Demixing can occur in binary hard-sphere mixtures with negative non-additivity

A binary fluid mixture of non-additive hard spheres characterized by a size ratio $\gamma=\sigma_2/\sigma_1<1$ and a non-additivity parameter $\Delta=2\sigma_{12}/(\sigma_1+\sigma_2)-1$ is considered in infinitely many dimensions. From the equation of state in the second virial approximation (which is exact in the limit $d\to\infty$) a demixing transition with a critical consolute point at a packing fraction scaling as $\eta\sim d 2^{-d}$ is found, even for slightly negative non-additivity, if $\Delta>-{1/8}(\ln\gamma)^2$. Arguments concerning the stability of the demixing with respect to freezing are provided.