3 Liquid Crystals By T. E. FABER Department of Physics The Cavendish Laboratory Cambridge C65 OH€ and G. R. LUCKHURST Department of Chemistry The University Southampton SO9 5NH 1 Introduction Liquid crystals are a state of matter existing between the crystal and the amorphous liquid phases. At the microscopic level the characteristic property of a liquid crystal is the existence of long-range orientational order in contrast with the short-range order typical of liquids. This ordered phase may be obtained from certain solids by heating and also destroyed by further heating; such liquid crystals are known as thermotropic. However various solids may also be made to yield an orientationally ordered phase by the addition of the appropriate solvent; the resultant phase is known as a lyotropic liquid crystal.Although these two classes of liquid crystals have certain features in common we shall be concerned largely with thermotropic liquid crystals in this Report. The behaviour of liquid crystals stems from molecular interactions both with each other and with external fields; in this sense all of their properties are molecular. However many of the bulk properties are adequately described by continuum mechanics which does not need to refer to the existence of molecules. This Report is therefore divided somewhat arbitrarily into two parts; the second of these deals with continuum studies of liquid crystals while the first considers other aspects which are loosely described as molecular. We shall not therefore describe their applica- tions in electro-optic display devices or as solvents in spectroscopy; neither will we refer to the synthesis of new liquid crystals.This is the first account of thermotropic liquid crystals to appear in Annual Reports even though this fascinating state of matter was discovered over eighty years ago. Consequently although much of the literature surveyed appeared in 1975,we shall also refer to earlier publications when this seems appropriate. A number of texts on liquid crystals are available; these include the pioneering account by Gray,’ the masterly treatise on their continuum behaviour by de Gennes,’ and the more recent volumes edited by Gray and win so^.^ The liquid-crystal state seems to have come of age for it now merits a series describing recent advances in both our G.W. Gray ‘Molecular Structure and the Properties of Liquid Crystals’ Academic Press New York 1962. P. G. de Gennes ‘The Physics of Liquid Crystals’ Oxford University Press 1974. ‘Liquid Crystals and Plastic Crystals’ Vols. 1 and 2 ed. G. W. Gray and P. A. Winsor Ellis Horwood Ltd. Chichester 1974. 31 32 T.E. Faberand G.R. Luckhurst understanding and application of liquid crystal^.^ Two general reviews of thermo-tropic liquid crystals have appeared re~ently.~.~ In addition the RCA Review contains a number of good in-depth accounts of many specialized areas of liquid crystals; these include the molecular field theories of nematic' and smectic A phases,' the hard-rod me~ophase,~ the determination of order parameters," con-tinuum theory," and their optical properties.'' The number of papers concerned with liquid crystals continues to grow and it would be impossible to review all of these in the space available. We have therefore selected those publications which we believe to be of particular importance as well as close to our own interests. PART I Molecular Behaviour 2 Orientational Order A complete description of both the orientational and spatial order in a liquid crystal is provided by a hierarchy of distribution functions which give the probability of finding clusters of molecules with particular positions and orientations. The simplest of these is the singlet orientational distribution function but even this has proved to be particularly elusive.The singlet distribution can be determined for a paramag- netic spin probe dissolved in a supercooled nematic me~ophase'~ but has not been obtained for a pure mesophase. The singlet distribution for uniaxial phases may be expanded in terms of spherical harmonics with expansion coefficients proportional to the orientational order parameters. For rigid cylindrically symmetric molecules this expansion reduces to a sum of Legendre functions where 0 is the angle between the molecular symmetry axis and the director. The series is slowly convergent and so many order parameters pLwould be required to yield the true distribution. Unfortunately only the first parameter 4 may be determined with any certainty; in principle the next order parameter P4may be obtained from Raman light-scattering experiments but as we shall see there is some controversy as to the reliability of these values.In the following sections we shall discuss the determination and interpretation of the order parameters p and P4for the various liquid-crystal phases. Nematics. Rigid Rod-like Molecules. Provided we are prepared to assume that the molecules constituting liquid crystals are rigid and possess a three-fold or highzr symmetry axis then a wide variety of techniques is available to determine P2. These include measurement of the partially averaged diamagnetic susceptibility 'Advances in Liquid Crystals' Vol. 1 ed. G.H.Brown Academic Press New York,1975. M. J. Stephen and J. P. Straley Rev. Mod. Phys. 1974,46,4.G.Durand and J. D. Litster Ann. Rev. Mafer. Sci. 1973 3 269. P. J. Wojtowicz RCA Rev. 1974,35 118. 8 P. J. Wojtowicz RCA Rev. 1974,35 388. 9 P.Sheng RCA Rev. 1974 35 132. 10 E.B.Priestley RCA Rev. 1974,35 144. P. Sheng RCA Rev. 1974 35,408. '2 E. B. Priestley RCA Rev. 1974 35 584. '3 P.Krebs and E. Sackmann Mol. Phys. 1972 23 437. Liquid Crystals tensor dielectric tensor and refractive index the order parameter then occurs in the relationship between these tensors and the appropriate molecular parameters. The form of this relationship is well defined for the diamagnetic susceptibility although there are considerable difficulties with the refractive index because the magnitude of the internal electric field is still the subject of some debate.There are those who prefer the Vuks formulation," whereas others prefer the more acceptable Neugebauer prescription,16 although the justification for either approach seems to be largely empirical. Despite the potential uncertainty in the order parameter extracted from measurements of refractive index there has been some effort to improve the accuracy with which the birefringence can be determined.'6y17 A more reliable technique is provided by observation of the dipolar splitting in the 'H n.m.r. spectrum although under low resolution only one order parameter may be determined." The results provided by these various methods may differ in the fine detail but all agree that at the nematic-isotropic transition pz is ca. 0.4 and it increases to approximately 0.7 at 40"C below the transition.The basic features of these results are adequately accounted for by the Maier-Saupe theory and the deviations from this theory are explained by employing a more general anisotropic intermolecular p~tential.~ This success is surprising because by analogy with normal liquids strong repulsive forces might be expected to play a dominant role in determining the molecular organization whereas the theory employs a relatively weak potential. The unimportance of the repulsive part of the anisotropic potential is further supported by approximate statistical mechanical theories such as the Onsager approach,' which show that although an order-disorder transition is predicted for hard particles the order parameter p2of 0.84at the transition is much too large.Other approxima- tions based on lattice models lead to similar discrepancies when compared to the behaviour of real nematics. Of course this failure could always be ascribed to the approximations rather than to the inappropriateness of the repulsive potential. The need for approximations in statistical mechanical calculations of dense ensembles may often be removed oy resorting to the powerful computer-simulation techniques. Thus Vieillard-Baron's Monte Carlo calculation for an ensemble of spherocylinders with length to breadth ratio of 3 is particularly important." Temperature plays no role in determining the static properties of hard particles and so the system is studied as a function of the density which is defined as the number density multiplied by the molecular volume.No evidence was found for an order-disorder phase transition even at densities as high as 0.54,whereas the scaled particle theory predicts a transition for a density of 0.518.20 The absence of a phase transition is particularly disappointing especially for the proponents of the repulsive potential. However it may be that the system has not come to equilibrium because the particles become locked in unrealistic configurations. It may therefore be significant that a preliminary molecular dynamics configuration for ellipsoids interacting with a continuous hard l4 Y.Poggi,J. Robert and J. Borel Mol. Cryst. Liquid Cryst. 1975. 29 311. Is R. Chang Mol. Cryst. Liquid Cryst. 197530 155. l6 H.S. Subramhanyam,C. S. Prabha and D. Krishnamurti Mol. Cryst. Liquid Cryst.,1974,28 201. W. Kuczynski and B. Stryla Mol. Cryst. Liquid Cryst. 1975,31 267. E. Boilini and S. K. Ghosh J. Appl. Phys. 1975,46 78. l9 J. Vieillard-Baron Mol. Phys. 1974,28 809. 2o M. A. Cotter and D. E. Martire J. Chem. Phys. 1970,52 1909. 34 T.E. Faber and G.R. Luckhurst potential does yield an orientationally ordered phase.’l However like the approxi- mate theories this calculation also gives an order parameter pzwhich is unrealisti- cally large. The molecular field approximation is most reliable when applied to the long-range properties of a mesophase; as a consequence there is some interest in seeing if the Maier-Saupe-like theories are as successful in predicting p4as in accounting for p2.Since most anisotropic molecular properties transform under rotation as second rank spherical harmonics the majority of experiments can only provide p2.However the intensity of light scattered in a Raman experiment is proportional to the mean- square polarizability and so depends on the order parameter p4as well as pz.The theoretical relationship between the scattered intensity and p4is quite straightfor- ward,22 although considerable precautions must be taken in the spectral analysis especially if the contribution of director fluctuations to the scattering is to be avoided. The technique has been applied to 4’-n-heptyl-4-cyanobiphenyl,using the cyano vibration which is well removed from other molecular vibration^.'^ The parameter p4is found to be lower than that predicted by either the Maier-Saupe theory or the Humphries-James-Luckhurst extension.The departure of P4 from the predicted values is still more marked for a mixture of 4-n-butyloxybenzylidene-4’-cyanoaniline and 4-methoxybenzylidene-4‘-n-butylaniline, where negative values of P4have been dete~mined.~~ The failure of theory is still unexplained but the discrepancy could stem from error in the experimental values of p4.This view is supported by linewidth variations in the e.s.r. spectra of a spin probe dissolved in the nematic mesophase of Merck Phase IVZ5which give values of p4in support of the Maier-Saupe theory. The order parameter P4is available from such measurements because the linewidths depend on the mean-square value of the relevant magnetic tensors which are also second rank.26’27 Clearly further experimental investigations employing both techniques are required preferably of the same system before we can be certain ofeither the success or failure of the simplest molecular field theories.Deviations from Cylindrical Symmetry. Of course the molecules of most mesogens are neither cylindrically symmetric nor rigid as is so often supposed. Since the molecules do not possess a three-fold or higher symmetry axis the single order parameter p2must be replaced with the Saupe ordering matrix defined by sab =(3 cos 0 cos 6 -6&)/2 (2) where 0 is the angle between the molecular axis a and the director. There has been one attempt to determine S for a nematogen,28 but this was not entirely satisfactory because it involved the combination of results from different experiments.The reason for this lack of detail is quite straightforward for techniques (such as measurement of the partially averaged refractive index tensor) only provide a single 21 J. Kushick and B. J. Berne J. Chem. Phys. 1976,64 1362. 22 E. B. Priestley and P. S. Pershan Mol. Cryst. Liquid Cryst. 1973 23 369. 23 J. P. Heger J. Phys. (Paris) 1975,36 L-209. 24 S. Jen N. A. Clark P. S. Pershan and E. B. Priestley Phys. Rev. Letrers 1973 31 1552. 25 G. R. Luckhurst and R. Poupko Chem. Phys. Letters 1974 29 191. 26 G. R. Luckhurst M. Setaka and C. Zannoni Mol. Phys. 1974,28,49. 