Point defects and transport in non-stoichiometric oxides: Solved and unsolved problems

I

PII: SOO22-3697(97)00205-9

Pergamon

POINT DEFECTS

Phys

Chem

SoLds

Vol59,

No. 4. pp 507-525. 1998 1998 Elsevier Science Ltd Pnnted I” Great Bntain All nghts reserved 0022-3697l98 $19.00 + 0 lM

AND TRANSPORT IN NON-STOICHIOMETRIC SOLVED AND UNSOLVED PROBLEMS

0

OXIDES:

RijDIGER DIECKMANN Department of Materials Science and Engineering, Cornell University, 228 Bard Hall, Ithaca, NY 14853-1501, USA Abstract-First, the basic concepts used in the area of defects and transport in non-stoichiometric oxides are reviewed. Their capabilities and shortcomings are addressed, including the problems of dealing appropriately with point defects at higher defect concentrations and with situations where more than one sublattice is available for one type of ion. As an example of where ideal solution point defect thermodynamics can be applied without much problems, the iron oxide magnetite, Fe3_ hod, is discussed. The cation diffusion in this material is also reviewed. Then some magnetite-basedsolid solutions are briefly discussed with regard to their defect chemistry and the transport of cations. Finally, two largely ignored topics are addressed: (i) the influence of the valence state of an ion on its diffusivity, and (ii) contributions from near-boundary regions to the variation of the oxygen content of nonstcuchiometric oxides. 0 1998 Elsevier Science Ltd. All rights reserved. Keywords: A. oxides, D. diffusion, electrical conductivity 1. INTRODUCTION

Defects in crystalline materials can be classified with regard to their dimensionality. Three-dimensional defects are, for example, pores and inclusions. Surfaces and interfaces are two-dimensional and dislocations are one-dimensional defects. Point defects are zerodimensional defects. For energy and entropy reasons, the only thermodynamically stable defects are point defects. However, this does not mean that one can really make materials that only contain point defects. It can never be avoided that one- and higher-dimensional defects form during the preparation and handling of materials, for example, due to stresses. High temperature annealing can reduce the concentration of these defects, but never to zero, because the driving forces for their removal are relatively small and the kinetics of their removal are slow. Consequently, practical materials always contain one- and higher-dimensional nonequilibrium defects in addition to point defects. In ionic crystals one has to consider ionic and electronic point defects. Ionic point defects are cation and anion vacancies and cations and anions on interstitial sites that are usually empty, while electronic point defects are electrons, e’, and holes, h’. Many properties of crystalline materials are defect related. Examples are the deviation from stoichiometry, e.g. A in Me, _*O, the electrical conductivity, the diffusivities of ions, the thermopower, the creep behavior, etc. In sufficiently pure materials, at sufficiently high temperatures, the influence of point defects on defectrelated properties prevails very often. In non-cubic materials there is an anisotropy in many properties, i.e. these properties are orientation dependent.

The principles of the relationships between point defect equilibria and properties are relatively well understood (see Fig. 1). In the bulk of sufficiently pure nonstoichiometric oxides, point defect equilibria which depend on the temperature, T, the total pressure, P, and on the chemical potentials of components, C(K,determine the concentrations of point defects. The concepts of point defect thermodynamics are used to formulate the relationships between the point defect concentrations and the values of the thermodynamic variables 7’. P and PK. Different properties depend on point defects (e.g. the electrical conduction, the diffusivity of ions and the nonstoichiometry). The electrical conductivity, u, ion tracer and the deviation from stoidiffusion coefficients, Dron, chiometry, 6, are quantities used to describe these properties. Theories on diffusion and on the electrical conduction in combination with point defect thermodynamics are used to analyze the dependence of u and Ok,, on the thermodynamic variables T, P and PK. Finally, processes that involve the simultaneous transport of matter and charge, such as the parabolic scale growth during oxidation or the point defect relaxation in ionic crystals after sudden changes in the value of a thermodynamic variable, can be quantitatively modeled based on data for u and ion tracer diffusion coefficients, DL,,by taking into account the coupling of the transport of matter and charge (i.e. by using theoretical concepts available for the migration of charged species in the case of the simultaneous presence of chemical ppotential and electrical field gradients). However, despite the available general understanding described above, problems exist in the quantitative understanding of these relationships, often due to a lack of appropriate data and/or concepts. In this article, several problems will be discussed that are 507

R. DIECKMANN

P&t Defect Equilibria (-fCT*P*ulo) 1 Point Defect Concentrations

__..____._._

Point Defect Thermodynamics

1 Defect Dapandent Propartias

a :...........: I I# C..____...__.-._.-..._._....__~ * . . . ..-...-.-_...-..-.--...-.---.--~..._.__.-..~~~--.

II II

II II ,

Transport in Electrochemical Potential Gradients Parabolic !ScalaGrowth

I

Point Defect Relaxation

I

0i5

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Fig. 1. Relationships between point defect equilibria, defect-dependent properties and transport in electrochemical potential gradients.

either solved, partially solved, or to a large extent unsolved. One of these problems is the appropriate thermodynamic treatment of point defects and associates or clusters formed between them. Only in the case of very dilute solutions of point defects in ionic solids can the defects be considered as being present in an ideal solution. At larger defect concentrations, the electrostatic interaction between charged point defects may become very important and, at very high concentrations, may lead to the formation of associates and defect clusters. If the electrostatic interaction is significant, it is inappropriate to ignore it by just using mass action laws; instead one has to introduce activity coefficients (the values of which change with the defect concentration) to account properly for the electrostatic interaction. It is very difficult to take into account simultaneously changes in activity coefficients and the formation of associates or clusters due to electrostatic interaction. A consequence is that at higher defect concentrations it is sometimes impossible to make unequivocal conclusions with regard to the predominant defects, even in the case of binary, non-stoichiometric compounds. Another problem is the distribution of ions and defects in binary, ternary and higher ionic crystals if more than one sublattice exists for cations and/or anions. There is a significant lack of secure knowledge on the distribution of ions and especially of defects on different lattice sites; this very often prevents a detailed quantitative analysis of defect-related data, for example, for the non-stoichiometry and the cation diffusion in spinels. Another largely ignored topic is the possibility that the

mobility of an ion may vary significantly with its charge state. Currently there is almost no knowledge on the influence of the charge state of an ion on its mobility. This is an unsolved problem for the analysis of diffusion data in cases where ions are present in more than one charge state, e.g. for the cation diffusion in magnetite, Fe3 _ 60,. A problem of importance for polycrystalline materials and also for thin films is that the energies required for the formation of point defects near boundaries may vary, as well as the mobilities of point defects, as a function of the distance to a boundary. How important is this in polycrystalline materials with different grain sizes and in thin films with different thicknesses? Is there a critical grain size or film thickness above which this is unimportant? At this time only relatively vague answers on these questions exist with much more work needed to reach a better understanding of this subject. 2. POINT DEFECTS AND DIFFUSION IN THE MODEL OXIDE Me, _ *O

To demonstrate the principles of pont defect thermodynamics and its application for predicting defect-related properties, it is useful to consider a model oxide of the type Me, _*O (i.e. a non-stoichiometric compound in which the O/Me-ratio varies due to component activitydependent point defect equilibria). For the sake of simplicity, it is assumed in the following that cation vacancies, (VMe2+)“,and holes, h’, are the majority defects formed by the reaction (Mez2 + )” + f O2 S (V,,Z+ )” + 2h’ + MeO(srg). (1)

Point defects and transportin non-stoichiometric oxides

o’-

p_

o*-

Me*+ O*-

Me”

o*-

02-

Me*+

509

i-r

02. mMe2+

)

bik

Fig. 2. Formation of a cation vacancy and of two holes by adding an oxygen atom to a model oxide of the type Me I-dO.

This is illustrated in Fig. 2. In eqn (1) and in the following text the ionic version of the Kroger-Vink notation for point defects [I] is used. This notation is preferred here because it offers a more unequivocal and complete description than the atomic version which is more commonly used. In the reaction described by eqn (1) a new molecule of Me0 is formed which is accommodated at a site of repeatable growth (srg), e.g. at a surface, a boundary or a dislocation. If the defect concentrations are sufficiently small, the electrostatic interaction between the charged defects is only small and therefore can be ignored. Then the equilibrium constant for the reaction described by eqn (1). K, , can be expressed as

where [i] denotes the concentration of the species i ( = (Vlule2+ )“, h’ and (Me:*+ )“) per lattice molecule of Met _ ,O, at&o is the thermodynamic activity of Me0 and uo, that of oxygen. In this article the oxygen activity is defined as ao, = Po,IP”, where PO2 is the oxygen partial pressure and P” a standard pressure of 1 atm. It is worth noting that sometimes other definitions are used. The latter equity in eqn (2) is because OMeo= 1 and ](Mezz + )“I = 1. The concentrations of holes and of cation vacancies are linked by the electroneutrality condition 2[(V,,z+)“] = [h’].

