A parabolic density profile for grafted polymers

We study the statistics of a grafted polymer brush, consisting of a set of monodisperse chains in solution, each attached irreversibly by one end to a flat surface. We use a self-consistent field method, valid in the limit of weak excluded volume and at moderately high surface coverage. Exploiting the fact that the chains are highly stretched, we map the problem (in the long-chain limit) onto one involving the motion of classical particles in an equal-time potential, which we can solve exactly. The resulting density profile for the brush takes a parabolic form.