Phase separation kinetics in mixtures of flexible polymers and low molecular weight liquid crystals

Transcription

1 Macromol. Theory Simul. 8, (1999) 409 Phase separation kinetics in mixtures of flexible polymers and low molecular weight liquid crystals Junxiang Zhu 1, Guoqiang Xu 1, Jiandong Ding* 1, Yuliang Yang 1, 2 1 Department of Macromolecular Science, Laboratory of Molecular Engineering of Polymers, Fudan University, Shanghai , China 2 Institute of Chemistry, Chinese Academy of Sciences, Beijing , China (Received: December 29, 1998; revised: March 10, 1999) SUMMARY: A dynamic Monte Carlo (MC) simulation is performed to investigate the phase behavior of mixtures of flexible polymers and low molecular weight thermotropic liquid crystals (LCs). The polymer is represented by three-dimensional self-avoiding lattice chains, while the LC is described by the Lebwohl- Lasher nematogen model. The initially homogeneous rod-coil mixture is, following a deep quench, separated into an isotropic phase rich in coils and a nematic phase rich in rods. The underlying spinodal decomposition (SD) process is then simulated and studied extensively. This is the first simulation of SD in a rod-coil mixture where the nematic ordering is included. Concentration fluctuations with a conserved order parameter are thus coupled with orientation fluctuations with a nonconserved order parameter. It is found that the early stage SD in the rod-coil mixture still exhibits the dominant spatial wavelength and that the scalar scattering functions in the late stage of SD obey the Furukawa scaling law. The kinetic difference between the so-called isotropic and anisotropic SD regions is, however, much less pronounced than predicted recently by the mean-field theory. Introduction Phase separation kinetics in various polymeric mixtures is a classic academic subject and meaningful for fabricating inhomogeneous composite materials 1). During the last decades, extensive investigations have been made on the morphology and pattern formation kinetics in immiscible flexible polymer blends and block copolymers, or other isotropic fluids 1 14). The Cahn-Hilliard linearized theory 6 8) successfully describes the early stage of the spinodal decomposition (SD), while the power-law growth 9 13) and the well-known Furukawa scaling relation for the scattering function 14) are found in the late stage of SD. In contrast, little work has been done on the kinetics of phase separation in the mixtures of flexible polymers and liquid crystals (LCs) 15). Recently, polymer dispersed liquid crystal (PDLC) materials have emerged as an interesting topic due to their prospective application in large-area electro-optical displays, etc ) In PDLC films, low molecular LCs are dispersed, usually in the form of nematic droplets, in an isotropic polymeric matrix during the process of phase separation. The phase separation in the PDLC formation is, in most of the cases, induced by polymerization, and sometimes by thermal quench and other methods. The study of the thermally induced phase separation in polymer-lc mixtures can form the basis to reveal the mechanism in the polymerization-induced phase separation and is also meaningful in its own right. Although the phase behaviors of rod-coil mixtures have been dealt with by several researchers 20 26), the phase diagram for the mixtures of flexible polymers and thermotropic low molecular weight LCs was proposed by Yang et al ) in the formalism of Flory-Huggins lattice theory 30) for isotropic mixing and Maier-Saupe theory 31) for nematic ordering. The theoretical results have partly been confirmed by corresponding experimental observations 29, 32, 33). One of the key arguments is that there are two spinodal curves in the phase diagram of a PDLC 34, 35), following an important finding by Dorgan 36) in the analysis of SD of an LC solution composed of liquid crystalline polymers and isotropic low molecular weight solvents. However, the SD kinetics in the PDLC formation composed of isotropic flexible polymers and anisotropic low molecular weight rods has not been explicitly known. During the phase separation process in a rod-coil mixture, the concentration fluctuation associated with a conserved order parameter must be coupled with the orientation fluctuation associated with a nonconserved order parameter, which is essential for a mixture containing mesogenic units, but has not been paid much attention to until recently 37 40) , 34 40) All of the preceding theoretical treatments were performed applying the mean-field formalism. As emphasized by Binder et al. 3, 4), the phase behavior in polymer blends might deviate from the predictions based on the mean-field theory to a large extent. Since there is no corresponding analytical theory free of mean-field approximations at the present time, computer simulation 41 43) becomes an alternative method besides theory and experiment. Unfortunately, as the first step of their Macromol. Theory Simul. 