Field indeced pattern simulation and spinodal. point in nematic liquid crystals. Chun Zheng Frank Lonberg Robert B. Meyer. June 18, 1995.

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1 Field indeced pattern simulation and spinodal point in nematic liquid crystals Chun Zheng Frank Lonberg Robert B. Meyer June 18, 1995 Abstract We explore the novel periodic Freeredicksz Transition found in nematic liquid crystals composed of very long particles in splay geometry. Good agreement has been reached between our simulations and the experimental results, including wavelength selection, the critical point and the cut-o point. We propose an articial-eld scheme to study pattern growth phenomenon at states away from the locally stable state and conrmed the existence of a spinodal point between the cut-o point and the normal Freedericksz Transition critical point. 1 Introduction Freeredicksz Transition, which normally refers to the homogeneous distortion of nematic liquid crystal layers in an external eld above certain critical point, has been known and studied for a long time. However, Meyer and Lonberg [1] had recently discovered by experiment that for polymer liquid crystals, this normal Freedericksz Transition is superceded by a periodic distortion with a lower critical eld value. Their further analysis predicts that for any nematic liquid crystal with an elastic constant ratio of K 1 =K 2 (splay elastic constant/twist elastic constant) larger than 3.3 this distortion has always a lower critical eld. Thus it becomes the ground state instead of the usual homogeneous distortion. This discovery has introduced a new eld of interest in nonlinear physics of liquid crystals and attracted widespread attentions. G. Srajer, F. Lonberg and R. B. Meyer [2] furthered the study of this system in 1

2 a wedged sample experiment and found a spinodal point between the pattern distortion and the normal homogeneous distortion. Similar systems have been studied since [3, 4]. Because of the strong nonlinearity involved in this pattern formation process, analytical solutions above the critical point are nearly impossible to obtain. Even with powerful computers and optimized algorithms, fully 3- dimensional calculation of the system is a formidable task. In this article, we report a series of 2-dimensional simulations with a \cut and try" approach. Our results seem to be in good agreement with previous experiments and yield some intuitive understandings of the underlying physics. A typical nematic liquid crystal is composed of rod-like molecules with orientational order. This orientational order can be broken by various distortions which raise the free energy of the system. When conned between two glass plates, all the molecules of the liquid crystal will be aligned by the surfaces through the anchoring force. However, because of the unisotropic magnetic susceptibilities possessed by these materials, the molecules can also be aligned by an external magnetic eld. When the orientation of the magnetic eld is dierent from that of the surfaces, a competition occurs. It is this competition that determines the critical point of normal Freedericksz Transition. Shown in Fig. 1 is a Freedericksz Transition simulation in splay geometry above H c. Compared to the undistorted state, while the elastic free energy of the system is raised by the splay distortion, the eld induced free energy of the system is lowered more and the total free energy of the system is actually reduced. A schematic phase diagram of pattern distortion (or periodic Freedericksz Transition) and normal Freedericksz Transition (the Homogeneous Distortion) is shown in Fig. 3, which has been observed in experiments and conrmed by our simulation. As with the normal Freedericksz Transition, the periodic Freedericksz is basically a second order phase transition at the critical point. However, the transition from the periodic distortion to the homogeneous distortion at the cut-o point has been demostrated as a rst order transition. 2

