Simple integral equation for the polymer brush

Employing a self-consistent mean field theory approach, we consider the density profile of a polymer brush. We express the self-consistent equation in terms of the monomer density and, by considering only a restricted set of polymer trajectories, reduce it to a relatively simple integral equation for the density profile. We find rather good agreement between the solution of our equation and that of the original one over almost the whole profile range. In particular, the profile reproduces well the smooth decay of the monomer density at the free surface of the brush and in this regard is an improvement over the classical parabolic profile.