Calculation of defect chemistry using the CALPHAD approach

Calculation of defect chemistry using the CALPHAD approach

Computer Coupling of Phase Diagrams and Thermochemistry 30 (2006) 33–41 www.elsevier.com/locate/calphad Calculation of defect chemistry using the CAL...

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Computer Coupling of Phase Diagrams and Thermochemistry 30 (2006) 33–41 www.elsevier.com/locate/calphad

Calculation of defect chemistry using the CALPHAD approach A. Nicholas Grundy ∗ , E. Povoden, T. Ivas, Ludwig J. Gauckler ETH Zurich, Department of Materials, Institute of Nonmetallic Materials, Swiss Federal Institute of Technology, Wolfgang-Pauli-Strasse 10, CH-8093 Zurich, Switzerland Received 4 October 2005; received in revised form 18 November 2005; accepted 23 November 2005 Available online 19 December 2005

Abstract The defect chemistry of the perovskite phase LaMnO3±d is modeled using the compound energy formalism and an associate model. In both cases the CALPHAD methodology is applied meaning that all thermodynamic and phase diagram data of the phase is simultaneously and consistently reproduced. The differences between the two modeling methods are discussed and the descriptions are submitted to a defect chemistry analysis. It is shown that the compound energy formalism is able to perfectly describe the defect chemistry of the perovskite phase whereas the associate model fails to correctly reproduce it. When using the associate model the choice of which associates to use plays a crucial role on how well the system can be approximated. As the associates are not physically meaningful entities this choice must be made arbitrarily. In the case of the compound energy formalism on the other hand a more physically realistic description of the system is achieved and fewer optimizing parameters are required. The reason for this is that the model description of the phase within the compound energy formalism is unambiguously constructed based on measured physical properties of the phase. The advantage of the associate model is that the model description is simple compared to the rather cumbersome expression obtained for the compound energy formalism. c 2005 Elsevier Ltd. All rights reserved. 

1. Introduction The LaMnO3±d perovskite is a very good candidate to compare different CALPHAD models to describe defect chemistry as it displays complex defect chemistry behaviour with both oxygen excess and oxygen deficiency, large defect concentrations and also a lot of experiments exist to validate the modeling results. It is largely due to the defects that doped LaMnO3±d has so many interesting properties such as giant magnetoresistivity [1] and is currently the most widely used material for solid oxide fuel cell cathodes [2]. In turn, it is due to these manifold properties and applications that so many experiments have been conducted on this material. The experimental data on the undoped LaMnO3 and (La, Sr)MnO3 perovskite are assessed in detail in two previous papers [3,4]. The observed defect chemistry is due to the fact that the Mn ion on the B-site can have the valence states 2+, 3+ and 4+ in the perovskite phase. Under reducing conditions an oxygen deficient perovskite, LaMnO3−d, is obtained by the reduction of Mn3+ to Mn2+ . Oxygen is liberated and oxygen ∗ Corresponding author. Tel.: +41 1 632 6431; fax: +41 1 632 1132.

E-mail address: [email protected] (A.N. Grundy). c 2005 Elsevier Ltd. All rights reserved. 0364-5916/$ - see front matter  doi:10.1016/j.calphad.2005.11.004

vacancies are formed thus maintaining charge neutrality. Under oxidizing conditions apparent oxygen excess, LaMnO3+d, is observed. In this case Mn3+ is oxidized to Mn4+ and equal amounts of vacancies are formed on A- and B-sites again maintaining charge neutrality. Oxygen excess should therefore more correctly be referred to as cation deficiency. A further important defect reaction that occurs is the charge disproportionation, or charge dismutation reaction by which Mn3+ partially disproportionates into Mn2+ and Mn4+ leading to good electronic conductivity even of stoichiometric LaMnO3 [5]. This reaction can be considered to be entropy driven and occurs to significant extent due to the relatively unstable electron configuration of Mn3+ [6]. In previous papers we have shown that the CALPHAD approach is well suited to model the defect chemistry of this phase. However, in the defect chemistry community there still remains some doubt as early attempts to model the LaMnO3 perovskite phase using the CALPHAD approach with an associate model led to a wrong reproduction of the defect chemistry [7]. In this paper we compare the calculated defect chemistry when modeling the phase using an associate model and the compound energy formalism. A review on the details of these models is given by Saunders and Miodownik [8].

