Strongly nonlinear dynamics of electrolytes in large ac voltages

We study the response of a model microelectrochemical cell to a large ac voltage of frequency comparable to the inverse cell relaxation time. To bring out the basic physics, we consider the simplest possible model of a symmetric binary electrolyte confined between parallel-plate blocking electrodes, ignoring any transverse instability or fluid flow. We analyze the resulting one-dimensional problem by matched asymptotic expansions in the limit of thin double layers and extend previous work into the strongly nonlinear regime, which is characterized by two features—significant salt depletion in the electrolyte near the electrodes and, at very large voltage, the breakdown of the quasiequilibrium structure of the double layers. The former leads to the prediction of “ac capacitive desalination” since there is a time-averaged transfer of salt from the bulk to the double layers, via oscillating diffusion layers. The latter is associated with transient diffusion limitation, which drives the formation and collapse of space-charge layers, even in the absence of any net Faradaic current through the cell. We also predict that steric effects of finite ion sizes (going beyond dilute-solution theory) act to suppress the strongly nonlinear regime in the limit of concentrated electrolytes, ionic liquids, and molten salts. Beyond the model problem, our reduced equations for thin double layers, based on uniformly valid matched asymptotic expansions, provide a useful mathematical framework to describe additional nonlinear responses to large ac voltages, such as Faradaic reactions, electro-osmotic instabilities, and induced-charge electrokinetic phenomena.

Figure 1

Sketch of 1D model problem. The electrolyte is confined between parallel-plate blocking electrodes separated by a gap of width $2L$, and a harmonic potential of $V_{\text{ext}}\left(t\right)=\pm V\text{sin}\left(\omega t\right)$ is applied to the left and right electrodes, respectively, so the overall potential drop across the cell is $2V\text{sin}\left(\omega t\right)$; this corresponds to a $4V$ peak-to-peak voltage or $\sqrt{2}V$ rms.

Figure 2

Numerical solution of the PNP equations for $V=30$, $\omega =0.3$, $\delta =0.3$, and $ϵ=0.001$. (a) Potential variation in time and space; solid black line shows the external potential on the electrodes $V_{\text{ext}}$. The inset zooms onto the rapid potential variation across the diffuse screening layer. (b) Zoom on cation concentration near the electrodes; anion concentration is identical but phase shifted one half period in time.

Figure 3

Numerical solution of the PNP equations for $V=120$, $\omega =0.3$, $\delta =0.3$, and $ϵ=0.001$. (a) Potential variation in time and space. (b) Zoom on cation concentration near the electrodes.

Figure 4

(Color online) Schematic picture of nested boundary layers in matched asymptotic expansion: the bulk outer region is connected via the middle diffusion layer to the inner diffuse layer. A compact (Stern) layer separates the electrolyte from the blocking electrode. At large voltage, local salt depletion in the diffusion layer can cause the double layer to change to a nonequilibrium structure, with an extended “space-charge” layer that is completely depleted of coions. Further, when the concentration in the diffuse layer approaches the steric limit, a condensed phase of ions forms at the electrode. The figure also indicates some of the variables and parameters introduced in the asymptotic analysis of the different layers.

Figure 5

(Color online) Quasisteady (dimensionless) accumulated charge $\tilde{q}$ in the double layer as a function of its voltage drop $\tilde{\Psi }=\tilde{\zeta }-\tilde{q}\delta$, plotted for different values of the Stern parameter $\delta$ with $\nu =0$ (solid line), and different values of the steric parameter $\nu$ with $\delta =0$ (dashed line), all in the weakly nonlinear regime with $\widehat{c}=1$.

Figure 6

(Color online) Distribution of dimensionless cell voltage, $V_{\text{ext}}=\overline{J}+\tilde{\zeta }-\tilde{q}\delta$ (dotted line), divided into contributions across the bulk electrolyte $\overline{J}$ (solid line), diffuse layer $\zeta$ (dashed line), and compact layer $-q\delta$ (dashed-dotted line), all scaled to the thermal voltage $kT/ze$, as a function of time for different values of the parameters $V$, $\omega$, $\delta$, and $\nu$. The panels show (a) Debye-Hückel limit, (b) Gouy-Chapman-Stern (GCS) model, (c) Gouy-Chapman (GC) model, and (d) Bikerman-Freise (BF) model.

