Energy conversion by surface-tension-driven charge separation

Abstract

In this work, the shear-induced electrokinetic streaming potential present in free-surface electrolytic flows subjected to a gradient in surface tension is assessed. Firstly, for a Couette flow with fully resolved electric double layer (EDL), the streaming potential per surface stress as a function of the Debye parameter and ζ-potential is analyzed. By contrast to the Smoluchowski limit in pressure-driven channel flow, the shear-induced streaming potential vanishes for increasing Debye parameter (infinitely thin EDL), unless the free surface contains (induced) surface charge or the flow at the charged, solid wall is permitted to slip. Secondly, a technical realization of surface-tension-induced streaming is proposed, with surface stress acting on the free (slipping) surfaces of a micro-structured, superhydrophobic wall. The streaming potential is analyzed with respect to the slip parameter and surface charge. Finally, the surface tension is assumed to vary with temperature (thermocapillarity) or with surfactant concentration (destillocapillarity). The maximal thermal efficiency is derived and compared to the Carnot efficiency. For large thermal Marangoni number, the efficiency is severely limited by the large heat capacity of aqueous solvents. By contrast, destillocapillary flows may reach conversion efficiencies similar to pressure-driven flow.

Appendix: Solution details of the Laplace equation

The velocity field between parallel plates containing periodic patches of no-slip and constant shear regions was estimated using the result by Philip, Eq. (30), for such a flow in an infinite half-plane, $y\ge 0$. No analytical result is known for a finite plate separation. However, the range of validity of this approximation can be assessed numerically. For this, the Laplace equation, $\nabla ^2 w =0$, was discretized using the finite element method as implemented in the commercial code Comsol Multiphysics. By symmetry, the computational domain can be restricted to a unit cell indicated by the gray area in Fig. 4. The boundary conditions are $w(x,0) = 0$ for $0<x<(1-a)W$ (at the solid wall) and $\eta \partial _y w(x,0) = -\tau$ for $(1-a)W<x<W$ (at the constant shear surface). On all other boundaries symmetry conditions apply, i.e., $(\mathbf {n} \cdot \mathbf {\nabla }) w = 0$ with $\mathbf {n}$ being the outward normal at the boundary. The average velocity, $(HW)^{-1}\int _0^H\!\!\int _0^W\!\! w\,{\mathrm{d}}x\,{\mathrm{d}}y$, normalized with the analytic value corresponding to Philip’s solution, $\beta _\Vert W \tau /\eta$, is tabulated in Table 1 for different values of the free-surface fraction, $a=B/W$, and aspect ratio, $H/W$. It is evident from the table that for $H/W=1$ the numerically obtained results deviate by only $\mathcal{O}(10^{-3})$ from the corresponding analytical result and even for $H/W=0.75$ the agreement is $\mathcal{O}(10^{-2})$.

Table 1 Numerical values for the normalized average velocity, $(\beta _\Vert HW^2 \tau /\eta )^{-1}\int _0^H\!\!\int _0^W\!\! w\,{\mathrm{d}}x\,{\mathrm{d}}y$, for different values of the free-surface fraction, $a=B/W$, and aspect ratio, $H/W$, as obtained using a finite element discretization

Fig. 6

Sketch of the integration contour, $\gamma$. Due to periodicity, the integration along sections (2) and (4) cancel

For the analysis presented in the main text, flow rates and line averages of the velocity field are needed. Normalized with the length or area of the integration region, these averages turn out to be identical, attesting the relevance of Table 1. In fact, even for finite H, one can show that

$$\begin{aligned} W^{-1}\int _0^W\! w(x,y_0)dx=W^{-1}\int _0^W\! w(x,y_1){\mathrm{d}}x \end{aligned}$$

(68)

for any $0\le y_0 < y_1 \le H$; thus, line averages and the flow rate are inherently linked. A sketch of a proof of this relation is as follows: Since $w$ is harmonic, $\nabla ^2 w=0$, it is the imaginary part of a holomorphic function $f(x+iy)=v(x,y)+iw(x,y)$ with $v,w:\mathbb {R}^2\rightarrow \mathbb {R}$, (Lawrentjew and Schabat 1967). By Cauchy’s integral theorem, ${\text {Im}}[\oint _\gamma f({\xi })d\xi ]=0$ for any closed path $\gamma$. Choose $\gamma$ as the rectangle with vertices $\xi =iy_0$, $2W+iy_0$, $2W+iy_1$, $iy_1$ as shown in Fig. 6. Due to the Cauchy–Riemann conditions, $v(x,H)$ is constant since $\partial _x v|_{y=H} =-\partial _y w|_{y=H}=0$; similarly, $v(0,y)$ and $v(2W,y)$ are constant since $\partial _x w|_{x=0}=0=\partial _x w|_{x=2W}$. Thus, not only $w(x,y)=w(x+2W,y)$ is periodic in $x$, but so is $v$ and thus $f$. The line integrals on the legs with constant x thus cancel, which completes the proof.