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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 23 — Nov. 8, 2010
  • pp: 23646–23656
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Monte Carlo simulation of the molecular distribution and optical properties of a nematic liquid crystal system with periodic surface gratings

C. Berlic and V. Barna  »View Author Affiliations


Optics Express, Vol. 18, Issue 23, pp. 23646-23656 (2010)
http://dx.doi.org/10.1364/OE.18.023646


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Abstract

We report Monte Carlo simulations based on the Lebwohl-Lasher model for characterizing the molecular director configuration in a nematic liquid crystal cell presenting periodical boundary anchoring conditions. We demonstrate the molecular orientation and spatial behaviour, while profiling the local order parameter distribution for the proposed confining geometry, as well as the boundary and interface interaction fields propagation through the namatic bulk for various temperatures in the proximity of the nematic-isotropic transition. Simulations were also performed concerning with the light passing through the planar and homeotropic periodical regions of the nematic cell and a mapping of the transmitted intensity was obtained for several ambient temperatures. The boundary constraints and the selected periodical geometry of the simulated system play an extremely important role for the demonstrated optical and orientational properties of the liquid crystalline material.

© 2010 OSA

1. Introduction

Liquid crystals represent a state of matter whose characteristics are in-between those of an isotropic liquid and those of a solid crystal. They have properties such as the flowing characteristic of liquids, as well as the anisotropy of optical, electrical and magnetic phenomena characteristic to the crystalline state, which are due to the partial ordering of molecules [1

1. P. G. de Gennes, and J. Prost, The Physics of Liquid Crystals (Oxford University Press, 1995).

3

3. P. Yeh, and C. Gu, Optics of Liquid Crystal Displays (Wiley, 2009).

].

It is well known that the main application of liquid crystals is that of display devices, due to the possibility of manipulation of the molecular director via magnetic and electrical fields, as well as by anchoring to the surface of contact [3

3. P. Yeh, and C. Gu, Optics of Liquid Crystal Displays (Wiley, 2009).

5

5. N. Scaramuzza, C. Berlic, E. S. Barna, G. Strangi, V. Barna, and A. Th. Ionescu, “Molecular Simulation of the Free Surface Order in NLC Samples,” J. Phys. Chem. B 108(10), 3207–3210 (2004). [CrossRef]

], either by periodic topologies [6

6. D. W. Berremann, “Solid Surface Shape and the Alignment of an Adjacent Nematic Liquid Crystal,” Phys. Rev. Lett. 28(26), 1683–1686 (1972). [CrossRef]

] or by anisotropic Van der Waals forces [7

7. M. Schadt, K. Schmitt, V. Kozinkov, and V. Chigrinov, “Surface-Induced Parallel Alignment of Liquid Crystals by Linearly Polymerized Photopolymers,” Jpn. J. Appl. Phys. 31(Part 1, No. 7), 2155–2164 (1992). [CrossRef]

].

The orientational order of liquid crystals is possible due to the fact that the interactions between molecules are anisotropic, but this fact makes the task of theoretically describing this class of materials in restrictive geometries much more difficult. Several theories based on the phenomenological treatment of Frank [8

8. F. C. Frank, “On the Theory of Liquid Crystals,” Discuss. Faraday Soc. 25, 19–28 (1958). [CrossRef]

], or on the minimization of the free energy have been proposed, but all of them involve especially complex calculus, making the use of approximations necessary.

Another way of describing the orientational order and the properties of liquid crystals is by using computer simulations [9

9. M. P. Allen, and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, 1989)

12

12. P. Pasini, C. Zannoni, and S. Zumer, Computer Simulations of Liquid Crystals and Polymers (Springer, 2005).

]. Since the advent of the computer technology and programming languages, Monte Carlo and molecular dynamics simulations have become widely used tools in describing properties of polymers and liquid crystals, including optical properties of these classes of materials [12

12. P. Pasini, C. Zannoni, and S. Zumer, Computer Simulations of Liquid Crystals and Polymers (Springer, 2005).

14

14. E. Gatin, D. Alexandreanu, A. Popescu, C. Berlic, and I. Alexandreanu, “Correlations between permeability properties and pore-size distribution of the porous media “hydron” useful as contact lenses,” Phys. Med. XVI, 13–19 (2000).

]. Much effort was also devoted to simulate the behaviour of liquid crystals displays [15

15. E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, and F. Semeria, “A Monte Carlo Simulation of a Twisted Nematic Liquid Crystal Display,” Int. J. Mod. Phys. C 6(1), 135–141 (1995). [CrossRef]

18

18. C. Chiccoli, P. Pasini, A. Sarlah, C. Zannoni, and S. Zumer, “Structures and transitions in thin hybrid nematic films: a Monte Carlo study,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 050703 (2003). [CrossRef] [PubMed]

], liquid crystals in confined geometries [5

5. N. Scaramuzza, C. Berlic, E. S. Barna, G. Strangi, V. Barna, and A. Th. Ionescu, “Molecular Simulation of the Free Surface Order in NLC Samples,” J. Phys. Chem. B 108(10), 3207–3210 (2004). [CrossRef]

,19

19. A. M. Smondyrev and R. A. Pelcovits, “Nematic Structures in Cylindrical Cavities,” Liq. Cryst. 26(2), 235–240 (1999). [CrossRef]

23

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

] and in using numerical methods of computing the light transmission [24

24. D. W. Berreman, “Liquid-Crystal Twist Cell Dynamics with Backflow,” J. Appl. Phys. 46(9), 3746–3751 (1975). [CrossRef]

]. Confined liquid crystals, due to the anisotropy of the material, play a major role in liquid crystal displays, based on the electro-optical effects which make light modulation.

