Mechanism of oxygen electrode delamination in solid oxide electrolyzer cells

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 5 ( 2 0 1 0 ) 9 5 2 7 e9 5 4 3

Available at www.sciencedirect.com

journal homepage: www.elsevier.com/locate/he

Mechanism of oxygen electrode delamination in solid oxide electrolyzer cells Anil V. Virkar Department of Materials Science & Engineering, 122 S. Central Campus Drive, University of Utah, Salt Lake City, UT 84112, USA

article info

abstract

Article history:

An electrochemical model for degradation of solid oxide electrolyzer cells is presented. The

Received 31 March 2010

model is based on concepts in local thermodynamic equilibrium in systems otherwise in

Received in revised form

global thermodynamic non-equilibrium. It is shown that electronic conduction through the

17 June 2010

electrolyte, however small, must be taken into account for determining local oxygen

Accepted 19 June 2010

chemical potential, mO2 , within the electrolyte. The mO2 within the electrolyte may lie out of

Available online 1 August 2010

bounds in relation to values at the electrodes in the electrolyzer mode. Under certain conditions, high pressures can develop in the electrolyte just near the oxygen electrode/

Keywords:

electrolyte interface, leading to oxygen electrode delamination. These predictions are in

Solid oxide electrolyzer cell

accord with the reported literature on the subject. Development of high pressures may be

Electrolysis

avoided by introducing some electronic conduction in the electrolyte.

Oxygen electrode delamination

ª 2010 Professor T. Nejat Veziroglu. Published by Elsevier Ltd. All rights reserved.

Non-equilibrium thermodynamics

1.

Introduction

1.1.

Solid oxide electrolyzer cell (SOEC) degradation

Considerable work has been reported on the use of solid oxide electrolyzer cells (SOEC) for electrolysis of H2O for hydrogen generation [1e17]. Typical SOEC consists of an oxygen ion conducting solid electrolyte such as yttria-stabilized zirconia (YSZ) sandwiched between two electrodes; steam-H2 electrode (which is the cathode in SOEC) made typically of nickel þ YSZ and oxygen electrode (which is the anode in SOEC) made typically of an electron (hole) conducting perovskite such as Sr-doped LaMnO3 (LSM) mixed with YSZ. The cell is typically operated over a temperature range from 800 to 900  C. Water vapor containing sufficient amount of hydrogen to prevent oxidation of Ni is circulated past the steam-H2 electrode. Externally applied DC voltage greater than the decomposition potential of H2O is applied across the cell such that H2O is decomposed at the steam-H2 electrode forming hydrogen with oxygen transported (as ions) through the electrolyte towards the oxygen electrode and electrons

transporting in the external circuit. The viability of SOEC for hydrogen generation as a practical system, however, has not been demonstrated in long term testing since SOEC cells do degrade over time, and it is known that the degradation rate is typically greater than solid oxide fuel cells (SOFC), which often are identical (or similar) cells but operated in the power generation mode. In SOEC, one of the modes of failure has been reported to be the occurrence of delamination of the oxygen electrode [6e10]. As an example of the morphology of oxygen electrode delamination in SOEC, Fig. 1 shows an SEM micrograph (Fig. 9 of Reference [10]) of a delaminated oxygen electrode from reference [10]. The principal objective of this manuscript is to propose a fundamental mechanism of degradation of solid oxide electrolyzer cells (SOFC), which explains the observed delamination of the oxygen electrode.

1.2.

Global non-equilibrium, local equilibrium

The mechanism is based on the very fundamentals of nonequilibrium thermodynamics and transport theory. The approach used in this manuscript is similar to the one used

E-mail address: [email protected] 0360-3199/$ e see front matter ª 2010 Professor T. Nejat Veziroglu. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijhydene.2010.06.058

9528

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 5 ( 2 0 1 0 ) 9 5 2 7 e9 5 4 3

Fig. 1 e SEM micrographs from Mawdsley et al. [10] showing oxygen electrode delamination along the oxygen electrode/electrolyte interface.

previously by the author for studies on transport through predominantly ionic conductors and degradation of solid oxide fuel cell stacks [18,19]. The approach is based on the incorporation of the local equilibrium criterion into the transport equations [18,20]. Much of the work on transport is based on linear, non-equilibrium thermodynamics. All reported work on transport through solid electrolytes and mixed ionic electronic conductors is based on either explicit or implicit assumption of local equilibrium, even though the implications of this very important assumption are rarely addressed. Recent non-equilibrium molecular dynamics (NEMD) simulations in several systems have shown that local equilibrium is applicable to a very vast majority of the systems that are in global thermodynamic non-equilibrium [21e23]. An important consequence of the existence of local equilibrium in solid electrolytes is that electronic conduction cannot be assumed to be identically zero even in a predominantly ionic conductor [18]. This is a subtle but a very important point. Yet in many studies, contradictory assumptions of the simultaneous existence of local equilibrium (often tacitly made or even apparently unknowingly made) and purely ionic conduction (no electronic transport) are made [24]. In studies in the general area of non-equilibrium thermodynamics, the concept of local equilibrium has been the cornerstone of many advances and has been extensively

discussed in the literature. The general conclusion of this enormous body of work on non-equilibrium thermodynamics is that local equilibrium is almost always valid, even in cases involving nonlinear, non-equilibrium thermodynamics. In fact, there are very few (possible) cases in which local equilibrium may be violated. These are very rare and not encountered in cases involving transport of matter by diffusion. Thus, insofar as transport is concerned, there appear to be no reported cases in which local equilibrium is violated. A possible violation of local equilibrium is synonymous with inability to uniquely assign a temperature to a point in the system [20]. As to how rapidly local equilibrium is typically achieved can be understood by an example of a quantity of a gas. Suppose molecules are assigned some arbitrary initial velocities that do not fit the Maxwell distribution. Molecular dynamics calculations show that within a few collisions the distribution becomes Maxwellian e and results in a locally well-defined temperature [20]. Similar reasoning is applicable to solids with atoms treated as linear harmonic oscillators. All known transport studies assume that there is a well-defined temperature at a point in the system at a given time, whether it is fixed in time (steady state) or varies with time (transient state). Thus, all transport studies assume local equilibrium. The basic concept of local equilibrium, extensively discussed in textbooks on irreversible thermodynamics, is that even in systems not in global equilibrium Gibbsian laws of thermodynamics are applicable to microscopic volumes suitably chosen [20]. This is the criterion of local equilibrium in an otherwise global non-equilibrium. In such cases, all extensive thermodynamic functions are replaced by their local densities [20]. This formulation dates back to the 1930s with the classic papers by Onsager. In what follows, we will examine the implications of local equilibrium in predominantly oxygen ion conductors in a globally non-equilibrium state, which forms the basis for the proposed electrode delamination model.

1.3. Non-equilibrium thermodynamics, local equilibrium and transport in predominantly oxygen ion conductors We consider an oxygen ion conductor with cation sub-lattice virtually immobile. In an oxygen ion conductor, the existence of local equilibrium with respect to oxygen is described by [25,26] ! 2 ! 1 r ; tÞ þ 2e0ð r ; tÞ4O ð r ; tÞ O2 ð! 2

(1)

which leads to 1 r ; tÞ þ 2me ð! r ; tÞ ¼ mO2 ð! r ; tÞ m ð! 2 O2

(2)

or 1 r ; tÞ þ 2~ me ð! r ; tÞ ¼ m ~ O2 ð! r ; tÞ (3) m ð! 2 O2 where mi ð! r ; tÞ’s are the local (function of position) chemical r ; tÞ’s are the local electrochemical potentials at time t, and m ~ i ð! ! r ; tÞ þ z FFð! r Þ for species i potentials given by m ~ ð r ; tÞ ¼ m ð! i

i

i

where zi is the valence (including sign), F is the Faraday constant and Fð! r Þ is the local electrostatic potential. There is no time dependence in Fð! r Þ since we are only considering

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 5 ( 2 0 1 0 ) 9 5 2 7 e9 5 4 3

cases wherein in the externally applied voltage is fixed. In such a case, the local Fð! r Þ is instantaneously established (which assumes cations are virtually immobile, anions exhibit rapid transport, and no interfacial space charges develop in response to applied voltage). If space charge develops, then the electrostatic potential may exhibit time dependence as the double layer capacitance builds up. In the above, ! r denotes the position vector. Local equilibrium also implies [25,26] 1 r ; tÞ þ 2d~ me ð! r ; tÞ ¼ d~ mO2 ð! r ; tÞ dm ð! 2 O2

(4)

where dX is a differential variation in X. In a typical oxygen ion conductor, the ionic current density is given by [25,26] s 2 ð! r ; tÞ s 2 ð! r ; tÞ r ; tÞ ¼  O r ; tÞ ¼ O r ; tÞ V~ mO2 ð! V~ mO2 ð! IO2 ð! ð  2ÞF 2F

(5)

r ; tÞ is the oxygen ion conductivity at ! r and time, where sO2 ð! t. If the material is predominantly an oxygen ion conductor, r ; tÞ >> se ð! r ; tÞ, where se ð! r ; tÞ is the electronic then sO2 ð! ! conductivity at r and time, t. Note that the electrical current (which is the rate of transport of electrical charge, Coulombs, r ; tÞ is in the opposite per unit time) due to oxygen ions, IO2 ð! r ; tÞ, as oxygen ions direction to the flux of oxygen ions, jO2 ð! are negatively charged. The same is the case with negatively charged electrons. In this manuscript we will quantitatively examine only the ‘true’ steady state. A brief description of an ‘apparent’1 steady state and its significance will also be presented. It is understood that local equilibrium is equally applicable in the vast majority of non-equilibrium cases involving transient states. In a ‘true’ or an ‘apparent’ steady state in a predominantly r ; tÞ, sO2 ð! r ; tÞ, mO2 ð! r ; tÞ are nearly oxygen ion conductor, IO2 ð! constant, independent of time, for given fixed externally applied conditions, such as a fixed applied voltage and results in a fixed total measured current. Also, in most such cases, r ; tÞ, is a constant and may be given by sO2 . Thus, we will sO2 ð! from now on drop the time dependence, except when the discussion explicitly concerns time dependence. In what follows, at places we will also drop the ! r , but still it is understood that we are referring to the evaluation of relevant properties at some position, ! r . Some discussion of ‘transient’ states will be presented later.

