Understanding the charge transfer processes at solid oxide fuel cell (SOFC) electrodes is critical to designing more efficient and robust materials. Activation losses at SOFC electrodes have been widely attributed to the ambipolar migration of charges at the mixed ionic–electronic conductor–gas interface. Empirical Butler–Volmer kinetics based on the transition state theory is often used to model the current–voltage relationship, where charged particles transfer classically over an energy barrier. However, the hydrogen oxidation/water electrolysis reaction H_{2(g)} + O^{2−} ⇌ H_{2}O_{(g)} + 2e^{−} must be modeled through concerted electron and proton tunneling events, where we unify the theory of the electrostatic surface potential with proton-coupled electron transfer kinetics. We derive a framework for the reaction rate that depends on the electrostatic surface potential, adsorbate dipole moment, the electronic structure of the electron donor/acceptor, and vibronic states of the hydrogen species. This theory was used to study the current–voltage characteristics of the Ni/gadolinium-doped ceria electrode in H_{2}/H_{2}O(g), where we find excellent validation of this novel model. These results yield the first reported quantification of the solvent reorganization energy for an SOFC material and suggest that the three-phase boundary mechanism is the dominant pathway for charge transfer at cermet electrodes.

## I. INTRODUCTION

The solid oxide fuel cell (SOFC) is a highly efficient chemical-to-electrical energy conversion technology compatible with both existing fuel (e.g., natural gas) and future fuel (e.g., renewably sourced hydrogen) infrastructures.^{1–3} These devices work through electrochemical RedOx reactions at the air and fuel electrodes, whereby chemical energy is used to drive external work. The Faradaic reactions at the fuel electrode can be given simply as H_{2(g)} + O^{2−} ⇌ H_{2}O_{(g)} + 2e^{−} (elementary steps are listed in Table I).^{4,5} The hydrogen oxidation/water hydrolysis reaction may occur via the two-phase boundary (2 PB) [Fig. 1(a)] or the three-phase boundary (3 PB) [Fig. 1(b)]. At the 2 PB, two neighboring polarons in the CeO_{2} surface will combine with two neighboring protons. This process is driven by a combination of the electrostatic potential and the concentration of electronic defects at the interface between mixed ionic–electronic conductors (MIEC) and the gas phase.^{2} Alternatively, in the presence of an electrocatalytic metallic phase, the electron–proton recombination may occur via a 3 PB mechanism. Electron density migrates from a polaronic state in the oxide phase to the metal; meanwhile, the proton is reduced as it migrates to the metal surface. The electrostatic nature of the 3 PB process is less well studied.

Elementary step . | Two-phase boundary . | Three-phase boundary . |
---|---|---|

Water adsorption | $H2O(g)+VO\u22c5\u22c5+OOx\u21cc2OHO\u22c5$ | $H2O(g)+VO\u22c5\u22c5+OOx\u21cc2OHO\u22c5$ |

PCET | $2OHO\u22c5+2CeCe\u2032\u21ccH2(g)+2OOx+2CeCex$ | $OHO\u22c5+CeCe\u2032+*\u21ccHNi+OOx++CeCex$ |

Hydrogen desorption | 2H_{Ni} ⇌ H_{2(g)} + 2* |

Elementary step . | Two-phase boundary . | Three-phase boundary . |
---|---|---|

Water adsorption | $H2O(g)+VO\u22c5\u22c5+OOx\u21cc2OHO\u22c5$ | $H2O(g)+VO\u22c5\u22c5+OOx\u21cc2OHO\u22c5$ |

PCET | $2OHO\u22c5+2CeCe\u2032\u21ccH2(g)+2OOx+2CeCex$ | $OHO\u22c5+CeCe\u2032+*\u21ccHNi+OOx++CeCex$ |

