Compaction of granular mixtures: a free volume model

We describe compaction, induced by weak tapping of a powder, as a process where a grain can jump into a hole, only if the hole is large enough. The distribution of hole sizes is taken to be the Poisson type, with a certain characteristic free volume. For macrodisperse powders, this leads to a classical logarithmic law of compaction, already derived by Knight et al. Here we focus our attention on the case of mixtures, between two populations: large grains (L) and small grains (S) with very different sizes, so that the (S) grains may fill the interstices of the (L) grains. Geometrically, these mixtures can exist as ''gravels'' (where the interstices are not completely filled) or ''puddings'' (where the L grains are not tightly packed). Dynamically, we expect a cross over curve between L-type compaction and S-type compaction, which is different from the geometrical boundary. This implies that, for certain material ratios rho = L/S, the plot of density versus number of tappings should show two distinct branches.