Combined temperature and density series for fluid-phase properties. I. Square-well spheres

Cluster integrals are evaluated for the coefficients of the combined temperature-and densityexpansion of pressure: Z = 1 + B-2(beta)eta + B-3(beta)eta(2) + B-4(beta)eta(3) + center dot center dot center dot, where Z is the compressibility factor,. is the packing fraction, and the Bi(beta) coefficients are expanded as a power series in reciprocal temperature, beta, about beta = 0. The methodology is demonstrated for square-well spheres with lambda = [1.2-2.0], where lambda is the well diameter relative to the hard core. For this model, the B-i coefficients can be expressed in closed form as a function of beta, and we develop appropriate expressions for lambda = 2-6; these expressions facilitate derivation of the coefficients of the beta series. Expanding the B-i coefficients in beta provides a correspondence between the power series in density (typically called the virial series) and the power series in beta (typically called thermodynamic perturbation theory, TPT). The coefficients of the beta series result in expressions for the Helmholtz energy that can be compared to recent computations of TPT coefficients to fourth order in beta. These comparisons show good agreement at first order in beta, suggesting that the virial series converges for this term. Discrepancies for higher-order terms suggest that convergence of the density series depends on the order in beta. With selection of an appropriate approximant, the treatment of Helmholtz energy that is second order in beta appears to be stable and convergent at least to the critical density, but higher-order coefficients are needed to determine how far this behavior extends into the liquid. (C) 2015 AIP Publishing LLC.