Electric Double Layer Composed of an Antagonistic Salt in an Aqueous Mixture: Local Charge Separation and Surface Phase Transition

We examine an electric double layer containing an antagonistic salt in an aqueous mixture, where the cations are small and hydrophilic but the anions are large and hydrophobic. In this situation, a strong coupling arises between the charge density and the solvent composition. As a result, the anions are trapped in an oil-rich adsorption layer on a hydrophobic wall. We then vary the surface charge density $\sigma$ on the wall. For $\sigma >0$ the anions remain accumulated, but for $\sigma <0$ the cations are attracted to the wall with increasing $|\sigma |$. Furthermore, the electric potential drop $\mathrm{\Psi }(\sigma )$ is nonmonotonic when the solvent interaction parameter $\chi (T)$ exceeds a critical value ${\chi }_c$ determined by the composition and the ion density in the bulk. This leads to a first-order phase transition between two kinds of electric double layers with different $\sigma$ and common $\mathrm{\Psi }$. In equilibrium such two-layer regions can coexist. The steric effect due to finite ion sizes is crucial in these phenomena.

Figure 1

Electric double layer containing small hydrophilic cations (${\mathrm{Na}}^{+}$) and large hydrophobic anions (${\mathrm{BPh}}_4^{+}$) in an aqueous mixture on a metal wall. There can be two kinds of ion distributions with a common potential drop $\mathrm{\Psi }$, which coexist in certain conditions (see Figs. 4, 5, 6). Gradation of solvent region represents water concentration.

Figure 2

Hydrophilic cations $n_1(z)$ and hydrophobic anions $n_2(z)$ next to hydrophobic wall for $\sigma =0$ with $n_0=4\times {10}^{-3}{v}_0^{-1}$. Bulk water composition ${\varphi }_{\infty }$ and interaction parameter $\chi$ are (a) 0.35 and 1.4, (b) 0.65 and 2.0, and (c) 0.5 and 1.9. Water profile $\varphi (z)$ is also shown (insets). Anions are richer in the oil-rich adsorption layer. Steric effect due to finite sizes of ions are accounted for (bold lines). Broken lines represent anion profiles without steric effect ($v_1=v_2=0$).

Figure 3

1D profiles of ions (left), $\varphi (z)$ (middle), and $e\psi (z)/T$ (right) for $\sigma =0.03$ (top) and $-0.2$ (bottom) near a hydrophobic wall, where ${\varphi }_{\infty }=0.35$ and $\chi =1.4$. These states are stable on the curve in Fig. 4. Broken lines are obtained without steric effect ($v_1=v_2=0$).

Figure 4

Potential drop $\mathrm{\Psi }$ vs surface charge density $\sigma$ (in units of $T/e$ and $e/a^2$, respectively) from 1D solutions for ${\varphi }_{\infty }=0.35$, where (a) $\chi =-0.4,-0.6,-0.8,-1,-1.2$, and $-1.4$ and (b) $\chi =1.4$. From Eq. (9), first-order phase transition occurs between two-layer states at $\sigma ={\sigma }_1$ and ${\sigma }_2$. Two colored regions in yellow have the same area in (a) and (b). Two points ($\circ$) in (b) on the curve represent two states at $\sigma =-0.2$ and 0.03 in Fig. 3. Dotted line in (b) represents $\mathrm{\Psi }$ without steric effect ($v_1=v_2=0$).

Figure 5

Coexistence curves in (a) $\sigma \text{−}\chi$ plane and (b) $\mathrm{\Psi }\text{−}\chi$ plane for ${\varphi }_{\infty }=0.35$, 0.4, 0.5, 0.55, and 0.65 at $n_0=4\times {10}^{-3}{v}_0^{-1}$. For each ${\varphi }_{\infty }$, two layers coexist inside the corresponding curve in (a), while $\mathrm{\Psi }$ is a field variable common in coexisting two layers. Critical points are marked (×).

Figure 6

Coexistence of two layers with $\sigma =0.02$ and $-0.14$ for ${\varphi }_{\infty }=0.35$ and $\chi =2$ (see Fig. 5). (a) $\varphi (x,z)$ and (b) $v_0[n_1(x,z)-n_2(x,z)]$ on the $zx$ plane. (c) $\sigma (x)a^2/e$ with $\overline{\sigma }=Q/S$ being ($A$) $-0.04$ and ($B$) $-0.07$ in units of $e/a^2$. (d) $[\omega (x)-\mathrm{\Psi }\sigma (x)]a^2/T$ for ($A$) and ($B$). Cross-sectional profiles of (e) $v_0{n}_1(x,a)$ and (f) $v_0{n}_2(x,a)$ at $z/a=0$, 1, 2, and 3, exhibiting small peaks at boundaries for $z\ge a$.