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J. Phys. Chem. B 2005, 109, 19234-19241
Effect of the Elastic Constant Anisotropy on Disclination Interaction in the Nematic Polymers Wenhui Song,*,†,‡ Houjie Tu,† Gerhard Goldbeck-Wood,† and Alan H. Windle† Department of Materials Science and Metallurgy, Cambridge UniVersity, Pembroke Street, Cambridge CB2 3QZ, U.K., and Wolfson Centre for Materials Processing, School of Engineering and Design, Brunel UniVersity, Uxbridge, Middlesex UB8 3PH, U.K. ReceiVed: June 1, 2005; In Final Form: July 14, 2005
In this work, disclination interaction behavior in relation to Frank elastic constant anisotropy in nematics has been studied. A large number of (+1/2, -1/2) disclination pairs are revealed by spontaneous band texture in a semiflexible copolyester. The pairs show no preferential relative orientation, with the intervening fields showing intermediate patterns. A two-dimensional tensor lattice model considering unequal elastic constants is applied to simulate the interaction behavior and patterns of disclination pairs in the presence of elastic anisotropy. Scaling laws for disclination density F(t) as a function of time step t with different elastic anisotropy are obtained as t-ν. The value of the exponent ν decreases as elastic anisotropy is increased. Obviously, elastic anisotropy slows the texture coarsening. The simulations also show that angular forces arise in the presence of elastic anisotropy and change the patterns of pairs during the texture coarsening. When disclination density is considerably decreased, some +1/2 disclinations start to rotate to the energetically favored patterns depending on the sign of the elastic anisotropy. As a result of the disclination rotation, the distribution of patterns of pairs continues to change during the annihilation. However, disclination pairs are influenced not only by elastic anisotropy but also by disclination interaction during the whole annihilation. Therefore, in a real system, the dependence of pairs on elastic anisotropy is not as strong as the theoretical prediction for an isolated pair, and the full pattern range of disclination pairs can be observed.
I. Introduction The microstructure and interaction of disclinations in the nematic state are fundamental to the properties and processing of liquid crystalline materials. The texture evolution in the nematic state of both small-molecular weight liquid crystals (SMWLCs) and liquid crystalline polymers (LCPs) is a process during which disclinations attract each other and then annihilate. This leads to a decrease in the number of disclinations and a coarsening of the texture. The configurational changes that disclinations induce on themselves and on their neighboring disclinations affect their evolution in a highly nonlinear way. Although the investigation of disclinations in liquid crystalline states has intermittently been pursued for more than a century, the behavior of disclination interaction is still beyond the current experimental techniques, and mainly studied in theory. Most of the results are about the interaction of an isolated defect pair on the assumption of equal elastic constants. The kinetics of texture coarsening in liquid crystals has been extensively investigated theoretically and numerically. The general results for a two-dimensional (2D) system showed that, in the diffusive regime, at high temperatures where Brownian motion dominates, the defect density F(t) decays according to the power law F(t) ∝ t-ν, where exponent ν ) 0.5.1,2 In the nondiffusive regime, the exponent can be from 0.75 to 1, depending on the theory or model that is adopted.1-4 In the experimental measurement, ν is usually around or less than 1. * To whom correspondence should be addressed. E-mail: wenhui.song@ brunel.ac.uk. † Cambridge University. ‡ Brunel University.
