The effects of ionic valency and size asymmetry on counterion adsorption

Abstract

We study the effect of asymmetry in solvent and ionic size on the equilibrium properties of multivalent ionic solutions near a charged surface. For a single ionic species in solution, we derive a generalized Grahame equation at the charged surface. For general size ratio between the ions and the solvent, we obtain analytical results for the concentration profiles as a function of the distance from the surface. For weak surface charge and small ion-to-solvent size ratio, the profile follows the classical Poisson-Boltzmann equation in dilute solution conditions. However, for high surface charge and large ionic size, the concentration profile saturates near the surface, leading to distinctive dependencies of the solution properties on the surface charge density and size asymmetry. Furthermore, the crossover between dilute and saturated regimes depends on the surface charge and ionic size asymmetry. We suggest that a solution containing multiple ionic species of different valencies and sizes stratifies close to the surface in the saturation regime. This leads to the formation of layers that are ordered according to the ions' valency-to-size ratio.

Figures (4)

• Figure 1: Counterion fraction at the charged surface $\phi_{\rm s}$ (solid blue line) as a function of the size ratio $v$ [Eq. (\ref{['phis_W']})]. (a) The weakly charged surface case; and (b) the strongly charged surface case. Convergence to the low $v$ limit (dashed purple line), Eq. (\ref{['grahame_PB']}), is seen in both cases, whereas in the high $v$ limit (dashed green line), Eq. (\ref{['W_approx']}) is a good approximation only for strongly charged surfaces where $\phi_{\rm s}$ is close to saturation. The vertical dash-dotted black line marks the crossover between the two limiting behaviors, $v^{*}=1/(1+\zeta)$.
• Figure 2: (a) Counterion volume fraction $\phi(x)=a^3v c(x)$, and (b) dimensionless electrostatic potential $\beta e\psi(x)$, as a function of distance from the surface, $x$, for different solvent sizes and size asymmetries. In all cases, the ionic size is held constant, $a^3v = 125$ Å$^3$, the ions are monovalent, $z=1$, and $\alpha=|z|/v=1/v$. The surface charge density is $\sigma{=}2e/{\rm nm^{2}}$. The dashed-dotted vertical line marks the value of $\ell^{*} = a^3\sigma /\alpha e=2.5\,$Å. The approximate parabolic profile of the electrostatic potential (dashed black line) for $v=1/8$ and $a=10$ Å is drawn in (b) from Eq. (\ref{['psi_lstar']}).
• Figure 3: Schematic of a stratified counterion system near a charged surface. The label $\alpha_k$ represents the dominating species in the layer $k$ away from the charged surface. And the stratified layers are characterized by $\alpha_1>\alpha_2>\cdots>\alpha_M$.
• Figure 4: The three regimes of counterion concentration profiles, plotted on the plane of surface charge ($\zeta\sim \sigma^2$) and size asymmetry ($v$). The crossover between the intermediate and saturation regimes occurs at $\zeta=1$, and the one between the dilute and the two other regimes at $v^{*}=1/(1+\zeta)$.