Modified Donnan model for ion binding to small soft nano particles

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, $V_{\mathrm{mD}}$, can be used to obtain an average particle potential,${\psi }_{\mathrm{mD}}$, but the physical background is somewhat obscure. Therefore, the $\mathrm{mD}$ model is discussed using the Poisson-Boltzmann theory. $V_{\mathrm{mD}}$ is determined by the specific volume of the ‘intra’ plus ‘extra’ particle diffuse charge, and it is shown that ${\psi }_{\mathrm{mD}}$ is a volumetric average of the intra particle potentials and the surface potential. Theoretical equations for $V_{\mathrm{mD}}$ are derived for plate, disk and spherical geometries. In general, $V_{\mathrm{mD}}=S{V}_{p}R$, where $S{V}_p$ is a specific soft particle volume, and R is a measure of the effect of extra particle diffuse layer on $V_{\mathrm{mD}}$. When limited information about the NP is available, the theoretical equation for $V_{\mathrm{mD}}$ can be approximated as $V_{\mathrm{mD}}=\alpha I^{-\beta }$, with adjustable parameters $\alpha$ and $-\beta$, where $I^{-\beta }$ is a measure of R and $\alpha$ an approximation of $S{V}_p$. A better understanding of the $\mathrm{mD}$ model will improve its future applications.