8, No. 5 i WILEY-VCH Verlag GmbH, D Weinheim /99/ $ /0

2 410 J. Zhu, G. Xu, J. Ding, Y. Yang work on this complicated system, the anisotropic characteristics of an LC, let alone the coupling between the concentration and orientation fluctuations, has not been taken into consideration by Lin et al ) at all, although PDLC is their central topic. A similar case exists in some experimental observations and theoretical treatments 15, 44, 45). In the present paper, a three-dimensional dynamic MC simulation 46) is performed to study the phase separation kinetics in the mixtures of flexible polymers and thermotropic low molecular weight LCs considering the nematic characteristics among rods for the first time. The anisotropic interaction between rods is reflected by the Lebwohl-Lasher (L-L) nematogen model 47). To our knowledge, although the phase separation behaviors in twodimensional 48, 49) and three-dimensional isotropic polymer blends 50) or polymer concentrated solutions 51) have been studied by the dynamic MC simulation, this method has never been extended to study the SD kinetics including both nematogens and polymeric coils. The present dynamic MC simulation is a powerful method to investigate the SD in rod-coil mixtures with the following advantages: first, it is free of mean-field approximations and based upon a molecular model; second, the nematic character of LC is embodied straightforwardly; third, some key physical quantities such as the coupling factors between the concentration and orientation fluctuations are inherently and reasonably included in the present simulation rather than set as a starting point as done in the other theory and simulation formalisms 37 40, 42 45). Model and simulation algorithm Lattice chain and Lebwohl-Lasher nematogen model The phase behavior of the polymer-lc mixture is simulated on simple cubic lattices in three dimensions. A twodimensional schematic presentation is sketched in Fig. 1, where a low molecular weight LC is described by a rod and the flexible chain is represented by x p successive connected subunits or beads. Every coarse-grained rod and chain bead are assumed to occupy one cubic lattice site. The nematic character of the LC is embodied by the L-L model 47) through an anisotropic attractive potential describing the interaction between nearest-neighbor (NN) rods. The anisotropic potential at position i resulting from LC-LC interaction is expressed as E LL ˆ ÿellx P 2 cos hij ˆ ÿellx j 7 NN j 7 NN cos2 hij ÿ1 1 where ell is a positive constant and reflects the maximal interaction energy between NN rods, P 2 (x) is the second Legendre polynomial, and hij is the angle between the rods i and j. Compared to the Maier-Saupe theory 31) for Fig. 1. Schematic presentation of the mixture of flexible lattice chains and low molecular weight nematogens. This figure is a simplified two-dimensional version of the lattice model the nematic-isotropic (NI) transition of LC, the L-L model is actually a potential of nearest-neighbor spinspin interaction on the lattices and thus quite suitable for efficient lattice simulation. Nevertheless, this potential can be simply regarded as a discretized version of the Maier-Saupe potential. In spite of its simplicity, the L-L model can characterize the NI transition of thermotropic LC 47), phase behaviors in mixtures containing LCs 27 29), Freedericksz transitions 52) and even the nonlinear rheological behavior of LCs 53). In our MC simulation, the NI transition temperature is T NI L 310 K for ell = 270k (k is the Boltzmann constant). The NI transition point zell/ (kt NI ) L 5.2 (z is the coordination number and equals to 6 for the cubic lattice) is higher than the theoretical value 