Fig. 3. Schematic plot of the concentrations of cation vacancies, [(V,,Z+)“], and of holes, [h’], in a model oxide of the type Me t-40 with cation vacancies and holes as the majority defects and very small point defect concentrations.

If the cations in Met _ AO diffuse only by one type of ionic defect then the cation tracer diffusion coefficient, Oh,, is related to the concentration of this defect, [defl, as follows %I, = !3&1er~e

]Me]

[deflf

Me = khkaiferMe[(VMe2+

)“I fM,. (6)

Here Gus is a geometrical factor, aMean elementary jump length, rMe an elementary jump frequency, [Me] the concentration of Me, and fMea correlation factor. The latter equity in eqn (6) holds if cation vacancies prevail in the cation tracer diffusion. The corresponding oxygen activity dependence of Db, for the latter case is shown in Fig. 4. If the electrical conduction in Me, _ 40 occurs exclusively by holes, the electrical conductivity, u, can be expressed as (7) where u,,. is the electrochemical mobility of the holes, zh their charge number ( = 1). F the Faraday constant, and V,,, the molar volume of MeO. A schematic plot of log u vs. log ao2 is shown in Fig. 5.

(3)

Insertion of the electroneutrality condition into eqn (1) leads to

The final result for the oxygen activity dependence of the concentrations of the considered point defects is therefore . [h ] = 2[(VHe2+)“I m &;.

(5)

Fig. 3 shows schematically the oxygen activity dependence of the concentrations of (VMe2+)” and h’.

Fig. 4. Schematic plot of the cation tracer diffusion coefficient, Oh,, in a model oxide of the type Me I_oO with cation vacancies and holes as the majority defects and very small point defect concentrations. It is assumed that only cation vacancies contribute to the cation diffusion.

R. DIECKMANN

510

L i

slope: l/S

Fig. 5. Schematic plot of the electrical conductivity,

u, in the

modeloxide of the type Me, _8Owith cation vacanciesand holes as the majority defects and very small point defect concentrations. It is assumed that only holes contribute significantly to the electrical conduction. 3. NON-IDEAL,

NON-DILUTE SOLUTION OF POINT DEFECTS

A general problem involved in the thermodynamic treatment of point defect equilibria in systems with charged species is the electrostatic interaction that occurs between charged point defects. An overview of what happens to point defects with increasing concentration is given in Fig. 6. At very low concentrations the defects are present in an ideal or an ideal dilute solution. Here the mass action law approach can be applied without any problems. With increasing defect concentration, the electrostatic interaction between the charged point defects increases. As a consequence, the value of the mean activity coefficient of the charged point defects changes; it decreases with increasing defect concentration. If this concentration increases further, associates may be formed by reaction between oppositely charged defects, with further concentration increases leading to larger defect aggregates (so-called defect clusters, e.g. containing cation vacancies, cation interstitials and electronic carriers). The main problems for the quantitative analysis of point defect equilibria, in the case of larger defect defect concentration:

status of defects:

concentrations, are missing concepts and in-situ structural information. Suitable and relatively easily applicable concepts for the quantitative treatment of electrostatic interactions between charged defects in solids are still missing. A few attempts have been made to develop suitable approaches based on statistical thermodynamics (see references [2-41) but without a real breakthrough. The resulting math is somewhat complex and difficult to handle. Therefore, for estimating the influence of electrostatic interactions among charged defects on thermodynamic activities one uses sometimes the Debye-Hiickel theory. This theory was originally developed to treat quantitatively the interactions between charged ions in very diluted aqueous solutions and is in principle not suitable to deal with the electrostatic interactions between charged species present in ionic solids in relatively high concentrations. Despite this problem, if defect interactions are likely to be present and the defect concentrations are not too high, applying the DebyeHiickel theory to ionic materials is still much better than simply ignoring all electrostatic interactions. Another problem, especially at high defect concentrations where associates and clusters may be present, is to identify unequivocally the types of associates and/or clusters which are present at thermodynamic equilibrium at high temperatures. For this purpose one needs, in principle, structural information from in-situ experiments which is for most ionic crystals not available. The use of observations made on quenched samples has the risk that the associates and/or clusters observed in such samples may have formed during quenching and, therefore, are not representative of the situation at high temperatures. 4. Me, -40 WITH CATION VACANCIES AND HOLES AS MAJORITY DEFECTS IN A NON-IDEAL SOLUTION If the electrostatic interaction between holes and cation vacancies in the model oxide Me, _ 40 cannot be ignored and changes of activity coefficients with increasing defect concentration become important, the equilibrium constant from eqn (2) has to be rewritten as K, =

[(”Me*+Y’l[h’12fh,o W$+Yl

4 incrsaslng 4

solution becomes non-Ideal, I.e., activity coefflclents of defects decrease: f,, < 1

4 IncreasIng 4

In formation of assoelates (complexes) between oppositely charged defects, e.g., v” + h’ + V’ in

high

formatlon of larger aggregates (clustsrs), e.g:l containing vacancies end interstttteta

Fig. 6. Overviewon the influence of the point defectconcentration on the status of the point defects.

ah:

Y13.e abl ’

4[(“&2+

=

(8)

where fe is the mean activity coefficient of the charged defects. The problem is to obtain appropriate values for f+ . Because f+ varies with the defect concentration, the oxygen activity dependence of defect concentrations becomes different from that in the case of an ideal or ideal dilute solution. As mentioned before, because there is no theoretical concept available that can be applied easily to calculate fz as a function of the defect concentration, one uses sometimes the Debye-Hiickel theory, originally developed for very dilute aqueous solutions of salts, for

Point defects and transport in non-stoichiometric oxldes

511

information on the type(s) of associate being present is usually not available. Therefore, even when the DebyeHiickel theory is used to estimate values forf, and the formation of associates (or clusters) is considered, the ,’ 2P)-3 I’ resulting modeling for the deviation from stoichiometry / T = 12OO’C ,r t / ,’ 1 is questionable to some extent. Very often point defect equilibria with consideration of electrostatic interaction have been analyzed by taking into account the formation of defect associates but not Ei=3%, , /’ v, = 11.6 cm3/mol /’ changes in the values of activity coefficients with chan/ Y, ging defect concentrations. This simplifies the treatment 0 -10 -5 -15 of point defect equilibria considerably. However, ignorlog 10 ao2 ing the changes off, with increasing defect concentraFig. 7. Comparison between cation vacancy concentrations cal- tions makes such defect modelings at least to some extent culated for a given value of the equilibrium constant K, (see eqn incorrect. Even if an excellent description of the uoZ(4) and eqn (8)) (i) by ignoring the variation of the electrostatic inreraction between the charged defects (dashed line), and (ii) by dependences of some defect-related properties can be taking this interaction into account using the Debye-Hiickel obtained, this is not sufficient proof that the use of the theory (solid line). concept is justified. This is demonstrated by the data for A making estimates forf,. Such an estimate has been made in Mn, _*O shown in Fig. 8(a,b) [5]. These figures show for a model oxide of the type Me, _ *O by assuming that the results obtained for the modeling of data for A in the following parameters apply: temperature T= 12OO“C, Mn, _ AO for temperatures between 900 and 1400°C by K, = 3.5 X lo-‘, relative permittivity e = 14 (similar as using two different approaches. The lines shown in found for COO), molar volume V,,, = 11.6 cm3 mol- ‘, Fig. 8(a) have been generated by assuming that only and the smallest distance between vacancies and holes at (Vhln2+)” and h’ exist as majority defects and by applying which no associate formation occurs is 3 A. Details about the Debye-Hiickel theory for estimating activity coeffithe equations used for such an estimate can be found in cients. The lines shown in Fig. 8(b) were derived most physical chemistry textbooks. In Fig. 7 it is demonby ignoring the fact that the values off+ vary with A strated how the estimated changes of the mean activity and by assuming that associates, ((V,,z+)“h’]’ and coefficient influence the oxygen activity dependence of ( (VMnz+)“2h’)“, are present in addition to isolated the cation vacancy concentration in the model oxide cation vacancies and holes. The lines shown in both Me, _aO. This figure shows that the concentration figures are practically identical, i.e. both of the curve bends upwards at high ao2 due to the electrostatic approaches described above allow one to describe forinteraction and that at higher defect concentrations the mally A as a function of uo2 equally well. However, this slope is significantly larger than that which one would does not answer at all the question of what the most obtain by ignoring the electrostatic interaction. Fig. 7 appropriate defect model is. Very likely none of the strongly suggests that ignoring the electrostatic interacmodels discussed above is entirely appropriate for tion by using the ideal mass action law becomes increasmodeling A in Mn, _ ,O; most probably isolated defects ingly inappropriate with increasing defect concentration. and associates will be present andf+-values decreasing It is worth noting that not only electrostatic interaction with increasing A have to be taken into account. Based on but also the formation of associates between oppositely the currently available experimental information and charged defects such as ( (VMe2+)“h’)’ ( = (VMez+)‘) and concepts, one cannot determine unequivocally the most ( (VMe2+)“2h’JX( = (VMe2+)“) can cause an upward appropriate quantitative model for A in Mn, _*O; this bending of log A vs. log ao, curves at higher defect statement holds also for the deviation from stoichiometry concentrations. As a consequence, in cases where in many other non-stoichiometric compounds. higher defect concentrations are present, a careful, quanIn view of the preceding discussion, how one can make titative analysis of experimental data for the deviation more unequivocal conclusions with regard to the prefrom stoichiometry with regard to the concentrations of dominant defects at higher defect concentrations is an different defect species must consider the electrostatic important question. First, one should not simply ignore interactions between charged defects as well as the electrostatic interactions between charged point defects, formation of associates (or clusters). Unfortunately, it is especially not at higher defect concentrations. Second, very difficult to do this in an unequivocal way. Three one should always try to include results for different major problems are (i) that the use of the Debye-Hiickel defect-dependent properties that have been obtained from theory allows only estimates of activity coefficients, (ii) in-situ studies into any modeling. Furthermore, one that reliable values for dielectric constants at high temshould avoid using any results from studies of defectperature are often missing, and (iii) that in-situ structural related properties performed at low temperatures on