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We show that the associate model is not able to correctly reproduce the defect chemistry of the perovskite phase. Also the choice of associates is crucial and strongly influences the obtained result. Finally a larger number of parameters are required to model the phase (at least one per associate). When using the compound energy formalism on the other hand, the actual defects that are present in the material are considered and it is thus possible to construct a model based directly on the measured physical properties of the phase. These include the crystal structure, the defect chemistry, conductivity data and many more. The model description might seem rather cumbersome but it can be constructed unambiguously when the principles outlined in this work are applied. The description requires fewer optimising parameters compared to the associate model and correctly reproduces the defect chemistry of the phase. The advantage of the associate model is the simplicity of the model. If only phase relations are of interest, as was the case in the work of Yokokawa et al. [7], the associate model is perfectly adequate. 2. Literature data used for optimisations A more comprehensive review of literature data on the La–Mn–O system is given elsewhere [3]. For the optimisation performed here, we only consider the following experimental data that we found to be the most reliable in our previous work. The Gibbs energy of the reaction 1/2La2 O3 + MnO + (1 − 2d)/4O2 = LaMnO3−d measured by Atsumi et al. [9] and Jacob and Attaluri [10] using EMF and Borlera and Abbatista [11] and Mizusaki et al. [12] using dissociation pressures were used together with the heats of formation measured by Laberty et al. [13], and Rørmark et al. [14] to fix the Gibbs energy of stoichiometric LaMnO3 as a function of temperature. The lanthanum deficiency of La1−x Mn1−y O3−d in equilibrium with MnOx and manganese deficiency in equilibrium with La2 O3 measured by van Roosmalen et al. [15] in air, by ZachauChristiansen et al. [16] under low oxygen partial pressure, and by Bosak et al. [17] in air and under low oxygen partial pressure were used. The measurements of the oxygen nonstoichiometry as a function of temperature and oxygen partial pressure by Mizusaki et al. [12] for LaMnO3±dLa0.95 MnO3±d and La0.9 MnO3±d were used as it is the most comprehensive data set and agrees quite well with other oxygen nonstoichiometry data. In addition the measurements of Alonso et al. were used as they were performed under high oxygen partial pressures giving valuable additional information. This oxygen nonstoichiometry data makes up the main bulk of experimental data. We further used the data on the average manganese valency in manganese deficient perovskites from Arulraj et al. [18] as it is the only data of this kind. 3. Comparison of models The aim of this paper is to compare the associate model and the compound energy model to describe defect chemistry in oxides. This work describes the LaMnO3 perovskite, but the principles are valid for all ionic materials. The thermodynamic

modeling is compared to the well established classic defect chemistry modeling using equilibrium constants of defect reactions. 3.1. Reduction We start by considering a simplified description of the perovskite phase in which only the reduction reaction is considered. The Mn ion on the B-site is only allowed to have the valence states Mn2+ and Mn3+ . This results in a very simple model description with which many of the concepts can be illustrated. 3.1.1. Classic defect chemistry modeling In Kr¨oger-Vink notation the reduction reaction of the perovskite phase LaMnO3−d is given by x x OO + 2MnMn ↔ V•• O + 2MnMn + 1/2O2 (g).

(1)

The lower index signifies the site where the ion sits (Lasite, Mn-site or O-site), the upper index indicates the charge compared to the normal charge of the crystal without defects, (x) is neutral, ( ) is −1 and (•) is +1. MnxMn and MnMn thus denote Mn3+ and Mn2+ sitting on Mn-sites with a net charge of zero and −1 respectively. OxO denotes O2− ions on oxygen sites with zero net charge and V•• O stands for oxygen vacancies with a net charge of 2+. For this defect reaction the equilibrium constant K r can be defined as follows [Mn Mn ]2 [V•• O ] pO2

1/2

Kr =

x 2 x [MnMn ] [OO ]

.

(2)

For small defect concentrations [MnxMn ]2 [OxO ] can be considered to be ∼1. In order to maintain charge neutrality the following relation must hold 2[Mn Mn ] = [V•• O ].

(3)

By substitution of Eq. (3) into Eq. (2) the following proportionality between defect concentration and oxygen partial pressure is obtained: −1/6

log[Mn Mn ] ∝ log pO2 ,

−1/6

log[V•• O ] ∝ log pO2 .

(4)

When modeling the defect chemistry, the equilibrium constant K r is adjusted to fit experimentally determined defect concentration as a function of oxygen partial pressure. The slope (in this case −1/6) is fixed by the chosen defect reaction. This is the standard method to model defect chemistry and has been applied to model the defect chemistry of (La, Sr)MnO3±d perovskites by Poulsen [19] and Nowotny and Rekas [20]. 3.1.2. Associated model A very simple way of modeling the nonstoichiometry of the perovskite is to consider it as consisting of a mixture of molecule-like associates. This approach was used by Yokokawa et al. [7] in an early attempt to model the perovskite phase. To model oxygen nonstoichiometry for the reduction reaction the following associates are used.

