Anisotropic Soft Pair Potentials and the Origin of the HGBLR Potential

The potentials just described possess spherical symmetry, and are therefore not suitable for the simulation of liquid crystals. In order to arrive at a model suitable for describing (microscopically or phenomenologically) liquid crystals we necessarily must include an anisotropic component in the intermolecular pair potential. Based upon calculations of the second virial coefficient of cylindrical molecules, Comer [15] proposed a general form of potential for such molecules as;

^Corner = e(Q )/r _{a(Q )}r \ [D.12]

The potential [11.12] depends on the relative orientations of the two molecules which is introduced through the parameter Q. For fixed Q however [11.12] remains spherically symmetric. As with the Lennard-Jones potential when written in the form of [11.10], s is a function that scales the energy well depth and / is a function depending on molecular separation r, but now both s and / depend additionally on the relative orientation of the particles.

Several potentials having the analytical form of [D.12] have been used in computer simulations of liquid crystals. By envisaging molecules represented as the rigid union of a set of ellipsoids, Beme and Pechukas [16] developed a soft non-spherical potential based on the Gaussian overlap model. The overlap integral of two Gaussian ellipsoids may be represented as;

The unit vectors

u,

represent the orientation of the principle symmetry axis of each ellipsoid of revolution, and

r

is vector joining their centres. By analogy with equation [D.12] we see that Si(u1,u2) and

a(ul5u2,r)

represent strength and range parameters respectively with / taking an exponential form. The unit vectors uz represent the orientation of the principle symmetry axis of each ellipsoid of revolution. For constant orientation of u, pi. 13] generates a series of ellipsoidal equipotentials in

r.

Berne and Pechukas obtained expressions for the strength and range parameters:

s081(u1,u2) = S o l - x 2(uru2)2

PL14]

a(u1,u2,f) = a0

_1-tX

(

A A A A \ 2 / A A A A \ 4

r-Uj + r-u2) , ( r u , - r u 2)

+ —— ~ . [n .i5 ] A A \2 V2 l

+ x(ur u2)

l - x ( n i - U2)

The terms s0 and c j0 are strength and range constants and % represents the

anisotropy of the ellipsoids;

x = (a i2 - a i ) / ( o f +<?!)> [n.16]

where Q| and a ± are the major and minor axes of the ellipsoidal Gaussians. Equations pi.14] and [II. 15] give an expression for the extent of overlap between two ellipsoidal Gaussians with respect to their relative orientations In fact, as Berne and Pechukas state [16], the overlap model gives an expression for the orientational dependence of molecular interactions, but it does not accurately reproduce the distance dependence. In order to achieve this, the authors suggest that the strength and range parameters of the Gaussian overlap model are used as the strength and range parameters of a simple atomic potential. For example

incorporation into the Lennard-Jones 12-6 potential, leading to the overlap potential of Berne and Pechukas;

rBp(ui,n2.r) = 4 s1(u1,u2)

C ( A A ^ \ \ 1 2 / / A A a \ \ 6

' a(ulsii2#r) 1 f d(u1,u2,r)

PI-17]

Equation [11.17] represents the interaction between two molecules represented as ellipsoids of revolution [16]. One graphical way of illustrating the behaviour of [II. 17], is to plot the distance dependence of the potential energy for a few select orientations. Common configurations include parallel end-end and side-side, X and T [17]. The distance dependence of the Beme-Pechukas potential for these four basic configurations is shown in figure [11.5].

Equation [11.17] has certain characteristics that make it particularly amenable to computer simulation studies [18]. It is a relatively simple function dependent on only three parameters, allowing for relative ease of calculation of the potential energy. Further the function is readily differentiable facilitating analytical computation of the forces and torques of a system of Beme-Pechukas particles. Variation of the parameter % (equation n.16) allows a range of molecular eccentricities, from veiy long prolate to flat oblate to be studied.

The Beme-Pechukas potential (equation [11.17]) was first used in computer simulations investigating the stability of the nematic phase and co-operative re­ orientation effects. For more details see chapter HI.

As can be seen in figure [II.5], the well depths of parallel configurations of Beme-Pechukas particles have equal magnitude. For prolate systems we would expect the side-side configurations of molecules to be favoured over end-end configurations. This is suggested by considering the potential energy curves exhibited by a linear array of Lennard-Jones centres [2] having an axial ratio of 3:1.

Subsequently, modifications of the Beme-Pechukas potential have been suggested [2, 18, 19] There are two principle changes. The first involves scaling the existing strength parameter with an additional function dependent on the intermolecular separation vector r;

s(u!, u2, r ) = e0e1v (fij, u2 )e£ (fii, u2, r). [E. 18] The scaling function 82(ui,U2,r) has the form of a(ui,u2,f ) / a 0; the exponents

v and p are treated as adjustable parameters that influence the relative well depths of different configurations (see for example Luckhurst and Simmonds, ref.

[

21

]).

Secondly (as can be seen in figure [II.5]), the Beme-Pechukas potential has the unrealistic feature that the well width is larger for end-end configurations, with respect to side-side configurations. This property is not exhibited by a four site linear Lennard-Jones array [2]. Gay and Berne suggested a shifted form of the range parameter for use within the Lennard-Jones function, yielding the Gay- Beme potential;

FGB(ui,U2,r) =

/ V2 f ^6

^ - a ( u „ u 2,f) + CT0J ^

[H.19] This is illustrated in figure

n.6.

