Sasaki - 2015 - Exchange Current Density of SOFC Electrodes - Theoretical Relations and Partial...

F136 Journal of The Electrochemical Society, 162 (1) F136-F152 (2015)

Exchange Current Density of SOFC Electrodes: TheoreticalRelations and Partial Pressure Dependencies Rate-Determinedby Electrochemical ReactionsT. Hosoi,a T. Yonekura,a K. Sunada,a and K. Sasakia,b,c,d,,zaDepartment of Hydrogen Energy Systems, Faculty of Engineering, Kyushu University, Nishi-ku,Fukuoka 819-0395, JapanbInternational Research Center for Hydrogen Energy, Kyushu University, Nishi-ku, Fukuoka 819-0395, JapancInternational Institute for Carbon-Neutral Energy Research (WPI-I2CNER), Kyushu University, Nishi-ku,Fukuoka 819-0395, JapandNext-Generation Fuel Cell Research Center (NEXT-FC), Kyushu University, Nishi-ku, Fukuoka 819-0395, Japan

As a theoretical consideration on electrode defect chemistry, general relations of exchange current density quantitatively representingSolid Oxide Fuel Cell (SOFC) electrode performance are systematically derived as a function of gas partial pressures, equilibriumconstants of adsorption and dissociation reactions on electrode surfaces, and electrochemical reaction rate constants for possibleelemental reactions at the cathode and the anode, in the case that an electrochemical reaction is the rate-determining electrode reaction.Simplified expressions are also derived, under the condition that one kind of neutral or charged adsorbed species is predominantat the electrode, to derive gas partial pressure dependence of exchange current density for given rate-determining electrochemicalreactions. Importance of considering elementary steps is highlighted to derive rate equations and to clarify various dependencies.Partial pressure dependencies of the exchange current density are compiled and discussed by simulating normalized exchange currentdensity values for given partial pressures. The applicability and limitation of the Butler-Volmer type expressions of exchange currentdensity for SOFC electrodes are carefully discussed. The Author(s) 2014. Published by ECS. This is an open access article distributed under the terms of the Creative CommonsAttribution Non-Commercial No Derivatives 4.0 License (CC BY-NC-ND, http://creativecommons.org/licenses/by-nc-nd/4.0/),which permits non-commercial reuse, distribution, and reproduction in any medium, provided the original work is not changed in anyway and is properly cited. For permission for commercial reuse, please email: [email protected]. [DOI: 10.1149/2.0561501jes]All rights reserved.

Manuscript received November 6, 2014. Published December 1, 2014.

Quantitative evaluations of SOFC performance are essential tocompare and develop electrode materials, to discuss the electrode re-action mechanisms, and to optimize the performance and durabilityof fuel cell stacks and systems. For the performance evaluation ofPolymer Electrolyte Fuel Cells (PEFCs), the Butler-Volmer equationand Tafel plots have been extensively applied to quantitatively ana-lyze their electrode characteristics.13 Computational fluid dynamic(CFD) analysis using exchange current density values obtained ex-perimentally and related electrochemical analysis have been widelymade.39 Exchange current density is the most fundamental parameterrepresenting electrode performance quantitatively. The value dependson the chemical composition and the microstructure of the electrodesand the operating conditions such as operating temperature and gascomposition. Therefore, dependencies of the exchange current den-sity on various operational parameters should be known to analyze andsimulate fuel cell performance for given electrodes. Such analysis hasalso been made for other types of fuel cells including Phosphoric AcidFuel Cells (PAFCs)10 and Molten Carbonate Fuel Cells (MCFCs).11

While SOFC operating at high temperatures is a promising efficientpower generation system for e.g. stationary applications, evaluationsof their electrode characteristics have been made typically based ontheir area-specific electrode resistance and electrode overvoltage mea-sured by the current interruption technique and/or the AC impedanceanalysis.1214 Reported values of exchange current densities are rela-tively limited,1521 which are however essential to describe the elec-trode performance quantitatively. Theoretical relations between theexchange current density values and the operational parameters havenot yet been systematically derived for various possible situations.The Butler-Volmer type equation has been applied in simulation stud-ies or in experimental analysis without verifying its applicability.22These could be the serious obstacle in applying various simulationtechniques, developed for other types of fuel cells, to SOFCs.

Electrode kinetics of SOFCs have been extensively studies sinceseveral decades.1214,2229 Generally speaking, three phase boundaries(TPB) are the electrode reaction sites where oxygen reduction at the

Electrochemical Society Active Member.zE-mail: [email protected]

cathodes and fuel oxidation at the anodes take place. Whilst the use ofmixed conducting oxides such as La-Sr-Co-Fe-oxides for the cathodesmay expand the electrode reaction area from the TPB to the mixedconducting cathode surfaces,12,13,2224 most of electrode materials in-cluding Ni metal and La-Sr-Mn-oxides are predominately electronicconductors where the electrochemical reactions may occur around theTPB. Many papers have dealt with SOFC electrode reaction mecha-nisms, but the typical porous electrodes have complicated microstruc-ture so that, in general, quantitative treatment of electrode reactionkinetics are rather difficult to be made. On one hand, considerable sci-entific efforts have therefore been made to overcome this fundamentalobstacle, e.g., by analyzing electrodes with a defined geometry,3034by quantifying the three-dimensional porous microstructure using thefocused ion beam technique,3441 and by applying quantum chemicalapproaches to simulate electrode processes on an atomic scale.42,43On the other hand, from the technological viewpoint, as real SOFCcommercialization has been just started requiring more accelerated de-velopment, simple and more rapid calculation tools and characteriza-tion procedures are increasingly desired e.g. to simulate and optimizesystem performance. In such simple treatments, electrode character-istics should be described for given electrodes without consideringcomplicated 3-dimensional electrode microstructures, for which theButler-Volmer type expression could be a simple and useful equationto represent overall electrode characteristics.

In simulation studies of SOFCs using computational fluid dynam-ics modeling (CFD software) where exchange current density valuesare the essential input parameters, Eqs. 1 and 2 are often used as typi-cal expressions to describe exchange current densities for the cathodeand the anode, respectively:

i0,c = c exp( Eact,c

RT

)(

pO2(C)pO2(C),re f

)A[1]

i0,a = a exp( Eact,a

RT

)(

pH2(A)pH2(A),re f

)B(

pH2O(A)pH2O(A),re f

)C[2]

where pO2(C), pH2(A), and pH2O(A) are partial pressure of oxygen atthe cathode, and hydrogen and water vapor at the anode, respectively.pO2(C),re f , pH2(A),re f , and pH2O(A),re f are reference partial pressure of

) unless CC License in place (see abstract). ecsdl.org/site/terms_use address. Redistribution subject to ECS terms of use (see 133.5.50.41Downloaded on 2015-06-15 to IP

Journal of The Electrochemical Society, 162 (1) F136-F152 (2015) F137

oxygen, hydrogen, and water vapor, respectively; c and a are rateconstants strongly depending on electrode materials and microstruc-ture for the cathode and the anode; A, B, and C are indexes ex-pressing partial pressure dependence; Eact,c and Eact,a are consideredas activation energy for the cathode reaction and the anode reac-tion, respectively, depending on electrode materials. These expres-sions are simple but empirical phenomenological relations assumingindependently the influence of gas compositions and operating tem-perature as the exchange current density is affected by these oper-ational parameters.44 However, Eqs. 1 and (2) are simply assuminga certain gas partial pressure dependence, based on no theoreticalconsideration about electrode reaction mechanisms. Reported partialpressure dependencies of exchange current densities are relativelyscattered.22,45

Defect chemistry has been extensively applied to describe bulkdefect concentrations and related physical properties such as electri-cal conductivity. Their partial pressure dependence can specify pre-dominant defect species in a solid state. The application of defectchemistry has been extended to interfaces such as grain boundariesand surfaces,28 as well as to lower temperatures for partially frozen-in states.46,47 In this study, defect-chemical treatments are made forSOFC electrode reactions. For such electrode defect chemistry, el-emental reaction processes at SOFC electrodes are first considered,and theoretical expressions for the exchange current density and theirapproximated expressions valid under each boundary condition arederived. We first consider the cathode reactions where oxygen only isinvolved in the electrode reactions. We then consider the anode reac-tions where both hydrogen and oxygen (water vapor) are involved inthe electrode reactions.

The theoretical expressions for the exchange current density andtheir approximated expressions valid under each boundary conditionare derived assuming that an electrochemical reaction is the rate-determining reaction in overall electrode kinetics while all other pro-cesses are in equilibrium. This means that this treatment may nothold in the case that surface exchange reaction or surface transportprocess on the electrodes is rate-determining instead of the electro-chemical reaction. For simplicity, we also assume that (i) the electrodeis a pure electronic conductor with negligibly low ionic conductivity,(ii) the electrolyte is a pure ionic conductor with negligibly low elec-tronic conductivity, (iii) the electronic conductivity of electrodes ishigher than the ionic conductivity of electrolytes and is independentof gas partial pressures, and (iv) the concentration overvoltage at theporous electrode is negligibly small, as are often the cases for typi-cal SOFCs.12,13 The relations derived are then used to reveal possiblegas partial pressure dependencies of exchange current density. Asthe Butler-Volmer type equation should be carefully applied to theSOFC electrode kinetics,22 the applicability and limitation of theseexpressions are critically evaluated and discussed, by carefully exam-ining the preconditions and specific boundary conditions to derive thetheoretical expressions in this study.