3 2 Theory and algorithms Although the free energy approach of Nematodynamics is more intuitive, we adopt the dissapative function approach for the convenience of numerical application. In polar coordinates, Lagrange theory gives: d dt _? F (1) where denotes polar coordinates or '. T is rotational kinetic energy, D is Rayleigh's dissapation function, and F is the generalized elastic force density. In our simulation, T is assumed to be 0, the term involving D is given n_ 1n _ where n is the director eld, 1 is the rotational viscosity and stands for summation over spatial Cartesian coordinates. No ow coupling is assumed. The generalized elastic force is given by @(@ ) where f is the elastic and magnetic free energy density. In explicit form, (3) f = 1 2 K 1[divn] K 2[n curln] K 3[n curln] 2? 1 2 a[n H] 2 (4) where a denotes the anisotropy of magnetic susceptibility. All these equations combined together expand to two torque equations with regard to the two degrees of freedom in (x) and ' (x). These two equations form the basis of our simulation. Since these two equations involve mostly trigonometric functions, they are two coupled second order partial dierencial equations with strong nonlinearity. We use iterational relaxation method to nd the equilibrium solutions [8, 9]. In each iteration, Newton- Raphson's method using derivatives is used together with a line-search algorithm to nd the root of each equation. Equilibrium state is assumed when both changes of and ' are smaller than 1:0 10?6 radian/sec. With a SiliconGraphis IRIS workstation, the CPU time would be from several minutes 3

4 to several days for one run depending on the nature of the task and the initial conditions. 3 Wavelength selection As shown in the phase diagram of periodic Freedericksz Transition (Fig. 3), pattern distortion response occurs in a long range of external eld strengthes from the critical point H cp to the cut-o point H t for nematic liquid crystals with a ratio of K 1 =K 2 larger than 3:3. For eld strength slightly above the critical point, pattern distortion is spontaneous. To simulate this phenomenon, we start the simulation from a uniform nondistorted state with the director aligned along the surfaces, then add uctuations of Monte Carlo distribution of both and ', and apply the magnetic eld to let the system evolve. For the ratio of K 1 =K 2 smaller than 3.3, nothing happens and the system stays at the undistorted state since the eld strength is below the normal Freedericksz Transition critical point. For the ratio of K 1 =K 2 larger than 3.3, the simulated result is a pattern response. This has been taken as part of the testing process of our program. Since we ignore the ow-coupling and use an accelaration scheme in the computation, the value of rotational viscosity 1 does not aect the nal result. Other parameters of importance in our simulation are chosen in consideration of previous experiments and listed below: elastic constant (10?7 dyn): K 1 = 12 K 2 = 0:65 K 3 = 24 anisotropy of magnetic susceptibility (emu/mol): thickness of sample (cm): a = 0:064 10?7 4

5 d = 82 10?4 Fig. 2 shows a typical picture of the equilibrium state simulation of the periodic distortion (for one period). Note that because of the limited scale of simulation, boundary conditions are important. While dierent boundary conditions give similar PD response, the variation of wavelengthes is substantial. The average wavelength for open boundary (where the spatial derivatives of and ' are 0) is larger than that for closed boundary (where the time derivatives of and ' are 0), and the two values approach the same when the length of the sample goes to innity. To nd the true value of the wavelength of PD response in equilibrium state, one would have to simulate an innitely long sample with either open or closed boundary. In practice however, this diculty can be overcome by the free energy calculation together with an imposed periodic boundary condition. For a periodic boundary, the equilibrium wavelength is determined by the length of the simulated sample. By varying this length and minimizing the corresponding free energy, one can nd the PD wavelength in equilibrium state. Fig. 4 shows the local minimum point of free energy density at eld strength H = 5:5(kG). In theory, the free energy density should develop a local minimum for wavelength at innity when the eld strength H exceeds the spinodal point. Apparently this is impossible to simulate in practice. What assures the consistency of our simulation is that we do nd the free energy density of uniform Freedericksz Transition being smaller than that of the long-wavelength limit of the PD Freedericksz Transition above the spinodal point. Fig. 5 is the simulation result of wavelength selection. The second order behavior of the wavelength curve has been explained previously. [1] The method we used in this simulation has a build-in systematic error of 5% which indicates a good agreement of our simulation with previous experiment (note there is a small dierence in the parameters we used in the simulation: the ratio of K 1 =K 2 is about 18 in our simulation but 15 in the experiment. The reason for this is to emphasize the second order behavior of the simulation curve. Choosing a smaller K 2 makes the curve deeper at the minimum point.) We also tried a dierent scheme which xes the grid size and varies the ratio 4x/4z. This approach can improve the accuracy innitely in theory but in 5