Nowadays, the twisted nematic liquid crystal display, based on the pioneering work of Schadt and Helfrich [25

25. M. Schadt and W. Helfrich, “Voltage-Dependent Optical Activity of a Twisted Nematic Liquid Crystal,” Appl. Phys. Lett. 18(4), 127–128 (1971). [CrossRef]

], is a widespread and relatively well known device. Nevertheless, it presents some drawbacks, such as the viewing characteristics problems due to the asymmetric alignment of the molecules [3

3. P. Yeh, and C. Gu, Optics of Liquid Crystal Displays (Wiley, 2009).

]. One of the solutions for overcoming this issue consists in using multi domain structures in which we have two or more orthogonal surface anchoring directions obtained by treatment of the layer surfaces via rubbing or photo exposure [26

26. M. Schadt, H. Seiberle, and A. Schuster, “Optical Patterning of Multi-Domain Liquid-Crystal Displays with Wide Viewing Angles,” Nature 381(6579), 212–215 (1996). [CrossRef]

]. In order to investigate this behaviour for a liquid crystalline system having two opposite anchoring directions, in our study we simulated an anchoring regime with half homeotropic and half planar boundary conditions, the cell electrodes having parallel striped layers which we denote as gratings.

2. Molecular model and simulation method

The simulation model was the well-known Lebwohl-Lasher model [27

27. P. A. Lebwohl and G. Lasher, “Nematic Liquid Crystal Order – A Monte Carlo Calculation,” Phys. Rev. A 6(1), 426–429 (1972). [CrossRef]

], where the liquid crystal molecules are considered as unit vectors (versors or spins), which occupy fixed positions in the sites of a cubic crystalline lattice. These versors are free to rotate in space and interact with each other through an orientation dependent energy:
Uij=εijP2(θij)
(1)
where εij is a positive constant, ε, for nearest neighbors particles and zero otherwise. P2 is the second rank Legendre polynomial and θij=sisj is the angle between the versors.

An important aspect of the model is, because the energy of interaction is invariant under a uniform rotation of all spins, the bend, splay and twist elastic constants of the liquid crystal are all equal [5

5. N. Scaramuzza, C. Berlic, E. S. Barna, G. Strangi, V. Barna, and A. Th. Ionescu, “Molecular Simulation of the Free Surface Order in NLC Samples,” J. Phys. Chem. B 108(10), 3207–3210 (2004). [CrossRef]

, 19

19. A. M. Smondyrev and R. A. Pelcovits, “Nematic Structures in Cylindrical Cavities,” Liq. Cryst. 26(2), 235–240 (1999). [CrossRef]

].

The fact that the molecules’ centers of mass are arranged in an ordered fashion does not contradict the fact that in a liquid crystal there is no position order of the molecules, because, in a real liquid crystal, the molecules arrange themselves in ordered domains. In fact, each spin represents an ordered domain encompassing many molecules whose centers of mass are disordered distributed [17

17. C. Chiccoli, S. Guzzeti, P. Pasini, and C. Zannoni, “Computer Simulations of Nematic Displays,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 360(1), 119–129 (2001). [CrossRef]

, 22

22. C. Berlic and V. Barna, “Nematic Director Distribution of a Liquid Crystalline System Presenting a Cylindrical Defect,” Journal J. Optoelectron. Adv. Mater. 12, 1427–1432 (2010).

].

Although the system is very simple, it was widely used to describe the properties of the liquid crystals displays [12

12. P. Pasini, C. Zannoni, and S. Zumer, Computer Simulations of Liquid Crystals and Polymers (Springer, 2005).

, 15

15. E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, and F. Semeria, “A Monte Carlo Simulation of a Twisted Nematic Liquid Crystal Display,” Int. J. Mod. Phys. C 6(1), 135–141 (1995). [CrossRef]

18

18. C. Chiccoli, P. Pasini, A. Sarlah, C. Zannoni, and S. Zumer, “Structures and transitions in thin hybrid nematic films: a Monte Carlo study,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 050703 (2003). [CrossRef] [PubMed]

], or nematic structures in various geometries [5

5. N. Scaramuzza, C. Berlic, E. S. Barna, G. Strangi, V. Barna, and A. Th. Ionescu, “Molecular Simulation of the Free Surface Order in NLC Samples,” J. Phys. Chem. B 108(10), 3207–3210 (2004). [CrossRef]

, 12

12. P. Pasini, C. Zannoni, and S. Zumer, Computer Simulations of Liquid Crystals and Polymers (Springer, 2005).

, 19

19. A. M. Smondyrev and R. A. Pelcovits, “Nematic Structures in Cylindrical Cavities,” Liq. Cryst. 26(2), 235–240 (1999). [CrossRef]

23

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

]. The main advantage of this simple model is the fact that spins' centers of mass are fixed, saving computer time during the simulation, comparatively with other models considering also translational degrees of freedom. On the other hand, the system is well studied and gives a realistic representation of a nematic liquid crystal, showing a first order phase transition at scaled temperature TNI*=kTNI/ε=1.1232±0.0006 [15

15. E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, and F. Semeria, “A Monte Carlo Simulation of a Twisted Nematic Liquid Crystal Display,” Int. J. Mod. Phys. C 6(1), 135–141 (1995). [CrossRef]

, 18

18. C. Chiccoli, P. Pasini, A. Sarlah, C. Zannoni, and S. Zumer, “Structures and transitions in thin hybrid nematic films: a Monte Carlo study,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 050703 (2003). [CrossRef] [PubMed]

].

