  We consider an electrolyte solution confined by semipermeable membranes in
contact with a salt-free solvent. Membranes are uncharged, but since small
counter-ions leak-out into infinite salt-free reservoirs, we observe a
distance-dependent membrane potential, which generates a repulsive
electrostatic disjoining pressure. We obtain the distribution of the potential
and of ions, and derive explicit formulas for the disjoining pressure, which
are validated by computer simulations. We predict a strong short-range
power-law repulsion, and a weaker long-range exponential decay. Our results
also demonstrate that an interaction between membranes does strongly depend on
the screening lengths, valency of an electrolyte solution, and an
inter-membrane film thickness. Finally, our analysis can be directly extended
to the study of more complex situations and some biological problems.
