Osmotic pressure acting on a semipermeable shell immersed in a solution of polyions

We study theoretically the osmotic equilibria for a shell immersed in a suspension of polyions (e.g., colloids, polyelectrolytes, etc.). The shell is treated as impermeable for polyions, but allowing free diffusion of counterions that permeate inside the shell. From the solution of linearized Poisson-Boltzmann equation, we obtain the distribution of a potential and concentration profiles for polyions and counterions. We then obtain an explicit formula for the excess osmotic pressure of a polyion solution exerted on the shell, which includes a quadratic term in order to provide a self-consistency of a linear theory. As a result this pressure is larger than given by a concentration of polyions at the outer shell boundary obtained within linearized theory. It is, however, always smaller than or equal to the bulk osmotic pressure. This difference is attributed to a repulsive electrostatic disjoining pressure due to an overlap of counterion clouds inside the shell. A comparison with molecular dynamics simulations is provided and demonstrates that although the concentration profiles obtained within a linear theory deviate from simulation data at large potential, the theoretical and simulation pressures are in surprisingly good harmony.