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On defect clustering in the wustite phase
ON DEFECT CLUSTERING IN THE WUSTITE PHASE Mieczy~aw RI~KAS and Stanisl~aw MROWEC Institute of Materials Science, Academy of Mining and Metallurgy,AI. Mickiewicza30, 30-059 Krak6w, Poland Received 15 April 1986 ; accepted for publication 29 May 1986
Defect structure in highly nonstoiehiometric ferrous oxide has been discussed in terms of defect thermodynamics based on mass action law. It has been shown that the best agreement between theory and experiment is obtained under the assumption that 4 : 1 clusters, suggested by Cheetham and Catlow, are the predominant defects in the wustite phase. The ionization degree of these extended defects increases with increasing temperature, reaching a maximum value at 1573 K. Relative partial onthalpy and entropy of oxygen in Fe t _yO were evaluated.
Continuous attention has focussed on ferrous oxide, Fel_yO , (wustite) for many years. One of the most extensively studied problems of this oxide concerns its defect structure. Wustite exhibits exceptionally large deviations from stoichiometry, varying from 5 at% (a/o) at the Fe/Fel_yO phase boundary up to about 15 a/o at the Fel_yO/Fe304 interphase. This nonstoiehiometry results from cation sublattice defects in the form of cation vacancies, the effective negative charge of which is compensated by trivalent Fe-ions being electron holes. In earlier works [ 1 - 5 ] it has been assumed that despite their large concentration, cation vacancies in ferrous oxide are randomly distributed in the crystal lattice and can be treated in terms of a simple point defect model. In such an approach the formation of defects in Fel_yO may be described by the following quasi-chemical reaction: I
02(g ) ~ V~~ + ih" + 0 0 ,
(1)
where i = O, i, 2 denotes the degree of defect ionization *. The nonstoichiometry of the wustite is then given by: r
tt
y = [V~e] + [VFe] + [Vl~e].
(2)
Applying the mass action law to this equilibrium and assuming that interaction among defects may be * Kx6ger and Vink [6] notation of defects is used throughout this paper. 0 167-2738/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
neglected, as well as supposing appropriate electroneutrality conditions, one arrives at the following relationship between nonstoichiometry and equili. brium oxygen pressure and temperature:
Y =y o P ~ n exp (Ey/R T ) ,
(3)
where l/n, or simply n, is the parameter depending on the ionization degree of cation vacancies, assuming 2, 4 and 6 values for neutral, singly- and doublyionized vacancies, respectively: Ey denotes the temperature coefficient directly related to the enthalpy of defect formation. First experimental results [ 7 - 9 ] concerning the dependence of nonstoichiometry on oxygen pressure support the above mentioned simple defect structure model. A linear relationship between logy and logPo: has been obtained with a slope of 1/6, suggesting that the predominant defects in the wustite phase are doubly ionized cation vacancies. However, further extensive thermogravimetric [10-16] and emf [ 1 7 22] studies have demonstrated that the dependence of nonstoichiometry in ferrous oxide on equilibrium oxygen pressure is much more complicated and cannot be explained in terms of a point defect model. In fig. 1 the most representative results of this dependence are collected. It follows from this plot that the parameter n in eq. ( I ) does not assume a constant value, even in the limited part of the phase field of Fel_yO , but changes continuously with nonstoiehiometry, ex-
186
M. Rfkas, S. Mrowec/Defect clustering in the wustite phase
-08
^,t--10
;o$
• *
+
o
~
•e
o
• •
4.
_¢
c~O
o
,p "
"
"12
,¢
,. 2o e"
o
~
1It
,
~ ."
,
I
•
• n
-1~
/ -
o"
•
al
-16
,~" ir
oo x,
.
mll
~t +
o,-,,"
•
ir
,~
qr
~ ,
•
,l •
"
~ ~ •
.
/ /n:5
/ 1/I[ Y:c°nst'Po 2" -
to n
A
-12
-10
-8
Ig P02 Fig. 1. Logy =f(logPo2 ) for several temperatures: (*) ref. [15], (u) ref. [42], (m) ref. [2], (A) ref. [21], (~) ref. [8], (e) ref. [43], (X ref. [18], (o) ref. [14], (4) ref. [16], (+) ref. [10].
ceeding considerably the maximum value of six, predicted by the point defect model. In addition, from fig. 1 it foUows clearly, that the nonstoichiometry, and consequently cation vacancy concentration, decreases with increasing temperature - negative A HF values - that is in contrast with simple defect structure. Rather, strong interaction and clustering of defects could have been predicted. This conclusion has been confirmed in subsequent X.ray [23,24] and neutron diffraction [25,26] studies. It has been shown that not only cation vacancies but also interstitial cations occur in the wustite sublattice. In fact, for each vacancy which is created as a result of nonstoichiometry (eq. (1)), a Frenkel defect pair is simultaneously formed: FeFe ~ Fe;'" + V~e ,
(4)
by passing of trivalent octahedral iron ion, representing electron hole, to tetrahedral interstitial site. This implies that the real concentration of cation vacancies is twice that resulting from nonstoichiometry, thus reaching enormous value of about 30 a/o. Due to the mutual interaction, vacancies and interstitials form
extended defects, or defect clusters [27-35]. Roth [25] suggested 2 : 1 clusters consisting of two doubly ionized octahedral cation vacancies and one trivalent Fe ion in tetrahedral interstitial position. Assuming that this type of defect predominates, Kofstad and Hed [36] treated this problem in terms of point defects thermodynamics based on the mass action law. Formation of the Roth's complexes may then be written as follows: FeFe + Vt + ½0 2 ~-(VFeFe . . . . . i. . VFe) + O o ,
(5)
where Vt represents an interstitial (tetrahedral) site available for occupancy by complex defect. The dissociation of this complex is given by (V~eFe~'" VI~'o)x ~ (Vl~e]Fe~''V~e)' + h ' .
(6)
In this case the nonstoichiometry of Fel_yO becomes i
e°l
tt
X
PI
oea
y = [(VFeFe i VFe ) ] + [(VFeFe i
II
l
VFe ) ] .
(7)
Two limiting cases may be considered. Case 1 (nonionized complexes predominate):
[(V~eFe/"'V~e) x ] >> [(V~eFe/"V~e)'] .
(8)
M. Rfkas, S. Mrowec/Defect clustering in the wustite phase
187
If the conditions (8) and (11) are not satisfied, one obtains:
10
y - h = K1P1J2(1 - h - 2 y ) ( 2 - 7y),
(14)
h2 =Ki(1-h-
(15)
2y)(y - h ) ,
where h = [h" ]. Eqs. (14) and (15) make it possible to determine the dependence o f y on oxygen pressure and temperature. However, the values of the equilibrium constants K 1 and K i must be known. These values have been calculated from data reported in refs. [36-38] :
e-
K 1 = 1.75 X 10 -9 exp(-332.2
kJ/mol/RT),
Kt = 9.13 X 102 exp (-74.9 kJ/mol/RT).
5'0
160 y.1000
Fig. 2. n = f ( y ) curve 1 nonionized, curve 2 ionized clusters
Application of mass action law to the reaction (5) yields
7y)]Po~/2 •
(17)
Using eqs. (14) and (15) a general relationship can be obtained, describing the dependence of the parameter n on temperature and nonstoichiometry:
150
respectively.
g 1 = [y/(1 - 2y) (2 -
(16)
=[ ~lny ~-i n ~OlnP02] T =2y
(9)
[yl_ h
2
+Ki(
+ 1-y-2h
1-4y+h X 2h+K/(1-y-2h)
1
1-h-2y
y
1_ )
] "
(18)
The parameter n in eq. (3) is given by
.
=(
a l n y ~-1
~alnP02] T
=2+14_-~y+2%.
(10)
It follows from this relation that the parameter n increases with nonstoichiometry, from, 2.6 up 5.3 (curve 1 in fig. 2). Case 2 (ionized complexes predominate): [(V~eFe/"'V~e)' ] ~, [(V~eFe/"'V~e) x
],
(11)
Applying the mass action law to the reactions (5) and (6) gives
K1Ki = Lv2/(1 -
3y)2(2 -
7y)]Polz/2 ,
(12)
and n=4+
14y +2-7y"
(13)
Also in this case the parameter n increases with nonstoichiometry (curve 2 in fig. 2) varying from 5.0 up to 9.8.
h
Fig. 3 shows this dependence on the background of theoretical curves for nonionized and ionized Roth's complexes, presented previously in fig. 2. It follows from this plot that all curves calculated with the use of eq. (18) are situated between both theoretical curves 1 and 2, suggesting the existence of both types of Roth's complexes in comparable concentrations. In addition, from fig. 3 it can be inferred that the concentration of ionized complexes increases with temperature. Finally, from fig. 4, it follows clearly, that the dependence of nonstoichiometry on oxygen pressure calculated by means of eqs. (14) and (15) for several temperatures is in fairly good agreement with experimental data. It could then be concluded that the defect structure of ferrous oxide may be satisfactorily described by 2 : 1 defect clusters (Roth's complexes). However the earliest diffraction studies of Roth [25] were performed on quenched samples. Much more detailed
M. Rckas, S. Mrowec/Defect clustering in the wustite phase
188
10
//~147_3K
//.,~1373K E
16o
y.lO00
Fig. 3. n = f ( y ) curve 1 nonionized, curve 2 completely
ionized clusters and partials ionized clusters for several temperatures.
