IIL3.1 Hard Particle Models; Hard Ellipses

The first attempted simulation of a liquid crystal with translational freedom can be ascribed to Vieillard-Baron [7]. Vieillard-Baron appreciated the limitations of

the Zwanzig model. The fact that in the limit of / -> oo and d -> 0, the volume of each parallelepiped is zero, led him to describe the system as artificial. Further in the Zwanzig model the density at which the orientation-disorientation transition occurs was found to be strongly dependant on the point at which the terms considered in the virial series expansion were truncated. In this vein, Vieillard-

Baron proposed a model for a more realistic liquid crystal, which consisted of a

system of hard ellipses, and used canonical MC to study the system. In order to evaluate whether two ellipses were in a condition of overlap, Vieillard-Baron introduced the contact function xFeiiiPses(I%Ui>U2,) which was dependent on the

orientation of the major axes U\ and u2 of a pair ellipses and the vector

describing their separation

r:

=

takes a non-zero value at other times. Evaluation of the contact function W provides a decision criterion when updating attempted MC moves, and enables

the pressure of the hard particle system to be computed. Further, Vieillard-Baron

showed that a comparison with the contact function of hard discs leads to an inequality which simplifies the calculation of the pressure within the system. The evaluation of the contact function of hard particle models often presents great difficulty, and it is for this reason that only geometrically simple hard particle models are considered in simulations.

In a study of N = 170 ellipses with axial ratio minor/major axes a /b = 1/6, Vieillard-Baron observed two first-order phase transitions. If a close-packed system of ellipses is expanded from the close-packed area (the equivalent parameter is the density in three-dimensions), Aq, then it undergoes a melting

transition at Am / Aq > 1 (4« being the area occupied by the system at the melting

transition), which Vieillard-Baron described as an increasing function of increasing axial ratio a lb . Initially the ellipses are oriented in a particular direction on the close-packed lattice. At the melting transition it is the centres of mass of the ellipses that first become translationally uncorrelated in the

a lb = 1/6 system, the ellipses maintaining their orientations. This is identified as the nematic phase, characterised by the directional order parameter M;

1

I N N

\

( Z E cos(2 0/ - 2ey))>

where 0 is the angle between the major axis of an ellipse with a given fixed direction (chosen by Vieillard-Baron as the original direction of the ellipses' symmetry axes at the beginning of the simulation). M is a positive rotationally invariant quantity. Clearly for all ellipses pointing along the same direction M -1; for a random orientation of the ellipses major symmetry axes M ~ 1 / N .

The directional order parameter is an example of a simple order parameter that may be used to quantify the degree of orientational order in the liquid crystal phase. The directional order parameter does not however give any information about the director orientation of the system.

In the absence of any precise information Vieillard-Baron speculated that the melting transition of the system of ellipses occurred at area Am / Aq < 1 • 15. This compares to the hard disc melting transition at area Am/ Aq <1-266. The

difference between these two transition densities may be due to the effective

single degree of translational freedom available to the ellipses in the dense nematic phase that contributes to the entropy of the phase, compared to the two- dimensional translational disorder of the hard disc system at the melting transition.

At a specific area of A /Aq = 1-40, Vieillard-Baron observed M=0*7±0*l.

For larger ratios the directional order parameter decays smoothly. A disorientation transition is indicated at Ad I A q - 1-775+0-025, with a

corresponding entropy change 0 • 05 < AS / NkB <0-12, which as pointed out by Vieillard-Baron is much smaller than the entropy change associated with the

melting transition of hard discs; AS/ NkB =0*36 [4]. Again the disorientation transition only effects the one degree of orientational freedom, and as such the nematic and isotropic branches of the isotherms are close and exhibit a large coexistence region.

Hard ellipses have been studied again more recently [16]. Constant-pressure MC simulations have been performed on a system of hard ellipses with aspect ratios k = 2, 4 and 6. Both latter systems exhibit three phases, isotropic, nematic and

solid. No nematic phase is indicated for ellipses of aspect ratio k = 2. Interestingly, while the isotropic-nematic phase transition appears first order for the k = 4 ellipse system, it appears to be continuous for the more eccentric k = 6. It should be noted however that in most cases only a system of approximately

N = 200 ellipses is simulated and so the order of phase transition observed may suffer from small system size effects.