Computational Materials Science 92 (2014) 238–243

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Computational Materials Science

journal homepage: www.elsevier .com/locate /commatsci

Effect of phase shift between geometrical and chemical patterningin nematic liquid crystal cells: A Monte Carlo study

http://dx.doi.org/10.1016/j.commatsci.2014.05.0370927-0256/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Tel.: +91 4066798104.E-mail address: jayad.sri@gmail.com (J. Dontabhaktuni).

Jayasri Dontabhaktuni a,⇑, Regina Jose b, K.P.N. Murthy b, V.S.S. Sastry b

a Centre for Modelling, Simulation and Design, University of Hyderabad, Hyderabad, Indiab School of Physics, University of Hyderabad, Hyderabad, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 31 December 2013Received in revised form 12 May 2014Accepted 16 May 2014

Keywords:Geometrically and chemically patternedsubstratesBiaxialityMonte Carlo methodsEntropic samplingWang-Landau algorithmConfined nematic liquid crystals

Thin films of nematic liquid crystals confined to geometrically as well as chemically patterned substrateat one end and a planar substrate with strong anchoring at the other are studied employing non-Boltzmann Monte Carlo methods. We investigate the effect of temperature on the director structuresas the system goes through the isotropic-nematic phase transition. The low temperature results show sig-nificant deviations from the phase diagram predicted within the continuum approximation, depicted as afunction of the tilt angle at the top substrate and the thickness of the cell. Onset of phase biaxiality isobserved at very low temperatures, and it increases as the tilt angle at the top substrate is increased,moving away from the normal to the substrate. A phase shift introduced between the geometrical andchemical patterns at the other end also enhances the phase biaxiality of the system to a fairly high value.This seems to provide a convenient experimentally tunable parameter for controlling the symmetry ofthe director structures.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

Nematic liquid crystals in contact with solid substrates areextensively used in display devices, like the twisted nematic con-figurations between flat substrate surfaces. The application of asmall external field is sufficient to change qualitatively the equilib-rium director configurations dictated by the substrates, and thusleading to profound changes in the optical properties of the nema-tic cell. Surface inhomogenities could occur as a natural result ofsurface treatments like rubbing, needed to induce the desiredboundary conditions. However such inhomogenities do not reflectin the bulk properties of the nematic cell as long as the lengthscales of the surface patterns are very small compared to the thick-ness of the cell as well as to the wavelength of the visible light.More recent developments however have demonstrated that sur-faces patterned with large periodicity are of considerable interestfrom technological point of view like, for example, in flat panel dis-plays with wide viewing angles ([1] and references therein). Inview of their potential technological importance, it is essential toexamine the anchoring effects in detail, and investigate possibletransitions between different equilibrium director structuresinduced by non-uniform surface interactions. Moreover, these

systems could prove to be very interesting if they were to be foundto exhibit bistable nematic states with different directororientations possible at the same energy [2–6]. Whereas manyexperimental and theoretical studies have concentrated on eithergeometrically [7,8] or chemically patterned surfaces [9–13], nema-tic liquid crystals confined to both geometrically structured andchemically patterned substrates have been objects of investigationonly recently [14–17]. For example, studies based on the contin-uum theory of nematic liquid crystals in contact with specific geo-metrically and chemically patterned substrates predictedtransition between two nematic director structures, the so-calledH and HAN phases [14,15,17], on varying the thickness of the cellor changing the anchoring angle, hD at the top substrate as depictedin Fig. 1. In such studies, the interaction of the nematic liquid crys-tal system in contact with sinusoidal grating with alternating pat-terns of homeotropic and planar anchoring is accounted for, basedon the Frank–Oseen model for distortion free energy [18,19],while the surface energy function is written in terms of theRapini–Papoular interaction [20].

Theoretical treatments normally proceed by reducing thedimensionality of the system for convenience of analysis, takinginto account the translational invariance along one of the direc-tions, and sometimes by performing a conformal mapping appro-priately to eliminate one more dimension. Further, a planarsurface inducing an effective anchoring angle is assumed to mimic

Fig. 1. Liquid crystal cell: (A and B) Schematics of liquid crystal cell with bothgeometrical and chemical patterns applied in-phase and shifted by pi=8 respec-tively. The head-less vectors indicate the molecular orientations at the gratedsubstrate. The liquid crystal molecules at the top substrate are oriented at tilt angleshD ranging from 0� to 45�. (C and D) Snap shots of the initial configuration of liquidcrystal cell of Model A and Model B respectively. The colour code indicates therelative orientations of the molecules with respect to the Z-direction. (Forinterpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)

J. Dontabhaktuni et al. / Computational Materials Science 92 (2014) 238–243 239

the geometrical pattern at the grooved substrate [14–17]. Theunderlying argument relies on the observation that the effect ofthe patterned surface does not extend into the film beyond thelength scale of the geometric structure and hence can be replacedby an effective free energy expansion. In all the cases the equilib-rium configurations under different distorting conditions areobtained by minimizing the effective free energy functional soproposed.

