Insertion of nanoparticles into polymer brush under variable solvent conditions

In this work, two-dimensional lattice-based self-consistent field theory is used to study the free energy cost associated with the insertion of a nanoparticle into a polymer brush. The nanoparticle is modeled as a cylinder and the self-consistent field equations are formulated on a cylindrical lattice. The use of two-dimensional formalism makes it possible to take into account the distortion of the brush density profile due to the embedded nanoinclusion. The insertion free energy penalty is analyzed as a function of the particle size, the brush grafting density, and the solvent quality. In agreement with the earlier simulation work, we find that the insertion free energy cost increases both with the particle size and the brush grafting density and decreases with deteriorating solvent quality. For nanoparticles located deep inside the brush, the insertion free energy is shown to scale with either squared (good solvent) or cubed (poor solvent) monomer volume fraction profile, in agreement with the earlier theoretical results. For shallow nanoinclusions located close to the brush surface, the insertion free energy is shown to scale linearly with the monomer volume fraction profile under both good and theta solvent conditions, again in agreement with the earlier work.