Nanoinclusions in dry polymer brushes

A theory of dry polymer brushes containing nanoinclusions is presented. Polymer brush-nanoparticle mixtures arise in various applications and in experimental systems where block copolymer materials, providing brushlike environments, organize nanoparticles to generate materials with novel properties. The ease with which a nanoinclusion enters a brush is measured by the free energy cost to introduce the inclusion, Delta F-inc. This depends strongly on particle shape and size b, as does the de-ree to which brush chain configurations are perturbed. For inclusions smaller than the typical chain fluctuation scale or blob size xi(blob), by extending the self-consistent mean field (SCF) theory for pure brushes, we show Delta F-inc = P(z)V-p for an inclusion of volume V-p a distance z from the grafting surface. Here P(z) is the quadratic SCF "pressure" field. Equilibrium particle distributions within a brush of chains of length N grafted at density or depend strongly on particle size: (i) particles smaller than a scale b* similar to sigma(-2/3) distribute uniformly, dominated by entropy, while (ii) larger inclusions penetrate the soft surface region of the brush in a layer of thickness delta approximate to h(b*/b)(3) where h is brush height and (iii) complete expulsion occurs for sizes above b(max) similar to (N/sigma)(1/4). Inclusions bigger than xi(blob) affect chain configurations much more strongly and require a different theoretical approach. We show Delta F-inc = beta P(z)V-p, where beta is a shape-dependent constant for which we obtain rigorous bounds. Vertically oriented cylinders achieve the minimum energy cost (beta = 1). Motivated by exact results for the approximate Alexander-de Gennes brush (chain ends fixed at brush surface), we argue that disk-shaped inclusions incur maximum energy cost (beta similar to t where t is the disk aspect ratio).