A.N. Grundy et al. / Computer Coupling of Phase Diagrams and Thermochemistry 30 (2006) 33–41

– LaMnO3 for the stoichiometric perovskite (denoted L3O) – LaMnO2.5 for the reduced perovskite containing only Mn2+ (denoted L2O). On forming a nonstoichiometric phase they are thought to mix randomly. It is clear that this is not what happens in reality; however from a modeling point of view such a description might be adequate. If the Gibbs energy of the two associates is GL3O and GL2O respectively then the Gibbs energy of the perovskite is given by an expression for a random solution of the associates giving = G Perov m

1

1 GL3O + GL2O x L3O x L2O   1 1 1 1 ln + ln + RT x L3O x L3O x L2O x L2O

(5)

where x L3O and x L2O are the molar fractions of the associates L3O and L2O. Additionally interaction parameters can be added to the above expression. The values of GL3O and GL2O optimised in this work are given in Table 1. We now analyse this modeling method using the classic defect chemistry treatment. The reduction reaction is then given by LaMnO3 ↔ LaMnO2.5 + 1/4O2(g).

(6)

Using the same considerations as above a slope of −1/4 is now obtained for the defect concentration as a function of the logarithm of oxygen partial pressure in contradiction to the slope of −1/6 obtained from the classic defect chemistry treatment. One might argue that the associates chosen above have been given an arbitrary size based on the usual way of writing the chemical formula of the perovskite phase. A more logical choice of associates might be to normalize their size to one oxygen ion giving the associates La1/3Mn1/3 O and La2/5 Mn2/5O. Now the defect reaction is given by La1/3 Mn1/3 O ↔ 5/6La2/5 Mn2/5O + 1/12O2(g)

(7)

and the equilibrium constant of this reaction is 1/12

Kr =

[La2/5 Mn2/5O]5/6 pO2 [La1/3 Mn1/3 O]

.

(8)

Now a slope of −1/10 is obtained. If one chooses the associates La2 Mn2 O6 and La2 Mn2 O5 then a slope of −1/2 is obtained. It can therefore be concluded that depending on the size of the chosen associates, completely different results are obtained for the dependence of the defect concentrations on the oxygen partial pressure and in the cases outlined above none of the chosen associates gives the same slope as the one expected from a classic defect chemistry treatment.

35

octahedral B-site coordinated by 6 oxygen anions and occupied by a smaller cation (Mn3+ or Mn2+ , in our case) and a face centred cubic lattice of O2− anions. The model we choose within the compound energy formalism for the LaMnO3−d phase is thus (La3+ )(Mn2+ , Mn3+ )(O2− , Va)3 . The Gibbs energy of the perovskite is given as a mixture of the four endmember perovskites: LaMn3+ O3 with a net charge of 0, LaMn2+ O3 with a net charge of −1, LaMn3+ Va3 with a net charge of 6+ and LaMn2+ Va3 with a net charge of 5+. The charged endmembers are of course hypothetical, nevertheless they need to be assigned a Gibbs energy. The Gibbs energies of these endmember perovskites are denoted as ◦ G Perov , ◦ G Perov , ◦ G Perov La3+ :Mn3+ :O2− La3+ :Mn2+ :O2− La3+ :Mn3+ :Va and ◦ G Perov . The Gibbs energy of any perovskite lying La3+ :Mn2+ :Va between these four endmembers is given as the weighed sum of these four endmembers plus a configurational entropy term arising from the mixing of Mn2+ and Mn3+ on the Mn sublattice and O2− and vacancies on the oxygen sublattice. As three of the four endmembers are charged the perovskite can only exist along the neutral line shown in Fig. 1. One endpoint of the neutral line corresponds to the stoichiometric perovskite LaMnO3, the other endpoint to the completely reduced perovskite LaMn2+ O2.5 . The Gibbs energy of these two perovskites are given by ◦

G Perov = GL3O = ◦ G Perov LaMn3+ O La3+ :Mn3+ :O2− 3



5 ◦ Perov G La3+ :Mn2+ :O2− 6   1 ◦ Perov 5 5 1 1 ln + ln + G La3+ :Mn2+ :Va + 3RT . 6 6 6 6 6

G Perov LaMn2+ O

2.5

(9)

= GL2O =

(10)

The last term in Eq. (10) takes into account the configurational entropy of random mixing of O2− and vacancies on the oxygen sublattice. By choosing ◦ G Perov as the reference and La3+ :Mn3+ :Va giving it the value ◦