27 G. R. Luckhurst R. Poupko and C. Zannoni Mol. Phys. 1975 30,499.28 R. Alben J. R. McColl and C. S. Shih Solid State Comm. 1972 11 1081. Liquid Crystals 35 piece of independent information and so cannot be expected to yield the five independent elements of the ordering matrix. Thus most experiments can give some average of S or if the appropriate molecular interaction is cylindrically symmetric one of its components. The complete matrix can be measured by n.m.r. spectros- copy and this is part of a standard procedure for determining the geometry of solutes dissolved in a liquid-crystal The same approach cannot be readily applied to a pure mesophase because the large number of dipolar interactions make it virtually impossible to resolve the complex ‘H n.m.r. spectrum. One solution to this problem is to replace certain protons by deuterons.Then the proton spectrum may be readily resolved after decoupling any deuteron-proton interactions. A successful analysis of the spectrum would then yield several partially averaged dipolar interac- tions from which S can be determined. Although the proton spectra of partially deuteriated mesogens have been reported there was no attempt either to simplify the spectra or to obtain a quantitative anal~sis.~’ Alternatively the ’H n.m.r. spectrum may be recorded; this is usually dominated by pairs of quadrupole-split lines from each deuterium with some secondary structure from dipolar splittings. This technique has been applied to perdeuterio-4,4’-dimethoxyazoxybenzene,and both the partially averaged quadrupole splittings and dipolar coupling between the ortho-deuterons were ~btained.~’ The analysis of these splittings provides useful geometrical information but cannot give the ordering matrix because the coupling between nuclei in different rings was too small to be resolved.Consequently only the local ordering matrix for each ring could be determined and these are found to be essentially cylindrically symmetric about the para axis presumably because of internal rotation about this axis. Since the molecules constituting nematogens are biaxial there is the possibility that the mesophase itself might also be biaxial. The factors influencing the transition from a uniaxial to a biaxial nematic phase have been studied in some detail using a variety of theoretical approaches.However such a transition has yet to be observed; nonetheless it is important to see how the molecular biaxiality will influence the properties of the uniaxial mesophase. Straley3’ has tackled this problem for an ensemble of hard rectangular particles but as we have seen the use of a repulsive potential may well invalidate the quantitative aspects of his calculation. A compar-able theory has been reported which employs a general expansion of the pair potential for molecules of arbitrary ~hape.’~ The number of parameters in the resultant pseudopotential is reduced by restricting the summation to second-rank interactions and further assuming that the remaining coefficients may be equated with those expected for dispersion forces. The pseudopotential parametrized in this way contains two arbitrary parameters; one of these is proportional to the nematic- isotropic transition temperature while the other is related to the deviation of S from cylindrical symmetry.By using the data available for the nematogen 4,4‘-dimethoxyazoxybenzene it is possible to predict the temperature dependence of the major element of S at constant volume in excellent agreement with ex~eriment.~~ 29 J. W. Emsley and J. C. Lindon ‘NMR Spectroscopy using Liquid Crystal Solvents’ Pergamon Press Oxford 1975. 30 J. J. Visintainer E. Bock R. Y. Dong and E. Tomchuk Cunad. J. Phys. 1975 53 1483. 31 P. Diehl and A. S. Tracey Mol. Phys. 1975,30 1917. 32 J. P. Straley Phys. Reu. (A) 1974 10 1881. 33 G. R. Luckhurst C. Zannoni P. L. Nordio and U.Segre Mol. Phys. 1975,30 1345. 36 T. E. Faber and G. R. Luckhurst Departures from predictions of the Maier-Saupe theory appear to be attributable to deviations from molecular cylindrical symmetry but more accurate measurements of the total ordering matrix are required before we can be certain of the success claimed for the theory. Nun-rigid Molecules. Although there is scant experimental support for deviations of S from cylindrical symmetry there is ample evidence for the profound influence of molecular non-rigidity. For example the flexible alkyl chains which form a vital part of many mesogens are known to be responsible for the alternation in the nematic- isotropic transition temperature along an homologous series. 1,3 This odd-even effect can be reconciled with the Maier-Saupe theory because the molecular interaction parameter is expected to alternate when averaged over the various chain configura- tions.However the theory cannot explain the observed alternation in the entropy of transition which is predicted to be constant. MarEelja was the first to develop a quantitative theory of such effects by specifically including the influence of the chain configurations on the intermolecular potential.34 He proposed a pseudopotential which contains terms similar in form to the Maier-Saupe potential where 0 is the angle between the director and the supposed symmetry axis of the ith unit. The order parameter for the jthunit is denoted by Py)and the coefficients vj are related to the strength of the interaction between various units; these are usually the rigid aromatic core and the flexible alkyl chains.MarEelja then includes another term in the total pseudopotential to represent the dependence of the internal energy on the chain configuration. The resulting single-particle pseudopotential is then employed to determine the various order parameters from the usual consistency equations as well as the Helmholtz free energy to obtain the transition temperature. Since the orientation of a unit in the alkyl chain depends on that of the rigid core the total number of distinct chain configurations is unmanageably large; this number is reduced by restricting the rigid core to either three or five configurations. Despite this drastic approximation the calculated alternation in both the transition tempera- tures and entropy of transition is in agreement with experiment.The complexity of MarEelja’s calculations tends to obscure those factors which are basically responsible for the various odd-even effects that are observed for most homologous series. This has prompted Pink to simplify MarEelja’s theory;35 the philosophy of his simplification is to treat the interactions of the alkyl chains both with each other and with the rigid aromatic core as perturbations. This has the advantage that averages over the chain configurations may be evaluated for a single particle in the absence of the molecular field; some influence of the field is in fact retained because configurations outside a cylinder generated by rotating an all-trans configuration of the chain are ignored.Despite some of the questionable assump- tions necessary for the simplified theory its predictions are comparable to those of the Mar5elja theory and hence in reasonable accord with expzriment. The two theories also predict that the order parameter P2for the rigid core evaluated at the transition temperature should exhibit an odd-even effect. N.m.r. spectroscopy provides virtually the only method for testing this prediction. For 34 S. Mareelja J. Chem. Phys. 1974 60 3599. 35 D. A. Pink J. Chem. Phys. 1975,63 2533. Liquid Crystals 37 example it the ordering matrix for the rigid unit is cylindrically symmetric then p2 could be determined from the dipolar splitting between the ~rtho-protons.~ An alternative route has been proposed by Pines and Chang;36 this involves measure- ment of the 13Cn.m.r.spectrum from which the dipolar interaction with protons has been removed by noise decoupling. The remaining spectrum is particularly simple because each nucleus in the carbon skeleton can only contribute one line whose frequency is determined by the partially averaged chemical-shift tensor and hence by the ordering matrix. This technique has been applied to the homologous series of 4,4’-di-n-alkyloxyazoxybenzenes; unfortunately no attempt was made to see if S for the rigid core was cylindrically symmetric and so only the order parameter at the transition was obtained for this series.37 The order parameter was found to exhibit an odd-even effect as predicted with the maximum value of 0.455 found for the ethoxy-derivative and the minimum value of 0.365 observed for propyloxy.These results reveal that the quantitative aspects of MarEelja’s theory are not quite so satisfactory since the predicted maximum and minimum values are 0.455 and 0.399.34 The MarEelja theory but not Pink’s simplification may also be used to calculate the order parameters of the methylene groups along an alkyl chain. As we might have anticipated these order parameters may be determined from the 2Hn.m.r. spectrum of the mesogen with deuteriated chain^.^^.^' The spectrum is dominated by a series of doublets each of which come from the methylene groups or the terminal methyl group. The unambiguous assignment of a splitting to a particular group is impossible in the absence of specifically deuteriated mesogens but it seems reason- able to suppose that the order parameter decreases along the chain.The results shown in Figure 1 for 4-cyan0-4’-n-pentylbiphenyl,~’ terephthalylidene-bis(4-n-b~tylaniline),~’ were obtained with and 4-n-butyloxybenzylidene-4’-n-octylaniline38 this assumption The behaviour of the chains in these compounds shows similar trends and is in qualitative agreement with MarEelja’s calculations for 4,4’-di-n- octyloxyazoxybenzene,34 although values are not yet available for the compounds studied. An alternative technique is to attach a paramagnetic group such as 2-N-oxyl-3,3-dimethyloxazolidine,at specific positions along the chain and to determine the ordering matrix for this group from the partially averaged g and hyperfine tensors.This approach has been used with several 4,4’-dialkyloxyazobenzenes dissolved in the smectic B and C phases of 4,4’-di-n-octadecyloxyazoxybenzene~o the order parameter is found to decrease along the chain in the manner similar to that shown in Figure 1. However the order parameters exhibit a more pronounced odd-even effect which may be attributed to the perturbing influence of the spin label on the configurations adopted by the chain. The orientational order in a mesophase makes it possible to observe a n.m.r. dipolar echo following an in-phase pulse sequence.41 The echo amplitude exhibits a gaussian dependence on the time 7between 90”pulses E(T,90”)=E(O,9Oo)exp {-iM272} (4) 36 A.Pines and J. J. Chang Phys. Rev. (A),1974,10 946; B. Clin Compr. rend. 1975,280 C 73. 37 A. Pines D. J. Ruben and S. Allison Phys. Rev. Letters 1974,33 1002. 38 B. Deloche J. Charvolin L. Liebert and L. Strzelecki,J. Phys. (Paris),1975,36 C1-21. 39 J. W. Emsley J. C. Lindon and G. R. Luckhurst Mol. Phys. 1975,30 1913. 40 F. Poldy M. Dvolaitzky and C. Taupin J. Phys. (Paris) 1975 36 Cl-27. 41 N. Boden Y. K. Levine D. Lightowlers and R. T. Squires Chern. Phys. Letters 1975 31 511; ibid. 1975,34,63. T.E. Faberand G.R. Luckhurst 0.2 sc-D 0.1 0 12345678 carbon number Figure 1 The orderparameters SC-, for the alkyl-chain segments in the mesogens 4-cyano-4'-n- pntylbiphenyl (O) terephthalylidene-bis(4-n-butylaniline) (0),and 4-n -butyloxybenzylidene-4'-n-octylaniline(0).where Mz is the second moment between pairs of strongly coupled protons. For example in an alkyl chain the methylene protons are strongly coupled and it is the interactions between protons in different methylene groups which contribute to M2. This second moment reflects the chain statistics but the extraction of detailed information concerning the orientational order of the chain is clearly a difficult task. However qualitative information is available from such experiments. For example the echo amplitude for the nematic mesophase of even members of the series of 4,4'-di-n-alkyloxyazoxybenzenesis found to follow a single decay whereas the decay for the odd members can only be represented by a sum of two decays.This observation is taken to imply a discontinuous flexibility gradient for the odd members of the series; although this information is potentially valuable experiments with simpler systems in which protons are replaced by deuterons are required to prove this conclusion. Cho1esterics.-The characteristic feature of the cholesteric mesophase is the helical arrangement of the director. Since the pitch of the helix is large compared with the range of the intermolecular potential responsible for the angular correlation the helical structure should not influence the magnitude of the order parameters. The constituent molecules of most cholesterogens contain a steroidal residue and are devoid of the aromatic groups found in most nematogens.It is likely therefore that the anisotropic forces responsible for the orientational order in the phases are quite different and that this difference would be reflected in the order parameter pz. Liquid Crystals 39 However the apparently important problem of determining p2for steroidal choles- terogens appears to have been neglected. There do not seem to be any measure- ments of p2for the pure phase even though the birefringence has been defe~mined.~~ The only other measurements are for the ordering matrix of aromatic solutes dissolved in a compensated mixture of steroidal cholester~gens.~~ SmecticA.-Experimental studies of the smectic A phase were much stimulated by the Kobayashi-McMillan theory.