Scaling relations for counterion release and attraction of oppositely charged surfaces

Two oppositely charged surfaces or membranes show a long-range attraction, even in the presence of neutralizing counterions. We predict, in the Poisson-Boltzmann limit of very small salt concentrations, the distance and charge dependence at which the intervening counterions are released to a reservoir with a small amount of salt or identical counterions. Surprisingly, this depends on two length scales: the Gouy-Chapman (GC) length, λ 0 , inversely proportional to the surface charge, and the Debye-Hückel (DH) screening length, λ D , in the reservoir. The characteristic distance at which the counterions are released scales with λ 0 log (λ D /λ 0 ). Since the amount of salt (or the concentration of counterions) in the reservoir can be very small, this logarithmic factor can greatly enhance the range over which the attractions between the surfaces can be significant. The attractive interaction between the surfaces is long-ranged, the pressure scaling inversely with the square of the distance for a wide range of length scales—much smaller than the DH screening length, but much greater than the GC length.