G Perov = ◦ G Perov − La3+ :Mn3+ :Va La3+ :Mn3+ :O2−

3◦ G O2 (gas) 2

(11)

and using the reciprocal relation ◦

G Perov + ◦ G Perov − ◦ G Perov La3+ :Mn2+ :O2− La3+ :Mn3+ :Va La3+ :Mn3+ :O2− − ◦ G Perov = G red rec La3+ :Mn2+ :Va

(12)

the following expressions are obtained for the remaining two endmember perovskites. 1 G Perov = GL2O + ◦ G O2 (gas) La3+ :Mn2+ :O2−   4 1 5 5 1 1 red + G rec − 3RT ln ln + ln 6 6 6 6 6 5 ◦ Perov G La3+ :Mn2+ :Va = GL2O − ◦ G O2 (gas) 4  5 5 1 1 5 red ln + ln − G rec − 3RT ln . 6 6 6 6 6 ◦

3.1.3. The compound energy formalism The idea of the compound energy formalism is to model the phase in accordance with its crystal structure and occupation of each distinct crystallographic site. In the case of perovskites with the general formula ABO3 three crystallographic sites can be distinguished: the A-site coordinated by 12 oxygen anions that is occupied by a large cation (La3+ , in our case), the

(13)

(14)

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Table 1 Thermodynamic parameters describing the perovskite phase LaMnO3±d Thermodynamic parameters for the compound energy description: Stoichiometric: (La3+ )(Mn3+ )(O2− )3 G LaMn3+ O = GL3O = 12 ◦ G A−La O + 12 ◦ G Mn O − 63 367 + 51.77T − 7.19T ln(T ) + 232 934T −1 2 3

3

2 3

Reduction: (La3+ )(Mn2+ , Mn3+ )(O2− , Va)3 : G LaMn2+ (O ,Va ) = GL2O = 12 ◦ G A−La O + ◦ G MnO + 27 672 2 3 5/6 1/6 3 Oxidation: (La3+ , Va) (Mn3+ , Mn4+ , Va) (O2− )3 G = GL4VO = 12 ◦ G A−La O + 43 ◦ G MnO − 91 857 + 20.31T 4+ La(Mn3/4 ,Va1/4 )O3

2 3

2

= GLV4O = 13 ◦ G A−La O + ◦ G MnO −53 760 2 3 2 2/3 ,Va1/3 3 Vacancy energy: (Va)(Va)(Va)3 G VaVaVa3 = GVVV = 6GL2O + 4GL4VO + 3GLV4O − 12GL3O − 254 212 G (La

)Mn4+ O

Associated description Stoichiometric: LaMnO3 G LaMn3+ O = GL3O = 12 ◦ G A−La O + 12 ◦ G Mn O −71 604 + 58.25T − 7.19T ln(T ) + 232 934T −1 2 3 2 3 3 Reduced: LaMn2+ O2.5   G LaMn2+ O = GL2O = 12 ◦ G A−La O + ◦ G MnO +5437 + 25.25T + 3RT 16 ln 16 + 56 ln 56 2 3

2.5

Oxidised: LaMn3/4 O3 , La2/3 MnO3 , La2/3 MnO3 G

LaMn4+ 3/4 O3

G La

= GL4VO = 12 ◦ G A−La O + 34 ◦ G MnO −94 737 + 28.3T + RT 2 3 2

4+ 2/3 Mn O3

G La



= GLV4O = 13 ◦ G A−La O + ◦ G MnO −57 899 + 12.36T + RT 2 3 2 

3+ 2/3 Mn O2.5

= G L V 3O = 13 ◦ G A−La O + 12 ◦ G Mn O +5000 + RT 2 3 2 3

1 ln 1 + 1 ln 1 4 4 4 4



 

1 ln 1 + 2 ln 2 3 3 3 3   1 ln 1 + 2 ln 2 + 3RT 1 ln 1 + 5 ln 5 3 3 3 3 6 6 6 6

Note: all parameters are given in SI units (J, mol, K, R = 8.31451 J/(mol K)). The parameters for the binary oxides are taken from our previous assessments of the Mn–O [23] and La–O [24] systems.