1.4. Electrochemical potential of O2 in predominantly 2 O ion conductors in terms of immeasurable and measurable parameters Assuming ionic conductivity to be independent of position and invariant with time (steady state), the m ~ O2 (for a one dimensional problem) is a linear function of position. The electrochemical potential of oxygen ions is given by [25,26] m ~ O2 ¼ mO2  2FF

(6)

The chemical potential of ions is not measurable although it is a useful concept for formulations of transport problems. Often, it is assumed that the chemical potential of oxygen ions 1

The meaning and significance of an ‘apparent’ steady state is discussed later.

9529

in the electrolyte, mO2 , is constant, independent of position. This is because the concentration of oxygen vacancies is much smaller than the concentration of oxygen ions in the vast majority of materials of interest, and thus effectively oxygen ion concentration is essentially fixed. For this reason, it is commonly assumed that mO2 is independent of position. This leads to a linear variation of F with position or VF ¼ constant (one dimensional case) for a fixed applied voltage. An important point to note, however, is that F is also not experimentally measurable. What one experimentally measures using an inert metallic probe and a voltmeter is 4 given by [27] m 4¼ e þF F

(7)

The 4, electric potential, is effectively negative of the reduced electrochemical potential of electrons given in units of volts [27]. In metals which have high free electron concentration (given here as number of free electrons per unit volume), the 4 and F (or more accurately the differences D4ð! r ;! r O Þ and ! ! ! ! DFð r ; r O Þ between two positions r and r O ) are treated as the same. In materials with very low electron concentration, however, it is important to recognize the distinction between 4 and F [27]. Using the local equilibrium concept, the electrochemical potential of oxygen ions is also given as 1 m ~ O2 ¼ mO2  2FF ¼ mO2  2F4 2

(8)

where both mO2 and 4 are the experimentally measureable potentials. That is, the definition of m ~ O2 can be given in terms of experimentally non-measurable potentials (mO2 ; F) or experimentally measurable potentials (mO2 ; 4). It can also be shown that electrochemical potential of oxygen ions, m ~ O2 , is in fact deducible from various experimental measurements, and thus is a measurable parameter. This is true of all ionic species. Thus, although the two definitions of m ~ O2 are equivalent (in terms of mO2 and F, or in terms of mO2 and 4), from an experimental standpoint the one in terms of measurable potentials is more useful. Much of the literature, however, is based on descriptions in terms immeasurable ionic chemical potentials and electrostatic potential.

1.5. Chemical potential of neutral oxygen and electric potential (reduced negative electrochemical potential of electrons) in predominantly oxygen ion conductors We now assume that electron and hole concentrations are very small, which is typical of an oxygen ion conductor with negligible electronic conductivity, for example yttria-stabilized zirconia (YSZ). Suppose, for example, electron concentration is w2  1010 cm3 at 800  C. Then for an assumed electron field mobility of w0.05 cm2 Volt1 s1, the electronic conductivity of YSZ would be w1010 Scm1, an exceptionally small value. Over a wide range of oxygen partial pressures, the oxygen vacancy concentration is constant (extrinsic region), say w1021 cm3, which is much larger than the electron concentration. The corresponding oxygen ion conductivity may be on the order of 102 to 101 S cm1, which is orders of magnitude larger than the electronic conductivity. Thus, the material is a predominantly ionic conductor with ion transport number of nearly unity.

9530

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 5 ( 2 0 1 0 ) 9 5 2 7 e9 5 4 3

The electron concentration, however, is a function of the local oxygen partial pressure, and thus is generally positiondependent. For example, the dependence may be of the type r ÞÞ1=4 , typical of many oxides [28]. Electron nð! r ÞwðpO2 ð! concentration may thus exhibit large relative changes subject to changes in the local oxygen partial pressure. Yet, the material may still be an essentially ionic conductor. For example, even if the electron concentration locally increases to 1014 cm3, ionic transport number remains close to unity. Assuming the dilute solution approximation for electrons, we r ÞfRTlnnð! rÞ or me ð! r Þf  RTlnpO2 ð! rÞ or have me ð! ! ! ! or m ~ e ð r Þ ¼ F4ð! r Þf  mO2 ð! rÞ or me ð r Þf  mO2 ð r Þ r Þ. This means, if we change mO2 ð! r Þ locally, the local 4ð! r ÞfmO2 ð! 4ð! r Þ will change proportionately. Alternatively, if we change r Þ will change proportionately. It is the latter 4ð! r Þ locally, the mO2 ð! aspect which is relevant to degradation. The preceding discussion suggests that all those factors which locally r Þ. Finally, this increase 4ð! r Þ, will also locally increase mO2 ð! means
 4F4ð! rÞ r Þfexp pO2 ð! RT

This is a very significant result. It states that even when r Þ, mO2 ð! r Þ, and Fð! r Þ are nearly independent of time, m ~ O2 ð! ! mO2 ð r ; tÞ and m ð! r ; tÞ þ Fð! rÞ 4ð! r ; tÞ ¼  e F

(11)

may indeed be time dependent. This represents an ‘apparent’ steady state e not a ‘true’ steady state. We now note that r Þ and Fð! r Þ are not experimentally measureable; mO2 ð! r ; tÞ and 4ð! r ; tÞ are experimentally measurable. however, mO2 ð! Thus, even in an ‘apparent’ steady state, the measurable therr ; tÞ and 4ð! r ; tÞ, may exhibit time modynamic parameters, mO2 ð! dependence, even when the externally measured voltage and current are virtually independent of time. This is an apparent steady state e not a true steady state. Returning to the local equilibrium involving oxygen ions, electrons and oxygen, note that

(9)

(which is routinely used but mainly for measurements made at electrodes), and that modest changes in 4ð! r Þ may locally r Þ orders of magnitude and could lead to change pO2 ð! mechanical cracking. In most studies, one often investigates r Þ or pO2 ð! rÞ 4 (dependent variable) as a function of mO2 ð! (independent variable). Here we treat 4ð! r Þ as the indepenr Þ or pO2 ð! r Þ as the dependent dent variable and mO2 ð! variable.

1.6.

r Þ ¼mO2 ð! r Þ  2FFð! r Þ ¼ mO2 ð! r Þ  2me ð! r ; tÞ  2FFð! rÞ m ~ O2 ð! 1 r ; tÞ ¼ mO2 ð! r ; tÞ  2F4ð! r ; tÞ ð10Þ þ 2me ð! 2

1 r Þ ¼ mO2 ð! r ; tÞ þ 2me ð! r ; tÞ mO2 ð! 2

(12)

which is the same as Eq. (2). In a typical oxygen ion conductor at a given temperature, mO2 is essentially fixed. Any changes that occur in the local chemical potential of electrons (Fermi level), r ; tÞ, affect the local chemical potential of oxygen, mO2 ð! r ; tÞ me ð! r ; tÞ. All those and thus affect the local oxygen pressure, pO2 ð! r ; tÞ will raise mO2 ð! r ; tÞ and raise factors which lower the me ð! r ; tÞ. Under some situations, this may lead to degradation. pO2 ð! This is the fundamental basis for the oxygen electrode delamination mechanism discussed in this manuscript.

‘True’ and ‘apparent’ steady states

Much of the discussion in this manuscript is on ‘true’ steady states. However, there are many situations in which the steady state is in fact not a ‘true’ steady state, but is an ‘apparent’ steady state. We consider for example a case wherein the externally measured voltage and current are essentially constant (independent of time). Since the electrolyte is a predominantly ionic conductor, it means at any position, ! r , the electrochemical potential of oxygen ions, r Þ, is also independent of time. Since, m ~ O2 ð! r Þ ¼ mO2 ð! r Þ  2FFð! r Þ it is also clear (and commonly m ~ O2 ð! r Þ and Fð! r Þ both are indeunderstood to be so) that mO2 ð! pendent of time (for a fixed applied voltage). An immediate and apparently obvious (and often incorrect) conclusion is that steady state is indeed established. However, we note that local equilibrium is valid and the material is a predominantly ionic conductor. Thus, the electron concentration is rather small. No guarantee exists that the electron concentration remains constant, however. Any slight changes in composir Þ or Fð! r Þ or tion that may occur locally do not affect mO2 ð! r Þ, but can affect me ð! r Þ (and 4ð! r Þ) and thus also affect m ~ O2 ð! r Þ. That is, the local Fermi level, me ð! r ; tÞ, may depend on mO2 ð! both position and time, even when an ‘apparent’ steady state is established, wherein the establishment of the steady state is judged on the basis of measured stationary voltage and measured stationary current. Note now that oxygen ion electrochemical potential is given by

2.

Analysis

2.1. Ionic and electronic currents through the cell and the measured current The typical materials of a solid oxide electrolyzer cell are as follows: (1) Electrolyte: yttria-stabilized zirconia (YSZ), a predominantly oxygen ion conductor. Typical composition is 8 mol% Y2O3e92 mol% ZrO2. (2) Electrode exposed to predominantly water vapor and H2, the steam-H2 electrode: Ni þ YSZ. (3) Electrode exposed to predominantly oxygen, the oxygen electrode: Typical composition is Sr-doped LaMnO3 (LSM) þ YSZ. Many other materials may be used for the various components. Thus, in what follows the terminology used will be generic; steam-H2 electrode, oxygen electrode, and electrolyte. In the fuel cell mode, the oxygen electrode is the cathode and the steam-H2 electrode is the anode. In the electrolyzer mode, however, the oxygen electrode is the anode and the steam-H2 electrode is the cathode. Since the focus of this manuscript is on the electrolyzer, the oxygen electrode is nominally the anode and the steam-H2 electrode is nominally the cathode. The electrode reactions in the electrolyzer mode are as follows:

9531

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 5 ( 2 0 1 0 ) 9 5 2 7 e9 5 4 3

At the steam-H2 electrode: H2 O þ 2e0/H2 þ O2

(13)

2-

O ions transport through the electrolyte to the oxygen electrode. At the oxygen electrode: O2 /1=2O2 þ 2e0

(14)

The 2e’ transport in the external circuit from the oxygen electrode to the steam-H2 electrode. The overall reaction is H2 O/H2 þ 1=2O2

(15)

An important point to note is that the above reaction can also be written simply as isothermal compression of oxygen 2 at the steam-H2 electrode to pOx from pStH O2 O2 at the oxygen electrode. Since it is compression, it represents work done on the system. Fig. 2 shows a schematic of the cell. The steam-H2 electrode is exposed to a mixture of H2 and H2O (with very low oxygen partial pressure) and the oxygen electrode is exposed to a mixture of O2 and H2O, in practice essentially pure oxygen (negligible hydrogen partial pressure).

2.2.

