Class of consistent fundamental-measure free energies for hard-sphere mixtures

In fundamental-measure theories the bulk excess free-energy density of a hard-sphere fluid mixture is assumed to depend on the partial number densities ${ρ_i}$ only through the four scaled-particle-theory variables ${ξ_α}$, i.e., $Φ({ρ_i})\toΦ({ξ_α})$. By imposing consistency conditions, it is proven here that such a dependence must necessarily have the form $Φ({ξ_α})=-ξ_0\ln(1-ξ_3)+Ψ(y)ξ_1ξ_2/(1-ξ_3)$, where $y\equiv {ξ_2^2}/{12πξ_1 (1-ξ_3)}$ is a scaled variable and $Ψ(y)$ is an arbitrary dimensionless scaling function which can be determined from the free-energy density of the one-component system. Extension to the inhomogeneous case is achieved by standard replacements of the variables ${ξ_α}$ by the fundamental-measure (scalar, vector, and tensor) weighted densities ${n_α(\mathbf{r})}$. Comparison with computer simulations shows the superiority of this bulk free energy over the White Bear one.

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