The influence of equilibrium potential on the hydrogen oxidation kinetics of SOFC anodes

Solid State Ionics 177 (2007) 3371 – 3383 www.elsevier.com/locate/ssi

The influence of equilibrium potential on the hydrogen oxidation kinetics of SOFC anodes Wolfgang G. Bessler a,⁎, Jürgen Warnatz a , David G. Goodwin b a

Interdisciplinary Center for Scientific Computing, Heidelberg University, Im Neuenheimer Feld 368, D-69120 Heidelberg, Germany b Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA Received 9 June 2006; received in revised form 11 October 2006; accepted 24 October 2006

Abstract Fundamental electrochemical relations predict that the kinetic properties of an electrochemical charge-transfer reaction depend on reactant and product concentrations due to electrical equilibrium-potential (Nernst potential) effects. This paper discusses the consequences for the interpretation of observed reaction rates and orders of the electrochemical hydrogen oxidation at solid oxide fuel cell (SOFC) Ni/YSZ anodes. A thermodynamic model of the three-phase boundary is developed that describes the coupling of electroactive intermediates with global gas-phase reactants and products. The model is used to study the behavior of various reaction pathways proposed before, including hydrogen spillover, oxygen spillover, and interstitial hydrogen transfer. The results are compared with literature experimental data. The well-established activating effect of water on the SOFC anode kinetics can be explained by equilibrium-potential effects alone, without the necessity of assuming any additional kinetic or catalytic effect. © 2006 Elsevier B.V. All rights reserved. Keywords: Solid oxide fuel cell; SOFC; Modeling; Simulation; Mechanism; Spillover; Ni/YSZ; Three-phase boundary

1. Introduction Hydrogen oxidation kinetics at solid oxide fuel cell (SOFC) Nickel/Yttria-stabilized zirconia (Ni/YSZ) anodes strongly depend on the gas-phase composition. It has been experimentally well established that the product water has an accelerating influence on the exchange current density [1–3]. This counterintuitive observation has been subject of considerable controversy. For example, Mizusaki et al. introduced an Eley–Rideal type gas-phase/surface reaction in order to interpret observed reaction orders [1]. Bieberle et al. proposed a catalytic effect of water due to increased protonic conduction on the YSZ surface [2]. Bessler et al. proposed a kinetic effect of Ni surface reactions involving adsorbed water [4]. Mogensen proposed that the uptake of water into impurity phases might enhance their ionic conductivity and therefore increase anode performance [5].

⁎ Corresponding author. Tel.: +49 6221 548252; fax: +49 6221 548884. E-mail address: [email protected] (W.G. Bessler). 0167-2738/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ssi.2006.10.020

In these discussions, it is not very well recognized that electrochemical reaction rates depend on the equilibrium electrical potential (the Nernst potential) and therefore on reactant and product concentrations. This general effect is discussed explicitly in some (but not all) electrochemistry textbooks [6–8], and some general conclusions include: (1) Kinetic experiments should be analyzed in relation to the absolute electrode potential versus a defined reference and not a relative overpotential. (2) Because the electrochemical equilibrium is a dynamical equilibrium between forward and backward reactions, kinetic investigations should be carried out far away from equilibrium (i.e., under polarization). In most investigations of SOFC anodes published to date, these conditions are not satisfied. Measurements are usually interpreted in terms of the overpotential η versus a reference electrode seeing the same gas composition as the working electrode. Investigations of varying gas composition or temperature are usually carried out under electrochemical

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equilibrium (η = 0). In the context of SOFCs, these issues have been pointed out by Mogensen and co-workers [3,9,10]. Indeed, Mizusaki et al. [1] observed that the anodic reaction rate is, in the limit of high polarization, independent of water partial pressure when plotting current versus absolute cell potential instead of overpotential. Goodwin and co-workers implicitly treated these effects in their kinetic modeling studies [11,12]. In the present paper, we analyze in more detail the interdependence of thermodynamic and kinetic properties of an electrochemical reaction in equilibrium. The resulting relations are applied to the hydrogen oxidation at an SOFC Ni/YSZ anode. A thermodynamic model of the H2/H2O/Ni/ YSZ chemical system is developed, and its predictions are compared to literature experimental data. Various reactions pathways and charge-transfer reactions are compared. 2. The equilibrium-potential effect 2.1. Background The following derivation can be found in similar form in electrochemistry textbooks [6–8]. It is included here in a condensed manner in order to provide a background for the following sections. Take the general form of an electrochemical charge-transfer reaction, n1 R1 þ n2 R2 þ N fm1 P1 þ m2 P2 þ N þ ze− ;

