Poisson-Helmholtz-Boltzmann model of the electric double layer: Analysis of monovalent ionic mixtures

In the classical mean-field description of the electric double layer, known as the Poisson-Boltzmann model, ions interact exclusively through their Coulomb potential. Ion specificity can arise through solvent-mediated, nonelectrostatic interactions between ions. We employ the Yukawa pair potential to model the presence of nonelectrostatic interactions. The combination of Yukawa and Coulomb potential on the mean-field level leads to the Poisson-Helmholtz-Boltzmann model, which employs two auxiliary potentials: one electrostatic and the other nonelectrostatic. In the present work we apply the Poisson-Helmholtz-Boltzmann model to ionic mixtures, consisting of monovalent cations and anions that exhibit different Yukawa interaction strengths. As a specific example we consider a single charged surface in contact with a symmetric monovalent electrolyte. From the minimization of the mean-field free energy we derive the Poisson-Boltzmann and Helmholtz-Boltzmann equations. These nonlinear equations can be solved analytically in the weak perturbation limit. This together with numerical solutions in the nonlinear regime suggests an intricate interplay between electrostatic and nonelectrostatic interactions. The structure and free energy of the electric double layer depends sensitively on the Yukawa interaction strengths between the different ion types and on the nonelectrostatic interactions of the mobile ions with the surface.

Figure 1

Scaled surface concentrations of cations $n_{+}\left(0\right)/n_0$ (solid lines) and anions $n_{-}\left(0\right)/n_0$ (dashed lines) as function of the asymmetry parameter $\alpha$. Two sets of curves are displayed, referring to ${\sigma }_n=+1/{\text{nm}}^2$ and ${\sigma }_n=-1/{\text{nm}}^2$ as indicated. No electrostatic interactions are present ($l_e=0$). The remaining parameters are $l_n=0.3\text{nm}$, ${\kappa }^{-1}=0.3\text{nm}$, and $n_0=0.06/{\text{nm}}^3$.

Figure 2

Free energy per unit area $F/\left(A{k}_{B}T\right)$ as function of ${\sigma }_n$ for $\alpha =0$ (solid line), $\alpha =1$ (dashed line), and $\alpha =-1$ (dotted line). The inset shows $F/\left(A{k}_{B}T\right)$ as function of $\alpha$ for two different values of ${\sigma }_n$ as indicated. No electrostatic interactions are present ($l_e=0$). The remaining parameters are $l_n=0.3\text{nm}$, ${\kappa }^{-1}=0.3\text{nm}$, and $n_0=0.06/{\text{nm}}^3$.

Figure 3

Electrostatic potential ${\Psi }_e\left(x\right)$ (upper three curves in the left diagram) and nonelectrostatic potential ${\Psi }_n\left(x\right)$ (lower three curves in the left diagram) and corresponding scaled ion densities $n_{+}\left(x\right)/n_0$ and $n_{-}\left(x\right)/n_0$ (right diagram) as function of the distance $x$ from the charged surface. The three different sets of curves are calculated for $l_n=0$ (dashed lines), $l_n=0.3\text{nm}$ (dotted lines), and $l_n=0.7\text{nm}$ (solid lines). All calculations are based on Eqs. (5) and the corresponding boundary conditions, with $l_e=0.7\text{nm}$, ${\kappa }^{-1}=0.3\text{nm}$, $n_0=0.06/{\text{nm}}^3$, $\alpha =0$, ${\sigma }_n=0$, and ${\sigma }_e=1/{\text{nm}}^2$. This implies ${\kappa }_e^{-1}=1.0\text{nm}$ and ${\kappa }_n^{-1}=0.9\text{nm}$. The horizontal solid line in the right diagram at $n_{+}\left(x\right)/n_0=n_{-}\left(x\right)/n_0=1$ marks the limiting behavior for $l_n\to \infty$.

Figure 4

Solid lines: scaled concentration profiles $n_{-}\left(x\right)/n_0$ of anions for ${\sigma }_n=0.5/{\text{nm}}^2$ (a), ${\sigma }_n=0$ (b), ${\sigma }_n=-0.5/{\text{nm}}^2$ (c), ${\sigma }_n=-1/{\text{nm}}^2$ (d), ${\sigma }_n=-2/{\text{nm}}^2$ (e), and ${\sigma }_n=-4/{\text{nm}}^2$ (f). The dashed line shows $n_{+}\left(x\right)/n_0$ for ${\sigma }_n=-4/{\text{nm}}^2$. All calculations are based on Eqs. (5) and the corresponding boundary conditions, with $l_e=0.7\text{nm}$, ${\kappa }^{-1}=0.3\text{nm}$, $n_0=0.06/{\text{nm}}^3$, $\alpha =0$, $l_n=0.3\text{nm}$, and ${\sigma }_e=1/{\text{nm}}^2$.

Figure 5

Scaled anion concentration $n_{-}\left(x\right)/n_0$ for three different values of the asymmetry parameter $\alpha$. The left and right diagrams are calculated for ${\sigma }_n=0$ and ${\sigma }_n=-2/{\text{nm}}^2$, respectively. The remaining parameters are $l_e=0.7\text{nm}$, ${\kappa }^{-1}=0.3\text{nm}$, $n_0=0.06/{\text{nm}}^3$, $\alpha =0$, $l_n=0.3\text{nm}$, and ${\sigma }_e=1/{\text{nm}}^2$.

Figure 6

Free energy per unit area $F/\left(A{k}_{B}T\right)$ as function of ${\sigma }_n$ for different values of $\alpha$ as indicated. The lower and upper sets of curves refer to ${\sigma }_e=0.35/{\text{nm}}^2$ and ${\sigma }_e=1/{\text{nm}}^2$, respectively. The remaining parameters are $l_e=0.7\text{nm}$, ${\kappa }^{-1}=0.3\text{nm}$, $n_0=0.06/{\text{nm}}^3$, and $l_n=0.3\text{nm}$. The inset compares $F/\left(A{k}_{B}T\right)$ for the linearized [upper curve; according to Eq. (13)] with the nonlinear model (lower curve), for both ${\sigma }_e=0.35/{\text{nm}}^2$ and $\alpha =0$ with the other parameters as above.

Figure 7

Scaled anion concentration as a function of the distance from the charged surface. The dashed line shows the result of the classical Poisson-Boltzmann theory (with $l_e=0.7\text{nm}$, $n_0=6\times {10}^{-4}/{\text{nm}}^3$, and ${\sigma }_e=1/{\text{nm}}^2$), the full line corresponds to the lattice model (with $a=0.5\text{nm}$), and the dotted line corresponds to the Poisson-Helmholtz-Boltzmann model (with $l_n=0.3\text{nm}$, ${\sigma }_n=0$, ${\kappa }^{-1}=0.3\text{nm}$).