  A system of patchy colloidal particles interacting with a solute that can
associate multiple times in any direction is a useful model for patchy
colloidal mixtures. Despite the simplicity of the interaction, because of the
presence of multi-body correlations predicting the thermodynamics of such
systems remains a challenge. Earlier Marshall and Chapman developed a
multi-body formulation for such systems wherein the cluster partition function
for the hard-sphere solvent molecules in a defined inner-shell (or coordination
volume) of the hard-sphere solute is used as the reference within the
statistical association fluid theory formalism. The multi-body contribution to
these partition functions are obtained by ignoring the bulk solvent, thus
limiting the applicability of the theory to low system densities. Deriving
inspiration from the quasichemical theory of solutions where these partition
functions occur in the guise of equilibrium constants for cluster formation, we
develop a way to account for the multi-body correlations including the effect
of the bulk solvent. We obtain the free energy to evacuate the inner-shell, the
chemistry contribution within quasichemical theory, from simulations of the
hard-sphere reference. This chemistry contribution reflects association in the
reference in the presence of the bulk medium. The gas-phase partition functions
are then augmented by a mean activity factor that is adjusted to reproduce the
chemistry contribution. We show that the updated partition function provides a
revised reference that better captures the distribution of solvent around the
solute up to high system densities. Using this updated reference, we find that
theory better describes both the bonding state and the excess chemical
potential of the colloid in the physical system.