In principle the reciprocal energy G red rec could be used as an adjustable parameter; it is strongly advisable to set the energy for the reciprocal reaction equal to 0 as this avoids the inevitable formation of miscibility gaps if it is allowed to attain large values. If the reciprocal energy is 0 one starts with an ideal description of the solid solution. Should nonideality be required it is advisable to use interaction parameters instead of changing the reciprocal energy. The interaction parameters consider interactions between cations on a single sublattice. Additionally reciprocal interaction parameters can be introduced that are however not identical to the reciprocal energy and can therefore not be used to compensate for a reciprocal energy. It is very convenient to see that the result obtained above for the strongly simplified description of the perovskite phase in which only reduction is considered is identical to the complete description of the LaMnO3 perovskite phase including both reduction and oxidation given in an earlier paper [3] in which a different reference was chosen. This shows that any reference can be chosen. The optimised values for the functions GL2O and GL3O, taken from our previous work [3], are given in Table 1.

Fig. 1. Geometrical representation of the reduced perovskite described using the compound energy model giving the reciprocal system LaMn3+ O3 – LaMn2+ O3 –LaMn3+ Va3 –LaMn2+ Va3 .

The equilibrium constant K r is

Submitting this modeling approach to a defect chemistry analysis gives the following defect reaction in the sublattice notation

Kr =

(La3+ )(Mn3+ )(O2− )3

Again considering only small defect concentrations the following proportionalities are obtained

→ (La3+ )(Mn2+ )(O2− 5/6 , Va1/6 )3 + 1/4O2 (g)

(15)

or in Kr¨oger–Vink notation x 1/2OO

x + MnMn



1/2V•• O

+ Mn Mn

+ 1/4O2 (g).

(16)

1/2 p [Mn Mn ][V•• O] O2

1/4

x ][O x ]1/2 [MnMn O

−1/6

.

(17)

−1/6

[Mn Mn ] ∝ pO2 , [V•• O ] ∝ pO2

(18)

which is in perfect agreement with the classic defect chemical analysis given in the preceding section.

A.N. Grundy et al. / Computer Coupling of Phase Diagrams and Thermochemistry 30 (2006) 33–41

3.2. Oxidation

Choosing these associates the Gibbs energy of the oxidized perovskite phase is given by

A unique feature of the LaMnO3 perovskite is that it shows not only a reduction reaction leading to oxygen deficiency at low oxygen partial pressures and/or high temperatures but also an oxidation reaction and oxygen excess at high oxygen partial pressures and/or low temperatures. In contrast to the reduction that is quite simple the oxidation reaction is somewhat more complex. We now describe only the oxidation of the perovskite. This means we only allow the valence states Mn3+ and Mn4+ giving the oxidized perovskite LaMnO3+d. In the following the basic concepts outlined above will be applied to the oxidation reaction of the perovskite. 3.2.1. Classic defect chemistry modeling The oxidation of the perovskite is accomplished by the oxidation of Mn3+ to Mn4+ . The extra charge is compensated by the formation of cation vacancies. This means that what is often written as oxygen excess, LaMnO3+d is in fact cation deficiency. The following defect reaction can be written for the oxidation of the perovskite phase 6 x 6 18 x 3 6 x x LaLa + MnMn + OO + O2 (g) → LaLa 7 7 7 14 7 1 6 · 1 x + VaLa + MnMn + VaMn + 3OO . 7 7 7 x 6/7 1/7 [Mn• ]6/7 [Va ]1/7 [O x ]3 ] [Va [LaLa La ] Mn Mn O 3/14

x ]6/7 [Mnx ]6/7 [O x ]18/7 p [LaLa Mn O O2

(19)

.

(20)

Again assuming only small defect concentrations we can ignore x ], [Lax ] and [O x ] as their concentrations are ∼1. Due [MnMn La O to charge neutrality the following relation must hold [Va La ]

=

[Va Mn ]

and

[Va La ], [VaMn ]

1 = [Mn•Mn ]. 3

(21)

By substitution we get the following proportionalities • [Va La ], [VaMn ], [MnMn ] ∝ pO2 . 3/16

1 G LaMnO3 x LaMnO3 1 1 + G La2/3 Mn4+ O3 + G LaMn4+ O 4+ 3/4 3 x x La2/3 Mn O3 LaMn4+ 3/4 O3   1 1 1  . (23) + RT ln  + + x LaMnO3 x La2/3 Mn4+ O3 x LaMn4+ O

= G Perov m

3/4

(22)

3.2.2. Associates When using the associate model a big difficulty is to make a reasonable choice of associates to use. In an early attempt to model the perovskite phase Yokokawa et al. [21] chose the following associates to model the perovskite phase LaMnO3 for stoichiometric perovskite LaMnO2.5 for the reduced perovskite containing only Mn2+ La2/3MnO3 for the La deficient perovskite containing only Mn4+ La2/3MnO2.5 for the La deficient perovskite containing only Mn3+ . In a later paper [7] he then found it necessary to additionally include the associate LaMn0.75O3 for the Mn deficient perovskite containing only Mn4+ .