neutron diffraction studies ofCheetham et al. [26,33] carried out at high temperatures, within the thermodynamically stable region of the wustite phase, have shown that the vacancy-to-interstitial ratio (RvI) is higher than Roth's value of two, suggesting that the Roth's cluster is an oversimplified model of the defect structure of Fel_yO. It has been found that the R v I ratio varies with nonstoichiometry between 3 and 4, reaching its highest value near the Fe/Fel_yO phase boundary, i.e., at the lowest nonstoichiometry. It follows from these results that at low concentrations of defects, 4 : 1 clusters can form, and with increasing defect concentration more complicated, larger aggregates of defects should prevail. This conclusion has been fully confirmed by Catlow et al. [39-41] in their elegant theoretical studies based on computational techniques. It has been shown that the binding energy of the 4 : 1 cluster (4V~e, Fe~") 5- is very high ( ~ 2 eV per defect) identifying thus this cluster as the basic unit of the defect structure of the wustite phase. As the defect concentration increases, these clusters aggregate, forming greater assembles as shown schematically in fig. 5. It was, then, reasonable to
,
-0.8
~
^~
~.,k-~'-b ~
E~
/
-16
-it,
46 Ig P02
Fig. 4. Logy = f(logPo2) theoretical curves for 2:1 cluster on the background of experimental results: (o) ref. [ 15], (o) ref. [42], (u) ref. [2], (A) ref. [21], (®) ref. [8], (®) ref. [43], (×) ref. [18], (o) ref. [14], (z~)ref. [16], (+) ref. [10].
M. R~kas, S. Mrowec/Defect clusteringin the wustite phase Y I0.I0-3
The condition of the constant value of the cation-toanion ratio becomes
1.0.10-2
5 [FeFe] + ~ (i+ 4)C i = [Oo] = 1. i=0 80"
OOK I \
[Vt]= 2 - ~ y .
~'
y ~\ /k
,o.. oj?IX\ -25 -50 Ig y
The set of eqs. (20)--(24) enables the calculation of the dependence of nonstoichiometry on oxygen pressure if all six equilibrium constants are known. In order to simplify the procedure it was assumed that in the given temperature and pressure range complexes with the effective charge - j predominate. Thus, eqs. (20)--(23) become
-15
Fig. 5. Scheme for Catlow [41] defect clusters.
analyze the dependence of nonstoichiometry on oxygen pressure and temperature in an analogous way as it has been done previously, taking into account all defect clusters suggested by Catlow [41]. As the first step, the formation of 4 : 1 clusters is considered and subsequently a general case, comprising greater defect assembles, will be discussed. The formation of 4 : 1 complexes may be written as follows: (i ÷ 1)FeFe÷ Vt + ~ 0 2 ~ [(VFe)aFei]i+ 30 0 + iFe~e,
(19)
where i = 0, 1 ... 5, denotes the degree of ionization of the cluster. Electroneutrality condition for this reaction is given by 5 h = [Fete ] = ~ iCi,
h =jC/,
(25)
y = 3Ci,
(26)
[FeFe ] + (4 +])Ci = 1,
(27)
[Vt] = 2 - ~ y .
(28)
Using these relationships the equilibrium constant, K/, can be obtained K/=(
Y ]/+1 a~ ?-3/2 3-(]+4)y'] 2 - ~ - y 02 '
(29)
where: a~ = 1 for/" = 0 and a~ =]/forj = 1,2 ... 5. The dependence of the parameter n in eq. (3) on stoichiometry calculated from eq. (29) is given by ] - 1 = T2[(j__+ 1) 3 - (/+ 3 4)y T. 6---S~yJ 23y "] ' n ={\bhaPo,]Tblny
(30) (20)
where Ci denotes the concentration of 4 : 1 dusters, having effective charge - i . The deviation from stoichiometry is, then, given by 5 y = 3 ~ Ci . i=0
(23)
Consequently, the equilibrium constant of the reaction (19) assumes the form: (24) Ki ={Cihi/[FeF e ] (i+ 1) [Vt] ) t,032/ 2.
,o J2/ -t,0-=-;5--3ff
(22)
The concentration of tetrahedral interstitial sites available for occupancy by defect clusters is given by the following equation:
k [
Acot.voc.1200K ~ ,. . . . . lt, OOK4\
189
(21)
Fig. 6 shows this dependence for six different ionizations of 4 : 1 defect clusters. Comparison of these relationships with the experimental values indicates that the best agreement is observed for completely ionized complexes (] = 5). In order to prove this conclusion, the equilibrium constant K 5 must be calculated from eq. (29). For these calculations, the equi-
M. Rckas, S. Mrowec/Defect clustering in the wustite phase
190
Fe Fel_yO 1673KI, 1573K~, 1473KI-
112
l i b r i u m o x y g e n pressure c o r r e s p o n d i n g t o a given value o f n o n s t o i c h i o m e t r y s h o u l d b e k n o w n as a funct i o n o f t e m p e r a t u r e . Most c o r r e s p o n d i n g results h a v e b e e n a c c u m u l a t e d f o r 1473 K. T h u s , f o r this t e m p e r a t u r e m e a n values o f P o 2 f o r several values o f y h a v e b e e n d i r e c t l y d e t e r m i n e d f r o m t h e p l o t p r e s e n t e d in fig. 1. F o r o t h e r t e m p e r a t u r e s i n t e r p o l a t i o n forrffulas r e p o r t e d in [ 1 6 , 2 2 ] h a v e b e e n u t i l i z e d in t h e s e calcul a t i o n s . I n a d d i t i o n , s u c h f o r m u l a s as t h e s e in [ 1 6 , 2 2 ] h a v e also b e e n used i n c a l c u l a t i n g PO2-values for 1473 K. Based o n all t h e s e results K 5 values h a v e b e e n o b t a i n e d as a f u n c t i o n o f t e m p e r a t u r e ( 9 7 3 1573 K ) a n d n o n s t o i c h i o m e t r y ( 0 . 0 5 ~< y ~< 0 . 1 4 5 ) . T h e s e d a t a are c o l l e c t e d in t a b l e 1. It follows f r o m this c o m p a r i s o n t h a t l o g K 5 values d o n o t p r a c t i c a l l y
Fe~ -I -t 4~/'j=5
1373K
1273K I"
1173K.
///j-~
lO73K
/
5'0
I00 y.1000
150
•
Fig. 6. n = f ( y ) for six 4 : 1 clusters of different ionization degrees.
Table 1 LogKs values (atm -a/2) for completely ionized 4 : 1 clusters in Fe I _yO. Temperature(K) 1
2
3
4
5
6
7
8
9
10
11
12
y X 1000
973
1073
1173
1273
1373
1423
1473
1473
1473
1523
1573
50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145
25.42 25.52 25.61 25.68 25.73 25.77 25.80 25.82
18.02 18.14 18.23 18.30 18.36 18.40 18.43 18.45 18.46 18.46 18.45 18.44 18.41 18.39 . . . . . .
15.24 15.35 15.44 15.51 15.56 15.60 15.62 15.64 15.64 15.64 15.63 15.61 15.58 15.55 15.51 15.47
12.90 12.99 13.06 13.11 13.15 13.21 13.24 13.25 13.25 13.24 13.23 13.20 13.17 13.14 13.10 13.05 13.00
11.85 11.93 11.99 12.03 12.05 12.12 12.14 12.16 12.16 12.15 12.13 12.11 12.08 12.05 12.00 11.96 11.91 11.85
10.85 10.95 11.03 11.09 11.14 11.17 11.19 11.19 11.19 11.18 11.16 11.13 11.10 11.06 11.01 10.96 10.90 10.84 10.78
10.99 10.96 11.06 11.09 11.06 11.08 11.07 11.12 11.13 11.15 11.13 11.10 11.11 11.04 11.08 11.05 10.99 11.00
10.90 10.98 11.04 11.08 11.11 11.12 11.14 11.16 11.17 11.16 11.15 11.13 11.11 11.08 11.04 11.00 10.95 10.90 10.85
10.07 10.14 10.18 10.21 10.21 10.20 10.23 10.24 10.25 10.24 10.23 10.21 10,18 10.15 10.11 10.07 10,02 9.97 9.92 9,86
9.17 9.25 9.30 9.34 9.36 9.31 9.34 9.35 9.36 9.36 9.35 9.33 9.31 9.28 9.24 9.20 9.16 9.11
logKs standard deviation
25.67
21.68
18.35
15.53
13.11
12.04
0.14
0.13
0.13
0.11
0.11
0.10
. . .
21.44 21.54 21.61 21.68 21.72 21.76 21.78 21.80 21.80 . . . . . .
. .