Equivalent circuits

Fig. 3(a) is an equivalent circuit for the electrolyzer cell in a ‘true’ steady state with an externally applied voltage source, EA.2 In Fig. 3(a), rci and rai are respectively the area specific ion charge transfer resistances at the steam-H2 electrode/electrolyte interface and the oxygen electrode/electrolyte interface. As-defined, these only refer to the physically sharp electrolyte/electrode interfaces and do not include the porous electrode contributions to the overall electrode reactions [18]. This is because one must then include simultaneous and parallel transport through both the solid (ions and electrons/ holes) and the porous (gas) regions of the electrodes. The resistances, rci and rai are effectively polarization resistances (excluding the porous electrodes) and may be described using ButlereVolmer type of phenomenological models. The rce and rae are area specific resistances for direct electron transfer across the steam-H2 electrode/electrolyte interface and oxygen electrode/electrolyte interface, respectively. In the semiconductor terminology, the rce and rae are area specific el contact resistances (non-ohmic contact). Finally, rel i and re are respectively the ionic and electronic area specific resistances of the electrolyte, given in terms of the respective conductivities and the electrolyte thickness. Note that rce , rae and rel e may be very large, but are not mathematically infinite. Fig. 3(b) is an equivalent circuit for the electrolyzer cell in an ‘apparent’ steady state. The ‘apparent’ steady state is characterized by a nearly constant voltage across the cell and a nearly constant externally measured current. Since EA is fixed, the voltage across the cell is fixed. However, the electronic current through the electrolyte, which is much smaller than the ionic current, may vary with position and time. This effectively reflects as time dependent individual internal 2 It is assumed here that the internal resistance of the external source with voltage EA is negligible. It can be easily included in the analysis.

Nernst potentials (with their sum still being a constant, EN) and also time dependent local electric potential, 4ð! r ; tÞ(with 4Ox  4SteamH2 fixed and equal to the applied voltage EA). The following discussion is restricted to the ‘true’ steady state depicted in Fig. 3(a). The applied voltage EA is in the opposite direction to the Nernst voltage EN created by differing oxygen partial pressures at the two electrodes; that is, the positive of the cell is connected to the positive of the external source and the negative of the cell is connected to the negative of the external source. When EA < EN, the cell does work on the external source (charging the externally connected battery). This is the fuel cell mode. When EA > EN, the external source does work on the cell. This is the electrolyzer mode. The Nernst voltage generated by differing oxygen partial pressures at the two electrodes is given by EN ¼

pOx RT O2 ln StH 4F pO2 2

! (16)

2 Note that pStH << pOx O2 O2 . Thus, as given EN > 0 and EA > 0. The Nernst potential can also be given in terms of the EMFs across the two interfaces and the bulk electrolyte by

pOx RT O ln a 2 EN ¼E þ E þ E ¼ pO2 4F ! pOx RT O2 ¼ ln StH 4F pO 2 a

el

!

c

paO RT þ ln c 2 pO2 4F

!

pcO2 RT þ ln StH2 4F pO2

!

ð17Þ

2

In terms of the chemical potentials of oxygen, Eq. (17) may also be given as 

a mOx O2  mO2

EN ¼E þ E þ E ¼   StH2 mOx O2  mO2 ¼ 4F a

el

c



4F

 þ

maO2  mcO2 4F

 þ

  2 mcO2  mStH O2 4F ð18Þ

Relationships between ionic and electronic current densities, internal EMFs, and electric potentials, 40 s are given by (by solving the circuit given in Fig. 3(a)) Ie rce ¼ Ec þ Ii rci ¼ 4SteamH2  4c < 0

(19)

Ie rele ¼ Eel þ Ii reli ¼ 4c  4a < 0

(20)

and Ie rae ¼ Ea þ Ii rai ¼ 4a  4Ox < 0

(21)

The electronic area specific resistances, rce and/or rae and/or rele are expected to be much larger than the ionic resistance for a typical electrolyzer cell. That is, rce and/or rele and/or rae >>> rci ; reli ; rai . Let us write Ri ¼ rci þ reli þ rai

(22)

and Re ¼ rce þ rele þ rae

(23)

These represent area specific ionic and electronic resistances of the cell which include electrolyte/electrode interfaces. Now, Re >> Ri. However, the electronic resistances cannot be mathematically infinite and thus cannot be

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 5 ( 2 0 1 0 ) 9 5 2 7 e9 5 4 3

µOOx2

µOa 2

H2

Low Oxygen Pressure Side

H2 + H2O Mixture

pOOx2 , pOx H 2O

jO 2 Ii Oxygen Electrode

pOSt2

µ Oc 2

Steam-H 2 Electrode

St H 2 pH ~ pHSt2OH 2 2

H2

pOx H2

O2 + H2O Mixture

High Oxygen Pressure Side

µOSt2

Electrolyte Membrane

9532

je Ie

_

+ I

Electron Flow

Current

EA

Fig. 2 e A schematic of a solid oxide electrolyzer cell, when operated in the electrolyzer mode (EA >EN ). Steam-H2 electrode is on the left; and oxygen electrode is on the right. The chemical potentials of oxygen in the electrodes close to the electrode/ 2 and mOx electrolyte interfaces are mStLH O2 . The chemical potentials of oxygen in the electrolyte close to the electrode/ O2 electrolyte interfaces are mcO2 and maO2 . The electrolyte is a predominantly ionic conductor. Thus, jIi j>>jIe j. The directions of the particle fluxes, jO2L and jeje are shown as well as the directions of ionic current, Ii and electronic current, Ie. dropped out of the analysis (and cannot be dropped out of the equivalent circuit) since this will violate local equilibrium and the chemical potential of oxygen will be indeterminate [18,29]. That is, Re is very large, but cannot be set to infinity a-priori. Note that the magnitude of the slope of VeI curve is Ri Re =ðRi þ Re ÞzRi . Thus, the Re is not reflected in typical experimental measurements, such as the voltage e current density plots or impedance spectra. Fig. 4 shows a schematic plot of voltage, EA, and current, I. The externally measured current is I ¼ Ii þ Ie. The point corresponding to EA ¼ EN strictly represents Ii ¼ 0. The externally measured current, I, is mathematically zero when the applied voltage is given by EA ¼ EN Re =ðRi þ Re Þ. If Re >> Ri, then the current is negligible when EA zEN and nearly corresponds to zero of the x axis in Fig. 4. Thus, when EA ¼ EN, strictly the measured current is I ¼ Ie. Over the entire range of the applied voltage across the cell, which is EA, the net electronic current has the same sign in both modes of operation (SOFC and SOEC) since we have selected EA > 0. The sign of the ionic current, Ii, however, depends upon the relative magnitudes of EA and EN3. The positive direction of the current axis corresponds to EA < ðEN Re =ðRi þ Re ÞÞzEN , and the negative direction of the

current axis corresponds to EA > ðEN Re =ðRi þ Re ÞÞzEN . When the cell is operated in the fuel cell mode, the ionic and the electronic currents through the cell are in opposite directions. When the cell is operated in the electrolyzer mode, the ionic and the electronic currents through the cell are in the same direction. In predominantly oxygen ion conductors, the electronic current is small in magnitude, yet it has a significant effect on chemical potentials [18]. Note that

3 In the SOFC stack degradation model described previously [19], the direction of the ionic current is always fixed. When cell imbalance occurs in a SOFC stack, the electronic current switches sign. Fundamental requirement for degradation to occur is that ionic and electronic currents through the electrolyte are in the same direction, regardless of the mode of operation, SOFC or SOEC [18]. In SOFC, this occurs in a stack as a result of cell imbalance [19]. In the electrolyzer degradation model examined in this manuscript, this condition occurs even in a single cell.

and

4Ox  4StH2 ¼ 4Ox  4a þ 4a  4c þ 4c  4StH2 > 0

(24)

is the measured voltage across the cell (which is also EA). For all cases considered in this manuscript, EA ¼ 4Ox  4StH2 > 0. From the equivalent circuit in Fig. 3(a), note that   Ie rce þ rele þ rae þ EA ¼ Ie Re þ EA ¼ 0

(25)

and   Ii rci þ reli þ rai  EN þ EA ¼ Ii Ri  EN þ EA ¼ 0

(26)

Thus, EA E ¼ A<0 Re rce þ rele þ rae

(27)

ðEA  EN Þ ðE  EN Þ ¼ A <0 Ri rci þ reli þ rai

(28)

Ie ¼ 

Ii ¼ 

as long as H2O decomposition is achieved, that is as long as (EA  EN) > 0. The measured current (in the external circuit) is I ¼ Ii þ Ie zIi . Thus, as defined, the measured current, I < 0 in the electrolyzer mode and I > 0 in the fuel cell mode.

9533

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 5 ( 2 0 1 0 ) 9 5 2 7 e9 5 4 3

a

Ox

Ec

EN

pOSt2

H2

pOc 2

rec

H2 + H2O

I

Ii

E el pOa 2

pOOx2 pOx H 2O

Ii

pOx H2

rea

Ie

_

Ii

Ea

ria

Ii

reel

Ie

Ie

pO RT ln St 2H 2 4F pO2

riel

Ii

pHSt2 H 2

a

Ec

ric

pHSt2OH 2

E el

O2 + H2O

Ie

+ EA

b

Ox

EN

pOSt2

H2

E c (t )

ric

pHSt2OH 2

rec (t ) I c (t ) e

H2 + H2O

I

Ii

I e (t )

pO RT ln St 2H 2 4F pO2

E el (t )

riel pOc 2

Ii

pHSt2 H 2

E c (t ) E el (t ) E a (t )

ria pOa 2

Ii reel (t ) I el (t )

Ii

rea (t ) I a (t ) e

e

_

Ii

E a (t )

pOOx2 Ox pH 2O

pOx H2

O2 + H2O

+ EA

Fig. 3 e (a): An equivalent circuit for the cell in a ‘true’ steady state. The positive directions of the ionic and the electronic currents are shown by arrows. In the fuel cell mode (EA < EN), the ionic current direction is shown by red arrows (Ii > 0). In the electrolyzer mode (EA > EN), the ionic current direction is shown by black arrows (Ii < 0). Thus, the sign of the ionic current depends upon the sign of EA L EN. For the cases considered here, EA > 0. Thus, the electronic current through the cell is always negative (Ie < 0). Note that generally jIi j>>jIe j. (b): An equivalent circuit for the cell in an ‘apparent’ steady state. The applied voltage is EA, which is fixed. The magnitude of the ionic current is much greater than the magnitude of the electronic current through the cell, consistent with predominant ionic conduction (jIi j>>jIe j). The net external current (measured) is also essentially fixed (within the accuracy of typical measuring instruments). However, the electronic resistances (and the electronic current) and internal EMFs are not fixed e they vary with time. Thus, this case represents an ‘apparent’ steady state. In many situations, this is probably the norm rather than the exception. Rest of the description of this figure is similar to Fig. 3(a).(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Since the electrolyte is a predominantly ionic conductor, Re >> Ri, it also means jIj >> jIe j for most situations (except when EA zEN ). The externally measured current (density) is thus I ¼ Ii þ Ie zIi for most cases (that is, as long as the applied voltage, EA, is sufficiently different from the Nernst voltage, EN).

