter representation, although perhaps less physically appealing, is computationally more convenient and was employed in most of the recent literature (see e.g. Brazovskii and Dmitriev 1975, Brazovskii and Filev 1978, Grebe1 er a1 1983a, 1984, Belyakov and Dmitrienko 1985). Below we will find it beneficial to use both representations interchangeably. It should nevertheless be pointed out that the various coefficients in the free-energy expansion (equation (4.2)) depend upon the chosen representation. (See Kleinert and Maki (1981) for a detailed comparison between the methods.)

In equation ( 4 . 4 ~ ) the polar and azimuthal angles {eu, &} are, for each (T = h2+ k 2 + l’, defined with respect to a local coordinate system, whose polar axis lies along (h, k, I ) ; ~ ~ ( a ) and $,(a) are the amplitude and phase associated with the particular state; and N denotes the multiplicity (i.e. the number of wavevectors of magnitude m1’2q) . Specifically N = 48/(2”0n1!), where no is the number of vanishing and n , the number of equal lhl, lkl, 111. As shown below, to calculate the third- and fourth-order contributions in the free-energy expansion, the spherical harmonics in equation ( 4 . 4 ~ ) need to be transformed to a common coordinate system, noting the dependence of these contributions upon the relative phases of the amplitudes &,,,(a). Within the Landau approach however, the explicit q dependence may be evaluated via the quadratic part of the free energy, which is straightforward to obtain as

F*= C N-’{u - mdg(T”’+[c,+C,(4-m’)/6]q2a}~Z,(a)/2. (4.5a) m u

The coefficient of each & ; ( U ) is now recognised as the energy associated with the particular mode.

Clearly, thermodynamic stability requires

c1> 0 Cl +2c,/3 > 0. (4.5b)

Setting

dF/dq = dF2/dq = 0

we find

where qo = d / f l c , . As discussed above, non-chiral systems (nematic or racemic) are characterised by

d = 0 and the transition from the isotropic is to a phase with q = 0. The corresponding free energy assumes the form

fN = t p y 4 - p3 + p4 (4.7)

682 T Seideman

where, following Grebel et a1 (1983a), the reduced temperature and order parameter are defined through

t = 12ay/p2 ( 4 . 8 ~ )

P ( r ) = E ( r ) / S s = p / & y (4.8b)

and the reduced free energy is

f = F/(P4/36y3) . ( 4 . 8 ~ )

As follows from equation (4.7), the line t = 1 marks the I-N thermodynamic phase boundary, while the corresponding reduced distance

&l= 12c, y / p 2

has the significance of a nematic phase correlation length. Turning to the cholesteric phase, we first note (equation ( 4 . 5 ~ ) ) that for d > 0 the

ground state is necessarily the m = 2 component. Using equation (4.6) the equilibrium value of the cholesteric phase wavevector is thus straightforward to express as

qc=aq0= d l c , . (4.9)

To obtain the correct structure in the racemic limit we now supplement the order parameter with the q = 0, m = 0 branch. Employing equations (4.2), (4.5u), (4.6), (4.8) and (4.9) as well as the orthogonality properties of the spherical harmonics, the (reduced) free energy takes the form

fc = t ~ 8 0 ) / 4 + ( t - ~ * ) ~ . : ( 2 ) / 4 +

where K , the chirality parameter, is defined via

- ~ P ~ ( O ) P L : ( ~ ) I + [ P @ ) + ~ 3 2 1 1 ~ ( 4 . 1 0 ~

K qc!$N* (4.10b

The transition temperature, given as the extremum of f c for each K , is now written

(4.11)

Using equations (4.10) the entropy change associated with the phase transition is expressed as

(4.12)

Clearly, for K 2 3 an I-c phase transition would be of the second order.

4.2. The Landau theory of the blue phases

In order to determine the thermodynamically stable phase at an arbitrary point (t, K )

in the temperature-chirality plane, the free-energy density is first minimised with respect to the order parameter E. In terms of the reduced quantities of Grebel et a1 (1983a) equation (4.2) is rewritten as

f = v-11 d r [ ( t P t + t2~aip~2, + p t ~ a i / l . v a i P b -2K52~ei,lpinalp,n)/4-~p,/1.,rPli + ( P ~ ) * I

(4.13)


where p = c2/ c 1 . Subtracting a term

V-' drA(r )p l , i so as to avoid the zero trace constraint pl l = 0, and differentiating, one obtains for the extrema

[ tplJ - t ia :plj - PS'N a / PI! + a l a / pJ1) / - Ktk( el,n + PI/ ) I/ -3&Pz/Pb +4PU:nFu, -Aat j = 0. (4.14)

By setting i = j and summing over i we find for the Lagrange multiplier

A = 6-21 + P& a la /P l / /6 . (4.15)

Resubstitution of equation (4.15) in (4.14) yields a set of five coupled non-linear differential equations for the order parameter. Determination of the stable solution requires substitution of the plJ thus obtained into equation (4.13), finally selecting the one which corresponds to minimal J:

As discussed in section 3, early experiments have shown the possibility of a first-order phase transition from the isotropic, occurring at t > t , -c. Clearly this requires the third-order term in the free energy to differ from zero, which, as will be explained below, is possible only if the associated wavevectors form one or more equilateral triangles (Brazovskii and Dmitriev 1975, Brazovskii and Filev 1978, Hornreich and Shtrikman 1979). It has been further shown experimentally that the anomalous phases characterise systems having a short pitch (corresponding to high K-see equation (4.10b)) in the helicoidal phase (Stegemeyer and Bergmann 1980, Johnson et a1 1980, Marcus 1981, Flack et a1 1982). Regarding the difficulty involved in minimising F in the most general situation (equations (4.13) to (4.15)), it is instructive to consider first the limit of high chirality. Equation ( 4 . 5 ~ ) implies that in this case any structure minimising the free energy has wavevectors equal in modulus to qc and m = 2. Thus, asymptotically, it is sufficient to take into account only the fundamental harmonic. The simplest three-dimensional structure of wavevectors in reciprocal space permitted by the above considerations is a regular tetrahedron. As shown in figure 17( a ) , choosing the spatial axes such that the wavevectors lie along the (1 10) directions generates an FCC structure in reciprocal space. The labelling of the wavevectors in terms of the

Figure 17. The (1 10) wavevectors.

684 T Seideman

Cartesian x, y, z coordinates is straightforward to derive by inspection of figure 17(a). The order parameter takes the form

6

E = c N - ’ ” E ~ ( ~ = ~ ) ( Y ~ , , ( B , , 4, ) exp[i(q, .r+$2(n)]+cc} (4.16) n = l

and the quadratic part of the free energy follows from equation ( 4 . 5 ~ )

( F 2 ) B p = ( a - d2/c1)&:(a = 2)/2. (4.17)

The cubic part is derived by considering all closed triangles of wavevectors as shown schematically in figure 18( a). Each such triangle has the significance of a non-vanishing integral resulting from raising equation (4.16) to the third power:

(4.18)

We will exemplify the discussion by analysing in detail only a single contribution, since the rest of the computation, although lengthy, involves no additional principles. Considering for instance the term satisfying

e x p [ i ( q l + r - q 2 . r - q , * r ) ] = 1

(figure 17) we have

( F 3 ) L P x -3!&:(2){exP[i($1 - $2- $6 ) i Y2,2(i1) y;,2($2) y;,2(46)+cc) (4.19)

where J

i1 = { e l , 4~ = { + , 4 4 }

4 2 = fez, 4 2 1 = {TI23

4 6 = I e 6 , 46}={7T,o} (see figure 17(b)). The three spherical harmonics must first be brought into a common reference frame by application of Wigner matrices. In the notation of Edmonds (1960)

