Mukherjee, P K

Nandi, B

Saha, J

Saha, M

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Indian J. Phys. 69 A (1), 121 -  126 (1995)I J P  A— an international loumalMonte-Carlo Simulation of-D isc Like Molecules ymth Gay-Berne Type Pair Interaction PotentialJ.Saha, B.Nandi, P.K.Mukherjee and M.Saha Department of Physics, University of Calcutta,92, A.P.C. Road, Calcutta - 700009, India.Abstract : This work reports a Monte-Carlo simulation in a system of discotic liquid crystal molecules interacting via Gay-Berne pair interaction potential. This simulation is able to generate columnar phase.Keywords : single site potential, thermotropic liqid crystal, columnar phase.FACS nos : 61.30, 64.70M1. In t r o d u c t i o nDispersive forces between molecules are believed to be primarily responsible for giving rise to liquid crystalline phases. But the anisotropy of the short range intermolecular repulsions also play important role in determining the phases. To take care of both attractive and repulsive forces in a phenomenological manner, at the earlier stage of computer simulations of aspherlcal molecules, several sites on each molecule were chosen between which Lennard-Jones potential could act. Later works avoided the computational© 1995 lACS69A(IM7122 ./ Salui, B Nandi. H K Mukherjee and M Saha inefficiency of this type of site-site  potential for large molecules by replacing it  with a single site potential. In an attempt to provide computationally simple re a listic  potential Gay and Berne [1] proposed a potential based on the Gaussian overlap model potential [2]. They modelled a single site potential for rod like molecules by fittin g  it  with the site -s ite  potential for a linear array of atoms.\ The potential is of the formu( ui G , r*) = 4 (u ,u. , r )I I I J.r ______,  r °  „  y  r  r “i r-o-UJ, ,tlz t r)«<r J [ r-ai'Q,  ,'u2 .r)*<r0Jwhere u ,^ u a are unit vectors giving the orientations of the two particles, 'r is the intermolecular vector and r is the molecular separation. £(11 .u^ , r) is the well depth and°^ ui distance of nearest approach which can beexpressed as, A<7(ul fA A= <rQ l - X  {A A, ^( r .u , +r ,o 2 )” H X ( G 1 . 0 Z )where the shape anisotropy parameter+. A A A A( r . U j - r . U a )  1 -JC( CTj . u z  ) >]-L /*X =  [(07, )2- 1 lII and i .  mean parallel and perpendicular to the Interwolecul* vector respectively.The well depthC ( u , , u2 , r ) = £ 0  £  ( u j , ^ ) £ ' ( u ,  . L ^ . r )Monte-Carlo simulation of disc tike molecules etc 123where( u t , Ov , r )£ (  u( r  . u t » r . u 2 ) ~l+ x ;  (U j . u z ) +A  A  A  . ^  1( r .u ,  - r . u2 ) i-x; (CT, -tli) .with the well depth anisotropy parameterwhere £  and £  refer to well depth with parallel and ii Xperpendicular relative orientations of the two molecules respectively and fx and are adjustable parameters.In the following we report the simulation of columnar phase for d is c -lik e  molecules Interacting via Gay-Berne type pair interaction potential using Monte-Carlo simulation technique. The parameters of the potential are so chosen that it  can approximate the behaviour of d is c -lik emolecules. Tor d isc -lik e  molecules*  = f(cre /  cr cr ) % 1 ](7^  is  the separation between two interacting moleculeswhen they are in end-end configuration just touching each other; <7^  is  the separation for side-side configuration. So, (Jj, and <T are equal to the diameter and width respectively. The value of is  negative for d is c -lik e  molecules. S im ilarly ,X ' = [ 1 - ( £ . / £ )  ] / [ 1 + ( £ / £ )  ]where £ .  