    EA rce ðEA  EN Þrci 2 ¼ mStH  þ 4F 4c  4StH2 O2 Re Ri ðEA  EN Þrci ð29Þ  Ri

2 þ 4F mcO2 ¼mStH O2

inside the electrolyte, just near the steam-H2 electrode, and

2.3. Chemical potential of oxygen, mO2 , (and oxygen pressure, pO2 ) inside the electrolyte, just near the electrodes From Eqs. (16) through (28), the chemical potentials of oxygen in the electrolyte, just near the electrode/electrolyte interfaces, are given by [18,19]

   Ox  EA rae ðEA  EN Þrai ¼ mOx  4  4a O2  4F Re Ri ðEA  EN Þrai  Ri

maO2 ¼mOx O2  4F

inside the electrolyte, just near the oxygen electrode.

ð30Þ

9534

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 5 ( 2 0 1 0 ) 9 5 2 7 e9 5 4 3

EA

power is generated by the electrochemical oxidation of hydrogen, with the power density given by jð4Ox  4StH2 ÞIj. Oxygen ions thus transport from the oxygen electrode through the electrolyte to the steam-H2 electrode. The oxygen partial pressures within the electrolyte just near the electrodes are given by

EN EA

EN

Electrolyzer Mode

Voltage

EA

EN




  ðEN  EA Þrci 4F  c StH2 2 2 pStH exp  4 þ pcO2 ¼ pStH 4 O2 O2 Ri RT

Fuel Cell Mode

(33)

and Current Density

Fig. 4 e A schematic plot of the measured voltage (which is also the applied voltage, EA) vs. the measured current density. In the SOFC mode, the measured current (density) is positive (plotted along the positive x axis). In the electrolyzer mode, the measured current is negative (plotted along the negative x axis). The corresponding oxygen partial pressures in the electrolyte, just near the electrode/electrolyte interfaces, are given by   4F EA rce ðEA  EN Þrci  Ri RT Re 
  ðEA  EN Þrci  4F 2 ¼ pStH exp 4c  4StH2  O2 Ri RT


2 pcO2 ¼ pStH exp O2

(31)

inside the electrolyte, just near the steam-H2 electrode, and   4F EA rae ðEA  EN Þrai ¼   Ri RT Re 
   ðEA  EN Þrai 4F ¼ pOx 4Ox  4a  O2 exp  Ri RT


paO2

(32)

Case I

0 < EA < EN: In this case, the cell operates in a fuel cell mode and Ii > 04. Thus, no electrolysis occurs. Rather, electrical 4

(34)

Now, 4c  4StH2 > 0, 4Ox  4a > 0 and EN  EA > 0. Thus, the signs of the terms in the curly brackets in Eqs. (33) and (34) are positive. Also, therefore, for this case (single cell only e not necessarily an unbalanced cell in a stack in which the terminal voltage across an unbalanced cell may switch sign 2 < mcO2 < maO2 < mOx and thus [19]) we must have mStH O2 O2 c a Ox 2 < p < p < p [19]. Fig. 5(a) shows a schematic of the pStH O2 O2 O2 O2 variation of 4 and mO2 through the cell (electrolyte). That is, in such a case the pO2 within the electrolyte is mathematically bounded by the values in the gas phase at the electrodes 2 , pOx (bounded by pStH O2 O2 ), and high pressures do not develop. Thus, no degradation (of the type discussed here) should occur.

2.3.2.

Case II

0 < EN < EA : This is the electrolyzer mode. The oxygen partial pressures are

pOx O2 exp

inside the electrolyte, just near the oxygen electrode. Note that the mO2 ’s (and the pO2 ’s) are described in terms of measureable transport parameters (rci ,rai and Ri); Nernst voltage, EN; applied voltage, EA; electric potentials at the two electrodes, 4StH2 and 4Ox ; and in principle measureable electric potentials in the electrolyte just near the electrolyte/ electrode interfaces, 4c and 4a [18,19]. All discussion here assumes a ‘true’ steady state. In a ‘true’ steady state, electronic and ionic current densities are uniform through all ! three segments of the equivalent circuit, that is / V: I i ¼ 0 ! and / V: I e ¼ 0. Thus in a true steady state, we must always have 4Ox > 4a > 4c > 4StH2 (for the case, 4Ox > 4StH2 , selected here, that is for EA > 0) [18]. The spatial variation of oxygen partial pressure within the electrolyte, however, depends upon the relative values of transport parameters and the magnitude of the applied voltage EA in relation to the Nernst voltage EN. We will now consider two cases.

2.3.1.


  ðEN  EA Þrai 4F  Ox pOx paO2 ¼ pOx 4  4a þ O2 exp  O2 Ri RT

In the earlier work on solid oxide fuel cells, the author had used a different convention; the oxygen electrode was on the left side and the fuel electrode was on the right side, for which the current was in the opposite direction; that is in the earlier work we had set Ii < 0 in the fuel cell mode since the oxygen electrode (cathode) was the left electrode [18,19].


  ðEA  EN Þrci 4F  c 2 pcO2 ¼ pStH exp 4  4StH2  O2 Ri RT

(35)

and 
  ðEA  EN Þrai 4F  Ox paO2 ¼ pOx 4  4a  O2 exp  Ri RT

(36)

which are the same equations as (31) and (32). Now 4c  4StH2 > 0, 4Ox  4a > 0, however, EA  EN > 0. Thus, 2 will depend whether pcO2 is greater than or smaller than pStH O2 upon the sign of the exponent (terms in the curly brackets in Eq. (35)); and whether paO2 is greater than or smaller than pOx O2 will depend upon the sign of the exponent (terms in the curly brackets in Eq. (36)). That is, depending upon the relative magnitudes of the various transport parameters, according to Eqs. (35) and (36), it is possible that mcO2 ðpcO2 Þ may decrease Ox 2 2 (pStH ) and/or maO2 ðpaO2 Þ may increase above mOx below mStH O2 O2 O2 (pO2 ). Thus, in the electrolyzer mode, the mO2 ðpO2 Þ within the electrolyte need not be mathematically bounded by values at the electrodes (by values in the gas phases) [18]. then Note, if fð4c  4StH2 Þ  ððEA  EN Þrci =Ri Þg < 0, StH2 c Ox 2 (pcO2 < pStH ) and if fð4  4a Þ mO2 < mO2 O2 a a Ox a Ox ððEA  EN Þri =Ri Þg < 0, then mO2 > mO2 (pO2 > pO2 ). Depending upon the transport parameters and the operating conditions, either one, both or none of the situations may occur. Fig. 5(b) a Ox shows a schematic wherein the maO2 > mOx O2 (pO2 > pO2 ). Recall that in the electrolyzer mode, the steam-H2 electrode (low oxygen partial pressure) is the cathode and the oxygen electrode (high oxygen pressure) is the anode. If the mcO2 ðpcO2 Þ

9535

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 5 ( 2 0 1 0 ) 9 5 2 7 e9 5 4 3

a pOSt2

a

pOOx2 , pOx H 2O

pOx H 2

µOOx2

Ii

µOa2

c

St H 2

µOSt2

je

H2

µOc 2

Ie

O2 + H 2O Mixture High Oxygen Pressure Side

Steam-H2 Electrode

H2 + H 2O Mixture Low Oxygen Pressure Side

Ox

jO 2

H2

Oxygen Electrode

pHSt2 H 2 , pHSt2OH 2

Fuel Cell Mode

b

Ox

a

H2

jO 2

µOOx2

Ii Steam-H2 Electrode

Low Oxygen Pressure Side

H2 + H 2O Mixture

c

St H 2

µOSt2

H2

µOa2

je

µOc 2

Ie

pOOx2 , pOx H 2O

pOx H2

O2 + H 2O Mixture High Oxygen Pressure Side

pOSt2

Oxygen Electrode

St H 2 St H 2 pH , p H 2O 2

Electrolyzer Mode Fig. 5 e (a): Schematic variations of electric potential (4) and oxygen chemical potential (mO2 ) through the electrolyte in the fuel cell mode (‘true’ steady state). Both the 4 and mO2 in the electrolyte are bounded by values in the electrodes. The directions of the particle fluxes, jO2L and je are shown as well as the directions of ionic current, Ii and electronic current, Ie. (b): Schematic variations of electric potential (4) and oxygen chemical potential (mO2 ) through the electrolyte in the electrolyzer mode. Note that while the 4 in the electrolyte is bounded by values in the electrodes (‘true’ steady state), the mO2 in the Ox electrolyte need not be bounded by values in the electrodes. This schematic shows a case wherein maO2 (paO2 ) exceeds mOx O2 (pO2 ). Delamination along the electrolyte/anode (oxygen electrode) is likely in such a case. The directions of the particle fluxes, jO2L and je are shown as well as the directions of ionic current, Ii and electronic current, Ie.

decreases to a value below the thermodynamic stability of YSZ with respect to metals (Y, Zr), local electrolyte decomposition is possible. If the maO2 ðpaO2 Þ increases to too high a value, electrode delamination (along oxygen electrode/ electrolyte interface) is possible.