(4.20)

where the rotation {ei, r$i} is the result of successive rotations first by {ai, pi , y i } and next by { e j , 4j}. Thus

Figure 18. Contributions to the free energy of the BCC O5 phase: ( a ) cubic, ( b ) - ( e ) quartic contributions.


where we used equation (2.5.6) of Edmonds (1960). (Alternatively all three functions may be transformed into a convenient frame, which would simplify the evaluation of angles while adding a third sum.) The integral over the three spherical harmonics may be performed analytically (Edmonds 1960)

(4.22)

The rotation matrices are tabulated functions, which, in principle, can be evaluated numerically for given arguments. Equivalently, employing the tensorial representation for the basis set (appendix 2), and following the method of Brazovskii and Dmitriev (1975), one obtains

(F3)Lp= -3!p12- ~~(2){exp[ i ($ , -$~-$ , ) ] Tr[M,(l)M,*(2)M,*(6)]+cc}. ( 4 . 2 3 ~ )

Evaluation of the scalar products yields

(F3);,= -3!p12-3'2~:(2){exp[i($, - G 2 - $6)][3 e ~ p ( i ( u , ) / 4 ] ~ + c c )

3 / 2 3

= -p~:(2)9&/128 C O S ( ~ C X , + $ ~ - $ , - $ ~ ) (4.23b)

where a. = -cos-'(1/3). The explicit computation, although straightforward in principle, is rather lengthy in practice using either of the two equivalent methods, particularly due to the large number of terms involved in the more general case.

Evaluation of the quartic part of the free energy follows along the same lines, by considering individually each of the different types of contributions depicted schematically in figures 18( b)-( e) (i.e. summation over all configurations satisfying the requirement that the vector sum of four wavevectors vanishes). The results, computed via the second method, were given in detail by Grebe1 et a1 (1983a). One finally obtains for the free energy

f~ = ( t - K 2 ) / . L U : ( 2 ) / 4 - ( 2 7 J 2 / 1 2 8 ) [ C O S ( 3 ( Y ~ + Ga) +COS(3a,+ $b) +COS(3(uof $=)

+ COS(3CYo- $a - $h - $c)]/.L;(2) + (79/64 + 25/ 1152)

[cos( $a+ $b) + cos( $b + $c) + cos( $c+ $ a ) l p i(2) *

The three phases appearing in equation ( 4 . 2 4 ~ ) denote the linear combinations

( 4 . 2 4 ~ )

$ , = $ , = 2 ( ~ = 2 , 1 ) - $ , = 2 ( ~ + = 2 , 2 ) - $ , = 2 ( ~ + = 2 , 6 )

$ h = $m=2((T=2, 2)-$,=2(C+=2, 3 ) - $ ~ = 2 ( ( + = 2 , 4)

$c = $m =2( (+ = 2,3) - $m = 2 ( C+ = 2 , I ) - 9, = 2 ( C+ = 2,5) (4.24b)

where it was noted that the phase factors appear in the cubic and quartic terms only in the form of one of these three combinations. This latter property is clearly a reflection of the invariance of the free energy with respect to an arbitrary translation of the origin. With cos(3a0) = 23/27, the free energy is minimised by choosing = $b = 4, = 0, indicating that the order parameter is invariant under the operations of the O5 (1432) space group (see appendix 1). Substituting in equation ( 4 . 2 4 ~ ) and differentiating to find the extremum gives

( 4 . 2 5 ~ ) fo5= ( t - ~ ~ ) / ~ : ( 2 ) / 4 - (23a/32)pU:(2) + (499/384)~;(2)

686 T Seideman

0 1 2 3

Is0

10 30 50 70 K x nematic

Figure 19. ( a ) Theoretical phase diagram when only I , c and O5 phases are allowed. (From Grebel et a1 (1983a).) ( b ) An early experimental phase diagram. (From Marcus and Goodby (1982).)

for the free-energy density, and

t I - 0 5 = 1587/1996+ ~ ~ = 0 . 7 9 5 + ~ ’ (4.25b)

for the thermodynamic phase boundary. Comparison with equation (4.11) now shows that for K 20.939 the transition from

the disordered to the BCC phase occurs at a higher temperature than that to the helicoidal one.

The theoretical phase diagram of figure 19(a) may be qualitatively compared with its experimental analogue (figure 19(b)) taken from the results of Marcus and Goodby (1982). (See also Yang and Crooker (1987) regarding the detailed interpretation of the experimental diagram.) The alternative two-dimensional hexagonal phase, first suggested by Brazovskii and Dmitriev (1979, was reconsidered in detail by Grebel et a1 ( 1 9 8 3 ~ ) and later by Wright and Mermin (1985). It was shown that, within the Landau theory framework, the isotropic + hexagonal phase transition is not expected to occur.

5. Theory versus experiment: higher harmonics

Many qualitative features of the BPS are well accounted for by the asymptotic model described in the previous section: the relative narrowness of the BP region (1-2 K), the narrowing with increasing pitch (decreasing K ) , and optical properties such as the absence of birefringence (Pelzl and Sackmann 1973, Demus et a1 1978), the optical activity (Meiboom and Sammon 1980, 1981) and gross features of the Bragg spectra (Pollmann and Scherer 1980, Goldberg and Schnur 1970, Meiboom and Sammon 1980, 1981, Johnson er a1 1980, Flack and Crooker 1981, Marcus and Goodby 1982, Marcus 1982a, b, c) and NMR spectra (Samulski and Luz 1980, Yaniv et a1 1983), which are common to all cubic phases. Nevertheless, the high-chirality (single spatial frequency) limit, which was initially treated exclusively (Hornreich and Shtrikman 1979, 1980a, b), is evidently inconsistent with a large number of the early experimental findings detailed in section 3: in particular, the appearance of more than one cubic structure (Bergmann and Stegemeyer 1979a); the red shift of the primary reflection (Bergmann et al 1979); the observation of a second strong Bragg reflection for both BPI and BPII (Goldberg and Schnur 1970, Pollmann and Scherer 1980, Meiboom and Sammon 1980, 1981, Johnson et al 1980), indicating as shown below that neither has an O5 structure; and


even more definitely the polarised light studies (Flack and Crooker 1981), which found the second Bragg line strongly sensitive to the sense of input light. The last result seems to rule out the O5 symmetry for which the c = 2 reflection is polarisation- independent in the back direction (see below).

In a series of later studies Hornreich and Shtrikman (1981a, b, c) and subsequently Grebe1 et a1 (1983a, 1984) and Belyakov et a1 (1983, 1985) extended the basic model described above by including additional spatial frequencies in E as indicated in equation (4.4). Generally speaking, the Landau theory is essentially an expansion into the ordered phase from the order-disorder phase boundary. Within a mean-field framework the necessity to take harmonics (and in principle higher-order invariants than those given in equation (4.2)) into consideration arises from the first-order nature of the transition. It may be expected that contributing terms (which should reduce the system’s free energy) are such that one can form third-order invariants with any two of the basic components. One can therefore envision terms such as:

(i) Y2,,(q) e x p ( d i q x ) + c c with wavevkctors (2 0 0), (ii) Y2,m(q,) exp(2iqJ- r ) + c c with wavevectors (2 2 0), and

(iii) Y2,m(q,) exp(iq; v ) +cc with wavevectors (1 0 0) but m = 0, -2. Qualitatively such contributions may be expected to:

(a) reduce the wavevector, thereby leading to a red shift of the Bragg reflection with

(b) lead to additional Bragg peaks, (c) split the quadrupolar N M R spectrum, and (d) shift the isotropic+ cubic phase transition to a higher temperature and increase

the discontinuity in the order parameter, which thereby increases the latent heat of transition.