Hydrogen desorption | 2H_{Ni} ⇌ H_{2(g)} + 2* |

Gadolinium-doped ceria (CGO) displays considerably fast oxygen transport kinetics and high electronic conductivity under reducing conditions.^{11–15} To enhance the overall electronic conductivity of the electrode and to increase the density of reactive 3 PB, it is beneficial to mix a metallic phase with the MIEC to create a cermet electrode, i.e., nickel/gadolinium-doped ceria (Ni/CGO).^{1,2,6,16–19} The operation of the Ni/CGO electrode under electrical bias has been studied for decades; however, a unifying model for hydrogen electro-oxidation or water electrolysis has not yet been agreed upon.^{18,20–30} Most charge transfer models utilize the empirical Butler–Volmer equation, which is convenient for fitting current–voltage data at low overpotentials.^{31,32} However, the Butler–Volmer equation suggests that the electrostatic potential driving the charge transfer reaction is located at the electrode–electrolyte interface. In our previous work, we have illustrated that the electrostatic potential modulated by the state of the electrode is located at the surface of the MIEC.^{1,2} The driving force for charge transfer should therefore be determined by the electrostatic potential at the MIEC–gas interface.

Herein, we build on these previous findings to model the kinetics of electron and proton transfer at MIEC–gas interfaces. By accounting for nuclear quantum effects of the transferring proton, the charge transfer at SOFC electrodes can be described within a concerted proton-coupled electron transfer (PCET) framework. Therefore, in this study, we aim to combine the theory of the electrostatic surface potential with the vibronically nonadiabatic PCET theory to develop a unifying kinetic model for charge transfer at MIEC–gas interfaces.^{1,2,33–36} This model is adapted to both 2 PB and 3 PB rate expressions that are fit to experimental data to discern the PCET mechanism at MIEC–gas interfaces.

## II. THEORY

### A. Thermodynamics

*μ*

_{i}, of a mobile species in an electrochemical system is expressed as

^{32}

^{,}

*k*

_{B}

*T*ln

*a*

_{i}is the activity potential, and

*ϕ*

_{i}is the electric potential. The activity (

*a*

_{i}=

*γ*

_{i}

*c*

_{i}) is the product of the concentration (

*c*

_{i}) and activity coefficient (

*γ*

_{i}), which is a measure of the non-ideality of the species.

^{37,38}A simplification can be made to express all the non-idealities within the excess chemical potential, $\mu iex$, which also includes contributions from the standard and electric potentials. Here, we use the definition of (electro)chemical potential, meaning that if the species of interest is charged, we find the electrochemical potential, and if the species of interest is neutral, we find the chemical potential.

*V*, this departure from the open circuit voltage,

*V*

_{0}, is described as the reservoir chemical potential,

*μ*

_{res}=

*e*(

*V*

_{0}−

*V*).

^{32,39–42}The difference between the internally controlled potential (

*μ*

_{h}) and the externally controlled potential (

*μ*

_{res}) is given by the reaction affinity,

*A*

_{r,o}=

*μ*

_{res}−

*μ*

_{h}, which controls the driving force of absorption of the species from the reservoir. In the case of electrochemical RedOx reactions, the affinity can also be expressed as the activation overpotential (

*η*),

^{32,39,42}

*n*is the number of electrons transferred in the Faradaic reaction.

^{43}If we consider the reduction reaction

*O*

^{n+}+

*ne*

^{−}→

*R*, the activation overpotential is described,

*μ*

_{R},

*μ*

_{O}, and

*μ*

_{e}represent the (electro)chemical potentials for the reduced, oxidized, and free electron states, respectively. We can use Eq. (1) to expand Eq. (3) to give the standard electrode potential (

*E*

^{0}), activity, and electrostatic potential,

*R*

_{ad}are the adsorbed states preceding and proceeding the rate determining charge transfer reaction, respectively. To derive the electrostatic surface potential, we assume that the rate determining step of the model system is the transfer of a single electronic charge, $Oad++e\u2212\u2192Rad$. As such, the activation overpotential is given as (derivation given in the supplementary material)

*χ*represents the electrostatic surface potential, which is an electric field formed by the difference in electrostatic potential of the electrode (

*ϕ*

_{e}) and the charged adsorbate (

*ϕ*

_{ad}),

^{1,16,33}

^{44}The notation

*χ*

^{gas−eq}refers to the surface potential when the RDS and therefore the electrode surface is in equilibrium with the gas phase and is expressed by the standard potential of the RDS and the activity at equilibrium,

^{1}

^{16,45,46}This key finding is routed in the mechanism by which the electrostatic surface potential is formed, whereby an adsorbate must form on the surface for an electrostatic surface potential to be induced; thus, the coverage of adsorbates on MIEC surfaces is a function of the activation overpotential.

