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The Poisson-Boltzmann (PB) theory is extensively used to gain insight oncharged colloids and biological systems as well as to elucidate fundamentalproperties of intermolecular forces. Many works were devoted in the past tostudy PB related features and to confirm them experimentally. In this work weexplore the properties of inter-plate forces in terms of different boundaryconditions. We treat the cases of constant surface charge, constant surfacepotential and mixed boundaries. The interplay between electrostaticinteractions, attractive counter-ions release, and repulsive van 't Hoffcontribution are discussed separately for each case. Finally, we discuss howthe crossover between attractive and repulsive interactions for constantsurface charge case is influenced by the presence of multivalent counter-ions,where it is shown that the range of the attractive interaction grows with thevalency.

Revisiting the Poisson-Boltzmann theory: charge surfaces, multivalent ions and inter-plate forces