* One intriguing prediction of the theory is that the nematic-smectic A transition should become second-order when it occurs at or below a certain reduced temperature.If the theory is restricted to a purely anisotropic intermolecular potential as in the Maier-Saupe theory then this tricriti- cal point is shown to occur when TS-N/TN-I equals 0.85. However the results of various experiments suggest that this estimate is too low. For example the order parameter of a spin probe dissolved in 4’-n-octyloxy-4-cyanobiphenylhardly changes at the nematic-smectic A transition which would therefore appear to be essentially second-~rder.~~ Since the transition occurs at a reduced temperature of 0.96 the theoretical prediction of 0.85 or less is clearly in serious error. A more refined test of the theory would be to vary the reduced temperature for the nematic-smectic A transition in order to locate the tricritical point exactly and two ingenious experiments have been devised to accomplish this aim.In the first study the transition temperatures TS-Nand TN-Ifor the mesogen 4-cyanobenzylidene-4’-n-nonylaniline were varied by increasing the pressure.45 The two transitions have different pressure coefficients and it was possible to decrease the reduced nematic- smectic A transition temperature while monitoring the order parameter with n.m.r. The discontinuity in p2was found to vanish when the reduced transition temperature was 0.92 corresponding to a pressure of 2.89 kbar. The transition temperature TS-N was varied in the second investigation by changing the composition of a binary mixture of 4-n-octyloxybenzylidene-4’-n-propylaniline and its ethoxy-hom01ogue.~~ The order of the transition was determined from the enthalpy of transition after due correction for the contribution from pretransitional effects.The tricritical point was then identified at a reduced temperature of 0.96. These high values of the reduced nematic-smectic A transition temperature can be explained if the scalar contribution to the intermolecular potential is included. The observed values of TS+/TN-,would then appear to indicate that this scalar term tends to dominate the potential. SmecticC.-The smectic C phase differs from the A phase because the constituent molecules prefer to be tilted away from the normal layer. Consequently any molecular theory of the smectic C phase must start from a pairwise potential which constrains the molecules to be inclined to the intermolecular vector in the minimum energy configuration.This contrasts with the situation in a nematic or smectic A where the minimum in the intermolecular potential occurs when the molecules are orthogonal to the vector. McMillan forces the molecules to tilt by the addition of 42 M. Evans R. Moutron and A. H. Price J.C.S.Faraday II 1975 71 1854. 43 E. Sackmann,P. Krebs,H. U. Rega,J. Voss and H. Mohwald,Mol. Cryst.Liquid Cryst. 1973,24,283. 44 G. R. Luckhurst and R. Poupko,Mol. Phys. 1975,29,1293. 45 T. J. McKee and J. R. McColl Phys. Rev. Letters 1975 34 1076. 46 D. L. Johnson C. Maze E. Oppenheim and R.Reynolds Phys. Rev. Letters 1975,34 1143. 40 T. E. Faber and G.R. Luckhurst electric dipoles which are not parallel to the molecular long axis.47 He then considers a particularly simple system of a layer of particles with this dipole-dipole interaction and using the molecular field approximations discovers a second-order transition from smectic A to a tilted smectic C phase. The rotational motion about the long axis is predicted to be quenched in the smectic Cphase but as yet there is no experimental evidence for such quenching. Indeed many experiments suggest that the rotational motion in the C phase is as unhindered as in the preceding smectic A phase.48 Wulf has proposed that it is the interactions between the alkyl chains which are responsible for the tilt in the smectic C phase because the chains are of necessity not parallel to the molecular long axis.49 The chains are assumed to be rigid entities and so the problem of chain statistics is avoided; their interaction is represented by an empirical term in the intermolecular potential.This is then treated in a manner analogous to that in the Kobayashi-McMillan theory of the smectic A phase. The theory successfully predicts a second-order phase transition between the smectic A and C phases. Unlike the smectic A phase a smectic C is predicted to be biaxial and the extent of the biaxiality is related to the tilt angle. In fact the deviation from cylindrical symmetry is predicted to be large and some evidence for this can be gleaned from a careful analysis of n.m.r.spectra of smectic C phases." 3 Molecular Dynamics The molecular structure of most mesogens is relatively complex and so there are a wide variety of motions which a molecule may execute. These include translation rotation vibration and internal rotation; the situation is further complicated by the possibility of coupling between these various modes. However the motions are invariably assumed to be independent and so we shall discuss them separately. TranslationalDiff usion.-The macroscopic anisotropy of a liquid crystal demands that the translation motion be described by a second-rank tensor D rather than a scalar as in normal liquids. For uniaxial systems such as a nematic or smectic A mesophase the diffusion tensor has cylindrical symmetry whereas for a smectic C it should be biaxial although the departure from cylindrical symmetry appears to be negligible for terephthalidene-bi~(4-n-butylaniline).~~ Of the several techniques available for the determination of D probably the most reliable involves direct observation of the motion in an aligned mesophase.For example Yun and Fre- dricks~n~~ monitored the diffusion of 14C-labelled4,4'-dimethoxyazoxybenzene at 122"Cand found Dll=4.0X cm2s-' with a ratio of Dll/D,equal to 1.25. These results clearly demonstrate the relative ease of diffusion parallel rather than perpendicular to the director although the difference is not as large as anticipated. Two other ingenious methods have been devised to monitor mass migration of a probe molecule in a mesophase; in one experiment the probe is a dye and so its progress may be observed dire~tly.~~ In the other the probe is optically active;54 this 47 W.L. McMillan Phys. Reu. (A) 1973,8 1921; R. J. Meyer and W. L. McMillan ibid. 1974 9 899. 48 Z. Luz R. C. Hewitt and S. Meiboom J. Chem. Phys. 1974,61 1758. 49 A. Wulf Phys. Reu. (A) 1975,11 365. 50 A. Wulf J. Chem. Phys. 1975,63 1564. st R. Blinc M. Burger M. Luzar J. PirS I. ZupanEiE and S. iumur Phys. Reu. Letters 1974 33 1192. 52 C. K. Yun and A. G. Fredrickson Mol. Cryst. Liquid Cryst. 1970 12 73. 53 F. Rondelez Solid State Cornm. 1974 14 815. s4 H. Hakemi and M. M. Labes J. Chem. Phys. 1974,61,4020. Liquid Crystals converts the nematic into a cholesteric phase with a pitch related to the probe concentration.The magnitude of the pitch is readily gauged from the distance between the Grandjean lines which gives the required time dependence of the concentration. The technique has been applied to a racemic mixture of (l),using one of the optical isomers as a probe," so removing the objection that the results are not directly relevant to the behaviour of the pure nematic phase. Here the enhanced molecular anisotropy results in a greater anisotropy in D;thus DI1/D1 ranges from 2.25 at 45 "C to 1.70 at 86 "C while Dllgoes from 0.5 x cm2s-' to 4.10X cm2s-' for the same change in temperature. N.m.r. spectroscopy may also be employed to determine the diffusion tensor by observing the influence of either a static or a pulsed gradient in the applied magnetic field on the form of the spin echo.The technique is difficult to apply to liquid crystals because of their complex spectra caused by the dipolar splittings although this difficulty has been circumvented in a number of ways. For example the structure may be removed by using a multiple-pulse experiment in conjunction with partial deuteriation; this procedure has been employed for 4-methoxybenzylidene-4'-n-butylaniline where Dll/D is found to be 1.4 at 25 "C with Dllequal to 6.9 X lo-' cm2s-'.~~ The smectic C and A phases of terephthalidene-bis(4-n-butylaniline) have been studied with the same technique,'l and in the A phase Dll is 14~ cm2s-' while Dll/D,is 0.3. This demonstrates the ease of motion within the smectic layers in contrast to movement across the layers; again the difference is not as large as might have been anticipated.An alternative procedure to spin decoupling is to align the mesophase with the director at the so-called magic angle (cos-' 1/J3)for then the dipolar splitting vanishes and the spectrum collapses to a single line. Kruger et ~11.~~ have employed this trick to study translational diffusion in the smectic A and B phases of 4-n-dodecanoylbenzylidene-4'-aminoazobenzene. At the start of the smectic A Dll/D,is only 0.6 but this decreases to 0.2 at the transition to the smectic B phase where the anisotropy in D increases still further. In principle incoherent quasi-elastic scattering of neutrons from a liquid crystal can also be used to determine the translational diffusion tensor.However the major difficulty and possibly the strength of this technique is that all kinds of molecular motion can contribute to the scattering. Indeed many of the earlier measurements of D using neutron scattering are now known to be in error because the scattering vector Q employed in these experiments was so large that there was rotational broadening of the quasi-elastic peak." It seems that the broadening comes entirely 55 H. Hakemi and M. M. Labes J. Chem. Phys. 1975,63,3708. 56 I. ZupanEiE J. PirS M. Luzar R. Blinc and J. W. Doane Solid State Comm. 1974 15 227. 57 G. J. Kruger J. Spiesecke and R. Weiss Phys. Letters (A) 1975 51 295. 58 J. Topler B. Alefeld and T. Springer Mol. Cryst. Liquid Cryst. 1974,26 297.42 T.E. Faberand G.R. Luckhurst from translational motion only if the scattering vector is less than 0.3 k’;under these conditions the width of the peak Aq is AW; = D,,COS’ e +D sin2 e (5) where 6 is the angle between the director and Q.59 The diffusion tensor determined for 4,4‘-dimethoxyazoxybenzene,for low Q values is found to be in good agreement with that determined from tracer experiments. Similar studies have also been reported for 4-n-pentyl-4’-cyanobiphenylusing material with and without the alkyl chain completely deuteriated.60 Since the scattering cross-section of deuterium is far less that that for a proton it is possible to see if the chain motions contribute to the broadening. In fact the linewidth was the same for both samples and so the broadening may be ascribed entirely to translational motion.The ratio D,,/D, determined from the slopes of a plot of Awa uersus Q2 was found to be 1.3 which is close to the value for 4,4’-dimethoxyazoxybenzene.The nematogen 4- methoxybenzylidene-4’-cyanoanilinewas also studied and at 112 “C the ratio Dll/D,was determined to be 2.2. The relatively large difference between these values was taken as further evidence for some sort of association in 4-n-pentyl-4’- cyano biphenyl. Despite the considerable effort on the part of experimentalists to determine the translational diffusion tensor for nematics and other liquid crystals there are remarkably few theoretical models to rationalize their results. One of the earliest attempts to develop a theory was made by Franklin,61 who modified the Kirkwood theory to allow for the anisotropy in the bulk viscosity of the nematic.Of course this is an over-simplification because in the Leslie formulation of the hydrodynamics of nematics there are five viscosity coefficients. Accordingly Franklid2 has modified the original theory to take account of all these coefficients and claims to find good agreement with the experimental temperature dependence of D for 4,4’-dimethoxyazoxybenzene. Although it is helpful to have a relationship between the translational diffusion tensor and the various Leslie viscosity coefficients a theory involving the molecular interactions might be more illuminating. Any such molecu- lar theory must start with the autocorrelation function of the momentum p since the scalar diffusion constant is .OD D = (k~/m7) p(0) -p(t) dt (6) J 0 This approach has been adopted by Chu and Mor~i,~~ although they only expand the correlation function for small times and then evaluate p’ and p’ by invoking the molecular-field approximation.The theory appears to work reasonably well although the assignment of various parameters occurring in the theory is not clear. A more reliable route to an understanding of the molecular factors determining D should be provided by the powerful computer-simulation technique of molecular dynamics. One calculation is available for highly anisotropic particles interacting sy K. Rokiszewski Acta Phys. Polon. (A),1972 41 549. 6o A. J. Leadbetter F. P. Temme A.Heidemann and W. S. Howells Chem. Phys. Letters 1975,34,363. W. Franklin Mol. Cryst. Liquid C.yst. 1971 14 227. 62 W. Franklin Phys. Rev. (A),1975,11 2156. 