2.4. Relative magnitudes of transport numbers and the development of high oxygen pressure in the electrolyte near the oxygen electrode The pOx O2 is on the order of 1 atm in the electrolyzer mode. While no definite information is available concerning above what internal pressures delamination is likely (along the oxygen electrode/electrolyte interface), let us assume that delamination will occur if paO2 exceeds 100 atm. Fracture mechanical analysis of brittle materials shows that this value of internal pressure at a subsurface defect (e.g. a pore) in the electrolyte just near the electrolyte/electrode interface is generally sufficient to cause significant enhancement in stress

due to stress concentration and free surface effects to cause local cracking [30]. This suggests that at a typical temperature of electrolysis of 800  C, for values of  EA rae ðEA  EN Þrai  <  0:106 V Re Ri

(37)

(corresponding to paO2 ¼ 100 atm) delamination may occur along the oxygen electrode/electrolyte interface. Actually, high pressure develops in the electrolyte just near the oxygen electrode. The electrolyte/electrode interface is essentially the weak link. Thus, delamination occurs along the oxygen electrode/electrolyte interface. This is quite a remarkable result since it shows that delamination may readily occur at the oxygen electrode/ electrolyte interface at an applied voltage beyond a certain value. Eq. (37) allows one to estimate what should be the magnitudes of transport parameters so that delamination does not occur. Note if fðEA rae =Re Þ  ððEA  EN Þrai =Ri Þg never becomes negative, delamination will not occur since then the

9536

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 5 ( 2 0 1 0 ) 9 5 2 7 e9 5 4 3

pressure just inside the electrolyte will be lower than pOx O2 . Note that we are referring to the values of applied voltage greater than the Nernst voltage; that is EA > EN. The preceding discussion thus shows that if  EA rae ðEA  EN Þrai 0  Re Ri

(38)

or if  a

 a  re rae Ri E  EN rc þrel þra ra  ¼ a ¼ e e e  A ri Re EA rai i re Re Ri

(39)

rci þrel þrai i

the charge transfer (polarization) resistance at the perovskite anode (oxygen electrode)/electrolyte interface, and increasing the electronic resistance at the same interface. Materials such as Sr-doped La(Co, Fe)O3 (LSCF) developed for SOFC oxygen electrode (cathode) may be ideal for the electrolyzer oxygen electrode (anode). However, one needs to make sure that the polarization resistance is low for the reaction O2 / 1 /2O2 þ 2e’ (oxidation) reaction, which is opposite to the SOFC cathode reaction (opposite to the oxygen reduction reaction, ORR). A point to note is that it is not at all obvious that the polarization resistance would be the same in both directions (oxidation vs. reduction) and may need to be separately determined.

or if

2.5. EA <

EN

ð1 

rae Ri rai Re



(40)

delamination will not occur. This means a relatively high value of rae and a relatively low value of Re are preferred. Note that rae =Re < 1 and rai =Ri < 1. How best to achieve this? The following discussion is presented assuming the electrolyte is YSZ. It is well known that YSZ has a very low electronic conductivity at 800  C. In fact, it is virtually immeasurable. Thus, Re is very high and it must be lowered to prevent high pressure buildup. If the YSZ electrolyte is doped with ceria (or some other oxide capable of creating some electronic conduction in YSZ) it will lower the rel e by introducing electronic conduction in YSZ and this should lower Re. If a small amount of CeO2 is added to YSZ, it virtually does not change its oxygen ion concentration. Thus, it also r Þ. However, a small does not appreciably change mO2 ð! amount of CeO2 will change electron concentration orders of magnitude (but still in the dilute solution limit), and thus will r Þ. Referring now to Eq. (12) increase the local Fermi level, me ð! r Þ is compensated for by a correnote that an increase in me ð! r Þ (and thus also a decrease in sponding decrease in mO2 ð! r Þ) to maintain local equilibrium. This is the role of CeO2 pO2 ð! as an additive to the electrolyte in lowering internal pressure and thus lowering the propensity for oxygen electrode delamination. If now a thin (may be a micron or even a fraction of a micron) layer of YSZ (or some other oxygen ion conductor with a very low electronic conductivity) is deposited on the electrolyte (on the oxygen electrode side), it will drastically increase rae . In this manner, it may be possible to achieve c a el c ðrae =Re Þ ¼ ðrae =ðrae þ rel e þ re ÞÞ close to one if re >> re ; re . It is also preferred that rai =Ri be as small as possible. That is the charge transfer resistance or the polarization resistance at the oxygen electrode/electrolyte interface should be as small as possible, while the electronic resistance at the oxygen electrode/electrolyte interface should be as large as possible. Finally, it may also be desirable to have a somewhat higher electrolyte resistance, rel i . This is an interesting result because it suggests that making too thin a YSZ electrolyte may actually slightly increase propensity to electrode delamination. Naturally, the electrolyte cannot be too thick otherwise the performance will be compromised. Also, too thick an electrolyte may increase tendency for electrode delamination related to operating conditions as will be discussed later. Thus, the main focus should be on lowering

Illustrative calculations

Sample calculations are given in what follows which will demonstrate the profound role of electronic conduction through the electrolyte/cell on the magnitude of oxygen pressure generated in the electrolyte just near the anode (oxygen electrode)/electrolyte interface in the electrolyzer mode (that is in the electrolyte near the oxygen electrode/electrolyte interface). No parametric data are available on the YSZ-based system to the author’s knowledge5. However, earlier work from the author’s group (Lim and Virkar [31]) gives some information on the various transport parameters for a gadolinia-doped ceria (GDC)-based cell tested as a fuel cell. Thus, these values are used for illustrative calculations in what follows.

2.5.1. Development of high internal pressures in the electrolyte just near the anode (oxygen electrode)/electrolyte (LSM þ YSZ/YSZ) interface Experimental work on GDC electrolyte, anode-supported fuel cells showed that the area specific resistances for both ion transfer and electron transfer across the Ni/Ni þ GDC interface are negligible [31]. In the electrolyzer mode, Ni þ GDC would be the cathode. Assuming the same parameters in the electrolyzer mode, we have rci z0 and rce z0. The area specific ionic and electronic resistances at the LSC þ GDC/GDC interface in the SOFC mode were measured as 0.3 U cm2 and 1.5 U cm2, respectively [31]. If the same values can be assumed in the electrolyzer mode, this would mean rai z0:3 U cm2 and rae z1:5 U cm2. The ionic area specific resistance of the elec2 trolyte was estimated as rel i z0:09 U cm [31]. The cell area specific ionic resistance is thus Ri ¼ 0.3 þ 0.09 þ 0.00 ¼ 0.39 U cm2. The above values are used in the following calculations. An additional parameter necessary is the electronic resistance of the electrolyte. This will be treated as a floating parameter which will be varied over a wide range. For the purposes of illustration, the electronic area specific resistance of the 2 2 el electrolyte is varied between, rel e ¼ 0 U cm and re ¼ 20 U cm 6 in the following calculations . For a 10-micron thick 5

Note that we need rae , rai , Re and Ri for estimating the parameter which are not known for LSM/YSZ/Ni or any other YSZbased system. The only data available are from [31] for a GDCbased system. 6 2 Note that even though the lowest value of rel e chosen is 0 U cm , this corresponds to total cell electronic resistance Re of 1.5 U cm2. That is, even in this case, the cell is a predominantly ionic conductor. rae Ri =rai Re ,

9537

0.2903 0.2903 9.50 21.50 8 20

0.39 0.39

0.205 0.091

0.96 0.982

1.117 1.117

0.158 0.07

1.275 1.187

87.6% (3.027  106) 94.1% (3.251  106)

104.6 2.3 3 104

No delamination No delamination No delamination No delamination Delamination Possible Delamination highly likely Delamination imminent 8.45  1017 4.64  104 2.67  105 1.44 8.63 (1.83  106) (2.248  106) (2.49  106) (2.86  106) (2.93  106) 52.8% 65% 72.2% 82.9% 84.8% 1.0 0.6 0.43 0.23 0.2 0.2903 0.2903 0.2903 0.2903 0.2903 1.50 2.50 3.50 6.50 7.50

0.39 0.39 0.39 0.39 0.39

1.3 0.78 0.55 0.3 0.26

0.793 0.865 0.899 0.943 0.95

1.117 1.117 1.117 1.117 1.117

2.117 1.717 1.547 1.347 1.317

Likelihood of delamination paO2 (atm)  100 Ii Þ ð2FIE A (Mols/Joule)

Ii I

I (A cm2) Ie (A cm2) Ii (A cm2) ti EA EN EA rae Ri rai Re

Ri (U cm2) Re (U cm2)

0 1 2 5 6

7 Note that the cell area specific ionic resistance, Ri, is 0.39 U cm2, which is much smaller than the cell electronic area specific resistance, Re. Thus, typical measurements by techniques such as impedance spectroscopy or voltage vs. current plots will reflect a somewhat smaller value than w0.39 U cm2.

2 rel e (U cm )

electrolyte, this corresponds to an electronic resistivity as 2 high as 2  104 Ucm (corresponding to rel e ¼ 20 U cm ). This means the Re 7 is varied between 1.5 U cm2 and 21.5 U cm2. Low values of electrolyte electronic resistance, rel e , in practice can be achieved by adding a small amount of a transition metal oxide or an oxide with cation exhibiting multiple valence states; e.g. CeO2 to the electrolyte. The temperature is selected as 800  C. The oxygen partial pressures at the electrodes are 2 ¼ 1020 atm and pOx selected as follows: pStH O2 ¼ 1 atm. These O2 correspond to a Nernst voltage of EN ¼ 1.0645 V. The applied voltage for this calculation is set at EA ¼ 1.5 V. Thus, the corresponding ðEA  EN Þ=EA ratio is 0.2903. The corresponding ionic current density is estimated as Ii ¼ ðEA  EN Þ= Ri ¼ 1:117 A cm2. This value is a measure of the electrolysis current (which corresponds to the hydrogen generation rate). We will define efficiency of electrolysis as IIi  100. Since the membrane has some electronic conductivity, the process is not 100% Faradaic. Note this value is different from the ionic transference number of the cell, which is given by ti ¼ Re =ðRi þ Re Þ. Finally, another measure of the efficiency of electrolysis is Ii =ð2FEA IÞ in units of moles of H2 produced per Joule of electrical energy supplied. These values are also listed in Table 1. Using Eq. (32), the paO2 is estimated for the various values of selected (and given as a function of rae Ri =rai Re ). Also estirel e mated is the ionic transference number for the cell, given by ti ¼ Re =ðRi þ Re Þ. Table 1 lists the results of calculations. As the electronic resistance of the electrolyte, rel e , is varied between zero and 20 U cm2, the cell electronic resistance varies between 1.5 U cm2 and 21.5 U cm2, the corresponding rae Ri =rai Re varies between 1.3 and 0.091 (the corresponding measured cell area specific resistance from voltage vs. current curves varies between w0.31 U cm2 ande0.383 U cm2), and the corresponding paO2 varies between w8.45  1017 atm (for ðrae Ri Þ=ðrai Re Þ ¼ 1:3 > ððEA EN Þ=EA Þ ¼ 0:2903) and w2.3  104 atm (for ðrae Ri Þ=ðrai Re Þ ¼ 0:091 < ððEA EN Þ=EA Þ ¼ 0:2903). This is a remarkable result which shows that changes in electronic conduction characteristics of the cell, which make modest changes in the overall cell resistance as would be measured from voltage vs. current density plots or impedance spectra (w0.383 U cm2 to w0.31 U cm2), can change the pressure generated in the electrolyte just near the oxygen electrode/electrolyte interface by several orders of magnitude. There is no conceivable way that the cell/material can withstand such high internally generated pressures. The conclusion is that under such conditions (high electronic resistance of the electrolyte), delamination along the oxygen electrode/electrolyte interface is imminent and cannot be avoided. In fact, it may be reasonably be expected that delamination will occur above some value of internal pressure (paO2 ) which is much lower than this value, perhaps say 100 atm. Details of the nature of the interface, defects present, electrode microstructure, and general mechanical properties will determine the pressure above which delamination will

Table 1 e Estimation of oxygen pressure within the electrolyte just near the oxygen electrode (anode)/electrolyte interface, paO2 , for various values of cell area specific electronic resistance, Re. The applied voltage is EA [ 1.5 V.