ð1Þ

where Ri are the reactant(s), Pi the product(s), ni and mi the (positive) stoichiometric factors for reactants and products, respectively, and z the number of electrons. For single-step elementary reactions, the forward and backward reaction rate coefficients kf and kr are given by [13,14] 
act 
Ef zF 0 D/ ð2Þ kf ¼ kf exp − exp a RT RT kr ¼

kr0 exp


act 
Er zF D/ ; − exp −ð1−aÞ RT RT

ð3Þ

is the free enthalpy of reaction. According to mass-action kinetics [14], the Faradaic current iF due to Reaction (1) is given by
 iF ¼ zF kf j½Ri ni −kr j½Pj mj ; ð7Þ i

j

where [R] and [P] are reactant and product concentrations, and i and j run over all reactants and products, respectively. Relations (2)–(7) are generally valid for any single-step elementary charge-transfer reaction, irrespectively of electrochemical equilibrium. Throughout the remainder of this section, we will assume electrochemical equilibrium (i.e. open circuit, given in the following by the superscript eq). At Δϕ = Δϕeq, iF must vanish. Setting iF in Eq. (7) to zero, substituting kf and kr with Eqs. (2) and (3), and taking advantage of the relations (4)–(6) yields the well-known Nernst equation for a single interface [13,14],
 DGR RT eq nj eq mi ln j½Pj  = j½Ri  ; D/ ¼ þ j i zF zF

ð8Þ

eq

which relates reactant and product concentrations [Req] and [Peq] to the open-circuit potential difference Δϕeq. Assuming equilibrium, Eq. (8) can be used to substitute Δϕ in Eqs. (2) and (3). It is immediately evident that this results in a dependence of the reaction rate coefficients k on reactant and product concentrations. In this paper, we call the resulting kinetic effect the equilibrium-potential effect. There are two quantities that are frequently used in electrochemistry to describe the kinetics of an electrochemical reaction: (a) the exchange current density and (b) the charge-transfer resistance. In the following, we derive general expressions for the reaction rates and orders of these quantities under consideration of the equilibrium-potential effect. (a.) The exchange current density i0 is usually used in a Butler–Volmer type equation, where the potential difference Δϕ is substituted by the activation overpotential ηact, gact ¼ D/−D/eq :

0

ð9Þ

act

where k are the preexponential factors, E the thermal activation energies, α the symmetry factor, and Δϕ the electrical potential difference between reactants and products. The preexponential factors and activation energies are not independent of each other, but are related through thermodynamic consistency conditions [14],

Using Eq. (9) to substitute Δϕ in Eqs. (2) and (3), expressing Δϕeq via Eq. (8), and inserting into Eq. (7) yields after some algebraic manipulation the familiar Butler–Volmer form, 

 azF ð1−aÞzF g gact ; iF ¼ i exp −exp − RT act RT

ð10Þ

0

Efact −Eract ¼ DHR ;

ð4Þ

kf0 =kr0 ¼ expðDSR =RÞ;

ð5Þ

where ΔHR is the reaction enthalpy, ΔSR the reaction entropy, and

with the exchange current density ð1−aÞni i0 ¼ zFkf0 expð−Efact =RT ÞexpðaDGR =RT Þ j½Req j½Pjeq amj : i  i

DGR ¼ DHR −T DSR

ð6Þ

j

ð11Þ

W.G. Bessler et al. / Solid State Ionics 177 (2007) 3371–3383

The derivation takes advantage of the thermodynamic consistency relation kf0 expð−Efact =RT ÞexpðaDGR =RT Þ ¼ kr0 expð−Eract =RT Þexpð−ð1−aÞDGR =RT Þ

ð12Þ

that can be easily verified using Eqs. (4)–(6) and that can also be used to re-express the middle terms of Eq. (11) in terms of reverse reaction coefficients. (b.) The charge-transfer resistance Rct is defined as Rct ¼ dU =dI ¼ ðdI=dU Þ−1 ¼ ðdiF =dD/Þ−1 :

ð13Þ

eq At electrochemical equilibrium, the dependence of Rct on reactant and product concentrations can be derived using either Eqs. (2)–(8) or Eqs. (10) and (11):

Req ct ¼

RT =z2 F 2

−ð1−aÞni j½Req j½Pjeq −amj : i 

kf0 expð−Efact =RT ÞexpðaDGR =RT Þ i

j

ð14Þ The relations (11) and (14) allow us to calculate the reaction eq with respect to reactant and product orders ν of i0 and Rct concentrations. For a given temperature, all parameters in Eqs. (11) and (14) except the concentrations are constant. Thus, for the exchange current density, mRðiÞ ¼

dlni0 ¼ ð1−aÞni dln½Req i 

mPðjÞ ¼

dlni0 ¼ amj ; dln½Pjeq 

ð15Þ

dlnReq ct ¼ −amj : dln½Pjeq 

ð16Þ

and for the charge-transfer resistance, mRðiÞ ¼

dlnReq ct ¼ −ð1−aÞni dln½Req i 

mPðjÞ ¼

Eqs. (11), (14), (15) and (16) represent the central result of this analysis: • The exchange current density and charge-transfer resistance at open-circuit depend not only on the kinetic constants (k0, Eact ), but also on the concentrations of reactants and products. This is a direct consequence of the influence of the electrical potential Δϕ eq on the rate coefficients (equilibrium-potential effect). • The reaction orders have the same sign for both reactant and product. Increasing either reactant or product concentration accelerates the charge-transfer reaction. The reaction orders furthermore depend on the symmetry factor α. It is important to realize a few assumptions (both implicit and explicit) made in this analysis: (1) Reaction (1) is an elementary charge-transfer step (validity of using the kinetic rate laws Eqs. (2) and (3)). (2) Reaction (1) is the only potential-determining process, and there are no other charge-transfer steps contributing to the electrode potential Δϕ (validity of using the electroactive species concentrations, i.e. the ones in-