63 K. S. Chu and D. S. Moroi J. Phys. (Paris) 1975 36 CI-99. Liquid Crystals 43 with a continuous but strongly repulsive potentiaL2' However the orientational order in this system of ellipsoids with the major axis three and a half times the minor axis was found to be much larger than that in a real nematic. Nonetheless the translational diffusional tensor can be scaled to allow for this high order according to the common (but suspect) rule where the superscript indicates the diffusion tensor for the completely aligned mesophase. Then the ratio Dll/D,,for an order parameter pzof 0.56 is calculated to be 3.35 which is considerably higher than the values determined experimentally.There are two possible explanations for this major discrepancy between theory and experiment. One is the difficulty of knowing whether the system has reached thermodynamic equilibrium because with highly anisotropic potentials it is possible for the system to become isolated in such metastable regions of configurational phase space. Alternatively the purely repulsive anisotropic interaction may not be appropriate for real liquid crystals. Rotation.-The molecular reorientation in a liquid crystal must be highly aniso- tropic for while rotation about the long axis is essentially unhindered the motion of the long axis is constrained by the long-range orientational order.Nordio and his colleagues have developed a theory of rotational diffusion to describe these motions by extending the Debye theory to allow for the torques experienced by a molecule as a consequence of the orientational order.64 The theory therefore relates the various rotational correlation times to the parameters in the orientational pseudopotential as well as to the components of the rotational diffusion tensor. However the theory does not attempt to describe those factors which determine this tensor and as we shall see these have yet to be assigned. The strong-collision model has also been applied to the evaluation of various rotational correlation functions involving liquid crystals; however its only merit would appear to be its mathematical simplicity for it is unable to relate the collisional correlation times to the order in the mesophase.26 Dielectric relaxation provides a valuable technique for investigating molecular reorientation in liquid crystals and has been applied to a variety of systems.Of course the permittivity is now a second-rank tensor and for most liquid crystals it possesses cylindrical symmetry. The component parallel to the director normally shows a low-frequency dispersion associated with long-axis reorientation as well as a high-frequency dispersion coming from rotation about the long axis. The low- frequency dispersion can normally be fitted to a single Debye rela~ation~~ or a narrow Fuoss-Kirkwood distribution,66 and the relaxation time obtained is iden- tified with the correlation time for rotation of the Iong axis.For example the low-frequency dispersion for 4-n-pentyl-4'-cyanobiphenyl can be explained in terms ofa single relaxation time which decreases from 8.5 X lo-' s at 14 "Cto 2.7 x lo-' sat 28 0C.65Similarly recent measurements for certain 4,4'-di-n-alkoxyazoxybenzenes can be interpreted with a single relaxation time.67 The temperature dependence of 64 P. L. Nordio and P. Busolin J. Chem. Phys. 1971,55,5485;P. L. Nordio G. Rigatti and U. Segre ibid. 1972,56,2117. 65 P. G. Cummins D. A. Dunmer and D. A. Laidler Mol. Cryst. Liquid Cryst. 1975 30 109. 66 V. K. Agarwal and A. H. Price J.C.S. Faraday Il 1974 70 188. 67 A. Mircea-Roussel and F. Rondelez J. Chem. Phys. 1975,63 231 1. 44 T.E.Faber and G.R.Luckhurst the rotational correlation times for this series was analysed with the aid of a simple Arrhenius plot and the resulting activation energies were employed to support the idea of significant pretransitional effects for those members of the series which possess a smectic C phase following the nematic.However considerable caution must be exercised when applying such a simple approach to the analysis of the temperature dependence of the correlation time because two quite different factors contribute to this time. The first is the long-range orientational order while the second is the rotational diffusion tensor. By employing Nordio's theory this first contribution may be removed from the correlation time to leave the diffusion tensor. For many normal liquids this tensor is proportional to the bulk viscosity and so there is now some effort to find which if any of the Leslie viscosity coefficients may be involved for a mesophase.It has been suggested that the twist viscosity coefficient may be important,68 but the determination of the rotational correlation time in the nematic and smectic A phases of 4-cyanobenzylidene-4'-n-octyloxyanilineseems to rule out this po~sibility.~' Thus the correlation time is continuous through the second-order nematic-smectic A transition whereas the twist viscosity coefficient diverges at the transition. Reorientation about the long axis in a mesophase has not received as much attention as motion of the long axis probably because the dispersion in the permittivity occurs in an experimentally difficult frequency range.In addition this motion is not significantly affected by the long-range order that is characteristic of a liquid crystal. However Evans et al.have studied this motion for cholesteryl oleyl carbonate in the isotropic cholesteric and smectic phases.42 They find that the correlation time is continuous at the isotropic-cholesteric transition but increases discontinuously when the smectic phase is formed. The far-i.r. spectra of the cholesteric and isotropic phases were also recorded; a broad absorption centred at ca. 75 cm-' was detected for both phases. By analogy with similar studies of 4-methoxybenzylidene-4'-n-butylan~l~ne70 this Poley-like absorption has been attri- buted to a librational motion of the rigid part of cholesteryl oleyl carbonate.These particular measurements probe the local structure of the system and it would appear that this does not suffer any major changes on going from the isotropic to the cholesteric mesophase. Of course the major difficulty of studying molecular rotation by dielectric relaxation is knowing the exact relationship between the frequency-dependent permittivity and the dipole-moment autocorrelation function. This problem is particularly severe for liquid crystals because both the permittivity E and the correlation function @( t) are second-rank tensors. However linear-response theory has been employed in an attempt to allow for this anisotropy and the following expression is obtained for the permittivity parallel to the director where n(w) is a frequency-dependent depolarization The autocorrelation 68 F.Rondelez and A. Mircea-Roussel Mol. Cryst. Liquid Cryst.,1974,28 173. 69 Y.Galerne Compt. rend. 1974,278 B 347. 70 M. Evans M. Davies and I. Larkin J.C.S. Faraday ZZ 1973,69 101 1. 71 G. R. Luckhurst and C. Zannoni Proc. Roy. SOC. 1975 A343,389. Liquid Crystals function @(t) is defined in terms of the dipole moment of an ellipsoidal cavity embedded in the mesophase. Consequently the problem of relating this to the single-particle autocorrelation function remains unless correlations between dipole moments in different molecules are ignored. Nonetheless this theory is an improve- ment over the relationships implicitly adopted for &(a), but in view of its complexity it remains to be seen if experimentalists consider that their data merit such an analysis.There are far fewer theoretical problems in obtaining rotational correlation times from the line broadening in electron resonance although this dynamic information relates to the motion of the spin probe and so unless the probe is carefully chosen day not reflect the behaviour of the pure mesophase. The theory governing the linewidths caused by the rotational modulation of the g and hyperfine tensors is reasonably well established for doublet-state spin probes.26 It has recently been extended to triplet-state species where the dominant interaction is the zero-field ~plitting,~’ The macroscopic anisotropy of a mesophase results in an angular dependence of the linewidths and this should be exploited to maximize the informa- tion available from line-broadening studies.These techniques have been employed to study the rotational motion of the spin probe (3-spiro-[2’-N-oxyl-3‘,3’-dimethyloxazolidine])-5a -cholestane in the smectic A phase of 3-N-(4’-ethoxybenzylideneamin0)-6-n-butylpyridine.~~ Using the diffusion model the ratio Dll/D,for the components of the rotational diffusion tensor was found to be about 50for an order parameter p2of 0.89. This ratio is extremely large and contrasts with a value of three predicted by a purely hydrodynamic model; departure from this prediction is taken to indicate essentially unhindered rotation about the long molecular axis. The same spin probe has also been employed to study the nematogen Merck Phase IV;25in this mesophase the order parameter p2for the spin probe is only 0.62 and this may account for the reduction of Dll/D,to 35.Most experimental and theoretical investigations of line broadening in e.s.r. spectra have been confined to the fast-motion limit partly because the spectral analysis is far simpler in this regime. The theory has been extended to the.slow-motion limit and employed to analyse the line broadening in the spectrum of perdeuteriated 2,2,6,6-tetramethyl- 4-piperidone- 1-oxyl dissolved in the nematic mesophase of Merck. Phase V.72 Because the spin probe is not cylindrically symmetric the diffusion model developed by Nordio and his c011eagues~~ for symmetric-top molecules was modified for asymmetric rotors.This complicates the spectral analysis which is also inhibited by the low degree of order found for the spin probe. However a detailed analysis is possible and it is found that parameters obtained in the fast-motion limit cannot be used to predict their values at lower temperatures. This discrepancy can be removed by replacing the Debye-like spectral densities j(w)= T/(I +w2~2) (9) by the semi-empirical expression j(w)= T/(I+w2T2) (10) The magnitude of E is rationalized in terms of coupling of the molecular reorienta- tion to other degrees of freedom of the spin probe’s environment. 72 C. F. Polnaszek and J. H. Freed J. Phys. Chem. 1975,79,2283. 46 T.E.Faber and G. R. Luckhurst There have been few studies of the rotational motion in liquid crystals by incoherent quasi-elastic neutron scattering possibly because the exact theory neces- sary to interpret the scattering is still being de~eloped.~’*~~.~~ The central problem is the evaluation of the intermediate scattering function Ii”c(Q t) =exp {-iQ * r(O)}exp {iQ -r(t)} (11) where r is the position vector of the scattering centre.Provided the various motions are uncorrelated this scattering function may be written as a product of scattering functions and for rotation If:,!(Q t) =exp {-iQ a(0)) exp {iQ -a(t)} (12) where a is the position vector of the scatterer with respect to the centre of rotation. Unlike the situation in magnetic resonance it is difficult to evaluate this correlation function except for very special situations.For example if the long molecular axis is fixed and the molecule undergoes rotational diffusion about this axis the correlation function can be evaluated as a series expansion.74 A Fourier transform in time then gives the so-called scattering law for rotation as where J is an 12‘’-order Bessel function of the first kind and D is the rotational diffusion constant. The angle between Q and the long molecular axis is denoted by 8. When 8 is 7r/2 the width of the quasi-elastic peak is found to pass through a maximum at Qa equal to about 3.9. Similar results are found for other models describing reorientation about the long axis.74 The assumptions employed in this derivation are likely to be realistic for a smectic H phase or a solid where the long molecular axes are completely ordered.Indeed such notions have been employed to study motion in the smectic H phase of terephthalidene-bis(4-n-butylaniline) and to show the absence of any correlation between the short molecular axes.75 The same mesogen has been studied in the solid phase using material with deuteriated chains; apparently the scattering is caused by chain motion and an analysis in terms of a jump-diffusion model gives a correlation time of ca. lo-’* s.76 However in a nematic phase and certain smectic phases the orientational order is not high but provided the rotation of the long axis can be ignored then the scattering law can be obtained by taking the appropriate average over 8. Such averaging complicates the resultant scattering function even in the simplifying situations with Q either parallel or perpendicular to the director.There is however a more serious problem with this approach for it is by no means certain that the motion of the long axis can be ignored when evaluating the scattering law. Indeed although e.s.r. studies confirm the expected anisotropy in the rotational diffusion tensor this is not sufficiently large as to justify the neglect of long-axis motion. It would appear that we must await further theoretical developments before incoherent quasi-elastic neutron scattering can be employed to study rotational motion in a nematic. 73 K. RoSciszewski Physica 1974,75 268. 74 A. J. Dianoux F. Volino and H. Hervet Mol. Phys. 1975 30 1181. ’5 H.Hervet F. Volino A. J. Dianoux and R. E. Lechner Phys. Rev. Letters 1975 34,451. 