^{33}Fleig theorized that the origin of the electrostatic surface potential was the dipole moment of the adsorbed gas, $\chi =\mu \u20d7\u22a5\rho 0cOad/\epsilon 0$, where $\mu \u20d7\u22a5$,

*ɛ*

_{0}, and

*ρ*

_{0}represent the dipole moment normal to the surface, vacuum permittivity, and the density of adsorption sites, respectively. This theory was later confirmed by Williams

*et al.*by illustrating excellent correlation between the magnitude of the electrostatic surface potential and the intrinsic dipole moment using the density functional theory.

^{1}By combining Eqs. (8), (10), and (11), we may relate the chemistry of the MIEC–gas interface with the activation overpotential,

### B. Electrode kinetics

^{32,39,42,47}The overall reaction rate can be written in terms of elementary processes as

*R*

_{r}and

*R*

_{o}represent reduction and oxidation rates, respectively. In the framework of nonlinear kinetics, we introduce terms that describe the probability of finding a particle in the desired location, the probability of occupation of the transition state, and the probability of a successful tunneling event once the transition state is occupied. The rate of electrochemical reactions is often described by the phenomenological Butler–Volmer equation, which was originally derived based on the transition state theory to model the rate of ion transfer (IT).

^{1,32,39,47–49}Here, the rate of ion migration over an activation barrier is determined through classical statistical thermodynamics composed of an attempt frequency and a success probability determined by the thermal energy of the system. The rate for a reduction reaction per surface site for the forward and backward reactions is therefore

^{32}

^{,}

*α*is the charge transfer coefficient 0 <

*α*< 1 and

*k*

_{0}is an overall pre-factor related to the curvatures of the excess free energy landscape near the initial, final, and transition states.

^{32}By moving the concentration terms outside of the exponent,

^{38,39,50}

*η*

_{f}=

*E*

^{0}′ −

*E*, where

*E*

^{0}′ is the formal electrode potential and

*E*is the applied electrode potential.

^{38,39}

### C. Electron transfer

^{51,52}The activation energy barrier is determined by the reorganization energy (

*λ*) of the solute and the surrounding solvent environment during charge transfer.

^{53,54}The forward and backward rates for outer-sphere electron transfer under the framework of the Marcus theory is given,

^{32,55}

*k*

_{T}is the electron tunneling probability and

*g*

_{r,o}is the work required to bring RedOx species to the reaction site.

^{39}

*ɛ*) in the electrode,

^{50}

^{,}

*R*

_{r,ɛ}and

*R*

_{o,ɛ}represent the reaction rate dependent on the position of the energy level

*ɛ*for the forward and backward reactions, respectively, and

*ρ*(

*ɛ*) represents the density of electronic states in the electrode.

^{39,50,56,57}Equation (19) describes electron transfer involving a continuum of states in a metallic electrode by integrating over the energy levels in the electrode. This integral is often described in the literature as the Marcus–Hush–Chidsey theory; however, Dogonadze and Kuznetsov also made pioneering contributions to the nonadiabatic ET theory.

^{58,59}In general, the tunneling probabilities

*k*

_{T}depend on

*ɛ*and are a function of the electronic coupling between single-electron states in the metal and the molecular orbitals of the RedOx species.

^{59,60}However, most theoretical treatments use a coupling matrix element associated with the entire metal d-band as in the Newns–Anderson model and the d-band theory of Hammer and Nørskov.

^{61}By neglecting variation in the density of states for a metallic electrode, we arrive at the electron transfer rate dependent on the position of the energy level

*ɛ*for the forward and backward reactions,

^{55}

^{,}

*f*(

*ɛ*) represents Fermi distribution function. Substituting Eqs. (20) and (21) into Eq. (19) yields an improper integral that is not ideal for numerical calculations. As such, Zeng

*et al.*derived a uniformly valid analytical approximation for the integrals by matching asymptotic approximations for large and small lambda,

^{39,59,63}

^{56}

## III. PROTON-COUPLED ELECTRON TRANSFER

By treating the proton quantum mechanically, we derive an expression for rates at the MIEC–gas interface in the framework of proton-coupled electron transfer (PCET). We will begin this section by summarizing the general theory of PCET through the derivation of vibronic states and charge transfer mechanisms. After which, we will look at where PCET is applied to electrochemistry specifically before integrating PCET into the framework of the MIEC–gas interface.

