Nanoinclusions in dry polymer brushes

Abstract

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, ΔF inc. This depends strongly on particle shape and size b, as does the degree to which brush chain configurations are perturbed. For inclusions smaller than the typical chain fluctuation scale or blob size ξ̄ blob, by extending the self-consistent mean field (SCF) theory for pure brushes, we show ΔF 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 σ depend strongly on particle size: (i) particles smaller than a scale b* ∼ σ -2/3 distribute uniformly, dominated by entropy, while (ii) larger inclusions penetrate the soft surface region of the brush in a layer of thickness δ ≈ h(b*/b) 3 where h is brush height and (iii) complete expulsion occurs for sizes above b max ∼ (N/σ) 1/4. Inclusions bigger than ξ̄ blob affect chain configurations much more strongly and require a different theoretical approach. We show ΔF inc = βP(z)V p, where β is a shape-dependent constant for which we obtain rigorous bounds. Vertically oriented cylinders achieve the minimum energy cost (β = 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 (β ∼ t where t is the disk aspect ratio).