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)})→Φ({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(1)ξ(2)/(1-ξ(3)), where y≡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)(r)}. Comparison with computer simulations shows the superiority of this bulk free energy over the White Bear one.