LATTICE SPIN SIMULATIONS OF TOPOLOGICAL DEFECTS IN NEMATIC FILMS WITH HYBRID SURFACE ALIGNMENTS

LATTICE SPIN SIMULATIONS OF TOPOLOGICAL

DEFECTS IN NEMATIC FILMS WITH HYBRID

SURFACE ALIGNMENTS

CESARE CHICCOLI* and PAOLO PASINI

INFN, Sezione di BolognaVia Irnerio 46, 40126 Bologna, Italy

*[email protected]

LUIZ ROBERTO EVANGELISTA and RODOLFO TEIXEIRA DE SOUZA

Departamento de F�{sica

Universidade Estadual de Maring�a, Avenida Colombo

5790-87020-900 Maring�a (PR), Brazil

CLAUDIO ZANNONI

Dipartimento di Chimica Fisica ed Inorganica

Viale Risorgimento 4, 40136 Bologna, Italy

Received 1 March 2011

Accepted 25 March 2011

We present a Monte Carlo study of the e®ects of elastic anisotropy on the textures of nematic¯lms with speci¯c random hybrid boundary conditions. The polarized microscopy images and

their evolution are analyzed in uniaxial systems for di®erent values of the elastic constants.

Keywords: Computer simulation; Monte Carlo; nematics; topological defects.

PACS Nos.: 61.30.Cz, 61.30.Gd, 61.30.Jf, 62.20.dq.

1. Introduction

Liquid crystals (LC) correspond to states of aggregation of matter intermediate

between liquids and solids.1 In particular, the main characteristic of the materials

forming the nematic LC state is the presence of long-range orientational order

combined with °ow properties similar to those of viscous liquids. The liquid crystals

molecules have in general an anisotropic shape and in the most common of the

compounds, called calamitic, have a rod-like form with one of the axes much longer

(typically at least three times) than the other two. In the nematic phase, these

elongated molecules have, on average, a preferred direction, denoted by n, known as

director. In many practical cases the director orientation will depend on the position

International Journal of Modern Physics C

Vol. 22, No. 5 (2011) 505�516

#.c World Scienti¯c Publishing CompanyDOI: 10.1142/S0129183111016403

505

inside the sample because of boundary conditions or external ¯elds and consequently

the free energy of the system will change with respect to that of a uniform state.

Elastic continuum theories, essentially based on the Frank free energy density1�3

which describes the distortions of the LC are normally appropriate to describe the

spatial variations of the director. To obtain a mathematical expression for the free

energy, it is possible to write it as a function of the deformations tensor ni;j (which

means the derivative of ni with respect to xj). If the order parameter is uniform, and

the deformations are small, the free energy can be written as a power series in ni;j ,

neglecting terms of order higher than 2. By taking in account the uniaxial nematic

phase symmetry around the director and with respect to inversion (n � �n) and by

organizing the remaining terms, it is possible to obtain2,3

fEL ¼ 1

2K1ðr � nÞ2 þ 1

2K2ðn � r � nÞ2 þ 1

2K3ðn�r� nÞ2; ð1Þ

where K1, K2 and K3 are known as splay, twist and bend elastic constants

respectively.

Another, somewhat complementary, approach to investigating the behavior of

these materials is based on performing computer simulations of microscopic LC

models.4 Monte Carlo simulations of lattice spin models, where a unit vector is taken

to represent a short range ordered cluster of molecules, provide a useful intermediate

step between the atomistic5 and molecular resolution6 approaches and macroscopic

level continuum theories2,3 and have proved very useful in investigating con¯ned

nematic systems.7 The simplicity of the models and the large number of particles that

can be treated on a lattice permit also to simulate observables which can be directly

monitored by experiments like, for example, NMR lineshapes and polarized

microscopy optical images.4,7 In particular, some of us have simulated a number of

di®erent systems ranging from thin ¯lms,10,11 droplets,12 simple twisted nematic

displays13 obtaining not only thermodynamics properties, but simulating optical

textures which for a su±cient number of pixels, satisfactorily reproduce the ones

obtained experimentally. The prototype model for this kind of investigations is the