4.54 predicted by Maier and Sauper 31) primarily due to the fact that the thermal fluctuation is neglected in the mean-field theory. For the convenience of comparison between the simulation outputs and the theoretical results, the reduced temperature T/T NI is therefore used in the subsequent paragraphs. The isotropic repulsive interaction between polymer segments and LCs, epl acts for every NN hetero-contact pair. For simplicity, the polymer chain is regarded as an extremely flexible coil and thus no anisotropic polymer- LC interaction is considered, nor is any intramolecular interaction within polymers except the excluded volume effect. Simulation algorithm In our MC simulations, periodic boundary conditions are used along the X, Y and Z directions. The simulation is performed as follows: first, randomly select a lattice site A(X,Y, Z), and then select one of its NN sites B(X 9,Y9,Z9) also randomly. If the two selected sites are both occupied

3 Phase separation kinetics in mixtures of flexible polymers and low molecular weight liquid crystals 411 by nematogens, a new orientation is generated for the site A by sampling a random vector uniformly distributed on the surface of the unit sphere. Then, Metropolis importance sampling 54) is applied for determining the acceptance or rejection of the new orientation. If the two selected lattices are both occupied by chain beads, the programme returns to the first step immediately. In the other case, if the two selected NN sites are occupied by a rod and a bead, the central site A and the selected NN site B try to exchange the positions. If this exchange does not lead to the intersection of the chain bond and is permitted by the micro-relaxation modes (the conventional bond fluctuation (chain twisting) 55) and end-bond rotation 56) modes), the Metropolis sampling is employed to determine whether this exchange is accepted. The above micro-relaxation modes are conventional and have been confirmed, based on which, the self-diffusions of both monomers and centers of mass as well as mutual diffusions are inherently included. The simulation continues by turning back to the first step no matter whether the trial move is accepted or not. A unit MC time or MC step is defined as that when every lattice site is, on average, selected once as site A. The initial mixing state for the SD study is generated by evolution for sufficiently long time in the athermal condition. The demixing kinetics after a sudden quench is then studied with the MC simulation. According to the kinetics interpretation of the Metropolis sampling by Binder 46), the dynamic MC simulation can, in principle, be employed to study the underlying kinetics since the number of MC steps is directly proportional to the physical time. The rod rotation takes place much faster than the bond fluctuation or end-bond rotation indeed. Just for this reason, the rod orientation is altered in the whole surface of the unit sphere in every elementary move within one MC step whereas we try to move a chain segment step by step. So, the time scale of an MC step is mainly determined by the diffusion of chain segments. All simulations for the SD kinetics in polymer-lc mixtures are performed on cubic lattices in the volume V = The corresponding system with V = exhibits similar dynamic behavior. Hence, such a system with L = 64 is large enough to alleviate the finite size effect for the problem studied in the present paper with a deep quench leading to relatively small domains. This guarantees, along with a short chain length, that the number of domains in the simulated system is large enough for statistics, and that the number of chains within one domain is, on average, also sufficient to avoid overstrong fluctuation in order to capture the physical essence of the well-known Ginzburg criterion for SD 57). The simulation outputs show that the physical quantities obtained from different initial homogeneous states or different evolution trajectories are similar to each other, which might arise from a strong self-average effect in a large system containing multiple particles sufficiently for statistics. It should be noted that similar phase separation behaviors in the present rod-coil mixtures were observed after we changed the probability of rod reorientation and that of segment hopping. Meantime, the justification