I/

I 1

R. DIECKMANN

512

quenched samples. The reason for this is that there is usually sufficient time for changes in the defect concentration and arrangement during quenching, even if performed relatively fast, and therefore many results obtained from studies on quenched samples do not refer to equilibrium defects. Desirable, but very difficult, is the

B----3

-

fit M basis of Fig. 9. FCC lattice of atoms (gray circles) with tetrahedral (open

circles) and octahedral interstices (filled circles). The cell size corresponds to l/8 of the oxygen sublattice of the elementaryunit cell of a spine].

Q---3

in-situ determination of the structure of the prevailing ionic defects. The problem in many cases is that, currently, there is still a lack of appropriate experimental techniques. An exception are defect clusters at large concentrations. Here, neutron scattering experiments performed at high temperatures under controlled atmospheres can be used to determine the type of clusters present. Their effective charge, however, usually remains unknown. For an example of such a study, see Ref. 161. -16

-1L

-12

-10

-8

-6

4

-2

5. COMPLICATIONS DUE TO THE OCCURRENCE OF MORE THAN ONJZSUBLATTICE-SPINELS The situation with regard to point defects becomes much

more complicated than discussed before if more than one sublattice occurs for one type of ion. Examples for oxides where two cationic sublattices occur are spinels which have the general formula ABzO4, e.g. magnesium aluminate, MgA1204. Many spinels have a cubic crystal structure, especially at high temperatures. The oxygen ions are present in the form of a face-centered cubic close packing, i.e. in a lattice with octahedral and tetrahedral interstices. One oxygen subcell, see Fig. 9. has 12 interstices, 1 + 1214= 4 octahedral interstices, and 8 tetrahedral interstices. The number of O-atoms is 8/8 + 612= 4. As a consequence, for 3 cations there exist 12 interstices, Fig. 8. The deviation from stoichiometry, A. in Mn,_,O as a function of oxygen activity between 900 and 14W’C [S]: (a) experimental data points and lines generated by fitting the experimental data to a model with doubly charged cation vacancies and holes and taking into account the electrostatic interaction between the defects using the Debye-Hiickel theory, and (b) experimental data points and lines generated by fitting the experimental data to a model with neutral, doubly and singly charged cation vacancies and holes and ignoring the variation of the defect activity coefficients with A.

Point defects and transport in non-stoichiometric oxides

i.e. 9 empty interstices are present. Often one has 2 threevalent cations and 1 two-valent cation. A spine1 unit cell involves eight subcells containing 32 O-ions, 32 octahedral interstices and 64 tetrahedral interstices. One half of the octahedral and l/8 of the tetrahedral interstices are filled with cations, i.e. 16 octahedral cations and 8 tetrahedral cations exist. Many unoccupied positions with three different coordinations (1 octahedral and 2 tetrahedral ones, the latter having different next-nearest neighbor coordinations) exist in the lattice. For the cation distribution there exist three different limiting cases: (i) normal (all 2 + -ions are located on tetrahedral sites and all 3 f -ions on octahedral sites); (ii) inverse (l/2 of the 3 + -ions are on octahedral sites and l/2 of them are on tetrahedral sites, and all 2 + -ions are on octahedral sites); (iii) random (A/B-ratio the same on both sublattices). To denote the cation distribution one very often uses the formula (A, _,B,)[A,B2 _,]O,; the brackets () are used to denote species in tetrahedral coordination and the brackets [] for octahedrally coordinated species. In the case of a normal spinel, x=0; and for an inverse spine], x= 1. Usually, x=f(Z’, P), i.e. the cation distribution changes with temperature, T, and total pressure, P. At constant total pressure all spinels randomize to some extent with increasing temperature. Besides the cation distribution, the point defect distribution is also very important when considering point defect-related properties. For the sake of simplicity, only cationic defects will be considered in the following. In the case of cation vacancies, one has to consider two species, vacancies on octahedral and on tetrahedral sites, Vo and Vr. For cation interstitials the situation is much more complicated. There exist octahedral and two types of tetrahedral interstitial sites, the latter two being different with regard to their next-nearest neighbor arrangement. As a consequence, each type of cation can occur on three different lattice positions, i.e. A::, A$, A$$, B:2. B3+ 171pA$, thus six different species have to be considered. A common problem for the analysis of defectrelated properties in crystals with different sublattices is the lack of knowledge of the in-situ distribution of ions and point defects between different sublattices. Some limited knowledge of in-situ cation distributions exists from in-situ Mossbauer spectroscopy studies (see reference [7]) and from thermopower measurements (see reference [8]). It must be mentioned that the data evaluation in the latter case is usually based on the assumption that the electrical conduction occurs only on one sublattice (in the case of iron-based spinels on the octahedral sublattice); experimental justification for such an assumption is usually missing. Practically no knowledge exists on the defect distribution. Presently, as a result of missing information, the defect and cation distribution is very often ignored in studies of defects and transport in

513

systems with more than one sublattice per ion; many details of defect-related properties remain unexplored and cannot completely be understood.

6. MAGNETITE, FeY_dOo

As a simple example of an oxide with two cation sublattices we will consider in the following the iron oxide magnetite, Fe3_a04. The cation and defect distribution will be ignored due to the problems discussed before. Cation vacancies can be formed by the reaction 3Fe2+ + 10 2 F! 2Fe3+ + V, + iFejO,(srg). 3

(9)

The formation of cation interstitials may occur as follows Fe”+ + V, 2 Fe;+ +VFe.

(10)

By using the concepts of point defect thermodynamics and ignoring the presence of different cation coordinations, one obtains the following oxygen activity dependences for the concentrations of cation vacancies and interstitials:

and [Fey+] = 4Klao,- 7J3= [I]“a&2/3,

(12)

where Kv and K, are the equilibrium constants for the reactions described by eqns (9) and (lo), respectively. [VI0 and [I]’ represent point defect concentrations which are normalized to a% = 1; they are sometimes called “defect constants”. The right part of eqn (11) is an approximation insofar that small changes in the Fe’+/ Fe*+ ratio which become detectable at the highest deviations from stoichiometry observed for magnetite are ignoted for a more detailed discussion see Ref. [9]. The latter two equities in eqn (11) are because in pure magnetite aklo4 = 1, [Fe3+] = 2 and [Fe’+] = 1. Due to the fact that magnetite has two different cation sublattices, Kv consists of two terms ( - vacancies on the octahedral and on the tetrahedral sublattices) and K, of six terms ( - ions in two charge states on three types of interstitial sites as discussed before). However, as long as no information exists on the different terms included in Kv and KI, it is useless to formulate them explicitly. The deviation from stoichiometry 6 in Fe3 _ 604 in terms of Kv, KI and ao2 is 6 = [VrJ - [Fe;+ ] = %$: = [V]Oag: - [I]“aG2u3.

- 4KIa&zU3 (13)

Based on this equation one expects an S-shaped curve in a 6 vs. log ao2 plot for constant temperature and total pressure if cation vacancies and interstitials

514

R. DIECKMANN 5

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I

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4-

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32-

’ I 04 ‘I ; $1

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Fe3_604 T = 1200 “C

-

calculated

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nlth Kv = 20

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0expermwntsl m= 1364mg data pomts

= 1200 “C

5

0 m=489Qmg

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+ reference Doint

1

0

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F

0 --10

T

-60

-6

-6 log

-4

-10.0

-

-10

a

I

I

I

-6

-6

-4

10 ao2

loglo

-

ao2

Fig. 10. 6 in FeJ_&Oo4 at 1200°C as a function of the oxygen activity [9, 1I].

