Electrohydrodynamics of binary electrolytes driven by modulated surface potentials

We study the electrohydrodynamics of the Debye screening layer that arises in an aqueous binary solution near a planar insulating wall when applying a spatially modulated ac voltage. Combining this with first order perturbation theory we establish the governing equations for the full nonequilibrium problem and obtain analytic solutions in the bulk for the pressure and velocity fields of the electrolyte and for the electric potential. We find good agreement between the numerics of the full problem and the analytics of the linear theory. Our work provides the theoretical foundations of circuit models discussed in the literature. The nonequilibrium approach also reveals unexpected high-frequency dynamics not predicted by circuit models.

Figure 1

A sketch of the system under study. The binary electrolyte is situated in the half space $x>0$. Below it, for $-d<x<0$, is a planar wall consisting of an insulating dielectric slab of thickness $d$ and below that, for $x<-d$, is a semi-infinite conductor. The top surface, $x=-d$, of the conductor is biased by a periodically modulated potential $V_{\mathrm{ext}}(y,t)$ of period $2\pi ∕q$ (dotted line), which gives rise to the formation of a Debye screening layer of thickness ${\lambda }_D$ in the electrolyte (dashed line).

Figure 2

The potential $\varphi$, pressure $p$, and velocity field $\mathbf{v}$. (a) shows a gray scale plot of the amplitude of the potential $\varphi$ as a function of $qx$ and $qy$, Eq. (25), and (b) the pressure $p$, Eq. (39). Notice the period doubling in the pressure compared to the electric potential. (c) shows a snapshot of the harmonically oscillating velocity field $\mathbf{v}$ in the bulk, Eq. (38), and (d) likewise in the Debye layer, Eq. (35). The flow pattern contains rolls, which are indicated by contours of constant velocity (dashed lines).

Figure 3

The real and the imaginary parts of the normalized effective capacitance $C_{\mathrm{eff}}(V_0,\omega )∕C_0$ versus the normalized frequency $\omega ∕{\omega }^{*}$ for four different values of the voltage amplitude $V_0∕V_T$. The dashed lines show the analytical results of the linear theory, Eq. (23c). The inset shows the relative deviation (below resonance) of the full numerical solution for $C_{\mathrm{eff}}\left(V_0\right)∕C_0$ from the Debye-Hückel result as a function of $V_0∕V_T$.

Figure 4

Numerical simulations of flow. (a) shows the maximal velocity of the harmonic flow ${\max }_{\mathbf{r},t}\left\{v(\mathbf{r},t)\right\}$ and the maximal velocity of the time-averaged flow ${\max }_{\mathbf{r}}\left\{{⟨v(\mathbf{r},t)⟩}_t\right\}$, both versus normalized frequency. The solid lines show exact results within the Debye-Hückel approximation [20] and the dashed line shows ${(\omega ∕{\omega }^{*}+{\omega }^{*}∕\omega )}^{-1}$ as suggested by Eq. (38). (b) shows an example of the time-averaged velocity field ${⟨\mathbf{v}(\mathbf{r},t)⟩}_t$.