Non-equilibrium thermodynamics of mixed ionic-electronic conductive electrodes and their interfaces: a Ni/CGO study †

Non-equilibrium thermodynamics describe the current – voltage characteristics of electrochemical devices. For conventional electrode – electrolyte interfaces, the local activation overpotential is used to describe the electrostatic potential step between the two materials as a current is generated. However, the activation overpotential for the metal/mixed ionic-electronic conducting (MIEC) composite electrodes studied in this work originates at the MIEC – gas interface. Moreover, we have studied the e ﬀ ects of non-equilibrium on the electrostatic surface potential and evaluated its in ﬂ uence over electrode kinetics. By investigating two phase (2PB) and three phase boundary (3PB) reactions at the Ni/Ce


Introduction
The solid oxide cell (SOC) is a highly efficient chemical-toelectrical and electrical-to-chemical energy conversion technology compatible with both existing fuel (i.e.natural gas) and future fuel (i.e.renewably sourced hydrogen) infrastructures.[3][4][5][6][7][8][9][10][11][12] We therefore aim to shed light on the water electrolysis reaction, and moreover all solid/gas phase charge transfer reactions, by studying the electrostatic phenomena at the interfaces of the Ni/CGO electrode.
Pristine and doped ceria have relatively large band gaps (5 eV), and as such, mobile electrons are observed as polaronic Ce 4f states found within the "forbidden region" of the band structure, rather than populating the conduction band. 13,14This results in the mixed ionic-electronic conducting (MIEC) properties of Ce 1Àx Gd x O 2Àd , where x ¼ 0.1-0.2(CGO): where O Â O , V O , M Â M and M 0 M represent an oxide ion at the anion site, a doubly charged oxygen vacancy, a metal ion on the cation site, and a small polaron (localised electron at the cation site) respectively.However, this reaction oen occurs via a fuel intermediate such as CO/CO 2 or H 2 /H 2 O, where the strength of the intrinsic dipole moment of the adsorbed gas species was understood to determine the performance of the electrode. 15,16wing to the relative ease of reduction of CGO under operational conditions, and therefore good electronic conductivity, biasing the electrode modulates the coverage of adsorbates and the concentration of electronic defects.Understanding these phenomena is critical to accurately model and design the electrodes.Therefore, in this study we will analyse and discuss the effects of non-equilibrium operation (under electrical bias) on the electrostatic potential at the interfaces of the Ni/CGO electrode.
The process of establishing the origin of the electrostatic surface potential involves analysis of all steps in the redox reaction and identication of the rate limiting step. 17As an example, we will take the water electrolysis reaction at the fuel electrode, Ce , which can occur via a two phase boundary (2PB) or three phase boundary (3PB) mechanism at the Ni/CGO fuel electrode.The overall reaction equilibrium can be broken down into the following steps depending on the mechanism (Fig. 1): where V yte and O yte 2À refer to the CGO electrolyte species.
The water adsorption step R 2PB 1 /R 3PB 1 is fast close to equilibrium and is only facilitated on the ceria surface and not on the nickel phase. 18 is widely considered to be rate limiting, where the 2PB and 3PB reaction mechanisms display stark thermodynamic and kinetic differences. 16,19Desorption of weakly physisorbed hydrogen gas is given by R 2PB 3 , while under the 3PB reaction mechanism, R 3PB 3 , two adsorbed hydrogen species on the nickel surface combine and desorb.To restore the defect composition of the CGO phase aer the hydrogen gas is produced, R 2PB 4 / R 3PB 4 and R 2PB 5 /R 3PB 5 are included to illustrate transport of oxygen vacancies from the electrolyte and electrons from the current collector, respectively.In solid oxide cells the electrolyte phase requires low electronic conductivity to avoid short circuiting the cell.
Via the 2PB mechanism, two hydroxyl ions and two surface polarons undergo a coupled electron transfer and proton migration process in a two-electron transfer step, resulting in weakly physisorbed hydrogen gas.Electron transfer may only be facilitated at the 2PB if bulk electronic conductivity in the oxide phase is high.In contrast, the 3PB mechanism involves proton migration to the nickel surface via the one-dimensional 3PB, followed by electron transfer from the oxide phase to the highly conductive nickel phase at the ceria-nickel interface.This mechanism is not dependent on high electronic conductivity in the oxide phase and is therefore most commonly used to model Ni/YSZ electrodes where polaron concentration is low relative to ceria. 20

Background thermodynamic theory related to solid oxide cells
In order to develop the non-equilibrium models as applied to Ni/CGO interfaces, it is rst necessary to briey consider the thermodynamic treatment of fuel cells, considering both the equilibrium and non-equilibrium cases.