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 5 ( 2 0 1 0 ) 9 5 2 7 e9 5 4 3

9538

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 5 ( 2 0 1 0 ) 9 5 2 7 e9 5 4 3

occur8. The key conclusion is that no matter what the mechanical properties are, high enough pressures will most certainly be developed to cause oxygen electrode delamination at values of electronic resistance sufficiently high. Indeed, experimental results in several studies have shown that often delamination occurs along the oxygen electrode/electrolyte interface in the electrolyzer mode [6e10]. This is an additionally remarkable result as it suggests that the intuitively obvious approach of trying to develop an electrolyte material with the highest possible ionic transference number may in fact be counterproductive from the standpoint of stability. From Table 1, note that delamination is likely for Re  9.5 Ucm2 for this set of calculations. For a 10 micron thick electrolyte, this corresponds to rel ¼ 8  103 Ucm or sel ¼ 1.25  104 Scm1. This means for a 10 micron thick electrolyte with electronic conductivity less than 1.25  104 Scm1, delamination is likely. Note that in YSZ, the electronic conductivity is much lower than 1.25  104 Scm1 at 800  C. Thus, electrode delamination is highly likely at 800  C with YSZ as the electrolyte for the operating parameters selected in these calculations. Thus, a small amount of electronic conduction through the electrolyte is actually beneficial. The illustrative calculations given here suggest that one should be able to achieve a reasonably high ionic transference number for the membrane while also ensuring stability against delamination/degradation. For the values selected 2 here, for example, an rel e ofe5 Ucm leads to a cell transference number of w0.943 (which means electrolysis efficiency defined here as ðIi =IÞ  100of 82.9%) and the corresponding paO2 is only w1.44 atm, and should not lead to degradation. Calculations given here are for illustrative purposes only. It is a straightforward matter to recognize that significant optimization of parameters is possible which not only will allow the attainment of high overall ionic transport number (high electrolysis efficiency), but will also ensure that high internal pressures are not developed under normal electrolyzer operating conditions, thus preventing electrode delamination/ degradation. If all relevant parameters are known (which can be experimentally measured), Eqs. (31) and (32) may be used to design membranes which will not degrade. The preceding discussion is given in terms of the required cell transport parameter, namely rae Ri =rai Re , which must be greater than a given operating conditions parameter, namely ðEA  EN Þ=EA , so that oxygen electrode delamination does not occur. This means, if a given set of operating conditions are selected, namely EN and EA, then those cells (materials and microstructures, for example) for which rae Ri =rai Re satisfies the required criteria will not degrade e and those that do not satisfy the required criteria, will likely degrade. Alternatively, for a given set of cells, there will be a critical value of ðEA  EN Þ=EA describing the operating conditions, which should not be exceeded. That is, for a given cell, one must

8 Some defects are almost always present. Note, however, degradation by delamination is expected even if no defects are present, since the cohesive strength of most materials is about 10% of the Young’s elastic modulus (which is typically w105 atm). Thus, internally generated pressures on the order of 104 atm (or greater) will cause cracking (delamination), regardless of the presence of any defects (which are always present).

have ðEA  EN Þ=EA  rae Ri =rai Re or EA < EN =ð1  ðrae Ri =rai Re ÞÞ to a prevent oxygen electrode delamination. Since Ri ¼ rci þ rel i þ ri a and Re ¼ rce þ rel þ r , note that multiple options exist to select e e the various transport parameters such that the delamination of oxygen electrode may be prevented. An important consequence of non-equilibrium thermodynamics is the occurrence of abrupt changes in ‘potentials’ (chemical and/or electrical) across interfaces [18,19,22,32]. If the interface is modeled as a thin transition region, perhaps a few nanometers or even a fraction of a nanometer, all changes in the transition region are naturally continuous. However, on a microscopic scale (or even at a submicroscopic scale), smooth changes occur in regions adjacent to the interfaces and sharp changes occur through the interfaces, as has also been shown by non-equilibrium molecular dynamics (NEMD) simulations [22,32]. Thus, the existence of a very high ! ! mO2 ð r Þ, which is maO2 (or pO2 ð r Þ, which is paO2 ) in the electrolyte r þ d! r Þ, which is just near the oxygen electrode; and low mO2 ð! ! ! Ox Ox mO2 (or pO2 ð r þ d r Þ, which is pO2 w 1 atm) in the oxygen electrode, just across the electrolyte/electrode interface, is perfectly reasonable. Here, d! r is the interface thickness. Fig. 6 shows a schematic of the abrupt change in oxygen pressure across an interface. The interface thickness is expected to be very small e perhaps on the order of a nanometer. The high pressure formed just inside the electrolyte links up with the near surface defects causing delamination. As soon as a delamination crack is formed, pressure within the crack rapidly reduces to that close to the pOx O2 . The initiation of delamination cracks can occur at multiple places, and they link up to form a macroscopic crack leading to electrode delamination. Fig. 7 shows possible locations at the electrolyte/electrolyte interface where high pressure may build up leading to delamination. The kinetics of this growth process, however, depends on the net flux of oxygen depositing in the crack, which depends upon the current and the various transport parameters and the operating conditions, as described in earlier work on related topics [30,33]. Also, if some interdiffusion occurs between the electrolyte and the oxygen electrode, it may alter local transport properties. Under such conditions, it is possible that the initiation of delamination may occur inside the electrolyte some distance from the interface [30].

2.5.2. Development of low internal oxygen pressure within the electrolyte just near the steam-H2 electrode/electrolyte interface The other possibility for degradation is local electrolyte decomposition if the pcO2 drops below the decomposition pressure for zirconia. At 1073 K, the standard free energy of formation of ZrO2 is 894.6 kJ/mol [34]. This corresponds to an equilibrium oxygen partial pressure ofe2.8  1044 atm. Thus, to ensure that decomposition does not occur, pcO2 must be above this value. The expected value of oxygen partial pressure in the hydrogen being formed at the steam-H2 electrode 2 w1020 atm. From Eq. (31), is assumed to be 1020 atm or pStH O2 note that 2 pStH RT O2 ln pcO2 4F

!

EA rce ðEA  EN Þrci ¼ þ Re Ri

(41)

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 5 ( 2 0 1 0 ) 9 5 2 7 e9 5 4 3

b

µOa 2

µOa 2 µO2

Electrolyte

Electrolyte

Electrode

µO2

µOOx2 On the order of Microns

Electrode

a

9539

µOOx2

On the order of Microns

On the order of Microns

Interface

On the order of Microns

Interface

Fraction of a Micron Or Nanometer Scale

Fraction of a Micron Or Nanometer Scale

Fig. 6 e (a) A schematic variation of oxygen chemical potential in the vicinity of the oxygen electrode/electrolyte interface when high oxygen pressure is developed within the electrolyte. The interface thickness is shown exaggerated. (b) The same schematic as in (a) showing the abrupt change across a very thin interface region.

Decomposition will occur at the steam-H2 electrode/electrolyte (YSZ) interface (into the electrolyte) provided the above exceeds 1.25 V at 800  C. Thus, to prevent decomposition, we must have EA

rci rc  e Ri Re



c c EN rci r r ¼ ðEA  EN Þ i  EA e  1:25 V Ri Ri Re

(42)

If the electrolyte is other than YSZ, Eq. (42) will be modified according to the corresponding free energy of formation of the

oxide. Eq. (42) shows that rci =Ri should be as small as possible and rce =Re should be as large as possible. This means the polarization resistance for the oxygen reduction reaction (ORR) at the steam-H2 electrode should be as low as possible, and the electronic resistance at the steam-H2 electrode/electrolyte interface should be as large as possible. The relative magnitudes of the voltage values show, however, that in general delamination at the oxygen electrode is the likely mode of degradation rather than decomposition at the steam-H2 electrode/electrolyte interface. This is in accord with the observations in several studies.

2.6.

Location of High Pressure And Delamination Cracks

Oxygen Electrode

Electrolyte

Particles

Fig. 7 e A schematic showing the locations where the high oxygen pressure builds up. Delamination cracks develop at multiple places and then link up to form a macroscopic delamination crack along the oxygen electrode/electrolyte interface.

SOFC vs. SOEC

As shown previously and discussed here, degradation of a single SOFC by delamination is not expected but that of a single SOEC is possible [18]. In an SOFC stack, if a cell imbalance occurs, it is possible to have both ionic and electronic currents in the same direction, which can cause electrode delamination/failure [18,19]. In this context, degradation of the electrolyzer (SOEC) is much more likely than that of a fuel cell (SOFC) since the ionic and the electronic currents through the cell are always in the same direction in the electrolyzer mode. In SOFC, degradation due to internally generated high pressures occurs due to cell imbalance in a stack [19]. In SOFC, it has been observed that delamination occurs along the electrolyte/fuel electrode interface [35]. Since the partial pressure of oxygen in the Ni þ YSZ fuel electrode in SOFC is w1020 atm, a large increase in mO2 is necessary for delamination to occur. By contrast, in SOEC, relatively modest increase in relation to adjacent gas phase partial pressure of oxygen is required for delamination to occur along the oxygen electrode/electrolyte interface. For this reason, degradation by delamination of the positive (oxygen) electrode is more likely

9540

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 5 ( 2 0 1 0 ) 9 5 2 7 e9 5 4 3

in electrolyzers as compared to the delamination of the negative (fuel) electrode in unbalanced fuel cells in a stack. Indeed, it has been reported that often the result of degradation of SOEC is delamination of the oxygen electrode [6e10]. Additionally, possible cell imbalance in an SOEC stack may further exacerbate the situation concerning delamination/ failure [19,35].

