Abstract
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.
Highlights
Porous electrodes in contact with aqueous solutions1–4͔ are found in many technological applications, such as the storage of electrical energy inelectrostatic double-layersupercapacitors5–13͔, capacitive deionizationCDIfor water desalination2,14–23͔, and the reverse process of CDI, namely, the capacitive extraction of energy from the salinity difference between different aqueous streams, for instance, river and sea water24͔
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: ͑iin 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, andiiin 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
In some of the applications, such as in CDI, the goal is to strongly modify the bulk salt concentration, but here we focus on modeling the dynamics of ions within the porous electrode, which in a later stage can be combined with more complicated models of the bulk solution, e.g., allowing for fluid flow and other geometries
Summary
Porous electrodes in contact with aqueous solutions1–4͔ are found in many technological applications, such as the storage of electrical energy inelectrostatic double-layersupercapacitors5–13͔, capacitive deionizationCDIfor water desalination2,14–23͔, and the reverse process of CDI, namely, the capacitive extraction of energy from the salinity difference between different aqueous streams, for instance, river and sea water24͔. The BTA analysis has since been extended by many authors, e.g., to polarizable particles in step electric fields26͔ ͑also taking into account tangential ion transport through the double layers34͔͒, electrolytes in large ac voltages27,28͔ ͑accounting for the imposed time scale and the transient formation of extended space charge35͔͒, and ac electro-osmotic flows32,36͔ ͑which couple ionic relaxation to fluid motion37͔͒ In all of these situations, the same two dynamical regimes can be identified, separated by a transition voltagearound 10 kT / ewhere counterion adsorption by the double layers begins to dominate capacitive charging. A key quantity controlling this dynamical transition is the charge efficiencyw / q in the notation of BTA, as coion expulsion at low voltages is replaced by additional counterion adsorption at high voltages We illustrate these basic principles by numerical simulations of CDI using the full, nonlinear mean-field theory and by analyzing reduced model equations for the limiting regimes
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