Fig. 11. Oxygen activity dependence of the iron tracer diffusion coefficient, DE,, in Fe,_6O4at 1200°C [lo, 111.

are the majority defects in Fe3-804. Results from thermogravimetric studies [9] have shown that this expectation is fulfilled. Fig. 10 shows data for 6 at 1200°C; the line shown has been generated by using

self-diffusion coefficient of Fe can be expressed as

eqn (13). It is worth noting that compounds like magnetite are special with regard to the electrostatic interaction between charged defects due to the existence of the redox equilibrium Fe*+ F! Fe 3+ +e’.

(14)

As a result of the almost constant Fe3+/Fe2+-ratio ( = 2), the electron concentration is also practically constant, i.e. this concentration is buffered and only insignificantly influenced by changes in the deviation from stoichiometry. Because electronic carriers are always very mobile compared to ions and the electron concentration is buffered, it is impossible to build up any long-range electrostatic interaction between charged species in compounds like magnetite. As a consequence, the electrostatic interaction between defects does not vary significantly with the defect concentration (at least as long as this does not become extremely large) and the point defects can be successfully treated with the concepts of point defect thermodynamics and by assuming an ideal solution of the point defects. This applies to all ionic compounds in which ions of one major component exist in two different charge states in similar concentrations, e.g. Mn304 and Cos04. Both cation vacancies and cation interstitials contribute to the cation diffusion in magnetite. Departing from the general formulation of diffusion coefficients,

DFe= = gFe(V)&(V)

zgFe(de‘,&(def) SF.(den(16) rvFel

[FelrFe(V)

+ ~Fe&e(I)

[Fe;+

fFel

1

l-Fe(I).

In this case two types of point defects, cation vacancies and cation interstitials, are expected to contribute to the cation diffusion. By taking into account the results discussed before for the concentrations of cation vacancies and cation interstitials and summarizing all terms believed to be independent of the defect concentration (i.e. g,, ai, I’;. [Fe], and correlation factors), the tracer diffusion coefficient of Fe can be expressed as

Based on this equation one expects V-shaped curves in log DC, vs. log uo2 plots if both cation vacancies and interstitials contribute significantly to the iron tracer diffusion. Experimental results have proven that this expectation is fulfilled [lo]. As an example data measured for 1200°C are shown in Fig. 11. In this figure there is a small deviation from the ug dependence visible at high oxygen activities. This deviation results from the fact that the [Fe2+]/[Fe3+] ratio decreases slightly at higher values of 6 which occur at high ao,; this subject has been taken properly into account when generating the curve shown in Fig. 11 [ 1I]. The constants D&) and D&t) are in principle relatively complex quantities. They involve the different terms for defect concentrations discussed before and also terms for the effective mobilities of all defects involved, i.e. jump frequencies and correlation factors.

where gi is a geometric factor, Qia jump length, and I’,(efr) 7. IRON-BASEDSPINEL SOLID SOLUTIONS an effective jump frequency which contains an elementary frequency, an availability factor ( -) linked to Even more complicated than magnetite are iron-based the defect concentration) and a correlation factor. The spine1 solid solutions of the type (Fe, _xMe,)3 -aO,, in

Point defects and transport in non-stoichiometric oxides

which iron ions are replaced by other cations, Me. The number of variables to be taken into account for modeling in detail the point defect chemistry and defect-related properties increases quickly by increasing the number of chemically different species, especially when these species occur in different charge states. For example, in the case of (Fe, _,Mn,)s _ 604 at least four different types of cations are present; two types of cation vacancies and at least 12 types of interstitial species may occur. What would be needed to perform a detailed analysis and modeling of the point defect chemistry and defectrelated properties of iron-based spine1 solid solutions? First, one would need data on the variation of the oxygen content with oxygen activity. Such information is available for many systems (e.g. for (Fe, _ XMe,), _ &04 with Me = Mn [12-141, with Me = Co [15], with Me = Co and Mn [16-181, with Me = Ti [19] and with Me = Cr [20]). To attribute defects to different locations one would need additional data from in-situ investigations, which allow unequivocal conclusions to be made with regard to the distribution of ions and defects instead of only allowing conclusions which are based on assumptions for which no direct experimental proof exists, like in the case of the use of thermopower data to derive cation distribution data. Furthermore, one would need results from in-situ investigations that provide information on the individual mobilities of different species. If at all possible, obtaining the type of data described above would be very tedious and time-consuming. First, suitable techniques would have to be developed that do not exist now. For example, it is unclear how to obtain experimentally information on the in-situ distribution of point defects which exist only in small concentrations. At present the author of this article does not have much hope that the data which would be needed will become available from experimental work in the near future. An interesting question is therefore whether the reliable information denoted above as being needed for a detailed analysis of the defect chemistry and defect-related properties could be obtained by theoretical calculations instead of from experimental work. Despite the problems existing for the detailed analysis of the defect chemistry and defect-related properties of iron-based spinels, the author’s group has performed many experimental studies on the deviation from stoichiometry and the cation tracer diffusion in such spinels (e.g. (Fe, _XMeX)3_604 with Me = Co [15], with Me = Mn[12, 13],withMe=CoandMn[16-18],withMe= Cr [20], with Me = Ti [ 191and, very recently and still in progress, with Me = Ni [21]). Qualitatively, the defect chemistry of magnetite-based spinels is similar to that of magnetite, i.e. cation vacancies and cation interstitials are the majority defects. The oxygen activity dependences of 6 observed for these spinels obey eqn (I 3). Examples for experimental data for 6 in (Fe, -.CoX),_604.

,

6[

515

I

I

I

I

I

5-

6

4-

T = 1200 “C

2-

0 data pomts + reference pomt

1 -

’

-1

-5

I

I

I

-3 log 10

-2

-1

1

5-

(Cro 33Feo 67k+a04 T = 1200 “C

4: 3

I

0

data

+

reference

0

aoz

6

4cl

I -4

I

I

I

I

pants po*nt

x stalchnometric

pant

2-

0

1 -

-1’

I -11-10

-9

I

I

I

I

-8

-7

-6

-5

log10

I -4

ao2

I I II I No .-l

1.0

I

I

I

I

I

I

I II I I II I I II I

-10.0

-9.6

-8.0

-8.5

-8.0

-7.5

-7.0

-I I

0

datapoint.a

+

reference

point

4

I ,

lOa

ao,

Fig. 12. 6 in (Fe I_IMe,)l_a04 at 1200°C as a function of the oxygen activity, a%: (a) Me = Co and x = 0.3 [15], (b) Me = Cr and x = 0.33 [20], and (c) Me = Ti and x = 0.2 [19].

(Fe, _rCrr)3_604r and (Fe, _,Ti,),_,O, are shown in Fig. 12(a-c), respectively. The lines shown in these figures were generated by fitting the experimental data to the latter part of eqn (13). It is interesting that it makes no difference for the type of oxygen activity dependences observed for 6 (and, as shown later, also for Dt) whether the ions replacing Fe ions in magnetite-based spinels have the valence state 4+ ( - Ti4+), 3+ ( - Cr3+) or 2+ (- coZ+). As mentioned above, qualitatively the point defect chemistry of magnetite-based spinels is very similar to that of magnetite and eqn (13), having been derived for pure magnetite, can also be used to describe the oxygen

R. DIECKMANN

516

-4

I M.Z=bkl

-6 0.0

I 0.2

11 0.4

I 0.6

, 0.8

1.0

X

Fig. 13. Values predicted for the defect constant [VI0 in (Fel_,Me,)~_a04 at 1200°C with Me = Me*+, Me3+ and Meti, respectively, in comparison with data for IV]’ derived from experiments for Me = Co, Mn, Cr and Ti. dependence of 6 in spinels of the type (Fe, _xMex)s _*04 containing Me ions with the valence states 2+, 3+ and 4+. It seems that the component activity-dependent disorder in spinels is essentially due to valence state changes of iron ions as a function of the oxygen activity. With this in mind, Aragon and McCallister [22] have proposed that values of the defect constants [VI0 and [I]’ for magnetite-based spinels can be predicted by making use of the equilibrium constants Kv and KI valid for pure magnetite and taking into account that the concentrations of Fe’+ and of Fe3+ ions change when iron ions are replaced by Me ions. Based on eqn (11) one can then write for [VI0 activity

tvlo=

[Fe2+13 [Fe3+]2

KV

(18)

Fe&‘4

where Kv refers to pure magnetite while [Fe’+], [Fe3+] and arelo4 refer to the solid solution (Fe, _ *MeJ3 _ a04. In an analogous way [I]’ can be expressed as (19) where KI refers again to pure magnetite while [Fe2+], [Fe3+l and a&oa refer to (Fe I_xMex)3_604.For making predictions it is necessary to relate the concentrations of Fe*+ and Fe3+ ions to the composition x of the solid solution. This can be achieved by considering the charge balance 2[Fe*+] + 3[Fe3+] + n[Me”+] = 4[0*-] = 8,