### A. Fundamental concepts of PCET theory

PCET reactions are prevalent in many biological, chemical, and electrochemical processes.^{34–37,57,62–70} A general reaction square scheme for heterogeneous electrochemical PCET involving the transfer of one electron and one proton is illustrated in Figs. 2(a)–2(c).^{67} Because the transferring electron and proton are treated quantum mechanically within this theoretical treatment, such a PCET reaction can be described in terms of the four diabatic states shown on the corners of the surface. As Figs. 2(a)–2(c) imply, the net PCET mechanism could be sequential with initial electron transfer (ET) or proton transfer (PT) producing a thermodynamically stable intermediate, or concerted, without forming such an intermediate.^{57}

^{34,57}PCET is related to the Marcus theory in that reactive electron–proton tunneling transitions can occur when the free energies of pairs of initial and final vibronic states become degenerate due to polarization of the solvent environment.

^{32,35,55,71}The center panel of Fig. 2(d) shows a schematic of the Marcus parabolas that represent the free energies of the initial (blue) and final (red) diabatic states along the collective solvent coordinate for concerted PCET. The probability of a PCET event occurring is proportional to the square of the vibronic coupling matrix (

*V*

_{μν}).

^{35,62,63}This coupling is defined as the Hamiltonian matrix element between the reactant and product vibronic wave functions.

^{57}In the high temperature regime, the quantum descriptions for the population distributions breakdown to the classical Boltzmann distribution function.

^{72}The Boltzmann population of vibronic state

*μ*is determined by the temperature and the splitting between

*μ*and the ground state at

*μ*= 0,

*Z*is the canonical partition function, which normalizes the exponential term,

### B. PCET at electrochemical interfaces

*et al.*

^{39}and follows the same framework combining nonequilibrium thermodynamics with the Marcus theory given by Bazant.

^{32}The key difference being that ion transfer is treated classically in CIET while proton transfer is also quantum mechanical in PCET. For inner sphere (or heterogeneous) electrochemical PCET, the electrode not only serves as a reservoir for electrons but also participates in chemical bonding.

^{67}For a metallic electrode in solution, the free energy of reduction (Δ

*G*

_{μν}) for the pair of reactant and product vibronic states

*μ*and

*ν*is the free energy difference between the minima of the corresponding electron–proton vibronic surfaces plus additional terms depending on the electrostatic potentials in solution and in the metal electrode as well as the energy level of the transferring electron in the electrode,

^{35,67,73}

*U*

_{μν}= (

*ɛ*

_{ν}−

*ɛ*

_{ν=0}) − (

*ɛ*

_{μ}−

*ɛ*

_{μ=0}) is the intrinsic energy bias. By modifying the rate of electron transfer dependent on the position of the energy level

*ɛ*given by Marcus in Eqs. (20) and (21), we arrive at

*k*

_{0}satisfies microscopic reversibility,

^{37,39}

*ρ*

_{F}is the density of states at the Fermi level of the metallic phase,

*P*

_{μ/ν}is the Boltzmann population of the vibronic state

*μ*/

*ν*given by the equation, and

*γ*

_{TS}is the activity coefficient of the transition state.

^{32}All vibrational information contained in the partial transition probability,

*w*

_{ij}(

*ɛ*,

*η*), of a given pair of initial (

*i*) and final (

*f*) proton vibrational states at high temperatures closely resembles the nonadiabatic expression for ET given in Eqs. (17) and (18),

^{57,67}

*V*

_{μν}defines the interaction between reactant and product vibronic states in the diabatic representation of the PCET reaction. The vibronic states are defined by the electronic states and proton vibrational states. One justification for a nonadiabatic model is that the proton vibrational wavefunction overlap between reactant and product states is typically small for PCET reactions.

