Imposed currents in galvanic cells
Introduction
Reversible galvanic cells, such as redox flow cells [1], [2], [3], [4], reversible solid oxide fuel cells [5], [6], [7], [8], [9], and rechargeable batteries [10], [11], [12] are electrochemical cells that can run both in a galvanic mode, thereby converting chemical energy in electric energy, and in the reverse, electrolytic, mode where electrical energy is returned into chemical energy. In the galvanic mode electrons flow spontaneously through an external load toward the more positive electric potential in the direction from anode to cathode, whereas in the electrolytic mode, the electrons are pumped toward the more negative potential of the cathode. Similar to rechargeable batteries, redox flow cells and reversible solid oxide fuel cells can be used to store electrical energy in the form of chemical energy. The main difference with rechargeable batteries is that the chemical energy is stored in a fluid phase which can be pumped to a secondary container, rather than being stored in a finite-capacity solid phase.
In this manuscript we develop a general mathematical model of electrochemical charge-transfer and ion transport in galvanic cells and apply it to a one-dimensional model cell operating in steady-state. We analyze not only the standard galvanic discharging regime in which the cell ‘generates electricity’ with positive power by spontaneously driving electrons through the external circuit [17], [18], but also two operating regimes which require external electrical forcing. The first is the electrolytic charging regime in which an applied voltage exceeding the open-circuit voltage drives current in the reverse direction to convert electrical energy into chemical energy [5], [6], [7], [8], [9], while the second is an unconventional ‘super-galvanic’ regime in which the current is forced beyond the maximum for a galvanic cell in a shorted circuit. These three modes of operation are illustrated in Fig. 1. Analysis of the full range of operation is important to better understand the efficiency and physical mechanisms of energy conversion, and to enable more sensitive experimental validation of mathematical models of galvanic cells.
In this manuscript we will present calculation results for the ion density and electrostatic potential profiles as function of charge-transfer kinetic rates and of the imposed current. Two models for the electrolyte are considered. In the first case we consider a membrane electrolyte material in which the countercharge has a fixed, constant, distribution in space, while only the reactive ion is mobile. This is the typical situation for a solid-state proton or oxygen-ion conducting electrolyte membrane. In the second case we analyze the situation where the inert (non-reactive) ion is mobile and is redistributed across the membrane. This situation is more typical for a fuel cell or redox flow cell operating in liquid (e.g., aqueous) solution.
In both cases we illustrate the general theory in calculations that assume a one-dimensional planar geometry, steady-state operation, and constant (time-independent) chemical potentials of the reduced forms of the reactive cation, whose values are different at the anode and the cathode to describe a galvanic cell. The assumption of a constant chemical potential in each electrode corresponds, for instance, to a hydrogen fuel cell where the cation is a proton which reacts to/from an absorbed hydrogen atom in equilibrium with a large gas phase of constant hydrogen pressure (higher in the anode compartment than in the cathode compartment). The same assumption can also describe the case that the ion reduces to a neutral atom upon arriving at the cathode and is incorporated into the electrode phase (and vice-versa for oxidation at the anode). In this situation, our calculations extend the analysis of Bazant and co-workers [13], [14], [15] on electrolytic cells, where the reactions occurring at the two electrodes are the same, only driven in opposite directions by an applied cell voltage (and thus the open-circuit voltage is zero). In such situations, the behavior of the model electrolytic cell is invariant with respect to the sign of the current, but this symmetry is broken in a galvanic cell because the conditions at the two electrodes are different.