of our main conclusions should be based upon the reasonable ratio of LC ordering and chain diffusion inherent in our MC algorithm, which will be examined further. Characteristics of SD kinetics in polymer-lc mixtures The phase diagram revealed by the MC simulation is presented in Fig. 2, together with that from the corresponding theoretical calculation based on the combination of Flory-Huggins theory and Maier-Saupe theory for a binary mixture of thermotropic LCs and flexible polymers 27 29, 34, 35). (The expression of the free energy and the method to calculate theoretical phase diagrams are explained extensively in ref. 35) ) The phase diagram for the binary polymer-lc mixture is highly asymmetry. The nematic phase resulting from phase separation is composed of almost pure rods and strongly excludes the flexible coils. Under a deep quench, a rod-coil mixture is separated into an isotropic phase rich in coils and a nematic phase rich in rods. The demixing region can be further divided into two parts by the NI transition curve (the short dashed curve in Fig. 2). Assuming that the probability for the neighboring sites occupied by nematogens equals to the volume fraction of LC, ulc, the concentration-related NI transition point for a polymer-lc binary mixture can, under the mean-field approximation, be rewritten as T*/T NI = ulc (2) where T NI and T* are the NI transition temperature for the bulk LC and that for the random-mixing polymer-lc mixture, respectively. On the left side of this curve, the LCs are isotropic in the phase rich in rods, while on the right side they are anisotropic 35, 40). From the left to right side of this curve the phase rich in LCs changes the state from isotropic to anisotropic. According to the assumption of the mean-field theory that the rod rotation leading to ordering is much faster than the translation motions for rods and coils leading to phase separation, whether the initial homogeneous phase immediately following a sudden quench is isotropic or anisotropic is completely determined by the position of the quenched state. The SD behaviors are, therefore, different on the left and right sides of the concentration-related NI transition curve shown in Fig. 2. Consequently, there are two spinodal curves, termed as the isotropic spinodal curve and the anisotropic spinodal curve, which constitutes an important

4 412 J. Zhu, G. Xu, J. Ding, Y. Yang Fig. 2. Phase diagram for a binary mixture of thermotropic low molecular weight LCs and flexible chains with the chain length x p = 5. The solid and blank circles are the simulated coexistence or binodal points and the simulated concentration-related NI transition points with the simulation parameters: elc = 270k and thus vll = zell/(kt) = 1620/T, epl = 72.3k and thus vpl = zepl/(kt) = 434/T, while the lines are the corresponding results from the mean-field theory with the calculation parameters: vll = 1620/T, vpl = 434/T following refs. 35) and 40). I and N denote an isotropic phase and a nematic phase. The solid, dotted and short dashed lines refer to the binodal, spinodal and concentration-related NI transition lines, respectively. The curves on the right side are the magnification of the theoretical phase diagram with ulc 7 [0.997, 1]. The stars denote the quenched states where the SD kinetics will be studied in the present paper characteristic of the polymer-lc mixture following refs. 28) and 35). However, the pronounced difference predicted by the mean-field theory has never been confirmed in the experiment, and thus seems necessary to be examined in the present simulation. Four quenched states in the I + N region are examined as marked in Fig. 2 representing different compositions, different quench depths and different SD regions. Coupling of concentration fluctuation and orientation fluctuation Theory 37 40) indicates that during an SD process in a rodcoil mixture, both component and orientation densities evolve as a function of position and time, and so it is necessary to introduce two order parameters in dealing with the problem. A series of typical snapshots of both concentration field and orientational order parameter field are shown in Fig. 3. The orientational order parameter is defined as pp 2 cos hijp. For each pixel in these graphics, the scalar order parameter is calculated with respect to the