Figure 7

(Color online) (a) Equivalent circuit representation for weakly nonlinear dynamics: compact and diffuse-layer capacitors in series with a bulk resistance. (b) Bode plot of the magnitude $\left|Z\right|$ and phase angle $\angle Z$ of the half-cell impedance for increasing driving voltage at $\delta =0.3$ and $\nu =0$. The characteristic frequency ${\omega }_o$, where $\angle Z$ passes through $-45°$ and $\left|Z\right|$ bends up, is marked with circles.

Figure 8

(Color online) Characteristic frequency ${\omega }_o$ vs driving voltage, plotted for different values of $\delta$ with $\nu =0$ (solid line), and different values of $\nu$ with $\delta =0$ (dashed line).

Figure 9

(Color online) Contour plot of $⟨\tilde{w}⟩$, the time-average excess salt concentration in the diffuse layer, as a function of driving frequency and voltage for $\delta =0$ and $\nu =0.01$. The dashed line marks the characteristic frequency ${\omega }_o$.

Figure 10

Sketch of the different dynamical regimes for a blocking cell in the space of applied voltage $V$ and nominal bulk volume fraction of ions, $\nu =2c^{\ast }/c_{\text{max}}$. Linear response holds for $V⪡1$ for any $\nu$ and diffuse-layer thickness $ϵ={\lambda }_D/L$. For thin diffuse layers $\left(ϵ⪡1\right)$ in electrolytes $\left(\nu ⪡1\right)$ there is a transition for $V>1$ to weakly nonlinear dynamics, where the diffuse layer acts as a voltage-dependent capacitor in series with a constant bulk resistance; at larger voltages, there is a transition to strongly nonlinear dynamics, which occurs first only with the oscillating diffusion layers (dashed-dotted line); at higher voltages there is another transition (dotted line) where the bulk solution becomes uniformly depleted by time-averaged mass transfer into the diffuse layers. The transition curves rise steeply with $\nu$. For ionic liquids and the molten salt limit $\nu \approx 1$, only the weakly nonlinear regime is possible since there is not enough volume available to compress significant numbers of ions in the diffuse layers, which approach the molecular scale.

Figure 11

(Color online) Strongly nonlinear response at $V=30$, $\omega =0.3$, $\delta =0.3$, $\nu =0$, and $ϵ=0.001$. (a) Distribution of the cell voltage, $V_{\text{ext}}=\overline{J}/{\overline{c}}_o+\tilde{\zeta }-\tilde{q}\delta$ (dotted line), onto the bulk electrolyte, $\overline{J}/{\overline{c}}_o$ (circles), diffuse layer, $\tilde{\zeta }$ (squares), and compact layer, $-\tilde{q}\delta$ (triangles). (b) Concentration ${\widehat{c}}_s$ at the inner edge of the diffusion layer. Symbols show results from the full numerical solution of the PNP equations, while the solid lines are predictions of our (much simpler) uniformly valid asymptotic approximations of Sec. 3, which are seen to be in excellent agreement.

Figure 12

(Color online) Contour plot of the minimal salt concentration ${\text{min}}_t{\widehat{c}}_s$ in the diffusion layer as a function of driving voltage and frequency for (a) $\delta =0.3$, $\nu =0$, and $ϵ=0.001$; (b) $\delta =0$, $\nu =0.01$, and $ϵ=0.001$. The dashed line marks the characteristic frequency ${\omega }_o$, the dotted line marks the critical voltage $V_c$ estimated by Eq. (70), and the shaded area marks a regime where ${\text{min}}_t{\widehat{c}}_s$ drops to zero and the double layer is driven out of quasiequilibrium. [For the lowest frequencies in the figure, the $O\left(\sqrt{ϵ/2\omega }\right)\approx 0.2$ wide diffusion layers extend across most of the bulk and no longer act as mathematical boundary layers.]

