Finite extensibility and density saturation effects in the polymer brush

We study the equilibrium statistics of flexible neutral polymer chains attached by one end onto a flat interface (the polymer « brush »). We account for the finite extensibility of the polymers, adopting a form for the free energy of a stretched chain that diverges as its fully-extended length is approached. Adopting also a Flory-Huggins equation of state for the local chemical potential, we obtain monomer density profiles and chain-end density distributions under various solvent conditions. We use an analytic self-consistent field approximation for long, highly stretched chains, as developed by Milner, Witten and Cates (Macromolecules 21 (1988) 2610), who found a parabolic density profile for brushes at « moderate densities » in a good solvent. Our work generalizes their approach to allow very high coverages and/or poor solvents to be considered ; in each case the density profile is much flatter than a parabola.