A scaling law for which ν ) 0.88 for MBBA was measured.5,6 Mason et al.7 measured a thin film of a twisted nematic liquid crystal and found ν ≈ 1. The dynamic behavior of defects in a long-chain nematic system with high viscosity and unequal elastic constants could be significantly different from that in SMWLCs. Generally, the very fine texture in the highly viscous polymeric melt makes experimental observation difficult. As a consequence, not much work has been done in this area. Only limited experiments8-10 and a computer simulation11 show that the disclination density obeys the power law F(t) ∼ t-0.70(0.02. A power law of the Schlieren texture coarsening was observed; F(t) ∝ t-0.7 for two thermotropic polyesters using optical microscopy.8,10 A similar result for which ν ) 0.74 for a semiflexible LCP was obtained by small angle light scattering (SALS).9 Rojstaczer and Stein found that the coarsening process was faster in a rigid polymer than in a semiflexible polymer and assumed that the time scale was proportional to γeff/k, where γeff is an effective viscosity coefficient and k is the elastic constant. These kinetic studies mentioned above indicate little difference between SMWLCs and LCPs as expected. On the other hand, LCPs are assumed to have strong elastic anisotropy; however, little is known about the defect annihilation in the presence of elastic anisotropy. Liu and Muthukumar4 first simulated the effect of elastic anisotropy on the kinetics of defect annihilation using a Monte Carlo method. Their simulation result suggests that the value of ν is independent of elastic anisotropy. This is a controversial result. In this work, a dependence of elastic anisotropy on defect annihilation has been found. Dafermos12 and Nehring and Saupe13 first considered the properties of disclination interaction between an ideal discli-
10.1021/jp052919g CCC: $30.25 © 2005 American Chemical Society Published on Web 09/22/2005
Disclination Interaction in the Nematic Polymers
J. Phys. Chem. B, Vol. 109, No. 41, 2005 19235 distortions almost unchanged. The configurations of disclinations in the real system are a result of competition between the disclination interaction and elastic anisotropy. In this work, the experimental observations of a large number of disclination pairs with a range of intermediate patterns bring up the issue of the behavior of disclination pairs and the relationship between disclination interaction and elastic anisotropy during the texture evolution. Numerical simulations18 have been applied again to interpret the process of disclination annihilation. The patterns of disclination pairs and their interaction behavior have been traced in different cases of elastic anisotropy. II. Classification of Disclination Pairs It would be necessary to classify the categories of disclination pairs studied here. Under the condition of equal elastic constants, an isolated (+1/2, -1/2) disclination pair, s1 at (x1, y1) and s2 at (x2, y2), can be described by employing the superposition principle:13
φ ) s1 tan-1
Figure 1. (a-e) Five typical director patterns for (+1/2, -1/2) disclination pairs in the nematic state created by applying the superposition principle. (f) Angle R is used to define the relative orientation and position of the neighboring disclination. All possible patterns of disclination pairs are in a range of 0° e R e 180°, which can be viewed as two categories. In quadrant I (0° e R < 90°), splay distortions are dominant in the central region of the pairs. In quadrant II (90° < R e 180°), bend distortions become dominant between the two disclinations. When R ) 90°, splay and bend distortions are equal. (g) Total, splay, and bend energies of disclination pairs plotted as a function of the relative orientation, R. The analysis is performed on a disclination pair with a separation d of 25 on a 100 × 100 lattice.
nation pair and proposed that like disclinations repel and unlike disclinations attract one another.12,13 The forces between the disclinations are additive and inversely proportional to distance. On the basis of continuum mechanics, Ranganath14 first suggested that the different patterns of (+1/2, -1/2) pairs are associated with elastic constant anisotropy. He suggested that an angular force of interaction between defects would arise in the presence of elastic anisotropy. The angular force would drive all disclination pairs to a certain configuration, either mainly bend or mainly splay in the central region of the pairs. According to his analysis, one of only two patterns of a disclination pair (Figure 1a,e) is energetically favored depending on the sign of the elastic anisotropy. The pairs with intermediate patterns (Figure 1b-d) are forbidden due to the effect of the angular force. One experimental observation showed that a disclination pair tended to rotate to a certain pattern during annihilation.15 This disclination rotation was suggested to be attributed to the strong elastic anisotropy of the lyotropic polymer. The preceding papers16,17 have investigated the effect of disclination interaction on the structure of +1/2 and -1/2 disclinations themselves. The results reveal that the splay and bend distortions of +1/2 disclinations vary with the relative orientation of disclination pairs, as shown in Figure 1g. Meanwhile, a -1/2 disclination changes its symmetry to adapt the interaction with its neighbors keeping the splay and bend
( )
( )
y - y1 y - y2 + s2 tan-1 + c0 x - x1 x - x2
(1)