[Fe*+]=1 -2.x and [Fe3+]=2, for n=3 [Fe*+]=1 and[Fe3+]=2-3xandforn=4[Fe2+]=1+3xand [Fe3+] = 2 - 6x. Due to the l/3 exponents in eqns (18) and (19). the influence of the activity of magnetite in the solid solution on [VI0 and [I]’ is only relatively small compared to that of the variation of x, and not very critical for the variation of [VI0 and [I]’ with the composition X. Fig. 13 and Fig. 14 show a comparison between values estimated for [VI0 and [I]‘, respectively, for n = 2,3, and 4 and values derived from experimental data for Me = Co, Cr, Mn and Ti. The dashed lines have been calculated by ignoring the variation of aPelo with x; only for calculating the dash-dotted lines for II =4 was this activity variation taken into account. Fig. 13 and Fig. 14 show that the predictions made for [V] ’ and [I] ’ and n = 2 are in relatively good agreement with the values derived for these defect constants from experimental data for (Fe, _XCo,)3_*04, especially for [VI’. In this oxide solid solution most of the cobalt ions are present as Co’+ ions. For (Fe,_rCrx)3_604. in which the Cr ions are very likely present predominantly as Cr3+ ions, there is no reasonable agreement observed between the curves predicted for [VI0 and [I]’ based on the approach proposed by Aragon and McCallister. The same is true for (Fe, _,MtQ3_604. The situation is better for (Fe, -.Ti,), _ 604. Here the agreement between predicted values and those obtained from experimental data for 6 is relatively good for [I]’ while for [VI0 there is only an agreement in the trends. The preceding comparison shows that estimates for the defect constants [VI0 and [I]’ made using the approach proposed by Aragon and McCallister only produce in some cases values for [VI0 and [I]’ which are close to the corresponding values derived from experimental data for 6. Therefore this approach is not useful for reliable defect concentration predictions in magnetite-based spinels. What are the reasons for this? First, when iron ions are replaced by other cations, changes may occur in the cation and defect distribution. This subject is ignored in Aragon and

(20)

where n denotes the charge of the ions of the type Me, and the site balance [Fe*+] + [Fe3+] +n[Me”+]

= 3.

(21)

In these balance equations, the deviation from stoichiometry, S, is ignored. It is obvious that the partition of Fe ions into Fe*+ and Fe3+ ions depends on the charge state of the Me ions. Solving for the concentrations of the differently charged iron ions gives for n =2

Fig. 14. Values predicted for the defect constant CI]” in (Fea_,Me,)3_~Ol at 1200°C with Me = Me*+, Me + and Me +, respectively, in comparison with data for [I]’ derived from experiments for Me = Co, Mn, Cr and Ti.

Point defects and transport in non-stoichiometric oxides

(COO

5 *Z o-100

follows from the oxygen activity dependences observed for the cation tracer diffusion coefficients at high and low oxygen activities, respectively. It is interesting that Cr (probably predominantly Cr3+) ions and Ti (probably predominantly Ti4+) ions diffuse much slower than Fe, Co and Mn ions. This observation cannot be related to the size of the ions. If the ion size were the most important factor for the diffusivity of ions, one would expect that smaller ions would move faster than bigger ones. However, Cr3+ and Ti4+ ions are similar in size or smaller than Fe, Co and Mn ions with the charge states 2+ and 3f. Therefore, it may be that this behavior is more related to the charge state of the ions during their motion than to the ion size.

3Feo&t-a%

T = 1200°C

3

25 -90 -

-

Y -110

I -1

I I -3 -2 log10 aoa

I -4

’ -5

517

-8 -9

-2 t-i--gj -lo

8. HOW DOES THE VALENCE STATE OF AN ION AFFECT ITS DIFFUSIVITY?

7:,-H

l z

z-12

2 - -13 -14’

5.

-8 log10

-10

-4

-6 ao2

-8

N$

-8

5 .E P

-9

-1001

/ -100

I -95

I -9 0 log10

I -85

I -8.0

I -7 5

sol,

Fig. 15. D&, in (Fe t_,Me,),_x04 at 1200°C as a function of the oxygen activity, a4 : (a) Mx = Fe, Mn and Co, Me = Co, x = 0.3 [15],(b) Mx = Fe, Co, Mn and Cr, Me = Cr and x = 0.33 [20], and (c) Mx = Fe, Co, Mn and Ti, Me = Ti and x = 0.2 [ 191.

McCallister’s approach. Second, the defect formation energies involved in the formation of cation vacancies and cation interstitials may change withx, another subject that is not considered. Examples for experimental cation tracer diffusion data for (Fe, _XCo,)3-604, (Fe, -XCrX)3-604, and (Fe, _ xTix)3_ &Ohare shown in Fig. 15(a-c), respectively. The lines shown in these figures for the tracer diffusion coefficients of different cations were obtained by fitting experimental data to eqn (17). As in magnetite, the cation diffusion in all magnetite-based spinels studied so far is governed at high oxygen activities by cation vacancies and at low oxygen activities by cation interstitials. This

An interesting question, also in view of the preceding discussion on the diffusion of ions in iron-based spinels, is whether the charge state of an ion influences its mobility and, if yes, how large this influence is. Different factors may influence the diffusivity of an ion on a lattice of an ionic crystal (e.g. point defect concentrations, the size of the moving ion, and the electrostatic interactions between the moving ion and the surrounding lattice). At a given charge state a decreasing size is expected to allow for faster diffusion. The size and the electrostatic interactions between a moving ion and the surrounding lattice depend on the charge state of the moving ion. Therefore, it must be expected that the mobility of an ion varies with its charge state. The influence of the charge state on the diffusivity of ions has been the subject of theoretical calculations by Sangster and Stoneham [23, 241. The results of these calculations suggest that the charge state influences frequencies and saddle point heights. and therefore the diffusivity. However, it remained unclear how Iarge the influence of the charge state on the diffusivity is and also in which charge state an ion diffuses faster if all other parameters involved are kept constant. The influence of the charge state of ions on their mobility is not only of academic interest but may also be important for the analysis of transport properties if ions of an element are present in an ionic crystal in different charge states (e.g. for the diffusion of transition metal ions in oxides and therefore for many practical oxidation problems). We were not aware of any previous experiments on the influence of the charge state of ions on their diffusivity that have provided information on the question whether there is any significant influence of the charge state or not, and, if yes, how large it is? Therefore, we have initiated experimental studies on this subject. One of these studies is now practically complete and the results obtained will be submitted for publication very

R. DIECKMANN

518

soon [25]. In the following, the background of this study and some results will be reviewed. The battle plan we have developed for obtaining information experimentally on the influence of the charge state of an ion on its diffusivity is to fix the point defect concentration in an inert material ( - MgO) by doping with another inert material ( - A1203) and then to vary the mean valence state of diffusing tracer ions ( - Fe-59) by varying the oxygen activity. In the case of Fe as diffusing ions in A1303-doped MgO, one expects that ion ions will be present as Fe*+ and Fe3+ ions and that the fraction of the iron ions present as Fe3+ ions will increase with increasing ao, , In the following the background of the analysis of the measured tracer diffusion data will be explained. The oxidation of Fe0 dissolved as tracer in Al2O3doped MgO (Al:MgO) to dissolved Fe203 ( - FeO1,s) can occur by the reaction 1 FeOAI:MgO + ;ioz s FeOi 3*,MgO.

(22)

The equilibrium constant for this reaction is K,= += uFeOao,

3+ [Fe IYF~O, J [Fe2+]yF~a~~’

The fraction of iron ions that are present as Fe3+, ~7,can therefore be expressed as 1

l+K,‘abl:‘=(~~~/a~,)“~+l’ (25)

where a& = 1/(K,‘)4 is the oxygen activity at which 7 = 0.5, i.e. [Fe3+]/[Fe2+] = 1. The oxygen activity a& is expected to vary with temperature. Eqn (25) suggests that a “master curve” exists for n as a function of log uo2 which has a shape that is independent of the actual thermodynamic conditions; with varying values of ah this curve shifts on the log ao2 axis. Fig. 16 shows a plot of n vs. log ao,/a&. The iron tracer diffusion coefficient, DE,, contains contributions from iron ions diffusing as Fe*+ and as Fe3+ ions. It can be expressed as

+ (l -r&c

2

rFe2+?Fe[(VMS2+

)“I,

5

0

-5

1 10

logto ao,/& Fig. 16. Oxygen activity dependenceof the fraction of the iron ions being present as Fe3+ions in a very dilute solution of iron ions in an inert oxide matrix with one cation sublattice; 7 =

[Fe3+]/([Fe2+]+ [Fe’+])and a& =ao, at which YJ= 0.5. where Dk2+ and D&1+ are the tracer diffusion coefficients of Fe*+ and Fe3+ ions, respectively, g is a geometric factor (g = l/6) and a an elementary jump length. rFe2+ and r&J+ are the jump frequencies of Fe*+ and Fe3+ ions, rCSpeCtiVely. jFe is a correlation factor and [(VM.$+)“] the cation vacancy concentration per lattice molecule of MgO. Rewriting eqn (26) and normalizing D& to the value of D& at 7 = 0.5 leads to

DK

of FeO, both dissolved in the MgO matrix. Because the dilute solution limit applies, the activity coefficients yFeo,~ and yr.$, are constants and one can write

Kc'&;

L -----?