11.05 0.13
%05 9,00
11.07
11.06
10.13
9.26
0.05
0.10
0.14
0.11
191
M. Rekas, S. Mrowee/Defect clustering in the wustite phase 1300
1100 1000 900
800
700"C
25
Table 2 Thermodynamic data for the formation of completely ionized 4:1 clusters~ Reaction
6FeFe+ Vt + ~O2 [(VFe),tFei]s - + 300+ 5Fete
Equilibrium constant Enthalpy of reaction Entropy of reaction Confidence interval (0.95 confidence level)
20
15 v
10
lnPo2 =
16
Fig. 7. LogKs =f(T-l).
change with nonstoichiometry which implies that these data represent real values of the equilibrium constants. Least standard deviation is observed in column 9 of table 1, because K 5 values collected in this column have been calculated from Poz-data obtained directly from the pressure dependence of nonstoichiometry plotted in fig. i and not from interpolation formulas for the same temperature (columns 8 [22] and 10 [16]. Fig. 7 shows the temperature dependence of the discussed equilibrium constant in the Arrhenius plot. As can be seen, the linear relationship is obtained with an extremely good correlation factor equal to 0.9999971. It implies that the enthalpy and entropy of the formation of completely ionized 4 : 1 clusters (eq. (19) w i t h / = 5) assuming constant values in the discussed experimental conditions. These data are collected in table 2. To prove the above discussed theoretical results, eq. (29) has been rearranged in order to get the suitable relation betweeny andPo2 f o r / = 5:
K = 3.96 X 10-Is exp(802.7 kJ/RT) AH.f = - 802.7 kJ/mol aS.f" = -333.2 J/mol • K
801.4 kJ/mol < - A H f < 804.0 kJ/mol 332.3 J/mol • K < - a S f < 334.1J/mol.K
32_(AHI AS1 RT R
6 ln(3/y - 9)
-in(2-~y)+in55).
(31)
Fig. 8 shows the calculated dependence o f y in Fel_yO on oxygen pressure on the background of experimental data for several temperatures. As can be seen the agreement is quite satisfactory. Eq. (31) can also be utilized for calculations of the relative partial free enthalpy of oxygen in the wustite phase: AG O = ½ R T l n P o 2 .
(32)
Taking into account the relation (33)
AG O = A H 0 - TAS---O ,
one obtains AH 0 = ~ AH£ = (-267.6 ± 0.4) kJ/mol,
(34)
and
- g / 3 (6 In ( 3 / ? - 9 ) - In(2 -
+ In 5 s ) .
(35)
As can be seen, the relative partial enthalpy of oxygen, AH0, is independent of comparison, which strongly supports the assumed defect structure model. Calculated &[/'0 and AS 0 data are summarized in table 3. Now, the general case will be considered concern-
M. R~kas, S. Mrowec/De[ect clustering in the wustite phase
192
_o.,! _~ -1.0 • /'
.;7 o °
Z.
0 °
-1.2
-1'6
-1¼
-12
-I0
-8
Ig P02 Fig. 8. Logy = f(logPo2 ) for several T experimental and theoretical curves; continuous lines for completely ionized (/= 5), dotted line for/= 4;(e) rof. [15], (u) tef. [42], (=) ref. [2], (a) ref. [21], (e) ref. [8], (,e) ref. [43], (×) ref. [18], (o) ref. [14], (~) ref. [16], (+) ref. [10].
where 0 ~< i ~< 2m - 3n. The electroneutrality condition for this reaction is given b y
ing the formation o f defect clusters built o f m cation vacancies and n interstitial iron ions. The formation o f such clusters may be written as follows: (n + i)FeFe + n V t +
2m-3n h=
m-n
~ i=1
(37)
iC i .
The deviations from stoichiometry, in turn, be-
X 0 2 ~ [(VFe)m (Fei) n ] i - + ( m - n)O O + i F e ~ e , (36)
comes
Table 3 Relative partial enthalpy, AHo, and entropy ASo of oxygen in the wustite phase (95% confidence level).
--AHo (kJ/moD
- A S o (J/reel • K)
y X 1000
this work
ref. [16]
ref. [22]
this work
ref. [16]
ref. [22]
57 65 74 83 91 107 i 15
267.7±0.4 267.6±0.4 267.6±0.4 267.6±0.4 267.6+0.4 267.6±0.4 267.6+0.4
256.6±2.3 259.9±2.0 264.7± 1.7 267.8±0.9 267.2±0.9 265.8±1.0 265.i± I.1
265.2 265.9 266.9 267.9 268.1 269.5 270.2
69.1±0.3 72.2+0.3 74.9±0.3 77.4±0.3 79.6±0.3 83.7±0.3 85.7+0.3
62.1±1.6 66.8+1.3 72.6±1.1 77.0+0.6 78.8+0.6 82.4±0.7 84.1+0.8
68.2 71.0 74,1 77.0 79.4 85.0 87.8
M. Rekas, S. Mrowec/Defect clustering in the wustite phase
y=(m-n)
2m-3n ~ Ci . i=0
(38)
Finally, the condition of constant value of the cationto-anion ratio can be written 2m-3n [FeFe ] + ~ i=0
(i + m ) C i = 1.
(39)
The concentration of tetrahedral interstitial sites available for defect cluster occupation is described by [Vt ] = 2 - aCi ,
(40)
where a denotes the number of tetrahedral sites close to cation vacancies involved in a particular cluster. The values of a for clusters to be considered are summarized in table 4. The equilibrium constant of the general reaction (36) assumes the following form:
Cihi
PO~(m-n)/21
193
better agreement appears for incompletely ionized defect complexes. This is conceivable if one considers the fact that, in general, the ionization degree increases with increasing temperature. Considering the standard deviations of logKi-values for particular clusters, it can be seen that the lowest deviations have been obtained for 4 : 1 dusters. This is most distinctly visible at the highest temperature. Fig. 9 shows the temperature dependence of equilibrium constants for completely ionized clusters. As can be seen, for all the clusters linear relationship in Arrhenius system of coordinates is observed. Calculated values of enthalpy and entropy of the formation of clusters under discussion are summarized in table 5. Following analogous procedure as for 4 : 1 cluster, relative partial enthalpy and entropy of oxygen can
(41)
Ki = [FeFe]n+i [Vt] n Considering a simplified electroneutrallty condition ( C / ~ O;Ci~ / = 0) eqs. (37)--(39) become:
h =jC/,
(42)
y = (m - n ) C / ,
(43)
[FeFe ] + (] +m)C/= 1 .
(44)
Combining eqs. (42)-(44) and (40) with eq. (41) yields y/+la/
Kj
m
[m - n - (1 + m)y]n+/,[2 - (ay/m - n)] PJ=(m-n)/2] -[ '
(45) where aj =//(m - n) n-1 f o r / ' ~ 0 anda/= (m - n) n - I f o r / = 0. Using the experimentally determined dependence o f y on oxygen pressure [16,22] the equilibrium constant K]. has been calculated for all possible j-values as a function of temperature of particular clusters. The results of these calculations are collected in table 4. It follows from this comparison that the best agreement with the mass action law at higher temperatures (1173-1573 K) is observed for completely ionized dusters, and at lower temperatures (973-1073 K)
70
7s
8o
go
+,0, Fig. 9. LogK = f(T -1) for completely ionized different clusters.
7 6 5 4 3
8 7 6 5
12 11 10 9
17 16 15 14 13 11
8:3 a = 46
10:4 a = 50
12:4 a = 55
16:5 a = 110
0
5 4 3 2 1
6 5 4 3 2
/
6:2 a = 32
4:1 a = 23
Kluster
98.26±0.33 98.97±0.23 99.29±0.22 99.78±0.08 100.25±0.14 101.41±0.26
71.14±0.20 71.50±0.09 71.92±0.08 72.40±0.17
53.48±0.14 53.91±0.05 54.42±0.10 55.02±0.20
44.24±0.15 44.66±0.05 45.16±0.08 45.78±0.18 46.52±0.27
34.89±0.14 35.29±0.04 35.82±0.06 36.50±0.16 37.23±0.28
25.67±0.14 26.03±0.04 26.53±0.07 27.27±0.15 28.24±0.24 29.83±0.34
973
Temperature(K)
83.97±0.27 84.29±0.16 84.64±0.16 85.06±0.15 85.51±0.25 86.66±0.39
60.46±0.14 60.81±0.09 61.21±0.15 61.67±0.27
45.47±0.10 45.88±0.07 46.38±0.16 46.95±0.27
37.58±0.12 37.97±0.05 38.46±0.13 39.05±0.24 39.76±0.36
29.56±0.12 29.94±0.04 30.43±0.13 31.04±0.24 31.83±0.35
21~8±0.13 22.02±0.04 22.50±0.12 23.16±0.22 24.18±0.30
1073
71.71±0.27 71.98±0.23 72.28±0.31 72.70±0.38 73.I1±0.53
51.50±0.19 51.79±0.27 52.13±0.42
38.76±0.12 39.12±0.22 39.55±0.37
32.01±0.12 32.35±0.15 32.68±0.38 33.30±0.44
25.11±0.12 25,42±0.14 25.86±0.29 26.42±0.45
18.35±0.13 18.64±0.10 19.06±0.26 19.67±0.39
1173
61.35±0.28 61.58±0.37 61.84±0.51 62.15±0.69
43.92±0.29 44.18±0.44
33.09±0.18 33.36±0.38
27.30±0.13 27.60±0.24
21.34±0.12 21.62±0.21 22.01±0.40
15.53±0.11 15.75±0.20 16.17±0.33 16.75±0.50
1273
Table 4 Log K values calculated by eq. (45) for several temperatures (95% confidence leveD.
