Inhomogeneous model colloid-polymer mixtures: Adsorption at a hard wall

We study the equilibrium properties of inhomogeneous model colloid-polymer mixtures. By integrating out the degrees of freedom of the ideal polymer coils, we derive a formal expression for the effective one-component Hamiltonian of the (hard sphere) colloids that is valid for arbitrary external potentials acting on both the colloids and the polymers. We show how one can recover information about the distribution of polymer in the mixture given knowledge of the colloid correlation functions calculated using the effective one-component Hamiltonian. This result is then used to furnish the connection between the free-volume and perturbation theory approaches to determining the bulk phase equilibria. For the special case of a planar hard wall the effective Hamiltonian takes an explicit form, consisting of zero-, one-, and two-body, but no higher-body, contributions provided the size ratio $q={\sigma }_p/{\sigma }_c<0.1547,$ where ${\sigma }_c$ and ${\sigma }_p$ denote the diameters of colloid and polymer respectively. We employ a simple density functional theory to calculate colloid density profiles from this effective Hamiltonian for $q=\mathrm{0.1.}\mathrm{}$ The resulting profiles are found to agree well with those from Monte Carlo simulations for the same Hamiltonian. Adding very small amounts of polymer gives rise to strong depletion effects at the hard wall which lead to pronounced enhancement of the colloid density profile (close to the wall) over what is found for hard spheres at a hard wall.