referred axis by diagonizing the associated second-rank tensor of the orientational order parameter and the concentrations of rods are averaged within the nearest lattices including the central lattice site itself. Fig. 3 demonstrates that the dynamic MC simulation can, although fine enough to reflect chain configuration, be used to study phase morphology and phase separation kinetics. This figure also indicates straightforwardly that the evolution of the concentration field must be coupled with the orientation fluctuation, which constitutes the most important characteristics of phase separation in rod-coil mixtures compared to that in coil-coil mixtures or polymer concentrated solutions. Dominant wavelength in the early stage of SD In order to make comparison with the present theories of phase separation kinetics such as Cahn-Hilliard theory, we calculate the structure factors or the scattering functions only for the concentration fluctuation, S u (q,t)

5 Phase separation kinetics in mixtures of flexible polymers and low molecular weight liquid crystals 413 Fig. 4. The scattering function for the concentration fluctuation S u (q) versus the scalar wave vector q in units of 2p/L for different times t. The quenched state is marked in the figure. The connected lines are shown only for guidance of eyes Fig. 3. Cross sections of snapshots for both concentration field and orientational order parameter field in an LC/polymer mixture resulting from phase separation at the indicated evolution time in units of MC steps recorded after a sudden quench from the corresponding athermal homogeneous state. V = ; x p = 5; ell = 270k, epl = 72.3k. The quenched state: ulc = 0.65, T/T NI = 0.8 S u q; t ˆ 1 V *( X V jˆ1 exp iq N r j t Š ) 2 + u L j; t ÿ u p j; t ÿ hu L ÿ u P iš 3 The average is made over the volume V or the lattice number. q is the scattering vector with the scalar scattering vector written as q. If the lattice j with the position vector r j (t) is occupied by a rod at time t, we set ul(j,t) = 1 and up(j,t) = 0, and vice versa. The contribution of the orientation of the rods to the scattering intensity is neglected. Fast Fourier transformation (FFT) 58) is employed to calculate the scattering function, which can enhance the calculation efficiency dramatically. The spherical average for the scattering vector is made. Fig. 4 shows the simulated time evolution of the structure factor. The shift of the scattering peak to the smaller values of q implies the domain coarsening during SD. The constant q max in the early stage of SD predicted by the linearized Cahn-Hilliard theory 6, 7) is not reproduced, and in fact, has not been observed in experiment for PDLC 15). According to the modification of the Cahn-Hilliard theory by Cook 8), the thermal noise can cause the scattering peak to slowly shift left from the beginning. However, the magnitude of the shift shown in this paper is too large to be simply accounted for this effect. The coupling of the concentration and orientation fluctuations is thus employed to explain the violation of the Cahn-Hilliard linearized theory in the rod-coil mixtures 37, 40). Since the different evolution trajectories exhibit similar q max at the same evolution time in our simulation, an inherent critical domain size or a dominant spatial wavelength indicated by Cahn-Hilliard theory seems still existing in the rod-coil mixtures.

6 414 J. Zhu, G. Xu, J. Ding, Y. Yang Fig. 6. Scalar structure factors in the late stage of SD. The simulation parameters are the same as those in Fig. 4. The dashed and dotted lines are the predicted results according to the Furukawa scaling laws (Eq. (5)) with respect to the critical and off-critical compositions, respectively Fig. 5. Scattering peak, q max and S u,max versus evolution time t. The simulation parameters are the same as those in Fig. 4. The slopes are marked in the figure Domain coarsening and power growth laws For the late stage of SD in a binary mixture of isotropic fluids, the scattering vector of the peak q max and the maximal scattering factor S u,max have the following power laws with the simulation time. q max l t a S u,max l t b (4) From theories 9 13), numerical simulation 5) and experiments 59 62), b A a A 0. It is believed that the exponents satisfy b A 3a in the intermediate stage and b = 3a in the late stage, although