Electrode thermodynamics
The (electro)chemical potential, m i (eV), of a mobile species in an electrochemical system is expressed as: where m i is the standard potential, k B T ln a i is the activity potential and ef i is the electric potential.The activity (a i ¼ g i c i ) is the product of the concentration (c i ) and activity coefficient (g i ), which is a measure of the non-ideality of the species. 18,21A simplication can be made to express all the non-idealities within the excess chemical potential, m ex i , which also includes contributions from the standard and electric potentials.Here, we use the denition of (electro)chemical potential, meaning that if the species of interest is charged, we nd the electrochemical potential, and if the species of interest is neutral, we nd the chemical potential.If we consider the rate limiting process of an electrode to be a Faradaic reaction whereby n electrons are transferred, then we nd the activation overpotential (h act ) as the chemical potential difference of a net reduction reaction: where m 2 and m 1 represent the (electro)chemical potentials for the initial (oxidised) and nal (reduced) states, respectively, and the delta symbol represents the difference between the state in equilibrium and non-equilibrium (Dm i ¼ m i À m eq i ).The activation overpotential therefore describes the electrostatic potential step at the electrode-electrolyte interface, such that h act ¼ Df cc À Df yte , where f cc is the electric potential of the metallic current collector and f yte is the electric potential of the electrolyte. 22,23

Electrode kinetics
The current density for charge transfer per surface site for the forward and backward reactions is given as: where n is the number of electrons transferred in the reaction, m r and m p represent the electrochemical potential of the reactants and products, respectively, and k .0 and k ( 0 represent the forward and reverse rate constants, respectively.We may assume that the attempt frequency for the forward and backward reactions are equivalent, such that k . . 21 Occupation of the transition state is short lived, therefore we can assume its concentration to be negligible.Thus, we only need to use the excess chemical potential term (m ex TS ), which is dened following Butler-Volmer kinetics: where b is dened as the symmetry factor 0 < b < 1, and is generally assumed to be 0.5. 21While b can be taken to be 0.5, it is retained in the subsequent expressions for generality.The equilibrium potentials under open circuit voltage (OCV) are utilized to formulate the exchange current density, j 0 , by substituting eqn (5) into eqn (4) (full derivation given in the ESI †): where v i is the stoichiometric coefficient of species i. a i is the activity of species i, and the exponential term m i is the standard chemical potential of species i.When an electrical bias is applied, an electrostatic potential step is created at the interface in which the rate-limiting step is located.The current density, j, is introduced as: where the exponent (Df r À Df p ) refers to the shi in electrostatic potential across the interface of interest under nonequilibrium conditions.To determine the origin of the activation overpotential at the Ni/CGO electrode, we will evaluate the electrostatic potential at the nickel-CGO and CGO-gas interfaces under non-equilibrium conditions, applying the models derived above.

Application to Ni/CGO electrodes
In this section we will apply the theory of non-equilibrium thermodynamics as detailed in Section 2 to the Ni/CGO electrode and elucidate the essence of the activation overpotential for the water electrolysis reaction.

