  We consider an osmotic equilibrium between bulk solutions of polyelectrolyte
bounded by semipermeable membranes and separated by a thin film of salt-free
liquid. Although the membranes are neutral, the counter-ions of the
polyelectrolyte molecules permeate into the gap and lead to a steric charge
separation. This gives rise to a distance-dependent membrane potential, which
translates into a repulsive electrostatic disjoining pressure. From the
solution of the non-linear Poisson-Boltzmann equation we obtain the
distribution of the potential and of ions. We then derive an explicit formula
for the pressure exerted on the membranes and show that it deviates from the
classical van't Hoff expression for the osmotic pressure. This difference is
interpreted in terms of a repulsive electrostatic disjoining pressure
originating from the overlap of counterion clouds inside the gap. We also
develop a simplified theory based on a linearized Poisson-Boltzmann approach. A
comparison with simulation of a primitive model for the electrolyte is provided
and does confirm the validity of the theoretical predictions Beyond the
fundamental result that the neutral surfaces can repel, this mechanism not only
helps to control the adhesion and long-range interactions of living cells,
bacteria, and vesicles, but also allows us to argue that electrostatic
interactions should play enormous role in determining behavior and functions of
systems bounded by semipermeable membranes.
