Ginzburg-Landau theory of solvation in polar fluids: Ion distribution around an interface

We present a Ginzburg-Landau theory of solvation of ions in polar binary mixtures. The solvation free energy arising from the ion-dipole interaction can strongly depend on the composition and the ion species. Most crucial in phase separation is then the difference in the solvation free energy between the two phases, which is the origin of the Galvani potential difference known in electrochemistry. We also take into account an image potential acting on each ion, which arises from inhomogeneity in the dielectric constant and is important close to an interface at very small ion densities. Including these solvation and image interactions, we calculate the ion distributions and the electric potential around an interface with finite thickness. In particular, on approaching the critical point, the ion density difference between the two phases becomes milder. The critical temperature itself is much shifted even by a small amount of ions. We examine the surface tension in the presence of ions in various cases.

Figure 1

Ionic fluid between parallel metal plates with space-dependent dielectric constant $\epsilon$ and charge density $\rho$. Surface charges $Q_L$ and $Q_0$ are fixed.

Figure 2

Normalized electric potential $U\left(z\right)=e\Phi \left(z\right)∕k_{B}T$ between parallel plates with $L=200a$, where we set $\Phi \left(0\right)=0$. The surface charge ${\sigma }_0(=-{\sigma }_L)$ is 0 (a), $2.5\times {10}^{-3}e∕a^2$ (b), and $-2.5\times {10}^{-3}e∕a^2$ (c). The curve of $\psi \left(z\right)=\varphi \left(z\right)-1∕2$ (broken line) indicates the presence of an interface in the middle and a wetting layer in contact with the wall at $z=L$.

Figure 3

Normalized ion densities $c_1\left(z\right)$ and $c_2\left(z\right)$ around an interface where $c_{1\alpha }=c_{2\alpha }=1.75\times {10}^{-3}$ and $c_{1\beta }=c_{2\beta }=0.29\times {10}^{-3}$.

Figure 4

Normalized electric potential $U\left(z\right)=e\Phi \left(z\right)∕k_{B}T$ for three cases of $c_{1\alpha }=1.75\times {10}^{-3}$, 0.018, and 0.037, where we set $\Phi \left(50a\right)=0$.

Figure 5

Order parameter profile $\psi \left(z\right)=\varphi \left(z\right)-1∕2$ for three cases of $c_{1\alpha }=1.9\times {10}^{-4}$ (a), 0.018 (b), and 0.037 (c). For the smallest $c_{1\alpha }$, $\psi \left(z\right)$ is almost equal to the profile without ions.

Figure 6

Normalized excess surface tension $a^2\Delta \gamma ∕k_{B}T$ vs $c_{1\alpha }$ for $\chi =3$ (a), 2.3 (b), and 2.05 (c) with $g_1=4$ and $g_2=2$ (solid lines), and for $\chi =3$ (d), 2.3 (e), and 2.05 (f) with $g_1=10$ and $g_2=5$ (broken lines).

Figure 7

Normalized integrand $\Gamma \left(z\right)$ of Eq. (4.41) for $\chi =2.3$, $g_1=4$, and $g_2=2$ (a), for $\chi =2.3$, $g_1=10$, and $g_2=5$ (b), and for $\chi =3$, $g_1=10$, and $g_2=5$ (c). Its integral with respect to $z∕a$ is equal to $a^2\Delta \gamma ∕k_{B}T$ in the approximate formula (4.40).

Figure 8

Normalized ion density $n\left(z\right)∕n_{\alpha }$, composition $\varphi \left(z\right)$, and image factor $F_{\mathrm{ima}}\left(z\right)$ for $\chi =2.3$, $g_1=4$, $g_2=2$, and $c_{1\alpha }={10}^{-3}$. The gray regions correspond to the two terms in Eq. (4.45).

Figure 9

Normalized ion density $n\left(z\right)∕n_{\alpha }$, composition $\varphi \left(z\right)$, and image factor $F_{\mathrm{ima}}\left(z\right)$ for $\chi =3$, $g_1=4$, $g_2=2$, and $c_{1\alpha }=2\times {10}^{-4}$.

Figure 10

Normalized ion density $n\left(z\right)∕n_{\alpha }$, composition $\varphi \left(z\right)$, and image factor $F_{\mathrm{ima}}\left(z\right)$ for $\chi =3$, $g_1=10$, $g_2=5$, and $c_{1\alpha }=2.1\times {10}^{-5}$. For these parameters, the role of the image factor on $n\left(z\right)$ is more dominant than that of the solvation interaction.

Figure 11

Composition profiles with varying $\chi$ for $g_1=4$, $g_2=2$, $A=4$, and $v_0\overline{n}=0.02$, for which ${\chi }_c^{\text{ion}}=1.91$.

Figure 12

Normalized ion density $c_1\left(z\right)$ and normalized charge density $c_1\left(z\right)-c_2\left(z\right)$ (multiplied by 100) with varying $\chi$ with the same parameter values as in Fig. 11.

Figure 13

Normalized electric potential $U\left(z\right)=e\Phi \left(z\right)∕k_{B}T$ (solid line) at small ion density $c_{1\alpha }=5\times {10}^{-4}$ with $\chi =2.3$, $g_1=10$, $g_2=5$, and $A=4$. Here $c_{1\beta }=5.9\times {10}^{-6}$ and ${\kappa }_{\beta }=2.5\times {10}^{-2}$. It is compared with the Poisson-Boltzmann solution, Eqs. (4.55, 4.56), without the image interaction. Also shown are composition profile $\varphi \left(z\right)$ and the scaled charge density ${10}^3[c_1\left(z\right)-c_2\left(z\right)]$.

Figure 14

Potential reduction ratio in the $\alpha$ region $R_p=[{\Phi }_{\alpha }-\Phi \left(z_{\mathrm{int}}\right)]∕({\Phi }_{\alpha }-{\Phi }_{\beta })$ versus $c_{1\alpha }$, where $\chi =2.3$, $g_1=4$, and $g_2=2$ (a), $\chi =2.3$, $g_1=10$, and $g_2=5$ (b), and $\chi =3$, $g_1=10$, and $g_2=5$ (c).

Figure 15

Normalized ion densities $c_1$ and $c_2$ around an interface, where $g_1=4$, $g_2=-4$, $\chi =2.3$, and $A=4$. The first species is hydrophilic and the second one is hydrophobic. Also shown are $c_1-c_2=a^3\rho ∕e$ and $c_1+c_2=a^{3}n$. A marked electric double layer is formed here.