Unformatted text preview: Ionics (2007) 13:127140 DOI 10.1007/s115810070087x ORIGINAL PAPER General model for the functional dependence of defect concentration on oxygen potential in mixed conducting oxides
Keith L. Duncan & Eric D. Wachsman Received: 4 January 2007 / Revised: 14 March 2007 / Accepted: 15 March 2007 / Published online: 13 June 2007 # SpringerVerlag 2007 Abstract To understand and engineer applications for mixed conducting oxides, it is desirable to have explicit, analytical expressions for the functional dependence of defect concentration and transport properties on the partial pressure of the external gas phase. To fulfill this need, general expressions are derived for the functional dependence of defect concentration on the oxygen partial pressure (PO2 ) for the mixed ionic electronic conductors. The model presented in this paper differs from expressions obtained using the popular Brouwer approach because they are continuous across multiple Brouwer regions. Keywords Defect distribution . Ceria . Analytical model . Mixed ionic electronic conductors . Defect equilibria Introduction In this work, models are developed for the functional dependence of defect concentration on oxygen partial pressure, PO2 , in mixed ionic electronic conductors (MIECs). First, previous attempts by other researchers [1 12] are reviewed, then a new model is developed. The model presented here is restricted to MIECs with either the fluorite (general chemical formula AO2) or the perovskite structure (general chemical formula ABO3 where A and B are trivalent1). This restriction is appropriate in this context because MIECs of interest (for fuel cell electrodes and gas separation membranes) typically have one of these structures or else a distorted variant. Suitable equations for the functional dependence of the concentration of defects as a function of PO2 have thus far been elusive for one reason: highorder polynomials. The usual method of solving the system of defect equations and the charge neutrality equation for the material generates fourthorder or higher polynomials. However, the roots of polynomials of orders greater than four (i.e., quintic or higher) are rather difficult to find. This is a long appreciated point and was recently discussed by Porat and Tuller [1] in relation to oxygenvacancyconducting MIECs and Poulsen [2, 3] in relation to protonconducting MIECs. Nevertheless, with reasonable simplifications, the system of defect equations may be reduced so that they generate cubic equations. When all the coefficients in the equation are real, cubic equations have at least one real root. However, the roots of such a cubic equation are not useful because they modulate between being real or complex depending on the PO2 value that is being used in the calculation. Hence, any analytical expression so derived would be severely limited to a narrow range of PO2 . This problem has inspired some novel solutions. However, their usefulness is compromised for a variety of reasons that are discussed briefly in the next section.
1 This requirement is imposed solely because including other perovskite structures in the models developed in this work would make it inordinately long. The principles employed in developing the models are generally applicable to any perovskite structure. K. L. Duncan : E. D. Wachsman (*) Department of Materials Science and Engineering, University of Florida, Gainesville, FL 326116400, USA email: [email protected] 128 Ionics (2007) 13:127140 Consequently, a new approach is needed to obtain a solution for the functional dependence of defect concentration on PO2 . We seek to provide such a solution in the model presented in this paper. A reduced version of this model was derived earlier [13] for acceptordoped ceria for the range of PO2 s where electrons, oxygen vacancies, and the acceptor dopant were the dominant defects. The defect concentrationPO2 expressions obtained thence enabled a more accurate calculation of the transport properties and material constants (from fitting the model to experimental data) of acceptordoped ceria than was previously obtainable with those derived [57] from the Brouwer approach [4]. Below, we present, for fluorite and perovskitestructured MIECs, a more general model, which provides a continuous function for the dependence of defect concentration on PO2 , spanning the entire defect equilibria from low to high PO2 . other. Additionally, the concentration of ionic defects is assumed to be constant throughout. This assumption is reasonable only in the electrolytic region when both ce and ch <<cion. Nevertheless, some ionic conducting oxides, like cubic stabilized zirconia, find useful application primarily in one regime, and in such cases, the Brouwer approach [4] provides appropriate expressions for the functional dependence of the concentration of defects. However, the range of application for many other materials encompasses more than one Brouwer regime. Accordingly, the dependences obtained from the Brouwer approach [4] are not usable. Porat and Tuller In recognition of the inadequacy of the Brouwer [4] approximations to sufficiently describe the concentration of defects that fall in between Brouwer regimes in MIECs, an approach to obtaining an analytical solution to the system of defect equations (described in the next section) describing the defect equilibria of an MIEC was developed in a simplified manner by Porat [8] and Porat and Riess [9]. Subsequently, Spinolo and AnselmiTamburini [10] developed the approach into a more general one. Then, Porat and Tuller [1] applied the approach to the MIECs Gd2(TiyZr1y)2O7 and CeyU1yO2 using the equation below:
1 2 PO2 Previous models Brouwer approach One of the earliest, most common, and useful approaches was developed by Brouwer [4]. In summary, the Brouwer approach [4] divides the defect equilibria into regions, called Brouwer regimes, where only a pair of oppositely charged defects (e.g., oxygen vacancies and electrons) dominate. The Brouwer approach [4] is successful in that it gives appropriate functional dependences in each limiting case, i.e., Brouwer regime. However, each Brouwer regime has a unique set of formulae for the defects. Hence, the equations are inaccurate and discontinuous near the boundary between two Brouwer regimes because there is no smooth transition from one equation to the other. For example, the Brouwer approach has been used to obtain [57] the conductivity expression: s tot s ion s e s h jzion ja1 jzion ja2 PO2 jzion ja2 PO2
1 2jzion jjaj 1 2jz jjaj ion 1 1 2 b1 b2 b2 2 4b1 b3 1
2 ! 2 1 where b1, b2, and b3 are functions of the defect equilibrium constants and defect concentrations. Equation 2 relates the external conditions (i.e., the PO2 ) to the concentration of defects in an oxide but it expresses the PO2 as a function of the defect concentration. However, description of pertinent phenomena, such as defect conductivity and flux, requires expressions for the defect concentration as a function of PO2 instead. Unfortunately, there is no readily apparent way of extracting an inverse function from Eq. 2 for defect concentration as a function of PO2 without oversimplification. Van Hassel et al. [11] also obtained expressions for the dependence of PO2 on defect concentration with a slightly different approach. where the subscripts "ion," "e," and "h" refer to ions, electrons, and holes, respectively; is conductivity; z is the charge equivalence number; a1, a2, and a3 are constants related to the defect reactions and doping levels; and F is Faraday's constant. =1 if the concentration, c, of the electronic species (ce or ch) is greater than zion times the concentration of the ionic species (cion). =0 if both ce and ch <<zioncion (i.e., in the electrolytic region). The discontinuous nature of the equations obtained from the Brouwer approach [4] is evidenced in Eq. 1 where the exponent, 1/2(jzi jae;h ), changes from one region to the Numerical methods Since the advent of powerful computing power in a desktop package, numerical modeling has become increasingly popular. In principle, the only requirements for numerical modeling are an accurate description of the system in consideration and computing power. Nevertheless, while the results of numerical modeling can yield tremendous Ionics (2007) 13:127140 129 quantities of data, it may be argued that analytical expressions give more insight and are inherently more tractable and, ultimately, more useful. Nevertheless, Poulsen (for LaMnO3 [2] and SrCeO3 [3]) and Bonanos and Poulsen (for BaCeO3 [12]) recently developed useful numerical models for defect generation and conductivity dependence on PO2 . The viabilities of numerical and analytical modeling both rely on the precision with which the modeled system is described. However, in comparison to analytical models, numerical models often yield less insight because their machinations, and nuances thereof, are often hidden in and subject to an algorithm. Accordingly, the accuracy of results produced by numerical models depends on the algorithm used to generate them. External equilibria The MIECs equilibrium with the gas phase occurs by exchange of oxygen between the crystal lattice and the gas as follows
1 O O $ VO 2e 1 O2 2 0 Kr 2 cVO c2 PO2 e 1 2 2 cOO Nc PO2 ;ref 4 where cVO is the concentration of oxygen vacancies and cOO is the concentration of oxygen ions in the oxygen sublattice2; PO2 ;ref is the reference partial pressure; and Gr is the equilibrium constant for the external equilibrium reaction. In accordance with current practice, it is assumed henceforth that PO2 ;ref 1 atm.3 Internal disorder AntiFrenkel disorder is the primary type of internal disorder found in fluorites and is given by: Defect equilibria Using KrgerVink notation, the defect equations governing fluorite and perovskite MIECs are given below. The first three equations apply generally to all MIECs, while the rest are categorized for the fluorite and the perovskite structure. Additionally, because a dilute solution is assumed, the activity of the host sublattice is assumed to be unity. All the Ks are equilibrium (mass action) constants for the adjacent reactions. In addition, the mass action expressions are normalized by the concentration of available sites for defect formation, thus allowing all equilibrium constants to be dimensionless. Intrinsic electronhole pair formation Electronhole pairs are thermally generated in MIECs and their formation is given by: null $ e0 h Ki Eg ce c h e kB T Nv Nc O $ VO Oi O 00 Kf cVO cOi c2 OO 5 where cOi is the concentration of oxygen interstitials and represents the available interstial sites, which, for the fluorite structure, is equal to cOO if we assume that oxygen ions migrate only to octahedrally coordinated interstial sites. Conversely, Schottky disorder is the dominant form of internal disorder found in perovskites, and is given by: Asite 2A 3O $ 2V A00 0 3VO A2 O3 s A O KsA c2 A c3 O V V c2 A c3 O A O 6 Bsite 2B 3O $ 2V B00 0 3VO B2 O3 s B O KsB 3 c2 B c 3 O V V c2 B c 3 O B O 7 Asite and Bsite Schottky disorders may be combined in a single equation as follows: A B 3O $ V A00 0 V B00 0 3VO ABO3 s A B O where Nv and Nc are the density of states in the valence and conduction bands, respectively; Eg is the band gap energy; kB is Boltzmann's constant; and T is temperature. Also, Nc 2 and m* kB T e Nv 2 22 !3