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volved in the charge-transfer reaction, in the Nernst Eq. (8)). (3) The reaction system is in electrochemical equilibrium. In the following discussion we will relax these assumptions. Furthermore, the question of the rate-limiting step will be addressed. 2.2. Multiple chemical steps, single charge-transfer step Consider the case of a global reaction consisting of multiple elementary chemical steps (e.g., adsorption/desorption reactions), but only one single charge-transfer step. The expressions derived in the previous section are still valid for the charge-transfer step and its involved electroactive intermediates. In addition, the Nernst Eq. (8) is also valid when using the global reactant and product concentrations, if all intermediates are in chemical equilibrium. The equilibrium half-cell potential Δϕeq can in fact be obtained using Eq. (8) from any combination of intermediates as long as the charge-transfer step is involved. In electrochemical experiments, reaction intermediates (e.g., surface-attached species) are often not known or not easily accessible. Reaction rates and reaction orders are therefore usually interpreted in dependence of global reactant or product concentrations [RiG ], [PiG] (which, in an experiment, are often set to a fixed value). At electrochemical equilibrium (open circuit), the concentrations of the electroactive species [Ri]eq and [Pi]eq that occur in Eqs. (11), (14), (15) and (16) depend in a generally nonlinear way on [RiG ]eq and [PiG]eq. This dependence is governed by the thermodynamic properties of all intermediate species. In order to evaluate the rate and orders of the chargetransfer step with respect to global reactant or product concentrations, knowledge of the thermodynamic properties of all intermediate species is required. This will be discussed in the next section for the particular case of SOFC anodes. 2.3. Multiple charge-transfer steps If multiple consecutive or parallel charge-transfer steps occur at the electrode, the kinetic and thermodynamic expressions, Eqs. (2)–(7), still apply to every single of these steps, with individual preexponential factors, activation energies, and thermodynamic parameters. However, all steps are subject to the same electrical potential difference Δϕ. The observed total Faradaic current iFtot is the sum of the individual currents given by Eq. (7). At electrochemical equilibrium, iFtot is zero. At the same time, thermodynamics require that all individual chargetransfer reactions need to be in equilibrium, i.e., iF of all reactions is zero, too. Therefore, the Nernst Eq. (8) is valid for all individual charge-transfer reactions, using individual values for ΔGR, z, [Ri] and [Pj], and the global value for the potential difference Δϕeq. Thus, Δϕeq can be obtained from the Nernst equation for any individual charge-transfer reaction, and it is equal to what is obtained from the Nernst equation for the global reactant and product concentrations, if all intermediates are in chemical equilibrium.

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For any individual charge-transfer step, we can now follow the derivation of the previous section, leading to Eqs. (11), (14), (15) and (16). Thus, the rates and orders of all charge-transfer reactions depend on their respective electroactive reactant and product concentration (equilibrium-potential effect). 2.4. No electrochemical equilibrium The expressions presented in the previous section were derived explicitly for electrochemical equilibrium (open circuit, iF = 0). If the electrode is polarized, the Butler–Volmer Eq. (8) remains valid only if the reactant and product concentrations stay constant, which is in general not the case. A convenient way to treat polarized electrodes is the direct use of Eqs. (2), (3) and (7), all of which hold independently of polarization). This set of equations represents an elementary kinetic view of charge transfer and fully describes the relationship between iF, Δϕ, [Ri] and [Pj]. As long as the kinetic coefficients are thermodynamically consistent (i.e., as long as they satisfy Eqs. (4) and (5) which, too, hold independently of polarization), Eqs. (2), (3) and (7) will actually allow predicting the equilibrium potential by setting iF = 0 and solving for Δϕ.

Table 1 Elementary charge-transfer reactions analyzed in this paper No.

Label (Figs. 2−3, 6–8)

Reaction

1 2 3 4 5

O spillover OH spillover H spillover to O H spillover to OH H interstitial

− O2− YSZ + []Ni f ONi + []YSZ + 2e − OHYSZ + []NifOHNi + []YSZ + e− − − HNi + O2− YSZ f OHYSZ + []Ni + e − HNi + OHYSZ f H2OYSZ + []Ni + e− Hxi Ni f HiU YSZ + e−

SOFC anodes, the analysis of surface double layers is out of the scope of the present work. In the remainder of this paper, surface double layers due to the presence of charged or partially charged adsorbed species are neglected. 3. SOFC hydrogen oxidation In the remainder of this paper, the concept of the equilibrium-potential effect will be applied to the electrochemical hydrogen oxidation at SOFC Ni/YSZ anodes. This reaction is believed to take place near the three-phase boundary of Ni, YSZ and the gas phase. It is formulated globally as H2 ðgasÞ þ O2− ðYSZÞfH2 OðgasÞ þ 2e− ðNiÞ:

ð17Þ

2.5. The rate-limiting step 3.1. Reaction pathways So far, no assumptions have been made about the absolute rates of the charge-transfer reactions, i.e., the magnitude of the preexponential factors and activation energies. While the kinetic behavior given by Eqs. (11), (14), (15) and (16) is generally valid, these relationships will only be observed experimentally if the charge-transfer reaction is rate-limiting. However, because any kinetic experiment requires a deviation from equilibrium, the observed global reaction rate may actually be governed by processes slower than the charge-transfer reaction. This includes chemical reactions or transport processes (diffusion, convection). In principle, knowledge of the thermodynamic and kinetic properties of all intermediate species and processes is required for a full understanding of these effects. 2.6. The electrical potential difference Δϕ In the elementary kinetic view of charge transfer described above, the term Δϕ that enters Eqs. (2) and (3) is the electrical potential difference between the initial and final locations of the transferred electron (for example, from a localized orbital of an adsorbate to the conduction band of the electrode). This value will in general be different to the (macroscopic) potential difference between the bulk of electrode and electrolyte because electrical double layers occuring at the involved surfaces and interfaces (e.g. due to charged adsorbates) can cause additional potential steps. In liquid electrochemistry, the Frumkin correction is often used to account for these effects [6–8,15]. In the context of SOFC cathodes using mixed electronic-ionic conductors, Fleig showed that surface double layers can have a significant effect on the electrochemical behavior [16]. Due to the complexity of this situation at three-phase boundaries of

There is quite some controversy as to the actual pathway and nature of the elementary steps of the global Reaction (17) (see [9,17–19] for a general discussion). In the following, various reaction pathways proposed previously are briefly summarized, focusing on the elementary-step charge-transfer reactions that are involved. The charge-transfer reactions are summarized in Table 1. Oxygen spillover. At the three-phase boundary line, oxygen ions (formally O2−) may hop from the YSZ surface to the Ni surface in an elementary charge-transfer reaction such as − O2− YSZ þ ½Ni fONi þ ½YSZ þ 2e ;

ð18Þ

where the subscripts denote the surface the species is attached to, and [] means a free surface site. This kind of species transition between different surfaces is usually referred to as “spillover”. Subsequent chemical reactions of hydrogen and oxygen species, including adsorption of molecular hydrogen and desorption of water, take place on the Ni surface. This general pathway was proposed by several authors [1,20–22]. It was also proposed for Pt/YSZ electrodes at both cathode [23] and anode [24,25]. Alternatively, spillover of hydroxyl ions (formally OH−) from the YSZ to the Ni surface is possible [26], OH−YSZ þ ½Ni fOHNi þ ½YSZ þ e− :

ð19Þ

Oxygen spillover is frequently discussed as being at the origin of electrochemical promotion in catalysis [27,28]. There is experimental evidence for oxygen spillover in Pt/YSZ electrodes from photoelectron emission microscopy [29] and cyclovoltammetry [30]. Also, because Reaction (18) does not

W.G. Bessler et al. / Solid State Ionics 177 (2007) 3371–3383

involve hydrogen species, it can be at the origin of electrochemical oxidation of CO [31]. Hydrogen spillover. A different possible pathway for Reaction (17) is the spillover of hydrogen from the Ni to the YSZ surface. The hydrogen atoms may hop to either an oxygen ion site in an elementary charge-transfer reaction such as − − HNi þ O2− YSZ fOHYSZ þ ½Ni þ e ;

ð20Þ

or to a hydroxyl site, HNi þ OH−YSZ fH2 OYSZ þ ½Ni þ e− :

ð21Þ

Both reactions may be active in parallel or consecutively. In this pathway, hydrogen adsorbs on the Ni surface, but water desorbs from the YSZ surface. The pathway was proposed by several authors [11,12,18,32]. Although there is, to the best of our knowledge, no direct experimental evidence of hydrogen spillover, the feasibility of this mechanism is supported by the fast surface transport of both, adsorbed hydrogen on the Ni surface [33] and protons on the YSZ surface [34]. Interstitial hydrogen transfer. At the typically high SOFC operating temperatures, both interstitial hydrogen atoms in bulk Ni and interstitial protons in bulk YSZ are known to be present in relatively high concentrations (in the 0.1 mol% range) with high enough diffusivities to support Reaction (17) [9,35–37]. Charge transfer may therefore take place at the twophase boundary of Ni and YSZ according to (Kröger–Vink notation) HxiNi fHUi YSZ þ e− :

ð22Þ

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operating conditions, all of these species are, in principle, present simultaneously. Their equilibrium concentrations depend on their respective thermodynamic properties and can be calculated for a fixed temperature and concentration of gasphase H2 and H2O if these properties are known. For this calculation, we consider a chemical system consisting of three gas-phase species (H2, H2O and O2), five species attached to the Ni surface (HNi, ONi, OHNi, H2ONi, and free Ni sites), four species attached to the YSZ surface (O2−YSZ, OH−YSZ, H2OYSZ, and free YSZ sites), two interstitial species (Hix Ni and HiU YSZ) x UU and two lattice YSZ species (OOYSZ and VOYSZ ). Based on experimental and theoretical data available in literature, we estimate the molar thermodynamic properties (molar enthalpies, molar entropies) of these species. This is discussed in detail in the Appendix. The data are given in Table 2 and form the basis of our thermodynamic model for the three-phase boundary of an SOFC Ni/YSZ anode. Equilibrium concentrations of all species are numerically calculated for fixed gas-phase H2 and H2O concentrations and fixed temperature. For these calculations, a number of chemical reactions were formulated between the various species. The rates of these reactions were set fast to ensure rapid equilibration. The rate equations describing this chemical system were then numerically integrated in time until steady state was reached. The thermodynamic model was implemented and solved using the software package Cantera [44]. 4. Results and discussion 4.1. Equilibrium concentrations