As can be seen in figure

H6,

sliding the Gay- Beme potential minimum position, rather than a simple scaling with intermolecular distance, removes the dilatory effect on the well depth width. Berne particles are now equal.

The Gay-Beme particles have been shown to provide a rich degree of polymorphism. They have been used successfully in simulations of calamitic and discotic liquid crystals by a number of workers, notably Luckhurst and co­ workers [21], and Rull and co-workers [22]. These simulations are discussed more fully in chapter m.

The Luckhurst-Romano potential is the sum of a simple anisotropic and a scalar pair potential, having the following form;

Thus the well depth widths for both end-end and side-side configurations of Gay-

Vlk = V0 + Va, p.20]

0.0

-0.5 * -

2.0

1

0 1

2

3 4 5

6

Figure IL5

Distance dependence of the Berne-Pechukas potential, equation 11.17 parameterised with a 0 = l, Sq = 1 and G | / a ± = 3. The different symbols correspond to the

configurations: closed squares, side-side; open squares, X-configuration; closed diamonds, T-configuration; open diamonds, end-end.

0.5 T -0.5 *

-2

-2.5 1 0 1

2

3 4 5

6

Figure IL6

Distance dependence of the Gay-Beme potential, equation 11.19 parameterised with <r0 = 1, s 0 = 1, g s / a e = 3 and es / s e = 5 with v = 1 and \i = 2. For a key to the

where Py is the angle between the principle symmetry axes of two rod-like particles, and P2 is the second Legendre polynomial. The anisotropic term is simple and has the same form as that used in the Maier-Saupe mean field theory of nematics [23, 24]. In an earlier form, this potential was used in a simulation of been used in simulations were the particles were not restricted to a lattice [26]. Although lattice models cannot hope, truly to closely represent a fluid phase, because of the obvious restriction of no translational motion, it is sometimes convenient to turn one’s attention to a small part of an altogether wider problem. In the case of lattice models, detailed studies of the effects of particle reorientation could be made. The Luckhurst-Romano potential was successfully used in an off lattice simulation of simple cylindrically symmetric particles [26] and more recently in an Monte Carlo simulation of a siloxane cyclic polymer system [27]. Again these are discussed more fully in chapter m.

For the simulations reported herein, we have used a modified form of the Luckhurst-Romano potential, formed by scaling it with part of the anisotropic well depth expression of Gay and Berne. The result is a fairly simple anisotropic pair potential, the hybrid Gay-Beme Luckhurst-Romano potential (HGBLR) which has the following form;

rod-like particles confined to the sites of a lattice [25], More recently the VA has

f/HGBLR(ui,U2>r) = s2(u1,u2,f){K0(r) + F/j(u1,u2,r)}. [11.22] Where:

^(«i,U2>r) = -4sA. [n.24]

S2(® l,«2,f)=l-

and;

X - {l / s 5} /{ l + 8e /Ej}. [11.26]

The HGBLR potential has a number of characteristics which may make it extremely useful in the task of simulating mesogens.

The inclusion of the part well depth scaling function of the Gay-Beme potential introduces a dependence on the intermolecular vector. Thus the HGBLR potential, in contrast to the Luckhurst-Romano potential correctly distinguishes between the parallel configurations, for example end-end and side-side.

The HGBLR potential has the advantage over the Gay-Beme potential of computational simplicity and therefore speed. In a comparative test with the Gay-Beme potential, the HGBLR potential was found to be an order of magnitude faster on a scalar processor [28]. This provides some motivation for developing the HGBLR potential as later several HGBLR sites would be joined together to provide more realistic models of mesogens.

The HGBLR potential with a disc-like and rod-like parameterization is presented in two-dimensions in figures IV.2 and IV.3 respectively. Inspection of these

figures reveals that the potential has a spherical hard core surrounded by an anisotropic attractive region. This is unrealistic for liquid-crystal forming molecules, which must necessarily deviate from spherical symmetry. Therefore in the simulation of single HGBLR centres, any observed liquid crystal phases would be due solely to the attractive anisotropic component in the potential, as we know that hard spheres do not form a liquid crystal phase. Later simulations, involving multiple HGBLR centres will provide a non-spherical hard core through the geometrical disposition of their centres.

The anisotropy parameter %' which represents the relative ratio of the side-side to end-end interactions can take a range of values, including negative values, enabling the anisotropic attractive part of individual HGBLR centres to be representative of disc-like or rod-like mesogens equally well.

The HGBLR potential has formed the basis of the MD simulations reported in chapters IV and V. As preliminary work, systems of individual HGBLR centres have been simulated, with calamitic and discotic parameterisations. This work is presented in chapter IV. Subsequently HGBLR sites have been rigidly joined to form multisite models of calamitic mesogens; the results of MD simulations of these models are presented in chapter V.

The remainder of this chapter will be concerned with carrying out the MD simulations and extracting useful information from them.

In document Molecular dynamics simulations of calamitic and discotic liquid crystals (Page 34-43)