Lebwohl�Lasher (LL) one8,9 which was the ¯rst successful lattice model put forward

to simulate the orientational properties of a nematic. One of the limits of the model

is that it corresponds to a case where the three elastic constants are equal

(K1 ¼ K2 ¼ K3) as the LL potential contains only one, scalar, interaction energy

term. We have shown that even in this one-constant approximation, very often used

also in theoretical analysis,2 some of the characteristics of nematics, like the creation

of topological defects in con¯ned systems, can be reproduced.10 However, while for

low molar mass liquid crystals the di®erences between the Ki constants are normally

low, the need for accounting in a simple way for di®erences in elastic constants is

particularly important for LC polymers,14,15 in LCs originated from long virus like

Tobacco Mosaic Virus (TMV)16 nanotubes17 or other nanocrystal suspensions18 or

even more simply for nematics approaching a smectic phase, where the bend elastic

constant is expected to diverge.1 Given the importance of relating elastic constants to

506 C. Chiccoli et al.

textures various numerical treatments have been put forward, particularly by

Windle and coworkers19 and Killian, Hess et al.20,21 To go beyond the one elastic

constant limit in the lattice Monte Carlo approach, it is convenient to employ a

pseudopair potential introduced by Gruhn and Hess22 and parametrized for simu-

lations by Romano and Luckhurst.23�25 This new potential directly depends on

elastic constants K1;K2;K3 and thus it can be very useful for investigating the e®ect

of the elastic anisotropy,26�28 e.g. in con¯ned systems. We have already applied the

Gruhn�Hess�Romano�Luckhurst (GHRL) pseudopotential coupled with Monte

Carlo simulations to study thin uniaxial ¯lms with random planar (Schlieren)

conditions.28

In the present work we wish to study a thin hybrid aligned nematic ¯lm (HAN),

i.e. a nematic ¯lm con¯ned between two surfaces with antagonistic (normal and

tangential) boundary conditions29 where the various elastic constant anisotropies

can produce di®erent topological defects in the polarized microscopy textures. In real

experiments a HAN ¯lm can be realized not only between suitably treated con¯ning

plates but also simply by placing a nematic on a isotropic substrate (for example

water or glycerin) which give a random planar alignment while the free surface at the

top induces homeotropic orientations.30

2. The Model Systems

The GHRL potential22,24 consists of a system of interacting centers (\spins") placed

at the sites of a certain regular lattice. The Hamiltonian is written as follows:

UN ¼ 1

2

X

i; j2Fi 6¼j

�ij þ JX

i2Fj2S

�ij ; ð2Þ

where F ; S are the set of particles in the bulk and at the surfaces, respectively,

and the parameter J models the strength of the coupling with the surfaces (assumed

to be the same). The particles interact through the second rank attractive pair

potential:

�jk ¼ ��f�½P2ðajÞ þ P2ðakÞ� þ �½ajakbjk � 1=9�þ �P2ðbjkÞ þ �½P2ðajÞ þ P2ðakÞ�P2ðbjkÞg; ð3Þ

where

� ¼ 1

3�ð2K1 � 3K2 þK3Þ;

� ¼ 3�ðK2 �K1Þ;� ¼ 1

3�ðK1 � 3K2 �K3Þ;

� ¼ 1

3�ðK1 �K3Þ;

Lattice Spin Simulations of Topological Defects in Nematic Films 507

with K1;K2;K3 are the elastic constants and � is a factor with the dimensions of

length. The scalars aj ; ak ; bjk are de¯ned as follows:

aj ¼ uj � s; ak ¼ uk � s; bjk ¼ uj �uk;

where s¼ r=jrj, r¼ xj �xk, with xj, xk being dimensionless coordinates of the jth

and kth lattice points; uj, uk are unit vectors along the axis of the two particles

(\spins") and P2 is a second rank Legendre polynomial. The spins represent a cluster

of neighboring molecules whose short range order is assumed to be maintained

through the temperature range examined.4 The one constant approximation case, i.e.