The role of higher harmonics was first alluded to by Hornreich and Shtrikman (1981a). Subsequently a more quantitative analysis was suggested (Hornreich and Shtrikman 1981b) relying upon the derivation of selection rules for scattering of optical light. The major distinguishing feature from the common crystallographic selection rules lies in the nature of light involved. While the highly energetic x-ray radiation is basically scattered by free electrons, at optical frequencies the BP structure factor has a tensor rather than a scalar character, i.e. all components of the tensor order parameter are involved in the light-matter interaction.

Much insight may be gained from symmetry arguments. Considering for instance the first example above with the (lowest-lying) m = 2 state and expressing explicitly the spherical harmonic, we have

respect to the cholesteric,

6 - 1 ’ 2 ~ 2 ( c = 4) (15/32~)”’ sin’ B{exp[2i4 +2iqx+i+,(c =4, l ) ]+cc} (5.1)

where the (2 0 0) polar axis is pointing along the $ direction. Since in Os (1432) there exists a fourfold symmetry axis parallel to 2 (see appendix l), an element that transforms (5.1) into its negative, the [2 0 0; m = 21 state, is forbidden in Os. It is, however, allowed in Os (14132), since in this symmetry there exists a fourfold screw axis parallel to 2, the application of which leaves (5.1) invariant. Thus, while in Os (1432) the (200) Bragg peak (which is due to the higher-lying m = 0 component) is expected to be weak and independent of the sense of polarised light, its intensity should be comparable to that of the primary (1 1 0) reflection in the O8 ( M132) space group. Moreover, due to the energetic difference between m = $2, -2 (see equation (4.5a)), this peak will be sensitive to the sense of the incident radiation. Similar considerations indicate that the [2 0 0; m = 21 state is allowed in the BCC T3 structure, while examination of various

T Seideman

sc structures leads to expecting two peaks of similar intensity for T’ and 02. Detailed optical selection rules were presented by Grebel et a1 (1983a), who discussed more fully their derivation and implications.

Based upon the above considerations, Grebel et a1 (1983a) computed the free energy and phase transition boundaries for the Os, the T3 and the O2 symmetries. Detailed analysis similar to that of the Os outlined above showed both the BCC (08) and the sc (02) to be more stable than Os at intermediate values of K , while the Os + T 3 transition temperature was found below t o 5 - C for the entire range of the chirality parameter. A similar approach was employed by Belyakov and Dmitrienko (1985). Comparison of the theoretical phase diagram as resulting from the inclusion of a single spatial harmonic in the order parameter with experimental data suggested the identification of BPI with O8 and BPII with the simple cubic O2 (see appendix 1).

The known BP properties are in fact in many respects compatible with this assignment. The variation of the dominance of the two phases with chirality is similar to the behaviour found by Marcus and Goodby (1982) (figure 19(b)); the polarised light results of Flack and Crooker (1981) and Flack et a1 (1982) are accounted for, as well as some of the morphological features found by Onusseit and Stegemeyer (1981). (A quantitative comparison of morphological details with Landau theory calculations would not be appropriate, however.) The intensity ratios of order-parameter components as predicted for O2 are in reasonable agreement with experimental observations (Meiboom and Sammon 1980, 1981), and a rough estimate of the O2 red shift agrees qualitatively with that measured for BPII. Nevertheless, several inconsistencies remain, particularly between the Os structure and BPI. The theoretical red shift is in this case much less than all experimental values, and the computed Bragg intensity ratios agree with experiment to a lesser extent than the corresponding O2 values.

Qualitative predictions regarding the significance of additional harmonics were first briefly mentioned by Grebel et a1 (1983a). A quantitative extension of the theory to include a number of different wavevector magnitudes was then presented by Belyakov et a1 (1983), Grebel et a1 (1984) and later by Belyakov and Dmitrienko (1985). Grebel et a1 (1984) proposed a modified technique, which avoided the time-consuming direct free-energy computation, by combining symmetry considerations with numerical techniques to perform the minimisation. In this approach, one first determines the phase factors from space-group symmetry and then combines components whose associated wavevectors have a common length, expressing the result in terms of a real second-order matrix. The third- and fourth-order contributions to the free energy are obtained for each combination of the amplitudes by multiplying and tracing the corresponding matrices, subsequently extracting the trace’s average value. Evaluating the matrix products and traces at points defined on a 3~ net within an appropriate portion of the unit cell, and averaging by taking the mean for each set of matrix products, Grebel et a1 attained high accuracy while significantly reducing the numerical effort. The inclusion of three harmonics in addition to the fundamental in the Fourier sum next resulted in the appearance of two additional locally stable Os structures, subsequently referred to as 0; and Oz. These results were shown to be convergent with respect to the number of harmonics. The 0; was found to be similar to the previously calculated 0’ (now termed 0;) from the symmetry point of view, 0: being quite different from the first two. The seven-structure theoretical phase diagram is shown in figure 20(a) and compared with its experimental counterpart (figure 20( b ) ) taken from the results of Yang and Crooker (1987). Based upon the augmented theory, one was now led to identifying 0; with BPI, retaining the identification of 0’ with BPII. With the new


lo1

0 1 2 K

0 1 2 3 4 5 6

42 2

42 0 - U e h

41 e

41 6 0 0 2 0 4 0 6 0 8 1 0

0 1 2 3 4 5 6 t I ‘ I . , ’ , ’ , . , . , 1

42 2

42 0 - U e h

41 e

41 6 0 0 2 0 4 0 6 0 8 1 0

Mole fraction chiral C E S

Figure 20. ( a ) Theoretical phase diagram when I , c, 02, Os, O:, 0; and 0; phases are allowed. (From Grebel er al (1984).) ( 6 ) The more recent experimental results of Yang and Crooker (1987).

results at hand, a much more detailed comparison of theory with experiment was possible (see figure 20( b ) ) . Quantitative comparison of theoretical polycrystalline Bragg intensities with the Meiboom and Sammon (1980, 1981) results was achieved by computing intensity ratios between allowed reflections in the limit of a thin scatterer, introducing various corrections to account for experimental deviations from ideality. A detailed comparison of the experimental red shifts with the theoretical values, calculated via equation (4.6), was also permitted. While both values supported the identification of 08 with BPI and O2 with BPII, for the former only a rough correlation with the overall measured trend was shown, while for the red shift quantitative agreement was reported. The experimental ratio between the maximum reflection wavelength and the cholesteric pitch was found in reasonable agreement with theoretical computations for both BPI and BPII (Belyakov and Dmitrienko 1985). However, while theory predicted a temperature-independent period, experiment found a strong enhancement for BPI (Meiboom and Sammon 1980, 1981) and a weaker enhancement (Meiboom and Sammon 1980, 1981) or virtually no change (Johnson et a1 1980) for BPII. Theoretical N M R spectra were presented by Grebel er a1 (1984) for 02, Os and the three Os structures, Although the overall appearance bears similarity to experimental BP N M R spectra (figure 1 l), comparison could not be carried to a more quantitia- t h e level in this case.