Interstitial hydrogen and protons are formed via surface adsorption and surface/bulk exchange from hydrogen on Ni and water on YSZ, respectively. This pathway was originally proposed by Chebotin et al. [38] and used by several authors for data interpretation [3,39]. It has the advantage of taking place at a 2D surface rather than a 1D line as the spillover pathways, thus potentially supporting higher currents. Furthermore, it was observed that segregated impurities may partially block the three-phase boundary and the YSZ surface [40,41]; in this case, the interstitial hydrogen transfer pathway is more likely than surface spillover [42]. Oxygen evolution. In a reverse reaction to what can take place at a Metal/YSZ cathode, molecular oxygen may be formed in the gas-phase above the anode and subsequently be reduced by hydrogen either through gas-phase reactions or through heterogeneous reactions on either of the involved surfaces [43]. On an elementary chemistry level, oxygen ions need to be oxidized at the three-phase boundary via Reaction (18) before recombining and dissociating from the Ni surface. Thus, for this mechanism, the same charge-transfer reaction as in the oxygen spillover mechanism is operating. 3.2. Thermodynamic modeling and simulation The various possible charge-transfer reactions (cf. Table 1) involve different electroactive intermediates. Under SOFC

The concentrations of gas-phase H2 and H2O were varied over five orders of magnitude (partial pressures from 1 to 105 Pa). The temperature was set to a fixed value of 973 K; this value was chosen to allow comparison to experimental data by Bieberle [17] and Mizusaki et al. [1]. Resulting equilibrium

Table 2 Thermodynamic data (molar enthalpies h and entropies s) at 973 K for the various species investigated (see Appendix for details) Species

h [kJ/mol]

s [J/K mol]

H2 H2O O2 ONi HNi OHNi H2ONi []Ni O2− YSZ OH−YSZ H2OYSZ []YSZ Hxi Ni U Hi YSZ x OO YSZ UU VO YSZ

19.9 − 216.9 21.8 − 222 − 31.8 − 193 − 273 0 − 236.4 − 282.5 − 273 0 33 16 − 236.4 0

155.6 221.8 232.8 39.0 40.7 106 130 0 0 67 98 0 − 97 − 102 0 0

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concentrations are shown in two-dimensional contour plots in Fig. 1. The strong and nonlinear dependence of the concentrations of all species on gas-phase composition is evident. Note that these concentrations will furthermore change with changing temperature. 4.2. Charge-transfer kinetics In electrochemical equilibrium, the rate of every single elementary charge-transfer reaction given in Table 1 depends on its respective reactant and product concentrations through equilibrium-potential effects, as discussed in the first section of this paper. In particular, Eq. (11) can be used to calculate the dependence of the exchange current densities i 0 on gasphase H2 and H2O concentration using the intermediate species concentrations given in Fig. 1. As the preexponential factors k and activation energies E act are not known, only the relative change of the exchange current density with the gasphase concentrations can be obtained from this analysis.

Using Eq. (11), assuming symmetry factors α = 0.5, and recognizing that n i and m j are unity for all reactions given in Table 1, the logarithm of the exchange current density is given by logi0 ¼

X i

0:5log½Req i þ

X

0:5log½Pjeq  þ c:

ð23Þ

j

For each individual charge-transfer reaction, the constant c is set such that the maximum current density in the investigated range of gas-phase concentrations corresponds to log i0max = 0 (i.e., we scale the data to their respective maxima). The results of this analysis for the five charge-transfer reactions given in Table 1 are shown as two-dimensional contour plots in Fig. 2a. The strong and nonlinear dependence of exchange current density on gas-phase concentrations is obvious. Most importantly, the different charge-transfer reactions show large differences in the behavior, both

Fig. 1. Calculated equilibrium concentrations (surface coverages θ and mole fractions x) of intermediate species included in the thermodynamic model (cf. Table 2) versus gas-phase hydrogen and water partial pressures at 973 K.

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Fig. 2. Calculated exchange current densities (a) and their reaction orders with respect to gas-phase hydrogen (b) and water (c) versus gas-phase partial pressures for the five elementary charge-transfer reactions (Table 1) at 973 K.

qualitatively and quantitatively. All mechanisms show an increase in exchange current density with increasing H2O concentration over a wide range of investigated parameters.

However, with increasing H2 concentration, the O and OH spillover reactions show decreasing kinetics, while the H spillover and H interstitial reaction kinetics is increasing.

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The reaction orders ν of the exchange current density with respect to the global H2 and H2O concentrations are calculated from the results shown in Fig. 2a according to mH2 ¼ dlogi0 =dlogpðH2 Þ

ð24Þ

mH2 O ¼ dlogi0 =dlogpðH2 OÞ:

ð25Þ

These results are shown in Fig. 2b and c. The global reaction orders strongly depend on gas-phase composition itself, and they can be both positive or negative. Again, they are distinctly different for the various charge-transfer mechanisms. Some mechanisms even show a change of sign of the reaction order. The obvious self-similarity of some of the plots is a consequence of the thermodynamic consistency of the model: every individual charge-transfer reaction has to fulfill the Nernst Eq. (8), and the variation of Δϕ eq with H2 and H2O concentration is identical for all individual charge-transfer reactions. Fig. 3 shows the dependence of exchange current density on gas composition for pH2 + pH2O = 1 bar, e.g., for a typical fuel utilization experiment. It demonstrates again the distinct behavior of the various charge-transfer mechanisms. For example, while the kinetics of OH spillover is strongly dependent on fuel utilization, the H spillover to O kinetics is almost constant over the complete range of hydrogen concentrations. To support interpretation of the results shown in Figs. 2 and 3, two particular cases are analyzed in more depth. The hydrogen interstitial transfer shows reaction orders of 1/4 with respect to both H2 and H2O over the entire range of conditions. It is straightforward to derive this dependence

Fig. 4. Ratio of exchange current densities for H spillover to OH calculated from either numerical simulations using the full model (Fig. 2a) or the simplified analytical expression given by Eq. (29), normalized to pH2 = pH2O = 0.5 bar. Values of ±0.2 represent deviations of ca. ±40% (logarithmic scale).

analytically. Both intermediates are formed from the gasphase species via the equilibrium reactions [35,36] H2 f2HxiNi ; UU

ð26Þ U

H2 O þ VOYSZ f2Hi YSZ þ OxOYSZ :

ð27Þ

Assuming ideal solutions of the interstitial species in the host bulk (cf. Appendix), application of the law of mass action leads to Hix Ni ∼ (pH2)1/2 and HiU YSZ ∼ (pH2O)1/2. Applying Eq. (11) with α = 0.5 yields the behavior observed in the numerical simulations, i0 fðpH2 Þ1=4 ðpH2 O Þ1=4 :

ð28Þ

The hydrogen spillover mechanism was applied by Zhu et al. in a modeling study of a methane-operated SOFC [12]. They analytically derived the following expression for the exchange current density:

i0 f

Fig. 3. Calculated exchange current densities versus fuel utilization (pH2 + pH2O = 1 bar) for the five elementary charge-transfer reactions (Table 1) at 973 K, normalized to x(H2) = 50%. The plot shows the results of Fig. 2a extracted along the solid line shown in Fig. 5. The solid line is calculated using the analytical formula, Eq. (29).

ðpH2 =p⁎H2 Þ1=4 ðpH2 O =101; 325 PaÞ3=4 1 þ ðpH2 =p⁎H2 Þ1=2

;

ð29Þ

where p⁎H2 is a nonlinear function of temperature [12]. The derivation of Eq. (29) was based on the following assumptions: (a) Reaction (21) (H spillover to OH) is rate-determining, (b) α = 0.5, (c) the YSZ surface is nearly fully covered with oxygen ions, (d) H is the only species present on Ni (no competitive adsorption of O, OH and H2O). Fig. 4 compares the predictions of Eq. (29), using parameters for the calculation of p⁎H2 given in Ref. [12], to the numerical simulations using our full thermodynamic model for Reaction (21), normalized to pH2 = pH2O = 0.5 bar. Calculations based on Eq. (29) are also shown as solid line in Fig. 3. The agreement is good only close to

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data show a slightly positive reaction order with respect to H2, and a strong positive reaction order with respect to H2O. While the predictions from the different charge-transfer mechanisms produce very different behavior, the experimental trend is reproduced quite well by the hydrogen spillover reaction towards hydroxyl groups (Table 1, no. 4). A comparison to selected data by Mizusaki et al. ([1], polarization resistance without electrolyte resistance contribution) is shown in Fig. 7. Here, the agreement between thermodynamic model and experiment is not as clear. While the data show quite some scatter, it seems that the H2O reaction order is smaller than in the Bieberle data. The data also shows no clear trend of the reaction order with respect to H2 ([1], Fig. 1). Coming closest to the experiments are model Fig. 5. Range of investigated conditions in the pattern anode experiments by Bieberle et al., Mizusaki et al., and DeBoer. The solid line indicates the conditions going from 100% hydrogen to 100% water (fuel utilization curve) at atmospheric pressure.

the normalization point; there are deviations towards both high and low water or hydrogen concentrations. When assuming a fuel utilization experiment (Fig. 3, pH2 + pH2O = 1 bar), the deviation is b 20% for 20% b x(H2O) b 75%. The deviation is mostly due to additional species competition on the Nickel surface (cf. Fig. 1) not included in the analytical model. 4.3. Comparison to experimental pattern anode data When comparing the predictions of the thermodynamic model to experimental data, it must be kept in mind that the model describes the kinetics of the charge-transfer reaction only. We therefore start from the hypothesis that one of the chargetransfer reactions is rate-determining, so that the observed experimental performance is dominated by that reaction. Furthermore, we restrict our analysis to pattern anode experiments [1,2,18,45]. This kind of experiment is believed to characterize the three-phase boundary chemistry only, while in porous (e.g. cermet) anodes, gas-phase transport may contribute to or even dominate the performance. In the experiments, typically either H2 or H2O partial pressure were varied while leaving the other parameter constant. The conditions investigated by Mizusaki et al., Bieberle et al., and DeBoer [1,2,18] are shown in Fig. 5. For comparing the model predictions, the simulated exchange current densities of each individual charge-transfer reaction are scaled to one experimental data point of each individual measurement series (chosen arbitrarily out of the middle of the investigated parameter range). Thus, again, we are only interested in the relative variation of exchange current density with gas-phase partial pressures. (Note that the absolute values vary for the different experimental studies.) Fig. 6 compares the thermodynamic model predictions with equilibrium electrode conductivities obtained by Bieberle et al. from impedance measurements ([2], polarization resistance without electrolyte resistance contribution). The experimental

Fig. 6. Comparison of experimental charge-transfer conductivities by Bieberle et al. (Ref. [2], 973 K) versus gas-phase hydrogen and water partial pressure to predictions for the five elementary charge-transfer reactions (Table 1). The simulated curves are scaled to one experimental data point in each panel. Note that some of the curves are overlapping.