An important element in our work is the consideration of diffuse charge, or polarization, effects near the electrodes. By including these effects in our calculations, we build on recent work for fuel cells [16], [17] that goes beyond the standard assumption of electroneutrality throughout the electrolyte phase, which is ubiquitous in the battery and fuel cell literature [5], [11], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27]. Though polarization of the electrolyte is confined to nanoscopic “diffuse charge” or “space charge” layers near the electrodes where ionic concentrations and the electrostatic potential rapidly vary, as shown in Fig. 2, the electrostatic potential across the polarization-layer can be a significant part of the overall cell voltage. An additional reason to consider the polarization-layer in detail is that the ion concentrations and field strength at the reaction plane (which we equate to the Stern, or Outer Helmholtz, plane, i.e. the boundary of the electrolyte continuum) strongly influence the electron charge-transfer rate. We emphasize in this work the importance of a final “Stern” boundary condition, which relates the Stern layer voltage difference to the field strength at the Stern plane [13], [14], [15], [38]. Considering these elements jointly, a complete mathematical model is developed which self-consistently and logically describes the effect of the charge stored in the polarization-layers on the electrochemical charge-transfer rates. In contrast, in standard models for electrochemical cells, local electroneutrality is implicitly assumed throughout the complete electrolyte phase, and the electron charge-transfer rate does not depend on the structure of the polarization-layer.
Polarization-layer effects in electrochemical cells have been included in a limited amount of previous work [13], [14], [15], [16], [17], [28], [29], [30], [39], [40] but except for Refs. [16], [17] these papers consider electrolytic operation where the open-circuit voltage (OCV) is zero and no current is generated spontaneously upon closing the electrical circuit. This class of problems includes the broad topic of electrodialysis, where ion transport is analyzed between ion-exchange membranes under the application of an external voltage. Related work has focused on super-limiting currents and hydrodynamic instability at large voltages [31], [32], but for this class of problems much simpler Dirichlet boundary conditions of constant ion concentrations and constant electrostatic potential are commonly used to describe the membrane electrolyte interface. In the present context of Faradaic charge-transfer reactions, some authors modeling diffuse charge in electrolytic cells [28], [29] have simplified the problem by taking the Gouy-Chapman limit of a zero Stern layer thickness, but this removes any local field strength-dependence from the charge-transfer reaction rate and thus predicts a reaction-limited current, which cannot be exceeded without negative ion concentrations in the model. With a Stern layer included, it can be shown that ion concentrations are always positive [13], [15]. In the case of a fuel cell with fixed countercharge, a more detailed galvanic cell model, which includes polarization-layer effects, has been developed by Franco et al. [16].
In our recent paper [17] a fuel cell model is presented for the case of fixed countercharge, where local electroneutrality in the bulk electrolyte is assumed and a quasi-equilibrium structure of the polarization-layer considered. This is the thin diffuse layer (‘thin-DL’) limit of the full model presented below. In Ref. [17] calculation results are presented, following Bazant et al. and Chu and Bazant [14], [15], as function of the parameter δ, the ratio of the effective Stern layer (or compact layer) thickness to the Debye screening length. In the “Gouy-Chapman limit” of δ = 0, the Stern layer potential difference is zero, while in the opposite “Helmholtz limit”, where δ = ∞, the Stern layer carries the complete voltage across the double layer, since in the Helmholtz limit the Stern capacity is assumed to be infinitely larger than that of the diffuse layer. The Helmholtz limit of the generalized Frumkin-Butler-Volmer (gFBV) equation (discussed below in detail) turns out to be equivalent to the standard Butler-Volmer (BV)-based model for fuel cell operation [18], [19], [20], [22], [26], [27]. Therefore, in Ref. [17] a mathematically simple methodology is presented to generalize the standard BV-model for fuel cells toward arbitrary values of the ratio δ. In the present work we build on Refs. [16] and [17] by also considering mobile countercharge, by analyzing in more detail the structure and charge sign of the polarization-layers as function of current, and by studying the effect of imposing negative and very positive currents on galvanic cell operation.
