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 density-expansion of pressure: Z = 1 + B₂(β) η + B₃(β) η² + B₄(β) η³ + ⋯, where Z is the compressibility factor, η is the packing fraction, and the B_i(β) coefficients are expanded as a power series in reciprocal temperature, β, about β = 0. The methodology is demonstrated for square-well spheres with λ = [1.2-2.0], where λ 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 β, and we develop appropriate expressions for i = 2-6; these expressions facilitate derivation of the coefficients of the β series. Expanding the B_i coefficients in β provides a correspondence between the power series in density (typically called the virial series) and the power series in β (typically called thermodynamic perturbation theory, TPT). The coefficients of the β series result in expressions for the Helmholtz energy that can be compared to recent computations of TPT coefficients to fourth order in β. These comparisons show good agreement at first order in β, 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 β. With selection of an appropriate approximant, the treatment of Helmholtz energy that is second order in β 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.