Time correlation functions of hard sphere and soft sphere fluids.

Abstract

We explore the transition between soft particle fluids of increasing steepness to the hard sphere limit. We analyze the analytic forms of the time correlation functions used in determining transport coefficients in Green-Kubo formulas for fluids composed of particles interacting through a repulsive r(-n) potential. We focus on the steeply repulsive n--> infinity limit where the potential tends to the hard sphere interaction. Dufty [Mol. Phys. 100, 2331 (2002)] developed a theoretical framework that can be used to characterize the transition from a steeply repulsive continuous potential toward the hard sphere potential for the shear stress time correlation function. This function was shown to consist of a rapidly decaying contribution (which is singular in the steeply repulsive limit) and a slowly decaying nonsingular part which can be reasonably well represented by Enskog's prediction on times of order and in excess of the mean collision time. We extend this treatment to the bulk viscosity and thermal conductivity. We focus on the bulk viscosity (pressure) correlation function as it is purely singular for hard spheres, and has no kinetic or cross term contributions in this limit. There is no relaxation of this correlation function on the mean collision or Enskog time scale for hard spheres. We show that it is not possible to represent the steeply repulsive behavior of this function entirely in terms of a sech function, i.e., C(B)(t)=sech(a(n)t/tau(n)), where a(n) is a numerical factor, t is time, and tau(n) is a relaxation time proportional to n(-1). An additional singular function, which we call w(t), is required to obtain the correct short-time behavior of C(B)(t) and the Enskog value for the bulk viscosity. With this additional function, the value of a(n) in the n--> infinity limit is a(n)=square root of 2 which is consistent with the second moment of the time expansion of the time correlation function. We compute this function for large n and extrapolate it to n--> infinity, determining one possible analytic form. The shear stress correlation function also gives a(n)=square root of 2 in the hard sphere limit for the singular part when the sech and w functions are used. This function has a nonsingular component, even in the hard sphere limit. We explore various forms for the crossover function X(t/tau(n)) introduced by Dufty, which weights the limiting singular and nonsingular contributions to C(S)(t) particularly at intermediate times. The qualitative behavior for the heat flux time correlation function (used to obtain the thermal conductivity) is much the same as the shear case. The w(t) derived by several self-consistent extrapolations appears, within the simulation statistics, to be the same for the bulk and shear viscosity, and for the thermal conductivity cases.