Irreducible rational approximants for the hard-sphere fluid

Thermodynamic and dynamical properties of the hard sphere system revisited by molecular dynamics simulation

Revised thermodynamic and dynamical properties of the hard sphere (HS) system are obtained from extensive molecular dynamics calculations carried out with large system sizes (number of particles, N) and long times. Accurate formulas for the compressibility factor of the HS solid and fluid branches are proposed, which represent the metastable region and take into account its divergence at close packing. Some basic second-order thermodynamic properties are obtained and a maximum in some of their derivatives in the metastable fluid region is found. The thermodynamic parameters associated with the melting-freezing transition have been determined to four digit accuracy, which generates accurate new values for the coexistence properties of the HS system. For the self-diffusion coefficient, D, it is shown that relatively large systems (N > 104) are required to achieve an accurate linear extrapolation of D to the infinite size limit with a D vs. N-1/3 plot. Moreover, it is found that there is a density dependence of the value of the slope in the linear regime. The density dependent correction becomes practically insignificant at higher densities and the hydrodynamic formula found in the literature is still accurate. However, with decreasing density the density dependence of the size correction cannot be neglected, which indicates that other sources of N-dependence, apart from those derived on purely hydrodynamic grounds, may also be important (and as yet unaccounted for). A detailed analytic representation of the density dependence of the HS self-diffusion coefficient and the HS viscosity, η, is given. It is shown that the HS viscosity near freezing and in the metastable region can be described well by the Krieger-Dougherty equation. Both D and η start to scale at high densities and in the metastable region in such a way that Dηp = const, where p ≃ 0.97, and D → 0 and η → ∞ at a packing fraction of 0.58, which coincides with some previous predictions of the HS glass transition density.

Monte Carlo simulations of fluids whose particles interact with a logarithmic potential

Monte Carlo simulations of a model fluid in which the particles interact via a continuous potential that has a logarithmic divergence at a pair separation of sigma, which we introduced in J. G. Powles et al., Proc. R. Soc. London, Ser. A 455, 3725 (1999), have been carried out. The potential has the form, phi(r)= -epsilon ln(fr), where epsilon sets the energy scale and fr=1-(sigma/r)m. The value of m chosen was 12 but the qualitative trends depend only weakly on the value of m, providing it is greater than 3. The potential is entirely repulsive and has a logarithmic divergence as approximately -ln(r/sigma-1) in the r-->sigma limit. Predictions of the previous paper that the internal energy can be computed at all temperatures using the standard statistical mechanics formula for continuous potentials are verified here. The pressure can be calculated using the usual virial expression for continuous potentials, although there are practical limitations in resolving the increasingly important contribution from the r-->sigma limit at reduced temperatures greater than approximately 5. The mean square force F2 and infinite frequency shear Ginfinity and bulk Kinfinity moduli are only finite for T*=kBT/epsilon<1. The logarithmic fluid's physical properties become increasingly more like that of the hard sphere fluid with increasing temperature, showing a sharp transition in the behavior of the mean square force and infinite frequency elastic constants at T*=1. The logarithmic fluid is shown to exhibit a solid-fluid phase transition.

Self-Diffusion Coefficient of the Hard-Sphere Fluid: System Size Dependence and Empirical Correlations

Molecular dynamics simulations have been used to calculate the self-diffusion coefficient, D, of the hard sphere fluid over a wide density range and for different numbers of particles, N, between 32 and 10 976. These data are fitted to the relationship D = D(infinity) - AN(-alpha) where the parameters D(infinity), A, and alpha are all density-dependent (the temperature dependence of D can be trivially scaled out in all cases). The value alpha = 1/3 has been predicted on the basis of hydrodynamic arguments. In the studied system size range, the best value of alpha is approximately 1/3 at intermediate packing fractions of approximately 0.35, but increases in the low- and high-density extremes. At high density, the scaling follows more closely that of the thermodynamic properties, that is, with an exponent of order unity. At low packing fractions (less than approximately 0.1), the exponent increases again, appearing to approach a limiting value of unity in the zero-density limit. The origin of this strong N dependence at low density probably lies in the divergence in the mean path between collisions, as compared with the dimensions of the simulation cell. A new simple analytical fit formula based on fitting to previous simulation data is proposed for the density dependence of the shear viscosity. The Stokes-Einstein relationship and the dependence of D on the excess entropy were also explored. The product Deta(s)p with p = 0.975 was found to be approximately constant, with a value of 0.15 in the packing fraction range between 0.2 and 0.5.

First derivative of the hard-sphere radial distribution function at contact

Molecular dynamics simulations have been carried out of the radial distribution function of the hard sphere fluid for a range of densities in the equilibrium fluid and just into the metastable region. The first derivative of the hard-sphere radial distribution function at contact was computed and its density dependence fitted to a simple analytic form. Comparisons were made with semi-empirical formulae from the literature, and of these the formula proposed by Tao et al (1992 Phys. Rev. A 46 8007) was found to be in best agreement with the simulation data, although it slightly underestimates the derivative at the higher packing fractions in excess of about 0.45. Close to contact, within a few per cent of the particle diameter, the radial distribution function can be represented well by a second order polynomial. An exponential function, which has some useful analytic features, can also be applied in this region.