Dimensional interpolation of hard sphere virial coefficients

Abstract

We examine the dependence on spatial dimension D of the Mayer cluster integrals that determine the virial coefficients Bn for a fluid of rigid hyperspheres. The integrals vary smoothly with D, and can be characterized analytically in both the low-D and high-D limits. Dimensional interpolation (DI) allows one to evaluate individual Mayer cluster integrals at D=2 and D=3 to within about 1%. The resulting low-order virial coefficients have an accuracy intermediate between those of the Percus–Yevick and hypernetted chain approximations. Much higher accuracy can be achieved by combining the DI and HNC approximations, using DI to evaluate those integrals omitted by HNC. The resulting low-order virial coefficients are more accurate than those given by any existing integral equation approximation. At higher order, the accuracy of the individual cluster integrals is insufficient to compute reliable virial coefficients from the Mayer expansion. Reasonably accurate values can still be computed, however, by taking partial sums of the Ree–Hoover reformulation of the Mayer expansion. We report hard disk virial coefficients through B15 and hard sphere values through B10; the maximum errors with respect to known values are about 1.2 and 4.3%, respectively. The new coefficients are in good agreement with those obtained by expanding certain equations of state which fail to diverge until unphysical densities (those beyond closest packing), and so help to explain the surprising accuracy of some of these equations. We discuss the possibility that the exact virial expansion has a radius of convergence which corresponds to an unphysical density. Several new equations of state with desirable analytic or representational characteristics are also reported.