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Interaction energies of vacancies in transition metal oxides with large concentration of vacancies
J. Phys.
Pergamon
Chem. Solids
Press 1962. Vol. 23, pp. 1463-1471.
INTERACTION
ENERGIES
TRANSITION
METAL
M. HOCH, Materials Science Program,
OF VACANCIES
OXIDES
CONCENTRATION
Printed in Great Britain.
WITH
LARGE
OF VACANCIES* and J. NBLKBN
A. S. IYBR
Graduate School, University of Cincinnati, (Received
IN
30 April
Cincinnati,
Ohio
1962)
Abstract-Equations are derived from statistical considerations to obtain vacancy interaction energies, En, Ezz (for vacancies with same sign), and Em (with opposite sign), from the variation of oxygen partial free energy with composition for FeO, TiO, and NbO. The oxygen partial free energy was determined through electromotive force measurements with solid electrolytes. Cells of the type Nb, NbO/electrolyte (0.8 ZrOs 0.2 CaO)/NbO, NbOs were studied. The measurements were carried out in vacuum, with tungsten electrodes and induction heating. In FeO, E22 = 2.74 f 0.10 kcal/mole. In TiO, El1 = E22 = 2.10 & 0.03 kcal/mole, El2 = -2.97 + 0.04 kcal/mole. In NbO, En = E22 = 17.9 kcal/mole, El2 = -25.3 k&/mole. The constancy of the vacancy interaction energies permitted the accurate placement of the boundaries of the TiO phase. The entropy of TiO should be increased by 1.68 e.u. to account for the randomness of vacant titanium and oxygen sites.
INTRODUCTION
SOMEof the transition metals of Groups IV, V, VII, and VIII form stable monoxides of the NaCl-type structure. Closer examination, however, shows that in all cases the structure is stable only in the presence of a large concentration of vacancies. The aim of the present work is to obtain some information about these vacancies. THEORY
The theory of compounds with small deviations
by WAGNER(~)and by FOWLER and GUGGENHEIM.(~) The theory was expanded by ANDERSON(3)to include larger deviations. However, he considered only the binary compounds AB where the defect is present solely in atom A, either as a vacancy or as an interstitial. In the compounds under present investigation, the defects occur only as vacancies, but both components can show these vacancies. The grand partition function, GPF, for a crystal of ideal composition AB with only vacancies as defects can be written as follows:t
(1)
N! X
from stoichiometric
cva vs!(N-
v2)!
composition
[a2K2(T)1(N-Y2) exp-(v&2+
has been given
* This research was supported in part by the Aeronautical Research Laboratory, Office of Aerospace Research, United States Air Force, under Contract AF 33(616)6299.
&sE2s+~&sE12)/kT
where N is the number of lattice sites of component 1 and also of component 2; VI and I’s are the vacancies of components 1 and 2; El and Es
1463
t Expansion of Fowler equation (1304).
and Guggenheim,
p.
552,
1464
M.
HOCH,
A. S.
IYER
are the energies necessary to create a vacancy of component 1 and component 2, respectively; Eli, E22, and El2 are the interaction energies per pair between 1 and 1,2 and 2, and 1 and 2 component vacancies; the S’s are the number of those pairs; al and LZ~are the absolute activities of components 1 and 2 in the crystal phase; and El(T) and Es(T) are contributions of an added atom of component 1 and 2, respectively, to the partition function for the normal vibrational modes of the crystal; K is the Boltzmann constant; T is the absolute temperature. Furthermore, if the crystal shows symmetrical behavior around the stoichiometric composition, then El1 = Eea is assumed. In the present investigation, the subscript 1 refers always to oxygen, 2 to metal. The equilibrium conditions are obtained by replacing the sum by its largest term,@) and differentiating the log of the partition function with respect to the number of vacancies, and equating to zero. During the differentiation, the number of metal atoms, i.e., (N- Vs) in equztion (l), is kept constant. Fe0 phase It is well known that the Fe0 phase exists only in the presence of excess oxygen, and that a large concentration of vacancies is present in the lattice. According to the measurements of JETTE and FOOTE(~) and of COLLONGUES,(~) the iron lattice contains vacancies whereas the oxygen lattice is, within experimental results (+ 1 per cent), completely filled. Su = &a = 0, &a = 2/Z (Vi/N), assuming that the interaction energy does not influence the random distribution of vacancies. The error in the evaluation of Eas introduced by this random assumption is small, as is shown in the NbO case below. z (the number of nearest neighbor iron sites around each iron site) = 12. Replacing the S’s in equation (1) and carrying out the differentiation gives :