φ is the angle of the director orientation with respect to the x axis, and c0 determines the relative orientation between the two disclinations. In the preceding paper,17 a full range of patterns of disclination pairs were drawn under the condition of equal elastic constants, as shown in Figure 1. The free energy analysis of all disclination pairs provides a clue about reclassifying their structures. Figure 1 is a schematic diagram showing the relative orientation of two disclinations (+1/2, -1/2) in a lattice and the corresponding total elastic free energy and energy attributable to the splay and bend distortions of the pairs as a function of R (R ) 180° + 2c0). As shown in Figure 1g, the probability of any type of a disclination pair is equal in the case of equal constants since the total elastic free energy of all types of pairs is fairly equal. On the other hand, the splay energy of the pairs decreases and the bend energy increases as R is changed from 0 to 180°. The difference between the splay and bend distortions reaches a maximum when R ) 0 or 180° (Figure 1a,e), which refers to the two extreme patterns of the pairs proposed by Ranganath.14 The pairs between these two extreme patterns are termed intermediate pairs. According to the variation of the splay and bend distortions with R, the patterns of disclination pairs can be divided into two categories. In the first quadrant, i.e., 0° e R < 90°, the splay distortion is predominant and continuously decreases when R is increased, while in the second quadrant, i.e., 90° < R e 180°, the bend distortion is predominant and continuously increases when R is increased. When R ) 90°, the splay and the bend distortions are equal. Thus, ideally, disclination pairs in the first quadrant are energetically favored when elastic anisotropy ) k11 - k33/k11 + k33 < 0, i.e., k11 < k33. If > 0, i.e., k11 > k33, the energetically favored pairs should turn to the second quadrant. III. Experimental Observations A spontaneous band texture provides rich information about disclination structure since the molecular director is perpendicular to the long axis of the bands.16,19-22 The orientation of the bands implies that the trajectories of the director field are continuous except at the disclination cores. In other words, there is no real boundary between domains in the polydomain structure. Each disclination is actually associated with several
19236 J. Phys. Chem. B, Vol. 109, No. 41, 2005
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Figure 2. Director patterns for (+1/2, -1/2) pairs of disclinations in nematic polyester Cl-6, decorated by band texture, and corresponding molecular director trajectories. There appears to be no preferential relative orientation of the pairs, with the intervening fields showing mainly splay distortions (a), mainly bend (b), and intermediate (c).
Figure 3. Near-annihilating pairs of (+1/2, -1/2) disclinations in nematic polymer Cl-6, decorated by band texture, and corresponding molecular director trajectories. In these cases, there is no preferential distortion dominating the central regions of the disclination pairs.
disclinations, forming a disclination network. Generally, a study of disclination pairs with opposite signs is considered the simplest way to deal with this complex network. Figure 2 shows three typical types of (+1/2, -1/2) disclination pairs revealed by bands in the nematics of a semirigid copolyester Cl-6. The formation of spontaneous band texture and the molecular structure of copolyester Cl-6 were reported previously.16,19,20 The corresponding molecular director trajectories normal to the long axis of the bands are obtained by the image analysis method described in ref 16. The pairs in panels a and b of Figure 2 belong to two extreme configurations. The central region between the two disclinations in Figure 2a is mainly splay, corresponding to the pattern where R ) 0° (Figure 1a), and mainly bend in Figure 2b, corresponding to the pattern where R ) 180° (Figure 1e). Again, they are the same as the patterns of (+1/2, -1/2) pairs proposed by Ranganath.14 Apart from these two types of disclination pairs, the pairs also show intermediate orientations within the intervening field of two disclinations, such as the pairs in Figure 2c, corresponding to one of the patterns within 0° < R < 180°. However, it is difficult to separate further the intermediate pairs into quadrants I and II because of many disclinations in the neighborhood. Each disclination is surrounded by a host of other disclinations, some with positive and some with negative signs. In many cases, a disclination in a pair which shows more splay distortions is often involved in another pair which shows more bend distortions. As a result of a survey, during which 342 pairs of (+1/2, -1/2) disclinations were observed, there are 238 pairs in the intermediate pattern (like in Figure 2c). Therefore, around 70% of the (+1/2, -1/2) disclination pairs appear to be intermediate in this polymer specimen. When disclination pairs approach each other, at a distance nearly close enough to annihilate, i.e., 0, the drag effect of elastic anisotropy is more significant than that in the case when < 0. The defects seem to be relatively stable in the nematic system with a high splay elastic constant. This may interpret the reason for high disclination density and slow texture coarsening in most polymeric nematics since LCPs, at least those without flexible spacers, are proposed to have a high splay elastic constant. It is noticed that this result disagrees with that obtained by using the Monte Carlo method,4 in which elastic anisotropy is found to exert no effect on the value of ν. B. Effect of Elastic Anisotropy on Disclination Pairs. An advantage of simulation is that the patterns of disclination pairs can be identified and traced during texture evolution. The simulations carried out in this work show that disclination pairs appear in the full pattern range of 0° e R e 180° during the whole simulated evolution in both cases of equal and unequal constants. Elastic anisotropy does have an impact on the pattern of disclination pairs, but the impact is not as drastic as the theoretical prediction for an isolated pair.14 The statistical data shown in Figure 5 demonstrate that the disclination pairs that are present in the larger proportion always show the pattern range which is energetically favored in the