3

Considering only the oxidation reaction of a stoichiometric perovskite the following defect reaction is obtained for the associate model 2LaMnO3 + 1/2O2(g) → La2/3 MnO3 + 4/3LaMn0.75 O3 . (24) This leads to the equilibrium constant of K Ox =

[La2/3 MnO3 ][LaMn3/4 O3 ]4/3 . [LaMnO3 ]2 pO1/2

(25)

2

Again we can assume that for small defect concentrations [LaMnO3] is ∼1 and we get the following proportionalities 3/14

The equilibrium constant of this reaction is K Ox =

37

[La2/3 MnO3 ], [LaMn3/4O3 ] ∝ pO2 .

(26)

This is close to the slope of 3/16 obtained from the classic defect chemistry analysis but is not identical. 3.2.3. Compound energy formalism The sublattice occupation of the oxidized perovskite is as follows (La3+ , Va)(Mn3+ , Mn4+ , Va)(O2− )3 . The Gibbs energy of the oxidized perovskite is given as a mixture of the following six endmembers: LaMn3+O3 with a net charge 0, LaMn4+ O3 with a net charge +1, VaMn3+ O3 with a net charge −3, VaMn4+ O3 with a net charge −2, LaVaO3 with a net charge −3 and VaVaO3 with a net charge −6. These six endmembers are displayed in Fig. 2. Similarly to the reduction reaction, where a neutral line was defined, we can now define a neutral plane on which the possible perovskites can lie. The plane is delimited by the three neutral perovskites LaMnO3, La2/3Mn4+ O3 and LaMn4+ 3/4O3 the energies of which are given by combinations of the endmember perovskites by the following relations ◦

G Perov = GL3O = ◦ G Perov LaMn3+ O La3+ :Mn3+ :O2− 3

(27)

2 ◦ Perov G 3+ 4+ :O2− 3 La :Mn  

 2 1 1 2 1 ln ln + ◦ G Perov +RT + (28) 4+ :O2− Va:Mn 3 3 3 3 3 3 ◦ Perov G = GL4VO = ◦ G Perov La3+ :Mn4+ :O2− O LaMn4+ 4 3/4 3    

3 1 1 ◦ Perov 3 1 ln + G La3+ :Va:O2− +RT + ln . (29) 4 4 4 4 4 ◦

G Perov = GLV4O = La Mn4+ O 2/3

3

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2 2 1 1 3 ln + ln − RT 2 3 3 3 3 1 ox2 − G ox1 rec − G rec 2 4 1 ◦ Perov G Va:Mn4+ :O2− = 2GLV4O − GL4VO + GVVV 3   3 

3 3 1 1 Gas 4 1 ln + G O2 + RT + ln 2 3 4 4 4 4    

2 1 1 1 2 ln + ln − G ox2 − 2RT rec . 3 3 3 3 3 Fig. 2. Geometrical representation of the oxidised perovskite described using the compound energy model.

The last terms in Eqs. (28) and (29) take into account the configurational entropy of random mixing of La3+ and vacancies and Mn4+ and vacancies on the A- and B-sites respectively. Now we choose ◦ G Perov as the reference and give it the Va:Va:O2− value ◦

G Perov = GVVV + Va:Va:O2−

3◦ G O2 (gas) 2

(30)

where GVVV is the “vacancy energy”, given in Table 1, as defined in the previous work [3]. Using the reciprocal relations ◦

G Perov + ◦ G Perov − ◦ G Perov La3+ :Mn3+ :O2− Va:Mn4+ :O2− La3+ :Mn4+ :O2− ◦

− G Perov = G ox1 rec Va:Mn3+ :O2− ◦ Perov ◦ Perov G Va:Va:O2− + G La3+ :Mn4+ :O2− − ◦ G Perov La3+ :Va:O2− ◦ Perov ox2 − G Va:Mn4+ :O2− = G rec

(31)

(32)

we finally obtain equations for the remaining four endmember perovskites 2 = GL4VO 3 1 1 1 + GLV4O − GVVV − G Gas O2 2 6 4 

  3 3 1 2 1 ln − RT + ln 3 4 4 4 4    

2 2 1 1 1 1 ln + ln + G ox2 − RT rec 2 3 3 3 3 6 1 3 ◦ Perov G La3+ :Va:O2− = 2GL4VO − GLV4O + GVVV 2  2 

3 1 3 Gas 3 1 ln + G O2 − 2RT + ln 4 4 4 4 4    

2 2 1 3 1 1 ln + RT + ln − G ox2 rec 2 3 3 3 3 2 ◦



G Perov = GL3O − 2GL4VO Va:Mn3+ :O2− 1 3 3 GLV4O + GVVV + G Gas O2 2 2   4 

3 1 3 1 ln + 2RT + ln 4 4 4 4

+

(36)