Electrostatics at the CGO-gas interface
As illustrated in Fig. 2a, under non-equilibrium conditions the electric potential of electrons in the nickel (current collecting, cc) phase (f cc ) will shi in accordance with the activation overpotential located at the bulk electrolyte contact . 16It must be noted that in this study a thin lm of dense ceria was used as opposed to a cermet electrode, therefore there was no distribution of the activation overpotential as a function of distance from the bulk electrolyte contact. 16However, charge transfer between ceria and the current collector, and between ceria and the electrolyte, was fast relative to charge transfer with the gas phase.Therefore, throughout the matrix of the Ni/CGO cermet it is rational to assume that the shi in electric potential of the ceria phase will be equal to the shi in the electric potential of the nickel phase (Df cc ¼ Df CGO ).Consequently, bias of the Ni/ CGO electrode does not result in the formation of an electric double layer at the Ni-CGO interface, as demonstrated in Fig. 2a.The process of charge transfer at the CGO-gas interface comprises the ambipolar exchange of ions and electronic species. 27The result of such a process causes charge separation and an associated electrostatic surface potential c: 17,18 where f CGO is the electrostatic potential experienced by mobile charge carriers in the electrode and f ad is the electrostatic potential experienced by the adsorbed species. 18Under bias, an electrostatic potential shi may be established at the surface (Dc ¼ c À c eq ), where an effective double layer is formed between the electrode surface and the adsorbed species (Fig. 2c  and d). 28We can therefore dene the non-equilibrium surface potential shi for a CGO surface in a H 2 O/H 2 gas atmosphere in eqn (8) to be: 18 The interaction between polar adsorbed species and the CGO surface was modelled by Williams et al. using rst principles calculation at the point of zero charge to accurately predict the coverage and electrostatic surface potential under nonequilibrium conditions. 18It was determined that the shi in electrostatic surface potential at the ceria (111) termination in H 2 O/H 2 gas at 500 C was a result of a change in coverage of polar adsorbates, such that c ¼ m .t r 0 3 0 q, where m .t , 3 0 , r0 and q represent the dipole moment normal to the surface, vacuum permittivity, the density of adsorption sites and the coverage of adsorbates, respectively.
We can derive the h-Dc relationship by realising that when R 2PB 2 /R 3PB 2 is rate limiting, the activation overpotential describes the state of non-equilibrium across the CGO-gas interface as: 29 where diatomic hydrogen adsorbed on the ceria surface is in equilibrium with the gas phase ða H2;ads ¼ a eq H2;ads Þ, thus Dm H2;ads ¼ 0. Eqn (10) shows that the activation overpotential is a function of congurational entropy (DS conf ) and electrostatic surface potential.If we consider the Ni/CGO cermet electrode illustrated in Fig. 3a, we are aware that there is no electrostatic potential shi between the CGO and metallic phases (Fig. 3b).Instead, we observe a shi in electrostatic surface potential which decays exponentially away from the electrolyte contact as the shi in electric potential of the adsorbate equilibrates with the CGO phase approaching the current collector contact, where lim y/cc We note that the potential of the metallic phase is constant throughout the electrode as the electronic resistance is assumed to be close to zero relative to the ionic resistance of the electrolyte phase.This theory contrasts with the Ni/YSZ model (Fig. 3c), where Faradaic reactions are driven by the difference in the electric potential shi between the current collector and electrolyte phases, h act ¼ Df cc À Df yte . 30

Defect concentration of the CGO phase
At the CGO-gas interface there is a collection of electrostatic phenomena that lead to an interface voltage, such as segregation of mobile defects at the surface, or gas adsorption forming polar adsorbates.The former will result in the formation of a space charge layer at the CGO surface, while the latter will be unscreened and will be highly dependent on the angle and magnitude of the intrinsic dipole moment of the adsorbate. 18,31,32The origin of the space-charge layer is the change in stability of defects at the surface relative to the bulk. 19,33nrichment of polarons at the ceria surface relative to the bulk has been observed by Haile et al. using X-ray absorption near edge spectroscopy (XANES), where the polaron fraction was found to be between 60-80% of the total cerium sites. 34he defect activities of the CGO surface can be expressed as a function of the activation overpotential: where the (full derivation is given in the ESI).† The activity of polarons in the Ce 4f state is modulated by the activation overpotential.Using the cation site constraint a Ce Â Ce ¼ 1 À x À a Ce 0 Ce and anion site constraint , where x is the concentration of gadolinium at cation sites, we can model the activity of polarons under bias. 33Here we assume the dilute solution approximation, as discussed in Section 2.1, such that g ¼ 1, and observe a generalised logistic function as the coverage of polarons increases with a negative activation overpotential as shown in Fig. 4. 35 This highlights that the CGO-gas interface dominates the activation overpotential in the Ni/CGO cermet case.

