Generalization of the model of Debye-Hückel according to a matrix approach. Application to the calculation of the potential of mean force in the case of electrolytes, polyelectrolytes and colloids

In this paper we have generalized on the basis of the superposition principle, the Debye-Hückel model in the case of a spatial distribution of “n” fixed ions surrounded by an ionic atmosphere. According to this approach, the mean electrostatic potential at a position M is written as a sum of individual screened Coulomb potentials: ∑ni = 1(Zie/ε)Aiexp(− χri)/ri. The electro-neutrality condition allows the obtaining of the exact expression of the superposition coefficients {Ai} in terms of the elements {Dij} of an explicit n-n matrix [D] which depend on the effective radii Ri of the ions and on their distances of separation dij. As applications: a) we have improved the expression of the PMF's approximation of Debye; b) we have simplified and resolved in the limits of this model, the BBGKY hierarchy, in order to obtain the PMF potential Wij(dij) and hence the screen parameter χ from the second Stillinger-Lovett restriction; c) we have also proposed an explicit expression for the electrostatic part of the DLVO potential; and finally, d) we have justified and improved the potentials of interaction of Yukawa between chain groups in the case of polyelectrolyte solutions.