� ¼ � ¼ � ¼ 0, reduces Eq. (3) to the well known LL potential which correctly

reproduces the orientational order characteristics of a nematic isotropic phase

transition.4 The bulk Nematic-Isotropic (NI) transition for the LL model occurs at a

reduced temperature9 T � � kT=� ¼ 1:1232. While in simulating bulk systems4 per-

iodic boundary conditions are employed, in the case of con¯nement the boundaries

are implemented by considering additional layers of particles, kept ¯xed during the

simulation, with suitable orientations chosen to mimic the desired surface align-

ment.4 Here we present an investigation of uniaxial nematic ¯lms with hybrid con-

ditions, homeotropic at the top and random planar at the bottom, already simulated

in the one-constant approximation.10 The starting con¯gurations of the lattice

were chosen to be completely aligned along the z direction and the evolution of the

system was followed according to the classic Metropolis Monte Carlo procedure.31

Polarizing microscope textures were simulated by means of a Müller matrix

approach,32 assuming the molecular domains represented by the spins to act as

retarders on the light propagating through the sample.33 The following parameters

were employed for computing the optical textures: ¯lm thickness d ¼ 5:3�m,

ordinary and extraordinary refractive indices no ¼ 1:5 and ne ¼ 1:66, and light

wavelength �0 ¼ 545 nm.

3. Simulations and Results

As mentioned before, HAN ¯lm has been simulated some years ago10 in the one-

constant approximation using the LL model. We found that the competition between

the alignments induced by the two surfaces is su±cient to create a stable point defect

when the lateral size of the system is much larger than the thickness, as can be seen in

Fig. 1. It is possible to observe strong and stable horizontal deformations associated

with topological defects. The defects are of strength m ¼ �1, i.e. the director ¯eld

undergoes a 2� rotation as one goes once around the defect core. The absolute value

jmj ¼ 1 seems to be the lowest possible topological charge of a defect in a HAN ¯lm

when K1 ¼ K2 ¼ K3. Here we have investigated the e®ect of changing the par-

ameters depending on the elastic constants in the GHRL potential.

At ¯rst we have considered the values of the elastic constants of p-zoxyanisole

(PAA) at 120�C as reported in the book of de Gennes and Prost1 and used by

Romano.23 The resulting textures are shown in Fig. 2.

508 C. Chiccoli et al.

Starting from these values we have then modi¯ed the relative strengths of K1;K2

and K3 to observe the e®ects on the textures. The resulting optical textures are

shown in Fig. 3. It is possible to observe that when the splay (K1) and bend (K3)

elastic constants are approximately similar and much larger than the twist one (K2)

there is the appearance also of half-integer strength defects (see Fig. 3, ¯rst row).

When (K1) is much larger than the other two, no point defects seems to appear

(Fig. 3, second row), while they are present when the bend deformation overcomes

the other two (Fig. 3, third row). To perform a more detailed analysis we have then

simulated various test cases varying K �i from 1 to 9 as shown in Table 1 where the

correspondent values of �; �; � and � are also reported.

The results of these test (Figs. 4 and 5) cases seem to con¯rm the observation

that:

(i) when the twist elastic constant (K2) is larger that the splay and bend ones no

point defects are observed in the textures of the HAN ¯lm;

(ii) when the twist elastic constant (K2) is much smaller than the other two of

similar values also defects with two brushes can appear;

(iii) if K1 or K3 are greater than the other two only defects with four brushes are

produced;

Fig. 2. Simulated optical patterns for a hybrid nematic ¯lm as obtained from a Monte Carlo simulation

of a GHRL potential for values of the three elastic constants corresponding to PAA, i.e. K1 ¼ 7� 10�12 N,K2 ¼ 4:3� 10�12 N, K3 ¼ 17� 10�12 N used by Romano and taken from the book by de Gennes and

Prost. The images are taken after 1000, 2000, 5000 and 10 000 MC cycles.