General ConsiderationIn this chapter, we consider the equilibrium and kinetics relations

for various possible elemental reactions and the conservation condi-tions at the cathode and the anode. Elemental reactions at an electrodemay include following processes: (i) adsorption and dissociation reac-tions on the porous electrode surface, (ii) surface transport processesfrom the electrode surface to the TPB where electrochemical reac-tions will take place, and (iii) electrochemical reactions around theTPB.12,13,2325,28 Here, we consider the cases in which an electrochem-ical reaction is rate-determining, so that surface transport processesare assumed to be fast enough. Concentration difference of the speciesbetween on the electrode surface and at the TPB can be therefore as-sumed to be negligible. Adsorption and dissociation reactions can beassumed as equilibrium reactions. In some cases these assumptionsmay be satisfactory, while in other cases these assumptions may not bevalid due to the fact that e.g. the O2 molecule has a binding energy ofca. 5 eV so that the dissociation of O2 could be rate-determining.43 If

such surface exchange reactions are rate-determining, exchange cur-rent density should be affected by such surface processes. In addition,we consider the case that oxygen partial pressure dependence of theelectrical conductivity of electrode materials is relatively small andthus negligible, as it is typically the case for the (heavily) Sr-dopedLaMnO3 cathode material and for the metallic Ni anode material,while it is not the case for some perovskite oxides such as (slightly)Fe-doped SrTiO3.28 In the following, species on the cathode, on theanode, and in the electrolyte are specified with (C), (A), and (E),respectively, in various reactions.

The reaction used to derive the rate equation must be an elementarystep. Using a non-elementary step definitely leads to different rateequations and different partial pressure dependencies. In this paper,we therefore consider the description of exchange current density in astep-by-step way. As a first step, in the following chapters and sections,(i) we consider possible major electrode reactions with non-chargedspecies. This is the simplest way, whilst most of the adsorbed speciesmay be charged on electrodes (especially on cathodes) of stronglyionic character and some of these reactions considered may not beregarded as elementary steps. (ii) We then examine more elementaryelectrode reactions with charged species, where each charge transferreaction step with only one electron involved is taken into accountin the reduction reaction of oxygen molecules. As the reaction usedto derive the rate equation must be an elementary step, (iii) we thusfurther consider other possible elementary steps. As many kinds ofpossible elementary steps could exist for both the cathode and anodereactions, more detailed theoretical and numerical studies are still inprogress for a separate paper.

Major reactions at cathodes. When mixed O2-N2 gas such asair is supplied to the cathode, oxygen gas molecules (O2g), adsorbedoxygen molecules (O2ad), and adsorbed oxygen atoms (Oad) may befirst taken into account as possible species involved in overall cathodereactions. As shown in Fig. 1, in the adsorption and dissociationreactions, oxygen gas molecules in the gas phase (O2g) may first causea non-dissociative adsorption reaction (to form O2ad), followed by adissociation reaction of adsorbed oxygen molecules on the cathodesurface and/or at the TPB (to form Oad). Equilibrium equations foreach adsorption and dissociation reaction on the cathode surface maybe described as follows:

O2g(C) + Vad(C) O2ad(C); KC AO2 O2(C)pO2(C) V(C)[3]

O2ad(C) + Vad(C) 2Oad(C); KC AO O(C)2

O2(C) V(C)[4]

where Ki is an equilibrium constant for an individual defect-chemicalreaction. The defect-chemical equilibrium reactions taken into ac-count are compiled in Table I. Here, the coverage of adsorbed speciesM (such as O2ad and Oad) on the cathode surface and at the cathodeTPB is denoted as M(C). The fraction of vacant adsorption sites is sim-ilarly denoted as V(C). The conservation condition of the adsorptionsites on the cathode surface and at the cathode TPB can be describedas follows:

O2(C) + O(C) + V(C) = 1 [5]Electrochemical reactions at the TPB can occur with adsorbed oxy-

gen atoms (Oad), adsorbed oxygen molecules (O2ad), and/or oxygengas molecules in the gas phase (O2g). Here, concentration (activity) ofelectrons in the cathode and oxygen ions in the electrolyte can be as-sumed to be high enough and regarded as almost constant or the unity.Therefore, for the three kinds of species mentioned above, followingelectrochemical reactions may be taken into account:

kC E OOad(C) + 2e(C) O2(E) + Vad(C)kC E O

[6]



CEOCEO2CEO2g

CAO2 CAO

Electrolyte

Cathode surface

Figure 1. Possible cathode reactions. Adsorption and dissociation reactions are shown as blue dotted dashed lines and electrochemical reactions as red solid lines.The subscripts g and ad denote gaseous species and adsorbed species, respectively.

kC E O2O2ad(C) + 4e(C) + VO (E) 2O2(E) + Vad(C)kC E O2

[7]

kC E O2gO2g(C) + 4e(C) 2O2(E)kC E O2g

[8]

where O2(E) denotes oxygen ion in the electrolyte. As a first step,one of the electrochemical reactions 6, 7, or 8 mentioned above maybe assumed to be the rate-determining reaction. These electrochemi-cal reactions, which could be rate-determining, are also compiled inTable II, where the (major) reactant is shown as the species involvedin electrochemical reactions.

Major reactions at anodes. Similarly to the cathode, when mixedH2-H2O fuel is supplied to the anode, hydrogen molecules in the gasphase (H2g), water vapor molecules in the gas phase (H2Og), adsorbedhydrogen molecules (H2ad), adsorbed water vapor molecules (H2Oad),adsorbed OH (OHad), adsorbed hydrogen atoms (Had), and adsorbedoxygen atoms (Oad) may be taken into account, as a first step, as pos-sible species involved in overall anode reactions. As shown in Fig.2, in the adsorption and dissociation reactions, hydrogen moleculesand water vapor molecules may first cause a non-dissociative adsorp-tion reaction followed by a dissociation reaction of these adsorbedmolecules. Equilibrium equations for each adsorption and dissocia-tion reaction on the anode surface may be described as follows:

H2g(A) + Vad(A) H2ad(A); K AAH2 H2(A)pH2(A) V(A)[9]

H2Og(A) + Vad(A) H2Oad(A); K AAH2O H2O(A)pH2O(A) V(A)[10]

H2ad(A) + Vad(A) 2Had(A); K AAH H(A)2

H2(A) V(A)[11]

H2Oad(A)+Vad(A) Had(A)+OHad(A); K AAO H H(A) OH(A)H2O(A) V(A)[12]

OHad(A) + Vad(A) Had(A) + Oad(A); K AAO H(A) O(A)OH(A) V(A)

[13]The defect-chemical equilibrium reactions taken into account are com-piled in Table I. Similarly to the cathode, the conservation conditionof the adsorption sites on the anode surface and at the anode TPB canbe described as follows:

H2(A) + H2O(A) + H(A) + OH(A) + O(A) + V(A) = 1 [14]Electrochemical reactions can occur with adsorbed hydro-

gen atoms (Had), adsorbed hydrogen molecules (H2ad), hydrogenmolecules in the gas phase (H2g), and/or adsorbed oxygen atoms(Oad) present at the TPB. Here, concentration (activity) of electronsin the anode and oxygen ions in the electrolyte can be assumed to behigh enough and regarded as almost constant or the unity. Therefore,following electrochemical reactions may be taken into account:

kAE H2Had(A) + O2(E) 2e(A) + Vad(A) + H2Oad(A)kAE H

[15]

Table I. Defect-chemical equilibrium reactions taken into account for the cathodes and the anodes.

Electrode Defect-chemical equilibrium reactions Equilibrium constants defined Equations

Cathode O2g(C) + Vad(C) O2ad(C) KC AO2 O2(C)pO2(C) V(C) (3)O2ad(C) + Vad(C) 2Oad(C) KC AO O(C)

2

O2(C) V(C)(4)

Anode H2g(A) + Vad(A) H2ad(A) K AAH2 H2(A)pH2(A) V(A) (9)H2Og(A) + Vad(A) H2Oad(A) K AAH2O H2O(A)pH2O(A) V(A) (10)

H2ad(A) + Vad(A) 2Had(A) K AAH H(A)2

H2(A) V(A)(11)

H2Oad(A) + Vad(A) Had(A) + OHad(A) K AAO H H(A)OH(A)H2O(A)V(A) (12)OHad(A) + Vad(A) Had(A) + Oad(A) K AAO H(A)O(A)OH(A) V(A) (13)



Table II. Electrochemical reactions with neutral species first considered for the cathodes and the anodes.