The nematic liquid crystal cell used in the simulation is of rectangular shape, having the dimensions NX×NY×NZ in lattice spacings and electrodes parallel with YOZ plane, situated at z=0 and z=NZ+1 as in Fig. 1
Fig. 1 Schematic representation of the initial state for the simulated liquid crystal system. The nematic liquid crystal molecules are confined in a glass sandwich type cell. A system of coordinates XYZ is assigned.
. For the left half of the cell, 1yNy2, the electrodes impose homeotropic conditions, and for the right one, Ny2+1yNy, planar conditions. In all other directions, we consider periodic boundary conditions.

The spins which are not placed in the immediate vicinity of the walls interact with each other through energy εBand are free to rotate inside their cells following a standard Monte Carlo procedure: a random spin is chosen and the interaction energy with its first neighbors is computed. This spin rotates randomly with a small angle, following the procedure described in [10

10. D. Frenkel, and B. Smit, Understanding Molecular Simulation: From Algorithms to Applications (Academic Press, 2001).

], and then the interaction energy for the new state is computed. This new state is accepted according to the Metropolis criterion [9

9. M. P. Allen, and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, 1989)

13

13. D. P. Landau, and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, 2000).

].

The anchoring of the molecules to the walls is realized by introducing fixed spins [15

15. E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, and F. Semeria, “A Monte Carlo Simulation of a Twisted Nematic Liquid Crystal Display,” Int. J. Mod. Phys. C 6(1), 135–141 (1995). [CrossRef]

23

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

]. These spins cannot rotate, but can interact with free spins through energy εS. In our model, the fixed spins are situated at z=1 and z=NZ.

As it can be seen from Fig. 1, the fixed spins situated in the left half of the cell, with y between 1 and Ny/2, are perpendicular to the electrodes. The fixed spins situated in the right side of the cell, with y between Ny/2+1 and Ny are parallel with the electrodes. Because this cell geometry, there is a competition between tendency of the left spins to align normally to the wall, and the right spins which tend to parallel align.

For such a system with complicated boundary conditions, instead of obtaining the bulk order parameter [1

1. P. G. de Gennes, and J. Prost, The Physics of Liquid Crystals (Oxford University Press, 1995).

, 2

2. S. Chandrasekhar, Liquid Crystals (Cambridge University Press, 1993).

], it is preferably to use the local tensor parameter [5

5. N. Scaramuzza, C. Berlic, E. S. Barna, G. Strangi, V. Barna, and A. Th. Ionescu, “Molecular Simulation of the Free Surface Order in NLC Samples,” J. Phys. Chem. B 108(10), 3207–3210 (2004). [CrossRef]

, 9

9. M. P. Allen, and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, 1989)

, 18

18. C. Chiccoli, P. Pasini, A. Sarlah, C. Zannoni, and S. Zumer, “Structures and transitions in thin hybrid nematic films: a Monte Carlo study,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 050703 (2003). [CrossRef] [PubMed]

, 21

21. C. Berlic, E. Barna, and C. Ciucu, “Monte Carlo Simulation of a Nematic Liquid Crystal Cell with a Hemispheric Defect on One Electrode,” J. Optoelectron. Adv. Mater. 9, 3854–3859 (2007).

23

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

]:
Qαβ=1nk=1n(32skαskβ12δαβ)
(2)
where α,β=x,y,z, δαβ is the Kronecker delta, <…> is the ensemble average and n is the number of spins. If n=N, the usual order parameter of the system is obtained from the largest positive eigenvalue of the order tensor Qαβ and the corresponding eigenvector is the liquid crystal director [9

9. M. P. Allen, and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, 1989)

]. If we choose n=1, the average is performed on Monte Carlo cycles only and Qαβ describes the local order [5

5. N. Scaramuzza, C. Berlic, E. S. Barna, G. Strangi, V. Barna, and A. Th. Ionescu, “Molecular Simulation of the Free Surface Order in NLC Samples,” J. Phys. Chem. B 108(10), 3207–3210 (2004). [CrossRef]

, 21

21. C. Berlic, E. Barna, and C. Ciucu, “Monte Carlo Simulation of a Nematic Liquid Crystal Cell with a Hemispheric Defect on One Electrode,” J. Optoelectron. Adv. Mater. 9, 3854–3859 (2007).

].

From the way in which the tensor order parameter is defined, it is noticeable that it has two important properties [21

21. C. Berlic, E. Barna, and C. Ciucu, “Monte Carlo Simulation of a Nematic Liquid Crystal Cell with a Hemispheric Defect on One Electrode,” J. Optoelectron. Adv. Mater. 9, 3854–3859 (2007).

]: it is a symmetric tensor, Qαβ=Qβα, and it has a zero trace, QXX+QYY+QZZ=0, meaning that only 5 of its 9 components are independent.

The diagonal components of the tensor order parameter represent the degree of order with respect the coordinate axis. The off-diagonal components Qαβ represent the bending of the director field in the corresponding plane αOβ [5

5. N. Scaramuzza, C. Berlic, E. S. Barna, G. Strangi, V. Barna, and A. Th. Ionescu, “Molecular Simulation of the Free Surface Order in NLC Samples,” J. Phys. Chem. B 108(10), 3207–3210 (2004). [CrossRef]

, 18

18. C. Chiccoli, P. Pasini, A. Sarlah, C. Zannoni, and S. Zumer, “Structures and transitions in thin hybrid nematic films: a Monte Carlo study,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 050703 (2003). [CrossRef] [PubMed]

]. The values of the tensor order parameter are between 0.5, the director is perpendicular to the corresponding direction, and 1, when the director in perfectly parallel with the direction [1

1. P. G. de Gennes, and J. Prost, The Physics of Liquid Crystals (Oxford University Press, 1995).

, 18

18. C. Chiccoli, P. Pasini, A. Sarlah, C. Zannoni, and S. Zumer, “Structures and transitions in thin hybrid nematic films: a Monte Carlo study,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 050703 (2003). [CrossRef] [PubMed]

]. Because the components of the tensor are obtained as statistical averages for each cell, their values are between these limits, with the value of 0 meaning the total disorder with respect the direction.