Fig. 1. Simulated polarized microscopy images of a LL (K1 ¼ K2 ¼ K3) uniaxial hybrid nematic ¯lm asobtained from Monte Carlo con¯gurations. The images are taken after 1000, 2000, 5000 and 10 000 MC

cycles with the sample between crossed polarizers. The system size is 100� 100� 12, the reduced tem-

perature is T� ¼ 0:4 and the anchoring coupling with the surfaces are J ¼ 1.

Lattice Spin Simulations of Topological Defects in Nematic Films 509

(iv) if K1 is much greater than the other two no point defects are observed;

(v) if K3 is much greater than the other two still defects with four brushes are

present.

Examining the pseudopotential in Eq. (3) it is apparent that it depends linearly

on Ki, so that a scaling factor � for which Ki ¼ �K 0i can be absorbed in �. This in turn

Fig. 3. Simulated optical patterns for a hybrid nematic ¯lm as obtained from a Monte Carlo simulation of

a GHRL potential for di®erent values of the three elastic constants. Here K �i ¼ Ki=10

�12 N.

Table 1. Values of elastic constants used in the simulations, K �i ¼ Ki � 1012 N and the

correspondent values of the parameters which appear in the potential. The values of �, � and �

are normalized to have � ¼ 1.

K �1 K �

2 K �3 � � � �

3 1 1 −4.0000 18.0000 1.0000 −2.00001 3 1 0.6667 −2.0000 1.0000 0.0000

1 1 3 −0.4000 0.0000 1.0000 0.40009 1 1 3.2000 −14.4000 1.0000 1.6000

1 9 1 0.8889 −2.6667 1.0000 0.0000

1 1 9 −0.7273 0.0000 1.0000 0.7273

3 3 1 0.2857 0.0000 1.0000 −0.28573 1 3 −2.0000 6.0000 1.0000 0.0000

1 3 3 0.3636 −1.6364 1.0000 0.1818

9 9 1 0.4211 0.0000 1.0000 −0.42119 1 9 −8.0000 24.0000 1.0000 0.0000

1 9 9 0.4571 −2.0571 1.0000 0.2286

510 C. Chiccoli et al.

means that the temperature as appearing in the Monte Carlo Boltzmann average

becomes T � ¼ kT=� ¼ �kT=ð��Þ ¼ T 0�. In other words, when all the elastic con-

stants are scaled by the same amount, the textures correspond to a similar system

with a di®erent ordering. For this reason even though the textures are to some extent

invariant to a scaling factor such as �, we give explicit values for all three Ki in the

¯gures. Apart from the ideal case values studied we have also performed simulations

for experimental elastic constant data taken from literature15,34 for pentylcyanobi-

phenil (5CB), 4-methoxybenzylidene-4′-n-butylaniline (MBBA) and TMV (Fig. 6).

Fig. 4. As in Fig. 3 for various test values of the elastic constants.

Lattice Spin Simulations of Topological Defects in Nematic Films 511

It is interesting to try to compare these observations with the results expected

from continuum elastic theory.10 To do this, a very useful and well known

approach for treating qualitatively the defect stability from the elastic point of

view consists in proposing a con¯guration for the defect and to compute its energy.

This energy is then compared with the one corresponding to the con¯guration

in the absence of a defect.10,30 This con¯guration is obtained by supposing

that the distortion in the plane perpendicular to the plates is not coupled with

the distortion in the perpendicular plane. In this approach, the director can be

Fig. 5. As in Fig. 4 for other test values of the elastic constants.

512 C. Chiccoli et al.

written as

n ¼ cos½m�� sin½�=2ð1� z=dÞ�iþ sin½m�� sin½�=2ð1� z=dÞ�jþ cos½�=2ð1� z=dÞ�k;

in which m is the strength of the defect. This con¯guration guarantees that for

z ¼ 0 the director lies in the polar plane and is perpendicular to the plate at z ¼ d.