Species involved in Kinetic constantsElectrode electrochemical reaction Reactions defined Equation

Cathode Oad(C) Oad(C) + 2e(C) O2(E) + Vad(C) koC E O ,koC E O (6)

O2ad(C) O2ad(C) + 4e(C) + VO (E) 2O2(E) + Vad(C) koC E O2,koC E O2 (7)

O2g(C) O2g(C) + 4e(C) 2O2(E) koC E O2g,koC E O2g (8)

Anode Had(A) 2Had(A) + O2(E) 2e(A) + Vad(A) + H2Oad(A) koAE H ,koAE H (15)

H2ad(A) H2ad(A) + O2(E) 2e(A) + H2Oad(A) koAE H2,koAE H2 (16)

H2g(A) H2g(A) + O2(E) 2e(A) + H2Og(A) koAE H2g,koAE H2g (17)

Vad(A) Vad(A) + O2(E) 2e(A) + Oad(A) koAE O ,koAE O (18)

kAE H2H2ad(A) + O2(E) 2e(A) + H2Oad(A)kAE H2

[16]

kAE H2gH2g(A) + O2(E) 2e(A) + H2Og(A)kAE H2g

[17]

kAE OVad(A) + O2(E) 2e(A) + Oad(A)kAE O

[18]

One of these electrochemical reactions may be assumed to be therate-determining reaction. These electrochemical reactions are alsocompiled in Table II.

Exchange Current Density at CathodesTheoretical relations. In this section, we now derive general ex-

pression of cathode exchange current density (i0,c) in the case that oneof the electrochemical reactions described in the previous section isthe rate-determining reaction. We first consider the situations whereneutral adsorbates are involved in electrode processes. Exchange cur-rent density may be formulated as a function of equilibrium constants,rate constants, coverages, and partial pressures.

As an example of such derivation, we derive a theoretical ex-pression of i0,c in the case that the reaction shown in Eq. 6 is rate-determining. Since electrons in the cathode material (e(C)) and oxy-gen ions in the electrolyte (O2(E)) are abundant during the reaction,Eq. 6 can be regarded as a pseudo-first-order 2-electron reaction.

Therefore, the forward reaction and the backward reaction are givenas,

C E O = 2F O(C) kC E O [19]and

C E O = 2F V(C) kC E O [20]respectively, where F is a Faraday constant. Reaction rate constantsare generally described as Arrhenius-type equations:

kC E O = AC E O exp(

EC E O

RT

)[21]

kC E O = AC E O exp(

EC E O

RT

)[22]

where AC E O and AC E O are pre-exponential factors for the forward re-action and the backward reaction, respectively. EC E O and EC E Oare the corresponding activation energy for the forward reaction andthe backward reaction, respectively. R and T denote gas constantand absolute temperature, respectively. The relations between theactivation energy and the electrode potential can be described asfollows:28,4852

EC E O = EoC E O + n F Ec [23]EC E O = EoC E O n F Ec [24]

where EoC E O and EoC E O are the activation energy at zero elec-trode potential for the forward reaction and the backward reaction,respectively. and are the transfer coefficient for the forward and

Electrolyte

Anode surface

AEOAEHAEH2AEH2g

AAH2

AAH

AAH2OAAOH

AAO

Figure 2. Possible anode reactions. Adsorption and dissociation reactions are shown as blue dotted dashed lines and electrochemical reaction as red solid lines.The subscripts g and ad denote gaseous species and adsorbed species, respectively.



Table III. Surface coverage of each surface species for the cathode and the anode.

Electrode Adsorbed species Surface coverage

Cathode O2ad(C) O2(C) = KC AO2 pO2(C) V(C)Oad(C) O(C) = KC AO2 12 KC AO 12 pO2(C)

12 V(C)

Anode H2ad(A) H2(A) = K AAH2 pH2(A) V(A)H2Oad(A) H2O(A) = K AAH2O pH2O(A) V(A)

Had(A) H(A) = K AAH2 12 K AAH 12 pH2(A)12 V(A)

OHad(A) OH(A) = K AAH2 12 K AAH2O K AAH 12 K AAO H pH2(A)12 pH2O(A) V(A)

Oad(A) O(A) = K AAH21 K AAH2O K AAH 1 K AAO H K AAO pH2(A)1 pH2O(A) V(A)

Where V(C) =(

KC AO2 pO2(C) + KC AO212 KC AO 12 pO2(C)

12 + 1

)1and V(A) =

(K AAH2 pH2(A) + K AAH2O pH2O(A) + K AAH2

12 K AAH 12 pH2(A)

12

+ K AAH212 K AAH2O K AAH 12 K AAO H pH2(A)12 pH2O(A) + K AAH21 K AAH2O K AAH 1 K AAO H K AAO pH2(A)1 pH2O(A) + 1)1

the backward reactions, respectively, where the general relation + = 1 is assumed to hold.1,2 Ec and n are the cathode potentialand the number of electrons involved in the electrochemical reaction,respectively. For the 2-electron reaction described in Eq. 6, it holdsthat n = 2. These relations simply assume that the activation energyof the forward reaction and the backward reaction is modified by theelectrode potential in the opposite directions with a symmetry factorof and , as described in e.g. Butler.2,4850 Net current density, whichcorresponds to the difference between the forward current density andthe backward current density, can be derived from Eqs. 1924:

ic = C E O C E O

= 2F O(C) koC E O exp(2 F Ec

RT

)

2F V(C) koC E O exp(

2 F EcRT

)[25]

where koC E O = AC E O exp(EoC E O/RT ) and koC E O = AC E O exp(EoC E O/RT ). Eq. 25 indicates that a Butler-Volmer typerelation2,22,28,51 can be derived for the SOFC electrode processes rate-determined by electrochemical reactions.

For ic = 0 at the equilibrium potential of Ec = Ec(Eq), currentdensities for the forward and backward reactions are the same. Thesecurrent densities correspond to the exchange current density, i.e.,

i0,c 0,C E O = 0,C E O [26]At this equilibrium, the forward current density 0,C E O and the back-ward current density 0,C E O are described as

0,C E O = 2F O(C) koC E O exp(2 F Ec(Eq)

RT

)[27]

0,C E O = 2F V(C) koC E O exp(

2 F Ec(Eq)RT

)[28]

Therefore, Eqs. 2628 give the following expression of exchangecurrent density as:28

0,c = i0,c+=0,C E O

0,C E O

= 2F koC E O koC E O

O(C) V(C) [29]

by using the relation, exp(2 F Ec(Eq)/RT ) exp(2 F Ec(Eq)/RT ) = 1, to cancel the exponentialterms in Eqs. 27 and 28. By typically assuming = = 0.5, i.e.,for the case of the symmetrical change in the activation energy inEqs. 23 and 24, i0,c in Eq. 29 can be rewritten as:

i0,c = 2F koC E O12 koC E O

12 O(C) 12 V(C) 12 [30]

Such derivation is also possible in the case that Eq. 9 or 10 is therate-determining electrochemical reaction.

Surface coverages can be derived as a function of gas partial pres-sures and equilibrium constants. O2(C) and O(C) can be derived byusing Eqs. 3 and 4 as:

O2(C) = KC AO2 pO2(C) V(C) [31]

O(C) = KC AO2 12 KC AO 12 pO2(C)12 V(C) [32]

The vacant adsorption site, V(C), can be derived by using Eqs. 31 and32. The conservation condition of the adsorption sites, Eq. 5, can berewritten by substituting O2(C) and O(C) in Eq. 5 with Eqs. 31 and 32as:

V(C) =(


12 + 1

)1[33]

Consequently, these cathode surface coverages are described withequilibrium constants and gas partial pressures, and i0,c can be de-scribed using Eqs. 30, 32, and 33 as a function of gas partial pressures,equilibrium constants, and rate constants as:


12 KC AO2 14 KC AO 14 pO2(C)

14

(


12 + 1

)1[34]

The surface coverages on the cathodes described in Eqs. 31 and 32are also compiled in Table III. The relations of the cathode exchangecurrent density i0,c derived for the electrochemical reactions in Eqs. 7and 8 can be derived in the same way. All these relations describingi0,c including Eq. 30 are compiled in Table IV.

Analytical relations and various dependencies. From the theo-retical expressions of i0,c derived and compiled in Table IV, it maybe possible to obtain approximated relations by comparing cover-ages of each species on the cathodes. As is often made in defectchemistry,28,46,47 the theoretical expressions of i0,c can be rewritten tomore simplified forms in the case that the coverage of one kind ofspecies is sufficiently high compared to other coverages. For exam-ple, if the reaction shown in Eq. 6 is rate-determining and adsorbedoxygen molecules are the predominant major species on the cathode(O2(C) = 1), the conservation condition of the adsorption sites at thecathode shown in Eq. 5 and thus Eq. 31 can be approximated to thefollowing relation:

O2(C) = KC AO2 pO2(C) V(C) = 1 [35]In this case, V(C) can be rewritten from Eq. 35 as

V(C) = KC AO21 pO2(C)1 [36]) unless CC License in place (see abstract). ecsdl.org/site/terms_use address. Redistribution subject to ECS terms of use (see 133.5.50.41Downloaded on 2015-06-15 to IP


Table IV. Cathode exchange current density, i0,c, where an electrochemical reaction is rate-determining with adsorbed oxygen atoms (Eq. 6),adsorbed oxygen molecules (Eq. 7), or oxygen gas molecules in the gas phase (Eq. 8).