very widely scattered a and b values were found in experiment for various polymeric mixtures. These relations are not always satisfied even in theory, for instance, Langer, Bar-on and Miller (LBM) suppose that b L 4a in the late stage of SD 9). The dependence of the scattering peak on the evolution time t is shown in Fig. 4. The q max and S u,max in Fig. 5 are obtained from the fitted Gaussian function around the scattering peak rather than directly from the apparent maximal point in Fig. 4. Although the rod-coil mixture is simulated in the present paper, the exponents seem somehow similar to the predictions of LBM 9), and Binder and Stauffer 10) for the isotropic fluids. Scaling of scattering functions with the Furukawa relation In the late stage of SD, the well-known Furukawa scaling law is universal for an isotropic binary mixture such as flexible polymer blends in d dimensions with the scaling function written as F x l x 2 0:5c x 2 c 5 Here, x and F(x) are defined as x = q/q max, F(x) = S u (q,t)q d max; c = d + 1 for an off-critical mixture and c = 2d for a critical mixture. In three dimensions, the value of c = 6 implies a percolating structure with self-similar evolution of the composition fluctuation, whereas the q 4 exhibiting Porod s law implies the existence of isolated phase-separated clusters and the appearance of a welldeveloped domain wall or interface. However, the scalar scattering functions in the off-critical rod-coil mixture with ulc = 0.5 obey the Furukawa law associated with the critical coil-coil mixture satisfactorily except the tails for large q (q A 2 in Fig. 6). This is not due to the inappropriate estimation for the so-called critical point in the rod-coil mixture, since all of the four quenched states studied in this paper (shown in Fig. 2) exhibit similar behaviors. What is more, both percolating structures (see, for instance, Fig. 3) and isolated phaseseparated clusters (see, for instance, Fig. 7) are reproduced from phase separation at these four quenched states, although the Furukawa relation with percolating structures describes the late-stage scaling of these four cases more satisfactorily. We cannot give a plausible explanation to this phenomenon at the present time. As a result, the Furukawa relation, which originally does not

7 Phase separation kinetics in mixtures of flexible polymers and low molecular weight liquid crystals 415 Fig. 7. Cross sections of snapshots for both concentration field and orientational order parameter field in an LC/polymer mixture resulting from phase separation at t = The quenched state: ulc = 0.5, T/T NI = 0.8 take into account any anisotropic effect, can describe the scaling law for a rod-coil mixture. Similar agreement was found experimentally in the literature from a scattering measurement on the polymerization-induced phase separation in a PDLC film but with the measurement temperature over the NI transition point of bulk rods 15). Fig. 8. The structure factors for the concentration fluctuation, S u (q) versus q at a given evolution time but for different compositions as marked in the figure. The connected lines are shown only for guidance of eyes Comparison of two SD regions The linear analysis of the corresponding time-dependent Ginzburg-Landau (TDGL) equation under the mean-field approximation 40) predicted that two regions separated by the concentration-related NI transition curve might exhibit greatly different SD behaviors. According to this prediction, the structure factors for the concentration fluctuation in the anisotropic SD region increase much faster than the counterpart in the isotropic SD region, so that it is difficult to clearly show the scattering curves in one figure for both cases at the same evolution time. This is, however, not true according to our MC simulation (Fig. 8). That is not surprising since the thermal fluctuations of concentration and orientation included intrinsically in the MC simulation are ignored in the mean-field treatment. The other possible reason might arise from the neglect of the local ordering in the mean-field theory. In our opinion, local ordering might exist even in the globally isotropic phase. It is sensitive to temperature and takes place rapidly after temperature is changed. So, rods may be locally ordered immediately after a sudden quench irrespective of