Electrokinetics at the CGO-gas interface via the double phase boundary (2PB)
Section 3.1 illustrated that the activation overpotential of the Ni/ CGO cermet electrode was located at the CGO-gas interface.To model the rate of charge transfer between the solid and gas phases, we must consider how the congurational entropy and electrostatic surface potential inuence the electrokinetics.Eqn (5) describes the (electro)chemical potential of the transition state, in which the activity coefficient (g TS ) describes the surface conguration required for a given reaction to proceed.This is an entropic effect, whereby the probability of a species at the transition state nding a correct conguration to undergo charge transfer is scaled by the availability of free sites. 26In other words, statistical thermodynamics states that a limited availability of free reaction sites would effectively increase the activation energy.This was demonstrated by Bazant et al. for the Li-ion intercalation process where the excess chemical potential of the transition state scaled with the concentration of lithium vacancies at the LiFePO 4 electrode surface. 26For the reaction R 2PB 2 , the transition state does not require a free surface site, and therefore activity coefficient of the transition state has no entropic constraint, thus g TS ¼ 1.The activation chemical potential difference for the forward reaction R 2PB 2 under electrical bias is therefore given as: Likewise, the reverse reaction can be expressed as: In realising eqn ( 12) and ( 13), we rationalise the importance of a large adsorbate dipole moment on the Faradaic reaction rate.The explanation is two-fold; rst, a stronger dipole moment will mean that the fraction of the activation overpotential being consumed by the congurational entropy is minimised according to eqn (10).This was rst discussed by Fleig, whereby the gradient vDc/vh approaches 1 when the intrinsic dipole moment approaches innity. 17Secondly, since congurational entropy scales logarithmically with respect to adsorbate coverage, a large intrinsic dipole strength is necessary for the thermodynamic driving force to be sustained over a reasonable overpotential range.In other words, a relatively weak dipole moment will result in saturation, or depletion, of adsorbates at the CGO-gas interface, thus limiting the current as the electrostatic potential plateaus. 36y combining eqn ( 12), ( 13) and ( 4) we can derive an expression for current density at the 2PB: 37 where the exchange current density is derived following eqn (6): Eqn ( 15) can be rearranged to express the exchange current density a function of equilibrium electrostatic surface potential at equilibrium (full derivation given in ESI †): where 1 Þ=kBT .The pre-exponential terms which are not derived for the conventional Butler-Volmer equation means that the current-voltage characteristics of this electrochemical process will likely be asymmetric under bias.Here we note another argument for designing a system which maximises the dipole moment of the adsorbate, since the exchange current density is highly dependent on the exponentially scaled surface potential, which is linearly proportional to the dipole moment of the adsorbate.

