Dressed counterions: Polyvalent and monovalent ions at charged dielectric interfaces

We investigate the ion distribution and overcharging at charged interfaces with dielectric inhomogeneities in the presence of asymmetric electrolytes containing polyvalent and monovalent ions. We formulate an effective “dressed counterion” approach by integrating out the monovalent salt degrees of freedom and show that it agrees with results of explicit Monte Carlo simulations. We then apply the dressed counterion approach within the framework of the generalized strong-coupling theory, valid for polyvalent ions at low concentrations, which enables an analytical description for salt effects as well as dielectric inhomogeneities in the limit of strong Coulomb interactions. Limitations and applicability of this theory are examined by comparing the results with simulations.

Figure 1

Schematic view of an impenetrable charged dielectric interface of surface charge density $-\sigma$ bathed by a monovalent 1:1 salt and polyvalent $q$:1 salt, with larger $q$-valent counterions. For the water-hydrocarbon interface one has $\varepsilon =80$, ${\varepsilon }^{\prime }=2$, which gives dielectric $\Delta =0.95$ that allows for an approximate treating as $\Delta =1$.

Figure 2

Comparison of explicit and implicit MC simulations for $\Xi =50$. (a) Counterion density profile at fixed bulk amount of counterions $\mathrm{\chi ̃}$ and two different salt concentrations $\mathrm{\kappa ̃}$. Implicit MC results (solid line) compare quite well with explicit MC results (dots). (b) Cumulative charge that follows from the results in (a). In the case of smaller screening ($\mathrm{\kappa ̃}=0.30$) the cumulative charge reaches positive values, which corresponds to overcharging of $3\hspace{0.5em}\%$. (c) “Phase diagram” showing the deviations between implicit and explicit MC simulations for $\Xi =50$. The agreement is better for smaller values of $\mathrm{\kappa ̃}$.

Figure 3

SC dressed counterion profiles for (a) various coupling parameters, (b) various screening parameters, and (c) various dielectric contrast parameters.

Figure 4

(a) “Phase diagram” showing the minimal amount of polyvalent salt $\mathrm{\chi ̃}$ that is required to achieve overcharging (solid lines) within the SC dressed counterion theory. Dotted lines correspond to the situation where $\beta W_{\mathit{cc}}^{\mathrm{mf}}=1$, Eq. (26), with the symbols and colors corresponding to the same parameter $\Xi$ as shown on the graph. Parameter space below dotted lines should be well described by the SC dressed counterion approach. (b) Rescaled density profile for dressed counterions for $\Xi =50$ and screening $\mathrm{\kappa ̃}=0.4$. The SC prediction is independent of $\mathrm{\chi ̃}$ and gives good results for low enough $\mathrm{\chi ̃}$. With increasing $\mathrm{\chi ̃}$ the implicit MC results (symbols) show gradual deviations from the theoretical prediction. (c) Cumulative charge, Eq. (16), as follows from the profiles in (b) for the SC theory (solid lines) and implicit MC simulations (symbols).

Figure 5

“Phase diagram” characterizing the overcharging of a charged surface. The threshold values of minimal $\mathrm{\chi ̃}$ neccessary for overcharging are obtained by all three different methods, i.e., explicit and implicit MC simulations as well as the SC dressed counterion theory.