-10

(23)

whereaFeO, 5 is the activity of FeO, 5and a rco the activity

[Fe3+] ‘=[Fe2+]+[Fe3+]=

0.0

(26)

D;,(v = 0.5) = (1-9)1+D*

2 re’+lD;=z+ 2

+17 The oxygen activity dependence of the iron tracer diffusion coefficient depends on the value of D&j+/D~,z+. If D;,+iD;, Z+> 1, one will observe an increase of D& with increasing ao2. If DE,,+ IDEel+ > 1, a decrease in D& with increasing ao2 will occur. The larger the ratios Feq+ respectively are, the larger D~e3+lD~,2+ and D~e2+iD* the variation in L& with ao, will be. For the diffusion experiments, samples of MgO doped with 0.5 mol% Al203 ( + Alo.~sMgo,&,~rsO) were prepared using a conventional ceramic method and also via a chemical synthesis departing from Al(N03)3+9H20 and Mg(N03)*.6H20 and using freeze drying. Because the concentrations of oxidizable impurities were negligibly small compared to the doping level, the point defect concentrations were fixed by the dopant concentration. It is believed that the following doping reaction applies: 2+ Al203 + 3(Mg,sz+ )’ Z 2(A1;2+ )’ + (Vwgz+)” + 3MgO. (28) As a consequence of the doping a constant cation vacancy concentration is expected to be established over the entire oxygen activity range studied at a given temperature. Then the iron diffusivity can only be altered by a change in the Fe valence state with ao2. The iron tracer diffusion experiments were performed by using carrier-free iron

Point defects and transport in non-stoichiometric oxides

0

20

40

60

60

100

Ax/w Fig. 17. Inverse of the error function of the argument 1 - A/A0 as a function of the thickness of the material removed from the sample, Ax, for a typical iron tracer diffusion experiment in AI:MgO. A is the radioactivity measured after removing material of the thickness AX, and A IS the initial radioactivity of the sample [25].

tracer (Fe-59). At 1200°C ao, was varied between 1 and

10-‘4.8 and at 1100°C between 1 and 10-‘5.9. Iron tracer diffusion coefficients were determined by using a residual radioactivity measurement technique, i.e. layers were stepwise removed from the sample and the residual radioactivity was measured after the removal of each layer. The applicable solution to Fick’s 2nd “law” for a thin film is

( >

A=l-et-f -

A0

hx

(29)

2m’

where A is the radioactivity after the removal of a layer of sample material with the thickness AXand A0 is the initial radioactivity before any layer removal. After plotting the inverse of the error function, erf-‘, of the argument (1 A/A’), defined as erf-’

hx

( > wm 1-s

=-

(30)

as a function of AX, one can determine from the slope a value for DA for a given diffusion time t. An example of

-

-1°.15 l’

-10.40

1’ -15

I -10 log10

-5

0

ao2

Fig. 18. Oxygen activity dependence of iron tracer diffusion coefficients, Dee,measured in AI:MgO at 1200°C [25J.

519

such a plot is shown in Fig. 17. The fact that the lines shown in this figure are straight and do not flatten at large values of Ax indicates that the diffusion along boundaries is negligible and that the results obtained reflect bulk diffusion. Experimental results for the iron tracer diffusion in (Ala,~sMgs,&.wrrO at 1200°C as a function of ao, are shown in Fig. 18. The data obtained for 1100°C follow a similar trend. At 1100 and 1200°C and a constant point defect concentration, D& increases with increasing oxygen activity. The data show the general trend observed in master curves generated for Dee,+ ID&,+ > 1 based on eqn (27) in combination with eqn (25) [25]. Due to the limited number of experimental data points we have not attempted to perform any data fitting. The line shown in Fig. 18 was generated by assuming D;=, + IDk, + = 1.6. A modeling of the tracer diffusion data obtained for 1100°C data requires a slightly larger D~e3+/D~e~+ ratio; a line generated with Die* + /Die1 + = 2.15 is close to the experimental data. This may suggest that the D&3+lD~,2+ ratio increases with decreasing temperature. In summary, the results of iron tracer diffusion measurements performed on samples of Al:MgO with constant vacancy concentration asf(ao,) show that D;e increases with increasing ao2. Because the variation of ao2 increases the mean valence state of the Fe-tracer ions, this observation suggests that Die,+ > Dk2+ in Al:MgO at 1100 and 1200°C. In the temperature range considered, the value of Dkl+ IDEe2+ is of the order of 2; there is a hint that the D~I+ IDie 2+ ratio may increase with decreasing temperature.

However, many questions remain unanswered. Is the increase of the iron tracer diffusivity due to a change in the electrostatic interaction between the moving ions and the lattice resulting from the increase of the charge from 2 + to 3 + , due to the decrease in the ion size when going from Fe2+ to Fe3+, or due to a combination of both? Is the observed increase in Die specific for Al:MgO, specific for simple oxides with the rock salt structure or a general phenomenon to be observed also in oxides with other structures? First results from an ongoing study on the diffusion of Fe-59 in MgA1204 with A1203 excess (Al:MgA1204) [26] show that the increase observed for L& in Al:MgO may be specific for Al:MgO. In the case of Al:MgAlr04. D& was found to decrease with increasing oxygen activity, just opposite to the behavior observed for D& in AI:MgO. However, the behavior of D& observed in AI:MgAlrO., could be due to the presence of two cation sublattices. It may be that iron ions change their coordination after oxidation, e.g. change from octahedral to tetrahedral coordination, and then move as Fe3+ ions relatively quickly on a different sublattice. In view of the current lack of understanding, further

R. DIECKh4ANN

520

investigations considering different structures are clearly needed. Another interesting but unaddressed question is how the influence of the valence state on the diffusivity varies in a given structure for different diffusion mechanisms, e.g. if the diffusion occurs by a vacancy mechanism or an interstitial (or interstitialcy) mechanism. An experimental study on this subject in the spine1 MgA1204 is underway.

9. CONTRIBUTIONS OF POINT DEFECTS IN NEARBOUNDARYREGIONS TO PROPERTIES OF POLYCRYSTALLINE

MATERIALS

In studies of defect-related properties at high temperatures it is very often (implicitly) assumed that the presence of near-boundary regions does not have any significant effect on the properties considered as long as the grain size is larger than a couple of micrometers. Some experimental results on the variation of the oxygen content in fayalite, FezSiOl, [27], in magnetite, Fe304, [28, 291 and in cuprous oxide, CunO [30], which will be discussed later suggest that such an assumption is invalid for these and possibly other materials, even at relatively large grain sizes. However, in the case of cobaltous oxide, COO,grain sizes in the micrometer range do not affect the deviation from stoichiometry 6 [31]. It is unclear why Co0 behaves differently from Fe$i04, Fe,Ol and CuZO with regard to the sensitivity of the oxygen content variation with oxygen activity on the grain size and also how other oxides behave with regard to this matter. The experimental observations for FeZSi04, Fe304 and CulO denoted above suggest that the point defect concentrations at near-boundary regions may be different from those established in the bulk. Important questions are (i) how large are these concentration differences, and, in cases where significant differences exist, (ii) how far do these differences extend from boundaries into grains, and finally, (iii) whether there exists for each material a critical grain size above which defects in near-boundary regions can be ignored? What are possible reasons for different point defect concentrations in near-boundary regions of ionic materials? One reason may be the presence of space charges at interfaces. In this case, the differences between point defect concentrations near boundaries and in the bulk will increase with decreasing distance from the boundary and vanish at a distance on the order of the Debye-length. At high temperatures and in oxides containing transition metals, the Debye-length usually does not exceed a couple of hundred Bngstriims. Therefore, space charges are very unlikely to influence significantly the variation of the oxygen content in relatively large-grained, polycrystalline oxides containing transition metals. Another factor that may cause differences in

point defect concentrations between near-boundary regions and in the bulk is a variation in the defect formation energy as a function of the distance from a boundary. Defect formation processes usually involve some relaxation of the crystal lattice around each defect formed. This relaxation is limited by the constraint of the surrounding lattice and therefore will occur to different degrees within the bulk, near boundaries and around dislocations. As a consequence, the Gibbs energy of point defect formation reactions will very likely be different from the bulk at locations near boundaries and dislocations. This energy difference will decrease with increasing distance from boundaries and dislocations. An interesting question is at which distance this difference will vanish. The number of results available for discussing the three questions posed earlier is very limited. Some insight is available from results for defect formation energy calculations [32-341 and some from the experimental results mentioned earlier. Defect energy calculations that consider the variation of this energy as a function of the distance to a boundary are relatively scarce. The author is aware of such calculations for NiO [32], for MgO [33] and for Cr203 [34]. The absolute defect energies calculated in these studies may not be very accurate due to the type of potentials available at the time when the calculations were performed. However, the results of all these calculations suggest that there is indeed a variation of defect formation energies as a function of the distance to a boundary in all the materials denoted above. Because the defect energy calculations mentioned before are all limited to relatively small numbers of lattice planes near boundaries, one cannot make any conclusions by how much the mean defect population changes in a polycrystalline material as a function of the grain size. In the following, the experimental results mentioned earlier which suggest that the point defect concentrations near boundaries are significantly different from those in the bulk will be reviewed. The first experimental observations of this kind made by the author of this article were on magnetite [28]. By using thermogravimetry, the change in the oxygen content for very fine-grained, polycrystalline magnetite, Fe3_604. was measured at 1000 and 1100°C as a function of the oxygen activity. The samples used were made by loosely pressing Fe203 powder together, sintering, and then reducing to Fe304. The result was samples with a significant amount of porosity; actual values for the porosity and the grain size were not determined. The experimental results for the variation of the oxygen content in such a magnetite sample at 1100°C are shown in Fig. 19 in comparison with data valid for single crystals; results for 1000°C look similar. The experimental data obtained for the oxygen content variation in the polycrystalline material suggest (i) that the