19238 J. Phys. Chem. B, Vol. 109, No. 41, 2005 presence of elastic anisotropy, either 0° e R < 90° or 90° < R e 180°. Furthermore, the proportions of the disclination pairs in different ranges vary with both elastic anisotropy and disclination density. Panels a-c of Figure 5 show the variation of the proportions of the disclination pairs in quadrants I and II and around R ) 90° as a function of elastic anisotropy at the initial, middle, and final stages of annihilation, respectively. In the beginning of the annihilation (at 50 time steps in Figure 5a), the number of pairs in quadrant I is slightly larger than that in quadrant II for < 0, and vice versa for > 0. As the annihilation continues, this tendency has been enhanced as shown in panels b and c of Figure 5. Panels b and c of Figure 5 show the distribution of the pair patterns when ∼40 and ∼60% of disclinations, respectively, annihilate with respect to the initial state at 50 time steps. Clearly, the preferred disclination pairs in the presence of elastic anisotropy gradually become dominant as the disclination density decreases. This result further suggests that the distribution of apparent elastic anisotropy becomes narrow and shifts up or down under the influence of elastic anisotropy, which was presented in ref 17. Nevertheless, the full range of patterns of disclination pairs still exists at the final stage of annihilation, despite strong elastic anisotropy. Besides, if the elastic constants are equal, the proportions of disclination pairs in quadrants I and II remain fairly equal during the whole evolution, as shown in Figure 5, which is in agreement with the result of free elastic energy analysis in ref 17. It would appear that as annihilation proceeds the distribution of pairs goes from favoring R ) 90° pairs to being in quadrants I and II, indicating that the favorable R ) 90° pairs annihilate more quickly. C. Rotation of Disclination. By analyzing an isolated disclination pair, Ranganath14 found that elastic anisotropy could induce an angular force, which would drive disclination pairs to rotate to an energetically favored pattern. In reality, isolated pairs are rare; each disclination is surrounded by many other disclinations. The annihilation is a complicated process with simultaneous movements of most defects. A close examination of disclination annihilation in the case of unequal constants indicates that the angular force emerges in the presence of elastic anisotropy, but there is competition between disclination interaction and the effect of elastic anisotropy during the evolution. At the first stage of annihilation, since disclinations are rather dense, the effect of disclination interaction overshadows the effect of elastic anisotropy. Disclination orientation is basically locked, showing mainly intermediate patterns. Therefore, the simulations show that the pattern of disclination pairs are in the full range of 0° e R e 180°, as shown in Figure 5a. As the disclination density decreases, the effect of elastic anisotropy becomes visible. Some disclinations are observed to rotate to the favored patterns in the presence of elastic anisotropy as given by Ranganath’s theoretical prediction.14 This proves that the angular force does exist. Figures 6 and 7 show series of close-ups of disclination rotation and corresponding total elastic free energy, the energies attributable to the splay and bend components and corresponding to the splay and bend distortions in the local areas. The rotation of disclinations results in the proportion of disclination pairs in either quadrant I or II increasing as the density of the disclination decreases, which may be associated with the distribution variation of pair patterns during the evolution shown in Figure 5. When < 0, for example, ) -0.3 as shown in Figure 6, disclination pairs in quadrant I are energetically favored. At 200 time steps, six neighboring disclinations annihilate, with a pair
Song et al.
Figure 6. (a) Snapshot of the director field quenched at 100 time steps from a simulation run on a 100 × 100 lattice with periodic boundary conditions and elastic anisotropy ) -0.3. Shown is a series of closeups showing the rotation, attraction, and annihilation of the disclination pair highlighted in the snapshot. Small filled boxes represent the disclination cores. (b) Total, splay, and bend energies of the local area in the series of close-ups in panel a vary with time step. (c) Proportions of the splay and bend distortions of this local area vary with time step.
Disclination Interaction in the Nematic Polymers
Figure 7. (a) Snapshot of the director field quenched at 450 time steps from a simulation run on a 100 × 100 lattice with periodic boundary conditions and elastic anisotropy ) 0.3. Shown is a series of closeups showing the rotation and lock state of the disclination pair highlighted in the snapshot. Small filled boxes represent the disclination cores. (b) Total, splay, and bend energies of the local area in the series of close-ups in panel a vary with time step. (c) Proportions of the splay and bend distortions of this local area vary with time step.