52.53±0.32 52.73±0A6 52.98±0.63
37.49±0.36 37.72 0.54
28.27±0.23 28.57±0.41
23.29±0.14 23.57±0.30
18.13±0.13 18.39±0.28
13.11~0.11 13.36±0.21 13.73±0.39
1373
48.50±0.37 48.70±0.55
34.54±0.43 34.75±0.63
26.07±0.27 26.34±0A8
21.46±0.16 21.72±0.35
16.67±0.15 16.90±0.33
12.04±0.10 12.24±0.24 12.60±0.43
1423
44.90±0.38 45.07±0.58 45.28±0.80
31.88±0.55 32.08±0.69
24.08±0.35
19.81±0.22 20.07±0.37
15.34±0.20 15.53±0.39
11.06±0.10 11.21±0.30 11.58±0.47
1473
41.47±0.59 41.66±0.73
29.41±0.59 29.58±0.81
22.24±0.37 22.48±0.60
18.29±0.22 18.51±0.45
14.12±0.20 14.32±0.42
10.13±0.11 10.30±0.32 10.62±0.53
1523
38.29±0.46 38.45±0.70 38~4±0.93 38.87±1.16 39.14±1.38 39.83±1.82
27.07±0.57 27.25±0.80 27.47±1.03 27.76±1.25
20.49±0.35 20.72±0.59 21.04±0.82 21.46±1.04
16.83±0.20 17.05±0.44 17.37±0.66 17.80±0.88 18.37±1.08
12.95±0.19 12.94±0.64 13.47±0.63 13.93±0.84 14.57±1.03
9.26±0.11 9.42± 0.31 9.75±0.52 !0.26±0.72 11.03±0.90 12.45±1.07
1573
N.
e*
r~
Thermodynamic data for the formation of different defect clusters. Cluster type
/
Temperature range (K)
Mean standard deviation of log K
_ AH (kJ/mol)
- ~S (J/tool • K)
4:1
5 4 3 2
973-1573 973-1173 973-1073 973-1073
0.12+0.01 0.06±0.03 0.10±0.04 0.19±0.05
802.7± 1.2 807.4±27.1 816.0±48.9 820.3±40.7
333.2± 0.9 331.2±25.4 330.3±46.0 330.8+38.4
6:2
6 5 4
973-1573 973-1173 973-1073
0.16±0.04 0.07+0.06 0.10+0.05
1073.5± 1.8 1081.4±22.0 1088.1±50.0
434.7+ 1.4 435.1±20.1 432.0±46.9
8:3
7 6 5
973-1573 973-1173 973-1073
0.16±0.04 0.08±0.06 0.11±0.04
1341.1± 2.41 1345.0± 35.6 1363.0±112.1
530.7± 1.9 526.9±33.3 535.4±105.3
10:4
8 7 6
973-1573 973-1073 973-1073
0.24±0.10 0.11±0.09 0.13±0.04
1614.2+ 2.7 1615.9±17.7 1624.5+28.7
634.3± 2.1 628.1±46.9 626.9±77.1
12:4
12 11 10
973-1573 973-1173 973-1073
0.37±0.18 0.15+0.10 0.12±0.07
2156.5± 3.9 2153.4±76.8 2163.1±94.6
853.0± 3.0 845.5±72.3 845.1±89.0
16:5
17 16 15 14 13
973-1573 973-1173 973"1073 973-1073 973-1073
0.36+0.11 0.21±0.04 0.19±0.04 0.12±0.05 0.20±0.08
2939.6a 13.4 2949.1+66.9 2951.0±104.8 2958.9±75.4 2965.3±87.3
1135.3a10.6 1135.3±63.0 1130.9±98.6 1129.8±70.9 1127.3±81.9
Table 6 Relative, partial enthalpy and entropy of oxygen in Fel _yO. Cluster
]
--AH 0 (kJ/moD
--AS0 (J/tool • K)y 0.055
0.065
0.075
0.085
0.100
0.115
0.130
0.145
4:1
5 4 3 2
267.6 269.1 272.0 273.4
68.6 71.3 75.2 80.6
72.1 74.2 77.5 82.3
75.3 76.8 79.5 83.8
78.2 79.1 81.3 85.2
82.2 82.4 83.9 87.1
85.9 85.3 86.2 88.9
89.4 88.2 88.4 90.5
92.8 90.9 90.6 85.4
6 :2
6 5 4
268.4 270.4 272.0
69.8 72.6 75.0
73.1 75.4 77.3
76.0 77.9 79.4
78.7 80.2 81.3
82.5 83.4 84.0
86.1 86.4 86.5
89.3 89.4 89.0
93.0 92.3 91.4
8 :3
7 6 5
268.2 269.0 271.6
69.8 71.3 75.5
72.9 74.0 77.9
75.8 76.6 80.1
68.9 78.9 82.1
72.7 82.3 85.1
76.4 85.5 87.9
89.5 88.6 90.6
93.2 91.9 93.5
10:4
8 7 6
269.0 269.3 270.7
70.8 71.7 73.7
73.8 74.4 76.0
76.6 76.9 78.2
79.2 79.1 80.2
82.8 82.4 82.3
86.3 85.5 85.9
89.7 88.6 88.6
93.1 91.6 91.3
12:4
12 11 10
269.6 269.2 270.4
71.3 71.7 73.0
74.4 74.5 75.6
77.1 77.0 77.9
79.7 79.3 80.0
83.2 82.6 83.0
86.5 85.7 85.8
89.8 88.6 88.5
92.9 91.5 91.1
16 : 5
17 16 15 14 13
267.2 268.1 268.3 269.0 269.6
69.1 70.0 70.6 71.5 72.3
72.3 73.0 73.4 74.1 74.8
75.1 75.7 75.9 76.5 77.1
77.8 78.2 78.3 78.8 79.1
81.6 81.8 81.7 81.9 82.1
85.2 85.2 84.9 85.0 84.9
88.8 88.6 88.1 88.0 87.8
92.5 92.1 91.4 91.0 90.7
M. Rgkas, S. Mrowec/Defectclusteringin the wustitephase
196
be calculated, using eqs. (32), (33) and (45): A H 0 = (m - n) - 1 A H f , A S 0 = (m - n) - 1 X ln(m
{ASf-
(46) R [(j + 1 ) l n y + l n a / - ( n + / )
- n - (] +m)y) - n ln(2-ay/(m -
n))] ) . (47)
As can be seen, also in the case o f greater clusters the relative partial enthalpy o f oxygen is independent o f crystal composition. These data, together with entropy values, are collected in table 6. Using eqs. ( 4 2 ) - ( 4 5 ) the relation between enthalpies o f ionic and electronic defects m a y be obtained AHh.=/
alnh
01ny ]
, al/RT ) po2 =( ~I/RT ]po: ~lny ~ - I A H = 2, / --(m - n) ~ ~--~--~]
ainy = 2 \a lnP02 IT AHO "
(48)
It follows from this relation that the enthalpy of electron holes formation is also independent o f nonstoichiometry. Summing up these considerations, the following conclusions m a y be formulated. A thermodynamic approach to the problem o f defect structure in highly defected compounds does not lead to definite solutions. In the discussed case o f nonstoichiometric ferrous oxide the only definite conclusion is that noninteracting randomly distributed point defects do n o t predominate in the crystal lattice o f this compound. However, the forgoing analysis has shown that the formation o f different defect clusters is not equally probable. The best agreement between theory and experiment has been found, namely, for 4 : 1 clusters as the predominant defects in the wustite phase. It has been shown also that the ionization degree o f these complexes increases with increasing temperature, reaching a maximum value at 1573 K.
References [1] C. Wagner and E. Koch, Z. Phys. Chem. B32 (1936) 439. [2] L.S. Darken and R.W. Gurry, J. Am. Chem. Soc. 68 (1946) 798.
[3] J. Aubry and F. Marion, C.R. Acad. Sci. (Paris) 241 (1955) 1778. [4] D.M. Smyth, J. Phys. Chem. Solids 19 (1961) 167. [5] G.H. Geiger, R.L. Lovin and J.B. Wagner Jr., J. Phys. Chem. Solids 27 (1966) 947. [6] F.A. Kri~gerand H.J. Vink, in: Solid state physics, eds. F. Seitz and D. Turnbuli, Vol. 3 (Academic Press, New York, 1956) p. 307. [7] L.S. Darken and R.W. Gurry, J. Am. Chem. Soc. 67 (1945) 1398. [8] K. Hauffe and H. Pfefffer, Z. Metall. 44 (1953) 27. [9] I. Bransky and D.S. Tannhausor, Trans. Met. Soc. AIME 239 (1967) 75. [ 10] P. Vallet and P. Raccah, Mere. ScL Rev. Metall. 62 (1965) 1. [ 11 ] R.J. Ackermann and R.W. Sandford Jr., Tech. Rept. ANL-7250 (1966) 46. [12] R.L. Levin and J.A. Wagner Jr., Trans. Met. Soc. AIME 233 (1965) 159. [13] P.F.J. Landler and K.L. Komarek, Trans. Met. Soc. AIME 236 (1966) 138. [14] B. Swaroop and J.B. Wagner Jr., Trans. Met. Soc. AIME 239 (1967) 1215. [15] I. Bransky and A.Z. Hed, J. Am. Ceram. Soc. 57 (1968) 231. [16] E. Takayama and N. Kimizuka, J. Electrochem. Soc. 127 (1980) 970. [17] G.B. Barbi, J. Phys. Chem. 68 (1964) 2912. [18] H.G. Sockel and H. Schmalzried, Ber. Bunsonges. Physik. Chem. 72 (1968) 745. [19] F.E. Rizzo and J.V. Smith, J. Phys. Chem. 72 (1968) 485. [20] H.F. Rizzo, R.S. Gordon and I.B. Cutler, Symp. on Mass Transport in Oxides (Natl. Bureau of Standards Special Publication 296, Washington, D.C., 1968). [21] B.E.F. Fender and F.D. Riley, J. Phys. Chem. Solids 30 (1969) 793. [22] A.A. Giddings and R.S. Gordon, J. Am. Ceram. Soc. 56 (1973) 111. [23] J. Smuts, J. Iron Steel Inst. 204 (1966) 237. [24] F. Koch and J.B. Kohen, Acta Crystall. 25 (1969) 275. [25] W.L. Roth, Aeta Crystall. 13 (1960) 140. [26] A.K. Cheetham, B.E.F. Fender and R.I. Taylor, J. Phys. C4 (1971) 2160. [27] J. Manene, J. Phys. Radiat. 24 (1963) 447. [28] Tchiong Tki Kong, A.D. Romanov, Ya.L. Shayovitch and R.A. Zvintehuk, Vestn. Leningr. Univ. Fiz. i Khim. 4 (1973) 144. [29] C. Le Corre, F. Jeannot, J.P. Morniroli and J.M. Genin, CR Acad. Sci. Fr. C 276 (1973) 1335. [30] B. Andersson and J.O. Sletnes, Acta Cryst. A31S3 (1975) S.86. [31] G. Shirane, D.E. Cox and S.L. Ruby, Phys. Rev. 125 (1962) 1158. [32] D.P. Johnson, Solid State Commun. 7 (1969) 1785. [33] P.D. Battle and A.K. Cheetham, J. Phys. C12 (1979) 337.