76 F. Volino A. J. Dianoux R. E. Lechner and H. Hervet J. Phys. (Paris) 1975,36 CI-89. Liquid Crystals 47 Internal Rotations. There have been relatively few investigations of the dynamics of internal motion for molecules within a liquid-crystal mesophase. However the situation might well get better because the improvement in spectrometer design has removed many of the experimental difficulties in measuring nuclear spin relaxation times. In addition preliminary observations suggest that deuterium relaxation times for the alkyl chains of the mesogen are dominated by internal rotations. At present neutron scattering has been employed to study alkyl-chain motions in 4-methoxybenzylidene-4'-n-butylaniline.77The correlation time governing the motion is said to be 3 X s but we have seen that the analysis of neutron-scattering experiments is fraught with difficulties.Ultrasonic absorption would appear to provide an alternative technique.78 Close to the order-disorder transition the attenuation is governed by critical fluctuations but at lower temperatures the absorption by internal modes is important. Thus for the nernatogen Merck Phase V a single absorption at 2 x 1O7 Hz is observed and attributed to trans-gauche isomeri-zation in the alkyl chains. PART11 Continuum Behaviour 4 Continuum Theory for Nematics and Cholesterics Curvature Elasticity.-A nematic specimen is never a perfect single crystal; the axis with respect to which the molecules are preferentially aligned never points in exactly the same direction throughout.Even if it is not deliberately distorted by strains imposed at its surfaces or by the application of electric or magnetic fields it may still be riddled by disclinations; and even if the disclinations are eliminated by careful annealing distortions are bound to arise through thermal agitation alone. Let us describe the local axis of alignment by a unit vector n known as the director bearing in mind that since the molecules in nematics never seem to distinguish up from down the states described by n and -n must be treated as equivalent. Now suppose we choose a Cartesian co-ordinate system (the director frame) such that n lies along the z-axis at the origin.To specify the degree of distortion around the origin we need to specify the local derivatives of n such as dn,/dx and the first objective of the continuum theory of nematics is to generate an expression for the free-energy density in powers of these. Much the same problem arises in the theory of elasticity of solids. There we describe the local strain by specifying the derivatives such as dX/dx of a displacement vector with components (X Y,Z),and any reader who is familiar with the subject should be able to convince himself that if the solid is uniaxial if its symmetry axis coincides at the origin with the z-axis and if the distortion is such that 2 is everywhere zero then the free-energy density is given in terms of the usual elastic constants by ax aY 77 J.A. Janik J. M. Janik K. Otnes and K. RoSciszewski PhySlCQ 1974 77 514. 78 S. Nagai P. Martinoty S. Candau and R. Zana Mol. Crysr. Liquid Cryst.,1975 31 243. T.E.Faber and G.R.Luckhurst The answer for nematics is almost identical if we replace dX/ax by dn,/dx and so on;; the first-order derivatives of n necessarily vanish because A is a unit vector which can vary in direction but not in length and that is why there was no need to include in equation (14)any terms involving 2. It seems that in the nematic case the coefficients of the fourth and fifth terms in the formula equivalent to equation (14) are not necessarily related but this is scarcely relevant because the fifth term can in any case be discarded. We discard it because from which it follows that an integral of the fourth term over any sheet that is everywhere normal to n is completely specified by the boundary conditions at the edge of that sheet.A term in the free energy that depends only on boundary conditions can have no influence on the direction of A somewhere in the interior* The starting point for the continuum theory of nematics is therefore the formula in the director frame or in any frame of reference f =fo+4Kl(div n)2+4Kz(n-curl n)’+$K3(n xcurl n)’ (16) The three Frank stiffness constants which feature in this formula are in practice very much smaller than the elastic constants of a typical solid which is why nematics distort so easily. It is also the reason incidentally why we may safely assume that unless the distortion is very marked indeed (e.g.in the core of a disclination) it has no effect on the degree to which the molecules are aligned i.e.on the local value of F2 and hence on other properties -such as the stiffness constants themselves -which in principle may depend upon the degree of alignment.In the materials so far investigated the stiffness constants lie in the range 10-7-10-6dyn; they decrease on heating and K33K1>K2.t K1 is known as the splay constant K2 as the twist constant and K as the bend constant. Left to itself a nematic specimen minimizes its total free energy by letting its director relax to the least distorted configuration that is consistent with the boundary conditions. In principle however we may prevent the director from relaxing by applying a suitable torque to each molecule from outside.The torque required may readily be calculated from equation (16) by application of the principle of virtual work and hence we may calculate the equal and opposite torque which is exerted on the molecules internally by the distortion of the director. The general expression is too complicated to be worth writing out here. Cholesteric liquid crystals closely resemble nematics but because they are com- posed of optically active molecules they like to adopt a configuration in which the * Nehring and Sau -P have shown that terms in the free energy of nematics depending on second-order derivatives such as d. n,/dyaz introduced by Oseen but lost sight of by most later authors may in practice be discarded for much the same reason.t Some theories suggest that K and K3 should become equal in the limit p2 + 0. Liquid Crystals director twists in a spiral fashion about an axis perpendicular to itself. For choles- terics the appropriate generalization of equation (16) is f=fo+iKl(div n)2+iKz(n'curl n*q0)'+$K3(n Xcurl n)* (17) where 27r/q0 is the pitch of spiral that the cholesteric likes best. The sign to be attached to qo in (17) depends upon whether the spiral is right- or left-handed. Viscosity.-Naturally enough extra stresses and torques develop in a nematic as soon as it begins to flow. Let us treat it as Newtonian and assume the stresses to be related in a linear fashion to the gradients of the flow velocity V=(u u w). If the director is rotating they may also be affected by the speed of its rotation but for the moment we shall suppose n to be fixed despite the flow.Symmetry considerations then allow us to write down equations for the shear stress components which once more in the director frame take the following form* au av fxy = fyx = 773 (G+Z) For an ordinary isotropic fluid the four coefficients introduced above would all be equal to the conventional shear viscosity q. In principle we can measure ql,q2,and q3by using a simple viscometer in which the flow is planar and only one component of the velocity gradient tensor is non-zero. Suppose for example that we fix n (e.g.with the aid of a large magnetic field) so that it lies along the direction of the flow and the aligned molecules are made to slide over one another in a lengthwise fashion then we shall measure ql.If we fix n so that it lies perpendicular to the flow planes and the molecules are made to slide over one another end to end then we shall measure qz.In the third principal configuration with n fixed in the flow planes but perpendicular to V we shall measure q3.It generally turns out in practice (for recent measurements on 4-methoxybenzylidene- 4'-n-butylaniline see ref. 79) that q2> q33 ql,which is perhaps hardly surprising. Incidentally the difference between q1and q3means that if we do the experiment with n in the flow planes but at some angle to V other than 0 or ~/2,then the tangential stress experienced by the viscometer must have a component perpendicu- lar to V.Furthermore if a nematic fluid is forced to undergo Poiseuille flow between flat plates while the director is held at an angle a transverse pressure gradient should develop.The latter effect reminiscent of the Hall effect perhaps has recently been demonstrated by Pieranski and Guyon." 79 1.w.Summerford J. R. Boyd and B. A. Lowry J. Appl. Phys. 1975,46 970. p. Pieranski and E. Guyon Phys. Letters (A),1974,49 237. * The reader conversant with elasticity theory may again find it helpful to explore the analogy of a uniaxial solid to persuade himself that equations (18)-(20) are correct. In elasticity theory the coefficients relating tzxto aX/azand rxz to aZ/ax are necessarily identical as can readily be proved by appealing to the law of conservation of energy.It can be proved by the methods of irreversible thermodynamics that the v4 which occurs in (18) is necessarily identical to the v4 which occurs in (19). T.E. Faberand G.R. Luckhurst The importance of q4 is apparent when we come to consider viscous torques. Given the shear stresses described by equations (18) and (19) it follows that a nematic fluid undergoing shear flow experiences a torque per unit volume the component of which about the y-axis say is given in the director frame by For an ordinary isotropic fluid this evidently vanishes. It need not do sofor a nematic fluid because any amount of torque can be absorbed as it were by whatever external agency is used to fix the director. Let us now decompose the fluid motion about the y-axis into a solid-body rotation with angular velocity my and a pure shear compo-nent ty;this means writing Then we may write Gy = YlOy -Y26y (23) where Y1 = 771 +7?2-2774 Y2= 11-12 (24) At this stage let us admit the possibility that the director is not fixed but is rotating about the y-axis with angular velocity say Ry.The appropriate generalization of equation (23) is clearly which ensures that in the absence of shear and when the whole specimen is rotating as a solid body director and all the viscous torque vanishes. The coefficient q4 therefore plays a role in determining the so-called torque coefficient or twist viscosity yl. This is of particular importance in experiments where for example a nematic sample is allowed to oscillate as a solid body in a magnetic field strong enough to hold n fixeds1782 or alternatively where the nematic is stationary and the magnetic field is changed in direction.It is y1which determines the decay rate of the oscillations in one case and the rate at which n relaxes to its new equilibrium configuration in the other. The twist viscosity has an additional significance for cholerestics because it controls the rate of permeation. Suppose we have a cholerestic specimen in which the axis of the cholerestic spiral points along the y-axis in the director frame and suppose that the fluid is in uniform motion in this direction with velocity V while for some reason the phase of the cholerestic spiral is unable to change. The fact that n is fixed in space means that in each element of fluid it is rotating with angular velocity 0,= Vq,.This implies a viscous torque per unit volume Gy=-yl Vq, and to provide the energy dissipated against this torque there must be a pressure gradient. A simple calculation shows that P. J. Flanders Mol. Cryst. Liquid Cryst. 1974 29 19. 82 S. Meiboom and R. C. Hewitt Phys. Rev. Letters 1975,34 1146. Liquid Crystals The smallness of the permeation coefficient A defined by equation (26)may explain as Helfrich originally pointed out the anomalously high apparent viscosities reported for cholerestics by some of the early experimenters who measured rates of flow through tubes. So far we have four independent viscosity coefficients for a nematic uiz.ql,q2,q3 and q4 or yl.The need for a fifth becomes apparent when we have occasion to consider the normal stress components in the director frame. If we regard the nematic as incompressible and therefore ignore the complications associated with bulk viscosity equations for the normal stresses may be written thus au av aw txx=-p+2q3-; tyy= -p+2q3-;JY tzz=-p+2q5-ax az where p is the pressure that would exist in the absence of velocity gradients. It may be added that many authors prefer to use a set of Leslie coefficients al to as rather than the q’sdefined above. The relationships between the a’s and the q’sare set out by Stephen and Straley.’ 5 Applications of Continuum Theory The ability of the theory which has been summarized above to explain a vast range of intriguing phenomena was amply demonstrated in the years before 1973 and little that is essentially new has emerged since then except in relation to flow instabilities and to the effects of smectic ordering.These matters will be dealt with in Sections 6 and 7 respectively. In the present section a survey of the pre-1973 work on nematics and cholerestics will be conducted and a number of elegant experiments that have been carried out since 1973 by way of extension of this work will be described. The reader should bear in mind that many of these experiments lead in the end to values for the stiffness and viscosity coefficients of the material under investigation. To improve our stock of information about these coefficients is a worthwhile objective by itself for any investigator since the information will no doubt be needed shortly for comparison with microscopic theories.FreederickszTransitions.