COLLONGUES(~)the mole fraction of oxygen, no is
N no =
lzl =
ZN-V2
Writing equation (2) for a second composition, (denoted by ‘), subtracting the two equations and replacing Vs and Vi in terms of nl and n;, yields = RTlnz
2?21- 1 , nl -----+RTlnal-RTlna;
-RTln
(3)
2n;-1
Measurement of the oxygen activity as a function of composition permits the evaluation of Ess from equation (3). TiO phase EHRLICH@) studied the structure and density of TiO. He concluded that at the stoichiometric composition 15 per cent of the titanium and oxygen lattice sites are empty and the vacancies are distributed randomly. In a titanium;rich material, more titanium sites are occupied and more oxygen sites are empty. The total number of vacancies is constant over the whole composition range of the TiO phase. Carrying out the differentiation, again assuming random distribution of vacancies, with z [the number of nearest neighbor titanium (oxygen) sites around each titanium (oxygen) site] = 12 and z’ (the number of nearest neighbor titanium sites around each oxygen site) = 6, gives RT,n
(N-
V1J2. V2
N2. Vl
-EI+&;
-2
RT In CZ#CIT]
X
N2 results
1--=
Vz”
_z,E12
Es
kT
of JETTE and
0
(2)
N2
FOOTE(~) and
(V2--Vl)N+KV2 [
1 1 (4)
V,2-2V2N+
Eu
[a1(KlT)1-:$
ZVsNa-
the
J. NELKEN
ZVlN+
N- V2 In y+ln
From
and
N2
Vz”
Writing equation (4) for a second composition (denoted by ‘), subtracting the two equations, and
INTERACTION
ENERGIES
OF
VACANCIES
replacing the V’s in terms of nr, gives (12-&l--6&a) x [nl-n;]
al-RTln
TRANSITION
u;]
[a+(1 +RTln
?2~(1*7nr- ‘7) l-l.7
nr
-RTln
METAL
OXIDES
1465
the number of atoms on the x1 sublattice is [a+ (1 + K)Nl- KVr], and the number of vacancies on the yl sublattices [r~-(l-K)Nl+kVr]. The mass action law requires
[2.89(n1+n;)-6.291 = -2[RTln
IN
n;a(1*7?2;-*7)
-K)Nr-KVr]
[a-_Nlf(l
-k)Vr]
= a2,
from which
l-1*7n;
Measurement of the oxygen activity as a function of composition permits the evaluation of (12&6Els) from equation (5).
a2 - VrNr
&a_
(5)
N+lr,
(N+'v;)2
Using only the first term in the expansion, and keeping in mind that the same conditions hold for the Nb lattice,
NbO phase
a &Gc=(6) the structure of NbO. By Nl+ Vl N2+V2 taking a NaCl-type cell and removing the eight corner atoms (e.g., niobium) and the atom at the The number of vacancies on each sublattice (the cube center (e.g., oxygen), the NbO structure is vacancies ,a;e distributed evenly among the yl, obtained. At the stoichiometric composition y1 , and y1 sublattices) is then given as : twenty five per cent of the niobium and oxygen on x1 sublattice: [N--N11 vacancies; on xa subsites are empty. lattice : [N- Na] vacancies In this structure the vacancies are ordered. To treat the vacancies statistically, the two interon y; sublattice : VI/3 vacancies ; on yb sublattice : penetrating f.c.c. lattices (containing the niobium [WI vacancies atoms and the oxygen atoms) are treated separately, in a fashion similar to the treatment of the AuCua on y;’ sublattice : VI/3 vacancies; on yg’ sublattice : lattice by GUGGENHEIM.(7) The oxygen sublattice [ v2/3] vacancies is divided into fou; interpenetrating simple cubic 1,, lattices, x1, y;, yr , y;“. The total number of on ~1 sublattice: Vr/3 vacancies; on y2“I suboxygen lattice sites is 4N (N on each sublattice). lattice: [v2/3] VaCanCieS. At the stoichiometric composition at 0°K the x1 The evaluation of $1, &a, and &a is done lattice contains all the vacancies, the yl sublattices according to GUGGENHEIM.(~) For Sir and for Saa all the atoms. At a finite temperature, however, a atoms are found on x1 and, correspondingly M each atom has x = 12 neighbors, of which onethird (i.e., f our ) are on each of the other three vacancies on the yr sublattices. To deviate from stoichiometric composition, Nl atoms and VI sublattices. For Sla each atom has x’ = 6 neighbors, of which one-third (i.e., two) are on each of vacancies can be added. Of the Nl atoms (1 -K) the other three sublatti:es. JTlr~s for $2 tk Nl will go on the xl sublattice, kNr on the yl subneighbors of x1 are,,on y2, y2 , y2 ; those of y2 lattices ; of the vacancies VI, (1 - K)Vl will go on are on x1, y1 and y1 .) the yr sublattices, KVl on the x1 sublattice. Thus BRAIJER(~~)studied
SIP = 3z[N-Nr]$+i
S22
[
T.$
1
3 = !$N_N1]V1+i
=;
[N-N2]V2+;