The rationale for these simplifications could have been the pro-hibitive effort involved in tackling the problem more exactlythrough numerical procedures within continuum approximation.Such an extremization procedure under the stated simplificationsinter alia however does not allow for possible role of thermaleffects which are indeed important in real nematic samples. Simu-lations also provide additional advantage of following the thermalbehaviour of the system through its IN transition. Keeping thesemerits in view, we perform Monte Carlo simulations of the modelliquid crystal system based on inter-molecular interactions amongthe molecules given by Lebwohl–Lasher potential [21], imposingsuitable boundary conditions to mimic geometrical as well aschemical patterns. We employ entropic sampling techniques[22,23,25] to investigate this confined system, as has been recog-nized earlier to be crucial for such studies [27].

In this article, we implement a specific geometrical and chemi-cal pattern which has been recently investigated analytically viafree energy minimization procedure, and study the effect of rele-vant control parameters on the formation of the film as the tem-perature is cooled, looking for stable structures at very lowtemperatures. We also examine the onset of phase biaxiality byintroducing a phase shift between geometrical and chemical pat-terns in the nematic liquid crystal cell.

2. Model and simulation methodology

Lebwohl–Lasher potential [21] describing the interactionbetween two nearest neighbouring liquid crystal molecules placedon a cubic lattice is given by:

H ¼ �Xhi;ji�ijP2ðcosðhijÞÞ ð1Þ

Here, �ij ¼ � if two nearest neighbours at i and j are liquid crystalmolecules; �ij ¼ x, the anchoring strength, if one of the nearestneighbours is a substrate molecule and x varies between 0 and 1.The summation hi; ji is over all the nearest neighbours in the lattice.The top layer is a solid substrate inducing strong anchoring at anangle, hD with respect to the homeotropic direction, as shown inFig. 1. A sinusoidal pattern is carved out of the bottom layers (say,with amplitude, A) along the x-axis. The thickness of the cell, Dalong the z-direction is taken to be large compared to the amplitudeof the sinusoid. The surface induced interaction is invariant alongthe y-direction. We choose the cell thickness to be much larger thanthe period of the sinusoidal grating, P at the bottom surface, and theamplitude of the sinusoid to be smaller than its wavelength. Now, inorder to implement this scenario on a lattice model, we choose acubic lattice of size 16� 16� 66, in the notation of x; y and zdimensions shown in Fig. 1. We introduce a sine wave extendingover 16 lattice points in the x-dimension, and a film thickness(z-direction) of 66 units. While the y-dimension could be arguedto be unimportant within the continuum approximation (since theproblem under simplifying assumptions is reducible to x–z plane),simulating at microscopic level necessitates extension into they-dimension. Accordingly, we consider the system to extend over16 lattice units in this direction as well. We find that it has someinteresting consequences on the development of bistability of thesystem. Periodic boundary conditions are applied along the x andy directions. The amplitude of the sinusoidal grating was initiallychosen to be 8 units. Such a choice may not implement accuratelythe geometrical constraints implied in [16], due to the limitationsintroduced by the discretization of space into lattice; it does how-ever capture essential features of the underlying model. The crucialassumption in the continuum treatment is that any arbitrary distri-bution of the anchoring direction over the grating period is essen-tially taken into account by introducing an effective tilt angle(say, haeff

) for purposes of predicting the bulk behaviour of the film.The focus of this simulation is thus more on the process of forma-tion of the director structures as the nematic phase forms, as wellas the realizability of phase biaxiality and its tuning by introducinga phase shift between geometrical and chemical patterning of thesubstrates at low enough temperatures.

We consider two model systems in the present work based onthe phase shift introduced between the chemical and geometricalpatterns on the bottom substrate, see Fig. 1A and B. In Model A,the chemical pattern is in-phase with the geometrical pattern.Homeotropic alignment is induced at crests and troughs of thesinusoidal wave each for a period of one fourth of the period ofgeometrical sine wave, while in the other regions locally planaralignment is imposed. In Model B, the chemical pattern is shiftedby one-eighth of the wavelength with respect to the geometricalpattern in contrast to Model A. The anchoring directions in thetwo models are shown in Fig. 1A and B. Initial configurations ofthe liquid crystal cell in both the models are illustrated in Fig. 1Cand D. Each ellipsoid represents the collective orientation of agroup of liquid crystal molecules positioned at that lattice point.Their relative orientations with respect to the z-direction are col-our coded in the figures: blue colour represents molecules parallelto z-axis while red represents perpendicular to the z-axis. Weemploy modified frontier sampling method [23] to study the sys-tem for different angles, hD in the nematic region. We computethe density of states (DoS) and hence collect entropic ensembleof microstates fairly uniformly distributed with respect to the sys-tem energy. We then extract the canonical ensemble at any desiredtemperature by appropriate re-weighting procedure [24]. Thus foreach value of hD, we compute equilibrium averages of different

