Simulation of the Liquid State

BY D.M. HEYES

1 Introduction

The molecular simulation of liquids is a now vast field of research, and as with many others in recent years it is becoming increasingly difficult to keep abreast of all of the significant developments that are taking place. The ready availability of fast computers has meant that there are many more researchers working in this ever expanding field of applications, producing ever larger amounts of work to assimilate! This poses something of a problem when it comes to writing a review, especially one with the rather ambitious title of 'Computer Simulation of Liquids'. I am not going to attempt to cover all the branches of this field. Rather, in my review of the developments between 1999 and 2001 I have restricted my discussion to a few areas that have interested me. The choice is inevitably somewhat subjective, but hopefully by adopting this approach I will have a better chance of producing a useful document, rather than a gallop through many topics with only the briefest of discussion about each, which I am sure would be of little use to the scientific community. There are a number of molecular simulation books that describe the standard techniques, and these are recommended as background material for the present article, e.g. refs. 1-7.

I am therefore not going to discuss the 'nuts and bolts' of molecular simulation, except to mention an often overlooked fact, which is the reason for much of the success of these approaches. Most simulations are carried out still typically for less than a thousand molecules, and if it was not for the use of periodic boundary conditions (PBC) it would not be possible to simulate bulk systems with this number of molecules. These systems would have such a high surface to volume ratio that the results would be dominated by surface effects. The PBC procedure is illustrated in Figure 1, which shows a two-dimensional square cell in which the molecules are surrounded by image cells. A molecule near the cell boundary interacts with the 'real' molecules in the central cell and with image cell molecules. Molecules leaving the cell re-enter through the opposite face with the same velocity.

2 Simple Liquids

The study of simple liquids can be said to be the beginning of Molecular Dynamics and Monte Carlo in the 1950s and 60s. Although the scope of molecular simulation, as a field or discipline, has widened dramatically since then, there is nevertheless a continual interest in simple liquids. In fact, this is partly due to the fact that the so-called 'simple' liquids are far from simple! One of the motivations for the continual interest in the simple liquids is that, because of the basic nature of the interparticle interactions, an improved understanding of these systems should lead to better theoretical models, which can be extended to more complex molecular liquids. Also, the rapid growth of interest in colloids and polymers (so-called 'complex' liquids) in recent years has provided new areas where the theories of simple liquids can be applied, especially those associated with local structure and thermodynamics. In the latter case, phase equilibria and the location of phase boundaries feature prominently. In this section, some of the recent advances in our understanding of simple liquids are covered.

2.1 Dynamics. - Of course, within the category of 'simple liquids' studied by statistical mechanics and molecular simulation, there are model liquids that are, strictly speaking, not found in nature. For example, the ubiquitous hard sphere fluid, where the pair potential has the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

is a case in point. The energy is infinite on contact of the spheres (at σ) and zero for larger separations. This is the energy of interaction which would approximate that of two macroscopically sized elastic spheres with high elastic modulus, say two snooker or billiard balls. In these cases the length-scale of the particle interactions is many orders of magnitude smaller than the particle diameter. One of the main features of hard spheres is the co-ordination number. It has been shown recently that for spheres at random close packing, the mean number of particle contacts is 4.8, which is somewhat lower than has often been assumed before (6, and even 12, have been used). Interestingly these authors also performed a computer 'simulation' in which they took a random test sphere, and placed immobile point contacts on its surface. They determined the mean number of points required on the surface of the sphere to eradicate the possibility of translation of the test particle, which was found to be 2D + 1, where D is the space dimension. Therefore, in 3D this 'co-ordination number' is 7, which is lower than the value of 4.8, indicating that in states where the contacts are 'correlated' (i.e. in a dense liquid or glassy state) translational diffusion can be removed by fewer contacts. The procedure for carrying out Metropolis Monte Carlo of hard spheres is particularly simple, as the Boltzmann factor does not require specific evaluation – just overlap detection. Jater proposed an improved Metropolis Monte Carlo algorithm to simulate hard core systems, in which they replaced the usual sequence of single particle trial displacements by a collective trial 'move' of a chain of particles.

The hard-sphere system is widely used in statistical mechanics as a reference state in theories of liquids and solids. Its uses have traditionally been quite broad, extending from equations of state, the structure of molecular liquids and dynamical properties. As mentioned already, it has also found a new lease of life as a model reference system for some colloids and granular materials. Simple molecules (e.g. water) interact with an appreciable attractive tail extending beyond the hard core, and which usually has more or less the same range as the core. It is not possible to find a simple molecule that does not have an appreciable attractive or van der Waals region as well as a hard repulsive core. In contrast, on the micron and larger scale, for these systems, the hard-sphere can be an even more realistic representation of the effective pair potential, which can be steeply repulsive and have a negligible attractive component. It must be borne in mind, however, that the hard-sphere particle potential is fundamentally unrealistic in that its pair potential is...