• M. Rfkas, S. Mrowec/Defect clustering in the wustite phase
[34] J.R. Gavarri, C. Carel and D. Wiegler,J. Solid State Chem. 29 (1979) 81. [35] C. Lebreton and L.W. Hobbs, Radiat. Eft. 74 (1983) 227. [36] P. Kofstad and A.Z. Hed, J. Electrochem. Soc. 115 (1968) 102. [37] M.S. Seltzer, A.Z. Hed, J. Electrochem. Soc. 117 (1970) 815. [38] J. Nowotny, M. R~kas and M. Wierzbicka, Z. Phys. Chem. (NF) 131 (1982) 191.
197
[39] C.RA. Catlow, in: Nonstoichiometric oxides, ed. O.T. SOrensen (Academic Press, New York, 1981) p. 61. [40] C.R.A. Catlow and B. Fender, J. Phys. 8 (1975) 3267. [41] S.M. Tomlinson, C.R.A. Catlow, J.H. Harding, in: Transport in non-stoichiometric compounds, eds. G. Simkovich and V.S. Stubican (Plenum Press, New York, London, 1985) p. 539. [42] W.K.Chen and N.L. Peterson, J. Phys. Chem. Solids 36 (1975) 1097, [43] C. Picard and M. Dod~, Bull. Soc. Chim. Fr. 7 (1970) 2486.
Defect structure in highly nonstoiehiometric ferrous oxide has been discussed in terms of defect thermodynamics based on mass action law. It has been shown that the best agreement between theory and experiment is obtained under the assumption that 4 : 1 clusters, suggested by Cheetham and Catlow, are the predominant defects in the wustite phase. The ionization degree of these extended defects increases with increasing temperature, reaching a maximum value at 1573 K. Relative partial onthalpy and entropy of oxygen in Fe t _yO were evaluated.
Continuous attention has focussed on ferrous oxide, Fel_yO , (wustite) for many years. One of the most extensively studied problems of this oxide concerns its defect structure. Wustite exhibits exceptionally large deviations from stoichiometry, varying from 5 at% (a/o) at the Fe/Fel_yO phase boundary up to about 15 a/o at the Fel_yO/Fe304 interphase. This nonstoiehiometry results from cation sublattice defects in the form of cation vacancies, the effective negative charge of which is compensated by trivalent Fe-ions being electron holes. In earlier works [ 1 - 5 ] it has been assumed that despite their large concentration, cation vacancies in ferrous oxide are randomly distributed in the crystal lattice and can be treated in terms of a simple point defect model. In such an approach the formation of defects in Fel_yO may be described by the following quasi-chemical reaction: I
02(g ) ~ V~~ + ih" + 0 0 ,
(1)
where i = O, i, 2 denotes the degree of defect ionization *. The nonstoichiometry of the wustite is then given by: r
tt
y = [V~e] + [VFe] + [Vl~e].
(2)
Applying the mass action law to this equilibrium and assuming that interaction among defects may be * Kx6ger and Vink [6] notation of defects is used throughout this paper. 0 167-2738/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
neglected, as well as supposing appropriate electroneutrality conditions, one arrives at the following relationship between nonstoichiometry and equili. brium oxygen pressure and temperature:
Y =y o P ~ n exp (Ey/R T ) ,
(3)
where l/n, or simply n, is the parameter depending on the ionization degree of cation vacancies, assuming 2, 4 and 6 values for neutral, singly- and doublyionized vacancies, respectively: Ey denotes the temperature coefficient directly related to the enthalpy of defect formation. First experimental results [ 7 - 9 ] concerning the dependence of nonstoichiometry on oxygen pressure support the above mentioned simple defect structure model. A linear relationship between logy and logPo: has been obtained with a slope of 1/6, suggesting that the predominant defects in the wustite phase are doubly ionized cation vacancies. However, further extensive thermogravimetric [10-16] and emf [ 1 7 22] studies have demonstrated that the dependence of nonstoichiometry in ferrous oxide on equilibrium oxygen pressure is much more complicated and cannot be explained in terms of a point defect model. In fig. 1 the most representative results of this dependence are collected. It follows from this plot that the parameter n in eq. ( I ) does not assume a constant value, even in the limited part of the phase field of Fel_yO , but changes continuously with nonstoiehiometry, ex-
186
M. Rfkas, S. Mrowec/Defect clustering in the wustite phase
-08
^,t--10
;o$
• *
+
o
~
•e
o
• •
4.
_¢
c~O
o
,p "
"
"12
,¢
,. 2o e"
o
~
1It
,
~ ."
,
I
•
• n
-1~
/ -
o"
•
al
-16
,~" ir
oo x,
.
mll
~t +
o,-,,"
•
ir
,~
qr
~ ,
•
,l •
"
~ ~ •
.
/ /n:5
/ 1/I[ Y:c°nst'Po 2" -
to n
A
-12
-10
-8
Ig P02 Fig. 1. Logy =f(logPo2 ) for several temperatures: (*) ref. [15], (u) ref. [42], (m) ref. [2], (A) ref. [21], (~) ref. [8], (e) ref. [43], (X ref. [18], (o) ref. [14], (4) ref. [16], (+) ref. [10].
ceeding considerably the maximum value of six, predicted by the point defect model. In addition, from fig. 1 it foUows clearly, that the nonstoichiometry, and consequently cation vacancy concentration, decreases with increasing temperature - negative A HF values - that is in contrast with simple defect structure. Rather, strong interaction and clustering of defects could have been predicted. This conclusion has been confirmed in subsequent X.ray [23,24] and neutron diffraction [25,26] studies. It has been shown that not only cation vacancies but also interstitial cations occur in the wustite sublattice. In fact, for each vacancy which is created as a result of nonstoichiometry (eq. (1)), a Frenkel defect pair is simultaneously formed: FeFe ~ Fe;'" + V~e ,
(4)
by passing of trivalent octahedral iron ion, representing electron hole, to tetrahedral interstitial site. This implies that the real concentration of cation vacancies is twice that resulting from nonstoichiometry, thus reaching enormous value of about 30 a/o. Due to the mutual interaction, vacancies and interstitials form
extended defects, or defect clusters [27-35]. Roth [25] suggested 2 : 1 clusters consisting of two doubly ionized octahedral cation vacancies and one trivalent Fe ion in tetrahedral interstitial position. Assuming that this type of defect predominates, Kofstad and Hed [36] treated this problem in terms of point defects thermodynamics based on the mass action law. Formation of the Roth's complexes may then be written as follows: FeFe + Vt + ½0 2 ~-(VFeFe . . . . . i. . VFe) + O o ,
(5)
where Vt represents an interstitial (tetrahedral) site available for occupancy by complex defect. The dissociation of this complex is given by (V~eFe~'" VI~'o)x ~ (Vl~e]Fe~''V~e)' + h ' .
(6)
In this case the nonstoichiometry of Fel_yO becomes i
e°l
tt
X
PI
oea
y = [(VFeFe i VFe ) ] + [(VFeFe i
II
l
VFe ) ] .
(7)
Two limiting cases may be considered. Case 1 (nonionized complexes predominate):
[(V~eFe/"'V~e) x ] >> [(V~eFe/"V~e)'] .
(8)
M. Rfkas, S. Mrowec/Defect clustering in the wustite phase
187
If the conditions (8) and (11) are not satisfied, one obtains:
10
y - h = K1P1J2(1 - h - 2 y ) ( 2 - 7y),
(14)
h2 =Ki(1-h-
(15)
2y)(y - h ) ,
where h = [h" ]. Eqs. (14) and (15) make it possible to determine the dependence o f y on oxygen pressure and temperature. However, the values of the equilibrium constants K 1 and K i must be known. These values have been calculated from data reported in refs. [36-38] :
e-
K 1 = 1.75 X 10 -9 exp(-332.2
kJ/mol/RT),
Kt = 9.13 X 102 exp (-74.9 kJ/mol/RT).