Species involvedin electrochemical Cathode exchange

reaction Current density current density

Oad(C) ic = 2F O(C) koC E O exp( 2F EcRT

) 2F V(C) koC E O exp

(2 F Ec

RT

)i0,c = 2F koC E O

12 koC E O

12 O(C) 12 V(C) 12

O2ad(C) ic = 4F O2(C) koC E O2 exp

( 4F EcRT

) 4F V(C) koC E O2 exp

(4 F Ec

RT

)i0,c = 4F koC E O2

12 koC E O2

12 O2(C)

12 V(C) 12

O2g(C) ic = 4F pO2(C) koC E O2g exp

( 4F EcRT

) 4F koC E O2g exp

(4 F Ec

RT

)i0,c = 4F koC E O2g

12 koC E O2g

12 pO2(C)

12

and O(C) is rewritten from Eqs. 32 and 36 as:

O(C) = KC AO2 12 KC AO 12 pO2(C)12 V(C) = KC AO2 12 KC AO 12 pO2(C)

12

[37]i0,c can be thus rewritten using Eqs. 30, 36, and 37 as:


12 KC AO2 34 KC AO 14 pO2(C)

34 [38]

which is also shown in Table V. As a result, the oxygen partial pressuredependency of i0,c is obtained as log i0,c/ log pO2(C) = 3/4.

In general, once the predominant species M is specified with thecathode surface coverage around the unity (M(C) = 1) the fraction(coverage) of the vacant adsorption sites (V(C)) shown at the bottomof Table III derived from the conservation condition in Eq. 5 canbe approximated, i.e., only one term becomes predominant and allother terms become negligible on the right hand side of this equationfor V(C). By substituting the equations of cathode exchange cur-rent density in Table IV with this simplified relation of V(C) andthe cathode surface coverage of other adsorbed species shown inTable III, approximated relations of cathode exchange current densityfor a given species involved in electrochemical reactions and a givenboundary condition can be derived, as shown in Table V. Table V andFig. 3 summarize the gas partial pressure dependencies of i0,c thathold under each specific condition.

Even though predominant adsorption species vary for differ-ent cathode materials and operating conditions etc., Table V andFig. 3 indicate that oxygen partial pressure dependence of i0,c varieswithin 3/4 log i0,c/ log pO2(C) 1/2 in the case that an electro-chemical reaction is rate-determining. Actually, the power of oxygenpartial pressure dependence has been reported from 1/4 to 1/4 ina cell with a ceria electrolyte in previous studies.27,28 Here, when theelectrochemical reaction shown in Eq. 8 is rate-determining whereO2g(C) is directly reduced to oxygen ions on the electrolyte surface,the partial pressure dependence has been derived not to be affected

by kinds of predominant adsorption species and thus the electrodeproperty depends directly on oxygen partial pressure.

On cathode exchange current density, following general tendenciescan be seen: (i) in the case that coverages of adsorbed oxygen speciesis relatively low so that most of adsorbed sites are vacant, it is expectedthat i0,c increases with increasing oxygen partial pressure (see the up-per side of Fig. 3). However, (ii) in the case that coverages of adsorbedoxygen species are relatively high so that most of adsorbed sites areoccupied, it is expected that i0,c decreases with increasing oxygenpartial pressure (see the lower side of Fig. 3). From Fig. 3, it may bepossible to consider the predominant adsorption species and electro-chemical reactions taking place from the dependence of i0,c on oxygenpartial pressure. However, we should carefully note that this oxygenpartial pressure dependence of i0,c is valid under the precondition thatthe corresponding electrochemical reaction is rate-determining. Thisprecondition becomes no longer valid, if adsorption and dissociationreactions or surface transport processes is rate-determining.

Exchange Current Density at AnodesTheoretical relations. Similarly to the treatment for the cath-

odes, anode exchange current density (i0,a) can be derived for = = 0.5, under the condition that one of the electrochemicalreactions shown in Eqs. 1518 is rate-determining. Surface coveragesfor anodes derived from Eqs. 913 are compiled in Table III. The rela-tions of the exchange current density derived for the electrochemicalreactions in Eqs. 1518 are compiled in Table VI.

Analytical relations and various dependencies. From the theo-retical expressions of i0,a, compiled in Table VI, it is possible to deriveapproximated expressions by comparing coverages of each species onanodes, as made for cathodes. Once the predominant species M is spec-ified with the anode surface coverage around the unity (M(A) = 1),

Table V. Cathode exchange current density, i0,c, where an electrochemical reaction is rate-determining with a given species under each boundarycondition of cathode surface.

Species involved in Boundary Cathode exchange current density for given specieselectrochemical reaction conditions and boundary condition

Oad(C) O2(C) = 1 i0,c = 2F koC E O

12 koC E O

12 KC AO2 34 KC AO 14 pO2(C)

34

O(C) = 1 i0,c = 2F koC E O12 koC E O

12 KC AO2 14 KC AO 14 pO2(C)

14

V(C) = 1 i0,c = 2F koC E O12 koC E O

12 KC AO2 14 KC AO 14 pO2(C)

14

O2ad(C) O2(C) = 1 i0,c = 4F koC E O2

12 koC E O2

12 KC AO2 12 pO2(C)

12

O(C) = 1 i0,c = 4F koC E O212 koC E O2

12 KC AO 12

V(C) = 1 i0,c = 4F koC E O212 koC E O2

12 KC AO2 12 pO2(C)

12

O2g(C) O2(C) = 1 i0,c = 4F koC E O2g

12 koC E O2g

12 pO2(C)

12

O(C) = 1V(C) = 1



0

all condions

Most of adsorpon sites on the cathode are vacant.

Cathode surface is almost fully covered by or .

Figure 3. Partial pressure dependence of cathode exchange current density(i0,c) on O2 partial pressure, for given species involved in the electrochemicalreaction (shown on the left hand side in the text boxes) and each boundarycondition of cathode surface coverage (shown on the right hand side in the textboxes).

the fraction (coverage) of the vacant adsorption sites (V(A)) shownat the bottom of Table III derived from the conservation condition inEq. 14 can be approximated, i.e., only one term becomes predominantand all other terms become negligible on the right hand side of thisequation for V(A). By substituting the equations of anode exchangecurrent density in Table VI with this simplified relation of V(A) and theanode surface coverage of other adsorbed species shown in Table III,approximated relations of anode exchange current density for a givenspecies involved in electrochemical reactions and a given boundarycondition can be derived, as shown in Table VII. Table VII andFig. 4 summarize the gas partial pressure dependencies of i0,a thathold under each boundary condition.

Even though predominant adsorption species may be different fordifferent anode materials and operational conditions etc., Table VIIand Fig. 4 indicate that the hydrogen and water vapor partial pres-sure dependencies of i0,a vary within 3/2 log i0,a/ log pH2(A) 5/2 and 3/2 log i0,a/ log pH2O(A) 1/2, respectively, if an

electrochemical reaction is rate-determining. Here, when the electro-chemical reaction shown in Eq. 17 is rate-determining where H2g(A)is directly oxidized by oxygen ions on the electrolyte, the partialpressure dependencies may not be affected by kinds of predominantadsorption species and thus the electrode property depends directlyon hydrogen and water vapor partial pressures.

On anode exchange current density, following general tendenciescan be seen: (i) in the case that coverages of adsorbed hydrogenatom and/or adsorbed hydrogen molecules are predominant occupy-ing most of surface adsorption sites, it is expected that i0,a decreaseswith increasing hydrogen partial pressure and increases with increas-ing water vapor partial pressure (see the lower-right side of Fig. 4). Onthe other hand, (ii) in the case that coverages of adsorbed water vapormolecules, adsorbed OH molecules, or oxygen atoms are predomi-nant, it is expected that i0,a increases with increasing hydrogen partialpressure and decreases with increasing water vapor partial pressure(see the upper-left side of Fig. 4). Rate-determining electrochemicalreactions and predominant adsorption species could be discussed withTable VII and Fig. 4 by analyzing the hydrogen and water vapor partialpressure dependencies of i0,a, experimentally obtained.

Exchange Current Density with Charged SpeciesIn this study, it has been assumed that the adsorbed surface species

(adsorbates) are regarded to be neutral in charge. However, the ad-sorbed species can be charged, especially on the cathode where per-ovskite oxides of ionic character are usually used.43,53,54,55 In orderto consider more elementary reaction steps where only one electronand only one or two species are involved, the following charged oxy-gen species, O2ad, O22ad, Oad, and O2ad , may be taken into account.Such cathode reaction model is schematically described in Fig. 5. Inthis consideration, by taking into account the charged adsorbates, theconservation condition of surface adsorption sites is now given as:

O2(C) + O2 (C) + O22 (C) + O(C) + O2(C) + V(C) = 1 [39]For simplicity, the interaction among adsorbates is neglected eventhough electrostatic (coulomb) interaction may become importantwith increasing concentration (coverage) of charged surface species.

Following three electrochemical reactions may be considered asmore elementary reaction steps:

kC E O2O2ad(C) + e(C) O2ad(C)kC E O2

[40]

kC E O2O2ad(C) + e(C) O22ad(C)kC E O2

[41]

kC E O Oad(C) + e(C) O2ad (C)kC E O

[42]

Table VI. Anode exchange current density, i0,a, where an electrochemical reaction is rate-determining with adsorbed hydrogen atoms (Eq. 15),adsorbed hydrogen molecules (Eq. 16), hydrogen molecules in the gas phase (Eq. 17), or vacant adsorption sites (Eq. 18).