Again it is comforting to see that these expressions are identical to the ones obtained in the complete description of the system given in the previous paper, again showing that this approach is general and unambiguous. The values of the functions GL3O, GLV4O, GL4VO and GVVV, again taken from our previous assessment [3], are given in Table 1. Based on the description using the compound energy formalism we may define the defect reaction as follows 1 2[(La3+ )(Mn3+ )(O2− )3 ] + O2 (gas) 2 4+ → [(La3+ , Va )(Mn )(O2− )3 ] 1/3 2/3 +

4 [(La3+ )(Mn43/4 , Va1/4 )(O2− )3 ] 3

(37)

translating this into Kr¨oger–Vink notation we get 1 x x x 2LaLa + 2MnMn + 6OO + O2 (gas) 2 1 2 x x → LaLa + VaLa + Mn•Mn + 3OO 3 3 4 x 1 x + LaLa + Mn•Mn + Va + 4OO 3 3 Mn

(38)

the equilibrium constant is then defined as follows K Ox =

G Perov La3+ :Mn4+ :O2−

(35)

x ]2/3 [Va ]1/3 [Mn• ][Ox ]3 [Lax ]4/3 [Mn• ][Va ]1/3 [Ox ]4 [LaLa La Mn La Mn Mn O O 1/2

x 2 x 2 x 6 [LaLa ] [MnMn ] [OO ] pO2

(39)

for small oxidation extent all defect concentrations are close to 1 except [VaLa ], [VaMn ] and [Mn•Mn ]. Using the following conditions that are necessary to maintain charge neutrality [Va Mn ] = [VaLa ]

(33)

and

• [Va Mn ], [VaLa ] = 1/3[MnMn ]

leads to the proportionality 1/3 1/3 pO2 ∝ [Va [Mn•Mn ][Mn•Mn ][Va La ] Mn ] 1/2

and

• [Va La ], [VaMn ], [MnMn ] ∝ pO2

3/16

(40)

which is exactly what is expected from the classic defect modeling. (34) 3.3. Complete description including oxidation and reduction The descriptions of the reduction and oxidation of the perovskite outlined above can be combined to give a complete description of the defect chemistry of the phase. Such descriptions have been published using all three approaches,

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Fig. 3. Oxygen nonstoichiometry in LaMnO3±d as a function of oxygen partial pressure for 873, 1073 and 1273 K modeled using the compound energy formalism (CEF) and the associated model compared to experimental data. The oxygen content is given in a linear scale.

39

Fig. 4. Oxygen deficiency in LaMnO3−d modeled using the compound energy formalism (CEF) and the associated model. The oxygen content is given logarithmically, where x(O) is the mole fraction of oxygen.

the compound energy formalism [3], an associated model [7] and classic defect chemistry [19,20] and the details of this modeling work will not be repeated here. As the associate model is not able to correctly model the defect chemistry even in the simplified description presented in this paper this model will also be unsuitable for a complete description of the perovskite phase. A comparison of the published defect chemistry modeling using the classic approach with equilibrium constants and the CALPHAD method applying the compound energy formalism for the perovskite phase including oxidation, reduction and Sr2+ doping will be given in a separate paper. 4. Results and discussion The optimised thermodynamic parameters for the associated and compound energy descriptions are given in Table 1. Fig. 3 shows the experimental data on oxygen nonstoichiometry as a function of oxygen partial pressure [12,22] together with the curves calculated using the compound energy formalism and the associated description. This figure clearly shows that the perovskite displays both oxygen excess and oxygen deficiency. When the defect concentrations are plotted linearly it seems as though both thermodynamic descriptions can reproduce the experimental data satisfactorily. However, when the defect concentrations are plotted logarithmically crucial differences are brought to light. 4.1. Reduction As in the previous section where we discussed the various models we will again start by considering the reduced perovskite only. Fig. 4 compares the calculated oxygen deficiency in LaMnO3−d together with experimental data in a logarithmic scale. Here the difference between the two model descriptions can clearly be seen. The description using the associated model displays the slope of −1/4 as calculated in the

Fig. 5. Oxygen excess in LaMnO3+d modeled using the compound energy formalism (CEF) and the associate model. The oxygen content is given logarithmically, where x(O) is the mole fraction of oxygen.