Point defects and transport in non-stoichiometric oxides

521

- -single crystals I

-5

I -12

-10

-8 bio

-6

I L -4

ao2

Fig. 19. Variation of 6 in single-crystal and polycrystalline

Fe9_&04at 1100°C.

overall variation of the oxygen content in the polycrystalline material is smaller than in single-crystal Fe3_604, (ii) that the oxygen activity dependence observed for the oxygen content variation in polycrystalline Fe3_a04 at high ao, is smaller than in single-crystal Fe,_804, and (iii) that the downward bending of the low oxygen activity branch observed for the oxygen content variation in single crystals in the polycrystalline material studied is practically absent. As discussed in Section 6, the bending of the low oxygen activity branch observed for magnetite single crystals is due to the presence of cation interstitials acting as majority defects. Its absence in polycrystalline magnetite suggests that the formation of iron ions on interstitial sites as point defects must be very strongly suppressed near boundaries and that the mean concentration of these defects near boundaries is significantly reduced. In the case of magnetite space charges cannot be responsible for the behavior described above because the Debye-length in this material is very small due to the Fe’+ F! Fe3+ + e’ equilibrium as well as the relatively large carrier mobility in this material. Due to the porosity present in the samples studied, it was for a long time unclear whether inner surfaces had significantly contributed to the oxygen activity dependence observed for the variation of the oxygen content in polycrystalline magnetite or not. To clarify this point, the author’s group has very recently prepared very dense magnetite with a grain size of 10 pm and experimentally shown that the bending of the low oxygen activity branch in an oxygen content variation vs. log ao2 curve observed for single-crystal Fe3_a04 at 850°C is practically absent in polycrystalline Fe3_a04 with a grain size of 10 pm [29]. This result suggests that the porosity present in the samples used in the earlier studies of the variation of the oxygen content in polycrystalline magnetite cannot be the main reason for the absence of the bending of the low oxygen activity branch. Consequently, a behavior of point defects near boundaries which is different from that in the bulk must be responsible for the observed difference in the oxygen

0 polycrystalline.

(Schmalzried

I

-10

I

-6 loglo

-6

[ISSZ]) I I

-4

aoz

Fig. 20. Oxygen activity dependence of the iron tracer diffusion coefficient, &, in single-crystal and polycrystalline Fel_h04 [35] at 1115°C.

content variation between polycrystalline and singlecrystal magnetite. This conclusion translates into defect formation energies that are in near-boundary regions and near dislocations different from those in the bulk. The currently available experimental data do not allow one to determine how large the differences in defect formation energies at different locations may be. For this purpose more systematic studies of the oxygen content variation in Fe3-604 as a function of the grain size are needed; such studies are underway. For magnetite films the experimental observation that the grain size in polycrystalline magnetite influences the oxygen content variation in this material suggests that the oxygen content variation in such films will very likely depend on the film thickness. There is also evidence from iron tracer diffusion studies that boundaries influence the defect population in magnetite very strongly. The oxygen activity dependence of the iron tracer diffusion coefficient in singlecrystal magnetite was discussed in Section 6 and is described by eqn (17). Fig. 20 shows a comparison between iron tracer diffusion coefficients, Dfi,, measured earlier by Schmalzried [35] by using fine-grained, polycrystalline magnetite samples with (interpolated) singlecrystal data for 1115°C. Different aoZ dependences of Dk result for single-crystal and polycrystalline Fe3_*04. The oxygen dependence of Dg, for polycrystalline magnetite at high a% is smaller than that for single-crystal Fej_*O,, qualitatively in agreement with what has been observed for the variation of the oxygen content at high oxygen activities. At low ao2, D& in single-crystal Fe3_a04 is larger than Die in polycrystalline Fes_&Or,an observation which is in contradiction with the usual expectation that the presence of boundaries enhances the diffusion of ions. If there is a contribution from boundary diffusion to the overall iron diffusion this contribution must be only very small. However, the influence of interstitials missing in near-boundary regions compared to the bulk seems to be very important for the overall cation diffusion in magnetite. It looks as if in the polycrystalline magnetite used by

R. DIECKMANN

522

o-o

2.0

a II

-50

100

- I I I I I

d

1.0

m

6 0.0

- 1 tjl 01 ,

-150 1 i

0

’ i -13

I I -11 -12 log10 aoa

I I -10

II

-1.0

’

I

-6

1

log10

Fig. 21. Experimental data for the oxygen content variation in

single-crystaland polycrystallinefayalite at 1130°C[27].

,

-4

-6

1

-2

ao2

Fig. 22. Variation of 6 in large-grained, polycrystalline CU~_~O (grain size > 1 mm) at 800°C[36]. the cation vacancies is

Schmalzried, interstitials practically do not contribute to the iron diffusion at all; otherwise one should see a minimum in I& as a function of the oxygen activity. It should be noted that more. experimental proof is desirable to prove further the correctness of the interpretation given above. This is because the diffusion coefficients reported by Schmalzried were determined by measuring the amount of radioactivity that has diffused after a certain time from a sample initially uniformly doped with radioactive ion into a sample originally free from any radioactive material and by assuming an ideal contact between the two samples during the diffusion anneal. Unfortunately, diffusion profiles were not determined. Therefore, there is no experimental proof whether Schmalzried’s assumption of an ideal contact is justified or not. Recently, the author’s group has studied the variation of the oxygen content in fayalite, FezSiOr, and found that the oxygen content variation in polycrystalline fayalite is significantly larger than that in single crystals [27]. Some results are shown in Fig. 21. Because the samples used were all very pure and dense, it is very likely that this observation is also related to a variation of defect formation energies in near-boundary regions. A more systematic study on the contribution of nearboundary point defects to the composition variation in oxides has been performed for cuprite, Cur-60. The variation of 6 in this oxide with the oxygen activity has been investigated earlier for very large-grained material (grain size > 1 mm) at temperatures between 800 and 1200°C [36]. In this study it was found that the majority of defects at high oxygen activities are neutral cation vacancies, i.e. associates formed between one hole and one cation vacancy, ( (Vcu+ )‘h’)‘( = (Vc-+ )“). The corresponding defect formation reaction is 2(Cu,+,+)” + f Ol(gas) Fi 2{ (Vcu+ )‘h’)’ + CuzO(srg). (31) The oxygen activity dependence of the concentration of

[(V,,+ )“I = l&;.

(32)

At low oxygen activities neutral oxygen vacancies, ( (Voz- )‘2e’)“( = (V,- )“), prevail. These defects are formed by the reaction (O$_ )” F? ( (Vo2- )“2e’}X+ 1 Oz(gas).

(33)

The oxygen activity dependence of the concentration of these defects is [(Vo2- )“I = a& 112. The deviation from stoichiometry, expressed as follows S=[(V,,+)“]

-2[(V,2-)“I

=Ktac

(34) 6, can then be - Kgo, '12,(35)

where K, and K2 are constants at a given temperature and total pressure. Experimental results for 6 in CU~._~O at 800°C as a function of ao2 are shown in Fig. 22. The line drawn corresponds to a fit performed by using eqn (35). For the sake of completeness it should be noted that it has been suggested [37] that neutral cation interstitials are the majority defects at low ao2 and not oxygen vacancies. Two facts speak against this suggestion: (i) the existing cation tracer diffusion data [38] show no hint for any contribution of cation interstitials to the cation diffusion and (ii) the experimental data for the oxygen content variation at low ao2 are much better described by an &z term than by an a:: term that would result for neutral cation interstitials as majority defects. More recently, the author’s group has performed measurements on the variation of 6 in Cu2_a0 as a function of the grain size at temperatures between 750 and 900°C [30]. The samples used in this study were prepared by sintering and had no open porosity and a density of -95%. Some results from this study for 800°C are shown in Fig. 23 in comparison with data obtained for very large-grained Cu2-60 (grain size > 1 mm) [36]. The trend of the results obtained for 750 and 900°C is similar.