The lack of more complete agreement with experiment is attributed to the combination of the limitations fundamental to the theoretical model with the deviations of reality from the idealised experiment. The consequences of multiwave scattering, which clearly can no longer be interpreted in terms of the single-process selection rules, and the question of imperfect or rapidly diffusing BP crystals are briefly reviewed in section 6.3.

6. Recent progress

6.1. The problem of BPIII

The existence of the highest-temperature blue phase, the so-called ‘fog’ or ‘grey phase’, was first suggested some nine years ago (Meiboom and Sammon 1981, Marcus 1981).

690 T Seideman

Convincing experimental evidence for the stability of this phase was obtained, as detailed in section 3 , only three years later by Collings (1984a, b) and Kleiman et al (1984). Experimental phase diagrams with all three BPS were subsequently obtained by polarising microscopy (Blumel and Stegemeyer 1984b), selective reflection (Demikhov et al 1985), optical rotatory dispersion (Collings 1986) and calorimetric measurements (Thoen 1988). The absence of an apparently amorphous phase such as BPIII from all computed BP diagrams (see figures 19(a) and 20(a ) ) is until the present day regarded as the major discrepancy of the theoretical framework.

An earlier study proposed that BPII I forms as a consequence of strongly developed fluctuations in the isotropic (Brazovskii and Filev 1978). This model was found consistent with the unusual circular dichroism observed in BPIII (Kizel’ and Prokhorov 1984). Nevertheless, later accumulated specific-heat data (Thoen 1988) have ruled out the pretransitional fluctuations model.

The emulsion theory developed by Finn and Cladis (1982) (see section 1) described BPIII in terms of an emulsion of randomly oriented cholesteric droplets, suspended in an isotropic background. This model requires a mixture of liquid crystals and is thus possible in the presence of a certain amount of impurity.

A third model, again phenomenologically motivated, was formulated by Collings (1984a) in light of his ORD measurements. It was suggested that BPIII consists of small cubic domains of BPII sc structure, suspended in an isotropic medium. Belyakov et al (1985) described a similar model, proposing, however, a BCC Os structure for the inner regions.

Two different approaches have attracted considerably more theoretical attention.

6.1.1. The uniaxial director representation. The particular interest in the BP model proposed by Meiboom and coworkers (see section 1 above) stems both from the physical insight gained and from the plausible explanation its extension provides to the occurrence of BPIII. As was shown in the past (Belyakov and Dmitrienko 1985, Hornreich and Shtrikman 1981c), the locally uniaxial configuration may also be derived via rigorous Landau theory arguments. The unifying features of the two approaches have already shed some light on our understanding of the cholesteric BPS, and in particular on the problem of BPIII .

On the basis of Landau free-energy calculations, Hornreich et a1 (1982) argued that the isotropic phase may become thermodynamically unstable with respect to a localised mode possessing cylindrical symmetry, while remaining stable with respect to the cholesteric. It was suggested that, in limited regions of the temperature-chirality plane, the ground state of certain cholesteric systems may be composed either of disordered solitons?, lacking long-range periodic structure, or, alternatively, of such soliton-like modes condensed into a cubic lattice. Either structure could, in principle, underlie the observed anomalous light scattering associated with BPIII (Demikhov et a1 1985). This model is compatible also with recent optical measurements of field- aligned BPIII samples (Yang and Crooker 1990). A similar explanation was proposed by Kleiman et a1 (1984), in light of their mechanical measurements (see section 3 ) . It was shown that the viscoelastic properties of BPIII evolve continuously from being similar to cholesterics at the BPII +. BPIII transition to being isotropic-like, suggesting that BPIII consists of similar disclination lines as BPII, which lose their positional order

t Note that the term ‘soliton’ is used to denote a local rather than a periodic solution. It does not imply a soliton in the topological sense.


while retaining the helicity at the transition point. Based on entropy change estimations, it was argued that in the immediate vicinity of the BPII + BPIII transition much of the cholesteric-like order is preserved. However, towards the B P I I I + isotropic transition, the double twist elements decrease continuously in size, thus accounting for the unusual width of the final transition (Kleiman et a1 1984). It should nevertheless be pointed out that similar behaviour has not been observed in other materials (see also Crooker 1989). The electron microscopy experiments of Zasadzinski et a1 (1986) provided supporting evidence for the randomly distributed cylinders suggestion. These authors presented images of the fracture surfaces of frozen cholesteric samples, obtained by quenching BPIII. It was proposed that a transition might occur from the high- temperature BPIII to a low-temperature, locally uniaxial amorphous phase of a qualitatively different structure. Both of the above suggestions, while interesting and doubt- lessly conceivable, must be considered cautiously.

6.1.2. Icosahedral ordering. Recently there have been a number of attempts to apply the quasi-crystal concept (Shechtman et a1 1984, Henley 1987) to the study of liquid crystals (Hornreich and Shtrikman 1986a, b, 1987a, Rokhsar and Sethna 1986, Filev 1986, 1987, Hornreich 1989). In addition to the general theoretical interest in the origins and consequences of quasi-crystal order, its relevance to the understanding of BPIII is particularly appealing. The corresponding space-group assignment would be 2 3 514 where 514 represents a hybrid rotation-phason symmetry element, analogous to the rotation-translation elements found in non-symmorphic space groups. The simplest 3~ quasi-crystalline structure is obtained by choosing a set of wavevectors comprising 15 linearly independent edges of a regular icosahedron, their negatives forming the remaining 15. Within the Landau mean-field approximation, analogous to the tetrahe- dral case outlined above, one minimises the quadratic part of the free energy by associating with each of these wavevectors a single Fourier amplitude of an m = 2 basis tensor, and choosing the common (reduced) magnitude of the wavevectors to be equal to K (see figure 21). This configuration was first considered by Kleinert and Maki (1981), who showed that the associated free-energy density was substantially higher than that of the cubic Os. Hornreich and Shtrikman (1986a) investigated a modified model having two wavevector sets of slightly different magnitude: ql ,n = (1 - S ) q , and q2,n = (1 + S ) q , with 6 = 0.025. Detailed analysis led, however, to no reduction of the icosahedral free energy. Alternative possibilities of incorporating harmonics into the order parameter were considered by Filev (1986) and by Rokhsar and Sethna (1986). While pointing out the crucial role of a large number of harmonics for a proper description at physical chiralities, Rokhsar and Sethna argued that their inclusion is unlikely to stabilise the icosahedral texture with respect to the cubic. Introducing as many as five harmonics into the order parameter, and minimising the corresponding average free energy with respect to the set {pz(c+i)}i=,,5, indeed showed that only two of the harmonics give appreciable free-energy contribution (Hornreich 1989). Maximisation of the ratiof:/f4 calculated within the two-harmonics approximation yielded for the phase transition temperature

t,-Ic = max(f:/f4) + K’ = 0.661 + K’ . (6.1) In equation (6.1) f3 denotes the third-order (negative) contribution to the reduced free energy, and f4 the corresponding fourth-order term. Although considerably higher than the transition temperature to the simple icosahedron, comparison with equation (4.25b) indicates that the augmented broken icosahedral structure is unstable with

692 T Seideman

l o )

Figure 21. ( a ) The icosahedral structure. ( b ) Plan view of the upper half of the icosahedron. (From Kleinert and Maki (1981).)

respect to the 05. Hornreich and Shtrikman (1987b) proposed supplementing the fundamental set of 30 edge wavevectors with a second set, so as to form wavector triangles each composed of two vectors from the original and one from the new set. Detailed computation yielded merely a slight destabilisation:

t l_ ,c = 0.654+ K * . (6.2)

Thus, within the present theoretical framework, cubic structures appear to remain energetically preferred throughout the temperature range of interest. Nevertheless, as stressed below, Landau theory free-energy calculations cannot give an unambigous determination of the structures of experimentally observed phases, particularly so in view of the small energy differences between the various structures.

