Nonlinear Poisson-Boltzmann solutions for charged parallel plates: When opposite charges repel

I present an exact solution of the Poisson-Boltzmann equation for two parallel plates and discuss the solution properties. I discuss in more detail plates with opposite charges: In this case, there are two critical separations, Lc,1 < Lc,2. For separations less than Lc,1, the force between plates is repulsive. It switches to attractive at Lc,1, but with the electric potential having the same sign on both plates. For L > Lc,2, the force remains attractive, and the potential at the plates has the same sign as the charge on each plate. I also describe charge regulation, determined by pKa, and provide formulas for both the critical distance where oppositely charged plates repel and their charging process. The implications of these results for the nanoparticle assembly, as driven by electrostatic interactions, are also discussed.