Species involvedin electrochemical Anode exchange current

reaction Current density density

Had(A) ia =2F H(A)2 koAE H exp(

2FEaRT

)2F V(A) H2O(A)

koAE H exp(2

FEaRT

)i0,a =2F koAE H

12 koAE H

12 H(A) V(A) 12 H2O(A)

12

H2ad(A) ia =2F H2(A) koAE H2 exp

(2FEa

RT

)2F H2O(A)

koAE H2 exp(2FEaRT

)i0,a =2F koAE H2

12 koAE H2

12 H2(A)

12 H2O(A)

12

H2g(A) ia =2F pH2(A) koAE H2g exp

(2FEa

RT

)2F pH2O(A)

koAE H2g exp(2FEaRT

)i0,a =2F koAE H2g

12 koAE H2g

12 pH2(A)

12 pH2O(A)

12

Vad(A) ia =2F V(A) koAE O exp(

2FEaRT

)2F O(A) koAE O exp

(2

FEaRT

)i0,a =2F koAE O

12 koAE O

12 V(A) 12 O(A) 12



Table VII. Anode exchange current density, i0,a, where an electrochemical reaction is rate-determining with a given species under each boundarycondition of anode surface.

Species involvedin electrochemical Boundary Anode exchange current density for given

reaction conditions species and boundary condition

Had(A) H2(A) = 1 i0,a = 2F koAE H

12 koAE H

12 K AAH2 32 K AAH2O 12 K AAH 12 pH2(A)

32 pH2O(A)

12

H2O(A) = 1 i0,a = 2F koAE H

12 koAE H


12 pH2O(A)

32

H(A) = 1 i0,a = 2F koAE H12 koAE H


12 pH2O(A)

12

OH(A) = 1 i0,a = 2F koAE H12 koAE H

12 K AAH2 32 K AAH2O 32 K AAH 32 K AAO H 2 pH2(A)

32 pH2O(A)

32

O(A) = 1 i0,a = 2F koAE H12 koAE H

12 K AT H2 52 K AAH2O 32 K AAH 52 K AAO H 2 K AAO 2 pH2(A)

52 pH2O(A)

32

V(A) = 1 i0,a = 2F koAE H12 koAE H


12 pH2O(A)

12

H2ad(A) H2(A) = 1 i0,a = 2F koAE H2

12 koAE H2

12 K AAH2 12 K AAH2O 12 pH2(A)

12 pH2O(A)

12

H2O(A) = 1 i0,a = 2F koAE H2

12 koAE H2


12 pH2O(A)

12

H(A) = 1 i0,a = 2F koAE H212 koAE H2

12 K AAH2O 12 K AAH 12 pH2O(A)

12

OH(A) = 1 i0,a = 2F koAE H212 koAE H2

12 K AAH2 K AAH2O 12 K AAH 12 K AAO H 1 pH2(A) pH2O(A)

12

O(A) = 1 i0,a = 2F koAE H212 koAE H2

12 K AAH2 32 K AAH2O 12 K AAH K AAO H 1 K AAO 1 pH2(A)

32 pH2O(A)

12

V(A) = 1 i0,a = 2F koAE H212 koAE H2


12 pH2O(A)

12

H2g(A) H2(A) = 1 i0,a = 2F koAE H2g

12 koAE H2g

12 pH2(A)

12 pH2O(A)

12

H2O(A) = 1H(A) = 1OH(A) = 1O(A) = 1V(A) = 1

Vad(A) H2(A) = 1 i0,a = 2F koAE O

12 koAE O

12 K AAH2 32 K AAH2O 12 K AAH 12 K AAO H 12 K AAO 12 pH2(A)

32 pH2O(A)

12

H2O(A) = 1 i0,a = 2F koAE O

12 koAE O


12 pH2O(A)

12

H(A) = 1 i0,a = 2F koAE O12 koAE O

12 K AAH21 K AAH2O 12 K AAH 1 K AAO H 12 K AAO 12 pH2(A)1 pH2O(A)

12

OH(A) = 1 i0,a = 2F koAE O12 koAE O

12 K AAH2O 12 K AAO H 12 K AAO 12 pH2O(A)

12

O(A) = 1 i0,a = 2F koAE O12 koAE O


12 pH2O(A)

12

V(A) = 1 i0,a = 2F koAE O12 koAE O


12 pH2O(A)

12

0

3

3

all condions

Anode surface is almost fully covered by , , or .

Anode surface is almost fully covered by or .

Figure 4. Partial pressure dependencies of anode exchange cur-rent density (i0,a) on H2 and H2O partial pressures, for givenspecies involved in the electrochemical reaction (shown on theleft hand side in the text boxes) and each boundary condition ofanode surface coverage (shown on the right hand side in the textboxes).



CEO2

Electrolyte

Cathode surface

CAO2 CEO2 CEOCAO

CIO

Figure 5. Possible cathode reactions. Adsorption and dissociation reactions, ion transfer reaction and electrochemical reactions are shown as blue dotted dashedlines, black dotted dashed lines and red solid lines, respectively. The subscripts g and ad denote gaseous species and adsorbed species, respectively.

We consider one of these three reactions as a rate-determining step,where these charged oxygen species are now involved. Other two re-actions will then be equilibrium reactions, compiled in Table VIII. Forthese cases with various rate-determining electrochemical reactions,the relations describing their surface coverage can be derived. Currentdensity and cathode exchange current density can be described forthese three kinds of species as compiled in Table IX.

The approximated relations can be derived in the same way, ascompiled in Table X, describing the major relations of cathode ex-change current density for given species involved in electrochemicalreactions and boundary conditions. Figure 6 also summarizes the par-tial pressure dependence of cathode exchange current density, i0,c. Bycomparing this figure (Fig. 6) for the charged oxygen species withFig. 3 for the neutral oxygen species, it can be found that the partial

pressure dependence of i0,c also varies by varying the kind of adsor-bates and boundary condition. Similarly to the dependence for theneutral adsorbates (Fig. 3), the partial pressure dependence for thecharged adsorbates ranges from 3/4 to 1/2 (Fig. 6). It is also distin-guished that a positive pO2(C) dependence is obtained for a low totalcoverage (V(C) = 1), which is also common for the cases with theneutral adsobates. In addition, identical pO2(C) dependence has beenobtained for the similar species with and without charge for the samegiven boundary condition. For example, for the boundary condition ofV(C) = 1, the same 1/2 power dependence is obtained for O2ad in Fig.3 and for O2ad and O2ad in Fig. 6. For this same boundary condition,the same 1/4 power dependence is also obtained for Oad in Fig. 3 andfor Oad in Fig. 6. By comparing the dependencies with and withoutcharge, these figures suggest the identical pO2(C) dependence, which

Table VIII. Defect-chemical equilibrium reactions with charged adsorbates, taken into account for the cathodes.

Species involved in electrochemical reaction Defect-chemical equilibrium reactions Equilibrium constants defined

O2ad(C) (and O2ad(C)) O2g(C) + Vad(C) O2ad(C) KC AO2 O2(C)

pO2(C) V(C)

O2ad(C) + e(C) O22ad(C) KC E O2 O22 (C)O2 (C)

O22ad(C) + Vad(C) 2Oad(C) KC AO O(C)2

O22 (C)V(C)

Oad(C) + e(C) O2ad (C) KC E O O2(C)O(C)

O2ad (C) O2(E) + Vad(C) KC I O V(C)

O2(C)

O2ad(C) O2g(C) + Vad(C) O2ad(C) KC AO2 O2(C)

pO2(C) V(C)

O2ad(C) + e(C) O2ad(C) KC E O2 O2 (C)O2(C)


O22 (C)V(C)

Oad(C) + e(C) O2ad (C) KC E O O2(C)O(C)


O2(C)

Oad(C) O2g(C) + Vad(C) O2ad(C) KC AO2 O2(C)

pO2(C) V(C)

O2ad(C) + e(C) O2ad(C) KC E O2 O2 (C)O2(C)

O2ad(C) + e(C) O22ad(C) KC E O2 O22 (C)O2 (C)


O22 (C)V(C)


O2(C)



Table IX. Cathode exchange current density, i0,c, where an electrochemical reaction with charged adsorbates is rate-determining with adsorbedoxygen molecules (Eq. 40), adsorbed charged oxygen molecules (Eq. 41), or adsorbed charged oxygen atoms (Eq. 42).