being in the anisotropic or isotropic SD region. Similar to the global ordering, the local ordering can also enhance the driving force of the phase separation in the rod-coil mixtures greatly. Hence, the mean-field theory where only global ordering is considered may underestimate the phase separation velocity in the so-called isotropic SD region for rod-coil mixtures. On the other hand, the theoretical assumption that global Fig. 9. Similar to Fig. 7 but with the quenched state: ulc = 0.85, T/T NI = 0.8 ordering takes place much faster than phase separation in the anisotropic SD region following a sudden quench seems not very realistic. According to our simulation, a striking ordering of the studied four quenched states always occurs along with the process of phase separation, rather than before phase separation or in the early stage of SD. So, the mean-field theory overestimates the phase separation velocity in the anisotropic SD region. As consequence, the difference between the two SD regions is exaggerated by the mean-field theory. The phase morphology for the quenched state (ulc = 0.85, T/T NI = 0.8) is shown in Fig. 9. In this case, the coilrich phase rather than the rod-rich phase is isolated, different from the counterpart for the quenched state shown in Fig. 7. However, it does not result from different SD regions, but from different compositions of these two quenched states in our opinion, because the comparison of the phase morphology at the quenched state (ulc = 0.65, T/T NI = 0.6) shown in Fig. 10 and that with the same composition (ulc = 0.65, T/T NI = 0.8) shown in Fig. 3 indicates that both of them are bi-continuous

8 416 J. Zhu, G. Xu, J. Ding, Y. Yang SD region and anisotropic SD region separated by the concentration-related NI transition curve, where the qualitatively different SD behaviors are predicted by the mean-field theory. It is, however, not found in our MC simulation. The pronounced thermal fluctuation and local ordering in our simulation along with the unrealistic assumption of the initial global ordering for the anisotropic SD region in the mean-field theory are assumed to be responsible for the discrepancy between theory and simulation. Fig. 10. Similar to Fig. 7 but with the quenched state: ulc = 0.65, T/T NI = 0.6 (maybe due to the fact that ulc = 0.65 is near the critical composition seen in Fig. 2), although these two states fall into different SD regions. It hence strengthens the argument that the mean-field theory of SD in rod-coil mixtures overestimates the difference between the so-called isotropic and anisotropic SD regions. It is obvious that the domains in Fig. 3 at t = 5000 are larger than those of Fig. 10. This demonstrates the decrease of the critical wavelength (domain size) for the concentration field with the increase of the quench depth (decrease of temperature), which has been found in the isotropic fluids for a long time 3, 4). Domain patterns of phase separation at other temperatures also verify this trendency. Conclusions In this paper, dynamic MC simulations have, for the first time, been employed to study SD kinetics in the binary mixture of flexible coils and low molecular weight thermotropic LCs where the nematic ordering is incorporated. The SD process in the I + N region under a deep quench is simulated and investigated on lattices. The concentration fluctuation is thus closely coupled with the orientation fluctuation. In our MC simulation, the SD process can be roughly divided into two stages exhibiting different growth exponents. It is found that the inherent dominant wavelength seems still existing in the early stage SD in the rod-coil mixtures. The dominant wavelength in the concentration field at the same evolution time after a sudden quench is reasonably increased with the decrease of the quench depth or the increase of the quench temperature for this model system. In the late stage of SD in rod-coil mixtures with different compositions, the scattering functions can be scaled by the Furukawa relation originally with respect to a critical composition in isotropic binary mixtures. According to the current literature, the I + N SD region can be further divided into the so-called isotropic Acknowledgement: This research was supported by NSF of China, the Doctoral Programme Foundation of Institution High Education from the State Education Commission of China, and the National Key Projects for Fundamental Research Macromolecular Condensed State from The State Science and Technology Commission of China. 