A number of the experimental observations on B P I I I provide supporting evidence for the assignment of icosahedral (5 3 2) point symmetry:


(a) Strong optical activity (Collings 1984a, b, 1986) is expected since the proposed icosahedral structures all lack an inversion centre.

(b) The lower shear elasticity of B P I I I (Kleiman et a1 1984) is consistent with the additional diffusive phason modes which are theoretically allowed in icosahedral structures.

(c) As a consequence of the (5 3 2) point symmetry the icosahedral structures are expected to be non-birefringent, in accord with experiment (Blumel and Stegemeyer 1984b).

(d) Qualitatively speaking one expects the latent heat associated with an I-IC phase transition to be similar to that of an isotropic-cubic transition, while considerably larger than that characterising a cubic-cubic phase transition. The latent-heat measurements described above agree with this feature (Kleiman et al 1984, Thoen 1988).

Contradicting experimental observations have also been reported. Bragg scattering work conducted by Demikhov et a1 (1985) have found a single broad peak for B P I I I ,

contrary to the pattern predicted for the icosahedral phase. Recent reflectivity studies on surface-aligned B P I I I samples (Yang et a1 1988) again detected no evidence of the lines which are allowed for a quasi-crystal. It may be concluded that the existing experimental data are not yet sufficient either to rule out conclusively the quasi-crystal model, or else to provide sufficient ground for its acceptance.

Clearly, the ‘fog phase’ deserves closer experimental and theoretical study. A suggested method of experimentally testing the nature of B P I I I is by the use of N M R

measurements. Theoretical N M R spectra were calculated for cubic (Grebe1 et a1 1983b) and icosahedral (Hornreich and Shtrikman 1987a, Filev 1987) structures in the slow- diffusion limit, finding the latter to differ substantially from the former. Interestingly, it differs also from the spectrum predicted for the ‘double twist’ cylinders model. It has been suggested that experimental N M R spectra may provide the lacking conclusive evidence to differentiate between the models.

6.2. External jield efects

The first theoretical study of the cholesteric phase under an applied field is again due to de Gennes (1968, 1974). The field-induced pitch variation was explained in an intuitively appealing manner, and the critical field for phase transition to the nematic was derived. A number of pertinent experiments were suggested and some interesting thermodynamic features were discussed. The associated free energy is written

where E , is the dielectric anisotropy, n is the director and an analogous expression applies in the case of a magnetic field. The field-matter coupling energy should in principle be supplemented by a distortion term (de Gennes 1968, 1974) given in the cholesteric case as

Fd = Kq,n curl n. (6.4) The effect of weak electric and magnetic fields on the cholesteric BPS was first investigated by Hornreich et al (1985a, b), who confined attention to the case of positive dielectric anisotropy. Within the weak-field approximation, these authors assumed the cholesteric and cubic order parameters to retain their zero-field form, showing the free energy of the helicoidal phase to be lowered by the field, while that of the cubic phase remains invariant. The two-dimensional hexagonal structure, first introduced by

694 T Seideman

U I 3 - e 2 - h

0

Brazovskii and Dmitriev (1975), was shown to become stable in a limited range between the disorderd and cubic BP regions (Hornreich et a1 1985a). At slightly stronger fields the new phase was predicted to appear between the isotropic and helicoidal phases (Hornreich et a1 1985b). The latter work introduced the director-representation point of view. This approach regards the field-induced hexagonal phase as an assembly of parallel cylinders, in each of which the director describes a curling mode configuration (see also Hornreich and Shtrikman 1981~) . The field-induced uniaxiality makes this representation particularly suitable for the present application. The weak-field approximation was analysed by rigorously calculating a lower bound to the threshold field for transition to the nematic. Such fields (near to which distortions in the periodic structure can obviously no longer be neglected) were claimed to be substantially stronger than the field strength at which the hexagonal phase becomes thermodynamically favourable.

Much work has been performed during the past year or so on the subject of field-induced BP transitions. The experimental work of Pieranski and coworkers (see section 3) has been paralleled by detailed theoretical calculations performed mostly by Hornreich and Shtrikman (1989,1990). The former work extended the field-dependent Landau model (Hornreich et a1 1985a, b) so as to include the experimentally observed (Pieranski et a1 1985) three-dimensional, as well as the two-dimensional, hexagonal structure. Possible H3D phases were modelled by an order parameter that is invariant under the space group P6*22. A set of wavevectors in the basal plane (qb), a perpendicular component (q,,) and an additional diagonal set ( q d ) were considered, which are equivalent to the experimentally reported (0 0 0 l), (1 0 i 0) and (1 0 i 1) reflection bands. In agreement with experiment (Pieranski et a1 1985, Jorand and Pieranski 1987), both the two- and the three-dimensional hexagonal phases were found thermodynamically stable in different regions of the phase diagram (figure 22). As shown by Pieranski and colleagues, H3D was found to occur in a lower field than H2D. It should nevertheless be noted that theoretical calculations have shown that two diferent three-dimensional hexagonal structures can appear under different experimental conditions (see figure 22). Since the two structures, HiD and HiD, differ merely in their phase factors, they could not be easily distinguished by the experiment of Pieranski and coworkers. Different possibilities of differentiating experimentally between the two three-dimensional hexagonal phases were discussed by Hornreich and Shtrikman (1989). In particular it was suggested that the NMR technique may serve as a convenient tool. Theoretical N M R spectra for the H2D, HiD and HiD phases were presented, showing qualitative difference between the three structures. These predictions still await experimental verification.

€,CO ~ E,>O -(--gsq Isotropic

Cholesteric Cholesteric

Cubic O2

I I , I I _


Inspired by the experimental investigation of negative-anisotropy materials (Heppke et a1 1989), a theoretical study of the E , < 0 case has been initiated (Hornreich and Shtrikman 1990). Preliminary results again show satisfactory agreement with experimental findings. In figure 22 we show a schematic phase diagram consisting of the entire theoretically explored temperature-anisotropy plane. Note that the experimentally detected tetragonal phase has not yet been computed theoretically. Such calculation, although feasible in principle, is extremely complicated in practice due to the large number of wavevectors required.

A detailed Landau theory analysis of the field-induced orientation and birefringence was given by Lubin and Hornreich (1987). In the presence of a uniform external field E, equation (4.13) is modified as

+ V-’ j d r [-V‘&,,P~~P~, + (3/2)’”pL,ele,I (6.5)

where e = (3y2 /2~p3E) l ’* and the last term expresses the (reduced) coupling energy of the cubic lattices to the external field. The order parameter (equation (4.16)) is now supplemented by the m = U = 0 term induced by the uniaxiality of the field. Thus

E = c N-l’z{(l - &,O)EZ((+) exP[i+*(U)l Y2,2(%, +U) U

+ ~u,o(2/3)1’2Eo(o) e x ~ [ i + ~ ( ~ ) I y2,0(eO, +o)I exp[iq(hx + ky + 12)1/2 (6.6)

and the equilibrium configuration is derived as above by minimisation of the free energy with respect to p.

