Nonlinear dynamics of capacitive charging and desalination by porous electrodes

The rapid and efficient exchange of ions between porous electrodes and aqueous solutions is important in many applications, such as electrical energy storage by supercapacitors, water desalination and purification by capacitive deionization, and capacitive extraction of renewable energy from a salinity difference. Here, we present a unified mean-field theory for capacitive charging and desalination by ideally polarizable porous electrodes (without Faradaic reactions or specific adsorption of ions) valid in the limit of thin double layers (compared to typical pore dimensions). We illustrate the theory for the case of a dilute, symmetric, binary electrolyte using the Gouy-Chapman-Stern (GCS) model of the double layer, for which simple formulae are available for salt adsorption and capacitive charging of the diffuse part of the double layer. We solve the full GCS mean-field theory numerically for realistic parameters in capacitive deionization, and we derive reduced models for two limiting regimes with different time scales: (i) in the “supercapacitor regime” of small voltages and/or early times, the porous electrode acts like a transmission line, governed by a linear diffusion equation for the electrostatic potential, scaled to the RC time of a single pore, and (ii) in the “desalination regime” of large voltages and long times, the porous electrode slowly absorbs counterions, governed by coupled, nonlinear diffusion equations for the pore-averaged potential and salt concentration.

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Figure 1

Sketch of the model problem. At $t=0$, a voltage ${\varphi }_1$ is applied to an ideally polarizable, porous electrode of thickness $L_e$, relative to a bulk electrolytic solution, from which it is separated by a stagnant diffusion layer of thickness $L_{\text{sdl}}$. The characteristic pore thickness $h_p$ (defined as the ratio of the pore volume to the pore area) is much larger than the Debye screening length, ${\lambda }_D$, so the pore space is mostly filled with quasi-neutral electrolyte, exchanging ions with a charged, thin double-layer “skin” on the electrode matrix. The volume-averaged potential $\varphi \left(x,t\right)$ and neutral salt concentration $c\left(x,t\right)$ in the pore space vary as counterions enter, and coions leave, the thin double layers, thus changing the mean surface charge density $q\left(x,t\right)$ and total excess salt density $w\left(x,t\right)$. The dimensionless coordinate $x$ used in Figs. 3, 4 is also indicated.

Figure 2

Sketch of the response of the blocking porous-electrode system in Fig. 1 to a suddenly applied voltage. (a) In the (“weakly nonlinear” [25]) “supercapacitor regime” of small voltages, or early times, the porous electrode charges both by adsorbing counterions and by expelling coions from the diffuse part of the double layers; meanwhile, the quasi-neutral solution phase maintains a nearly constant salt concentration everywhere, both inside the pores and in the external solution. (b) In the (“strongly nonlinear” [25]) “desalination regime” of large voltages and long times, the coions are mostly expelled from the double layers, as counterions adsorb at high concentration; as a result, the quasi-neutral solution becomes depleted of salt, both inside the pores and in the outside solution.

Figure 3

Equivalent-circuit interpretation of the linearized charging dynamics of a porous electrode, Eq. (19), as a transmission line [23], where the quasi-neutral solution in the pores acts as a series of resistors coupled to the electrode by parallel double-layer capacitors.

Figure 4

Capacitive response of porous electrode to step change in electrode potential as function of time and position. Arrow shows the direction of time, $0.001<t<0.5$. Parameter settings in the text. (a) Electrostatic potential, $\varphi$. (b) Salt concentration, $c$.

Figure 5

(Color online) (a) Ion current profiles in the electrode as function of time. Arrows show the direction of time ($0.001<t<0.5$; first solid curves, next dashed curves). Ion current at position $x=0$ equals the total current. (b) Total current versus time $t$, and salt transport rate evaluated at the outer edge of the SDL.

Figure 6

Sketch of a macroscopic volume element of volume $V$ containing a pore space of volume $V_p$ and internal surface area $A_p$, which intersects the volume element with cross-sectional area $A_v$.