Fig. 2. Layer-wise angles: (A and B) Local director orientations in each layer along the z-direction (averaged over x–y plane) as a function of the tilt angle at the top substrate,hD ¼ 0—45� in the case of A and B models at the reduced temperature, T ¼ 0:6. The layer-wise angles in the case of Model B for tilt angles hD ¼ 55—90� are shown in the insetin (B). The corresponding director orientations at temperature, T ¼ 1:2 are shown in (C and D). All the angles on y-axis are represented in degrees. (E and D) depict snapshotsof the liquid crystal cell rotated by 30� about z-axis for Model A and B respectively at T ¼ 0:6.

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relevant physical observables as a function of temperature. Thedetails of the simulation and the process of obtaining thermody-namic averages are given elsewhere [25]. The relevant physicalproperties in this study, that describe adequately the symmetryof the director field arising from orientational alignment of uniax-ial molecules, are the degree of uniaxial order, S and the possibleloss of cylindrical symmetry around the principal director repre-sented by the biaxiality parameter, B [26]. In order to calculatethe order parameters, a symmetric and traceless ordering matrixis constructed with the unit vectors ei representing the orientationof liquid crystal molecule at ith site as shown below.

Q ¼ 1N

XN

i¼1

eðiÞx eðiÞx � 1=3 eðiÞx eðiÞy eðiÞx eðiÞz

eðiÞy eðiÞx eðiÞy eðiÞy � 1=3 eðiÞy eðiÞz

eðiÞz eðiÞx eðiÞz eðiÞy eðiÞz eðiÞz � 1=3

0BB@

1CCA ð2Þ

The matrix Q is diagonalized by appropriate rotational transforma-tions. The eigen values and the corresponding eigen vectors are cal-culated. The second rank scalar order parameter, S is given byS ¼ 3

2 kmax, where kmax is the maximum eigen value. The absolutevalue of the difference of the other two eigen values gives a mea-sure of the phase biaxiality, B.

Fig. 3. Broadening and branching of microstates: Uniaxial and biaxial order parameters as a function of the microstates collected from the entropic ensemble for both themodels at hD ¼ 0� and 45� respectively as indicated in the figures.

Fig. 4. Effect of thickness in the y-direction: Uniaxial order parameters collected asa function of the energy of microstates collected varying the cell thickness iny-direction, t = 3, 4, 6 and 8 units, respectively, at hD ¼ 0� for Model B.

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3. Results and Discussion

The local director orientations of each layer along the z-direc-tion averaged over x–y plane are depicted in Fig. 2. In the nematicphase (at the reduced temperature T ¼ 0:6, scaled by the prefactor� in Eq. (1)), the director at the patterned substrate reorients to thehomeotropic direction in Model A (at L ¼ 0, see Fig. 1) independentof the tilt angle, hD at the top substrate (at L ¼ 66), as shown inFig. 2A. Moreover, the director orientation profiles are symmetricabout the mid-layers for smaller tilt angles. As the tilt angle atthe top substrate, hD reaches 45�, the molecules gradually orientfrom 45� at top substrate to 0� at the structured substrate, as seenclearly also from Fig. 2E which is a snapshot of the liquid crystalcell (rotated by an angle 30� about z-axis for visualisation pur-poses). In the case of Model B, however this symmetry about themid-layers is broken while relaxing to homeotropic direction nearthe structured substrate ðL ¼ 0Þ. The snapshot in Fig. 2F of therotated cell for hD ¼ 45� shows that the molecules gradually reori-ent themselves to the homeotropic direction near the mid-layersand tilt again to about 20� before becoming normal to the surfaceagain near the structured substrate (see Fig. 2B). For tilt angleshD ¼ 55� to 90� the variation of directors in the mid-layers is sim-ilar, with the qualitative difference that they become planar at thestructured substrate, as seen in the inset of Fig. 2B. At higher tem-peratures (isotropic phase), the layer-wise local directors reorientto an angle 58� at the mid-layers in both the models. As theyapproach the structured substrate the directors align homeotropi-cally in Model A, while they make an effective angle 45� in the caseof Model B.