6 practice error is inevitable because of the limitation in computation time and digital representation of data. Fig.6 is a comparison of free energy density between the HD and PD states at dierent eld strengthes. The free energy density of the PD state is calculated for the equilibrium wavelength described above. The cut-o point is where the free energy density of the HD state crosses that of the PD state. This has been proven to be a rst order phase transition, while both HD and PD Freedericksz Transitions are second order phase transitions. Note that both the critical points for HD and PD phase transitions can be extrapolated from this graph. 4 Spinodal point A spinodal point has been found to exist above which the homogeneous distortion state is metastable with regard to the pattern distortion state. To study this spinodal point experimentally, G. Srajer, F. Lonberg and R. B. Meyer [2] used a wedged sample to simulate a continuous range of eld strengthes. To understand this, it is useful to express the relationship of the sample thickness and the critical eld in H c = 1 d s K a (5) where K is the appropriate elastic constant (here K 1 ) associated with the given geometry. Thus a varying thickness corresponds to a varying critical eld, which results in a varying eective eld strength when the sample is placed in a uniform external eld. In the experiment, the nucleated pattern distortion was seen as stripes growing from spontaneous PD region to homogeneous distortion region with point defects ajusting the wavelength. This technique was innovative and ecient, but dicult to duplicate with the computer simulation. A 3-dimensional approach would have to be necessary to simulate a sample of varying thickness, and the complicated boundary conditions and generation of defects would make the computation impractical in our case. A natural approach to tackle the spinodal phenomenon is to nd the path for the system connecting the HD state and the PD state. Adding uctuations of random distribution to a uniformly distorted state, the system would evolve to the periodic distorted state if the eld strength is below the spinodal point. 6

7 We found that at H = 5:5kG PD state almost always grows out of the corresponding HD state with an average amplitude of uctuation modes in the order of 10?3 but at H = 5:8kG the PD modes do not grow even with an average amplitude of uctuation modes 100 times larger. This is a good indication that the spinodal point lies between these two values. However, zeroing in to the exact value is complicated by the generation of initial uctuations. Also, this approach lacks deterministic certainty. Thus, nding the growing mode for the PD state becomes an important and interesting step in understanding the spinodal phenomenon. In order to eliminate the complication of random uctuation modes, Robert B. Meyer suggested that we add to Equation (4) an articial magnetic eld term of 4th order f 0 =?(n H 0 ) 4 (6) where the direction of H 0 is along the direction of local directors in the HD state. The fourth order term is the lowest term which provides a stronger binding than the original second order term and keeps the same symmetry.. When the external magnetic eld strength is below the spinodal point, the PD state is the only stable state of the system. With the introduction of the fourth order eld interaction, this stable state is shifted toward the HD state. By adjusting the strength of the articial eld, and starting from the HD state with uctuation of random distribution, one would be able to study the states in the vicinity of the local equilibrium state for external magnetic eld strengthes below the spinodal point. When the articial eld is shut o after steady state is reached, the system will then evolve to the PD state. The importance of this \articial eld" scheme is in its ability to track down the \growing mode" for the PD state. An intuitive guess of this growing mode is the nal mode of the PD state which happens to be true in our case, but this is not necessarily true for nonlinear phenomena. Justication has to be made such as the approach we adopted here. Another interesting idea was suggested by Frank Lonberg which is to run the program in a reversed time order to force the system from the PD state back to the HD state. While this process is impossible in experiment, it can be carried out in computer simulation. Before the system fully relaxes to the HD state, the residue mode should be the growing mode for the PD state. For actual eld strength above the spinodal point, one would have to 7

