Static properties and time correlation functions of fluids with steeply repulsive potentials

There has been much interest in recent years in the properties of fluids composed of particles that interact through potentials with a 'tunable' softness, particularly for highly repulsive particles close to the hard-sphere limit. Much of this interest is driven by the granular media and colloid communities, but there are fundamental issues concerning the physics of fluids in general that can be addressed with such model systems. In this report we continue our series of investigations into the properties of an example of such a fluid, the so-called inverse power or soft-sphere fluid which is composed of particles interacting through a pair potential, phi(r) = epsilon(sigma/r)(n), where n measures the steepness or stiffness of the potential. We review the current state of our knowledge of the properties of such fluids and point out some still unresolved areas. We present the results of new computations for a range of n values and densities. Interest is focused on the time correlation functions as exemplified by the force autocorrelation function, C-F(t), and the shear stress correlation function, C-s(t). Powles and Heyes [POWLES, J. G. and HEYES, D. M., 2000, Molec. Phys., 98, 917.] showed that at short times the shear stress autocorrelation function fits quite closely to the analytic form, C-s(t)/C-s(0) = sech (2(1/2)x) = 1 - x(2)+ (5/6) (x(4)) -... where x is a reduced time that incorporates the effects of n and temperaure, x = (T)*(1/2) nt*. T* and t* are the reduced temperature and time in particle units consisting of particle mass and the pair potential parameters, epsilon and sigma. We give further supporting evidence for this functional form, both for C-s and C-F. However, we still lack a theoretical explanation for this behaviour. A formal time expansion of C-F(t) with similar approximations to those used in our previous publications for the O(t(2)) coefficient predicts a Gaussian analytic form for C(t), therefore posing an unresolved paradox at present. We present a general procedure by which we can derive formally exact expressions for static properties (such as the interaction energy, pressure and infinite frequency elastic moduli) that can be conveniently reduced to relatively simple approximate expressions involving n and an equivalent hard-sphere density and equation of state, which become more accurate in the hard-sphere limit. We also derive a simple (approximate) analytic formula for the mean square force on a particle in terms of n and the hard-sphere equation of state, which is accurate in the hard-sphere limit. We examine the radial distribution functions of these fluids for various n and density, and compare their forms close to contact with the hard-sphere radial distribution functions with appropriate diameters.