Figure 13

(Color online) Strongly nonlinear response at $V=120$, $\omega =0.3$, $\delta =0.3$, $\nu =0$, and $ϵ=0.001$. (a) Distribution of the cell voltage, $V_{\text{ext}}=\overline{J}/{\overline{c}}_o+\breve{\Phi }+\tilde{\zeta }-\tilde{q}\delta$ (dotted line), onto the bulk electrolyte, $\overline{J}/{\overline{c}}_o$ (circles), diffuse layer (dashed line), space-charge layer (dashed-dotted line), and compact layer, $-\tilde{q}\delta$ (triangles); also shown is the sum of the space-charge and diffuse-layer voltages, $\tilde{\zeta }+\breve{\Phi }$ (squares). (b) Concentration ${\widehat{c}}_s$ at the inner edge of the diffusion layer (triangles) and extent $y_o$ of space-charge layer (crosses). Solid and broken lines show results from our asymptotic model, and symbols show the full numerical solution of the PNP equations.

Figure 14

(Color online) Concentration profiles at $V=120$, $\omega =0.3$, $\delta =0.3$, $\nu =0$, and $ϵ=0.001$; “frames” cover one-half period in time starting from $t=0$. Open and filled circles show the cation and anion concentrations, respectively, according to the full numerical solution from Fig. 3, and solid lines show uniformly valid approximations based on the asymptotic analysis [Eq. (118)] with very good qualitative and quantitative agreement. Black arrows mark the extent $y_o\left(t\right)$ of the space-charge layer (if any) according to the asymptotic model.

Figure 15

(Color online) Strongly nonlinear response for MPB model with relatively large and small bulk ion volume fractions, $\nu =0.01$ and 0.0001, respectively, at $V=120$, $\omega =0.3$, $\delta =0$, and $ϵ=0.001$. (a) and (c) Distribution of the cell voltage, $V_{\text{ext}}=\overline{J}/{\overline{c}}_o+\breve{\Phi }+\tilde{\zeta }$ (dotted line), onto the bulk electrolyte, $\overline{J}/{\overline{c}}_o$ (solid line), diffuse layer (dashed line), and space-charge layer (dashed-dotted line). (b) and (d) Concentration ${\widehat{c}}_s$ at the inner edge of the diffusion layer (solid line) and extent $y_o$ of space-charge layer (dashed line).

Figure 16

(Color online) Peak voltages on space-charge layer, ${\text{max}}_t\breve{\Phi }$ (solid line), diffuse layer, ${\text{max}}_t\tilde{\zeta }$ (dashed line), and compact layer, ${\text{max}}_t\tilde{q}\delta$ (dashed-dotted line), for different values of the capacitance ratio $\delta$ and nominal ion volume fraction $\nu$. (a)–(c) show results at $\omega =0.3$ as a function of $V$, and (d)–(f) show results at $V=120$ as a function of $\omega$. For the PB model, (a) and (d) show that the compact-layer voltage $-\tilde{q}\delta$ dominates at $\delta =0.3$, although $\breve{\Phi }$ becomes significant at large voltage. For the MPB model, (b) and (e) show that the diffuse-layer voltage $\tilde{\zeta }$ dominates at $\nu =0.01$, while in (c) and (f) the space-charge layer voltage $\breve{\Phi }$ dominates at $\nu =0.0001$.

Figure 17

(Color online) Solution in Smyrl-Newman transition layer in terms of Painlevé transcendents. (a) Rescaled electric field $P\left(z\right)$ (solid line) and leading-order terms from asymptotic approximation in the space-charge layer, $z⪡-1$, and diffusion layer, $z⪢1$, (dotted line). The dashed line shows a solution $\tilde{P}$ to Eq. (A8) with b.c. $\tilde{P}={\tilde{P}}_o=-10$ applied at $z=-z_o=-5$. (b) Potential variation $\varphi =-{\int }_0^{z}P\left(z^{\prime }\right)d{z}^{\prime }$ in transition layer (solid line), and leading-order asymptotics (dotted line). Again, the dashed line shows the result for the solution $\tilde{P}$. (c) Rescaled charge distribution ${\partial }_{z}P$ (solid line) and leading-order asymptotics (dotted line). Symbols show individual ion concentrations [cf. Eq. (A12)]. (d) Excess voltage $\tilde{\zeta }$ as a function of $z_o$ with ${\tilde{P}}_o=-10$ (solid line), and leading-order terms from asymptotic approximation (dotted line). The circle marks $z_o=5$ corresponding to $\tilde{P}$ from (a) and (b), and the dashed and dashed-dotted lines show the approximation by Eqs. (A22, A27), respectively.