-While liquid crystals still cost El per gram or more there is some incentive to devise experiments that can be done on small samples and in fact many of the experiments to be described in this section were done on thin films Contained between two glass (or similar) slides. In a thin film the boundary conditions for n are naturally of overwhelming importance. The boundary condi- tions depend upon the treatment of the glass. If it has been carefully rubbed over a sheet of paper always in the same direction then where the liquid crystal makes contact with the glass n is obliged to lie parallel to the surface and in the rubbing direction.The same condition can be achieved more easily and reproducibly by evaporating onto the glass at a slanting angle a thin film of say silicon monoxide. If the glass has been treated with suitable surfactants then n is obliged to lie perpen- dicular to the surface. Thus we can obtain between two glass slides a film of nematic that is almost a single crystal aligned either in a planar (nparallel) or a homeotropic (n perpendicular) sense. T.E. Faber and G.R. Luckhurst Now suppose that we subject such a film to a magnetic or electric* field. Because the susceptibility of a nematic is anisotropic the field adds a term of the form -&ll(n H)’-ixL(n XW2 to the free-energy density in the magnetic case and something similar in the electric case.Now the difference (xrl-xl),denoted by xa,is almost invariably positive for nematics. Hence in the presence of a magnetic field a nematic can lower its free-energy density by an amount $xaHZby allowing n to point along H rather than perpendicular to it and for intermediate orientations the nematic experiences a magnetic torque. This means that if we start with the planar nematic field shown in Figure 2 with n oriented initially along the x-axis and apply a Figure 2 Cross-section through a planar nematic film above the Freedericksz transition induced by the application of a magnetic field along the y-axis. Cylinders are drawn to indicate schematically the local orientation of the director. magnetic field along the y-axis the single-crystal configuration becomes unstable at a certain critical field Hc.Above this critical field it pays for n in the interior of the film to swing round into the y-direction even though this introduces some twist distortion in the neighbourhood of the surfaces. If the twist is confined to two thin layers of thickness 6 small compared with the thickness of the film d then the free energy per unit area should be something like Fo-haH2(d -6) + where Fois the free energy per unit area in the untwisted state. From this expression it is easy to see that above the transition we should expect 6 (K~/x~H’)’ (28) in equilibrium and that ~c (~2/xad~P (29) In a film of thickness 20 pm the critical field might be ca. 10kG. This type of transition is known as a Freedericksz transition.The reader must be left to imagine for himself the variety of different geometries in which similar transitions can be studied using planar or homeotropic films and parallel or perpendicular fields and to consult de Gennes’s book’ concerning the possibilities that exist for cholerestics where a large enough magnetic field is capable of unwinding the spiral. In many of these geometries of course the distortion of n near * A.c. electric fields with frequencies in the kHz range are to be preferred in this context. Low-frequency or d.c. fields may trigger off the electrohydrodynamic instabilities discussed in Section 6. Liquid Crystals 53 the surface of the film involves splay or bend rather than twist and H,therefore depends upon K1 or K3 rather than K2.When an electric field is employed rather than a magnetic one it is of course the anisotropy of the dielectric constant which matters and it should be noted that this quantity (E = ql-E~) is sometimes positive and sometimes negative. Most Freedericksz transitions are easy to detect (e.g.with a polarizing microscope) but the one illustrated in Figure 2 is not readily spotted,83 which is why K2is less often determined than K1 and K3. Recently however methods have been devised involving tilted fields,84 or crossed magnetic and electric fields applied simultane- ~usly,~~ whereby all three stiffness constants can be found. Other recent works6 concerns the dynamics of the transition which is of technical importance in relation to display devices.Under this heading may be included an experiment where the transition is stimulated by application of a continuously rotating magnetic field,87 from which y1can be deduced. Finally something very like a Freedericksz transition can be brought about in planar films of some nematics by twisting one surface with respect to the other in the absence of any field. When the twist exceeds a critical value that may be as high as 6~, n in the interior of the film suddenly tilts into the perpendicular configuration and the twist can then relax. A careful study of this effects8 suggests that very close to the nematic-isotropic transition 2K2+ (K2+K3)in these materials. Disc1inations.-Disclinations are line defects somewhat resembling the edge and screw dislocations that occur in solid lattices.In a nematic film that has recently been subjected to some mechanical disturbance they are normally to be seen in great abundance and they are indeed the threads or nema which give the nematic phase its name. A dislocation is defined of course by its Burgers vector. The equivalent quantity for a disclination is an angle namely the angle by which n would appear to a Maxwell demon to rotate during a journey round the disclination and back to his starting point. The disclination may be classified by an index s which is *i if the angle is &T,f1if the angle is *2~, and so on. In addition we need to distinguish two principal types of disclination depending upon whether the rotation occurs about an axis parallel to the disclination or perpendicular to it.Some authors refer to these loosely perhaps as screw and edge disclinations respectively. Figure 3 may help the reader to visualize some of the disclinations that arise most frequently in practice. It was originally thought that all disclinations had some sort of core in which perhaps the liquid was isotropic since the simple two-dimensional models illustrated in Figure 3 suggest singularities in div n or curl n on the axis. It is now recognized however that round disclinations of integral s n usually tilts into the third dimension and thereby adopts a configuration that is everywhere non-~ingular.~~-~~ The nature of the core in s = * disclinations remains an interesting question about which very little is known.83 R. Dreher Z. Naturforsch.,1974,29a 125. 84 H. J. Deuling M. Gabay E. Guyon and P. Pieranski J. Phys. (Paris),1975,36 689. 85 H. J. Deuling E. Guyon and P. Pieranski Solid State Comm. 1974 15 277. 86 D. W. Berreman Appl. Phys. Letters 1974 25 12. 87 F. Brochard L. Uger and R. B. Meyer J. Phys. (Paris) 1975,36 C1-209. 88 R. Turner and T. E. Faber Phys. Letters (A) 1974,49,423. 89 S. I. Anisimov and I. E. Dzyaloshinskii Souiet Phys. (J.E.T.P.),1973 36 774. 90 R. Turner Phil. Mag. 1974,30 13; 31 719. 91 C. Williams and Y. Bouligand J. Phys. (Paris) 1974,35 589. T.E. Faber and G.R. Luckhurst I I I I I 1 0 0 I Figure 3 Orientation of the director around some common disclinations. In each case the disclination itself is perpendicular to the plane of the diagram.(a) screw s = ++;(b) screw s = -3; (c) edge s =$; (d) screw s =+l;(e) screw s =+l. Continuum theory may be used to make predictions about the line tension of disclinations about the way they should interact with each other and with solid surfaces and about the direction and speed at which they should move when the nematic is sheared or subjected to a magnetic field. A start has been made on the task of verifying these predictions e~perimentally.~~,~~ To classify the line defects that may arise in cholesterics is a more complicated task. Because a cholerestic has a structure with layers in it dislocations can occur as well as disclinations similar to the dislocations in smectics that will be mentioned in Section 7.Their topology has been discussed in detail by B~uligand~~ and Ra~lt.~~ Flow Alignment.-If a nematic specimen containing no disclinations is undergoing steady planar shear flow as shown in Figure 4 in which direction will the director choose to set itself boundary conditions permitting? Presumably in some direction such that the viscous torque vanishes. One such direction is along the y-axis in Figure 4 but there is another possibility suggested by equation (24). Suppose we choose our z-axis to lie at an angle 8 to the flow velocity (see Figure 4). Then clearly ry= -my for 8 =0 and cy= +o for 8 = ~/2.In general ty= -my cos 28 which means that Gy vanishes when cos 28 =-Yl/Y2 = 1 -2h-771)/(772-771) (30) 92 J.A. Guerst A. M. J. Spruijt and C. J. Gerritsma Phys. Letters (A) 1973,43 536. 93 G. Malet J. Marignan and 0.Parodi J. Phys. (Paris),1975,36 L-317. 94 Y. Bouligand J. Phys. (Paris) 1974 35 215 959. 95 J. Rault Phil. Mag. 1974,30,621. Liquid Crystals Figure 4 Stable orientation of the director in a nematic undergoing uniform planar shear flow. Arrows indicate the flow velocity. It turns out on closer examination and has now been demonstrated experimen- tally,96 that it is the second of these possibilities which represents the stable configuration. The experimental results are rather well described by a microscopic theory due to For~ter,~~ which suggests -71/72 = 3p2/(p24-(31) where (Y depends upon the shape of the molecules. For molecules which can be treated as rigid ellipsoids with a length to breadth ratio of a/b (Y = (1+2b2/a2)/(1-b2/a2),so the theory turns out to imply that tan 8 = b/a for perfect alignment (p2= 1) 8+n/4 for poor alignment (F2+0) Thus in most nematics at most temperatures 8 is a modest angle a typical value being in the region of 15”.We shall discover in Section 7 however that in a nematic which is on the point of going smectic qlmay become very large and once it becomes larger than q4there ceases to be any real solution for equation (30).What is thought to happen in that event is that the director adopts a non-uniform configuration in which the angle 8 changes continuously from one shear plane to the next. The viscous torque is then balanced locally by a torque due to the bend and splay in the nematic.This ‘tumbling’ pattern of steady flow (not to be confused with the convective patterns to be discussed in Section 6) may have been observed recently by Cladis and T~rza.~* Ultrasonic Attenuation.-A theory for the attenuation of both longitudinal and transverse sound waves in nematics has been worked out and by comparing it with 96 P. Pieranski and E. Guyon Phys. Rev. (A),1974 9 404. 97 D. Forster Phys. Rev. Letters 1974 32 1161. yu P. E. Cladis and S. Torza Phys. Rev. Letters 1975 35 1283. T.E. Faber and G.R. Luckhurst experimental results it is possible to obtain values for the viscosity coefficient^.^^-^'^ The theory is not straightforward because the shear component of the flow pattern in the sound wave is liable to cause some realignment of n,quite apart from complica- tions associated with thermal conduction and bulk viscosity.When sound is propa- gated along the spiral axis of a cholesteric it may affect the pitch of the spiral in a periodic fashion and when the wavenumber of the sound wave coincides with qo there is a 'Brillouin zone' effectlo' and a discontinuous jump in the attenuation ~oefficient."~ Surface Ripples.-The ripples of short wavelength or Rayleigh waves which are excited thermally on the surface of any liquid and which can be studied by light-scattering techniques are of considerable interest in liquid crystals. A ripple is liable to introduce splay or bend (depending on the boundary conditions at the free surface) and potential energy may be stored in this distortion as well as in the more usual surface-tension term.For nematics the extra energy is likely to be negligible but it could be detectable in cholesterics and ~mectics.'~~~~~~ The most recent experimental work in this field is described in ref. 106. Rayleigh Scattering.