Atomistic molecular dynamics simulations of a composite consisting of an ungrafted or a grafted spherical silica nanoparticle embedded in a melt of 20-monomer atactic polystyrene chains have been performed. The structural properties of the polymer in the vicinity of a nanoparticle have been studied. The nanoparticle modifies the polymer structure in its neighborhood. These changes increase for higher grafting densities and larger particle diameters. Mass and number density profiles show layering of the polymer chains around the nanoparticle, which extends to similar to 2 nm. In contrast, the increase in the polymer's radius of gyration and other induced ordering (alignment of the chains parallel to the surface and orientation of backbone segments) are shorter-ranged. The infiltration of free polystyrene chains into the grafted chains region is reduced with increasing grafting density. Therefore, the interpenetration of grafted and free chains at high grafting densities, which is responsible for the mechanical anchoring of nanoparticles in the polystyrene matrix, is less than what would be desirable for a well-reinforced composite.