1) T. Hashimoto, in Current Topics in Polymer Science, R. Ottenbrite, L. A. Utracki, S. Inoue, Eds., MacMillian, New York ) C. C. Han, Polym. Eng. Sci. 26, 3 (1986) 3) K. Binder, J. Chem. Phys. 79, 6387 (1983) 4) A. Sariban, K. Binder, J. Chem. Phys. 86, 5859 (1987) 5) A. Chakrabarti, R. Torai, J. D. Gunton, M. Muthukumar, J. Chem. Phys. 92, 6899 (1990) 6) J. W. Cahn, J. E. Hilliard, J. Chem. Phys. 28, 258 (1958) 7) J. W. Cahn, J. Chem. Phys. 42, 93 (1965) 8) H. E. Cook, Acta Metall. 18, 297 (1970) 9) J. S. Langer, M. Bar-on, D. Miller, Phys. Rev. A 11, 1417 (1975) 10) K. Binder, D. Stauffer, Phys. Rev. Lett. 33, 1006 (1974) 11) E. Siggia, Phys. Rev. A 20, 595 (1979) 12) I. M. Lifshitz, V. V. Slyozov, Chem. Phys. Solids 19, 35 (1961) 13) K. Kawasaki, T. Ohta, Prog. Theor. Phys. 59, 362 (1978) 14) H. Furukawa, Physica A 123, 497 (1984) 15) J. Y. Kim, C. H. Cho, P. Palffy-Muhoray, M. Mustafa, T. Kyu, Phys. Rev. Lett. 71, 2232 (1993) 16) P. S. Drzaic, J. Appl. Phys. 60, 2142 (1986) 17) J. W. Doane, N. A. Vaz, B. G. Wu, S. Zumer, Appl. Phys. Lett. 48, 269 (1986) 18) J. L. West, Mol. Cryst. Liq. Cryst. 157, 427 (1988) 19) J. Ding, Y. Yang, Jpn. J. Appl. Phys. 31, 2837 (1992) 20) J. R. Dorgan, D. S. Soane, Mol. Cryst. Liq. Cryst. 188, 129 (1990) 21) S. Lee, A. G. Oertli, M. A. Gannon, A. J. Liu, D. S. Pearson, H. W. Schmidt, G. H. Fredrickson, Macromolecules 27, 3955 (1994) 22) A. Nakai, T. Shiwaku, W. Wang, H. Hasegawa, T. Hashimoto, Macromolecules 29, 5992 (1996) 23) T. Shimada, M. Doi, K. Okano, J. Chem. Phys. 88, 7181 (1988) 24) R. Holyst, M. Schick, J. Chem. Phys. 96, 721 (1992) 25) F. Brochard, J. Jouffroy, P. Levinson, J. Phys. (Paris) 45, 1125 (1984) 26) A. J. Liu, G. H. Fredrickson, Macromolecules 26, 2817 (1993) 27) Y. Yang, J. Lu, H. Zhang, T. Yu, Polym. J. (Tokio) 26, 880 (1994)

9 Phase separation kinetics in mixtures of flexible polymers and low molecular weight liquid crystals ) H. Zhang, F. Li, Y. Yang, Sci. China B 38, 412 (1995) 29) H. W. Chiu, T. Kyu, J. Chem. Phys. 103, 7471 (1995) 30) P. J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca ) W. Maier, A. Saupe, Z. Naturforsch. A 13, 564 (1958) 32) W. Ahn, C. Y. Kim, H. Kim, S. C. Kim, Macromolecules 25, 5002 (1992) 33) M. Pracella, B. Bresci, C. Nicolardi, Liq. Cryst. 14, 881 (1993) 34) C. Shen, T. Kyu, J. Chem. Phys. 102, 556 (1995) 35) H. Zhang, Z. Lin, D. Yan, Y. Yang, Sci. China B 40, 128 (1997) 36) J. R. Dorgan, Liq. Cryst. 10, 347 (1991) 37) J. R. Dorgan, J. Chem. Phys. 98, 9094 (1993) 38) J. R. Dorgan, D. Yan, Macromolecules 31, 193 (1998) 39) A. J. Liu, G. H. Fredrickson, Macromolecules 29, 8000 (1996) 40) Z. Lin, H. Zhang, Y. Yang, Macromol. Theory Simul. 6, 1153 (1997) 41) B. Lin, P. L. Taylor, Polymer 37, 5099 (1996) 42) J. M. Jin, K. Parbhakar, L. H. Dao, Macromolecules 28, 7939 (1995) 43) P. I. C. Teixeira, B. M. Mulder, J. Chem. Phys. 105, (1996) 44) J. Y. Kim, P. Palffy-Muhoray, Mol. Cryst. Liq. Cryst. 203, 93 (1991) 45) G. W. Smith, G. M. Ventouris, J. L. West, Mol. Cryst. Liq. Cryst. 213, 11 (1992) 46) K. Binder, D. W. Heermann, Monte Carlo Simulation in Statistical Physics, Springer-Verlag, Berlin ) P. A. Lebwohl, G. Lasher, Phys. Rev. A 6, 426 (1972) 48) A. Baumgartner, D. W. Heermann, Polymer 27, 1777 (1986) 49) Y. Yang, J. Lu, D. Yan, J. Ding, Macromol. Theory Simul. 3, 731 (1994) 50) A. Sariban, K. Binder, Polym. Commun. 30, 205 (1989) 51) G. Xu, J. Zhu, J. Ding, Y. Yang, Sci. China B 41, 424 (1998) 52) J. Ding, Y. Yang, Liq. Cryst. 25, 101 (1998) 53) J. Ding Y. Yang, Polymer 37, 5301 (1996) 54) N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, J. Chem. Phys. 21, 1087 (1953) 55) R. G. Larson, L. E. Scriven, H. T. Davis, J. Chem. Phys. 83, 2411 (1985) 56) P. H. Verdier, W. H. Stockmayer, J. Chem. Phys. 36, 227 (1962) 57) V. L. Ginzburg, Sov. Phys. Solid State 2, 1824 (1960) 58) R. W. Ramirez, The FFT Fundamentals and Concepts, Prentice-Hall, Englewood Cliffs ) T. Hashimoto, M. Itakura, N. Shimidzu, J. Chem. Phys. 85, 6773 (1986) 60) F. S. Bates, P. Wiltzius, J. Chem. Phys. 91, 3258 (1989) 61) T. Hashimoto, M. Itakura, H. Hasegawa, J. Chem. Phys. 85, 6118 (1986) 62) J. Lal, R. Bansil, Macromolecules 24, 290 (1991)