Species involvedin electrochemical Cathode exchange current

reaction Current density Cathode exchange current density

O2ad(C)(and O2ad(C))

ic = F O2(C) koC E O2 exp

( F EcRT

) F O2 (C)

koC E O2 exp(

F EcRT

)i0,c = F koC E O2

12 koC E O2

12 O2(C)

12 O2 (C)

12

O2ad(C) ic = F O2 (C) koC E O2 exp

( F EcRT

) F O22 (C)

koC E O2 exp(

F EcRT

)i0,c = F koC E O2

12 koC E O2

12 O2 (C)

12 O22 (C)

12

Oad(C) ic = F O(C) koC E O exp

( F EcRT

) F O2(C)

koC E O exp(

F EcRT

)i0,c = F koC E O

12 koC E O

12 O(C)

12 O2(C)

12

Table X. Cathode exchange current density, i0,c, where an electrochemical reaction with charged adsorbates is rate-determining with a givenspecies under each boundary condition of cathode surface. Only the cases, where non-charged species are predominant, are listed in this Table, asa high coverage for charged adsorbates may be less probable than for neutral adsorbates, due to mutual electrostatic repulsion.

Species involvedin electrochemical Boundary Cathode exchange current density for given

reaction conditions species and boundary condition

O2ad(C) O2(C) = 1 i0,c = F koC E O2

12 koC E O2

12 KC AO2 12 KC E O2

12 KC AO

12 KC E O 1 KC I O 1 pO2(C)

12

(and O2ad(C)) V(C) = 1 i0,c = F koC E O2

12 koC E O2


12 KC AO


12

O2ad(C) O2(C) = 1 i0,c = F ko

C E O212 ko

C E O212 KC AO2 12 KC E O2

12 KC AO


12

V(C) = 1 i0,c = F koC E O212 ko

C E O212 KC AO2 12 KC E O2

12 KC AO


12

Oad(C) O2(C) = 1 i0,c = F koC E O

12 koC E O


14 KC E O2

14 KC AO

14 KC I O 12 pO2(C)

34

V(C) = 1 i0,c = F koC E O 12 koC E O


14 KC E O2

14 KC AO

14 KC I O 12 pO2(C)

14

0

Figure 6. Partial pressure dependence of cathode exchange current density(i0,c) on O2 partial pressure, for given species involved in the electrochemicalreaction (shown on the left hand side in the text boxes) and each boundarycondition of cathode surface coverage (shown on the right hand side in the textboxes).

is independent of the charge of the adsorbates. It should be notedhere that the coverage of charged absorbates could not approach theunity due to their electrostatic interactions so that the cases with suchboundary conditions may not be probable.

As similar derivations can be made for given species and boundaryconditions, it is desired to extent such treatments to other electrodeprocesses including the anode reactions. Adsorbed species on the Nianodes could be charged, whilst some DFT calculation studies5658have considered neutral adsorbates. We should also note that, forcharged adsorbates, the mass action law also contains a contributionby the surface potential drop,54 which gives further complications.This is quite intuitive, because the adsorption reaction comprises thetransfer of electron(s) over this surface potential step.

As there are many possible elementary reactions for the cathodeand even more for the anode with multiple elements involved, weshould carefully consider such steps. To mention a few, even for thereduction reaction of oxygen molecules mentioned in this section,we could consider other pathways with other possible elementaryreactions than described in Eqs. 40 to 42, e.g.;

Adsorption of a neutral oxygen molecule associated with one elec-tron transfer:

O2g(C) + e(C) O2ad(C) [43]Dissociation of a doubly-charged oxygen molecule (O22ad) withoutcharge transfer:

O22ad(C) 2Oad(C) [44]Dissociation of a singly-charged oxygen molecule (O2ad) with oneelectron transfer:

O2ad(C) + e(C) 2Oad(C) [45]For a given rate-determining step considered, one rate equation forthe step (with a forward reaction term and a backward reaction term)



0

all condions

Figure 7. Partial pressure dependence of cathode exchange current density(i0,c) on O2 partial pressure, for given species involved in the electrochemicalreaction (shown on the left hand side in the text boxes) and each boundarycondition of cathode surface coverage (shown on the right hand side in the textboxes). = 0.25( = 0.75).

and equations for all other fast preceding and fast succeeding equi-libria will give the corresponding rate equations for exchange currentdensity and its dependencies. It should be noted that for the fastpreceding and succeeding equilibria, several steps can be condensedinto one equilibrium reaction, but not for the rate-determining reac-tion. As the universal elementary rate-determining reaction for SOFCelectrodes has not yet been experimentally specified,22 more carefulanalysis is surely required in considering elementary rate-determiningsteps and such study should be coupled with experimental results. Assuch comparison is beyond the scope of this theoretical study, furtherinvestigation is now in progress to experimentally examine partialpressure dependencies, first for the cathodes, coupled with detailedtheoretical and numerical analysis, where various possible elementarysteps mentioned in this study are taken into account.

Transfer Coefficient Dependence

In this study, the transfer coefficient has been assumed as = = 0.5. In order to consider the cases with a different transfer co-efficient, partial pressure dependence is derived in this section for = 0.25 ( = 0.75) and = 0.75 ( = 0.25). For the case of = 0.25, Eq. 29 will be rewritten as:

i0,c = i0,c+ = 0,C E O

0,C E O

= 2F koC E O koC E O

O(C) V(C)

= 2F koC E O34 koC E O

14 O(C) 34 V(C) 14 [46]

Various partial pressure dependencies of exchange current densityhave been derived, as shown in Fig. 7 for = 0.25 and in Fig. 8for = 0.75, through the identical procedure in the derivations. Thesame procedure has also been applied for the anode exchange currentdensity, as shown in Fig. 9 for = 0.25 and in Fig. 10 for = 0.75.These figures reveal that the partial pressure dependence has slightly

0

all condions

Figure 8. Partial pressure dependence of cathode exchange current density(i0,c) on O2 partial pressure, for given species involved in the electrochemicalreaction (shown on the left hand side in the text boxes) and each boundarycondition of cathode surface coverage (shown on the right hand side in the textboxes). = 0.75 ( = 0.25).

deviated, while the general tendency for different predominant speciesstill remains identical to that for = 0.5 described in Figs. 3 and 4.The power of oxygen partial pressure dependence ranges from 5/8to 3/4 for = 0.25 and from 7/8 to 1/4 for = 0.75 for thecathode exchange current density, i.e., the power of oxygen partialpressure dependence tends to shift more negative with increasingthe transference coefficient. For the anode exchange current density,hydrogen partial pressure dependence also tends to shift more negativewith increasing the transference coefficient (from Fig. 9 to Fig. 10),while the power of water vapor pressure dependence varies in theopposite direction.

It should be noted that, as stated by Fleig,53 the pre-factor of thefield effect on the rate can differ from the classical Butler-Volmervalues, for which the sum ( + ) may not be necessarily the unityany more.

Empirical Phenomenological RelationsApproximated expressions of the exchange current density for the

cathode and the anode derived in the previous section are consistingof the product of the rate constants for the forward and backward elec-trochemical reactions, the equilibrium constants of the adsorption anddissociation reactions on the electrode surface, and gas partial pres-sures. Here, an equilibrium constant of the adsorption and dissociationreactions, Ki , can be expressed by using an enthalpy Hi and a pre-exponential factor K oi originating from entropy of the reactions:1,2,28

Ki = K oi exp(Hi

RT

)[47]

For example, if Eq. 47 is substituted to the approximated expressionof i0,c in the case that the electrochemical rate-determining reactionoccurs with the adsorbed oxygen atoms at the cathode (Oad(C)) under



0

3

3

all condions

Figure 9. Partial pressure dependencies of an-ode exchange current density (i0,a) on H2 andH2O partial pressures, for given species involvedin the electrochemical reaction (shown on the lefthand side in the text boxes) and each boundarycondition of anode surface coverage (shown onthe right hand side in the text boxes). = 0.25( = 0.75).

the boundary condition of V(C) = 1, as shown in Table V, we canthen obtain the following relation:


12 KC AO2 14 KC AO 14 pO2(C)

14

= 2F AC E O12 AC E O

12 K o

14

C AO2 K oC AO14

exp(

12

EoC E O + 12

EoC E O + 14 HC AO2 + 14 HC AO

RT

)

pO2(C)14 [48]

where K oC AO2 and K oC AO are the pre-exponential factors of the corre-sponding equilibrium constants shown in Eqs. 3 and 4, respectively.

0

3

3

all condions

Figure 10. Partial pressure dependencies of anodeexchange current density (i0,a) on H2 and H2O par-tial pressures, for given species involved in the elec-trochemical reaction (shown on the left hand side inthe text boxes) and each boundary condition of an-ode surface coverage (shown on the right hand sidein the text boxes). = 0.75 ( = 0.25).



HC AO2 and HC AO are the enthalpy of the equilibrium constantsshown in these equations, respectively. By compiling the constantterms, Eq. 47 above can be rewritten as:

i0,c = total,C E O exp(Etotal,C E O + Htotal,C E O

RT

) pO2(C)O2

[48]where total,C E O , Etotal,C E O , Htotal,C E O , and O2 are constants.This equation has actually a similar form to the empirical equation ofi0,c shown in Eq. 1 explained in the Introduction. It is generally possi-ble to apply this kind of derivations for the approximated expressionsof i0,c and i0,a derived in this study. The empirical phenomenolog-ical equations, Eqs. 1 and 2, represented by the product of partialpressures, pre-exponential factors, and exponential terms includingtemperature and energy terms are therefore compatible with the the-oretical relations based on defect-chemical considerations derived inthis study.

