Depletion interaction between spheres in an ideal equilibrium polymer fluid: Exact asymptotic results

We use a continuum chain model and develop an analytical theory for the interaction between two spheres immersed in a fluid of ideal equilibrium polymers. The theory can be applied to both adsorbing and nonadsorbing spheres. Here we focus on two nonadsorbing spheres and determine the classical depletion interaction between them. Compact, and exact, results are derived for the asymptotic behavior of the depletion interaction, which has a Yukawa form. We show also that in the limit of large spheres (and large surface to surface separation) the Derjaguin approximation is valid. We compare our asymptotic expression with numerical solutions of an ideal equilibrium polymer fluid consisting of discrete chains. Our asymptotic approximation accurately predicts long-range interactions between small spheres. For large spheres it predicts the interaction very well over most of the separation range. We also consider a single sphere immersed in the polymer fluid and show how our results can be generalized to treat polydisperse polymer fluids, where the polydispersity is described by a Schulz-Flory distribution.