The symmetry-reducing effect of the field was discussed in the context of two different models (Lubin and Hornreich 1987). In both cases attention was confined to the case of a perfect single crystal that is free to rotate, and to the weak-field limit in which the symmetry of the system remains virtually unperturbed from its field-free classification. The first, strain-free model assumed the wavevectors and basis functions to retain their zero-field value, symmetry breaking being induced merely via the changes in order-parameter amplitudes. These changes measure the primary electro-optic effect. The free energy f ( ~ , t, e, { p } ) can thus be evaluated explicitly, and its numerical minimisation with respect to the set of amplitudes { p } yields the stable orientation of the free crystal with respect to the field, as well as the induced birefringence. The second, improved model allows for a mechanical distortion of the cubic unit cell. This is achieved by introducing a distortion parameter d, which multiplies the wavevector components parallel to the field, yielding a new set of unit vectors whose directions are field-dependent. The free energy f = f ( ~ , f, e, {p } , d, r ) is in this case minimised with respect to d and r as well as the set of order-parameter amplitudes. (The r dependence, where r = q / qo, is due to the fact that this ratio can no longer be computed using equation (4.6) and is thus treated as a parameter.) In addition to the information attained via the strain-free model, the equilibrium values of r and d now yield directly the distortion of the unit cell.

Detailed analysis of the 05, O2 and Os structures (assuming in all cases a positive dielectric anisotropy), led to the following conclusions:

(a) The unit cell in general is compressed in reciprocal space (i.e. elongated in real space), thus leading to the observed red shift of the Bragg back-scattered light (Porsch and Stegemeyer 1986, Heppke et a1 1983, Pieranski and Cladis 1987a).

696 T Seideman

(b) The magnitude of the shift decreases with increasing temperature and chirality, being quadratic in the field. For Os it changes sign at very high chiralities. In experimental units the calculated shift corresponds to roughly a few nanometres.

(c) The induced birefringence is typically small and, again, proportional to e2. Turning to experiment, we note the qualitative agreement between the computed

(Lubin and Hornreich 1987) and observed (Pieranski et a1 1986) orientation of single crystals in weak electric fields, when Os and O2 are assigned to BPI and BPII respectively. This finding thus further substantiates the generally accepted assignment of the two BPS, ruling out the Os structure, for which no such agreement was found. A more quantitative comparison does not seem appropriate at the present status of theoretical and experimental research. Red shifts of typically 60 nm were reported by Heppke and coworkers for a field strength of E = 30 kV cm-'. The slightly lower values reported by Porsch and Stegemeyer (1986) and the shift of 33 nm observed by Pieranski and Cladis (1987a) are still an order of magnitude larger than the theoretical prediction. Moreover, in all three measurements the field dependence of the shift was not quadratic in E but rather close to linear.

Very recently the effect of electric fields on B P I I I samples was measured experimentally (Yang and Crooker 1990). Evidence was presented both in favour of the double twist model and in favour of the BCC O5 domains structure. Nevertheless, no firm conclusions were attained and the quasi-crystal model was not ruled out.

Evidently, the theory and experiment of external field effects contain much structural information. Future development of both aspects is nevertheless mandatory before any more detailed assessments could be made.

6.3. Multiwave scattering

The phenomenon of multiple scattering, characterising thick samples, is of theoretical interest in that it can be used, in analogy to the case of x-ray diffraction, to determine the relative phases of the harmonics in the Fourier expansion of the order parameter. The experimental observation of multiwave scattering (Marcus 1982b) indicates its practical significance for the study of BPS. Moreover, a number of the most puzzling findings in the experimental study of BPS (Tanimoto and Crooker 1984, Keyes 1987, Yang and Crooker 1987, Jerome et a1 1988) have been offered an explanation in terms of a multiple scattering event. Possible experimental effects include rotation of the plane of polarisation of light, Pendellosung beats, linear birefringence and dichroism (contrary to expectation for cubic space groups) and frequency dependence of the polarisation of diffracted beams. The last effect may result also in some depolarisation even in perfect samples. It should be stressed that since polarisation features are determined from the results of relative measurements, they depend crucially on how well multiple scattering effects have been avoided experimentally, or alternatively on how accurately they have been included in the theoretical analysis. The theory of multiwave scattering was briefly considered by Belyakov et a1 (1982) and by Belyakov and Dmitrienko (1989, who formulated a dynamic model. The possibility of explaining the inconsistency of experiment with the generalised selection rules in terms of a multiwave process was first put forward by Crooker (1985). Although the major aim of the calculation has been to reinterpret the Bragg scattering results of Tanimoto and Crooker (1984), the analysis is of interest in that it provides a general framework for calculating the polarisation of multiply scattered light, which should be applicable to other situations. The Bragg spectrum of C B I ~ / E ~ as measured by Tanimoto and Crooker

7'he liquid-crystalline blue phases 697

(1984) is shown in figure 8. Keeping in mind the selection rule that forbids the (1 1 1,2) line for sc (02) (Grebe1 er a1 1983a), the observation of polarised back-scattering from the third Bragg line in C B I ~ / E ~ led to the conclusion that for this material BPII is BCC

(Tanimoto and Crooker 1984). Nevertheless as noted by Crooker (1985), the high symmetry of the cubic lattice together with the large scattering power of C B ~ S / E ~ may well allow for multiple scattering in addition to the direct event. In particular, although the coefficient E * * ( 1 1 1) vanishes in sc space groups, a back-reflection having the same Bragg wavelength may still be produced by any combination of allowed scattering vectors. One such example is shown in figure 23 where the double scattering via the allowed E&*( 110) and ~ ~ ~ ( 0 0 1 ) coefficients is depicted schematically. An explicit calculation of the multiply scattered polarisation, resulting from each of the manifold of double and triple scattering possibilities, was performed (Crooker 1985), assuming a coherent multiwave process. It was nevertheless shown that in all cases the lowest- order m = -2 multiple reflections have exactly the same polarisation properties as the (1 1 1) single reflection. The obvious conclusion to emerge from those calculations is that, for the material studied by Tanimoto and Crooker, BPII is in fact of the BCC space group for which the third line (2 1 1,2) is strongly allowed. Furthermore, later studies of the second BP in another material (Yang and Crooker 1987) have found again a strongly polarisation-dependent third back-reflection. Following his Bragg scattering experiments in ( + )-2-methylbutyl-p- [ (p-methoxybenzylidene)amino]cinnamate (MBMBAC), a compound that exhibits all three BP, Keyes (1987) proposed for BPII an FCC structure of the O3 or O4 symmetry. It was argued that this assignment applies to the second BP in many other materials, implying that space-group symmetry is not necessarily universal for both BPS in different systems. Inconsistency with the theoretical assignment of BPII was again evidenced by Jerome er a1 (1988).

kx

Figure 23. Schematic representation of double scattering. q I l 1 is the scattering vector for the direct (1 1 1) process. qool and q l l 0 are the scattering vectors for a (0 0 1) - (1 1 0) double scatter. (From Crooker (1985).)