Simulated Dependenceies of Exchange Current Densities onPartial Pressures

Cathode exchange current density. The partial pressure depen-dencies of i0,c are examined by simulating normalized exchange cur-rent densities for various species involved in electrochemical reactionsand boundary conditions. Such simulation has been made for the typ-ical cases, namely, for the neutral adsorbates with = = 0.5. Thesimulation results for different electrochemical rate-determining re-actions at cathodes are shown in Fig. 11. In this figure, it is assumedthat air utilization is 0% at the oxygen partial pressure (pO2(C)) of0.21 atm in the ambient air, and i0,c is normalized to be the unity(one) at this oxygen partial pressure. We can see that partial pres-sure dependence varies for different predominant adsorption speciesand different boundary conditions, even if the same electrochemicalreaction takes place.

The coverages of oxygen species may be low and thus vacant sitesare predominant (V(C) = 1) at a lower oxygen partial pressure near airoutlet especially at a higher operational temperature. In such cases,it is expected that i0,c decreases with increasing air utilization, i.e.,with decreasing oxygen partial pressure (as shown in Fig. 11(a)11(c)for V(C) = 1). On the other hand, the coverages of adsorbed oxygenatom or molecules could be predominantly high at higher oxygenpartial pressure near air inlet, at a lower operational temperature,and/or at a higher total pressure. In such cases, it is expected that i0,cincreases with increasing air utilization, i.e., with deceasing oxygenpartial pressure because of an increase in the number vacant activesites (see Fig. 11(a) for O2(C) = 1 and O(C) = 1, and Fig. 11(b) forO2(C) = 1). In general, since SOFCs operate at a high temperature,it should be noted that a large negative entropy of adsorption willusually lead to lower adsorbate coverages5659 so that the case withthe coverage around the unity will be exceptional.

Anode exchange current density. Similarly to cathodes, the par-tial pressure dependencies of i0,a are examined by simulating nor-malized exchange current densities for various species and boundaryconditions. The simulation results for different electrochemical rate-determining reactions at anodes are shown in Fig. 12. In this figure,it is assumed that fuel utilization is 0% at the water vapor partialpressure (pH2O(A)) of 0.20 atm (i.e., pH2(A) = 0.8 atm), which is a typ-ical value of water vapor concentration obtained just after the steamreforming of methane-based fuels, and i0,a is normalized to be theunity at these water vapor and hydrogen partial pressures. As in thecase of the cathodes, we can see that partial pressure dependencevaries for different predominant adsorption species, even if the sameelectrochemical reaction takes place.

The coverages of species may be low and thus vacant sites are pre-dominant for weakly-adsorbed species and/or at a higher operationaltemperature. In such cases, it is expected that i0,a has a maximumvalue around pH2(A) = pH2O(A) = 0.5 atm when adsorbed hydrogen

pO2(C) / atm0.0 0.2 0.4 0.6 0.8 1.0

Nor

mal

ized

i 0,c

/ -

0

1

2

3

4

5

Air utilization / %-350-300-250-200-150-100-50050100

pO2(C) / atm0.0 0.2 0.4 0.6 0.8 1.0

Nor

maliz

ed

i 0,c

/ -

0

1

2

3

4

5

Air utilization / %-350-300-250-200-150-100-50050100

(a)

(b)

pO2(C) / atm0.0 0.2 0.4 0.6 0.8 1.0

Nor

maliz

ed

i 0,c

/ -

0

1

2

3

4

5

Air utilization / %-350-300-250-200-150-100-50050100

, ,

(c)

Figure 11. Normalized cathode exchange current density as a function of O2partial pressure and air utilization for different boundary conditions in electro-chemical reactions where the species involved in electrochemical reaction are(a) Oad(C), (b) O2ad(C), and (c) O2g(C), respectively.



pH2O(A) / atm0.0 0.2 0.4 0.6 0.8 1.0

Nor

maliz

ed

i 0,a

/ -

0

1

2

3

4

5

Fuel utilization / %

-20 0 20 40 60 80 100

pH2(A) / atm0.00.20.40.60.81.0

pH2O(A) / atm0.0 0.2 0.4 0.6 0.8 1.0

Nor

maliz

ed

i 0,a

/ -

0

1

2

3

4

5


-20 0 20 40 60 80 100

pH2(A) / atm0.00.20.40.60.81.0

(b)

(a)

pH2O(A) / atm0.0 0.2 0.4 0.6 0.8 1.0

Nor

maliz

ed

i 0,a

/ -

0

1

2

3

4

5


-20 0 20 40 60 80 100

pH2(A) / atm0.00.20.40.60.81.0

3

pH2O(A) / atm0.0 0.2 0.4 0.6 0.8 1.0

Nor

maliz

ed

i 0,a

/ -

0

1

2

3

4

5


-20 0 20 40 60 80 100

pH2(A) / atm0.00.20.40.60.81.0

(c)

(d)

Figure 12. Normalized anode exchange current density as a function of H2O and H2 partial pressure and fuel utilization for different boundary conditions inelectrochemical reactions where the species involved in electrochemical reaction are (a) Had(A), (b)H2ad(A), (c) H2g(A), and (d) Vad(A), respectively.

atom and adsorbed hydrogen molecules are involved in the electro-chemical reactions shown in Eqs. 1517. Such a maximum can beseen in Fig. 12(a)12(c). In the case that electrochemical reactionincluding adsorbed oxygen atom in Eq. 18 is predominant, it is ex-pected that i0,a increases with increasing fuel utilization (i.e., withdecreasing hydrogen partial pressure and with increasing water vaporpressure). On the other hand, the coverages of adsorbed hydrogen atomor molecules could be relatively high and thus predominant (H(A) = 1or H2(A) = 1) at a higher hydrogen partial pressure near the fuel inlet.In such cases, it is expected that i0,a increases with increasing fuel uti-lization (see Fig. 12(a), 12(b), and 12(d)). In contrast, the coverages ofadsorbed water vapor molecules, adsorbed OH molecules, or oxygenatoms could be high and thus predominant (H2O(A) = 1, OH(A) = 1,or O(A) = 1) at a higher water vapor partial pressure near the fueloutlet. In such cases, it is expected that i0,a decreases with increasingfuel utilization (see Fig. 12(a), 12(b), and 12(d)).

Common features. In both for the cathodes and for the anodes,higher value of forward and backward reactions are required to ob-tain higher exchange current densities which represent the activityof the electrodes. Therefore, as summarized in Table IV and TableVI, the product of current density for the forward and the backwardreactions should be high for more active electrode. Since these fac-tors include the coverages, partial pressure dependence of exchangecurrent densities varies for different degree of the coverages. The cov-erages could vary depending on operational temperature, gas partialpressure, electrode and electrolyte materials involved in the electro-chemical reaction, elementary reaction species and chemical state ofoutermost surfaces of these materials, and impurities covering elec-trode surfaces to prevent adsorption of certain species. In addition,defect concentrations appearing in the rate equations have certain oxy-gen partial pressure dependencies as well known in defect chemistryfor ionic and electronic defects. Such pO2 dependence may even differ



between the bulk and the surface.55 Scattered exchange current densityvalues reported and their partial pressure dependencies can be wellexpected and understood based on these theoretical considerations.

Applicability and Limitation of the Relations ExpressingExchange Current Densities

The actual electrode reactions in SOFC are considered to be muchmore complicated. More detailed modeling and careful considerationsmentioned below have to be required upon analyzing the electrodereaction kinetics, taking into account the validity of the relationsof exchange current density, and applying the models to real SOFCelectrodes.

1. Rate-determining reaction: In this study, electrochemical reac-tions at the electrodes are assumed to be rate-determining, forwhich the Butler-Volmer type expression could be derived. Basedon this expression, approximated equations have been derived.However, it is of course possible that the adsorption and dis-sociation reaction and/or the surface transport process on theelectrode surface are rate-determining instead. Approximated ex-pressions of exchange current density derived in this study onlyhold under the assumption that an electrochemical reaction israte-determining. If this precondition is no longer valid, the the-oretical expressions of exchange current densities derived in thisstudy may no longer be applicable. For the surface exchangerate-determining case, the exchange current density may dependon surface exchange coefficient. For the surface transport rate-determining case, the exchange current density may depend onsurface diffusion coefficient.

2. Mixed-conducting electrode: In this study, mixed conductivity ofthe electrode and electrolyte materials is not taken into account. Ifa mixed-conducting material is used for the electrodes, adsorptionand dissociation reactions on the electrode surface followed bythe bulk diffusion from the electrode surface into the electrodematerial may dominate the whole electrode reactions.23,24,6063Exchange current density values should be reconsidered for thissituation where surface exchange coefficient or bulk diffusioncoefficient will be rate-determining.

3. Reactions with CO and hydrocarbons: In the actual SOFCsystems, fuels containing CO and various hydrocarbons as-sociated with internal reforming can be used on the anodeside. Internal reforming reaction and shift reaction (CO (g) +H2O (g) CO2 (g) + H2 (g)) should be then taken into account.In addition, if CO(g) is involved in the electrochemical reac-tions, the corresponding reactions with CO should be taken intoaccount.