8 combine the above ideas to nd the full path connecting the PD and HD states. Starting from the PD state one can reach to the HD state if the articial eld is strong enough. Since the articial eld does not aect the HD state and is not accounted for in the free energy calculation, it merely provides a convenient way of nding the intermediate states. We suggest that the two ideas explained here can be used as a general method in exploiting spinodal phenomena. Fig. 7 shows two realizations of above ideas. Ideally, each point on the graph should represent an equilibrium state corresponding to a specic articial eld strength. Here we simply set the articial eld strength large enough to bring the system over the bump from the PD state to the HD state. This gives one of the many routes bridging the PD and HD states (note that this route could be unique). The important fact is that dierent routes give consistant results ( another example is shown in Fig. 12 for eld strengeh of 5.8 kg). Note that the number of states between the PD and HD states is innite and those states plotted on the graph are merely chosen arbitrarily. The position of the bump and the slope of the curve can change in any way, but the overall shape of the curve should always be the same. The structure of the growing mode of the PD state is found by applying the articial eld method. Starting from the HD state for a given eld strength below spinodal point, and adding a rather large uctuation eld of random distribution, the system evolves to an equilibrium state which is away from the HD state because of the articial eld. This equilibrium state is a superposition of the HD state and the growing mode of the PD state. By Fourier analysing it, we can reconstruct the growing mode of the PD state. Fig 9 (a) and (b) are showing the structures of the components of this growing mode in both amplitude and phase shift. The important thing is relationship of the relative amplitudes and phase shifts but not the absolute value of the amplitudes and the phase shifts. To nd the spinodal point accurately, we choose the initial growth time of PD modes as an index. When the applied magnetic eld approaches the spinodal point from below, the initial growth time of PD state from HD state diverges. For each given eld strength, we nd the HD state rst. Then a minimal growing mode (of order of 10?5 ) of the PD state is added. The initial growth time is dened as the time for the average change for and ' to exceed certain value (0.2). This result is shown in Fig. 8. We use the 8

9 notation of fm,ng to represent the components of the growing mode: (x; z) = X m;n amp(m;n) e 2mx+2nz+ 0 (7) '(x; z) = X m;n ' amp(m;n) e 2mx+2nz+' 0 (8) where m, n are wave numbers in x and z directions respectively. The spinodal point value found in our simulation is H sp =5.61 kg which does not match the experiment (in which H sp is found to be about 6.9 kg) well (the small dierence in values of K 2 does not account for this discrepency). While the exact reason for this is not clear, we believe that the geometry of the wedge sample used in the experiment is an important factor to consider. When the sample is quenched from a high eld to a low eld, pattern response starts from the spontaneous pattern formation region (the boundary between this region and the HD region corresponds to the spinodal point). However, as soon as the eld is lowered, the nucleation process starts. By the time the stable pattern becomes visible, the nucleation process could have pushed the boundary far from the true value. This results in a higher spinodal point value which is consistent with our result. In searching for the path between the PD and the HD states, we also studied the free energy densities associated with dierent elastic constants. Fig. 10 indicates that for eld above the critical point, free energy of bend geometry is actually larger than that of twist geometry. The eld induced free energy for the same process is shown in Fig. 11 and the total in Fig. 12. Again, dierent pathes may give dierent results for these values, but the basic physics behind remaines the same. 5 Conclusion Our 2-dimensional simulations of the periodic Freedericksz Transition have been self-consistent and successfully explained the experimental observations. The articial eld approach we used for the spinodal study proves to be ecient as well as reliable. Our simulations demonstrated the possibility of studying a large and complex system with a relatively small grid size and modest computation capability. 9

10 References [1] F. Lonberg and R. B. Meyer, Phys. Rev. Lett. 55, 718 (1985). [2] G. Srajer, F. Lonberg and R. B. Meyer, Phys. Rev. Lett. 67, 1102 (1991). [3] G. Srajer, S. Fraden and R. B. Meyer, Phys. Rev. A 39, 4828 (1989). [4] Axel Kilian, Phys. Rev. E 50, 3774 (1994). [5] P. G. de Gennes, The Physics of Liquid Crystals (Clarendon Oxford, 1974). [6] G. Vertogen and W. H. de Jeu, Thermotropic Liquid Crystals, Fundamentals (Springer-Verlag Berlin 1988). [7] G. Srajer, Ph.D thesis, Brandeis University, [8] W. H. Press at. Numerical Recipes in C (Cambridge University press, 1992). [9] Ames, W.F. 1977, Numerical Methods for Partial Dierential Equations, 2nd ed. (New York, Academic Press). 10

