Concentration and solvency effects on the excess amount and surface free energy of a colloidal particle in a solution of nonadsorbing polymer

Analytical expressions are derived for the polymer excess amount and the grand potential (surface free energy) of flat and spherical surfaces immersed in a solution of nonadsorbing polymer chains in the mean-field approximation. We start from a recent mean-field expression for the depletion thickness δ which takes into account not only the effect of the chain length N but also that of the polymer concentration φb and the solvency χ. Simple expressions are obtained for the interfacial properties at a colloidal surface, using both the adsorption method and the osmotic route. For a sphere of radius a, the excess amount can be separated into a planar contribution Γ = -φbδ and a curvature correction Γ_c = -(π²/12)φbδ_c ²/a, where δ_c is a "curvature thickness" which is close to (but smaller than) δ. The grand potential has a planar contribution ω = (2/9)_φb/δ and a curvature part ω_c = (π/18)φb/a. We test the results against numerical lattice computations, taking care that the boundary conditions in the continuum and lattice models are the same. We find good agreement up to a polymer segment volume fraction of 10%, and even for more concentrated solutions our simple model is reasonable. For spherical geometry we propose a new equation for the segment concentration profile which excellently agrees with numerical lattice computations. The results can be used as a starting point for the pair interaction between colloidal particles in a solution containing nonadsorbing chains, which is discussed in the following paper.