2 m* kB T e 22 !3 2 8 Ks cVA cVB c3 O c2 c3 c2 c3 1 V V V V V 2A 3O 2B 3O 3 cAA cBB cOO cAA cOO c BB c O O * where m* and mh are the effective masses of the electrons e and holes, respectively, and is Planck's constant. 2 External equilibria for the cation sublattice is ignored because it is typically thermodynamically unfavorable. The usual assumption is made that the concentration of oxygen atoms remains approximately constant. 3 130 Ionics (2007) 13:127140 where cV B bcBB cVA cAA =(KsB /KsA )1/2; cVA , and cVB are the Asite and Bsite cation vacancy concentrations, respectively, and cAA and cBB are the Asite and Bsite cation concentrations, respectively. The chemical activities of A2O3, B2O3, and ABO3 are all assumed to be unity. Doping Only fixedvalenceacceptor doping is considered for the fluoritestructured oxide MIECs because it predominates in their applications, and it is given by:
AO M2 O3 2 2MA VO 3O ! O
0 9 Conversely, perovskites are doped in many ways; however, we will restrict ourselves to fixedvalenceacceptor doping on the Asite: Asite
ABO MO 3 MA 2V VB O ! O O
0 000 10 Small polaron formation A polaron is a defect in an ionic crystal that is formed when excess charge at a point polarizes the lattice in its immediate vicinity [14, 15]. Polarization of the lattice results in a reduction of the energy of the system, and the carrier is then assumed to be localized in a potential energy well. When small polarons are formed, the conduction mechanism involves the "hopping" of electronic defects between adjacent ions of usually, but not necessarily, the same type and with multiple (at least two) oxidation states. This type of conduction is most often observed in oxides containing transitionmetal cations because of the ease by which their cations can change their oxidation states. For convenience, the formation of a small polaron is treated like a chemical reaction with an associated equilibrium constant. While useful, this treatment may be misleading because, as Madelung [15] points out, the formation of a small polaron is not dependent on thermal energy. Nevertheless, it is used here because it harmonizes with the treatment of the other defect equations. The correct treatment of small polaron formation is discussed by Duncan and Wachsman [13]; however, at present, the approach used below suffices. The formation of small polarons consisting of electrons localized on a cation is given by: MM e 0 ! MM
0 and isovalent doping on the Bsite, which is done in synthesis with a stoichiometric addition of A2O3, Bsite
2ABO M2 O3 A2 O3 3 2A 2MB 6O ! A O Kspe cspe c1 Nc c1 e M 12 11 If the acceptordopant is fully ionized, then its concentration, cD, is constant. Typical dopant concentrations and unit cell volumes for common MIECs are given in Table 1. Table 1 Dopant concentrations and unit cell volumes for common oxide MIECs Composition Structure cD Unit cell volume (3) (1027 m3) where MM is a multivalent metal (dopant or host) ion in its ' normal oxidation state, MM is an "ntype" small polaron (i.e., an electron trapped on a cation), cspe is the concentration of ntype small polarons, and cM* is the concentration of metal ions in their normal oxidation state (i.e., MM ). In the interest of brevity, we will restrict our discussion of small polaron formation in perovskites to the Bsite (i.e., for perovskites, one would replace the subscripts M and M* with BB and B , respectively). B The formation of small polarons consisting of holes localized on a cation is given by: MM h ! MM Fluorites Ce0.8Sm0.2O1.9 Zr0.8Y0.1O1.95 Bi0.8Er0.2O1.5 Perovskites La0.6Sr0.4Co0.2Fe0.8O3 Pr0.7Sr0.3Co0.2Mn0.8O3
a b Ksph csph c1 Nv c1 h M 13 Cubic Cubic Cubic 161 136b 181c a 5.0 4.7 22d 1.0 1.4 Rhombohedral 349e Orthorhombic 229f where is a small polaron consisting of a hole trapped on a cation, i.e., "ptype" small polaron, and csph is the concentration of ptype small polarons. Finally, a more general expression for small polaron formation may be obtained by summing Eqs. 12 and 13 with the result: 2MM e0 h ! MM MM
