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 ${\rho_i}$ only through the four scaled-particle-theory variables ${\xi_\alpha}$, i.e., $\Phi({\rho_i})\to\Phi({\xi_\alpha})$. By imposing consistency conditions, it is proven here that such a dependence must necessarily have the form $\Phi({\xi_\alpha})=-\xi_0\ln(1-\xi_3)+\Psi(y)\xi_1\xi_2/(1-\xi_3)$, where $y\equiv {\xi_2^2}/{12\pi \xi_1 (1-\xi_3)}$ is a scaled variable and $\Psi(y)$ 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 ${\xi_\alpha}$ by the fundamental-measure (scalar, vector, and tensor) weighted densities ${n_\alpha(\mathbf{r})}$. Comparison with computer simulations shows the superiority of this bulk free energy over the White Bear one.