^{57}When the free electron involved in the RedOx reaction comes from/goes into a metallic phase, the rate of PCET dependent on the position of the energy level

*ɛ*is expressed by combining Eqs. (28) and (29) with Eqs. (30) and (31),

^{47,74}

^{39}If the free electron involved in the RedOx reaction comes from a polaronic state, the density of states can be approximated by a Dirac delta function around the localized energy level

*ɛ*

_{0}; Eqs. (32) and (33) become

^{39}

### C. PCET at the CeO_{2}–H_{2}/H_{2}O interface

_{2}–H

_{2}/H

_{2}O interface plus the net result of the two reactions,

*R*

_{1}and

*R*

_{2}represent the ion transfer (or water adsorption) and PCET reactions, respectively, and

*R*

_{o/r}is the net chemical reaction. The affinities of the above three reactions are

^{37}

^{,}

*A*

_{r,o}is the affinity of the net reaction. Since the adsorption step (

*R*

_{1}) is fast relative to the PCET step and in a state of quazi-equilibrium, we assume

*A*

_{1}= 0 and

*A*

_{2}=

*A*

_{r,o}(derivation in the supplementary material). To calculate the concentration of surface species and the electrostatic surface potential as a function of overpotential, Eqs. (40) and (41) can be expanded,

^{38}Equation (42) was conditioned to experimental data in Fig. 3(a), showing an exponential overpotential dependency on polaron concentration; Eq. (43) was conditioned to experimental data in Figs. 3(a) and 3(b), illustrating a linear overpotential dependency on hydroxyl concentration and surface potential.

^{1,16}If the electron involved in the RedOx reaction at the MIEC–gas interface was polaronic, we do not integrate over the band structure of the metallic phase. As such, we can derive the solution for the rate of PCET at MIEC–gas interfaces via the 2 PB,

*R*

^{3PB}) can be derived from Eqs. (32) and (33). By taking the equilibrium concentration outside the integral, the electrostatic surface potential shift is isolated according to Eq. (12),

*et al.*to give the final solution for the rate of PCET at MIEC–gas interfaces via the 3 PB,

^{63}

The transition states requires a Ce atom to occupy the cation site at the 3 PB. Therefore, we can express the activity coefficient of the transition state as *γ*_{TS} = (1 −x)^{−1}, where x is the concentration of aliovalent dopants since the dopants cannot participate in RedOx reactions.^{39} When conditioning Eq. (47) to the electrochemical impedance spectroscopy data, we find an excellent fit in Fig. 3(d). When compared to the Butler–Volmer equation [Eq. (14)] and 2PB-PCET equation [Eq. (44)], we find that the 3PB-PCET model fits the experimental data far better. This suggests that the mechanism involves PCET at the 3 PB and that the kinetics of this PCET reaction can be accurately captured using a nonadiabatic model. It is important to note that the vibronic coupling and Boltzmann population are difficult to calculate. Goldsmith *et al.* illustrated limited change in the contribution from each vibronic coupling relative to the exponentially scaled electrostatic component.^{75} Therefore, we assume that we do not observe their effects in the current–voltage experiment. From this fitting, we were able to calculate the reorganization energy *λ* = 18*k*_{B}*T* (1.35 eV at 873 K). This is large relative to that estimated for lithium ion intercalation into LiFePO_{4} $(\lambda LiFePO4=8.3kBT)$ and graphite $(\lambda LiC6=5kBT)$.^{31,76} DFT calculations carried out by Castleton *et al.* approximated the diabatic barrier for polaron migration in bulk CeO_{2−δ}.^{77} The result that agreed closest to our system was the local density approximation + U (LDA + U), which yielded *λ*_{LDA+U} = 1.28 eV. However, there was a wide range of results as the barrier was heavily dependent on the functional used; moreover, the polaron migration process is not entirely representative of the PCET process.