0 MM Calculated from lattice parameter data in Inaba and Tagawa [17]. Calculated from lattice parameter data in Fonseca and Muccillo [18]. c Calculated from lattice parameter data in Shuk et al. [26]. d Based on the concept of the Bi3+ cation as a dopant in the fluorite (cation sublattice) structure, which normally consists of cations with a 4+ charge (see Wachsman and Duncan [27]). e Calculated from lattice parameter data in Kostogloudis and Ftikos [28]. f Calculated from lattice parameter data in Kostogloudis et al. [29]. Ksp Kspe Ksph 14 csph cspe c1 c1 Nc Nv c2 csph cspe c2 Ki1 e h M M Because some of the cations participate in forming small polarons, they are consequently distributed between two Ionics (2007) 13:127140 131 valence states. Hence, a mass balance equation, below, must be invoked to account for conservation of the number of cations (i.e., to keep track of the cations); cM cspe csph cM 15 A summary of all the defect equations is given in Table 2 for easy reference. Analytical model for the functional dependence of defect concentration on PO2 Having summarized the defect equations for fluorite and perovskite MIECs, they may now be used to obtain the functional dependence of defect concentrations on PO2 . However, as we discussed earlier, the traditional method of solving the system of defect equations and the charge balance equation yields analytically unsolvable polynomial equations, hence the need to model the system instead. Furthermore, the most popular method used to simplify the system of equations, the Brouwer approach [4], produces results that, while accurate in each Brouwer regime, are discontinuous going from one Brouwer regime to another. This often renders equations derived from the Brouwer approach inapplicable when determining material properties of many MIECs of practical interest, such as conductivity and flux. The Brouwer approach [4] simplifies the charge balance equations Eqs. 18 and 19 to dominant defect pairs (one negative and one positive) for each Brouwer regime. In the model we will develop, the charge balance equations are simplified to dominant defect triads instead. As we will show, this is equivalent to modeling two Brouwer regimes at a time instead of one. This offers significant advantages because most MIECs of interest find application and/or are stable in no more than two Brouwer regimes. Moreover, the problem of discontinuity between regimes is avoided. In this section, models are developed for the functional dependence of defect concentration on PO2 in three regions. The expressions obtained thereby are subsequently combined to yield a general expression for each defect that spans the entire PO2 range. where cM is the total concentration of cations. Furthermore, given Kspe =Nc/cM and Ksph =Nv /cMthe formation of small polarons is electrostatic in nature [13, 14, 15] and, therefore, Kspe and Ksph are not true (i.e., functions of temperature only) equilibrium constantsthen for ce >>ch, Eq. 12 or 14 yields: cspe >> csph cspe ce =1 ce =cM 16 and for ce <<ch, Eq. 13 or 14 yields: csph ch =1 ch =cM cspe << csph 17 Charge balance MIECs are assumed to be electrically neutral. Accordingly, a charge balance equation may be written for each MIEC family as follows: Fluorites ce cD 2cOi 2cVO ch Perovskites ce cD 3cVA 3cVB 2cVO ch or, from Eq. 8 ce cA 31 cBB =cAA cVA 2cVO ch 20 19 18 Table 2 Summary of defect equations for fluorites and perovskites General Electron hole Ext. equilibria Small polarons Mass balance Int. equilibria Charge balance
1 1 Ki ce ch Nc Nv 1 2 c VO c 2 P O e 2 Kr 1 2 P2 cOO Nc O ;ref 2 Ksp csph cspe c2 Ki1 M Fluorites Perovskites cM cspe csph cM Kf cVO cOi c2 OO ce cD 2cOi 2cVO ch
V Ks cAA cBB c3 O A B OO cV cV c3 ce cD 3cVA 3cVB 2cVO ch 132 Ionics (2007) 13:127140 Low PO2 We will begin with what we call the lowlow PO2 region, which encompasses two Brouwer regimes defined by the dominant defect pairs: ce, cVO and cD, cVO , i.e., Fluorites cOi << ce ; cD and ch << cVO Equations 4 and 21 show (in accordance with the Brouwer approach [4] for that region) that, in the low PO2 region, as PO2 0, 2cVO ce and cVO ; ce >> cA Similarly, Eqs. 4 and 21 show that as PO2 2cVO cD and cVO ; cD >> ce where "PO2 " corresponds to PO2 approaching the upper limit of the low PO2 region. The lower limit (boundary) for cVO may be incorporated as a boundary condition for Eq. 24. Thus, Eq. 24 may now be solved by integration Z PO Z cV 1 1 1 5 2 O 2 2 Kr2 cOO Nc PO24 dPO2 8 cVO dcVO
1 1 2 cD Perovskites cVA ; cVB << ce ; cD and ch << cVO Therefore, in this region, the charge balances for fluorites Eq. 18 and perovskites Eq. 19 both reduce to: ce cA 2cVO Combining Eqs. 4 and 21 yields (PO2 ;ref 1 atm):
1 1 1 1 21 with the result cVO PO2 1 1 1 3 2 2 Kr Nc cOO PO24 4 1 2 cD 3
2 !2
3 25 2 2 Kr2 cOO Nc cVO PO24 cD 2cVO 22 Combining Eq. 25 with Eqs. 36 yields: ! 1
3 If we assume that cOO >>cVO , cOi (i.e., the system is dilute with respect to ionic defects) and, therefore, cOO remains approximately constant with respect to cVO and PO2 ; then, differentiating Eq. 22 with respect to cVO yields: dPO2 =dcVO 1 1 1 5 2 1 2 4 8Kr 2 cOO Nc cVO PO2 ce PO2 1 1 1 1 1 1 2 2 Kr2 Nc cOO PO24 3 Kr2 Nc cOO PO24 4 1 2 cD 3
2 26
1 1 1 1 1 1 2 4 2 Kr 2 Ki Nv cOO PO2 3 Kr2 Nc cOO PO24 4 ch PO2 2PO2 c1 VO 23 1 2 cD 3
2 !1
3 27 Fluorites cOi PO2 1 1 1 2 Kf c2 O 3 Kr2 Nc cOO PO24 O 4 2 4 For PO2 >> PO2 , where PO2 44 Kr c2 O Nc c6 , Eq. 23 O VO reduces to: 1 2 cD 3
2 ! 2
3 28 dPO2 =dcVO % 1 1 1 5 2 1 2 4 8Kr 2 cOO Nc cVO PO2 24 Perovskites cVA PO2 cA A cV PO2 bcBB B
