We investigate variants of the Asakura–Oosawa (AO) model for colloid-polymer mixtures, represented by hard classical particles interacting via their excluded volume. The interaction between the polymers is neglected but the colloid-polymer and colloid-colloid interactions are present and can be condensed into an effective depletion interaction among the colloids alone. The original AO model involves hard spherical particles in three spatial dimensions with colloidal radii R and the so-called depletion radius δ of the polymers, such that the minimum possible center-to-center distance between polymers and colloids allowed by the excluded-volume constraints is R + δ. It is common knowledge among physicists that there are only pairwise effective depletion interactions between the colloids if the geometric condition is fulfilled. In this case, triplet and higher-order many body interactions are vanishing and the equilibrium statistics of the binary mixture can exactly be mapped onto that of an effective one-component system with the effective depletion pair-potential. Here we rigorously prove that the criterion is both sufficient and necessary to guarantee the absence of triplet and higher-order many body interactions among the colloids. For an external hard wall confining the system, we also include a criterion which guarantees that the system can be exactly mapped onto one with effective external one-body interactions. Our general formulation also accounts for polydisperse mixtures and anisotropic shapes of colloids in any spatial dimension. In those cases where the resulting condition is only sufficient, we further demonstrate how to specify improved bounds.
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