Diffusion of hard spheres a binary quenched-annealed mixtures

In quenched-annealed (QA) mixtures of hard spheres (HS), a given fraction of the particles is
mobile (volume fraction $\phi_m$), while the rest are fixed and act as obstacles
(volume fraction $\phi_o$). Typically, the phase diagram of QA mixtures
in the $(\phi_m,\phi_o)$ plane features a region where the mobile particles are diffusive,
separated from a localized phase by the void percolation line, where the diffusion
becomes anomalous.\par
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\indent In this work, we have investigated the self-diffusion coefficient of mobile HSs
in binary QA mixtures for different values of the ratio $\alpha$ between the diameter of the
mobile HSs and that of the obstacles. We find that the motion is always diffusive, with a diffusion coefficient
that decreases linearly with increasing values of $\alpha$. For large volume fraction of the obstacles
we find a more complex decrease, that can be approximated with two successive
linear trends with different slopes.
In the $\alpha \to 0$ limit we recover the well-known mean-field result.
Furthermore, we show that in the limit $\alpha\to 1$
the diffusion coefficient of monodisperse QA mixtures as a function of the obstacle packing fraction
behaves as an order parameter for the mobile-to-localized transition, allowing one to
recover the void percolation threshold
at the same value of $\phi_o$ as predicted by mode-coupling theory.