preceding section, that is in contradiction to the expected slope of −1/6 derived using the defect chemistry considerations. The curve calculated using the compound energy description shows the expected slope of −1/6. The experiments on the other hand show the expected slope of −1/6 at low temperatures only. At higher temperatures the slope gradually changes and at 1273 K the experiments correspond far better to the slope of −1/4 of the associated description. This behaviour does not mean that the defect reactions considered are wrong and the associate model is more appropriate but has a different origin as will be explained below. 4.2. Oxidation Fig. 5 compares the two model descriptions of the oxidation of the perovskite. As explained above the expected slope from

40

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the defect chemistry considerations is 3/16 and is obtained using the CEF description. The associate model shows a slope of 3/14 which is quite close (but not identical!) meaning that both descriptions are quite similar. Here the agreement between the experiments and the modeling is significantly poorer than in the case of the reduction. This is most probably due to the experimental difficulties encountered in these experiments as the perovskite with oxygen excess shows cation deficiency which means that cation diffusion is necessary and this is a very slow process, especially at the comparatively low temperatures of 873 K. 4.3. Complete description of oxidation and reduction As mentioned above, the experiments of the oxygen deficiency as a function of oxygen partial pressure only display the expected slope of −1/6 at a comparatively low temperature of 873 K. At high temperatures the slope gradually becomes steeper reaching almost −1/4 at 1273 K. The reason for this is that if both reduction and oxidation are considered there is the following additional defect reaction that needs to be considered

Fig. 6. Oxygen deficiency in LaMnO3−d modeled using the compound energy formalism (CEF). The dashed line is calculated with a model that includes only reduction, the solid line includes both oxidation and reduction. The oxygen content is given logarithmically, where x(O) is the mole fraction of oxygen.

x → Mn Mn + Mn•Mn 2MnMn

with the equilibrium constant K Disp =

[Mn Mn ][Mn•Mn ] . x 2 [MnMn ]

(41)

This reaction is called charge disproportionation or charge disminuation. Due to this reaction the assumption that was made in the defect chemistry analysis of the reduction, that the concentration of [MnxMn ] is ∼1, is no longer valid. At low temperatures the degree of disproportionation is low and the experiments show almost the expected slope of −1/6 as [MnxMn ] is close to 1; however at higher temperatures the disproportionation becomes more pronounced and [MnxMn ] decreases. This leads to the gradual change of slope. When both the oxidation and reduction of the perovskite is modeled using the compound energy formalism this disproportionation reaction is automatically considered. It can be seen in Fig. 6 how the slope of the complete description of the perovskite deviates from the slope when only reduction is considered. Now an almost perfect description of the experimental data is obtained. It is this change of slope as a function of temperature that fixes the degree of charge disproportionation. The same behaviour can be seen for oxidation in Fig. 7. Here the trend is not as clearly visible due to the scatter of the experimental data. But it can be seen that the three experimental points as the lowest oxygen excess at 873, 973 and 1073 K are clearly better described when both oxidation and reduction are considered. Fig. 8 finally compares the degree of charge disproportionation at 873 and 1273 K obtained in this manner. At 873 K the stoichiometric perovskite contains 1.9% Mn2+ and Mn4+ and 96.2% Mn3+ . At 1273 K the disproportionation increases to 4.5% Mn2+ and Mn4+ and 91% Mn3+ . This significant degree of charge disproportionation is higher that in perovskites with other B-site cations due to the relative instability of the 3d4 electron configuration of the Mn3+ cation and is the reason

Fig. 7. Oxygen excess in LaMnO3+d modeled using the compound energy formalism (CEF). The solid line is calculated with a model that includes only reduction, the dashed line includes both oxidation and reduction. The oxygen content is given logarithmically, where x(O) is the mole fraction of oxygen.

for the high electronic conductivity observed in stoichiometric LaMnO3 perovskites. 5. Conclusions We describe the defect chemistry of the LaMnO3±d perovskite with two different CALPHAD models. We show that the associate model does not reproduce the defect chemistry correctly, as it is based on a physically false assumption. The physically more realistic model based on the compound energy formalism gives a far better description and is perfectly in line with the well established defect chemistry analysis. It is further shown how the charge disproportionation of the stoichiometric

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[9]

[10] [11] [12]

[13]

[14]

Fig. 8. Calculated fractions of the species Mn2+ , Mn3+ and Mn4+ on the Bsites of the perovskite LaMnO3±d at 873 and 1273 K as a function of oxygen partial pressure modeled using the compound energy formalism (CEF). The Mn3+ content never reaches 100% but is always partially disproportionated into Mn2+ and Mn4+ .

[15]

[16]

perovskite can be quantified by carefully optimising the defect chemistry data using the compound energy model.

[17]

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[18]

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