The system described above was used to simulate the behaviour of a liquid crystal cell with boundary order imposed by electrodes. The number of spins used during the simulation was kept at values Nx=16, Ny=60 and Nz=24, which means that we had 23,040 spins. Because spins located at electrodes are fixed and, because we have used periodic conditions along the OX and OY directions, the number of free spins is 21,120 and a Monte Carlo cycle consisted of 21,120 attempted moves.

The Monte Carlo procedure was a standard one: we randomly picked a spin and rotated it. We calculated the energies in the old and in the new state and the move was accepted using the Metropolis acceptance criterion [8

8. F. C. Frank, “On the Theory of Liquid Crystals,” Discuss. Faraday Soc. 25, 19–28 (1958). [CrossRef]

13

13. D. P. Landau, and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, 2000).

]. The mechanism was repeated for 160,000 Monte Carlo cycles with 60,000 of them used for equilibration.

The big numbers of spins and Monte Carlo cycles were chosen in order to avoid finite geometrical effects and to have good statistical averages.

The coupling interaction constant between free spins was εB=1 and the interaction between a free spin and a fixed one was εS=1.5. It is known the fact that a liquid crystal display is commanded as an electro-optical device, using an electric field [15

15. E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, and F. Semeria, “A Monte Carlo Simulation of a Twisted Nematic Liquid Crystal Display,” Int. J. Mod. Phys. C 6(1), 135–141 (1995). [CrossRef]

23

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

]. However, in our work we were interested in only the effects of the boundary conditions on the liquid crystal cell microscopic and optical properties, so we have chosen a somewhat big value for this parameter, meaning a strong anchoring regime.

In order to obtain the optical properties of the simulated liquid crystal display, we used the standard Müeller matrix approach which has been used in other numerical calculations [24

24. D. W. Berreman, “Liquid-Crystal Twist Cell Dynamics with Backflow,” J. Appl. Phys. 46(9), 3746–3751 (1975). [CrossRef]

] and Monte Carlo simulations of optical textures [15

15. E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, and F. Semeria, “A Monte Carlo Simulation of a Twisted Nematic Liquid Crystal Display,” Int. J. Mod. Phys. C 6(1), 135–141 (1995). [CrossRef]

18

18. C. Chiccoli, P. Pasini, A. Sarlah, C. Zannoni, and S. Zumer, “Structures and transitions in thin hybrid nematic films: a Monte Carlo study,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 050703 (2003). [CrossRef] [PubMed]

, 20

20. E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, and F. Semeria, “Computer simulations of nematic droplets with bipolar boundary conditions,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(4), 2929–2939 (1994). [CrossRef] [PubMed]

]. The main assumptions we did were that we may use ray optics and each cell of the simulated liquid crystal is a simple linear retarder described by a Müeller matrix which mainly depends on the orientation of the director, as well as optical properties of the molecule [20

20. E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, and F. Semeria, “Computer simulations of nematic droplets with bipolar boundary conditions,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(4), 2929–2939 (1994). [CrossRef] [PubMed]

,28

28. J. A. Schellman, “Polarization Modulation Spectroscopy”, in Polarized Spectroscopy of Ordered Systems, B. Samori’ and E.W. Thulstrup, eds. (Kluwer, Dordrecht, 1988).

].

Thus, the light beam crossing the liquid crystal display is then retarded by a matrix resulting for a product of Müeller matrix corresponding to each cell. Due to the geometry of our system, we considered the light traveling in the OZ direction. The light modulated by the liquid crystal display is retarded and polarized and it is represented by the Stokes vector [15

15. E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, and F. Semeria, “A Monte Carlo Simulation of a Twisted Nematic Liquid Crystal Display,” Int. J. Mod. Phys. C 6(1), 135–141 (1995). [CrossRef]

18

18. C. Chiccoli, P. Pasini, A. Sarlah, C. Zannoni, and S. Zumer, “Structures and transitions in thin hybrid nematic films: a Monte Carlo study,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 050703 (2003). [CrossRef] [PubMed]

, 20

20. E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, and F. Semeria, “Computer simulations of nematic droplets with bipolar boundary conditions,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(4), 2929–2939 (1994). [CrossRef] [PubMed]

]:
S=POUTi=2Nz1MiPINSIN
(3)
where SIN is the Stokes vector for the incoming unpolarized light, S is the Stokes vector for the light passed through liquid crystal display, Mi is the Müeller matrix of the site i in a column of cell parallel with OZ. As in real liquid crystal optics experiments, the light is observed with the help of crossed polarizers placed on each side of the cell, PIN and POUT.

In the above relation, the product is taken only for spins with z=2 to z=Nz1, because these spins are mobile, the spins standing on the electrode being only an artifact to simulate anchoring.