Fig. 3 depicts the distribution of microstates with respect toenergy. The microstates have been collected from the entropicensemble in Model A and Model B at two imposed tilt angleshD ¼ 0� and 45�, sorted as per their uniaxial and phase biaxial orderparameter values. Noting that the extent of distribution of thesestates over the corresponding order parameters at a given energy(bin) value (variance) is a measure of their susceptibility, it maybe noted that in Model A homeotropic anchoring at the top sub-strate (hD ¼ 0�) has a tendency to form a fairly pure uniaxial phaseas shown in Fig. 3A: the fluctuations in this case with respect touniaxial order diminish noticeably at low temperatures whilethere is no significant distribution of microstates in this systemwith respect to the biaxiality parameter (Fig. 3B). In contrast, forthe same Model at hD ¼ 45�, Fig. 3E, the uniaxial order has higher

degree of fluctuations as seen from the broadening, while permit-ting microstates to be distributed, fairly widely, over a finite rangeof biaxiality parameter (Fig. 3F). This suggests an onset of the phasebiaxiality with relatively shallow free energy profiles allowing forlarger excursions in the order parameters. In Model B, on the otherhand, similar scenario (corresponding to Model A at hD ¼ 45�)already exists at hD ¼ 0� (Fig. 3C and D). With increase of hD to45�, this model acquires a much wider distribution of microstateswith respect to both the order parameters at any given energyFig. 3E and F), and more prominently the biaxiality parameterhas a higher average value with longer range of fluctuations. Theseobservations provide an insight into the observed macroscopicthermal behaviour of the two order parameters and their suscepti-bilities, shown in Fig. 5.

We now examine the role of the thickness of the cell in they-direction on the simulation results, seen as arising from finite tem-perature manifestations. We investigate the microstates collectedfrom the entropic ensemble systematically for different thicknesses,

Fig. 5. Order parameters: (A and B). Uniaxial order parameter, S as a function of temperature for hD ranging from 0� to 45� for Models A and B respectively. The correspondingbiaxial order parameter, B is shown in (C and D).

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t = 3–8 lattice units in the y-direction. Even though the structuredsubstrate affects the layers uniformly along this direction, weobserve that for very thin cells (with respect to the y-direction) themicrostates spread out over a rather broader range sometimesaccompanied by branching along the order parameter axes (at thesame energy bin), as seen in Fig. 4. This is clearly the consequenceof increased averaging over the neighbouring interactions in thisdirection, which plays a role in finite-temperature simulations.One should expect to reach an asymptotic limit as this thickness isincreased, overcoming finite size effects, and the present simula-tions bring this consequence.

Fig. 5A and B show the uniaxial and biaxial order parameters asa function of temperatures for Models A and B, respectively. Weobserve that while the uniaxial order parameter, S in Model Aremains unchanged as we increase the tilt angle hD from 0� to45� (see Fig. 5A), the order parameter in the case of Model Breduces from 0.8 to 0.68 as hD varies from 0� to 45�. In the caseof homeotropic alignment (Model A, hD ¼ 0�), biaxiality increasesat the isotropic to nematic transition and reduces to near zero asthe temperature is reduced. As the hD is increased from 0� to 45�,there is an onset of biaxiality at very low temperatures, seeFig. 5C. In the case of Model B, there is an onset of biaxialityð0:04Þ in the system even for hD ¼ 0�. Biaxiality increases to a fairlyhigh value of 0.16 as hD increases to 45� (typical values observed inbiaxial systems [28,29]). We would like to add here that the simu-lated observables at tilt angles, hD ¼ 50� to 90� are found to be thesame as those at hD ¼ 45� to 0�, and hence are not shown here.

4. Conclusions

In the present work we simulate a film of liquid crystals con-fined to structured substrate interacting via a simple lattice modelby employing entropic sampling based method. Our studies showthat the continuum predictions (Model A) are an asymptotic limitof simulation results at very low temperatures. In this context, this

study provides an insight on how such a limit is achieved and therole of thermal fluctuations in realistic nematics. We also obtainthe phase transition behaviour on this model. We further investi-gate a new model (B) for which continuum predictions are notavailable. The onset of high degree of phase biaxiality in Model Bis interesting from applications point of view. The techniqueemployed allows us to see the distributions of microstates in thetwo-dimensional space of order parameter and energy, providinga vivid description of the free energy topology. In this context,comparing the distributions in Models A and B, under otherwiseidentical conditions, provides an insight into understanding thechanges in free energy landscape as the system is subjected to ahigher degree of frustration by introducing the relative phase shift(Models A and B).

Acknowledgments

All the simulations were performed at Centre for ModellingSimulation and Design of the University of Hyderabad. J. D. wouldlike to acknowledge Department of Science and Technology, NewDelhi for grants under Project No. UH/CMSD/HPCF/2006–7 andDST PURSE grant.

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