Theory of the collapse of the polyelectrolyte brush

Both an asymptotic analytical analysis for chain length N --> infinity and exact numerical calculations for finite chain lengths were applied to the structural properties of polyelectrolyte brushes under poor solvent conditions in a self-consistent field framework. We extend previous work on polyelectrolyte brushes and find evidence for a structural phase transition caused by internal phase separation in the polyelectrolyte brush upon a decrease of the solvent quality. In the limit of long chains, when a local electroneutrality approximation is exact, we find that the transition in the brush is continuous and tends to be second order. In the numerical calculations which employ the full Poisson-Boltzmann equation, a parameter window is found in which the structural phase transition is first-order. This is proven by the existence of a hysteresis loop in various properties of the brush, such as the degree of dissociation, the average height, the electrostatic potential profile, and the overall and end segment-density profiles. Apart from this difference as to the order of the transition, we find extremely good correspondence between the numerical calculations and the analytical asymptotic analysis for long polymer chains. The structure of the internally phase-separated layer is characterized by a condensed phase near the surface, a dilute swollen layer extending far into solution, and a thin interface between the two regions.