Electrokinetics at the CGO-gas interface via the triple phase boundary (3PB)
For conventional cermet electrodes like Ni/YSZ, charge transfer with the gas phase is thought to occur exclusively at the triple phase boundary (3PB) due to the limited electronic conductivity of the YSZ phase. 1 The relatively high electronic conductivity of  16 The fit for Ce 0 Ce (magenta) and Ce Â Ce (cyan) are found by solving eqn (11).
CGO under reducing conditions means that charge transfer with the gas phase may occur at the CGO-gas interface, hence the rationale behind Section 3.3.However, charge transfer at the 3PB is still possible and must be considered in the model even though the availability of appropriate 3PB sites is substantially less than that of 2PB sites. 38t the 3PB, the shi in free energy under bias is expressed as: Here we see that the driving force for charge transfer is still expressed in terms of the electrostatic surface potential at the CGO-gas interface even through the reaction occurs at the 3PB, making eqn ( 10) and ( 17) equivalent.Note that hydrogen adsorbed on the nickel is in equilibrium with the gas phase ða VNi ¼ a eq VNi Þ.This makes sense when we recall that there is no double layer at the Ni-CGO interface and that all processes are fast relative to charge transfer, thus the electrostatic potential will likely be spread out across the CGO surface.The transition states for R 3PB 2 require an unoccupied nickel site at the 3PB, thus we can express the activity coefficient of the transition state as g TS ¼ (aV Ni )À1. 26The activation chemical potential defence for the reduction reaction R 3PB 2 under electrical bias is therefore given as: Likewise, the reverse reaction can be expressed as: showing the effects of hydrogen affinity on the free energy of the reaction and the size of the energy barrier, where metals with a low affinity for hydrogen (black line) destabilise the final state of the reaction and increase the size of the activation barrier according to eqn (5).
Combining eqn ( 18) and ( 19) with ( 4) we can derive an expression for current density at the 3PB: which is similar to the result given by eqn ( 14) for the 2PB system.This only differs by the number of electrons transferred and the exchange current density, which is derived following eqn (6): Eqn ( 21) can be rearranged to include the equilibrium electrostatic surface potential at equilibrium at the 2PB:  V Ni e Àbec eq k B T Eqn ( 21) can be generalised as j 0 3PB $ ð1 À cÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cð1 À cÞ p , where c ¼ a eq HNi is the concentration of lled sites on the nickel surface.This is equivalent to the derivation for coupled ion-electron transfer given by Bazant et al. 26,39 The metallic phase has two main purposes: lowering the electrical resistance of the cermet matrix and catalysing the 3PB reaction. 301][42][43] From an entropic perspective, free nickel sites are desired to enhance the probability of a successful "jump" across the 3PB, as expressed by the activity coefficient of the transition state (Fig. 5a-d). 26However, lowering the affinity for adsorbed hydrogen on the nickel surface will increase the free energy of the nal state, thus increasing the energy barrier for the R 3PB 2 , as expressed in eqn (5) (Fig. 5e).Eqn (20) states that the electrostatic surface potential located at the 2PB inuences the rate of Faradaic reactions at the 3PB.The origin of this phenomenon is rooted in the dipole-dipole interactions, whereby the coverage of adsorbates at the 3PB will inuence the adsorption enthalpy at the 2PB.For example, in water electrolysis mode, electron transfer is driven by the adsorbates 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 adsorbates.If electron transfer occurs via a 2PB mechanism, the adsorbate-free region will be lled as the adsorbates reorganise to maximise congurational entropy and minimise dipole-dipole interactions.The same principle applies to the 3PB mechanism, where a vacant hydroxyl site close to the 3PB will stimulate adsorbate reorganisation across the entire CGO surface, resulting in the relaxation of the electrostatic surface potential at the CGO-gas interface.On calculating which of the mechanistic pathways will dominate the total current density, we can see that the current-voltage relation is identical (eqn ( 14) and ( 20)), and only differs due to the exchange current density.However, one must also consider the availability of reaction sites and the energy barrier for the reaction to progress.

Outlook on other systems
To date we have only applied the theory of the electrostatic surface potential to discuss the interaction between Ni/CGO and H 2 /H 2 O. Chueh et al. demonstrated that the electrostatic surface potential at the Sm 0.2 Ce 0.8 O 1.9 -(CO/CO 2 ) interface was invariant under an applied overpotential (Fig. 6a). 32This observation could be a result of the adsorbed carbonate forming a weak intrinsic dipole moment normal to the surface, or that the adsorbate is neutral and therefore has no electric potential. 32,44Nevertheless, the electrode polarisation is signicantly increased for the CO/CO 2 system compared with the H 2 /H 2 O system (Fig. 6b). 15,16,45Analysis of eqn (7) makes it clear that if there is no electrostatic potential driving the charge transfer reaction of the CO/CO 2 system, then the respective current density will be severely impeded. 36Given that an exponentially scaled electrostatic driving force is absent for the CO/CO 2 system, one may question why the current-voltage relationship in Fig. 6b still exhibits Tafel-like behaviour.This phenomenon is attributed to eqn (11) where the ceria defect concentration is exponentially scaled by the overpotential, such that the pre-Fig.6 (a) Dc-h and (b) j-h relationship at the Sm 0.2 Ce 0.8 O 1.9 -gas interface in 1 : 8 : 4 H 2 : H 2 O : Ar (blue) and 2 : 25 CO/CO 2 (magenta) measured at 500 C and collected by APXPS. 15,16,45xponential factor in eqn (7) is the essence of the observed Tafel-like behaviour. 36

