Excess free energy of nanoparticles in a polymer brush

We present an efficient method for direct determination of the excess free energy ΔF of a nanoparticle inserted into a polymer brush. In contrast to Widom's insertion method, the present approach can be efficiently implemented by Monte Carlo or Molecular Dynamics methods also in a dense environment. In the present investigation the method is used to determine the free energy penalty ΔF(R, D) for placing a spherical particle with an arbitrary radius R at different positions D between the grafting plane and the brush surface. Deep inside the brush, or for dense brushes, one finds ΔF ∝ R3 whereas for shallow nanoclusions ΔF ∝ R2, regardless of the particle interaction (attractive/repulsive) with the polymer.The pressure and density fields around spherical nanoinclusions in a polymer brush are also investigated. Extensive Monte Carlo simulations show that the force, exerted on the particle by the surrounding brush, depends essentially on the proximity of the nanocolloid particle to the brush surface not only in strength but also with respect to its angular distribution. For shallow nanoinclusions close to the brush surface this angular distribution is shown to result in a growing buoyant force while deep inside the brush this effect is negligible.