2.7. Comparison of the available experimental results from the literature with the predictions of the model Several studies have shown that oxygen electrode delamination occurs in solid oxide electrolyzers under certain operating conditions [6e10]. There also are some reports of relatively stable operation for several hundred hours [36e38]. The objective of the following discussion is to determine if there were significant differences among the various reported studies in the context of the model described here. Quantitative comparison of the model with the published literature is not possible since the required information on the various transport parameters (rae ,rai ,Re, and Ri) is not available. Some information, however, is available which allows for a qualitative comparison. Hauch et al. [36] observed stable performance and no delamination was observed. In their work, the temperature was typically 850  C and in some tests as high as 950  C. At higher temperatures, the electronic conductivity of YSZ is expected to be higher, which would tend to mitigate high pressure generation. The long term test at 850  C was conducted in their work at a current density of 0.5 Acm2. The corresponding EN and EA were 0.855 V and 1.14 V. Thus the corresponding ðEA  EN Þ=EA was 0.25. In most tests, increase in the overall cell resistance was noted. The various contributions to the cell resistance were obtained by de-convolution of impedance spectra. These authors attributed increase in resistance to the degradation of the steam-H2 electrode. In the studies by Brisse et al. [37] tests were conducted at 800 and 900  C. The long term test (160 h) was conducted at 800  C under an applied voltage of 1.05 V. The corresponding EN was 0.78 V and the ratio ðEA  EN Þ=EA was 0.14. In both of these studies, the ratio ðEA  EN Þ=EA was lower than used in the calculations presented here (0.2903). While no information is available on rae Ri =rai Re in either of these tests, lower values of ðEA  EN Þ=EA are consistent with relatively ‘benign’ electrolysis N was conditions, especially the long term test for which EAEE N 0.14. Thus, the observation that no delamination of the oxygen electrode occurred in their studies, appears to be consistent with the low ðEA  EN Þ=EA ratio in these studies, indicating relatively benign electrolytic conditions. In another study Doenitz et al. [38] tested cells at 995  C at a cell voltage EA of 1.07 V, a Nernst voltage EN of 0.83 V, and the ratio ðEA  EN Þ=EA of 0.22. No delamination was reported. In all of their studies, it is thus the expectation that the operating parameter ðEA  EN Þ=EA was less than rae Ri =rai Re . That is, the conditions were rather benign and no delamination was observed. In studies conducted by O’Brien et al. [9], by contrast, delamination of the oxygen electrode was observed at the conclusion of the 1000 h test. Throughout the test, increase in cell resistance was observed (from initial w0.6 Ucm2 toe1.3

Ucm2 after 1000 h). The current density at which the test was conducted was not given in the paper. However, the test data given in their [9] Figs. 8 and 9 show that ðEA  EN Þ=EA values may have been more than 0.35 and as high as 0.4. This means electrolysis conditions used in the work by O’Brien et al. [9] were far more severe than used in the studies by Hauch et al. [36] and Brisse et al. [37]. In the studies by O’Brien et al. [9], the cells were thick, electrolyte supported. Higher applied voltage, EA, was probably needed to obtain a reasonable electrolysis current because of the large electrolyte thickness, effectively increasing ðEA  EN Þ=EA and thus increasing propensity for oxygen electrode delamination. Thus, the observation that oxygen electrode delamination occurred is in accord with the mechanism described here. In the work by Guan et al. [8] also, significant delamination of the oxygen electrode was observed with LSM þ YSZ as the oxygen electrode. The cell voltage EA was 1.3 V, the Nernst voltage EN was 0.855 with the corresponding ðEA  EN Þ=EA of 0.34. Again, comparison with the calculations given here shows that test conditions were rather severe, and the observed electrode delamination is consistent with the predictions of the mechanism presented here.

2.8.

Transient effects

This section describes the possible role of transient effects in the context of the proposed model. The preceding analysis assumes the existence of a ‘true’ steady state. Often, the actual state may be an ‘apparent’ steady state and in some cases even a clear transient state may exist, as judged on the basis of the time dependence of the externally measured parameters (current and voltage). That is, if a clear transient state exists, then the experimentally measurable parameters such as voltage and current will exhibit time dependence. A number of phenomena such as electrode microstructure changes, interconnect oxidation, seal degradation, etc. will likely occur over time. For example, it has been observed that many changes such as chromium migration from the interconnect into the electrodes occurs over time. Such changes will alter the transport parameters such as rae ,rai ,Re, and Ri, thus altering the tendency for electrode delamination by the mechanism described here. That is, many transient effects will conspire to produce conditions leading to ðEA  EN Þ=EA > rae Ri =rai Re and eventually creating conditions ripe for delamination. If an ‘apparent’ steady state exists, however, no detectable changes in the measured voltage and current may be observed. That is, the system will appear to have reached a steady state. However, if one independently measures 4ð! r ; tÞ within the electrolyte, such as by embedded probes [31], one should be able to observe that it changes with time, even when the measured voltage and current do not change with time. Also, when an ‘apparent’ steady state exists, the 4ð! r ; tÞ within the electrolyte need not be bounded by values at the electrodes (4I and 4II ). Indeed, studies on YSZ-based solid oxide fuel cells have shown that under certain testing conditions, 4ð! r ; tÞ is often not bounded by values at the electrodes [39]. In an apparent steady state, Eqs. (19), (20) and (21) respectively become

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 5 ( 2 0 1 0 ) 9 5 2 7 e9 5 4 3

Ice ðtÞrce ðtÞ ¼ Ec ðtÞ þ Ii rci ¼ 4StH2  4c ðtÞ ¼ D4StH2;c ðtÞ

(43)

Iele ðtÞrele ðtÞ ¼ Eel ðtÞ þ Ii reli ¼ 4c ðtÞ  4a ðtÞ ¼ D4c;a ðtÞ

(44)

and Iae ðtÞrae ðtÞ ¼ Ea ðtÞ þ Ii rai ¼ 4a ðtÞ  4Ox ¼ D4a;Ox ðtÞ

(45)

such that the sum of the three D4’s still corresponds to the measured (fixed) voltage across the cell, namely D4StH2;c ðtÞ þ D4c;a ðtÞ þ D4a;Ox ðtÞ ¼ 4StH2  4Ox ¼ EA ¼ V

(46)

since V ¼ EA ¼ 4StH2  4Ox

(47)

is independent of time, and the measured current, which is essentially Ii, is also nearly independent of time, consistent with an ‘apparent’ steady state. This case is depicted in Fig. 3(b).

2.9. Time required to attain steady state and/or the condition of criticality All of the analysis given in this manuscript is based on the assumption of a ‘true’ steady state. Within the typical experimental time frame, a ‘true’ steady state may not be easily attained. An important factor concerns the time required to attain the ‘true’ steady state e or at least the time required to reach the condition of criticality (as judged on the basis of critical pressure generated to reach a steady state). This time depends on transport parameters and other parameters related to deviations from stoichiometry. These are not known for any system to a reasonable level of accuracy. Nevertheless, the general approach necessary to solve the relevant differential equations has been previously described for a couple of cases [19,30,33]. The general conclusion is that the greater the overall electronic conductivity, the faster are the kinetics of achieving the steady state [33]. However, as described here, by a suitable choice of transport parameters, it is possible to achieve the final steady state which is subcritical e that is, it does not lead to degradation. If the electronic conductivity is very low, greater time is required to attain steady state. This will reflect as a larger incubation time before significant degradation may become experimentally detectable (such as increase in the overall resistance). The incubation time depends in a complicated way on a number of transport parameters [30,33]. However, critical condition may be reached long before the steady state is reached, leading to failure. Thus, from the standpoint of long term stability, it is preferable to introduce some electronic conductivity so that steady state corresponding to subcritical condition is reached and high pressures are not generated.

9541

1) The model shows that degradation/failure of SOEC will primarily manifest as delamination along the oxygen electrode/electrolyte interface. Oxygen electrode delamination occurs as a result of the formation of high internal oxygen pressure within the electrolyte, just near the oxygen electrode/electrolyte interface. 2) The higher the electronic conductivity of the electrolyte (cell), the lower is the tendency for the formation high internal pressures. Preliminary calculations show that modest changes in electronic conduction can cause orders of magnitude changes in oxygen pressure, paO2 . Thus, the present analysis shows that a small amount of electronic conduction through the electrolyte is actually preferred from the standpoint of stability. This is a significant result as it suggests that an oxygen ion conductor of the highest possible ionic transport number (negligible electronic transport number) may be more prone to degradation (oxygen electrode delamination) and thus is not the desired material as an electrolyte. In this context, YSZ may not be the ideal material as electrolyte for SOEC, especially at 800  C and lower temperatures. 3) Addition of a small amount of a transition metal oxide or other oxides with cations exhibiting multiple valence states, such as ceria, to the electrolyte should increase its electronic conductivity and decrease the tendency for the delamination of oxygen electrode. It should be possible to suitably tailor electronic transport through the cell to minimize tendency for high pressure buildup and simultaneously also ensure a relatively high ionic transport number for the cell and thus a high electrolyzer efficiency. This can be achieved by depositing a very thin layer of a purely oxygen ion conductor with a high electronic resistance on the base electrolyte on the oxygen electrode side. 4) The overall propensity for oxygen electrode delamination can be described by a parameter given in terms of the various transport parameters, namely, rae Ri =rai Re . The higher the rae Ri =rai Re , the smaller is the propensity for oxygen electrode delamination. In terms of the operating parameters, delamination should not occur as long as ðEA  EN Þ=EA < rae Ri =rai Re . Alternatively, the applied voltage must satisfy the following condition, namely EA < EN =ð1  ðrae Ri =rai Re ÞÞ, to prevent oxygen electrode delamination. Experimental methods can be devised in principle to measure the required cell parameters, namely rae , rai , Ri and Re; and identify safe operating regime for SOECs.

Acknowledgements This work was supported by Idaho National Laboratory (Battelle) under Contract No. 00088561.

3.

Summary

Based on the model for oxygen electrode delamination in solid oxide electrolyzer cells (SOEC) presented in this manuscript, the following are summary statements which describe essential features of the model, its predictions, and implications concerning the design of robust solid oxide electrolyzer cells.

Nomenclature

rci

Area specific ionic charge transfer (polarization) resistance at the cathode (steam-H2 electrode)/ electrolyte interface, U cm2.