1 1 3 1 1 1 4 3 2 2 4 Kr Nc cOO PO2 In the low PO2 region, 2cVO cD and cD 1027 m3. However, because cVO and PO2 are both variables, the validity of the simplification may be assessed for a given cVO and PO2 . PO2 is the lowest PO2 , for which the approximation is valid and is dependent on the material parameters of the MIEC. For example, for Ce0.8Sm0.2O1.9, a conservative value of PO2 =1.61023 atm may be obtained using cVO % 0:5cD 2:5 1027 m3 and KrcOO 2 Nc 1072 m9 [13]. In practice, PO2 should be even lower because cVO >0.5cD at that point. 2 b 2 Ks2 cAA cOO 1 2 cD 3
2 !1 29 Intermediate PO2 The next region is the intermediate PO2 region, which encompasses two Brouwer regimes defined by the dominant defect pairs cD, cVO and cD, ch; i.e., Ionics (2007) 13:127140 133 Fluorites ce ; cOi << cD Perovskites ce ; cVA ; cVB << cD Hence, in this region, the charge balances for fluorites Eq. 18 and perovskites Eq. 19 both reduce to: cD 2cVO ch 30 cVA;B (i.e., the total concentration of cation vacancies cVA cVB ), ch, for perovskites, i.e., Fluorites cOi ;cD >> ce and ch >> cVO Perovskites cVA ;cVB ;cD >> ce and ch >> cVO Therefore, in this region, the charge balance for fluorites Eq. 18 reduces to cD 2cOi ch 37 Combining Eqs. 4 and 30 yields the quadratic equation:
2 2 2c2 Kr Ki2 Nv cD ch c1 PO2 h OO 1 31 whose sole meaningful solution is
ch PO2 1 1 1 2 2 1 4 4 Kr Ki Nv cOO PO2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8Kr Ki2 Nv2 cOO cD 1 2 PO2 1 4 PO2 ! while the charge balance for perovskites Eq. 19 reduces to: cD 31 cBB =cAA cVA ch 38 32 Furthermore, combining Eq. 32 with Eqs. 36 yields:
ce PO2 4Kr Ki1 Nc Nv1 cOO PO24 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 2 4 8Kr Ki2 Nv2 cOO cD PO2 PO2 1 !1 For fluorites, we combine Eqs. 4, 5, and 37 with the result:
1 2 2 cD 2Kr Ki2 Kf Nv cOO PO2 c2 ch h 1 33
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cVO PO2 1 1 2 2 1 16 Kr Ki Nv cOO 1 39 2 4 2 8Kr Ki2 Nv cOO cD PO2 PO2 1 !2 If we assume that cOO ) cvO ; cOi (i.e., the system is dilute with respect to ionic defects) and, therefore, cOO remains approximately constant with respect to ch and PO2 , then differentiating Eq. 39 with respect to ch yields:
2 2 dPO2 =dch Kr Ki2 Kf1 Nv c1 PO2 c2 4PO2 c1 OO h h 1 34 Fluorites
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cOi PO2 2 16Kr Ki2 Kf Nv c3 O O 40 8Kr Ki2 Nv2 cOO cD 1 2 PO2 1 4 PO2 !2 1 2 4 For PO2 << PO2 , where PO2 16Kr Ki4 Kf2 Nv c6 c2 , h OO Eq. 40 reduces to: 2 2 dPO2 =dch % Kr Ki2 Kf1 Nv c1 PO2 c2 OO h 1 35 Perovskites cVA PO2 cAA cV PO2 cBB B
3 1 64Kr2 Ks2 c3 O cAA O 1 3 3 2 Ki Nv 41 36 !3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 8Kr Ki2 Nv cOO cD 1 2 PO2 1 4 PO2 High PO2 The final region modeled is the high PO2 region, which encompasses two Brouwer regimes defined by the dominant defect pairs: cD, ch and cOi , ch, for fluorites; cD, ch and In the high PO2 region, ch cD and cD 1027 m3. However, because ch and PO2 are both variables, the validity of the simplification may be assessed for a given c h and PO2 . PO2 is the highest PO2 for which the approximation is valid and is dependent on the material parameters of the MIEC. For example, for Ce0.8Sm0.2O1.9, a conservative value of PO2 % 10121 atm may be 2 27 3 obtained using ch % cD 5 10 m , K r cOO Nc % 1072 m9 , 26 6 2 42 6 KiNcNv 10 m and Kf cOO 10 m [13, 16]. In practice PO2 should be even higher because ch >cD at that point. 134 Ionics (2007) 13:127140 Equations 4, 37, and 39 show (in accordance with the Brouwer approach [4] for that region) that, in the high PO2 region, as PO2 ! 0 cD ch and cD ;ch >> cOi where "PO2 ! 0" corresponds to PO2 approaching the lower limit of the high PO2 region. Similarly, Eqs. 4, 37, and 39 show that, as PO2 ! 1 2cOi ch and cOi ;ch >> cA The lower limit (boundary) for ch may be incorporated as a boundary condition for Eq. 41. Thus, Eq. 41 may now be solved by integration remains approximately constant with respect to ch and PO2 , then differentiating Eq. 46 with respect to ch yields:
3 4 dPO2 =dch 4 cAA cBB 1 2 Kr2 Ki3 Ks 2 Nv PO2 c3 4PO2 c1 h h 9 1 3 1 1 47
2 4 F o r PO2 << 93 cAA cBB 3 Kr Ki4 Ks 3 Nv ch3 , Eq. 47 reduces to: 4 4 3 2 2 16 1 1 1 3 dPO2 3 4 % 4 cAA bcBB 1 b 2 Kr2 Ki3 Ks 2 Nv PO2 c3 h 9 dch 48 Z PO2 0 2 PO22 dPO2 Kr Ki2 Kf1 Nv c1 OO 1 Z ch cD c2 dch h with the result ch PO2 1 1 2 2 6Kr Ki2 Kf Nv cOO PO2 !1 c3 D
3 In the high PO2 region, ch cD and cD 1027 m3. However, because ch and PO2 are both variables, the validity of the simplification may be assessed for a given ch and PO2 . Equations 4, 38, and 46 show (in accordance with the Brouwer approach [4] for that region) that, in the high PO2 region, as PO2 ! 0, cD ch and cD ;ch >> cVA ; cVB 42 Combining Eq. 42 with Eqs. 35 yields:
1 2 ce PO2 Ki Nc Nv 6Kr Ki2 Kf Nv cOO P 2 c3 D 1 ! 1
3 43 where "PO2 ! 0 corresponds to PO2 approaching the lower limit of the high PO2 region. Similarly, Eqs. 4 and 39 show that, as PO2 ! 1, 3cVA 3cVB ch and 3cVA ; 3cVB ; ch >> cA 2 1 2 2 cVO PO2 Kr Ki2 Nv cOO PO22 6Kr Ki2 Kf Nv cOO PO2 c3 D 1 1 !2
3 44 The lower limit (boundary) for ch may be incorporated as a boundary condition for Eq. 41. Thus, Eq. 41 may now be solved by integration cOi PO2 1 1 2 2 Kr Ki2 Kf Nv cOO PO2 1 1 2 2 6Kr Ki2 Kf Nv cOO PO2 ! 2 c3 D
3 45 cAA cBB 1 2 Z PO2 0 PO24 dPO2 dch 1 Z ch 3 1 4 2 3 2 3 c3 h 9 Kr Ki Ks Nv cD To obtain an analogous equation set for perovskites we combine Eqs. 4, 5, and 38 to obtain:
3 4 cD 3cAA bcBB b 2 Kr 2 Ki3 Ks2 Nv PO2 c3 ch h 1 3 1 3 with the result !1 c4 D