More recently an alternative multiwave scattering model has been put forward in an attempt to reinterpret the Tanimoto-Crooker experiment as well as the newly accumulated data (Birman er a1 1989). The novel feature of the present scheme is that it allows for wholly or partly incoherent scattering processes. While a coherent event is described by summing all possible products of matrices corresponding to the individual routes, essentially relating the amplitudes of the reflected and incident electric fields, an incoherent process is described as a relation between intensities. Following

T Seideman

Birman et al (1989), the properties of the incident and reflected light are characterised by Stokes vectors, which are then related by a Mueller matrix. Reconsidering the double and triple processes described by Crooker, these authors have shown the experimental data (Tanimoto and Crooker 1984) to be consistent with an sc assignment, provided an incoherent multiple scattering mechanism is assumed. It was, moreover, suggested that the results reported by Keyes (1987) may also be interpreted in terms of an incoherent multiwave process. Similar conclusions were attained via a different approach by Belyakov and Dmitrienko (1988).

Clearly, for scattering from a perfect single crystal the coherent rather than the incoherent event should dominate. Nevertheless, a real sample is mostly non-ideal and as a consequence the waves diffracted by different regions may have, generally speaking, different polarisations. This holds true also in the case of ordinary crystals. For blue-phase samples we may expect the effect to be further accentuated due to the rapid diffusion characteristic of liquid crystals. Thus, the possibilities of both coherent and incoherent scattering should in principle be taken into account in interpreting the experimental data.

7. Discussion

As detailed above, the Landau theory of phase transitions in cholesterics has explained most observed properties of the anomalous phases. A number of restrictions of the model, as well as some remaining unanswered questions, should, however, be pointed out.

Clearly, a complete solution of the BP structure problem demands knowledge of the spatial molecular distribution function, which cannot be derived uniquely from space-group symmetry. Moreover, being in essence a mean-field theory, the Landau model neglects fluctuation contributions to the entropy changes (see also de Gennes 1971, Brazovskii and Filev 1978). It cannot, for instance, be quantitatively compared with experimental latent-heat ratios involving an order-disorder transition. (Theoreti- cal relative latent heats associated with Of + C and 02+ Of transitions, i.e. involving only ordered phases, were found to be in better agreement with specific-heat measurements.) By restricting analysis to m = 2 Fourier components, the theory avoided consideration of subgroups such as T' (P23), T3 (123) and T5 (1213). Although the m # 2 amplitudes are expected to be experimentally small, they may be significant in certain respects, for example the exact location of the BPI + BPI] phase boundary. This quantity can also be affected by the neglect of higher-order terms in the free-energy density. Quantitative comparison of a theoretical phase diagram with an experimental one will therefore not be appropriate within the present theoretical framework. Further, owing to the extremely small differences in free energy between the various phases, shown both theoretically (Grebe1 et a1 1983a, 1984, Belyakov and Dmitrienko 1985) and experimentally (Bergman and Stegemeyer 1979a, b, Onusseit 1983, Kleiman et a1 1984, Thoen 1988), a universal phase diagram is not to be expected. Figures 19 and 20 should be regarded accordingly. A number of more fundamental points of discrepancy are to be mentioned.

Experimentally an O5 phase such as appears in the theoretical phase diagram for K 5 1 has not been found. Since with very small free-energy changes O5 appears only for much larger K , while, as shown in figure 20(b) , experiment detected amorphous


BPIII in the same region of the phase diagram, it was suggested (Grebe1 et a1 1984) that for intermediate chiralities this phase could have lower free energy than 05.

While the BPI structure has been reasonably well understood since the early days of research into the BP properties, that of BPII remained controversial for a long period. Although consistent with a range of experimental findings (see e.g. Meiboom and Sammon 1980, 1981, Marcus and Goodby 1982, Kizel’ and Prokhorov 1983, 1984, Kleiman et a1 1984, Thoen 1988), the assignment of 0 2 ( s c ) to this phase appears incompatible with a number of others (Tanimoto and Crooker 1984, Yang and Crooker 1987, Keyes 1987). Moreover, this symmetry assignment implies that BPII should survive on decreasing the chirality, contrary to experiment (Tanimoto et al 1985, Collings 1986, Yang and Crooker 1987) (see in particular figure 20( b ) ) . Nevertheless, recent advances of the theoretical understanding (section 5.3) now seem to provide sufficient ground for acceptance of the sc (02) space-group assignment.

As detailed above, experimental and theoretical studies of electric-field effect on the BP have already yielded detailed information regarding the field-free symmetry and space-group assignment of the two lower-temperature BP. However, only a very rough correlation of theory with experimental work has so far been possible, due partly to the inherent limitations of the theory (Lubin and Hornreich 1987) and partly to some inconsistencies in the experimental literature. (Compare for instance Pieranski et a1 (1986) with Porsch and Stegemeyer (1986).) Extension of the theory and further experimental studies are likely to resolve a number of controversial issues. Specifically, an experimental measurement with free crystals may clarify the discrepancy between theory and experiment regarding the magnitude and field dependence of the Bragg scattering red shift (Porsch and Stegemeyer 1986, Heppke et a1 1983, Pieranski and Cladis 1987a). Further data on the field-induced orientation of freely rotating single crystals should also be valuable. For field-parallel observations the induced birefringence in BPI was found to increase with field strength (Porsch et a1 1984). Generally, the extent of deformation should increase with the field via either a second- or a third-order electro-optic effect. The detailed behaviour thus reveals important information regarding the symmetry of the undeformed phase. Birefringence measurements in non-parallel observation have not been conducted yet. Such experiments would yield complementary data and allow a direct comparison with theoretical computations.

A number of experimentally observed details remain unclear. For instance, the temperature independence of the BPII + c critical field strength (Stegemeyer et a1 1986) has not been explained. Another open question regards the existence of threshold field strength required in order to obtain a deformed BP structure. Finally, the experimentally observed field-induced tetragonal BP (Pieranski and Cladis 1987a) requires additional theoretical analysis. Further field-free Bragg intensity data should also be beneficial in clarifying points in contention, and in particular results on other than cholesteric mixtures.

As discussed above, an experimental field that may be expected to contribute to the more detailed understanding of BP structure includes the various NMR techniques (Samulski and Luz 1980, Yaniv et al 1983). The experiment is, however, notoriously difficult to perform, primarily due to diffusion effects (Samulski and Luz 1980). Thus, in the limit of fast (liquid-like) molecular motion, the spectrum reduces to a single sharp line, whereby the entire structural information is lost. Diffusion effects may be reduced by ingenious choice of the system studied. With large enough molecules it may be hoped that substantial information will be retained. To date, however, all attempts to repeat the successful measurements reported by Samulski and Luz (1980)

700 T Seideman

and by Yaniv et a1 (1983) have been in vain. It should be pointed out that, even when diffusion is not fast enough to average out the spectral details, comparison with theory will yield the desired information only after the latter has taken into account the effect of partial averaging. Such computations have not yet been performed.

Future theoretical advances may be expected to proceed by a number of independent routes. Within the Landau theory framework, extension of the theory to include higher-order terms in the free-energy expansion, contributions from m # 2 Fourier components to the order parameter and/or further spatial harmonics is feasible using the methods outlined above (Grebe1 et a1 1984). Such generalisation would permit a closer analysis of experimental phase diagrams. The Landau model should also allow the calculation of interface energies between a BP and the disordered phase. Crystallite shapes may thus be calculated theoretically and compared with experiments (Blumel et a1 1983, Blumel and Stegemeyer 1984a, Stegemeyer et a1 1986). The computation is, however, very difficult and has not been carried out yet.