and £  are well depths at side-side and end-end5 8.configuration. For the s 1 de-side configuration of d isc-H k e  molecules ut|| u^  .u^ J- r. This gives ctu^ . Sowe take <re= o^for our calculation.124 J Snhti, B Nandi, P K Mukherjee and M Saha2. Monte-Carlo simulationTor Monte-Carlo simulation we have chosen a 10 10 10 lattice  system. In the in it ia l  configuration a ll  the molecular short axes are parallel to laboratory 2 -ax is and the centres of mass of molecules are placed on regular square lattice points in each plane parallel to laboratory X-Y plane. The intermolecular distance between a molecule and Its nearest neighbour along X or Y axis are set equal to 1 but along Z 1t is equal to a = a y T h i s  value of 'a' assures uniform density along the three directions when a columnar phase is stabilised. Metropolis algorithm is  used to equilibrate the system. Cut the translationaV motion of the molecules are controlled 1n such a way that it  cannot go out of the cell it  belongs l .e . the neighbours of the molecules are kept fixed. This restriction  is  more or less valid in a very dense system like d isco tic  liquid  crystal which has isotropy in one direction only in columnar phase. All the molecules are treated in the same manner and we get the final configuration of the la ttice  as a whole. The total procedure constitutes an Iteration. Iterations are performed t i l l  equilibrium Is reached. At the production stage of simulation we f irs t  get random position co-ordinates (Figl) by finding an equilibrium Isotropic d istrib u tio n  at a very large T-ioo. Then we lowered the temperature considerably to T 0 .1. The distribution of molecular centers of mass on XZMonte-Carlo simulation of disc like molecules etc 125plane (Mg2) shows that columns are formed along Z direction. Centre of mass positions on XY plane shows rectangular la tt ic e  structure (F1g3). The values of parameters V and fj. are taken equal to 2 and 1 respectively. We take <r /  c r -1/3  and f /£  = 1/5 ; diameter of a molecule r =0.9. Thec s s csimulation was performed at constant volume condition. AfterT  • S •Fif 2. Square irrinjetent of •olecules oi H  pi me (Z is pinllel to the director) it T=8.1each Iteration the orientational order parameter 7)=<p^(cos0)> are calculated. The criterion for equilibrium  1s that the dlfferece in average value of order parameters for successive 1000 configurations is  stable within 5%.: — r  • • • • •  •  •s  •  • ••  I  •  •• « • ft ft• ft • •• • • • ft • ft ft '• •• • !  • •• • • • • • •  ^ 1 • •» •• ft • • • • • {• ft • • • % • *•••• • •  •ft ft • • • •fif 1. Arrujeoeit of coUales01 H  pIlM SIMMS colooi forutloe doeo Z uis.Fij 1. Rudoa distriketioo of the centre of iiss of eclectics it 1:100.126 J Saha, B Nandi, P K Mukherjee and M Saha3. ConclusionThis computer simulation shows that the Gay-Berneinteraction sta b ilize s to a columnar phase with SQuare arrangement of molecular centers of mass in planesperpendicular to the columnar phase. This type of columnarphase is seen in lyotropic system and not in thermotropic\\d is c -lik e  mesogens where the columnar phases are rectangular or hexagonal. The same conclusion was reached by Cmerson et al [3] from their molecular dynamics study of disc lik e  Gay Berne mesogens. The present work supports their findings. The structure obtained in the present work as well as in the molecular dynamics study is  believed to be due tt) strong correlation between discs 1n neighbouring columns with molecules fitt in g  into the grooves of a neighbour. These grooves are the result of the e ll ip t ic a l cross-section of molecules and interdigitation is  a hindrance towards forming a hexagonal phase, further studies are indeed called  for reproducing a rectangular or a hexagonal phase.References[1] J.G.Gay and B.J.Berne, J.Chem.Phys., 74,.3316 (19 8 1).[2] B.J.Berne and P.Pechukas, J.Chem.Phys., 56, 4213 (1972); J.Kushlck and B.J.Berne, J.Chem.Phys., 64, 1362 (1976).f3] A.p.j.Emerson, Ph.D. thesis, Univ. of Southampton, U.K.

Monte-Carlo Simulation of Disc Like Molecules with Gay-Berne Type Pair Interaction Potential

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Monte-Carlo Simulation of Disc Like Molecules with Gay-Berne Type Pair Interaction Potential

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