5'0
160 y.1000
Fig. 2. n = f ( y ) curve 1 nonionized, curve 2 ionized clusters
Application of mass action law to the reaction (5) yields
7y)]Po~/2 •
(17)
Using eqs. (14) and (15) a general relationship can be obtained, describing the dependence of the parameter n on temperature and nonstoichiometry:
150
respectively.
g 1 = [y/(1 - 2y) (2 -
(16)
=[ ~lny ~-i n ~OlnP02] T =2y
(9)
[yl_ h
2
+Ki(
+ 1-y-2h
1-4y+h X 2h+K/(1-y-2h)
1
1-h-2y
y
1_ )
] "
(18)
The parameter n in eq. (3) is given by
.
=(
a l n y ~-1
~alnP02] T
=2+14_-~y+2%.
(10)
It follows from this relation that the parameter n increases with nonstoichiometry, from, 2.6 up 5.3 (curve 1 in fig. 2). Case 2 (ionized complexes predominate): [(V~eFe/"'V~e)' ] ~, [(V~eFe/"'V~e) x
],
(11)
Applying the mass action law to the reactions (5) and (6) gives
K1Ki = Lv2/(1 -
3y)2(2 -
7y)]Polz/2 ,
(12)
and n=4+
14y +2-7y"
(13)
Also in this case the parameter n increases with nonstoichiometry (curve 2 in fig. 2) varying from 5.0 up to 9.8.
h
Fig. 3 shows this dependence on the background of theoretical curves for nonionized and ionized Roth's complexes, presented previously in fig. 2. It follows from this plot that all curves calculated with the use of eq. (18) are situated between both theoretical curves 1 and 2, suggesting the existence of both types of Roth's complexes in comparable concentrations. In addition, from fig. 3 it can be inferred that the concentration of ionized complexes increases with temperature. Finally, from fig. 4, it follows clearly, that the dependence of nonstoichiometry on oxygen pressure calculated by means of eqs. (14) and (15) for several temperatures is in fairly good agreement with experimental data. It could then be concluded that the defect structure of ferrous oxide may be satisfactorily described by 2 : 1 defect clusters (Roth's complexes). However the earliest diffraction studies of Roth [25] were performed on quenched samples. Much more detailed
M. Rckas, S. Mrowec/Defect clustering in the wustite phase
188
10
//~147_3K
//.,~1373K E
16o
y.lO00
Fig. 3. n = f ( y ) curve 1 nonionized, curve 2 completely
ionized clusters and partials ionized clusters for several temperatures.
neutron diffraction studies ofCheetham et al. [26,33] carried out at high temperatures, within the thermodynamically stable region of the wustite phase, have shown that the vacancy-to-interstitial ratio (RvI) is higher than Roth's value of two, suggesting that the Roth's cluster is an oversimplified model of the defect structure of Fel_yO. It has been found that the R v I ratio varies with nonstoichiometry between 3 and 4, reaching its highest value near the Fe/Fel_yO phase boundary, i.e., at the lowest nonstoichiometry. It follows from these results that at low concentrations of defects, 4 : 1 clusters can form, and with increasing defect concentration more complicated, larger aggregates of defects should prevail. This conclusion has been fully confirmed by Catlow et al. [39-41] in their elegant theoretical studies based on computational techniques. It has been shown that the binding energy of the 4 : 1 cluster (4V~e, Fe~") 5- is very high ( ~ 2 eV per defect) identifying thus this cluster as the basic unit of the defect structure of the wustite phase. As the defect concentration increases, these clusters aggregate, forming greater assembles as shown schematically in fig. 5. It was, then, reasonable to
,
-0.8
~
^~
~.,k-~'-b ~
E~
/
-16
-it,
46 Ig P02
Fig. 4. Logy = f(logPo2) theoretical curves for 2:1 cluster on the background of experimental results: (o) ref. [ 15], (o) ref. [42], (u) ref. [2], (A) ref. [21], (®) ref. [8], (®) ref. [43], (×) ref. [18], (o) ref. [14], (z~)ref. [16], (+) ref. [10].
M. R~kas, S. Mrowec/Defect clusteringin the wustite phase Y I0.I0-3
The condition of the constant value of the cation-toanion ratio becomes
1.0.10-2
5 [FeFe] + ~ (i+ 4)C i = [Oo] = 1. i=0 80"
OOK I \
[Vt]= 2 - ~ y .
~'
y ~\ /k
,o.. oj?IX\ -25 -50 Ig y
The set of eqs. (20)--(24) enables the calculation of the dependence of nonstoichiometry on oxygen pressure if all six equilibrium constants are known. In order to simplify the procedure it was assumed that in the given temperature and pressure range complexes with the effective charge - j predominate. Thus, eqs. (20)--(23) become
-15
Fig. 5. Scheme for Catlow [41] defect clusters.
analyze the dependence of nonstoichiometry on oxygen pressure and temperature in an analogous way as it has been done previously, taking into account all defect clusters suggested by Catlow [41]. As the first step, the formation of 4 : 1 clusters is considered and subsequently a general case, comprising greater defect assembles, will be discussed. The formation of 4 : 1 complexes may be written as follows: (i ÷ 1)FeFe÷ Vt + ~ 0 2 ~ [(VFe)aFei]i+ 30 0 + iFe~e,
(19)
where i = 0, 1 ... 5, denotes the degree of ionization of the cluster. Electroneutrality condition for this reaction is given by 5 h = [Fete ] = ~ iCi,
h =jC/,
(25)
y = 3Ci,
(26)
[FeFe ] + (4 +])Ci = 1,
(27)
[Vt] = 2 - ~ y .
(28)
Using these relationships the equilibrium constant, K/, can be obtained K/=(
Y ]/+1 a~ ?-3/2 3-(]+4)y'] 2 - ~ - y 02 '
(29)
where: a~ = 1 for/" = 0 and a~ =]/forj = 1,2 ... 5. The dependence of the parameter n in eq. (3) on stoichiometry calculated from eq. (29) is given by ] - 1 = T2[(j__+ 1) 3 - (/+ 3 4)y T. 6---S~yJ 23y "] ' n ={\bhaPo,]Tblny
(30) (20)
where Ci denotes the concentration of 4 : 1 dusters, having effective charge - i . The deviation from stoichiometry is, then, given by 5 y = 3 ~ Ci . i=0
(23)
Consequently, the equilibrium constant of the reaction (19) assumes the form: (24) Ki ={Cihi/[FeF e ] (i+ 1) [Vt] ) t,032/ 2.
,o J2/ -t,0-=-;5--3ff
(22)
The concentration of tetrahedral interstitial sites available for occupancy by defect clusters is given by the following equation:
k [
Acot.voc.1200K ~ ,. . . . . lt, OOK4\
189
(21)
Fig. 6 shows this dependence for six different ionizations of 4 : 1 defect clusters. Comparison of these relationships with the experimental values indicates that the best agreement is observed for completely ionized complexes (] = 5). In order to prove this conclusion, the equilibrium constant K 5 must be calculated from eq. (29). For these calculations, the equi-
M. Rckas, S. Mrowec/Defect clustering in the wustite phase
190
Fe Fel_yO 1673KI, 1573K~, 1473KI-
112
l i b r i u m o x y g e n pressure c o r r e s p o n d i n g t o a given value o f n o n s t o i c h i o m e t r y s h o u l d b e k n o w n as a funct i o n o f t e m p e r a t u r e . Most c o r r e s p o n d i n g results h a v e b e e n a c c u m u l a t e d f o r 1473 K. T h u s , f o r this t e m p e r a t u r e m e a n values o f P o 2 f o r several values o f y h a v e b e e n d i r e c t l y d e t e r m i n e d f r o m t h e p l o t p r e s e n t e d in fig. 1. F o r o t h e r t e m p e r a t u r e s i n t e r p o l a t i o n forrffulas r e p o r t e d in [ 1 6 , 2 2 ] h a v e b e e n u t i l i z e d in t h e s e calcul a t i o n s . I n a d d i t i o n , s u c h f o r m u l a s as t h e s e in [ 1 6 , 2 2 ] h a v e also b e e n used i n c a l c u l a t i n g PO2-values for 1473 K. Based o n all t h e s e results K 5 values h a v e b e e n o b t a i n e d as a f u n c t i o n o f t e m p e r a t u r e ( 9 7 3 1573 K ) a n d n o n s t o i c h i o m e t r y ( 0 . 0 5 ~< y ~< 0 . 1 4 5 ) . T h e s e d a t a are c o l l e c t e d in t a b l e 1. It follows f r o m this c o m p a r i s o n t h a t l o g K 5 values d o n o t p r a c t i c a l l y
Fe~ -I -t 4~/'j=5
1373K
1273K I"
1173K.
///j-~
lO73K
/
5'0
I00 y.1000
150
•
Fig. 6. n = f ( y ) for six 4 : 1 clusters of different ionization degrees.
Table 1 LogKs values (atm -a/2) for completely ionized 4 : 1 clusters in Fe I _yO. Temperature(K) 1
2
3
4
5
6
7
8
9
10
11
12
y X 1000
973
1073
1173
1273
1373
1423
1473
1473
1473
1523
1573
50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145
25.42 25.52 25.61 25.68 25.73 25.77 25.80 25.82
18.02 18.14 18.23 18.30 18.36 18.40 18.43 18.45 18.46 18.46 18.45 18.44 18.41 18.39 . . . . . .