When ions can be considered as point charges, without excluded volume, the structure of the electrolyte including the diffuse (polarization) layer that forms on the electrodes is described using the full, non-equilibrium Poisson-Nernst-Planck (PNP) model for the transport rates of all mobile ions through the electrolyte [32]. At equilibrium, or in the thin-DL limit, the diffuse part of the double layer can be described by Poisson-Boltzmann (PB) theory (Gouy-Chapman equation). These microscopic models for the polarization-layer must be coupled to a suitable expression for the electron charge-transfer rate and combined with an additional boundary condition on the electrostatic potential, to replace the macroscopic assumption of bulk electroneutrality. Here we will use a generalized Frumkin-corrected Butler-Volmer (gFBV) equation which, in our view, extends in a relevant way more familiar formulations based on the assumption of local equilibrium of the polarization-layer. The gFBV-formulation explicitly considers the compact (or Stern) layer potential difference as the driving force for electron transfer and the concentrations of reactive ions directly adjacent to the electrode, i.e., at the Stern or reaction plane, and does not a-priori assume the existence of a quasi-equilibrium Boltzmann distribution of the polarization-layer. For the additional electrostatic boundary condition, we model the Stern layer as a uniform dielectric, and this effectively gives the forward and backward reaction rates an Arrhenius dependence on the normal electric field at the electrode. The generalized formulation completes the mathematical description without arbitrary assumptions such as local equilibrium or electroneutrality of the electrolyte or for instance a prescribed, constant surface charge, and can be applied in such situations as thin electrolyte films (where diffuse layers overlap and/or the bulk electrical field is a significant portion of the field strength in the polarization-layer), operation at large, super-limiting currents [15] or large AC frequencies, which are all situations where the diffuse charge distribution loses its quasi-equilibrium structure.
FBV-formulations for the charge-transfer rate (either generalized, and those that assume quasi-equilibrium for the polarization-layers) differ from the standard unmodified BV-approach in which the ion concentrations outside the polarization-layer (in the quasi-neutral bulk solution) are used and the interfacial overpotential is based on the total potential difference across the interface (thus over the Stern layer plus the diffuse layer) or more heuristically is based on the electrode potential relative to that of a reference electrode. When the BV equation is used, one is forced to combine this with a model for ion transport that neglects the possibility of charge separation (as occurs in the polarization-layer). Instead, the gFBV equation which is based on local conditions at the Stern plane (ion concentration, and field strength, which determines the Stern layer potential difference) can be unequivocally and transparently combined with the PNP model which describes ion concentration and potential profiles both in the electrolyte bulk, as well as in the diffuse layers, all the way up to the reaction planes. The resulting PNP-gFBV model can be generally used, for the equilibrium situation, as well as for steady-state and fully dynamic transport problems.
Section snippets
History of modeling electrochemical charge-transfer including polarization-layer effects
The mathematical description of charge-transfer reactions is at the heart of any model for electrochemical cells, so we begin with a brief historical overview of various contributions where polarization-layer effects are included in charge-transfer rate modeling, a development initiated by Frumkin [33] who first included diffuse-charge effects in a Butler-Volmer framework. In this section we describe what we call the “generalized Frumkin-Butler-Volmer” (gFBV) equation, and compare with related
Theory
In this section we discuss the complete mathematical model, which combines the Poisson-Nernst-Planck (PNP) model for the ion transport flux in the electrolyte with the generalized Frumkin-Butler-Volmer (gFBV) rate equation and the Stern boundary condition. The theory can be generally applied both for the case where the countercharge is mobile and for the case where it is immobilized, such as will be assumed in an example calculation for a solid electrolyte used in a hydrogen fuel cell. In the
Solid electrolyte
First we show results for the cell voltage-current characteristic of a fuel cell with immobilized countercharge. We take the example of a hydrogen concentration cell in which mobile protons in the electrolyte are electrochemically exchanged with gaseous hydrogen at the electrodes (via an assumed equilibrium of H2 in the gas phase with adsorbed hydrogen atoms near the charge-transfer site) [16], [24]. To generate a positive cell voltage, the anode compartment hydrogen pressure pH2,A is larger
Conclusions
We have developed a general mathematical framework to describe the steady response of galvanic cells to the full range of spontaneous and imposed currents, which yields simple analytical results in the asymptotic limit of thin double layers, as well as a convenient system for numerical simulation. These results should be widely applicable to fit and interpret experimental current-voltage measurements in galvanic cells, such as fuel cells and batteries, and to test different theoretical
Note added in proof
The second part of Eq. 1 can also be found in L.I. Antropov, “Kinetics of electrode processes and null points of metals,” Council of Scientific & Industrial Research, New Delhi (1960), p. 13.
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