During the Monte Carlo simulation, beside tensorial order parameter calculation, we performed the product from relation (3) and we recorded the first component of the Stokes vector of the light passed thorough the liquid crystal display, quantity that represents the intensity of the light [20

20. E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, and F. Semeria, “Computer simulations of nematic droplets with bipolar boundary conditions,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(4), 2929–2939 (1994). [CrossRef] [PubMed]

, 28

28. J. A. Schellman, “Polarization Modulation Spectroscopy”, in Polarized Spectroscopy of Ordered Systems, B. Samori’ and E.W. Thulstrup, eds. (Kluwer, Dordrecht, 1988).

].

3. Results and discussions

3.1 Director distribution

Because the periodic boundary condition along OX axes, we have obtained the components of the tensor order parameter by averaging in OX direction.

We performed a first set of Monte Carlo simulations at reduced temperature T*=0.9 which is very deep in the nematic phase. The components of the tensor order parameter in the middle of the cell, z=12, far enough from the electrodes, are depicted in Fig. 2
Fig. 2 Components of the tensor order parameter for Z=12 at T*=9. Error bars sizes are of dimension of the symbols and were omitted. Lines are only guide to the eye.
. From this graph, we may conclude that Qxx has a value close to 0.5, which means that the director does not lies along the OX axis, as it was normal, because the system's boundaries impose direction parallel with OZ in the left side of the cell, and with OY in the right ones. This observation is sustained by the facts that the values for Qxy and Qxz are approximately 0, meaning that we have no rotations of the director in the XOY and XOZ planes.

On the other hand, the Qzz and Qyy have a more interesting behaviour. In the middle of the left side of the cell, Qzz is approximately 0.7, which is an excellent agreement with the value of the bulk order parameter found for the Lebwohl-Lasher model [15

15. E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, and F. Semeria, “A Monte Carlo Simulation of a Twisted Nematic Liquid Crystal Display,” Int. J. Mod. Phys. C 6(1), 135–141 (1995). [CrossRef]

20

20. E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, and F. Semeria, “Computer simulations of nematic droplets with bipolar boundary conditions,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(4), 2929–2939 (1994). [CrossRef] [PubMed]

]. In the same point, the value of Qyy is 0.5. We may thus conclude that, in the left side of the cell, the director is mainly parallel with the OZ axis, as it was imposed by boundary conditions. In the right side of the cell, the situation is absolutely symmetric: in the middle of this region Qyy0.7 and Qzz0.5, meaning that the director is in that case parallel with the OY axis, as it was imposed by boundary conditions for y60.

The rotation of the director take place in the YOZ plane, as it can be noticed from the behaviour of the Qyz, which minimum is reached in the points were Qzz and Qyy have their own maximum points and the maximum is fory=0, where QYY=QZZ. As it is expected, the maximum value for the components of tensor order parameter that describe the rotation is achieved near the central region of the display, where boundary conditions jumps from hometropic to planar.

From the behaviour of tensor order parameter, we noticed that the director rotation take place in the YOZ plane, thus we considered that it is interesting to plot in Fig. 3
Fig. 3(a) (a) Map of QYY in the YOZ plane at T*=0.9.
Fig. 3(b) (b) Map of Qzz in the YOZ plane at T*=0.9.
the complete map of Qyy and Qzz in this plane.

We may notice the left-right and up-down symmetry of order parameters in each half of the cell, symmetry that is imposed by boundary conditions. From these two maps, we may also discern the presence of the transition region situated in the middle of the cell and to margins, because of the periodic conditions. It also worth to mention the left-right mirroring of QYY and QZZ. For example, the behaviour of the QYY in right region of the cell, where we have planar boundary conditions, it is similar with the behavior of QZZ in the left region of the cell, where are homeotropic ones.

In Fig. 4
Fig. 4 Qyy components of the tensor order parameter for z=12 at different temperatures. For T*=1.3 we have QYY=0, meaning that there is no order. Error bars sizes are of dimension of the symbols and were omitted. Lines are only guide to the eye.
, we show the Qyy component as function of y, at different temperatures. Once more, in order to avoid the values for layers too close to the electrodes, we used the layer situated in the middle of the cell, for which z=12.

From Fig. 4, we may notice that the shape of Qyy is maintained, but the order decrease with temperature, as it is expected, because of the increased thermal fluctuations. For T*=1.3, temperature situated in the isotropic phase, Qyy=0, as well the all other components of the tensor order parameter, meaning the absence of the nematic order, as it is expected.

For all the temperatures used for simulation we systematically check if QXX+QYY+QZZ=0 for all the cells in the liquid crystal display, and we found a total agreement with the theory.

3.2 Light traveling simulations

We performed simulation of light traveling through a liquid crystal cell filled with a nematic liquid crystal having ordinary refractive index n0=1.5 and the extraordinary refractive index ne=1.7, similar of the 5CB liquid crystal [13

13. D. P. Landau, and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, 2000).

, 18

18. C. Chiccoli, P. Pasini, A. Sarlah, C. Zannoni, and S. Zumer, “Structures and transitions in thin hybrid nematic films: a Monte Carlo study,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 050703 (2003). [CrossRef] [PubMed]

, 29

29. G. W. Gray, K. J. Harrison, and J. A. Nash, “New Family of Nematic Liquid Crystals for Displays,” Electron. Lett. 9(6), 130–131 (1973). [CrossRef]

].