To analyze the defect stability, we compute the energy by direct integration of the

Eq. (1) in a cylindrical region with radius R and height d, where an internal

Fig. 6. Simulated optical patterns for a nematic ¯lm in a hybrid geometry as obtained from a MonteCarlo simulation of a GHRL potential for di®erent values of the three elastic constants as taken from

experimental results. The values correspond to 5CB and MBBA from Ref. 34 (¯rst and second row,

respectively), MBBA and TMV from Ref. 25 (third and fourth row, respectively). The images are taken

after 2000, 5000 and 10 000 MC cycles with the sample between crossed polarizers. The system size is100� 100� 12, the reduced temperature is T� ¼ 0:4 and the anchoring coupling with the surfaces is

J ¼ 1:0. The values of the elastic constants Ki ¼ K �i � 10�12 N taken into account are reported in the ¯rst

three columns.

Lattice Spin Simulations of Topological Defects in Nematic Films 513

concentric cylinder with radius r, in which is contained the defect core, is removed to

avoid divergences. For the cases m ¼ 0; 1=2; 1, the energy gm can be written as

g0 ¼ � �3ðK1 þK3Þðr 2 � R2Þ16d

; ð4Þ

g1=2 ¼�d 2ð4K1 þK2 þ 3K3Þ ln½R=r� þ 4�3ðK1 þK3ÞðR2 � r 2Þ

64d; ð5Þ

g1 ¼�f8d 2K1 ln½R=r� � �ðr � RÞ½�ðK1 þK3Þðr þ RÞ � 8dK1�g

16d: ð6Þ

When g1 is compared with g0, as reported in Ref. 10, we conclude that the case

with m ¼ 1 is energetically more stable when R d, for any physical value of the

constants. Note that in both g0 and g1, the twist term is absent. From the comparison

of g1=2 with g1 and g0, we deduce that there are no physically meaningful values of

the constants able to make this con¯guration less energetic than the other two.

Moreover, if g�1 and g�1=2 are computed, the energy represented by g1 is also the

lowest one. This seems to suggest that, for all values of the elastic constants, only a

defect corresponding to m ¼ 1 should be found in the HAN cell. The result is in

agreement with the one reported in Ref. 30, which a±rms that the defect with half-

integer strength is forbidden for this kind of boundary conditions. However, these

conclusions are in con°ict with our ¯ndings, which clearly show that defects with

m ¼ 1 and m ¼ 1=2, as well as no defects, can be found in the HAN cell for appro-

priate values of the elastic constants. The con°ict can be due in part to the

assumption of small distortions characterizing the Frank approximation for the

elastic energy. For thin HAN cells the distortions are hardly small and corrections to

the elastic free energy density may be unavoidable. Moreover, the e®ects of a non-

uniform order parameter should eventually be taken into account in a more realistic

approach. Anyway, the analytical treatment involves a more sophisticated math-

ematical problem, and Monte Carlo simulations seem to contemplate many features

of the elastic anisotropy and can be surely considered a more useful tool in tackling

this kind of problems. Nevertheless, the mathematical problem involving HAN

con¯gurations is under investigation in the framework of the continuum theory and

the results will be published elsewhere.

4. Conclusions

We have performed a detailed simulation study of a nematic ¯lm with hybrid

boundary conditions (random planar on one surface and homeotropic on the other)

by using a simple GHRL pseudopotential which takes into account the elastic ani-

sotropy of the liquid crystal. We have considered di®erent combinations of the splay,

twist and bend elastic constants to verify their relative importance on the formation

of various optical patterns. The results provide in some cases a challenge to the

standard elastic continuum type treatment and point to the need of their general-

ization, e.g. to allow for nonuniform order parameters across the ¯lm.

514 C. Chiccoli et al.

Acknowledgments

LRE and RTS are grateful to Brazilian agencies CAPES, CNPq and INCT-FCx, CC

and PP acknowledge support by INFN Grant No. I.S. BO62.

References

1. P. G. De Gennes and J. Prost, The Physics of Liquid Crystals, 2nd edn. (Oxford Univ.Press, Oxford, 1995).

2. G. Barbero and L. R. Evangelista, An Elementary Course on the Continuum Theory forNematic Liquid Crystals (World Scienti¯c, Singapore, 2000).