J. Phys. Chem. B, Vol. 109, No. 41, 2005 19239 (R ≈ 46°) left in this local area. At this moment, the +1/2 disclination starts to rotate clockwise around its own core axis to the pattern with R ) 0 showing predominantly splay distortions. The rotation of +1/2 disclination has been reported during the annealing of a lyotropic polymer.15 The energy profile of this local area in Figure 6b shows that the total elastic free energy and the energies attributable to the splay and bend components gradually decrease during the rotating and attracting processes, and drop dramatically after the pair annihilates. It seems that both the rotation and attraction of disclination may contribute to the total energy decay. The differences between the proportions of the splay and the bend distortions reach a maximum at the end of rotation at around 350-400 time steps, as shown in Figure 6c. As expected, the rotation of the +1/2 disclination results in an increase of the splay distortions and a decrease of the bend distortions when ) -0.3. Similar rotation behavior of some +1/2 disclination happens when > 0. The rotation angle is found to depend on the local environment. As a result, the pairs show a preferred range of patterns instead of a specific pattern. Figure 7 shows a disclination rotation when ) 0.3. In the local area, no other defects are in the vicinity of this pair after 600 time steps. The series of close-ups shows that rotation happens with very little attraction involved. The +1/2 disclination stops rotating at an intermediate pattern when R ≈ 110°, and the pair is locked after rotation. During the rotation, the total energy of this local area does not obviously decrease, as shown in Figure 7b. Therefore, the rotation of disclination mainly rearranges the proportions of splay and bend distortions, as shown in Figure 7c. The energy decay during the interaction is mainly contributed by the disclination attraction. Not all disclination pairs sensitively depend on elastic anisotropy. For some pairs, the rotation does not occur in the final stage of annihilation, even though the patterns are energetically unfavored in the presence of elastic anisotropy. For example, Figure 8 shows the annihilation of two pairs when ) -0.3. Pair A belongs to quadrant II, which is energetically unfavored, pair B belongs to quadrant I, which is favored. During pair A annihilation, the +1/2 disclination does not rotate to the favored orientation; on the contrary, it rotates slightly to the orientation showing more bend distortion at 400 time steps, as shown in Figure 8. This causes the bend distortion to be slightly increased during annihilation even though ) -0.3, as seen in Figure 8c. Thus, angular forces must be balanced out for the +1/2 disclination in pair A by other neighbors. As expected, the energy gradually decays as the disclinations in each pair approach each other, and eventually falls off when two pairs collapse in Figure 8a. Disclination attraction often overshadows the angular force. As a result of competition between interaction forces and angular forces, disclination pairs annihilate in different ways, depending on the local environment. Some pairs attract and annihilate directly due to strong interaction, especially in the areas of a high disclination density, regardless of the relative orientation of pairs. Meanwhile, some pairs attract and rotate simultaneously, and annihilate finally in a favorable orientation. Occasionally, in the areas of a low disclination density, some pairs rotate into a favorable orientation before they can attract. It has been demonstrated that elastic anisotropy slows disclination annihilation: the stronger the elastic anisotropy, the slower the annihilation. The angular force induced by elastic anisotropy may account for the drag effect of elastic anisotropy. The angular forces drive some disclinations to rotate to energetically favored orientations during or before attraction.