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predictions from hydrogen spillover or from the interstitial mechanism (Table 1, nos. 4 and 5). The experiments by de Boer ([18], dominating process no. 3 from equivalent circuit fits) were performed at higher temperature (1123 K). They are compared to simulations in Fig. 8. The experiments do not show a consistent trend over the complete range of investigated concentrations. Looking at Figs. 7 and 8, one may be tempted to propose a change of mechanism within the investigated concentration ranges. However, given the uncertainties in both experiment and model (see Discussion below), we believe that there is currently no base for such a detailed interpretation. Sukeshini et al. [45] made only few measurements with varying H2/H2O gas composition, and there is no general trend as to its influence on anode kinetics.

Fig. 8. Comparison of experimental charge-transfer conductivities by DeBoer (Ref. [16], 1123 K) versus gas-phase hydrogen and water partial pressure to predictions for the five elementary charge-transfer reactions (Table 1). The simulated curves are scaled to one experimental data point in each panel. Note that some of the curves are overlapping.

4.4. Discussion Several conclusions can be drawn from the comparison of experiment and model:

Fig. 7. Comparison of experimental charge-transfer conductivities by Mizusaki et al. (Ref. [1], 973 K) versus gas-phase hydrogen and water partial pressure to predictions for the five elementary charge-transfer reactions (Table 1). The simulated curves are scaled to one experimental data point in each panel. Note that some of the curves are overlapping.

(1) The strong dependence of electrode performance on gasphase H2 and H2O concentrations are qualitatively, in some cases even semi-quantitatively, reproduced by the thermodynamic model. This finding supports the hypothesis that charge transfer is the rate-determining step in pattern anodes. (2) The equilibrium-potential effect, as presented in the first part of this paper, is fully capable of explaining the acceleration of anode kinetics with increasing H2O concentration. No additional kinetic or catalytic assumptions such as surface adsorption and reaction proposed by

W.G. Bessler et al. / Solid State Ionics 177 (2007) 3371–3383

(3)

(4)

(5)

(6)

Mizusaki et al. [1], protonic conduction on the YSZ surface proposed by Bieberle et al. [2], or uptake of water into impurity phases proposed by Mogensen [5] are required. The model is unable to consistently interpret all literature experiments simultaneously. This may be due to both scatter in the experiments and shortcomings of the model, as discussed below. The study therefore does not allow identifying unambiguously a single active reaction pathway. The observations presented here are most consistent with a surface hydrogen spillover reaction (Table 1, no. 4) as rate-determining step. Transport processes in the gas phase, solid bulk or on the surface are necessarily involved in the three-phase boundary electrochemistry. These processes are not included in our model. Furthermore, Mogensen et al. have shown that segregated impurity phases may block the three-phase boundary [41,42], presumably influencing the active reaction pathway. To further analyze these processes and their impact on SOFC performance, full kinetic models are required. These kind of models are actively being developed by the authors [11,22,46]. All experiments were performed at different conditions (gas-phase composition, temperature, cf. Fig. 5), so that comparing the experiments among themselves is difficult. They furthermore disagree in such basic observations as the number and shape of impedance arcs [1,2,18,45]. None of them analyzed the potential presence of impurity phases. Thus, we must conclude that the currently available experimental data base is not sufficient to support detailed mechanistic studies such as this one. We therefore emphasize the need of performing additional pattern anode experiments on the H2/H2O/Ni/YSZ system. These experiments should cover a larger range of conditions (e.g., Fig. 2 predicts a high sensitivity of different charge-transfer reactions on reaction orders at high H2O and low H2 concentrations) and temperatures. They furthermore need to take into account the more recent observations of potential impurity presence. The thermodynamic data used for the current investigations is compiled from different literature resources (see Appendix). Although it yields, for the first time, an overall consistent description of the chemical system investigated here, it needs to be considered tentative as long as it is not validated independently. For this purpose, data from both theory (e.g., quantum chemical calculations) and experiment are urgently needed.