-By Rayleigh scattering is meant the quasi-electric scattering of laser light that is such a marked feature of nematics and cholesterics -hence their milky appearance. It is not to be confused with inelastic Brillouin scattering. Both types of scattering are due to fluctuations in the body of the specimen and in both cases the fluctuations can be described theoretically in terms of a set of periodic modes thermally excited and with a mean-square amplitude proportional to temperature as required by the law of equipartition of energy.But the modes responsible for Brillouin scattering are the ordinary Debye modes of the system i.e. sound waves. The modes responsible for Rayleigh scattering are distortion modes in which the fluid remains almost stationary while it is the direction of n which varies in a periodic fashion. Because nematics and cholesterics are strongly birefringent any periodic variation in the direction of n is likely to mean a periodic variation in the refractive index seen by a plane-polarized light beam passing through the specimen and it is this that causes the strong scattering. By measuring the scattered intensity as a function of angle for various geometries one may determine the mean-square amplitude of each mode and by analysing the frequency spectrum of the scattered beam one may determine the frequency spectrum of the modes.One of the principal achievements of continuum theory has been the detailed explanation of data obtained from experiments of this kind (for a recent example see ref. 107). The modes are of course to be labelled by a wave-vector q and for each value of q there are two possible polarizations; the distortion is a mixture of splay and bend in one case and of twist and bend in the other (becoming pure splay or pure twist when 99 J. C. Bacri J. Phys. (Paris) 1974 35,601. loo K. Miyano and J. B. Ketterson Phys. Rev. (A) 1975,12,615. S. E. Munroe G. C. Wetsel M. R. Woodard and B. A. Lowry J. Chem.Phys. 1975,63 5139. lo* J. D.Parsons and C. F. Hayes Solid Stare Comm. 1974,15,429. lo3 I. Muscutariu S. Bhattacharya and J. B. Ketterson Phys. Rev. Letters 1975,35 1584. J. D. Parsons and C. F. Hayes Phys. Rev. (A) 1974 10 2341. lo5 A. Rapini Canad. J. Phys. 1975 53 968. Iw D.H.McQueen and V. K. Singhal J.Phys. (D), 1974,7 1983. lo' H.Fellner W. Franklin and S. Christensen Phys. Rev. (A) 1975,11 1440. Liquid Crystals q is perpendicular to n and pure bend when these two vectors are parallel). The mean-square amplitude predicted by the theory expressed as the mean-square angle through which n is rotated when the mode is thermally excited is where K is an appropriate combination of K1 K2,and K3. Naturally all three stiffness constants can be deduced from the scattering data if they are not already known.The modes are heavily overdamped by viscosity of course the rotational inertia of the molecules being negligible in virtually every context such as this. Their frequency spectra are therefore Lorentzian centred about o=0 and with a width of order Kq2/q (much less than the frequency qu of a Debye mode of the same wavelength). Scattering experiments can therefore yield values for the viscosity coefficients as well. Nuclear Magnetic Relaxation.-The distortion modes discussed briefly in the previ- ous section are also relevant to the theory of the anomalous relaxation times that are observed in n.m.r. experiments on protons in nematics and cholesterics these relaxation times Tl and T2,are generally a good deal shorter than for the same material in its isotropic phase and they become shorter still as the magnetic field in which the experiment is done is reduced.There may of course be a number of relaxation processes at work but the one that is normally blamed for the anomalies just described has to do with the dipole-dipole interaction between protons attached to adjacent carbon atoms in the same molecule. As the molecule librates and the angle between the line joining the two protons and the magnetic field changes the strength of this interaction varies. In the isotropic phase it varies rapidly with a coherence time that may be lO-'Os or less and its frequency spectrum is therefore a rather broad Lorentzian. Out of this frequency spectrum only the components in two narrow bands centred on oLand 2uL(where oLis the resonant Larmor frequency always much less than 10" s-') contribute to the spin relaxation rate l/Tl.The shorter the coherence time the less intensity there is in these bands and the slower the relaxation rate. We have the usual motional narrowing situation where Tl is long and where -because the Lorentzian spectrum has a flat top -it is independent of wLand therefore of the strength of field used. The situation is different for a nematic because the librations of the molecule are constrained. If we think first of the case of perfect alignment (p2= l),the only motions allowed to each molecule are rapid rotations about the local direction of n and the relatively slow co-operative rocking motions which correspond to excitation of distortion modes.This was first pointed out by Pincus who found that the rocking motions should contribute to the relaxation rate a term proportional to [see equation (32) and the remarks that follow it]. If the upper limit to this integral may be taken as infinity the result is proportional to @it. T.E. Faberand G. R. Luckhurst Later authors have tried to extend this theory to the case of imperfect alignment and have obtained a result of the form 1/T1=A+Bp22 (34) where the frequency-independent term A takes care of other relaxation processes. This has been fitted to some sets of experimental data with fair success but there are serious discrepancies. Something of a controversy has therefore blown up between those who argue that a Pincus-type theory modified perhaps by application of a cut-off to the integral in equation (33)at a finite wavenumber qC,lo8 is ~ufficient~~~*~~~ and those who believe that an altogether different frequency-dependent relaxation mechanism involving diffusion of the molecules is at work.'11 To the present author it seems clear that there must be a cut-off qc,and that its magnitude can be established by the same argument that is used to establish the cut-off in the Debye theory one cannot have more distortion modes than there are degrees of freedom in the system. But by specifying the amplitude and phase of every mode up to this cut-off one provides a complete description (within the framework of a continuum model) of the misalignment of the molecules and it is quite inappropriate to suppose that there is enough order left on the microscopic scale to justify the addition of a factor p2*to the Pincus formula.A more elaborate treatment is required. It leads to formulae for l/Tl and 1/T2which include terms proportional to and In oLas well as the conventional term proportional to wLf.It looks as though they may explain the data rather 6 Dynamic Instabilities Thermal Convection.-Readers may be familiar with the Benard convection cells which are liable to develop in an ordinary fluid when it is heated from below with the not-unrelated cells that may develop between two coaxial cylinders rotating at different angular velocities and with the phenomenon of turbulence in shear flow.These can all be classed as dynamic instability effects. Liquid crystals also become unstable in many circumstances and much ingenuity has been devoted to attempts to discover just when and how they do so. Some very curious results have been established. To take a particularly spectacular example it is now known that a nematic layer can be set into convection by heating it from the top. In a layer a few mm thick a temperature difference between top and bottom of only one degree may be sufficient. The effect was first predicted and observed for layers with homeotropic alignment (i.e. with n vertical) of any nematic material whose thermal conductivity is aniso- tropic in a positive sense (ie. KII>Kl). The explanation for it is indicated by Figure 5(a).In this Figure the rods which are used to indicate the local alignment of n show that this has acquired a small tilt in the central region initially as the result of a chance fluctuation.Because the thermal conductivity is anisotropic the heat current in this central region is not entirely vertical; it has a component from left to right. As a J. W. Doane C. E. Tarr and M. A. Nickerson Phys. Rev. Leners 1974,33,620. 109 S. D. Goren C. Korn and S. B. Marks Phys. Reu. Letters 1975 34 1212. 110 W. Wolfel F. Noack and M. Stohrer Z. Naturforsch. 1975,3Oa 437. R. Blinc M. Vilfan and V. Rutar Solid State Comm. 1975 17 171. 112 T. E. Faber to be published. Liquid Crystals result heat is being drained from the regions marked '-' in the figure and is accumulating in those marked '+'.The resultant density difference has started a circulation of the liquid as suggested by a dotted line. The flow associated with this circulation is exerting a clockwise torque on the director and is therefore enhancing the fluctuation with which the process began. Space does not permit further discussion of the thermal convection problem but at least three interesting papers on the subject describing results for cholesterics as well as nematics have been published in the past two years.'13-l15 hot cold Figure 5 Convection in nematic films. (a) Thermal convection induced by heating the top of a homeotropic film; + and -indicate regions which are hot and cold respectively. (b) EHD convection induced by an a.c.voltage across a planar film; + and -now indicate space charge in the fluid during the half cycle when the top of thefilm'is positive with respect to the bottom; the circulation originates from the force exerted on the space charge by the field. (c) Convection induced by uniform shearpow in a homeotropicfilm ; the top surface of the film is moving along the y-axis into the paper. (N.B.In this instance the co-ordinate system used is not the director frame.) 60 T.E. Faber and G.R. Luckhurst Electrohydrodynamic Convection.-Rolling convective cells of much the same appearance as the ones caused by temperature gradients may be set up in planar nematic and cholesteric films by the application of transverse electric fields either d.c.or audiofrequency a.c. for reasons that may be apparent from Figure 5(b). The effect is known as electrohydrodynamic or EHD convection and the cells are known as Williams domains. The a.c. effect is easier to make sense of because under d.c. conditions charge-injection from the electrodes may complicate the situation. With a.c. there is a threshold voltage of the order of 10V independent of film thickness which increases up to some critical frequency of the order of 100Hz. Above this frequency a different type of instability is found where the domains are narrower and the liquid within any one domain chooses to move up and down with the frequency of the field rather than roll continuously in the same sense. The threshold conditions for both types of EHD convection are now well under~tood."~~"~ They depend not only on the anisotropy of the electrical conduc- tivity cr but also on the sign and magnitude of E, and the theoretical predictions have been checked for a range of materials in which crll/crl and E take widely different values."*,' l9 Above the threshold however some curious and beautiful domain patterns have been observed especially with cholesterics,'20,'21 which are still not fully understood.Nor is it known why the Williams domains in a nematic film rock slowly to and fro when the voltage is above threshold or why the narrow domains in the high-frequency regime adopt a herringbone or chevron structure. Well above threshold moreover one sees a gradual transition to a generally turbid state of motion which has received little attention as yet though it is in this turbid state that dynamic-scattering display devices work.Convection Induced by Shear Flow.-Circulation patterns resembling Williams domains are also liable to arise when a nematic is sheared as has recently been demonstrated in some elegant experiments by Pieranski and Guy~n.~~,'~* In one of these experiments a planar nematic film between flat plates treated so as to favour alignment in the x direction is uniformly sheared by moving one plate in the y direction. Were it not for the boundary conditions the shear would make n swing round into the favoured orientation (see Section 5)at a small angle 8 to the y-axis in the yz plane and in fact n does start to swing round in the centre of the film; it is prevented from swinging very far however by the twist that develops in the structure [see Figure 5(c)].Now remember that planar shear in a nematic generates a sideways component in the shear stress if A is aligned more or less in the flow planes but at some angle to the flow velocity other than 0 or ?r/2 (see Section 4). In the present instance there are bound to be shear stresses t, acting in the x-direction. Because C. R. Carrigan and E. Guyon J. Phys. (Paris) 1975,36 L-145. I14 E. Dubois-Violette Solid State Comm. 1974 14 767. I15 J. D. Parsons J. Phys. (Paris) 1975,36 1363. I16 P. A. Penz Phys. Rev. (A),1975,10 1300. I. W. Smith Y. Galerne S. T. Lagerwall E. Dubois-Violette and G.Durand J. Phys. (Paris) 1975,36 C1-237.M. I. Barnik L. M. Blinov M. F. Grebenkin S. A. Pikin and V. G. Chigrinov Phys. Letters (A),1975 51 175. lI9 M. Goscianski and L. Uger J. Phys. (Paris) 1975; 36,C1-231. H. Amould-Netillard and F. Rondelez Mol. Cryst. Liquid Cryst.,1974,26 11. 121 M. de Zwart and Th.W. Lathouwers Phys. Letters (A) 1975,55 41. 