4. Impurity poisoning: The poisoning species such as sulfur con-tained in the practical fuels strongly affect the surface reactions,electrode reactions, shift reactions, and internal reforming reac-tions. If such poisoning / degradation phenomena are associatedwith certain microstructural changes of the electrodes,6466 theapparent exchange current density values will be changed.

5. Electrode microstructure: Exchange current density values de-pend strongly on the microstructure of the individual porousSOFC electrodes. The identical cells made in the same man-ufacturing procedures may have similar apparent exchangecurrent density values, useful for e.g. system simulations. How-ever, it is surely important, from the fundamental research view-point to compare and discuss exchange current density valuesper TPB length by taking into account the three-dimensionalstructure.33,6770

More fundamental theoretical and atomistic aspects should be care-fully considered:

6. Applicability of the Butler-Volmer type expression: It may beimpossible to conclude universal reaction mechanisms of SOFCelectrodes as pointed out by Mogensen et al.,22 partly because

electrode kinetics are affected to a large degree by e.g. purity ofthe raw materials of the electrode and the electrolyte, manufac-turing history, microstructure, impurities present on the surfaceand at the TPB, and various operating conditions. The distri-bution of overvoltage may not be negligible depending on thedistance from the bulk electrolyte surface into the porous elec-trode structure having a certain thickness. Therefore, it may bedifficult to explain all of the electrode reactions using the Butler-Volmer equation which assumes one rate-determining reaction.22Indeed, partial pressure dependencies of exchange current densitytheoretically derived in this study suggest that exchange currentdensity values can vary strongly for different rate-determiningelectrochemical reactions and predominant adsorption specieseven for a given rate-determining electrochemical reaction, assummarized in Figs. 3, 4, and 6 to 10. The values of exchangecurrent density experimentally determined and reported in theliterature1821,44 are relatively scattered also indicating that ex-change current density values can be affected by many factors, asmentioned above. Experimental data on SOFC exchange currentdensity have been compiled in Ref. 44, where it was generallydifficult to derive or conclude electrode reaction mechanisms andelementary reactions.

We should also note that the Butler-Volmer equation, derivedfor the situation in liquid electrochemistry cannot simply be trans-ferred to SOFC electrodes.53 In liquid electrochemistry, a sup-porting liquid electrolyte solution is present. However, at SOFCelectrodes, no such electrolyte is available at the gas-solid elec-trode. The electrical potential drop at the surface, which mayaffect the kinetics of rate-determining steps, is thus not necessar-ily identical to the applied overpotential.53,71 Charged adsorbateswill create an additional electrical potential drop at the surface.Due to such additional complications, the physical meaning ofextracted parameters such as the transfer coefficient may differfrom the simple Butler-Volmer theory, whilst the Butler-Volmerexpression can phenomenologically represent electrode behaviorespecially in a limited bias range. Such fundamental questions onelectrochemistry in solid versus liquid state have been importantscientific issues since decades,71 which should be discussed andclarified in the future.

7. Transfer coefficient: In this study, transfer coefficient, , inthe Butler-Volmer type expression is first considered to be 0.5.Eq. 29 clearly suggest that a deviation of from 0.5 leads toa change in the partial pressure dependencies. Such deviationshave clearly been simulated for the different values in Figs. 7to 10. Physical meaning of such change in transfer coefficientshould be then considered in the Bulter-Volmer type expressionfor SOFC electrodes.

8. Charged adsorbates: In this study, adsorbed (oxygen) speciesare regarded to be neutral or charged. It is well possible thatsuch species are actually charged. Density functional theorystudies have revealed that adsorbed oxygen species carry somenegative charge.43 In addition, the O-O bond in the neutral O2molecule has a binding energy of ca. 5 eV which decreases onlywhen electrons are donated into antibonding orbitals to makethe dissociation possible.72 Especially for the perovskite-basedcathode materials of ionic character, the formation of chargedadsorbates should be considered, for which charged species haveto be taken into account as discussed in this study.

9. Adsorption isotherm: The simplest adsorption isotherm, theLangmuir-type, assumes that adsorption energy (enthalpy) isindependent of the coverages. This general assumption has alsobeen made for the adsorption reactions model in this study.More sophisticated models will become necessary in the casethat adsorption energy depends on the surface coverages. Indeed,surface reaction models of the Temkin-type have been appliedin considering sulfur poisoning on the Ni anode surface.64 Inaddition, if charged adsorbates are predominate on the elec-trode surfaces, the electrostatic interaction among such charged



species may no longer be negligible, affecting the dependencies,especially at higher surface adsorbate coverages.

10. Elementary rate-determining steps: The reactions of e.g. Eqs.7 and 8 are discussed as possible rate-determining steps in thisstudy. In chemical kinetics, however, the concept of the rate-determining step implies that this step is an elementary reactionstep, and that all other preceding or following steps are faster(thus in quasi-equilibrium). However, Eqs. 7 and 8 (and severalother similarly complex reactions e.g. Eq. 15 where two hydro-gen atoms and one oxygen ion would have to meet) are overallreaction equations comprising far too many particles (e.g. 4 elec-trons) to be an elementary reaction step. Two electron transfersmay typically occur in two separate steps. For the rate expres-sions to be meaningful, the rate-determining step must be anelementary reaction. This also applies to the reverse reaction. Ifit contains more than 3 particles, it may be also highly improba-ble to be elementary.55 In order to consider this, more elementaryprocesses have to be considered as for the dissociation of oxy-gen molecules, as described for the charged oxygen species inEqs. 40, 41, and 42. In addition, for considering the electrodeprocesses on cermet composite anodes, spillover reactions maybe important to be taken into account.73

Although the expressions obtained in this study are relatively sim-ple for much more complicated electrode kinetics, these expressionsbased on defect chemistry for electrode processes may be helpful, asan initial step, to quantitatively evaluate practical SOFC electrodes.As exchange current density value is coupled with other electrochem-ical parameters such as area-specific electrode resistance, overpoten-tial, and electrode conductivity, the expressions in this study can beextended to analyze such electrode-related electrochemical parame-ters. Experimental studies for comparison are important and thus nowpartly in progress to carefully measure exchange current density andrelated experimental values, by using e.g. impedance analysis,22,28,71to derive the partial pressure dependencies and to compare them withthe dependencies theoretically derived.

ConclusionsRelations describing cathode and anode exchange current density

have been derived, by taking into account possible elementary elec-trode reactions. The exchange current density of SOFC electrodeshas been formulated as a function of equilibrium constants of ad-sorption and dissociation reactions, rate constants of electrochemicalreactions, and gas partial pressures. Such theoretical relations indi-cate that partial pressure dependence of exchange current densitiescan vary for different predominant adsorption species on the elec-trodes, rate-determining electrochemical reactions, and the transfercoefficient considered. Empirical equations of exchange current den-sity have been verified as simplified forms of the theoretical relations.

For the SOFC electrode processes rate-determined by electro-chemical reactions, Butler-Volmer type expressions have been defect-chemically derived to describe the electrode exchange current den-sities. While more detailed electrode models would be needed, therelations of exchange current density values derived in this study couldbe useful to discuss electrode reaction mechanisms under the precon-dition that electrochemical reactions are rate-determining. Similarlyto the bulk defect chemistry to describe e.g. electrical conductivityof inorganic materials, the approach through electrode defect chem-istry may well describe various high-temperature electrochemical pro-cesses including exchange current density, while its applicability andlimitations should be carefully examined.

Even if the electrode reaction is rate-determined by an electro-chemical reaction rather than surface exchange, surface transport, andmixed conduction, the influence of various aspects has to be taken intoaccount, including gas composition, temperature, impurities, reform-ing and shift reactions, chemical nature of surfaces, and electrodemicrostructure. More fundamental scientific aspects should also beconsidered, including the fundamental difference in electrochemistry

between the liquid-solid interfaces and the gas-solid interfaces, phys-ical meaning of the transfer coefficient, degree of charge of variousadsorbates, adsorption isotherms, the interactions among adsorbates,rate-determining and elementary steps. In particular, the elementarysteps of the SOFC electrode reactions should be specified experimen-tally or theoretically in a future to construct more decisive theoreticalrelations on exchange current densities and to reveal their dependen-cies. The scattering of experimental data of exchange current densityand related values in the literature may be caused by the wide varietiesof dependencies theoretically derived and the existence of many otherfactors affecting such dependencies.

AcknowledgmentFinancial support by JSPS Grant-in-Aid for Scientific Research

(S) (No. 23226015) is gratefully acknowledged. One of the authors(K.S.) (Sasaki) thanks Prof. Dr. M. Mogensen for helpful discussionon the reference 22.

List of Symbolsig(C) gas species i on the cathode sideig(A) gas species i on the anode sideiad(C) adsorbed species i on the cathode surfaceiad(A) adsorbed species i on the anode surfaceVad(C) vacant adsorption site on the cathode surfaceVad(A) vacant adsorption site on the anode surfacee(C) electron in the cathodee(A) electron in the anodeO2(E) oxygen ion in the electrolytepi(C) partial pressure of gas species i on the cathode surface

(atm)pi(A) partial pressure of gas species i on the anode surface

(atm)i(C) coverage of adsorbed species i on the cathode surface

()i(A) coverage of adsorbed species i on the anode surface ()

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