15.24 15.35 15.44 15.51 15.56 15.60 15.62 15.64 15.64 15.64 15.63 15.61 15.58 15.55 15.51 15.47
12.90 12.99 13.06 13.11 13.15 13.21 13.24 13.25 13.25 13.24 13.23 13.20 13.17 13.14 13.10 13.05 13.00
11.85 11.93 11.99 12.03 12.05 12.12 12.14 12.16 12.16 12.15 12.13 12.11 12.08 12.05 12.00 11.96 11.91 11.85
10.85 10.95 11.03 11.09 11.14 11.17 11.19 11.19 11.19 11.18 11.16 11.13 11.10 11.06 11.01 10.96 10.90 10.84 10.78
10.99 10.96 11.06 11.09 11.06 11.08 11.07 11.12 11.13 11.15 11.13 11.10 11.11 11.04 11.08 11.05 10.99 11.00
10.90 10.98 11.04 11.08 11.11 11.12 11.14 11.16 11.17 11.16 11.15 11.13 11.11 11.08 11.04 11.00 10.95 10.90 10.85
10.07 10.14 10.18 10.21 10.21 10.20 10.23 10.24 10.25 10.24 10.23 10.21 10,18 10.15 10.11 10.07 10,02 9.97 9.92 9,86
9.17 9.25 9.30 9.34 9.36 9.31 9.34 9.35 9.36 9.36 9.35 9.33 9.31 9.28 9.24 9.20 9.16 9.11
logKs standard deviation
25.67
21.68
18.35
15.53
13.11
12.04
0.14
0.13
0.13
0.11
0.11
0.10
. . .
21.44 21.54 21.61 21.68 21.72 21.76 21.78 21.80 21.80 . . . . . .
. .
11.05 0.13
%05 9,00
11.07
11.06
10.13
9.26
0.05
0.10
0.14
0.11
191
M. Rekas, S. Mrowee/Defect clustering in the wustite phase 1300
1100 1000 900
800
700"C
25
Table 2 Thermodynamic data for the formation of completely ionized 4:1 clusters~ Reaction
6FeFe+ Vt + ~O2 [(VFe),tFei]s - + 300+ 5Fete
Equilibrium constant Enthalpy of reaction Entropy of reaction Confidence interval (0.95 confidence level)
20
15 v
10
lnPo2 =
16
Fig. 7. LogKs =f(T-l).
change with nonstoichiometry which implies that these data represent real values of the equilibrium constants. Least standard deviation is observed in column 9 of table 1, because K 5 values collected in this column have been calculated from Poz-data obtained directly from the pressure dependence of nonstoichiometry plotted in fig. i and not from interpolation formulas for the same temperature (columns 8 [22] and 10 [16]. Fig. 7 shows the temperature dependence of the discussed equilibrium constant in the Arrhenius plot. As can be seen, the linear relationship is obtained with an extremely good correlation factor equal to 0.9999971. It implies that the enthalpy and entropy of the formation of completely ionized 4 : 1 clusters (eq. (19) w i t h / = 5) assuming constant values in the discussed experimental conditions. These data are collected in table 2. To prove the above discussed theoretical results, eq. (29) has been rearranged in order to get the suitable relation betweeny andPo2 f o r / = 5:
K = 3.96 X 10-Is exp(802.7 kJ/RT) AH.f = - 802.7 kJ/mol aS.f" = -333.2 J/mol • K
801.4 kJ/mol < - A H f < 804.0 kJ/mol 332.3 J/mol • K < - a S f < 334.1J/mol.K
32_(AHI AS1 RT R
6 ln(3/y - 9)
-in(2-~y)+in55).
(31)
Fig. 8 shows the calculated dependence o f y in Fel_yO on oxygen pressure on the background of experimental data for several temperatures. As can be seen the agreement is quite satisfactory. Eq. (31) can also be utilized for calculations of the relative partial free enthalpy of oxygen in the wustite phase: AG O = ½ R T l n P o 2 .
(32)
Taking into account the relation (33)
AG O = A H 0 - TAS---O ,
one obtains AH 0 = ~ AH£ = (-267.6 ± 0.4) kJ/mol,
(34)
and
- g / 3 (6 In ( 3 / ? - 9 ) - In(2 -
+ In 5 s ) .
(35)
As can be seen, the relative partial enthalpy of oxygen, AH0, is independent of comparison, which strongly supports the assumed defect structure model. Calculated &[/'0 and AS 0 data are summarized in table 3. Now, the general case will be considered concern-
M. R~kas, S. Mrowec/De[ect clustering in the wustite phase
192
_o.,! _~ -1.0 • /'
.;7 o °
Z.
0 °
-1.2
-1'6
-1¼
-12
-I0
-8
Ig P02 Fig. 8. Logy = f(logPo2 ) for several T experimental and theoretical curves; continuous lines for completely ionized (/= 5), dotted line for/= 4;(e) rof. [15], (u) tef. [42], (=) ref. [2], (a) ref. [21], (e) ref. [8], (,e) ref. [43], (×) ref. [18], (o) ref. [14], (~) ref. [16], (+) ref. [10].
where 0 ~< i ~< 2m - 3n. The electroneutrality condition for this reaction is given b y
ing the formation o f defect clusters built o f m cation vacancies and n interstitial iron ions. The formation o f such clusters may be written as follows: (n + i)FeFe + n V t +
2m-3n h=
m-n
~ i=1
(37)
iC i .
The deviations from stoichiometry, in turn, be-
X 0 2 ~ [(VFe)m (Fei) n ] i - + ( m - n)O O + i F e ~ e , (36)
comes
Table 3 Relative partial enthalpy, AHo, and entropy ASo of oxygen in the wustite phase (95% confidence level).
--AHo (kJ/moD
- A S o (J/reel • K)
y X 1000
this work
ref. [16]
ref. [22]
this work
ref. [16]
ref. [22]
57 65 74 83 91 107 i 15
267.7±0.4 267.6±0.4 267.6±0.4 267.6±0.4 267.6+0.4 267.6±0.4 267.6+0.4
256.6±2.3 259.9±2.0 264.7± 1.7 267.8±0.9 267.2±0.9 265.8±1.0 265.i± I.1
265.2 265.9 266.9 267.9 268.1 269.5 270.2
69.1±0.3 72.2+0.3 74.9±0.3 77.4±0.3 79.6±0.3 83.7±0.3 85.7+0.3
62.1±1.6 66.8+1.3 72.6±1.1 77.0+0.6 78.8+0.6 82.4±0.7 84.1+0.8
68.2 71.0 74,1 77.0 79.4 85.0 87.8
M. Rekas, S. Mrowec/Defect clustering in the wustite phase
y=(m-n)
2m-3n ~ Ci . i=0
(38)
Finally, the condition of constant value of the cationto-anion ratio can be written 2m-3n [FeFe ] + ~ i=0
(i + m ) C i = 1.
(39)
The concentration of tetrahedral interstitial sites available for defect cluster occupation is described by [Vt ] = 2 - aCi ,
(40)
where a denotes the number of tetrahedral sites close to cation vacancies involved in a particular cluster. The values of a for clusters to be considered are summarized in table 4. The equilibrium constant of the general reaction (36) assumes the following form:
Cihi
PO~(m-n)/21
193
better agreement appears for incompletely ionized defect complexes. This is conceivable if one considers the fact that, in general, the ionization degree increases with increasing temperature. Considering the standard deviations of logKi-values for particular clusters, it can be seen that the lowest deviations have been obtained for 4 : 1 dusters. This is most distinctly visible at the highest temperature. Fig. 9 shows the temperature dependence of equilibrium constants for completely ionized clusters. As can be seen, for all the clusters linear relationship in Arrhenius system of coordinates is observed. Calculated values of enthalpy and entropy of the formation of clusters under discussion are summarized in table 5. Following analogous procedure as for 4 : 1 cluster, relative partial enthalpy and entropy of oxygen can
(41)
Ki = [FeFe]n+i [Vt] n Considering a simplified electroneutrallty condition ( C / ~ O;Ci~ / = 0) eqs. (37)--(39) become:
h =jC/,
(42)
y = (m - n ) C / ,
(43)
[FeFe ] + (] +m)C/= 1 .
(44)
Combining eqs. (42)-(44) and (40) with eq. (41) yields y/+la/
Kj
m
[m - n - (1 + m)y]n+/,[2 - (ay/m - n)] PJ=(m-n)/2] -[ '
(45) where aj =//(m - n) n-1 f o r / ' ~ 0 anda/= (m - n) n - I f o r / = 0. Using the experimentally determined dependence o f y on oxygen pressure [16,22] the equilibrium constant K]. has been calculated for all possible j-values as a function of temperature of particular clusters. The results of these calculations are collected in table 4. It follows from this comparison that the best agreement with the mass action law at higher temperatures (1173-1573 K) is observed for completely ionized dusters, and at lower temperatures (973-1073 K)
70
7s
8o
go
+,0, Fig. 9. LogK = f(T -1) for completely ionized different clusters.