In the present study, we have focused on a variation of gas composition at a constant temperature of 973 K. The equilibrium-potential effect will have an additional influence on the temperature dependence of charge-transfer reactions. It follows from Eq. (11) that the exchange current density is temperature-dependent not only through the activation energy Eact, but also via the free enthalpy of reaction and, possibly dominating, the temperature-dependent equilibrium concentrations of the electroactive species. These effects need to be

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considered when interpreting activation energies in terms of observed global temperature behavior. The influence of equilibrium potential on charge-transfer kinetics was investigated for the particular case of H2 oxidation at SOFC Ni/YSZ anodes. However, the equilibrium-potential effect is generally valid for any kind of electrochemical chargetransfer reaction (Section 2), including different SOFC anode materials, SOFC cathodes, or other electrochemical systems. Mechanistic interpretations of observed electrochemical kinetics, including reaction orders with respect to global species concentrations, need to take this effect into account. 5. Summary and conclusions Fundamental electrochemical relations predict that the kinetic properties of an electrochemical charge-transfer reaction (exchange current density, polarization resistance, reaction orders) depend on reactant and product concentrations due to a change in equilibrium potential (Nernst potential). The consequences include the acceleration of the charge-transfer kinetics when increasing either reactant or product concentrations. When the charge-transfer reaction is part of a multi-step mechanism, its equilibrium kinetics relative to global reactant or product concentrations is additionally coupled to the thermodynamic properties of the intermediate electroactive species. The strong interdependence of these effects requires the use of detailed models to interpret observed electrochemical reactivity. We use here the term equilibriumpotential effect to describe these processes. A tentative thermodynamic model was established in order to investigate the influence of equilibrium-potential effects on the hydrogen oxidation kinetics at SOFC Ni/YSZ anodes. Three different reaction pathways (oxygen spillover, hydrogen spillover, interstitial hydrogen transfer), based on five different elementary charge-transfer reactions, were studied. All chargetransfer reactions show a strong and highly nonlinear dependence of their kinetics on gas-phase hydrogen and water concentration due to equilibrium-potential effects. The behavior is distinctly different for the various mechanisms. The simulation predictions were compared to literature experimental pattern anode data. The model is fully capable of explaining the acceleration of anode kinetics with increasing H2O concentration. This is due to equilibrium-potential effects only, and no additional kinetic or catalytic processes proposed before [1,2,4,5] need to be assumed. Although the various experimental sources are not consistent, they agree mostly with the interpretation that the rate-determining step is charge transfer during hydrogen spillover from the Ni surface to a hydroxyl group on the YSZ surface. The study shows that identification of reaction pathways and charge-transfer steps should in principle be possible by analyzing such basic experimental properties as the dependence of exchange current density on gas-phase composition. The need of additional experiments on the H2/H2O/Ni/YSZ system is emphasized. Acknowledgements The authors thank Anja Bieberle-Hütter (ETH Zürich), Mogens Mogensen (Risø National Laboratory) and Olaf

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Deutschmann (Karlsruhe University) for helpful discussions. Partial support from the Office of Naval Research under contract N00014-05-1-0339 is also gratefully acknowledged. Appendix A. The thermodynamic model The thermodynamic model of the SOFC H2/H2O/Ni/YSZ three-phase boundary consists of three gas-phase species (H2, H2O and O2), five species attached to the Ni surface (HNi, ONi, OHNi, H2ONi, and free Ni sites), four species attached to the YSZ surface (O2−YSZ, OH−YSZ, H2OYSZ, and free YSZ sites), two interstitial species (Hix Ni and HiU YSZ) and two lattice YSZ x UU species (OO YSZ and VO YSZ). The development of the thermodynamic data (molar enthalpies and entropies) of these species, summarized for T = 973 K in Table 2, is detailed in this appendix. Since only differences in thermodynamic properties are required to compute equilibrium constants, some species properties may be assigned arbitrarily, and the others defined relative to these. It is common to set the enthalpies of the gasphase elemental species at T = 298 K to zero. Here, we additionally set the enthalpy and entropy values for the empty UU surface and bulk sites, []Ni, []YSZ, and VO YSZ, to zero. Thermodynamic data for the gas-phase species are taken from the NIST-JANAF thermodynamical tables [47]. Data for the Ni-attached species are taken from Hecht et al. [48]. These authors have developed and validated a kinetic reaction mechanism for steam reforming and partial oxidation of methane over Nickel. The thermodynamic database underlying this mechanism can be downloaded from [49]. The data for the YSZ-attached species are obtained as 2− follows. The properties of surface oxygen ion OYSZ are set to x the same values as the bulk ion OO YSZ (see below), so that the 2− relative concentrations OYSZ / []YSZ are equal to the bulk ratio UU x OO YSZ / VO YSZ. This follows from the notion that the YSZ surface is simply the outermost layer of the bulk oxygen sublattice. Surface water molecules H2OYSZ are assumed to be − physisorbed species, while surface hydroxyl ions OHYSZ are assumed to represent the chemisorbed form of water 2− − (following from H2OYSZ + OYSZ f 2 OHYSZ ). The thermodynamic data of these two species are obtained from the experiments of Raz et al. [34] by re-fitting their measured thermogravity data of adsorbed water mass versus temperature to a two-species model (physisorbed H2OYSZ and chemi− sorbed OHYSZ species). x The thermodynamic properties of bulk oxygen ions OO YSZ are unknown. They can in principle be calculated from the absolute electrical potential difference between Ni and YSZ for a global x − single-electrode reaction (e.g., OO YSZ f 1/2 O2 + 2 eNi). However, this potential difference involves two different solid phases and is therefore inaccessible. Here, we arbitrarily set the potential difference at 101325 Pa O2 to 0.5 V and calculate the x resulting thermodynamic properties of OO YSZ from the Nernst equation. Note that the choice of this value does not influence the results of the thermodynamic calculations. The bulk concentraUU x tions of OO YSZ and VO YSZ are set independently to the fixed value given by the stoichiometry of ZrO2 doped with 8 mol%

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