Because we have considered that the light beam is parallel with the OZ axis, we recorded the intensity of light after it passed the liquid crystal display, obtaining the map of intensities as in Figs. 5a
Fig. 5(a) (a) Map of the intensity of the transmitted light for T*=0.9 and λ=545nm.
Fig. 5(b) (b) Map of the intensity of the transmitted light for T*=1.1 and λ=545nm.
and 5b, where each pixel of the display correspond to a lattice point in the XOY plane. For this set of simulations, we used a wavelength of 545 nm, as in reference [20

20. E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, and F. Semeria, “Computer simulations of nematic droplets with bipolar boundary conditions,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(4), 2929–2939 (1994). [CrossRef] [PubMed]

]. The temperatures used in the simulations where T*=0.9 to T*=1.1. As expected, we notice a sharper transition region corresponding to the lower temperature, due to the higher order degree at T*=0.9. This statement is also sustained by the higher values we found for the order tensor components at this temperature, Fig. 4.

Analyzing the intensity map of the transmitted light, it is easy to see that the intensity is, with good approximation, constant along the OX axis, because of the imposed boundary conditions.

The small values of the intensity of the transmitted light in the region where homeotropic boundary conditions are imposed are easy to understand, because it is known that that under crossed polarizers for normal incidence, the homeotropic texture is always black, which is a characteristic of this texture [30

30. T. Scharf, Polarized Light in Liquid Crystals and Polymers (John Wiley & Sons, Inc., Hoboken, New Jersey, 2007), Chap. 8.

]. The deviation from zero of the intensity may be explained by the presence of the thermal fluctuations, which, distorting the perfect homeotropic alignment, altered the angle of incidence, and we finally have be transmission due to the birefringence at off-normal angles.

For the region with planar boundary conditions, the intensity is somewhat bigger than in the region with homeotropic ones, but also small in absolute value. In [30

30. T. Scharf, Polarized Light in Liquid Crystals and Polymers (John Wiley & Sons, Inc., Hoboken, New Jersey, 2007), Chap. 8.

] is stated that for a nematic slab of thickness d, pretilt angle θ and planar conditions, the transmitted intensity is
I=I02sin2(2φ)sin2[πdλ(nenone2sin2θ+no2cos2θno)]
(4)
where φ is the angle between the projection of the director in the plane perpendicular to the direction of light and the OX axis. Since in our simulation the light was traveling in the OZ direction and the planar structure imposed by the boundary condition is parallel with the OY axis, the φ angle is close to π2. In that situation, sin2(2ϕ) is close to zero and the intensity given by Eq. (4) should be very small, as we actually obtained from Monte Carlo simulations.

Another important feature is that the maps of intensities also change with temperature. In Fig. 6
Fig. 6 Intensity of the transmitted light averaged along the OX axis.
, we plotted the transmitted light averaged along the OX axis.

We observe that the intensity of the transmitted light is always bigger in the region where planar boundary conditions are imposed. As temperature increases, the intensity in the homeotropic region increases, because larger fluctuations mean a higher depart of the director from a perfect hometropic alignment. Same explanation can be given for the region with planar boundary conditions: at small temperatures, the projection of the director is mainly parallel with the OY direction, but with the temperature increasing it deviates from this position. Note that, if the system temperature exceeds the nematic-isotropic transition temperature, the intensity decreases, so that we may expect a more complex behaviour. It is also interesting to see that the homeotropic region is more sensitive to temperature changes.

In all the simulations, we noticed that the transition between hometropic and planar conditions was both smooth in light intensities and tensor order parameter components. From Fig. 6, we may also notice that in the hometropic region, the influence of the temperature is more pronounced than in the planar one.

4. Conclusions

In the present paper, we have investigated a nematic liquid crystal cell having periodic surface gratings by means of Monte Carlo simulations for the well known Lebwohl-Lasher model. We have obtained a complete mapping for the tensor order parameter components of the liquid crystal at various temperatures below the isotropic-nematic transition temperature. We found that for a relatively strong anchoring regime, the order imposed by the gratings situated on the electrodes propagates deep in the nematic bulk resulting in a complex behaviour for the components of the molecular tensor. The order level of the nematic liquid crystal has a strong influence on the optical properties as well. By using the Müeller matrix approach during the simulations, we acquired the optical textures of the nematic system, while recording the intensity of the light transmitted through the cell at normal incidence. Because we have not used an electric or magnetic field to orientate the molecules, the intensity was fairly small, whereas all the optical effects are the result of the interplay between thermal fluctuations and propagation in the bulk of the order imposed by the complex boundary conditions. We also found that the system temperature plays an extremely important role in tailoring the optical characteristics of the simulated cell. We also believe that the investigated model is useful not only in understanding the operation mode of liquid crystal cells with complex geometries but also in optimizing the design for novel LCD displays.

Acknowledgements

This research work was supported by CNCSIS – UEFISCSU PNII grant “IDEI”, no. 450-1902 /2008.

References and links

1.

P. G. de Gennes, and J. Prost, The Physics of Liquid Crystals (Oxford University Press, 1995).

2.

S. Chandrasekhar, Liquid Crystals (Cambridge University Press, 1993).

3.

P. Yeh, and C. Gu, Optics of Liquid Crystal Displays (Wiley, 2009).

4.

A. L. Alexe-Ionescu, A. Th. Ionescu, E. S. Barna, N. Scaramuzza, and G. Strangi, “Role of Surface Order on the Total Electric Conduction in NLC Samples,” J. Phys. Chem. B 107(23), 5487–5490 (2003). [CrossRef]

5.

N. Scaramuzza, C. Berlic, E. S. Barna, G. Strangi, V. Barna, and A. Th. Ionescu, “Molecular Simulation of the Free Surface Order in NLC Samples,” J. Phys. Chem. B 108(10), 3207–3210 (2004). [CrossRef]

6.