7 6 5 4 3
8 7 6 5
12 11 10 9
17 16 15 14 13 11
8:3 a = 46
10:4 a = 50
12:4 a = 55
16:5 a = 110
0
5 4 3 2 1
6 5 4 3 2
/
6:2 a = 32
4:1 a = 23
Kluster
98.26±0.33 98.97±0.23 99.29±0.22 99.78±0.08 100.25±0.14 101.41±0.26
71.14±0.20 71.50±0.09 71.92±0.08 72.40±0.17
53.48±0.14 53.91±0.05 54.42±0.10 55.02±0.20
44.24±0.15 44.66±0.05 45.16±0.08 45.78±0.18 46.52±0.27
34.89±0.14 35.29±0.04 35.82±0.06 36.50±0.16 37.23±0.28
25.67±0.14 26.03±0.04 26.53±0.07 27.27±0.15 28.24±0.24 29.83±0.34
973
Temperature(K)
83.97±0.27 84.29±0.16 84.64±0.16 85.06±0.15 85.51±0.25 86.66±0.39
60.46±0.14 60.81±0.09 61.21±0.15 61.67±0.27
45.47±0.10 45.88±0.07 46.38±0.16 46.95±0.27
37.58±0.12 37.97±0.05 38.46±0.13 39.05±0.24 39.76±0.36
29.56±0.12 29.94±0.04 30.43±0.13 31.04±0.24 31.83±0.35
21~8±0.13 22.02±0.04 22.50±0.12 23.16±0.22 24.18±0.30
1073
71.71±0.27 71.98±0.23 72.28±0.31 72.70±0.38 73.I1±0.53
51.50±0.19 51.79±0.27 52.13±0.42
38.76±0.12 39.12±0.22 39.55±0.37
32.01±0.12 32.35±0.15 32.68±0.38 33.30±0.44
25.11±0.12 25,42±0.14 25.86±0.29 26.42±0.45
18.35±0.13 18.64±0.10 19.06±0.26 19.67±0.39
1173
61.35±0.28 61.58±0.37 61.84±0.51 62.15±0.69
43.92±0.29 44.18±0.44
33.09±0.18 33.36±0.38
27.30±0.13 27.60±0.24
21.34±0.12 21.62±0.21 22.01±0.40
15.53±0.11 15.75±0.20 16.17±0.33 16.75±0.50
1273
Table 4 Log K values calculated by eq. (45) for several temperatures (95% confidence leveD.
52.53±0.32 52.73±0A6 52.98±0.63
37.49±0.36 37.72 0.54
28.27±0.23 28.57±0.41
23.29±0.14 23.57±0.30
18.13±0.13 18.39±0.28
13.11~0.11 13.36±0.21 13.73±0.39
1373
48.50±0.37 48.70±0.55
34.54±0.43 34.75±0.63
26.07±0.27 26.34±0A8
21.46±0.16 21.72±0.35
16.67±0.15 16.90±0.33
12.04±0.10 12.24±0.24 12.60±0.43
1423
44.90±0.38 45.07±0.58 45.28±0.80
31.88±0.55 32.08±0.69
24.08±0.35
19.81±0.22 20.07±0.37
15.34±0.20 15.53±0.39
11.06±0.10 11.21±0.30 11.58±0.47
1473
41.47±0.59 41.66±0.73
29.41±0.59 29.58±0.81
22.24±0.37 22.48±0.60
18.29±0.22 18.51±0.45
14.12±0.20 14.32±0.42
10.13±0.11 10.30±0.32 10.62±0.53
1523
38.29±0.46 38.45±0.70 38~4±0.93 38.87±1.16 39.14±1.38 39.83±1.82
27.07±0.57 27.25±0.80 27.47±1.03 27.76±1.25
20.49±0.35 20.72±0.59 21.04±0.82 21.46±1.04
16.83±0.20 17.05±0.44 17.37±0.66 17.80±0.88 18.37±1.08
12.95±0.19 12.94±0.64 13.47±0.63 13.93±0.84 14.57±1.03
9.26±0.11 9.42± 0.31 9.75±0.52 !0.26±0.72 11.03±0.90 12.45±1.07
1573
N.
e*
r~
Thermodynamic data for the formation of different defect clusters. Cluster type
/
Temperature range (K)
Mean standard deviation of log K
_ AH (kJ/mol)
- ~S (J/tool • K)
4:1
5 4 3 2
973-1573 973-1173 973-1073 973-1073
0.12+0.01 0.06±0.03 0.10±0.04 0.19±0.05
802.7± 1.2 807.4±27.1 816.0±48.9 820.3±40.7
333.2± 0.9 331.2±25.4 330.3±46.0 330.8+38.4
6:2
6 5 4
973-1573 973-1173 973-1073
0.16±0.04 0.07+0.06 0.10+0.05
1073.5± 1.8 1081.4±22.0 1088.1±50.0
434.7+ 1.4 435.1±20.1 432.0±46.9
8:3
7 6 5
973-1573 973-1173 973-1073
0.16±0.04 0.08±0.06 0.11±0.04
1341.1± 2.41 1345.0± 35.6 1363.0±112.1
530.7± 1.9 526.9±33.3 535.4±105.3
10:4
8 7 6
973-1573 973-1073 973-1073
0.24±0.10 0.11±0.09 0.13±0.04
1614.2+ 2.7 1615.9±17.7 1624.5+28.7
634.3± 2.1 628.1±46.9 626.9±77.1
12:4
12 11 10
973-1573 973-1173 973-1073
0.37±0.18 0.15+0.10 0.12±0.07
2156.5± 3.9 2153.4±76.8 2163.1±94.6
853.0± 3.0 845.5±72.3 845.1±89.0
16:5
17 16 15 14 13
973-1573 973-1173 973"1073 973-1073 973-1073
0.36+0.11 0.21±0.04 0.19±0.04 0.12±0.05 0.20±0.08
2939.6a 13.4 2949.1+66.9 2951.0±104.8 2958.9±75.4 2965.3±87.3
1135.3a10.6 1135.3±63.0 1130.9±98.6 1129.8±70.9 1127.3±81.9
Table 6 Relative, partial enthalpy and entropy of oxygen in Fel _yO. Cluster
]
--AH 0 (kJ/moD
--AS0 (J/tool • K)y 0.055
0.065
0.075
0.085
0.100
0.115
0.130
0.145
4:1
5 4 3 2
267.6 269.1 272.0 273.4
68.6 71.3 75.2 80.6
72.1 74.2 77.5 82.3
75.3 76.8 79.5 83.8
78.2 79.1 81.3 85.2
82.2 82.4 83.9 87.1
85.9 85.3 86.2 88.9
89.4 88.2 88.4 90.5
92.8 90.9 90.6 85.4
6 :2
6 5 4
268.4 270.4 272.0
69.8 72.6 75.0
73.1 75.4 77.3
76.0 77.9 79.4
78.7 80.2 81.3
82.5 83.4 84.0
86.1 86.4 86.5
89.3 89.4 89.0
93.0 92.3 91.4
8 :3
7 6 5
268.2 269.0 271.6
69.8 71.3 75.5
72.9 74.0 77.9
75.8 76.6 80.1
68.9 78.9 82.1
72.7 82.3 85.1
76.4 85.5 87.9
89.5 88.6 90.6
93.2 91.9 93.5
10:4
8 7 6
269.0 269.3 270.7
70.8 71.7 73.7
73.8 74.4 76.0
76.6 76.9 78.2
79.2 79.1 80.2
82.8 82.4 82.3
86.3 85.5 85.9
89.7 88.6 88.6
93.1 91.6 91.3
12:4
12 11 10
269.6 269.2 270.4
71.3 71.7 73.0
74.4 74.5 75.6
77.1 77.0 77.9
79.7 79.3 80.0
83.2 82.6 83.0
86.5 85.7 85.8
89.8 88.6 88.5
92.9 91.5 91.1
16 : 5
17 16 15 14 13
267.2 268.1 268.3 269.0 269.6
69.1 70.0 70.6 71.5 72.3
72.3 73.0 73.4 74.1 74.8
75.1 75.7 75.9 76.5 77.1
77.8 78.2 78.3 78.8 79.1
81.6 81.8 81.7 81.9 82.1
85.2 85.2 84.9 85.0 84.9
88.8 88.6 88.1 88.0 87.8
92.5 92.1 91.4 91.0 90.7
M. Rgkas, S. Mrowec/Defectclusteringin the wustitephase
196
be calculated, using eqs. (32), (33) and (45): A H 0 = (m - n) - 1 A H f , A S 0 = (m - n) - 1 X ln(m
{ASf-
(46) R [(j + 1 ) l n y + l n a / - ( n + / )
- n - (] +m)y) - n ln(2-ay/(m -
n))] ) . (47)
As can be seen, also in the case o f greater clusters the relative partial enthalpy o f oxygen is independent o f crystal composition. These data, together with entropy values, are collected in table 6. Using eqs. ( 4 2 ) - ( 4 5 ) the relation between enthalpies o f ionic and electronic defects m a y be obtained AHh.=/
alnh
01ny ]
, al/RT ) po2 =( ~I/RT ]po: ~lny ~ - I A H = 2, / --(m - n) ~ ~--~--~]
ainy = 2 \a lnP02 IT AHO "
(48)
It follows from this relation that the enthalpy of electron holes formation is also independent o f nonstoichiometry. Summing up these considerations, the following conclusions m a y be formulated. A thermodynamic approach to the problem o f defect structure in highly defected compounds does not lead to definite solutions. In the discussed case o f nonstoichiometric ferrous oxide the only definite conclusion is that noninteracting randomly distributed point defects do n o t predominate in the crystal lattice o f this compound. However, the forgoing analysis has shown that the formation o f different defect clusters is not equally probable. The best agreement between theory and experiment has been found, namely, for 4 : 1 clusters as the predominant defects in the wustite phase. It has been shown also that the ionization degree o f these complexes increases with increasing temperature, reaching a maximum value at 1573 K.
References [1] C. Wagner and E. Koch, Z. Phys. Chem. B32 (1936) 439. [2] L.S. Darken and R.W. Gurry, J. Am. Chem. Soc. 68 (1946) 798.
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