3. I. W. Stewart, The Static and Dynamical Continuum Theory of Liquid Crystals (Taylorand Francis, London, 2004).

4. P. Pasini and C. Zannoni (eds.) Advances in the Computer Simulations of Liquid Crystals(Kluwer, Dordrecht, 2000).

5. G. Tiberio, L. Muccioli, R. Berardi and C. Zannoni, Chem. Phys. Chem. 10, 125 (2009).6. C. Zannoni, J. Mat. Chem. 11, 2637 (2001).7. C. Chiccoli, I. Feruli, P. Pasini and C. Zannoni, in Defects in Liquid Crystals: Computer

Simulations, Theory and Experiments, eds. O. D. Lavrentovich, P. Pasini, C. Zannoniand S. Žumer (Kluwer, Dordrecht, 2001) Ch. 4.

8. P. A. Lebwohl and G. Lasher, Phys. Rev. A 6, 426 (1972).9. U. Fabbri and C. Zannoni, Mol. Phys. 58, 763 (1986).10. C. Chiccoli, O. D. Lavrentovich, P. Pasini and C. Zannoni, Phys. Rev. Lett. 79, 4401

(1997).11. C. Chiccoli, P. Pasini, I. Feruli and C. Zannoni, Mol. Cryst. Liq. Cryst. 398, 195 (2003).12. E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini and F. Semeria, Chem. Phys. Lett. 197,

224 (1992).13. E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini and F. Semeria, Int. J. Mod. Phys. C 6,

135 (1995).14. M. Kleman, in Liquid Crystallinity in Polymers. Principles and Fundamental Properties,

ed. A. Ciferri (VCH, New York, 1991), p. 365.15. W. H. Song, H. J. Tu, G. Goldbeck-Wood and A. H. Windle, Liq. Cryst. 30, 775 (2003).16. A. J. Hurd, S. Fraden, F. Lonberg and R. B. Meyer, J. de Physique 46, 905 (1985).17. W. H. Song, I. A. Kinloch and A. H. Windle, Science 302, 1363 (2003).18. L. S. Li, J. Walda, L. Manna and A. P. Alivisatos, Nano Lett. 2, 557 (2002).19. J. Hobdell and A. H. Windle, Liq. Cryst. 19, 401 (1995); 23, 157 (1997).20. A. Sonnet, A. Kilian and S. Hess, Phys. Rev. E 52, 718 (1995).21. A. Kilian and S. Hess, Z. Naturforsch. 44a, 693 (1989).22. T. Gruhn and S. Hess, Z. Naturforsch. A51, 1 (1996).23. S. Romano, Int. J. Mod. Phys. B 12, 2305 (1998); S. Romano, Phys. Lett. A 302, 203

(2002).24. G. R. Luckhurst and S. Romano, Liq. Cryst. 26, 871 (1999).25. P. J. Le Masurier, G. R. Luckhurst and G. Saielli, Liq. Cryst. 28, 769 (2001).26. R. K. Goyal and M. A. Denn, Phys. Rev. E 78, 021706 (2008).27. Z. D. Zhang and Y. J. Zhang, Phys. Lett. A 372, 498 (2008).28. C. Chiccoli, P. Pasini and C. Zannoni, Mol. Cryst. Liq. Cryst. 516, 1 (2010).29. C. Chiccoli, P. Pasini, A. Sarlah, C. Zannoni and S. Žumer, Phys. Rev. E 67, 050703

(2003).30. O. D. Lavrentovich and V. M. Pergamenschik, Int. J. Mod. Phys. B 9, 2389 (1995).

Lattice Spin Simulations of Topological Defects in Nematic Films 515

31. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, J. Chem.Phys. 21, 1087 (1953).

32. A. Killian, Liq. Cryst. 14, 1189 (1993).33. R. Ondris-Crawford, E. P. Boyko, B. G. Wagner, J. H. Erdmann, S. Žumer and J. W.

Doane, J. Appl. Phys. 69, 6380 (1991).34. D. A. Dunmur, in Physical Properties of Liquid Crystals: Nematics, eds. D. A. Dunmur,

A. Fukuda and G. R. Luckhurst (INSPEC, IEE, London, 2001), p. 216.

516 C. Chiccoli et al.