Modified Donnan model for ion binding to small soft nano particles

Abstract

The charge density of soft nano particles (NPs) can be described with a site binding model in combination with an electrostatic model. When the ionic strength, I, is dominated by a moderate 1-1 electrolyte concentration, the electrostatic interaction can be approximated with the classical Donnan model for large soft NPs. For small soft NPs, the empirical modified Donnan (mD) model, based on a modified Donnan volume, V_mD, can be used to obtain an average particle potential,〖 ψ〗_mD, but the physical background is somewhat obscure. Therefore, the mD model is discussed using the Poisson-Boltzmann theory. V_mD is determined by the specific volume of the ‘intra’ plus ‘extra’ particle diffuse charge, and it is shown that ψ_mD is a volumetric average of the intra particle potentials and the surface potential. Theoretical equations for V_mD are derived for plate, disk and spherical geometries. In general, V_(mD )=SV_p R, where SV_p is a specific soft particle volume, and R is a measure of the effect of extra particle diffuse layer on V_(mD ). When limited information about the NP is available, the theoretical equation for V_mD can be approximated as V_mD=αI^(-β), with adjustable parameters α and -β, where I^(-β) is a measure of R and α an approximation of SV_p. A better understanding of the mD model will improve its future applications.