D. W. Berremann, “Solid Surface Shape and the Alignment of an Adjacent Nematic Liquid Crystal,” Phys. Rev. Lett. 28(26), 1683–1686 (1972). [CrossRef]

7.

M. Schadt, K. Schmitt, V. Kozinkov, and V. Chigrinov, “Surface-Induced Parallel Alignment of Liquid Crystals by Linearly Polymerized Photopolymers,” Jpn. J. Appl. Phys. 31(Part 1, No. 7), 2155–2164 (1992). [CrossRef]

8.

F. C. Frank, “On the Theory of Liquid Crystals,” Discuss. Faraday Soc. 25, 19–28 (1958). [CrossRef]

9.

M. P. Allen, and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, 1989)

10.

D. Frenkel, and B. Smit, Understanding Molecular Simulation: From Algorithms to Applications (Academic Press, 2001).

11.

M. E. J. Newman, and G. T. Barkema, Monte Carlo Methods in Statistical Physics (Oxford University Press, 1999)

12.

P. Pasini, C. Zannoni, and S. Zumer, Computer Simulations of Liquid Crystals and Polymers (Springer, 2005).

13.

D. P. Landau, and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, 2000).

14.

E. Gatin, D. Alexandreanu, A. Popescu, C. Berlic, and I. Alexandreanu, “Correlations between permeability properties and pore-size distribution of the porous media “hydron” useful as contact lenses,” Phys. Med. XVI, 13–19 (2000).

15.

E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, and F. Semeria, “A Monte Carlo Simulation of a Twisted Nematic Liquid Crystal Display,” Int. J. Mod. Phys. C 6(1), 135–141 (1995). [CrossRef]

16.

C. Chiccoli, P. Pasini, S. Guzzeti, and C. Zannoni, “A Monte Carlo Simulation of In-Plane Switching Liquid Crystal Display,” Int. J. Mod. Phys. C 9(3), 409–419 (1998). [CrossRef]

17.

C. Chiccoli, S. Guzzeti, P. Pasini, and C. Zannoni, “Computer Simulations of Nematic Displays,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 360(1), 119–129 (2001). [CrossRef]

18.

C. Chiccoli, P. Pasini, A. Sarlah, C. Zannoni, and S. Zumer, “Structures and transitions in thin hybrid nematic films: a Monte Carlo study,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 050703 (2003). [CrossRef] [PubMed]

19.

A. M. Smondyrev and R. A. Pelcovits, “Nematic Structures in Cylindrical Cavities,” Liq. Cryst. 26(2), 235–240 (1999). [CrossRef]

20.

E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, and F. Semeria, “Computer simulations of nematic droplets with bipolar boundary conditions,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(4), 2929–2939 (1994). [CrossRef] [PubMed]

21.

C. Berlic, E. Barna, and C. Ciucu, “Monte Carlo Simulation of a Nematic Liquid Crystal Cell with a Hemispheric Defect on One Electrode,” J. Optoelectron. Adv. Mater. 9, 3854–3859 (2007).

22.

C. Berlic and V. Barna, “Nematic Director Distribution of a Liquid Crystalline System Presenting a Cylindrical Defect,” Journal J. Optoelectron. Adv. Mater. 12, 1427–1432 (2010).

23.

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

24.

D. W. Berreman, “Liquid-Crystal Twist Cell Dynamics with Backflow,” J. Appl. Phys. 46(9), 3746–3751 (1975). [CrossRef]

25.

M. Schadt and W. Helfrich, “Voltage-Dependent Optical Activity of a Twisted Nematic Liquid Crystal,” Appl. Phys. Lett. 18(4), 127–128 (1971). [CrossRef]

26.

M. Schadt, H. Seiberle, and A. Schuster, “Optical Patterning of Multi-Domain Liquid-Crystal Displays with Wide Viewing Angles,” Nature 381(6579), 212–215 (1996). [CrossRef]

27.

P. A. Lebwohl and G. Lasher, “Nematic Liquid Crystal Order – A Monte Carlo Calculation,” Phys. Rev. A 6(1), 426–429 (1972). [CrossRef]

28.

J. A. Schellman, “Polarization Modulation Spectroscopy”, in Polarized Spectroscopy of Ordered Systems, B. Samori’ and E.W. Thulstrup, eds. (Kluwer, Dordrecht, 1988).

29.

G. W. Gray, K. J. Harrison, and J. A. Nash, “New Family of Nematic Liquid Crystals for Displays,” Electron. Lett. 9(6), 130–131 (1973). [CrossRef]

30.

T. Scharf, Polarized Light in Liquid Crystals and Polymers (John Wiley & Sons, Inc., Hoboken, New Jersey, 2007), Chap. 8.

OCIS Codes
(160.1190) Materials : Anisotropic optical materials
(160.3710) Materials : Liquid crystals
(160.4760) Materials : Optical properties
(230.3720) Optical devices : Liquid-crystal devices
(260.1440) Physical optics : Birefringence
(350.2770) Other areas of optics : Gratings

ToC Category:
Optical Devices

History
Original Manuscript: July 13, 2010
Revised Manuscript: September 28, 2010
Manuscript Accepted: October 13, 2010
Published: October 27, 2010

Citation
Berlic C. and Barna V., "Monte Carlo simulation of the molecular distribution and optical properties of a nematic liquid crystal system with periodic surface gratings," Opt. Express 18, 23646-23656 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-23-23646