122 E. Guyon and P. Pieranski J. Phys. (Paris) 1975,36,C1-203. Liquid Crystals the orientation of n depends on z these tranverse stresses also vary with z which means that a layer of the liquid of thickness Az experiences a finite force per unit area of (dt,,/dz)Az tending to move it sideways. The force acts in the +y direction in one half of the film and in the -y direction in the other half. The result is to set the nematic rolling about the direction of flow as suggested by the dotted circles in Figure 5(c).It is already clear that there are a great variety of flow convection patterns of this sort to be explored. It is also clear that as the shear rate is increased the rolling domains become unsteady as the Williams domains do in EHD convection and that turbulence ensues at Reynolds numbers far below those which mark the onset of turbulence in normal fluids. One may speculate that when a nematic fluid is forced at any substantial speed through a tube or past an obstacle turbulence is inevitable and that disclinations are created in abundance. The next challenge facing theoretical physicists or applied mathematicians in this area is surely to develop a statistical theory to describe the flow properties of a bulk nematic permeated by disclina- ti on^.'^^ The next challenge for the experimentalist is to devise experiments to prove the theory inadequate! 7 Smectics Continuum Theory of the Smectic A Phase.-Our picture of the smectic A phase in its undistorted state is of molecules arranged in regular layers of thickness t say each layer being perpendicular to the director n -which still describes as in nematics the axis with respect to which the molecules are preferentially aligned.It seems intuitively likely that (a) to tilt n away from the layer normal or (b) to change t without tilting n,will cost a lot in free energy and the continuum theory of smectics in its most primitive form is based on the assumption that both these operations are forbidden.Granted this assumption it follows that we may use the line integral t-’jIn *dl to measure the number of layers that separate B from A. Since the answer must be the same whatever route between these two points we choose it follows that the integral of (n -dl) round any closed contour must vanish and hence by Stokes’ theorem that curl n must vanish. Thus [see equation (16)] a’primitive smectic can sustain no twist or bend*. This is equivalent to saying that K2and K3are infinite in the smectic A phase. We are then left with the possibility of splay distortion and a distortion free- energy which depends only on (div n)’. Now imagine an initially flat layer rep- resented by the equation z =0 in the local director frame which is bent about the origin in some way until z = l(x y).Since n coincides everywhere with the layer normal it follows that in the neighbourhood of the origin n = -dl//ax and n = -dc/ay so that div n = -(d2[/dx2+a2[/dy2). Hence we can write the free-energy density of a primitive smectic in terms of the principal radii of curvature of the layers in the region of interest thus f =fo++&(R;+R ;I)’ (35) Left to themselves therefore the layers in a smetic prefer either to lie flat or to curve in opposite directions like a saddle so as to make R1= -R2. * If operation (a)is forbidden but not (b),then twist is forbidden but not bend. 62 T.E. Faber and G.R. Luckhurst The assumption that (a) or (b) are totally forbidden is rather a drastic one and many situations can be envisaged where the director has no option but to tilt a little or the layers to contract or expand given rigid boundary conditions.It can be shown however that in thermal equilibrium the resultant twist or bend is necessarily confined to regions near the surfaces of the sample or to the cores of disclinations and dislocations and that it should not infect the bulk. It is nevertheless desirable to add to equation (35) a term to allow for the finite compressibility of the layers of the form where t is the local layer thickness and to its equilibrium value. This determines for example the velocity of second sound in smectics a type of wave motion detectable by Brillouin scattering experiments in which t varies in a periodic fashion while the density stays almost constant.The theory of second sound in smectics has been discussed by Br0~hard.l~~ It is believed that permeation can occur in smectics i.e.that if a pressure gradient is established across the layers the liquid will flow through them at a steady rate described by equation (26). The permeation coefficient A must be much smaller than in cholesterics however and no experiment has yet been devised to measure it. The mechanism for the process is presumably akin to the mechanism of vacancy diffusion that in principle allows a crystal which is acted on by a pressure gradient to advance at one end and retreat at the other even when there is no motion of the lattice as a whole. Finally what can be said about the viscosity coefficients in the smectic A phase? The layers can slide easily enough over one another so presumably q2is no larger than in the nematic phase.Likewise q3 should be quite small; there is little to prevent the layers from shearing in their own plane though in this respect a smectic B phase could be very different. But the type of shear flow associated with q1can only take place if n tilts away from the layer normal. If K2and K3 are to be regarded as infinite in a smectic then ql and y1should be too. Pre-Smectic Behaviour in Nematics.-It is now known that if a nematic liquid is liable to transform into a smectic on cooling the transition is always heralded by a marked increase in just those coefficients which we expect to be infinite in smectics namely K2 K3 and yl.A dozen papers have been published since 1973,too many to be listed here reporting experiments in which such an increase has been observed.The subject has attracted much interest because it is one to which the Landau theory of phase transitions can be applied. This theory has already been shown to give a good account of pre-transitional effects in isotropic liquids just above an isotropic- nematic transition.. There have in fact been two varieties of Landau theory developed for the nematic-smectic transition. One due primarily to McMillan 124 is of the classical mean-field type. It predicts that K3 and y1should diverge as the temperature of the transition is approached like (T-T*)-$. The other due originally to de Gennes but carried further by Jahnig and Bro~hard,'~~ is based on an analogy with theories **3 F.Brochard Phys. Letters (A) 1974,49 315. 124 W. L. McMillan Phys. Rev. (A) 1974,9 1720. 125 F. Jahnig and F. Brochard J. Phys. (Paris) 1974,35 301. Liquid Crystals 63 established for superconductors and liquid helium. It predicts that K3should diverge like (T-T*)-3and y3like (T- T*)-3. Which of them agrees best with experiment is not yet clear. It is difficult for the experimentalist to determine an exponent unambiguously from his results because there is often a non-divergent but neverthe- less temperature-dependent term in the measured quantity to be allowed for and because he is never quite sure what to choose for T*. An added complication is that the exponents observed experimentally seem to depend very much on purity.'26 One matter for controversy is whether the transition from nematic to smectic A is second-order or weakly first-order if it is second-order then r* should coincide with the transition temperature T, but otherwise r* may be slightly below T,.The material most often studied is 4-cyanobenzylidene-4'-n-octyloxyanilineand for this at any rate the most direct evidence available from thermal measurement~'~' and density measurements,'28 indicates a weak first-order transition. That was a surprise to some until it was pointed that the twist and bend which are present due to thermal fluctuations in the nematic phase above the transition and which must be frozen out when the material turns smectic can play much the same role that a magnetic field does with a superconductor.By the same token L~bensky'~~ argues that the cholesteric-smectic transition should always be first-order because of the intrinsic twist in the cholesteric phase. Experiments on cholesteryl oleyl carbonate under pressure seem to falsify this prediction ho~ever.'~~,'~~ Perhaps the argument fails because the cholesteric spiral has unwound itself before the transition is reached. Certainly the pitch has been observed in two cases to Dislocations and Flow in Smectics.-It is much easier to persuade a smectic specimen to adopt what is virtually a single-crystal configuration than to do the same for a nematic. One has only to hold a bottle full of 4-cyanobenzylidene-4'-n-octyloxyaniline for example in a magnetic field and allow it to cool slowly through the nematic-smectic transition to obtain a sample in which the director is so well aligned that the milkiness which characterizes the nematic phase is almost com- pletely removed.If the bottle is now slightly tilted the sample will be seen to flow -a little reluctantly perhaps but enough to convince most observers that they are looking at a proper liquid. By what processes does the flow occur? There is of course no problem in understanding the type of flow in which the smectic layers slide over one another but suppose they lie at right angles to the direction of shear flow as shown in Figure 6(a),what then? The work of Williams and Klerna~~'~~ has provided most of the answers. To begin with (as the upper surface of the smectic film shown in Figure 6 is moved to the right) the layers may deform elastically as suggested by Figure 6(b);the thickness of each layer is now necessarily 126 P.E. Cladis Phys. Letters (A) 1974,48 179. 12' D. Dyarek J. Baturif-Rub~E and K. FranuloviE Phys. Rev. Letters 1974 33 1126. S. Torza and P. E. Cladis Phys. Rev. Letters 1974 32 1406. 129 T. C. Lubensky J. Phys. (Paris),1975,36 C1-151. I3O P. H. Keyes H. T. Weston W. J. Lin and W. B. Daniels J. Chem. Phys. 1975 63 5006. 131 R. Shashidhar and S. Chandrasekhar J. Phys. (Paris) 1975.36 C1-49. 132 C.-C. Huang R. S. Pindak and J. T. Ho Phys. Letters (A),1974 47 263. 133 R. S. Pindak C.-C. Huang and J. T. Ho Phys. Rev. Letters 1974 32 43. 134 C. E. Williams and M.KICman J. Phys. (Paris),1975,36 C1-315. T.E.Faber and G.R.Luckhurst Figure 6 Layer structures in a smectic specimen; the orientation of n is suggested by double arrows. (a) An undistorted film. (6) and (c) The same film after shear; edge dislocations have appeared in (c). (d) A bulk specimen initially oriented as in (a) which has been stretched horizontally; between the dotted lines there are focal conic walls. less than to in the interior of the film and n may not everywhere be normal to the layers so there is excess free energy stored in the layers. To release this free energy the smectic layers strive to return to toin thickness which is possible if and only if the number of layers in the specimen changes. Somehow by a process that must involve permeation but has not yet been studied adjacent layers must coalesce in the interior to leave the configuration sketched in Figure 6(c).There are now edge dislocations regularly spaced along the upper and lower surfaces of the film and running at right angles to the direction of flow. Williams and KlCman have shown (see also refs. 135-137) that once dislocations have been nucleated it is energeti- cally favourable for them to run together to form the type of linear defect that is illustrated in Figure 7 -equivalent to an edge dislocation of large Burgers vector but strictly speaking a pair of disclinations (s = &) Where the smectic layers are strongly curved in the neighbourhood of these defects div n is large. It therefore pays for the layers to buckle (to make R1=-R2).The defects curve into ellipses and a focal conic structure develops of the sort that has been understood for many years. The layers may now be almost parallel to the shear direction in the interior and further flow can proceed without impediment. 135 M. Kkman J. Phys. (Paris) 1974,35 595. 136 P. S. Pershan J. Appl. Phys. 1974,4S 1590. l3’ P. S. Pershan and J. Prost J. Appl. Phys. 1975 46 2343. Liquid Crystals Figure 7 The layer structure around two disclinations of opposite sign corresponding to an edge dislocation of Burgers vector 8t0. Related things probably happen when a smectic single crystal is stretched along its director a type of strain that can of course be regarded as a combination of two shear strains at 45" to the director.Very small extensions suffice to distort the layers in a periodic fashion and one can imagine this distortion developing into the structure sketched in Figure 6(d),with focal conic walls separating each domain. What happens when a smectic is squeezed along its director is a different question. Presumably the layers must get fewer in number as they grow in area and nucleation of edge dislocation loops must be required. The process has been discussed in a paper by members of the Orsay groups.138 This paper incidentally contains some intriguing speculations about flow phenomena dominated by permeation which might be observable in smectic systems where nucleation of dislocations is very difficult. The speculations have yet to be confirmed experimentally.13* Orsay Group on Liquid Crystals J. Phys. (Paris) 1975,36 C1-305. 139 J. Nehring and A. Saupe J. Chem. Phys. 1971,54,337. 140 B.I. Halperin and T. C. Lubensky Solid State Comrn. 1974,14 997. 141 P.G.de Gennes J. Phys. (Paris),1974,35 L-217.