Point defects and transport in non-stoichtometric oxides

0 large-graikd 3 (Xue, Dieckmann

observation was also made for the oxygen diffusion in Cur0 [40]. In the latter case, however, the diffusion is via minority defects. All of these observations suggest that the defects present near boundaries and in the bulk are the same. However, their concentrations near boundaries may be significantly different from the bulk. Based on this observation it has been assumed for the modeling of 6 in polycrystalline Cur-~0 that neutral oxygen vacancies and neutral cation vacancies are the majority defects in the bulk and also in near-boundary regions. 6 can then be expressed as

Q’.,( [ 19901) ,/d,o

o-

I

-1

-8

I

T = 800 “C 1

-6 logto aoz

-4

-2

6 = ci(KIagt - K2a~z”2) + (1 - cr)(K3ag: - K4aGl”2),

Fig. 23. Variation of 6 in polycrystalline Cu~_~Owith different grain size at 800°C (data for large grains from [36], grain size > 1 mm: data for smaller grains from Ref. [30]).

The experimental results clearly demonstrate that in CU~_~Othere is a very significant influence of the grain size on the variation of the oxygen content, i.e. that boundary-near regions contribute differently to the oxygen content than the bulk. The reason for this behavior is unknown. Two approaches can be used to model formally the influence of the grain size on the variation of 6 in polycrystalline Ct1-~0. In the first of these approaches, polycrystalline Ct1-~0 is considered as an assembly of approximately spherical grains of one size in which each grain has a core with bulk defect properties and a surrounding shell with different defect properties. In the second approach excess concentrations at boundaries are introduced instead of considering a shell with different defect properties. In the following only the first approach will be discussed further. Based on the assumptions made for using this approach, the deviation from stoichiometry in polycrystalline CU~-~O can be expressed in terms of contributions from the bulk, Bbu,and from near-boundary regions, bnb, i.e. 6=&,“+(1

_(lL)d”b,

(36)

where CYis the fraction of the material that has bulk properties. For sphere-like grains (Y is related to the grain diameter, do, and the thickness of the shell, ( = near-boundary region), Ad/2, as follows a=

volume of bulk

523

(37)

It is very unlikely that6 in samples with grain sizes > 1mm has any significant contribution resulting from nearboundary regions, i.e. the d-values measured for such samples can be considered to be valid for &. In diffusion studies it has been observed that the oxygen activity dependences of the cation tracer diffusion along boundaries and in the bulk are very similar in Fe3-604 [39] and also in NiO at high oxygen activities [39]. A similar

(38) where K3 and K4 are constants related to the formation of neutral cation vacancies and neutral anion vacancies near boundaries, respectively. A non-linear least square fitting of the experimental data obtained for 6 in polycrystalline Cuz-aO to eqn (38) was attempted. Unfortunately, fits providing simultaneously meaningful values for KS, K4 and Adcould not be obtained. However, it was possible to determine which values the ratios K3/K1 and KJK2 would have for different values of Ad. The values obtained for the ratio KdK2 scatter much more than those for KJK,; this is no surprise in view of the small contribution of the last term in eqn (38) to the curves shown in Fig. 23. Here only the data for KdK, are discussed further. If Ad/2 should be equal to the maximum value of the Debyelength (expected to be present at the Cu/Cu20 equilibrium; value estimated based on different data from the literature, see [30]), KdK, would be of the order of 200 in the entire temperature range investigated. The Debyelength at larger values of 6 (i.e. at high oxygen activities which are of interest here) is on the order of lo-’ cm. At this length K3/KI would be on the order of lOOfl,i.e. very large. If the ratio KdK,, however, should be only of the magnitude of 10, then Ad/2 would be on the order of 0.1 pm at 900°C and even larger at lower temperatures, e.g. about 0.25 pm at 750°C. We do not see any physical reason why the ratio KdKI should be as large as 100 or even larger; such large ratios are also not supported by any results from the calculations denoted earlier [32-341. Therefore, we believe that Ad has relatively large values at all temperatures investigated; this is unexpected and very surprising. The real physical meaning of Ad is still unclear. As has been shown above, in FeZSiOd, Fe3_604 and Cu2_60, the grain size influences very significantly the variation of the oxygen content at relatively large grain sizes while this phenomenon was not observed in COO. Currently there are only guesses for the reasons behind this phenomenon, with no real understanding. The author believes that the influence of the grain size on the overall defect population and defect-related properties is a topic where properly performed defect energy calculations

R. DIECKMANN

524

could be extremely helpful to reach a better understanding. Regardless of the current non-understanding of this matter, the experimental finding that in certain oxides the grain size influences the oxygen content variation and therefore the point defect population at relatively large grain sizes is somewhat scary. Resulting questions are: (i) which of the experimental data reported in the literature on the oxygen content variation in other oxides do not truly refer to bulk properties, (ii) which of the experimental data reported for other defect-related properties have the same problem, and (iii) which conclusions based on these data are erroneous due to using the non-justified assumption that the contributions of near-boundary material can be ignored?

10.

CONCLUDING REMARKS

In this article, some solved and some unsolved problems in the area of point defects and transport in nonstoichiometric oxides were discussed. Many more problems exist that were not mentioned. It was the intent of this paper to ask critically what we really know and understand about a few, selected topics and not to give a comprehensive overview. It was discussed that the principles of the relationships between point defect equilibria and defect-related properties in non-stoichiometric oxides are relatively well established. However, it was also demonstrated that things are not always as easy as assumed in many publications. The appropriate treatment of the electrostatic interaction between charged point defects is still a great problem, and therefore probably very often ignored in the literature. Some conclusions reported in the literature which were made by using “simple defect chemistry”, i.e. by ignoring the consequences of the electrostatic interaction between charged defect species, may be completely or at least to some extent erroneous. Another largely unsolved problem is to identify unequivocally the defect species present at thetmodynamic equilibrium, especially at larger point defect concentrations. Experimental methods suitable for determining the type of defect species being present under in-situ conditions are almost completely missing. In principle, one could think to use results of defect structure determinations performed on quenched samples at low temperatures to identify properly the defects of importance. Unfortunately, such results are always questionable because the concentrations and the local arrangement of defects may change during quenching, the latter even in the case of very fast quenching. Another topic discussed was the fact that the analysis of the relationships between defects and defect-related properties becomes very complicated when more than one sublattice exists for one type of ion. This is the case for cations in spinels. One would like to know how the

cations and the point defects present are distributed between different sites. For the distribution of the cations one can obtain some experimental information, e.g. from thermopower or Mossbauer studies. In order to extract the cation distribution information from the experimental data, sometimes assumptions are used that are not experimentally proven. This is a problem. However, a much more severe problem is that, to date, there is no experimental method available that the author knows which allows one to obtain information experimentally on the distribution of point defects. As a consequence, it is presently still impossible to perform any detailed modeling of point defects and defect-related properties in spinels and similar compounds. Therefore, considerations of defects and defect-related properties in such materials usually ignore the existence of different sublattices. This was discussed for magnetite, Fe3-&04, and formagnetite-basedspinelsofthetype(Fe, _,Me,)3_604. Finally two topics were addressed which are barely understood. The first one was the question of how the charge state of an ion affects its diffusivity. The second one was the question of how important the influence of boundaries in polycrystalline oxides on the overall oxygen content of such materials is. Results were presented for the diffusion of radioactive iron tracer ions in Al:MgO and AI:MgA1204 that show that the valence state of the iron ions has an influence on the mobility of these ions. It is still unclear what the real physical reason for this influence is. For reaching a better understanding of this topic much more experimental and also computational work is needed. The same holds true for the influence of boundaries on the oxygen content of polycrystalline oxides. Experimental evidence has shown that the variation of the oxygen content of the oxides FeZSi04, Fe3_a04 and Cur-60 may be very significantly influenced by the presence of boundaries, certainly in the latter two cases even at relatively large grain sizes. Such a sensitivity of the oxygen content variation to the grain size at relatively large grain sizes may also exist for other oxides. If this should be true, many experimental results that have been reported for the deviation from stoichiometry and also for other defect-related properties of nonstoichiometric compounds that were so far believed to reflect bulk properties may in reality not do so and, consequently, many conclusions based on such data may be incorrect. To find out for which oxides this could be a problem, new experimental work is needed that must include looking for possible grain size effects. authorthanks the U.S. Department of Energy for funding most of the research discussed in this article under grant DE-FG 02-88ER 45357. He is also grateful to the NSF-sponsored Cornell Materials Science Center (Grant DMR9 12 1654) for the use of central facilities.Finally, he thanks all his present and former undergraduate and graduate students, postdots and visiting scientists for their very significant contributions to this research.

Acknowledgements-The

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