4 46 ch PO2 12cAA bcBB b 1 2 If we assume that cM >>cVO , cVA , cVB (i.e., the system is dilute with respect to ionic defects) and, therefore, cM 3 1 3 3 4 Kr 2 Ki3 Ks2 Nv PO2 49 Ionics (2007) 13:127140 Table 3 Comparison of Brouwer approach and the approach developed in this work
Region I ce, cVO (ce = 2cVO) Brouwer approach Region IIa Region IIb cA, cVO cA, ch (cD = ch) (cD = 2cV )
O 135 Dominant defect pairs (Charge balance) Region III Fluorites: cOi, ch (2cOi = ch) Perovskites: cVA,B, ch (3cVA,B = ch) New approach LowPO2 IntermediatePO2 HighPO2 ce, cD, cVO (ce + cD = 2cVO) Dominat defect triads (Charge balance) cA, cVO ch (cD = 2cVO + ch) Fluorites: cD, cOi , ch (cD + 2cOi = ch) Perovskites: cD, cVA,B, ch (cD + 3cVA + 3cVB = ch) Combining Eq. 49 with Eqs. 3, 4, and 6 yields:
ce PO2 Ki Nc Nv 12cAA cBB 1 2 3 1 3 3 4 Kr 2 Ki3 Ks2 Nv PO2 ! 1 c4 D
4 50
2 cVO PO2 Kr Ki2 Nv cOO PO22 1 3 1 3 4 Kr 2 Ki3 Ks2 Nv3 PO2 12cAA cBB 1 2 !1 c4 D
2 51
1 3 cAA K 3 Ks2 Nv cAA 3 4 cVA PO2 cVB PO2 i PO2 1 3 cBB 2 2 Kr 12cAA cBB 1 2 3 1 3 3 4 Kr 2 Ki3 Ks2 Nv PO2 ! 3 c4 D
4 52 The features of our approach, presented above, and the Brouwer approach [4] are compared and summarized in Table 3. Generalized defect equations We will now generalize the equations for the functional dependence of defect concentration on PO2 i.e., Eqs. 25 to 29, 32 to 36, 42 to 45, and 49 to 52 over the entire PO2 range, i.e., remove the constraints of low, intermediate, and high PO2 . In contrast to the Brouwer approach, such generalization is possible because of the overlap between the low, intermediate, and high PO2 regions defined above (and synopsized in Table 3). A similar strategy may not be used with Brouwerderived equations, however, because they lack this feature. The overlapping of the low, intermediate, and high PO2 regions also means that each defect concentration equation will converge to identical values at the boundaries4 between those regions. Consequently, to avoid duplication, the concentration value at the point of convergence must be subtracted from the total. As an example, to get cVO (P) over the complete PO2 range requires the summation of Eqs. 25, 34, and 44 for fluorites (or Eqs. 25, 34, and 51 for perovskites). However, Eqs. 25 and 44 both converge to cD/2 at their upper and lower limits, respectively (both limits reside in region IIa; see Table 3). Therefore, to avoid duplication, cD/2 must be subtracted from the sum. Similarly, because Eqs. 34 and 441 (or Eq. 51 for perovskites) both converge to Kr Ki2 Nv2 cO c2 PO 2 at their upper and lower limits, respecD tively (both limits reside in region IIb; see Table 3), 1 2 Kr Ki2 Nv cO c2 PO 2 must also be subtracted to avoid duplication. D Hence, by applying this strategy, cVO (P) may be generalized with the result
O 2
O 2 cVO PO2 1 4 3 4 PO2 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 81cOO cD 1 1 2 PO2 2 cD 3
2 !2
3 1 1 1 16 1 cOO 1 4 PO2 !2 53 2 1cOO PO22 611 Kf cOO PO2 c3 D 1 !2
3 1 cD 1cOO c2 PO22 D 2
4 As defined, the boundaries between any two regions are smeared over the range of PO2 s common to both. 1 136 Ionics (2007) 13:127140 for fluorites and cVO PO2 1 4 3 4 PO2 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 81cOO cD 3 1 2 PO2 2 cD !2 3
3 2 1 4 PO2 1 Perovskites: ce PO2 54 !1 2 4 cD 1 1 PO24 3 PO24 4 1 1 1 2 41 2 cOO PO24 !2 1 2 cD 3
2 ! 1
3 1 1 1 16 1 cOO rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 81cOO cD 3 2 1 2 PO2 1 4 PO2 !1 1 1cOO PO 2
2 4 121 2 cAA cBB 2 Ks2 PO2 1 3 1 cD 1cOO c2 PO22 D 2
2 and 1 Kr Ki2 NV . for perovskites, where a The above strategy may likewise be applied to Eqs. 26, 33, 43, and 50 for ce(P); Eqs. 27, 32, 42, and 49 for ch(P); Eqs. 28, 35, and 45 for cOi (P); and Eqs. 29, 36, and 52 for cVA (P) (or cVB (P)) to generalize for other defects, the functional dependence of defect concentration on PO2 over the entire PO2 range with the results: Fluorites: ! 1 1 1 1 3 3 4 3 4 ce PO2 PO2 4 PO2 2 cD 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !1
2 41 2 cOO PO24 1 1 2 1 2 cOO 1 1 1 2 4 81cOO cD PO2 PO2 1 2 K f cO O P O 2 1 1 1 1 1 2 1 2 cO O 121 cAA cBB 1 2 1 3 4 Ks2 PO2 ! 1 c4 D
4 1 1 2 Kr2 Nc cOO 1 1 1 1 2 1 cD 2 PO24 1 2 cOO c1 D 2 58 ch PO2 1 1 1 1 2 4 1 2 cOO PO2 3 PO24 4 1 1 1 1 4 4 1 cOO PO2 1 2 cD 3
2 !1
3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 81cOO cD 1 2 1 2 PO2 1 4 PO2 ! 121 3 2 cAA cBB 1 1 1 1 3 4 Ks2 PO2 !1 c4 D
4 2 4 1 2 cOO PO2 1 2 cD 1
2 cD 59 61 1 ! 1 c3 D
3 1 1 1 1 2 1 cD 2 PO24 1 2 cOO c1 D 2 55
ch PO2 1 1 1 1 2 4 1 2 cOO PO2 3 PO24 4 1 cA cV PO2 Ks2 cAA cVA PO2 A B 1 cBB 2
8 2 > 3 > > > c2 4 > OO > <
3 1 4 4PO 1 2 cD 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 1 3 !1
3 cD 3
2 31 25 3 P4 " 4 1 11 c1 PO2 OO 4 1 2 611 Kf cOO PO2 2 4 81cOO cD PO2 PO2 1 ! > > > 64c3 > > OO > : 3 1 2 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 81cOO cD 1 2 PO2 3 12 O2 12cAA cBB 1 3 1 2 1 2 Ks 2 1 4 PO2 3 4 PO2 # 3 9 4 > > > > > c4 > D = 3
2 !3 2cOO cD !1 c3 D
3 56 3 4 PO 2 3 1 2 c3 D > > > > > > ; 60 Finally, rigorous analysis of the defect equations i.e., Eqs. 25 to 29, 32 to 36, 42 to 45, 49 to 52, and 53 to 60 show that ci(PO2 )0 (i=VO, e, h, Oi, VA, VB) for all allowed 1 values of PO2 (44 Kr2 c2 Nc4 c6 << PO << 16Kr2 Ki4 Kf2 Nv4 c6 c2 for O V h O 4 4 3 3 fluorites and 44Kr2c2 Nc4c6 << PO << 93cA cB 3 2 Kr2 Ki4Ks2Nv4c16 for 3 O V h perovskites).
