Wedge wetting by electrolyte solutions

The wetting of a charged wedgelike wall by an electrolyte solution is investigated by means of classical density functional theory. As in other studies on wedge wetting, this geometry is considered as the most simple deviation from a planar substrate, and it serves as a first step toward more complex confinements of fluids. By focusing on fluids containing ions and surface charges, features of real systems are covered that are not accessible within the vast majority of previous theoretical studies concentrating on simple fluids in contact with uncharged wedges. In particular, the filling transition of charged wedges is necessarily of first order, because wetting transitions of charged substrates are of first order and the barrier in the effective interface potential persists below the wetting transition of a planar wall; hence, critical filling transitions are not expected to occur for ionic systems. The dependence of the critical opening angle on the surface charge, as well as the dependence of the filling height, of the wedge adsorption, and of the line tension on the opening angle and on the surface charge are analyzed in detail.

Figure 1

Schematic depiction of the studied system. The two unit vectors ${\vec{e}}_u$ and ${\vec{e}}_{u^{\prime }}$ are parallel to the two walls, which meet at the opening angle $\theta$. An arbitrary location $\vec{r}$ can be specified by the lateral and the normal components ${\vec{r}}_u,{\vec{r}}_v$ or ${\vec{r}}_{u^{\prime }},{\vec{r}}_{v^{\prime }}$ with respect to the walls. The parallelogram close to the wedge apex indicates the geometry of the unit cells by which the space in between the walls is tiled.

Figure 2

Bulk phase diagram in terms of the solvent chemical potential ${\mu }_0^{*}$ and the temperature $T^{*}$ for fixed ionic strength $I$. The solid red line represents the liquid-gas coexistence line for the salt-free case ($I=0$), which is given by the analytical expression ${\mu }_0^{*}=-\frac{1}{T^{*}}$. The black crosses indicate points of the liquid-gas coexistence curve for the case $I=5\mathrm{mM}$. The shift is small, which also holds for all ionic strengths used in this work (up to $I=100\mathrm{mM}$).

Figure 3

(a) Effective interface potential $\beta \omega$ as a function of the film thickness $l$ of a liquid film between a planar wall and the gas bulk phase and (b) equilibrium total packing fraction profile ${\varphi }_{\text{tot}}\left(v\right)$, with $v$ denoting the distance from the wall (see Fig. 1), corresponding to the minimum of $\beta \omega \left(l\right)$ for ionic strength $I=100\mathrm{mM}$, temperature $T^{*}=0.43$, wall-fluid interaction strength $h=0.09327$, decay length $\lambda =2\mathrm{d}$, and surface charge density $\sigma =0.03\mathrm{e}/{\mathrm{d}}^2$. The inset in panel (b) shows the corresponding ion packing fractions ${\varphi }_{+}$ and ${\varphi }_{-}$ as functions of the distance $v$ from the wall. Panel (a) identifies the system exhibiting partial wetting for the present configuration.

Figure 4

Excess adsorption $\mathrm{\Gamma }$ [see Eq. (9)] for (a) an electrically neutral planar wall ($\sigma =0$) and decay length $\lambda =1\mathrm{d}$ as function of the wall-fluid interaction strength $h$ and (b) three different sets of wall-fluid interaction strength $h$ and decay length $\lambda$ as function of the wall charge density $\sigma$. Both panels exhibit an increase of the excess adsorption $\mathrm{\Gamma }$ upon approaching critical values $h_C$ or ${\sigma }_C$, respectively, at which the system undergoes a wetting transition. The discontinuity of $\mathrm{\Gamma }$ at the critical values identifies the wetting transition to be of first order. This can also be verified by considering the effective interface potential $\beta \omega \left(l\right)$, as shown in the inset in panel (a) for conditions slightly above the wetting transition. The barrier separating the local minimum at small film thickness l from the global minimum at large film thickness l proves the first-order nature of the wetting transition.

Figure 5

Critical opening angle ${\theta }_C$ of the wedge, at which the filling transition occurs, as function of the wall charge density $\sigma$ for decay lengths $\lambda \in \{1\mathrm{d},2\mathrm{d}\}$. For $\theta >{\theta }_C$ the wedge is macroscopically empty, whereas for $\theta <{\theta }_C$ it is filled by liquid. The values ${\theta }_C$, derived via Eqs. (1) and (12), increase with increasing wall charge $\sigma$. At wall charge density $\sigma ={\sigma }_C$ the critical angle of the filling transition is ${\theta }_C={180}^{\circ }$, i.e., the filling transition is actually the wetting transition of a planar wall (see Sec. 3b).

Figure 6

Equilibrium packing fraction profiles ${\varphi }_{\text{tot}}(x,y)$ inside a wedge with opening angle (a) $\theta ={180}^{\circ }$ and (b) $\theta ={180}^{\circ }$ for ionic strength $I=100\mathrm{mM}$, wall-fluid interaction strength $h=0.09327$, decay length $\lambda =2\mathrm{d}$, and wall charge density $\sigma =0.03\mathrm{e}/{\mathrm{d}}^2$. Far away from the symmetry plane of the wedge the packing fraction profiles coincide with those at planar walls [see Fig. 3]. Upon decreasing the opening angle $\theta$, an increase of the density close to the tip of the wedge occurs [see panel (b)].

Figure 7

Wedge adsorption $\mathrm{\Delta }$ [see Eq. (13)] as function of the opening angle $\theta$ of the wedge and of the wall charge density $\sigma$. In panel (a) the decay length of the wall-fluid interaction potential is $\lambda =1\mathrm{d}$, whereas in panel (b) it is $\lambda =2\mathrm{d}$. Similar to the filling height $l_w$ shown in Fig. 8, the wedge adsorption $\mathrm{\Delta }$ increases for increasing wall charge density $\sigma$ as well as for decreasing opening angle $\theta$. The limits of $\mathrm{\Delta }$ upon approaching the filling transition, $\theta \searrow {\theta }_C$, are finite, which signals a first-order filling transition. To highlight this, the inset in panel (a) shows a double-logarithmic plot of the wedge adsorption $\mathrm{\Delta }$ as function of the distance $\theta -{\theta }_C$ from the filling transition.

Figure 8

Filling height at the symmetry plane $l_w$ as function of the opening angle $\theta$ of the wedge and of the wall charge density $\sigma$. In panel (a) the decay length of the wall-fluid interaction is $\lambda =1\mathrm{d}$, whereas it is $\lambda =2\mathrm{d}$ in panel (b). The dashed black curve in both panels corresponds to the thickness of the first layer of cells on the symmetry axis. The comparison of this curve with the curves of the filling height $l_w$ shows, that the increase of $l_w$ close to the critical opening angle $\theta \gtrsim {\theta }_C$ stems from the increasing interactions close to the tip of the wedge. Furthermore, the filling height $l_w$ increases with both an increasing wall charge density $\sigma$ as well as a decreasing opening angle $\theta$. The finite limits for $l_w$ upon $\theta \searrow {\theta }_C$ point to a first-order filling transition. Similarly in Fig. 7 the inset in panel (a) shows a double-logarithmic plot of the filling height $l_w$ as function of the distance $\theta -{\theta }_C$ from the filling transition to verify its first-oder nature.

Figure 9

Line tension $\tau$ as function of the opening angle $\theta$ and of the wall charge density $\sigma$ for decay lengths (a) $\lambda =1\mathrm{d}$ and (b) $\lambda =2\mathrm{d}$. For small wall charge density $\sigma$, the line tension $\tau$ is negative for all opening angles $\theta$, whereas for sufficiently large $\theta$ and $\sigma$ positive values of $\tau$ may occur.