The first statistical mechanical computer simulations of model liquids were performed in the early 1950s on the MANIAC computer at Los Alamos, New Mexico, USA. This was the pioneering work of Metropolis et al [1] which established a version of the standard Monte Carlo method for solving many dimensional integrals [2]. The time saving technique used importance sampling and allowed for a simulation of 2-dimensional hard spheres [1], with Rosenbluth and Rosenbluth [3] continuing the work to look at hard spheres in three dimensions. Later in that decade, the molecular dynamics technique of Alder and Wainwright [4] was used to simulate hard spheres. Evaluation, discussion and comparison of the two simulation methodologies began immediately [see for example 5; 6]. Many rudimentary techniques introduced in this period are still in
use today and some of these have been described in chapter n.
Both Metropolis Monte Carlo (MC) and molecular dynamics (MD) provided physicists and chemists with powerful tools with which to probe the liquid state at the microscopic level. These were a great asset because unlike the harmonic theory of solids, and the ideal gas law, there is no effective reference starting point in a general theory of fluids. Hard and soft sphere fluids were studied extensively [6]; nevertheless, at least fifteen years elapsed before the first
In this chapter we shall discuss some of the important simulations that have been performed to date employing the anisotropic potentials necessary to describe liquid crystalline phases. Early simulators were plagued initially by (a), mathematical problems, e.g. trying to find an effective way to compute the contact function of two hard bodies, and (b), by practical problems, e.g. low speed and small memory of early computers. Progress in both these areas has been made over the last four decades and in this chapter we shall detail some of
the simulations of liquid crystals performed to date. These simulations can be roughly divided into four main categories:
hard particle models; continuous potential models;
lattice models;
realistic atom-atom potential models.
For our purposes here we shall divide the following sections broadly according to the classification above. The order above does not represent any prejudice on the importance of these models. Most simulations, however have been performed on hard particle models, so we shall begin with these. A review of single site soft particle models follows. Although there are no other published works relating to the hybrid Gay-Beme Luckhurst-Romano (HGBLR) potential with the exception of those subsequently reported herein [8], this section is particularly important as
it most closely relates to the HGBLR potential. We mention lattice models only briefly. As the computational power of modem computers rapidly increases, simulations of realistic models of liquid crystals are perhaps just becoming attainable. We present a review of the realistic simulations performed to date.
As will be seen, each of the above have their particular uses, either in providing insight into real liquid crystal mechanisms, or providing a test of theoretical approaches. Before we begin examining the results of computer simulations we shall begin with a brief discussion of two contrasting theories ascribing the formation of orientationally ordered phases to (a), geometric effects and (b), long range anisotropic dispersion forces respectively. Comparison of theories, with computer experiments and with experiments on real liquid crystals have enabled us to gain valuable insight into the formation of liquid crystal phases.
III.2 Anisotropic Repulsive Forces vs Anisotropic Attractive
Dispersion Forces.
At the time of the first simulations of liquid crystals the nature of the isotropic- nematic phase transition had, on the one hand, been hypothesised to be strongly dependant on geometric effects as first described by Onsager [9]. In attempting to describe the formation of lyotropic liquid crystal phases in suspensions of anisometric particles, Onsager had developed a theory which attributed the formation of the nematic phase of a system of long hard rods to excluded volume effects alone. Onsager calculated the first two virial coefficients for a system of long rods which he considered to be made up of many groups of rods, each group with a specific orientation. By minimising the available free energy of the system, Onsager arrived at a distribution function describing the orientations of the molecular long axes. He further showed that if the system was compressed to a sufficiently high density a transition to an orientationally ordered phase occurs. Isihara [10], applied the Onsager theory to rigid molecules of different shapes, including ellipsoidal and cylindrical molecules. Zwanzig [11] extended Onsager’s rod work to include higher order terms in the virial expansion of the equation of state of a system of rectangular parallelepipeds length /, and square
base of side d. In the limit that / -» a>, and d -» 0, and with the constraint l2d = constant, Zwanzig computed, exactly, the first seven virial coefficients for a small number of specific orientations of the parallelepipeds (Onsager theory was originally based on a continuous distribution of the orientations of the rods considered there). Zwanzig’s analysis showed that at every order of the virial expansion considered, the system exhibited a van der Waals loop, associated with an order-disorder phase transition, at an appropriate density. A comparison with the Onsager work showed differences in the properties of the isotropic phase, but calculations truncated at the second virial coefficient were within 10% to 2 0% of
those evaluated by Zwanzig (however, it should be noted that the third virial coefficient for an isotropic system of long thin rods is negligible [9]). This suggested that merely a second virial coefficient treatment may be sufficient to yield some valuable approximations when more complicated systems of rods are considered, e.g. the effects of external fields or the effects of allowing attractive forces to act between the rods [11].
At the same time, and on the other hand, Maier and Saupe [12] had demonstrated within the mean field approximation the existence of an orientational order- disorder transition in a system of cylindrically symmetric rigid molecules, dependent only on long range anisotropic attractive components of the pair potential. Maier and Saupe solved their system within the molecular field approximation. In these so called mean field theories the equilibrium configurational partition function is reduced to a product of single particle partition functions, one for each molecule each of which necessarily only depends on the coordinates of the molecule. This is the case for an ideal gas where the molecules of the gas may be considered to act independently of each other. The interaction of a molecule within the mean field is represented by a
pseudo-potential which is an effective one particle potential. There are many ways of arriving at a description of the pseudo-potential, and Luckhurst has listed
some of these and also presents a derivation based on a hierarchy of molecular distribution functions [13].
In the Maier-Saupe theory the effective singlet orientational potential is;