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Song et al. The rotation of disclinations slows the annihilation process and changes the distribution of patterns of disclination pairs. In summary, the numerical studies have produced a clear picture of the effect of elastic anisotropy on disclination interactions, which are thought to be important in understanding the evolution of the microstructure of LCPs. Unfortunately, the simulation results cannot compare satisfactorily with the experimental observations due to the unknown elastic anisotropy of the material and the experimental difficulty of observing disclination interaction during the texture evolution in polymeric melts. However, there is no doubt that a number of disclination pairs of intermediate patterns can be observed in a real system because of the competition between disclination interaction and elastic anisotropy. V. Conclusions Experimental observations show that intermediate patterns of (+1/2, -1/2) disclination pairs are a characteristic feature in a semirigid LCP. A coupling rule of disclination interaction is suggested. During the interaction, the relative orientation of the two disclinations depends on their local environment, with the directors along the central line of the two disclinations tending to be parallel to each other. The simulations show scaling laws for disclination density F in terms of time step t as F(t) ∝ t-ν with different elastic anisotropy. The power-law exponent ν decreases as the elastic anisotropy is increased. The effect of elastic anisotropy is to slow the disclination annihilation process. The drag effect of elastic anisotropy is more dramatic when > 0. This may correspond to the tight texture and slow texture coarsening usually observed in polymeric nematics. The simulation results also show the distribution of a full pattern range of (+1/2, -1/2) disclination pairs with both equal and unequal constants. In the case of equal elastic constants, the distribution of patterns of disclination pairs appears to be random. Most disclination pairs keep their intermediate configurations until they annihilate. In the presence of elastic anisotropy in the system, there is competition between elastic anisotropy and disclination interaction. Therefore, the dependence of the patterns of disclination pairs on the sign of elastic anisotropy is not as strong as the theoretical prediction for an isolated pair. The larger proportion of the pairs showing either more splay or bend distortions depends on the sign of the elastic anisotropy, and will increase as the disclination density is decreased. The simulation results prove the existence of angular force induced by elastic anisotropy, which drives some disclinations to rotate to an energetically favored orientation. The rotation of disclinations often occurs in an area with low disclination density, resulting in the slow disclination annihilation and changes in the local distribution of the splay and bend distortions. Nevertheless, the total elastic free energy decay during the whole annihilation seems to be mainly contributed by disclination attraction. References and Notes
Figure 8. (a) Snapshot of disclinations quenched at 150 time steps from a simulation run on a 100 × 100 lattice with periodic boundary conditions and elastic anisotropy ) -0.3. The series of close-ups shows attraction and annihilation of two disclination pairs highlighted in the snapshot. Small filled boxes represent the disclination cores. (b) Total, splay, and bend energies of the local area in the series of closeups in panel a vary with time step. (c) Proportions of the splay and bend distortions of this local area vary with time step.
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Disclination Interaction in the Nematic Polymers (8) Shiwaku, T.; Nakai, A.; Hasegawa, H.; Hashimoto, T. Polym. Commun. 1987, 28, 174. (9) Rojstaczer, S.; Hsiano, B. S.; Stein, R. S. Polym. Prepr. 1988, 29, 486. (10) Shiwaku, T.; Nakai, A.; Wang, W.; Hasegawa, H.; Hashimoto, T. Liq. Cryst. 1996, 19, 679. (11) Hudson, S. E.; Thomas, E. L. Chemtracts: Macromol. Chem. 1991, 2, 73. (12) Dafermos, C. M. Q. J. Mech. Appl. Math. 1970, 23, s49. (13) Nehring, J.; Saupe, A. J. Chem. Soc., Faraday Trans. 2 1972, 68, 1. (14) Ranganath, G. S. Mol. Cryst. Liq. Cryst. 1983, 97, 77. (15) Wang, W. Liq. Cryst. 1995, 19 (2), 251. (16) Song, W.; Fan, X.; Windle, A. H.; Chen, S.; Qian, R. Liq. Cryst. 2003, 30, 765. (17) Song, W.; Tu, H.; Goldbeck-Wood, G.; Windle, A. H. Liq. Cryst. 2003, 30, 775.
J. Phys. Chem. B, Vol. 109, No. 41, 2005 19241 (18) Tu, H.; Goldbeck-Wood, G.; Windle, A. H. Phys. ReV. E 2001, 64 (1), 1704. (19) Chen, S.; Du, C.; Jin, Y.; Qian, R.; Zhou, Q. Mol. Cryst. Liq. Cryst. 1990, 188, 197. (20) Chen, S.; Song, W.; Jin, Y.; Qian, R. Liq. Cryst. 1993, 15, 247. (21) Hoff, M.; Keller, A.; Odell, J. A.; Perces, V. Polymer 1993, 34, 1800. (22) Song, W.; Chen, S.; Jin, Y.; Qian, R. Liq. Cryst. 1995, 19, 549. (23) Zheng-Min, S.; Kle´man, M. Mol. Cryst. Liq. Cryst. 1984, 111, 321. (24) Se, K.; Berry, G. C. Mol. Cryst. Liq. Cryst. 1987, 153, 133. (25) De’Neve, T.; Kle´man, M. Liq. Cryst. 1995, 18, 67. (26) Zapotocky, M.; Goldbart, P. M.; Goldenfeld, N. Phys. ReV. E 1995, 51 (2), 1216. (27) Hobdell, J. R.; Windle, A. H. Liq. Cryst. 1997, 23 (2), 157.