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References

  1. P. G. de Gennes, and J. Prost, The Physics of Liquid Crystals (Oxford University Press, 1995).
  2. S. Chandrasekhar, Liquid Crystals (Cambridge University Press, 1993).
  3. P. Yeh, and C. Gu, Optics of Liquid Crystal Displays (Wiley, 2009).
  4. A. L. Alexe-Ionescu, A. Th. Ionescu, E. S. Barna, N. Scaramuzza, and G. Strangi, “Role of Surface Order on the Total Electric Conduction in NLC Samples,” J. Phys. Chem. B 107(23), 5487–5490 (2003). [CrossRef]
  5. N. Scaramuzza, C. Berlic, E. S. Barna, G. Strangi, V. Barna, and A. Th. Ionescu, “Molecular Simulation of the Free Surface Order in NLC Samples,” J. Phys. Chem. B 108(10), 3207–3210 (2004). [CrossRef]
  6. D. W. Berremann, “Solid Surface Shape and the Alignment of an Adjacent Nematic Liquid Crystal,” Phys. Rev. Lett. 28(26), 1683–1686 (1972). [CrossRef]
  7. M. Schadt, K. Schmitt, V. Kozinkov, and V. Chigrinov, “Surface-Induced Parallel Alignment of Liquid Crystals by Linearly Polymerized Photopolymers,” Jpn. J. Appl. Phys. 31(Part 1, No. 7), 2155–2164 (1992). [CrossRef]
  8. F. C. Frank, “On the Theory of Liquid Crystals,” Discuss. Faraday Soc. 25, 19–28 (1958). [CrossRef]
  9. M. P. Allen, and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, 1989)
  10. D. Frenkel, and B. Smit, Understanding Molecular Simulation: From Algorithms to Applications (Academic Press, 2001).
  11. M. E. J. Newman, and G. T. Barkema, Monte Carlo Methods in Statistical Physics (Oxford University Press, 1999)
  12. P. Pasini, C. Zannoni, and S. Zumer, Computer Simulations of Liquid Crystals and Polymers (Springer, 2005).
  13. D. P. Landau, and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, 2000).
  14. E. Gatin, D. Alexandreanu, A. Popescu, C. Berlic, and I. Alexandreanu, “Correlations between permeability properties and pore-size distribution of the porous media “hydron” useful as contact lenses,” Phys. Med. XVI, 13–19 (2000).
  15. E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, and F. Semeria, “A Monte Carlo Simulation of a Twisted Nematic Liquid Crystal Display,” Int. J. Mod. Phys. C 6(1), 135–141 (1995). [CrossRef]
  16. C. Chiccoli, P. Pasini, S. Guzzeti, and C. Zannoni, “A Monte Carlo Simulation of In-Plane Switching Liquid Crystal Display,” Int. J. Mod. Phys. C 9(3), 409–419 (1998). [CrossRef]
  17. C. Chiccoli, S. Guzzeti, P. Pasini, and C. Zannoni, “Computer Simulations of Nematic Displays,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 360(1), 119–129 (2001). [CrossRef]
  18. C. Chiccoli, P. Pasini, A. Sarlah, C. Zannoni, and S. Zumer, “Structures and transitions in thin hybrid nematic films: a Monte Carlo study,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 050703 (2003). [CrossRef] [PubMed]
  19. A. M. Smondyrev and R. A. Pelcovits, “Nematic Structures in Cylindrical Cavities,” Liq. Cryst. 26(2), 235–240 (1999). [CrossRef]
  20. E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, and F. Semeria, “Computer simulations of nematic droplets with bipolar boundary conditions,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(4), 2929–2939 (1994). [CrossRef] [PubMed]
  21. C. Berlic, E. Barna, and C. Ciucu, “Monte Carlo Simulation of a Nematic Liquid Crystal Cell with a Hemispheric Defect on One Electrode,” J. Optoelectron. Adv. Mater. 9, 3854–3859 (2007).
  22. C. Berlic and V. Barna, “Nematic Director Distribution of a Liquid Crystalline System Presenting a Cylindrical Defect,” Journal J. Optoelectron. Adv. Mater. 12, 1427–1432 (2010).
  23. P. Pasini, and C. Zannoni, eds., Advances in the Computer Simulations of Liquid Crystals (Kluver, Dordrecht, 2000).
  24. D. W. Berreman, “Liquid-Crystal Twist Cell Dynamics with Backflow,” J. Appl. Phys. 46(9), 3746–3751 (1975). [CrossRef]
  25. M. Schadt and W. Helfrich, “Voltage-Dependent Optical Activity of a Twisted Nematic Liquid Crystal,” Appl. Phys. Lett. 18(4), 127–128 (1971). [CrossRef]
  26. M. Schadt, H. Seiberle, and A. Schuster, “Optical Patterning of Multi-Domain Liquid-Crystal Displays with Wide Viewing Angles,” Nature 381(6579), 212–215 (1996). [CrossRef]
  27. P. A. Lebwohl and G. Lasher, “Nematic Liquid Crystal Order – A Monte Carlo Calculation,” Phys. Rev. A 6(1), 426–429 (1972). [CrossRef]
  28. J. A. Schellman, “Polarization Modulation Spectroscopy”, in Polarized Spectroscopy of Ordered Systems, B. Samori’ and E.W. Thulstrup, eds. (Kluwer, Dordrecht, 1988).
  29. G. W. Gray, K. J. Harrison, and J. A. Nash, “New Family of Nematic Liquid Crystals for Displays,” Electron. Lett. 9(6), 130–131 (1973). [CrossRef]
  30. T. Scharf, Polarized Light in Liquid Crystals and Polymers (John Wiley & Sons, Inc., Hoboken, New Jersey, 2007), Chap. 8.

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