Conditional existence of Donnan potential in soft particles and surfaces: Dependence on steric effects mediated by electrolyte ions and structural charges

Abstract

When a charged layer decorating a particle or a macroscopic surface is equilibrated with an electrolyte solution, a constant Donnan potential is established through that layer due to charge-driven accumulation of counterions and companion exclusion of coions. This situation arises when the thickness of the surface layer well exceeds the screening Debye length, a condition derived from mean-field Poisson-Boltzmann theory within point-like charge approximation. Herein, we revisit this condition underlying the applicability of Donnan electrostatic representation with the account of steric effects mediated by the sizes of the electrolyte ions and structural layer charges. A transcendental equation is derived for the Donnan potential as a function of sizes and valences of anions and cations, electrolyte concentration and size of the layer charges, and a closed-form expression is provided for symmetrical electrolytes. Therefrom we evidence that the existence of a Donnan potential is conditioned not only to large values of the layer thickness compared to a here-defined Debye length operative within the shell, but to additional verification of a criterion that involves space charge density of the layer, solution ionic strength and electrolyte nondiluteness parameter. Illustrative computational examples show how the existence and magnitude of the Donnan potential depend on the key molecular descriptors of the electrolyte and soft interface, and they further quantify the deviations from predictions based on classical Donnan potential expression valid for dilute electrolytes.