O O 2 O O O 2 A B 1 1 1 1 2 4 1 2 cOO 1 cD 2 PO2 2 cD cOi PO2 Kf c2 O O 1 3 PO24 4 2 cD 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 81cOO cD 1 1 2 PO2 1 3 ! 2
3 1 4 PO2 !2 161Kf c3 O O Discussion ! 2
3 2 2 11 Kf cOO PO2 611 Kf cOO PO2 c3 D 1 2 2Kf c2 O c1 Kf 11 cOO c2 PO2 O D D 1 57 In this section, the expressions derived above for the functional dependence of defect concentration on PO2 are examined. Plots of defect concentration vs PO2 defect equilibrium diagrams (DEDs) are generated from Eqs. 53 to 60 and are compared with the predictions of the Brouwer approach [4]. Ionics (2007) 13:127140 137 Figure 1 shows a comparison of our model with the Brouwer approach and the method of Porat and Tuller for a fluorite MIEC in the low PO2 (Fig. 1a) and high PO2 (Fig. 1b) regions. IntermediatePO2 is omitted because no additional assumptions (beyond that required by its definition) were made for that region. As mentioned earlier, the method of Porat and Tuller gives an accurate solution to the defect equations for PO2 as a function of defect concentration. However, for most applications, it is desirable to have defect concentration as a function of PO2 . Nevertheless, for the purpose of comparison, it provides a useful gauge to ascertain the accuracy of the model against the traditional Brouwer approach. As expected, Fig. 1a shows that the model is more accurate than the Brouwer approach except in the limit that ce >>cD (i.e., when ce 2cVO ). Similarly, Fig. 1b shows the model is again more accurate than the Brouwer approach except in the limit that cOi ) cD (i.e., when 2cOi % ch ). More importantly, however, Fig. 1 shows that the model is continuous over the entire defect equilibria from low to high PO2 . This is significant because many MIEC properties e.g., oxygen flux are often calculated using Brouwerderived expressions, although the oxygen potential gradient across the MIEC places it in more than one Brouwer region. The PO2 range covered by a Brouwer region is dependent on a combination of equilibrium constants and cation, anion, and dopant concentrations [16]. Hence, the regions depicted in Fig. 1 could, theoretically, occur in any PO2 range. However, our earlier studies [13], as well as that of Poulsen [2], suggest that typical oxide MIECs (especially those with an appreciable acceptor dopant concentration, >10%) will not penetrate very far into regions I or III. One reason for this is that many oxide MIECs cannot stably support large concentrations of ionic defects and they decompose instead. Another explanation, related to the first, is that the structure of the MIEC is unstable with respect to the cation charge (changed due to small polaron formation) in regions I or III, precipitating a phase change. Moreover, as defect concentration increases, the probability of associated defects becomes nonnegligible and the defect equilibria must be modified accordingly; see Duncan and Wachsman [13]. The discontinuity in equations derived form the Brouwer approach [4] arises from the reduction of the charge balance expressions Eqs. 18 and 19 to a pair of defects that dominate in each Brouwer region. This strategy is successful over some of a given region but becomes increasingly inaccurate as the boundary between two regions is neared. The failure is due to the presence of nonnegligible concentrations of a third defect species in the neighborhood of each boundarysee Figs. 2, 3, and 4. Conversely, the model presented here is successful because, for each Fig. 1 Comparison of a oxygen vacancy b electron hole concentration dependence on PO2 as predicted by our model (solid lines), the Brouwer approach [4, 16] (broken lines), and the method of Porat and Tuller [1] (circles) for a fluoritestructured MIEC with Kr =2.11015, Ki =109, and Kf =106 Fig. 2 Defect concentration dependence on PO2 for a typical fluorite MIEC (at 800 C, Kr =1014, Ki =108, Kf =106, cM =21028 m3, cOO =4.751028 m3, cD =51027 m3, and Nc =Nv =1029 m3) 138 Ionics (2007) 13:127140 Fig. 3 Defect concentration dependence on PO2 for a typical perovskite MIEC (at 800 C, Kr =1015, Ki =106, Ks =1028, cM =41028 m3, cOO 5:951027 m3, cD =1027 m3, Nc =Nv =1029 m3, =0.01, and cAA cBB ) modeled region (i.e., effectively two Brouwer regions), the charge balance expression Eq. 18 or Eq. 19 is reduced to a triad, rather than a pair, of dominant defects; see Table 3. Consequently, the resulting expressions are continuous across the Brouwer regions contained therein. The model developed herein is therefore quite useful because, in many practical applications, MIECs operate in one to three regions. For example, for a typical PO2 range, cubic stabilized zirconia (1022 atm<PO2 <1 atm) and cubic stabilized bismuth oxide5 (1013 atm<PO2 <1 atm) operate exclusively in region IIa. Conversely, in that same PO2 range, acceptordoped ceria typically operates in Brouwer regions I and IIa. Furthermore, La1x Srx Co1y Fey O3 6 may be subjected to PO2 gradients that span two to three Brouwer regions (IIa, IIb, or III) for oxygen separation or partial oxidation of natural gas applicationswhich serves to emphasize the need for continuous equations to describe the dependence of defect concentration on PO2 . In summary, the Brouwer approach [4] gives suitable results for the functional dependence of defect concentration on PO2 if the MIEC operates in only one Brouwer regime. However, if the MIEC operates in more than one Brouwer regime, the functional dependence of defect concentration on PO2 is better described by the expressions developed in this work. Figure 2 shows the DED of a typical fluorite and Figs. 3 and 4 show DEDs of typical perovskite MIECsfor = 0.01 (i.e., Asite Schottky defects dominate) and =100 (i.e., Bsite Schottky defects dominate), respectively. The DED in Fig. 2 was plotted using Eqs. 53 and 5557. Likewise, the DEDs in Figs. 3 and 4 were plotted using Eqs. 54 and 5861. The continuity of the defect concentration expressions is readily observed in all three figures. The plots are usually divided into four Brouwer regions as shown (see also Table 3). In region I, oxygen vacancies and electrons dominate. In region III, either oxygen interstitials and holes (fluorites) or cation vacancies and holes (perovskites) dominate. Between these two regions is region II, where defect species that are enhanced by the acceptor dopant dominate. This region may further be subdivided into region IIa where vacancies dominate and region IIb where holes dominate. If the MIEC were donordoped, region IIb would be dominated by oxygen interstitials (fluorites) or cation vacancies (perovskites) and region IIa by electrons. The width (in terms of PO2 ) of region IIa called the electrolytic region is critical to the performance of solid oxide fuel cells and electrocatalytic reactors. For these applications it is desirable that region IIa be as wide as possible so that the MIEC is predominantly an ionic conductor (i.e., an electrolyte). Conversely, MIECs used in membrane or electrode applications, where high electronic conductivity is required, often have region IIb (sometimes called the semiconducting region) as the wider region. Of the MIECs of interest, those with the fluorite structure typically have a wider region IIa than the perovskites, which typically have a wider Region IIb (one notable exception being LaGaO3). This circumstance is