Of central importance is the nature of B P I I I . Here further study, both theoretical and experimental, is evidently called for. In spite of the number of discouraging theoretical computations (see e.g. Rokhsar and Sethna 1986, Hornreich and Shtrikman 1986a, Hornreich 1989), the possibility of characterising this anomalous phase by an icosahedral structure cannot be ruled out yet, particularly so in view of the range of supporting experimental evidence (Collings 1984a, b, 1986, Blumel and Stegemeyer 1984b, Thoen 1988). Other suggestions (Hornreich et a1 1982, Kleiman er a1 1984) are nevertheless plausible, and it may well be that future research will offer a different and more consistent model.

Although this study was largely devoted to the Landau theory formalism of the BP problem, we stress the role and future potential of alternative approaches to the subject in substantiating our understanding of the BP structure and providing a complementary theoretical framework. Most notably, the originally phenomenological model advanced by Meiboom and Sammon (Meiboom er a1 1981, 1983a, b, Sammon 1982) and by Berreman (1984) (see figure 2), although less general than the foregoing analysis, offers in our view of a deeper understanding of the physical factors responsible for the appearance of the BPS. It is interesting to note that the results derived via the two different approaches are compatible in most respects. A number of recent theoretical studies discussed alternative approaches to the geometrical frustration induced by the double twist condition. Hornreich and Shtrikman (1988b), reinvestigating the 05, 0’ and O* structures, have considered the constraints imposed on the tensor order parameter by each particular space-group symmetry. The topological features of the distributions corresponding to minimum Landau free energy were explored and their consequences were noted. The idea of considering the local constraint on another Riemannian manifold, first proposed by Sethna (1983a, b), was followed by the work of Pansu et a1 (1987), who applied it to the S3 sphere. Of considerable interest is the possibility of direct experimental observation of defects. Such have so far been reported only for the field-induced 3~ hexagonal BP (Jorand and Pieranski 1987).

Acknowledgments

I am indebted to Professors R M Hornreich and S Shtrikman for reading the manuscript, for their invaluable help and advice and for many stimulating discussions. This work would never have come to light without continuous help and encouragement from


Professor Z Luz, to whom I am also grateful for reading the manuscript and for many helpful comments. I would like to express my gratitude also to Professor P P Crooker for reading the manuscript and for his valuable comments and advice. I am indebted to Professors P G de Gennes, D Demus, H Kleinert, J Thoen and Z Luz for permission to reproduce figures from their past work, and in particular to Professors H Stegemeyer, P J Collings, P P Crooker and R M Hornreich for providing me with preprints of their recent work. Finally, I would like to thank Mr D Mohar for reading the manuscript and for his helpful comments.

Appendix 1. Crystallographic notations and space-group symmetry

A number of somewhat technical details seem necessary for a clear understanding of the theory, and in particular its relation to the experimental measurement.

Crystals of different symmetry are referred to seven different sets of crystallographic axes of reference (edges of the unit cells of the Bravais lattice). A more fundamental division is that into 32 crystal classes on the basis of point-group symmetry. The cubic 4 3 2 crystal class discussed above is a non-centrosymmetric, optically active one, possessing the indicated symmetry elements (and the additional elements implied).

The principal position refers to rotation/inversion along (1 0 0) axes, the secondary to rotation/inversion along (1 1 I) axes, and the tertiary implies rotation/inversion along the (1 1 0) axis; 4 ] , 4* denote screw axes symbols.

Regarding the lattice symbols conventions: I signifies a body-centred (BCC) cell, P a primitive (sc), and F a face-centred (FCC) cell.

More specifically, for 14132 (0') the origin is on 3, at 1/4 translation from each of three non-intersecting 4] axes and three non-intersecting 2 axes. General conditions limiting possible reflections are:

for ( h k l ) h + k + l = 2 n

for ( c y 0 0) only a =4n (a = h, k, 1 ) .

In the I432 (0') the origin is at 43 and the general reflection conditions are:

( h k U h + k + 1 = 2n

( a 00) a = 2 n ( a = h, k, I ) ,

In P4*32 (02) the origin is at 23 and again, for ( a 0 0) type reflections only the a = 4 n are allowed.

Further conditions for particular sets of equivalent positions are detailed for each of the space groups in International Tables f o r X - R a y Crystallography (1965), where all symmetries are listed and the notations more fully explained.

Appendix 2. Dyadic operator formulation

A dyadic U is defined as a set of nine components A,, each of which transforms from one coordinate system to another according to the rule

(A2.1)

7 02 T Seideman

where h are scale factors and the prime denotes quantities in the new coordinate system (see e.g. Morse and Feshbach 1953). The most useful property of dyadics for the present application is that they combine with a vector to form a vector by contraction

U * B = a^,A,,B, m,n

(a^, being unit vectors along the three directions of a right-handed coordinate system). They may thus be regarded as operators that transform a vector into another, the transformation being represented by the nine components A,.

Brazovskii and Dmitriev (1975) regard the (symmetrised) quadratic part of the free energy as a sum of dyadic operators acting in the space of the symmetric traceless tensors E , ~

(A2.2)

in fact, the inverse of the correlation function:

The mathematical problem is thus formulated in terms of a sum of eigenvalue equations

R :; ( 4 1 M zp ( 4 ) = ( 1 4 I ) M 7 8 ( 4 1 tzMyp/2+ tZ;M;,/2- tgM;a’,6,p/3= rmMzp.

m = 0, . . . , 4 (A2.3)

Although the authors reach an erroneous result, namely that for c2 < 0 the cholesteric phase ground state would belong to the m = 1 branch, we found the formalism of interest in that it permits an elegant and compact notation, and moreover a natural coordinate transformation of the eigenvectors to a common reference frame.

Owing to the structure of the t z ( h k 1 ) matrix, the eigenvectors may be expressed in the form

6-”*[ 3 U^, ( nu) U^@ (nu) - amp] m = O ri,(nu)rip(nu) m = l

[M,(nu)lep = U^:(nu)U^g(nu) m = 2 (A2.4) i2-’/2[ U^, ( n u ) t p ( nu 1 + i p ( n f l ) 2 ( nu) 1 m = 3 I i2-1’2[~2( n u ) f p (nu) + ;p*( m)f, ( n a ) ] m = 4

where U ^ = ( i+ i i j ) /&, and the real unit vectors i, ij and f = ( h i + ky*+ lz^)/u1’2= q/ lq l form a right-handed set. For clarity we employed the notation of Grebe1 et al (1983a) to denote the tensor equivalents of the spherical harmonic representation (equations (4.4) and below). The U^ may be recognised as the basis vectors of circular polarisation, eigenvectors of L,, = ieep,qY/lql. It is because of the fact that [M,(nu)Iep depend only through a common phase factor on the way of choosing i and 4 that their scalar products (in terms of which all observable quantities are expressed) are independent of this choice and may be represented in an invariant form. A method of calculating the scalar quantities was described by Brazovskii and Dmitriev (1975):


To evaluate U , u 2 , i2 and f j 2 a‘,e roiated about the t2 = (h , f + k , j + 12i) direction until f j 2 and 4, coincide, whereby g2+ 5: and u2+ u2 exp(ia,). Since il * &=cos e, where e is the angle between q1 and q2, it follows that

U , u2 = 2-”*(cos e - I ) exp(-ia,). (A2.5)

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