Summary
In this work, the origin and physics of the activation overpotential for a Ni/CGO fuel electrode has been explained.We have analytically demonstrated the effects of non-equilibrium thermodynamics on the concentration of electronic defects and the adsorbate induced electrostatic surface potential.Moreover, we used our understanding of non-equilibrium thermodynamics to formulate analytical expressions for the current-overpotential characteristics of water electrolysis at the Ni/CGO electrode via the 2PB and 3PB mechanism.This rigorous mathematical description allowed the prediction of the origin of the activation overpotential of Ni/CGO electrode and postulates that the shi in electrostatic surface potential was the driving force for current density via both mechanisms.This is a signicant 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 theories put forward in this work are not conned to fuel electrodes and can be extended to other elds where charge transfer occurs at a solid-gas interface, such as nitrogen reduction and in lithium air batteries.
At intermediate temperatures and under open circuit voltage (OCV), the coupled ion-electron transfer step (ET) R 2PB 2 /R 3PB 2

Fig. 1
Fig. 1 Schematic diagram of the water electrolysis reaction via the two-phase boundary (left) and three-phase boundary (right) mechanisms.
h act ¼ Df ecc À .Using operando X-ray Photoelectron Spectroscopy (XPS), Chueh et al. demonstrated that the shi in electric potential of the ceria phase (f CGO ) was also equal to the activation overpotential lim y/yte h act ¼ Df CGO

Fig. 2
Fig. 2 (a) Electrochemical equilibrium at a Ni-CGO interface where the electrochemical potential of electrons in the two phases achieve equilibrium by electron density migrating to the CGO phase, resulting in an interfacial voltage in accordance with the Nernst equation (filled line).Under non-equilibrium conditions, the difference in shift between the electrical potential of electrons in Ni and CGO phases shifts with a magnitude proportional to the activation overpotential.(b) Partial density of states for the nickel phase where the Fermi energy at equilibrium (filled line) shifts to a higher value (dashed line) under an applied negative overpotential, and CGO phase where the position of the band centre and integrated area of the Ce 4f state (red) illustrates the shift in the electrical potential and increase in polaron concentration, respectively.(c) Electrochemical equilibrium at a CGO-gas (H 2 O/H 2 ) interface where the electrochemical potential of electrons in the two phases achieves equilibrium, resulting in gas adsorption and the formation of an electrostatic surface potential in accordance with the Nernst equation (filled line).Under non-equilibrium conditions, the electrical potential of oxygen vacancies in the CGO phases shifts, causing a shift of the electrostatic surface potential with a magnitude explained by eqn (10) (dashed line).(d) Partial density of states of the adsorbate at the CGO surface shows both chemical and electrostatic contribution in non-equilibrium (red) under an applied negative overpotential.An increase in coverage will populate the adsorbate DOS while a change in the electrostatic potential will shift the position of the band centre.The charge density difference schematic illustrates the adsorbate induced electrostatic potential at the ceria (111) termination.

Fig. 3
Fig. 3 (a) Illustration of the porous Ni/CGO cermet electrode comprised of CGO (blue), pore (white) and current collector (grey) phases.(b) Distribution of electrostatic potential shifts as a function of distance from the electrolyte (y), where Df CGO is equal to Df cc and out of phase with Df ad within the active region of the electrode.(c) Illustration of the porous Ni/YSZ cermet electrode comprised of electrolyte (red), pore (white) and metallic (grey) phases.(d) Distribution of electrostatic potential shifts as a function of distance from the electrolyte (y) where Df yte is out of phase with Df cc within the active region of the electrode.30

Fig. 4
Fig. 4 Surface cation site coverage at the CGO-gas interface as a function of overpotential.Circles represent Ce 0 Ce data collected by APXPS on a Sm 0.2 Ce 0.8 O 1.9 -gas interface as a function of overpotential at 500 C in 1 : 8 : 4 H 2 O : H 2 : Ar atmospheres (Chueh et al.).16The fit for Ce

Fig. 5
Fig. 5 (a) Normalised current density as a function of hydrogen coverage on the nickel surface according to eqn (22). 26,39(b and c) Illustrate the entropic constraint caused by filling the nickel surface and blocking viable reaction sites at the 3PB.(d) Schematic diagram of the excess chemical potential landscape exploring reaction R 3PB 2 using metals with high affinity for hydrogen (black) and low affinity for hydrogen (red), whereby increasing the effective activation barrier by using metals with a higher affinity for hydrogen.(e) Schematic diagram of the excess chemical potential landscape exploring reaction R 3PB2