5 6 Fig. 4 Defect concentration dependence on PO2 for a typical perovskite MIEC (at 800 C, Kr =1015, Ki =106, Ks =1028, cM =41028 m3, cOO 5:95 1028 m3, cD =1027 m3, Nc =Nv =1029 m3, =100, and cAA cBB ) Generally, bismuth oxide decomposes at PO2 <$ 1013 atm at 600 C. Generally, La1x Srx Co1y Fey O3 decomposes at PO2 <$ 1014 atm. Ionics (2007) 13:127140 139 attributable, in part, to the more complex structure of perovskites, the presence of (multivalent) transition metal cations, and their higher metaltooxygen ratio. Figures 3 and 4 also show the effect that has on the DED of the modeled perovskite MIEC (all other material constants are the same in both plots). A comparison of the figures and analysis of Eqs. 6, 19, and 60 shows that >1 favors cations vacancies on the Asite and <1 favors cations vacancies on the Bsite. Figure 5 compares the total concentration of electronic defects from Eqs. 58 and 59 with the concentration of small polarons in a typical perovskite MIEC (obtained by substituting Eqs. 16 and 17 into Eqs. 58 and 59, respectively). The plots show that the small polaron concentration is limited by the number of available cation sites. This is an important feature because, very often, the small polaron "hopping" mechanism for electronic conduction is assumed to be always applicable. In other words, it is assumed that ce is always equivalent to cspe. This may not always be true, however, and the effect of nonlocalized electrons (if any) on the transport properties of MIECs should be investigated. Finally, many acceptordoped perovskite MIECs of interest become unstable when cspe > csph [17, 18], and the intersection of the cspe and csph curves may be viewed as a stability boundary for such MIECs. To demonstrate the efficacy of the approach developed above, we applied the model to conductivity data for 10mol% fluoritestructured, gadoliniadoped ceria (GDC), as shown in Fig. 6. The PO2 range over which the data were acquired is limited to the low PO2 region (Brouwer regions I and IIa); hence, it is sufficient to use the equations from that section alone because the complete defect concentration expressions would not contribute to the analysis. Because the dominant mobile defects are electrons Fig. 6 Dependence of electrical conductivity on PO2 for GDC; circles, data from Wang et al. [19]; line, model and oxygen vacancies, then (incorporating Eqs. 25 and 26) the electrical conductivity is given by: s tot zv quVO cVO ze que ce zV quV
1 1 3 2 4 k r PO2 4 1 2 cD 3
2 !2
3 1 1 1 1 2 2 ze que kr PO24 3 k r PO24 4 1 2 cD 3
2 ! 1
3 61 Fig. 5 Dependence of small polaron concentration on PO2 for a typical perovskite MIEC (at 800 C, Kr =1015, Ki =106, Ks =1028, cM =4 1028 m3, cOO =5.951028 m3, cD =1027 m3, Nc =Nv =1029 m3, =10, and cAA cBB ) where z is the charge on the point defect, q is the elementary electronic charge and u is electrical mobility, 2 and k Kr cOO Nc . We hasten to point out that we are certainly not the first to derive conductivity dependence on PO2 for oxides. However, as discussed earlier, previous models rely on relationships derived from the Brouwer approach, which are discontinuous across Brouwer regimes as demonstrated in Eq. 1. The novelty of the model developed herein is that it provides continuous equations for the point defect concentration on PO2 in the region of interest, which, for GDC, spans two Brouwer regimes. Accordingly, physical properties modeled using this relationship will also be continuous in the same range. Figure 6 shows a plot of Eq. 61 fitted to conductivity data from Wang et al. [19] for GDC. The excellent fit to the conductivity data demonstrates the accuracy of the model in predicting the effect of point defect concentration on electrical conductivity. The good fit was obtained using =1.31081 atm1/2 m9 and uv 1:3108 V1 m2 s1 as reported by Steele [20]. The electron mobility, ue, which was adjusted to obtain the best fit to the data, is 1.45 103 V1 cm2 s1. This value for ue is a bit lower than other reported values [20, 21], and this is most likely because 140 Ionics (2007) 13:127140 here GDC was not constricted to a single Brouwer regime. It is easy to show (Eq. 1) that limiting GDC to the electrolytic1 regime, 1i.e., Brouwer region IIa, (where 1 1 2 e 22 ze qca 2 Kr ue PO24 ) will yield higher ue values. Conversely, limiting GDC to the highly reducing regime, i.e., 1 1 1 Brouwer region I, (where e 23 ze qKr3 ue PO26 ) will give lower ue values for any given e. Typically, the former is assumed in the literature [20, 21], thereby yielding misleading results. In closing, the model has also been applied to the thermogravimetric, thermomechanical, and thermochemical properties of GDC and other fluorites [21, 2325]. Conclusion In this work, we have presented, with minimal assumptions, a model that describes, through analytically derived formulae, the functional dependence of defect concentration on PO2 in oxide MIECs with fluorite and perovskite structures. We were able to confirm the model's completeness through its ability to fit the numerically obtained DED. Unlike expressions derived from the Brouwer approach [4, 16], expressions produced by our model are continuous for the entire defect equilibria from low to high PO2 . To achieve this continuity, the model sacrifices some accuracy at (high and low) extreme PO2 values (see Fig. 1), where it deviates from numerical solutions. However, we anticipate that this is not a problem for most current practical applications of oxide MIECs because they are often not used in such extreme situations. We demonstrated the efficacy of the model by applying it to the electrical conductivity (which is a material property that is controlled by point defect concentration) of GDC. In closing, we would like to point out that, because of the general nature of the defect equilibria for oxide MIECs, the model may be applied to oxide MIECs beyond the categories of fluorites and perovskites as defined above.
Acknowledgements We wish to acknowledge the support of the US Department of Energy (contract # DEAC0576RL01830). References
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