Interesting, the electrostatic driving force in Eqs. (47) and (44) is the electrostatic surface potential located at the MIEC–gas interface, even if the PCET event occurs at the 3 PB.^{2} The origin of this phenomenon is rooted in the dipole–dipole interactions, whereby the coverage of adsorbates at the 3 PB will influence the adsorption thermodynamics across the MIEC–gas interface. Under water electrolysis mode, PCET is driven by the charged particles (adsorbates and polarons) seeking to establish thermodynamic equilibrium with the gas phase, resulting in the release of hydrogen, relaxation of the electrostatic surface potential, and a spatial region free of charged particles. When PCET occurs at the 2 PB, a charge-free region will be filled as the charged particles reorganize to maximize configurational entropy and minimize columbic interactions. The same principle applies to the 3 PB mechanism, where a charge-free region close to the 3 PB will stimulate reorganization across the entire MIEC surface, resulting in the relaxation of the electrostatic surface potential at the CeO_{2}–H_{2}/H_{2}O interface. We can account for the asymmetry observed in the current–voltage plot in Fig. 3(d) to dipole–dipole depolarization. This occurs when a dipole is exposed to the electrostatic field of all other dipoles, shrinking in the hydroxyl bond length, thereby reducing the size of the electric field normal to the surface. Dipole–dipole depolarization increased at negative overpotential as the coverage of hydroxyls on the electrode surface increases [Fig. 3(b)].^{1,2,16} A decrease in the adsorbate dipole moment reduces the magnitude of the surface potential shift and therefore increases the resistance of the PCET process at negative overpotentials relative to positive overpotentials.

## IV. CONCLUSION

In this work, the origin and physics of the activation overpotential for the Ni/CGO fuel electrode has been explained. Previously, we demonstrated the effects of nonequilibrium thermodynamics on the concentration of electronic defects and the adsorbate induced electrostatic surface potential. Here, we derive a kinetic relationship between the overpotential and surface potential via the 2 and 3 PB mechanism. In the framework of the proton-coupled electron transfer (PCET) theory, electronic and protonic species were treated as quantum particles whereby the probability of a concerted tunneling event was controlled by an external voltage. To account for the electronic structure of the metallic phase, variation in the density of states was neglected to give an analytical expression for the hydrolysis reaction via the 3 PB. Using the current–voltage data, we found the best fit was given by the 3 PB PCET equation vs the Butler–Volmer equation and the 2 PB PCET equation. This suggested that the dominant pathway for charge transfer at the Ni/CGO electrode includes the electrocatalytic metallic phase. Understanding the mechanism of ambipolar charge transfer at the electrode–gas interface is a significant consideration for the design and operational conditions of SOC electrodes as the strength of the intrinsic dipole moment of the adsorbed gas species is shown to have a profound effect on the magnitude of the activation overpotential. The theory of the electrostatic surface potential put forward in this work is not confined to PCET processes at SOFC electrodes and can be extended to other fields where charge transfer occurs at a solid–gas interface, such as nitrogen reduction and in lithium air batteries.

## SUPPLEMENTARY MATERIAL

Supplementary material notes: derivations of the electrostatic surface potential, affinity of the water electrolysis reaction, and the de Donder relation.

## ACKNOWLEDGMENTS

This work was supported by Ceres Power Ltd. I.D.S. and S.J.S. acknowledge the EPSRC for funding through the award of Grant No. EP/R002010/1. R.E.W. acknowledges start-up funding from the Case School of Engineering at Case Western Reserve University.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Nicholas J. Williams**: Conceptualization (lead); Data curation (lead); Methodology (lead); Writing – original draft (lead). **Robert E. Warburton**: Formal analysis (supporting); Validation (supporting); Writing – review & editing (supporting). **Ieuan D. Seymour**: Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). **Alexander E. Cohen**: Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). **Martin Z. Bazant**: Supervision (equal); Writing – review & editing (supporting). **Stephen J. Skinner**: Supervision (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.

## REFERENCES

_{2}O dissociation on reduced CeO

_{2}(111)

_{2-x}/Gas interface during electrochemical oxygen insertion reactions

_{2}-H

_{2}O atmosphere

*Modern Thermodynamics: From Heat Engines to Dissipative Structures*

*Electrochemical Methods: Fundamentals and Applications*

*p*(O

_{2}) dependence of oxygen surface coverage and exchange current density of mixed conducting oxide electrodes: Model considerations

*Encyclopedia of Electrochemistry*

_{2}: A density functional theory characterization

*Nanoscale Energy Transport and Conversion: A Parallel Treatment of Electrons, Molecules, Phonons, and Photons*

*Electron Transfer in Chemistry and Biology: An Introduction to the Theory*

_{2}