11 H z x Figure 1: Homogeneous Distortion (Freedericksz Transition) of a nematic liquid crystal layer in an external magnetic eld (H > H c ). Note the short bar on the end of directors is not physical, it is used to keep track of the molecular orientations. (Simulation result.) 11

12 Figure 2: Pattern Distortion of a nematic liquid crystal layer in an external magnetic eld (periodic boundary). See Figure 1 for reference. K1/K2 < 3.3 no distortion Homogeneous Distortion 0 Hcp Hc Hsp Ht H no distortion Pattern Distortion (spontaneous) Pattern Distortion (nucleated) Homogeneous Distortion K1/K2 > 3.3 Figure 3: : Schematic diagram of phase transitions of nametic liquid crystals in external magnetic elds. K 1 and K 2 are splay and trist elastic constants respectively. H c is the critical point for Homogeneous Distortion (Freedericksz Transition); H cp is the critical point for Pattern Distortion, H t is the cut-o point for PD, and H sp is the spinodal point of PD. 12

13 0-2 f (arbitrary unit) λ /d Figure 4: Free energy density corresponding to dierent wavelengthes. The free energy density is in arbitrary unit, and the wavelength is scaled to the thickness of the sample. The curve approaches a limit when the wavelength becomes large, and this is due to the accuracy of the computation. When the eld is above the spinodal point, we would expect the curve to have some ne structure for wavelengthes at the vicinity of innity (but this is impractical with computer simulation). 13

14 λ /d H (kg) Figure 5: Simulation result for equilibrium wavelengthes corresponding to dierent external eld strengthes. Note there is a systematic error of 5in for this simulation due to the grid size and the algorithm we used. 14

15 f (arbi. unit) PD response HD response H (kg) Figure 6: Comparison of free energy densities for PD response and HD response. The free energy density of PD response is calculated for the equilibrium wavelength. The curve indicates a well dened cut-o point. The free energy density is in arbitrary unit. 15

16 f free energy density PD state intermediate states HD state (a) -2-4 f free energy density PD state intermediate states HD state (b) Figure 7: Demostration of free energy barrier for eld above spinodal point (which is 5.61 kg). The left-most state is the PD state and the right-most state is the HD state. We used the articial eld method to force the system from the PD state back to the HD state. Intermediate states are taken randomly. (a) below the spinodal point (5.4 kg), there is no barrier between the PD state and the HD state; (b) above the spinodal point (5.8 kg), the free energy barrier is obvious (the inlet picture is a blow-up of the last 5 points in the big picture. 16

17 t (s) H (kg) Figure 8: When the eld strength approaches the spinodal point from below, the initial growth rate of uctuation diverges. 17

18 1 0.8 amplitude 0.6 \theta \varphi n (2π) α ο (deg) \theta \varphi Figure 9: Structure of growing modes in amplitude. Shown on the graphes are modes f1,ng for ' with n ranging from 1 to 10. Graph (a) is the composition of these modes in the initial growing stage after a few steps from a random starting conguration; (b) is the distribution of these modes in nal equilibrium state. The initial growing stage is basically in searching this composition of modes. n 18 (2π)

19 35 30 f (arbi. unit) splay trist bend PD state intermediate states HD state Figure 10: Free energy density changes associated with dierent elastic constants along the route from the PD state to the HD state f PD state intermediate states HD state Figure 11: Field indeced free energy density (in the same unit as used in Fig. 10) 19

20 f PD state intermediate states HD state Figure 12: The total free energy density (in the same unit as used in Fig. 10) for all the intermediate states along the path from the PD state to the HD state at H=5.8 kg. 20