Geometric effects on electrocapillarity in nanochannels with an overlapped electric double layer

Unsteady filling of electrolyte solution inside a nanochannel by the electrocapillarity effect is studied. The filling rate is predicted as a function of the bulk concentration of the electrolyte, the surface potential (or surface charge density), and the cross sectional shape of the channel. For a nanochannel, the average outward normal stress exerted on the cross section of a channel $\left({\overline{P}}_{zz}^{}\right)$ can be regarded as a measure of electrocapillarity and it is the driving force of the flow. This electrocapillarity measure is first analyzed by using the solution of the Poisson-Boltzmann equation. From the analysis, it is found that the results for many different cross sectional shapes can be unified with good accuracy if the hydraulic radius is adopted as the characteristic length scale of the problem. Especially in the case of constant surface potential, for both limits of $\kappa h\to 0$ and $\kappa h\to \infty$, it can be shown theoretically that the electrocapillarity is independent of the cross sectional shape if the hydraulic radius is the same. In order to analyze the geometric effects more systematically, we consider the regular $N$-polygons with the same hydraulic radius and the rectangles of different aspect ratios. Washburn's approach is then adopted to predict the filling rate of electrolyte solution inside a nanochannel. It is found that the average filling velocity decreases as $N$ increases in the case of regular $N$-polygons with the same hydraulic radius. This is because the regular $N$-polygons of the same hydraulic radius share the same inscribing circle.

Figure 1

The surfaces of the control volume. The $S_1$ surfaces are parallel to the channel wall and the $S_2$ surfaces are parallel to the cross section of the channel.

Figure 2

Various cross sectional shapes with the same hydraulic radius $h$: (a) regular polygons and (b) rectangles.

Figure 3

Electric potential ${\psi }^{*}$ distributions with $\kappa h=1$. (a) Two-dimensional electric potential distributions for the case of $V^{*}=1$. (b) Electric potential from the center to the perpendicular foot for the case of $V^{*}=1$. (c) Electric potential from the center to the perpendicular foot for the case of $V^{*}=0.1$. The top solid line is for the analytic solution in Eq. (22).

Figure 4

The dimensionless average total outward stress exerted on the channel wall, $P_{\mathrm{wall}}^{*}=P_{\mathrm{wall}}/n_b{k}_{B}T$, with $V^{*}=zeV/k_{B}T=1$.The solid line is for the analytic solution obtained with the linearized PB equation in Eq. (23).

Figure 5

Plot of ${\overline{P}}_{\mathrm{wall}}^{*}$ vs $V^{*}$ with $\kappa h=1$, where $P_{\mathrm{wall}}^{*}=P_{\mathrm{wall}}/n_b{k}_{B}T$. The solid line is for the analytic solution in Eq. (11) with $\kappa R=1$.

Figure 6

Plot of ${\overline{P}}_{zz}^{*}$ vs $\kappa h$ for $V^{*}=1$, where $P_{zz}^{*}=P_{zz}/n_b{k}_{B}T$. The solid line is for the analytic solution in Eq. (24).

Figure 7

Plot of ${\overline{P}}_{zz}^{*}$ vs $\kappa h$ for $V^{*}=1$, where $P_{zz}^{*}=P_{zz}/n_b{k}_{B}T$. The red solid line is the analytic solution in Eq. (24) and the blue solid line is from Eq. (12) in Ref. [30].

Figure 8

Plot of ${\overline{P}}_{zz}^{*}$ vs $V^{*}$ with $\kappa h=1$, where $P_{zz}^{*}=P_{zz}/n_b{k}_{B}T$. The solid line is for the analytic solution in Eq. (24) with $\kappa R=1$.

Figure 9

Dimensionless electric potential ${\psi }^{*}$ for the constant surface charge density ${\sigma }^{*}=1$. (a) Two-dimensional electric potential distributions for regular polygons with $\kappa h=1$; (b) maximum electric potential at the surface $V_{max}^{*}$. Solid line is for the analytic solution in Eq. (27).

Figure 10

The repulsive pressure exerted on the channel wall is plotted against $\kappa h$ for the case of ${\sigma }^{*}=1$ (a) and against ${\sigma }^{*}$ for the case of $\kappa h=1$ (b). As we can see from the figure, the repulsive pressure can be well represented by the hydraulic radius if $\kappa h\ge 2$ and ${\sigma }^{*}\le 1$.

Figure 11

The dimensionless average total outward normal stress exerted on the surface parallel to the cross section ${\overline{P}}_{zz}^{*}$ in the case of constant surfacecharge. (a) ${\overline{P}}_{zz}^{*}$ against κh for ${\sigma }^{*}=1$; (b) ${\overline{P}}_{zz}^{*}$ against ${\sigma }^{*}$ for $\kappa h=1$. Solid line is for the analytic solution in Eq. (28).

Figure 12

The dimensionless average velocity in the $z$ direction for the regular $N$-polygons.

Figure 13

Dimensionless shape function $f^{*}/C{a}_{\pi }$ with $V^{*}=1,\kappa h=1,\theta ={90}^{\circ }$. (a) Three-dimensional visualization of interfacial shapes; (b) shape functions plotted from the center to the perpendicular foot. Top solid line is for the analytic solution in Eq. (40).

Figure 14

Regular polygon of hydraulic radius $h$.