Multi-scale entropic depletion phenomena in polymer liquids

We apply numerical polymer integral equation theory to study the entropic depletion problem for hard spheres dissolved in flexible chain polymer melts and concentrated solutions over an exceptionally wide range of polymer radius of gyration to particle diameter ratios (Rg/D), particle-monomer diameter ratios (D/d), and chain lengths (N) including the monomer and oligomer regimes. Calculations are performed based on a calibration of the effective melt packing fraction that reproduces the isobaric dimensionless isothermal compressibility of real polymer liquids. Three regimes of the polymer-mediated interparticle potential of mean force (PMF) are identified and analyzed in depth. (i) The magnitude of the contact attraction that dominates thermodynamic stability scales linearly with D/d and exhibits a monotonic and nonperturbative logarithmic increase with N ultimately saturating in the long chain limit. (ii) A close to contact repulsive barrier emerges that grows linearly with D/d and can attain values far in excess of thermal energy for experimentally relevant particle sizes and chain lengths. This raises the possibility of kinetic stabilization of particles in nanocomposites. The barrier grows initially logarithmically with N, attains a maximum when 2Rg ∼ D/2, and then decreases towards its asymptotic long chain limit as 2Rg ≫ D. (iii) A long range (of order Rg) repulsive, exponentially decaying component of the depletion potential emerges when polymer coils are smaller than, or of order, the nanoparticle diameter. Its amplitude is effectively constant for 2Rg ≤ D. As the polymer becomes larger than the particle, the amplitude of this feature decreases extremely rapidly and becomes negligible. A weak long range and N-dependent component of the monomer-particle pair correlation function is found which is suggested to be the origin of the long range repulsive PMF. Implications of our results for thermodynamics and miscibility are discussed.