9542 rai

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 5 ( 2 0 1 0 ) 9 5 2 7 e9 5 4 3

Area specific ionic charge transfer (polarization) resistance at the anode (oxygen electrode)/ electrolyte interface, U cm2. c Area specific electronic charge transfer (direct re electron transfer) resistance at the cathode (steamH2 electrode)/electrolyte interface, U cm2. a Area specific electronic charge transfer (direct re electron transfer) resistance at the anode (oxygen electrode)/electrolyte interface, U cm2. Area specific ionic resistance of the electrolyte, rel i U cm2. el Area specific electronic resistance of the electrolyte, re U cm2. c a Ri ¼ ri þ rel i þ ri Area specific ionic resistance of the cell, U cm2. c a Re ¼ re þ rel e þ re Area specific electronic resistance of the cell, U cm2. Flux of oxygen ions, molcm2 s1. jO2 Flux of electrons, molcm2s1. je ! IO2 ð r ;tÞ Oxygen ion current density as a function of position and time, A cm2. Ii ¼ IO2 Ionic (oxygen) current density, Acm2. Electronic current density, Acm2. Ie I Load current (density), Acm2. ! r Position vector, cm. t Time, s . ti ¼ Re =ðRi þ Re Þ Ionic transport number of the cell including interfaces F Faraday constant, Cmol1. R Gas constant, Jdeg1mol1. T Temperature, deg Kelvin. Nernst voltage, V. EN a Ea ¼ ðmOx O2  mO2 Þ=4F Nernst potential across electrolyte/oxygen electrode interface, V. Eel ¼ ðmaO2  mcO2 Þ=4F Nernst potential across the electrolyte (excluding interfaces), V. 2 Þ=4F Nernst potential across electrolyte/ Ec ¼ ðmcO2  mStH O2 steam-H2 electrode, V. Applied voltage, V. EA 2 Oxygen partial pressure in the gas phase at the pStH O2 steam-H2 electrode, atm. or Pa. Oxygen partial pressure in the gas phase at the pOx O2 oxygen electrode, atm. or Pa. Oxygen pressure in the electrolyte just near the paO oxygen electrode (anode)/electrolyte interface, atm. or Pa. Oxygen pressure in the electrolyte just near the pcO2 steam-H2 electrode (cathode)/electrolyte interface, atm. or Pa. Valence of species i. zi Chemical potential of species i, Jmol1. mi Ox Chemical potential of oxygen at the oxygen electrode mO2 (gas phase), Jmol1. 2 Chemical potential of oxygen at the steam-H2 mStH O2 electrode (gas phase), Jmol1. a Chemical potential of oxygen in the electrolyte near mO2 the oxygen electrode/electrolyte interface, Jmol1. c Chemical potential of oxygen in the electrolyte near mO2 the steam-H2 electrode/electrolyte interface, Jmol1. Electrochemical potential of species i, Jmol1. m ~i

sO2 Oxygen ion conductivity, Scm1. Electronic conductivity, Scm1. se F Electrostatic potential, V. 4 ¼ ~ me =F ¼ ðme =FÞ þ F Electric potential (reduced negative electrochemical potential of electrons), V.

references

[1] Doenitz W, Schmidberger R, Steinheil E, Streicher R. ‘Hydrogen production by high temperature electrolysis of water vapour’. Int J Hydrogen Energy 1980;5:55e63. [2] Doenitz W, Dietrich G, Streicher R. ‘Electrochemical high temperature technology for hydrogen production or direct electricity generation’. Int J Hydrogen Energy 1988;13(5):283e7. [3] Doenitz W, Erdle E. ‘High-temperature electrolysis of watervapour e status of development and perspectives for application’. Int J Hydrogen Energy 1985;10:291e5. [4] Isenberg AO. ‘Energy conversion via solid oxide electrolyte electrochemical cells at high temperatures’. Solid State Ionics 1981;3-4:431e7. [5] Jensen SH, Larsen PH, Mogensen M. ‘Hydrogen and synthetic fuel production from renewable energy sources’. Int J Hydrogen Energy 2007;32:3253e7. [6] Herring JS, O’Brien JE, Stoots CM, Hawkes GL, Hartvigsen JJ, Shahnam M. ‘Progress in high-temperature electrolysis for hydrogen production using planar SOFC technology’. Int J Hydrogen Energy 2007;32:440e50. [7] Momma A, Kato T, Kaga Y, Nagata S. ‘Polarization behavior of high temperature solid oxide electrolysis cells (SOEC)’. J Ceram Soc Japan 1997;105:369e73. [8] Guan J, Ramamurthi B, Rudd J, Hong J, Riley P, Weng, D and Minh N, ‘High performance flexible reversible solid oxide fuel cell’, GE Global Research Center Report DE-FC36e04GO14351, May 15, 2007. [9] O’Brien JE, Stoots CM, Herring JS, Condie KG, Housley GK, ‘The high-temperature electrolysis program at the Idaho National Laboratory: observations on performance degradation’, OSTI ID: 961915, DE-AC07e99ID-13727. Report No. INL/CON-09-15564. [10] Mawdsley JR, Carter JD, Kropf AJ, Yildiz B, Maroni VA. ‘Posttest evaluation of oxygen electrodes from solid oxide electrolysis stacks’. Int J Hydrogen Energy 2009;34:4198e207. [11] Doenitz W, Schmidberger R. ‘Concepts and design for scaling up high temperature water vapour electrolysis’. Int J Hydrogen Energy 1982;7:321e30. [12] Quandt KH, Streicher R. ‘Concept and design of a 3.5 MW pilot plant for high temperature electrolysis of water vapor’. Int J Hydrogen Energy 1986;11:309e15. [13] Maskalick NJ. ‘High temperature electrolysis cell performance characterization’. Int J Hydrogen Energy 1986;11:563e70. [14] Iwahara H, Uchida H, Yamasaki I. ‘High-temperature steam electrolysis using SrCeO3-based proton conductive solid electrolyte’. Int J Hydrogen Energy 1987;12:73e7. [15] Kreuter W, Hofmann H. ‘Electrolysis: the important energy transformer in a world of sustainable energy’. Int J Hydrogen Energy 1998;23:661e6. [16] Shin Y, Park W, Chang J, Park J. ‘Evaluation of the high temperature electrolysis of steam to produce hydrogen’. Int J Hydrogen Energy 2007;32:1486e91. [17] Erdle E, Doenitz W, Schamm R, Koch A. ‘Reversibility and polarization behavior of high temperature solid oxide electrochemical cells’. Int J Hydrogen Energy 1992;17:817e9. [18] Virkar AV. ‘Theoretical analysis of the role of interfaces in transport through oxygen ion and electron conducting membranes’. J Power Sources 2005;147:8e31.

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 5 ( 2 0 1 0 ) 9 5 2 7 e9 5 4 3

[19] Virkar AV. ‘A model for solid oxide fuel cell (SOFC) stack degradation’. J Power Sources 2007;172:713e24. [20] Kondepudi D, Prigogine I. ‘Modern thermodynamics: from heat engines to dissipative structures’. New York: John Wiley & Sons; 1998. [21] Tennebaum A, Ciccotti G, Gallico R. ‘Stationary nonequilibrium states by molecular dynamics: Fourier’s law’. Phys Rev A 1982;25(No. 5):2778e87. [22] Rosjorde A, Fossmo DW, Bedeaux D, Kjelstrup S, Hafskjold B. ‘Nonequilibrium molecular dynamic simulations of steadystate heat and mass transport in condensation’. J Colloid Interface Sci 2000;232:178e85. [23] Bedeaux D, Kjelstrup S, Rubi JM. ‘Nonequilibrium translational effects in evaporation and condensation’. J Chem Phys 2003;119(17):9163e70. [24] Armstrong RD, Todd M. In: Bruce PG, editor. Solid State Chemistry. Cambridge, UK: Cambridge University Press; 1995. p. 264. [25] Wagner C. ‘Equations of transport in solid oxides and sulfides of transition metals’. Prog Solid-State Chem 1975;10 (1):3e16. [26] Heyne L. Mass transport in oxides. NBS Special Publication; 1968. 296 pp. 149e164. [27] Hebb MH. ‘Electrical conductivity of silver sulfide’. J Chem Phys 1952;20:185e90. [28] Kofstad P. ‘Nonstoichiometry, diffusion, and electrical conductivity in binary metal oxides’. New York: John Wiley & Sons; 1972. [29] Virkar AV. ‘Theoretical analysis of solid oxide fuel cells with two-layer, composite electrolytes: electrolyte stability’. J Electrochem Soc 1991;138(5):1481e7.

9543

[30] Virkar AV, Nachlas J, Joshi AV, Diamond J. ‘Internal precipitation of molecular oxygen and electromechanical failure of zirconia solid electrolytes’. J Am Ceram Soc 1990;73 (11):3382e90. [31] Lim H-T, Virkar AV. ‘Measurement of oxygen chemical potential in thin electrolyte film, anode-supported solid oxide fuel cells’. J Power Sources 2008;180:92e102. [32] Kjelstrup S, Bedeaux D. ‘Non-equilibrium thermodynamics of heterogeneous systems’. In: Series on advances in statistical mechanics, vol. 16. New Jersey: World Scientific; 2008. [33] Virkar AV. ‘A model for internal pressurization in cationic solid electrolytes’. J Mater Sci 1985;20:552e62. [34] Barin I. ‘Thermochemical data on pure substances: parts I and II’. Weiheim, Germany: VCH Publication; 1993. [35] Lim H-T, Virkar AV. ‘A study of solid oxide fuel cell stack degradation by inducing abnormal behavior in a single cell test’. J Power Sources 2008;185:790e800. [36] Hauch A, Ebbesen SD, Jensen SH, Mogensen M. ‘Solid oxide electrolysis cells: microstructure and degradation of the Ni/ yrrtia-stabilzied zirconia electrode’. J Electrochem Soc 2008; 155(1):B1184e93. [37] Brisse A, Schefold J, Zahid M. ‘High temperature water electrolysis in solid oxide fuel cells’. Int J Hydrogen Energy 2008;33:5375e82. [38] Doenitz W, Erdle E, Streicher R. In: Wendt H, editor. ‘Electrochemical hydrogen technologies. Electrochemical production and combustion of hydrogen’. Amsterdam: Elsevier; 1990. p. 213e59 [chapter 3]. [39] Lim H-T, ‘Ionic and electronic transport across interfaces in thin electrolyte film, anode-supported solid oxide fuel cells’, PhD dissertation submitted to the University of Utah, (2009).