1” [1 k
[$I
1466
M. HOCH, s12=
A. S.
IYER
3,;
[;I
g,-,,;.
and J. NELKEN
[Iv-Nz].
3,;
[T]
[q]
.9
(7)
In the calculation of S1l, &a, and S1s it is assumed that the vacancies are distributed randomly on each sublattice (e.g., V1/3 vacancies distributed randomly on the y;’ sublattice). To carry out the evaluation of equation (1) relations between N1, Na, VI and Va are required. From equation (6) V1+N1
=
v2+N2
Furthermore, it is assumed number of atoms is constant, 3N+N1-
(64
that:
V1+3N+Ns-
(1) the total
Va = 3N;
(8)
and, (2) when a change of composition takes place, half of the change is done by introducing atoms (N1 or Na), the other half by introducing vacancies (sE11-z’E12)(2.25(fl1+n;)-5*25)(~1-n;)
+&ln(l+2n1)3(3-2n1)(6nl-l)5i3 4
(5 -
(1+2n;)s(3_2n;)(6+1)513
Again, the measurement of oxygen activity as a function of composition permits the evaluation of (16&l- 9-&s) from equation (10). To obtain an estimate of the error introduced by the assumption of random distribution of vacancies (in Fe0 and TiO, and on each sublattice in NbO), in the evaluation of E1l and E12, the NbO phase can be treated as the TiO phase, i.e., all vacancies uniformly distributed over all sublattices. In this case, VI+ Vs = 0.50 N and the final equation corresponding to equation (5) is -2[RTln
al-RTln
+ RTln
-
(9)
N2 = VI
In writing the partition function, the distribution of vacancies on each sublattice must be considered, and an equation similar to equation 7.21.5 of GUGGENHEIM(7) is obtained. Carrying out the differentiation, combining the resulting equation with that at another composition (designated by ‘) and replacing the V’s by the mole fraction of oxygen no = 121= (3N+N1V1)/6N results in: (16-k-
g.%) (nl= -(RTIn
n;)(nl+
a;)
(l-
s-.5) 1.5n1)
(l-
(111 \- -I
1*5n;>
As in the NbO phase the vacancies are almost completely ordered, the error in the evaluation of El1 and E1s by equation (11) will indicate a much larger error (perhaps tenfold) than is introduced by the random assumption in Fe0 and TiO and on each sublattice in NbO As the interaction energy of vacancies is coulombic in nature, and is inversely proportional to the distance, and in both TiO and NbO the closest distance for similar charges is d2 times that for opposite charges, it will be assumed that EIZ = -1/(2)-C. APPARATUS
n; - 2)
al-RTln
n$5
a;]
n;s(1*5 .;--5)
RTln
--
N1 = vs
(lo)
(5-64513
=
(Vs or VI). Thus,
6n1)5/3
The
evaluation
AND EXPERIMENTAL PROCJXWRE
of
the
vacancy
interaction
INTERACTION
ENERGIES
OF
VACANCIES
IN
TRANSITION
METAL
OXIDES
1467
bottom electrode. The crucible measures Q in. in diameter, 4 in. wall thickness, and 2 in. high. It rests on a tungsten tripod. One leg of the tungsten tripod is connected to a thin tungsten wire which passes through the bottom plate of the vacuum system. The second electrode is a h in. tungsten rod, which enters the vacuum system through a Wilson type seal in the top plate. The lid of the crucible, which has a &in. hole in the center, is mounted on the tungsten rod in such a fashion
energies by equations (3), (5), (10) and (11) requires the difference in partial free energy of oxygen at two compositions. This is obtained by e.m.f. (electromotive force) measurements using solid electrolytes. The galvanic cell technique using solid electrolytes has been developed by KIUKKOLA and WAGNER.@) Cells of the type Nb, NbO/electrolyte/NbO, NbOs are used. This cell measures the difference between the partial free energies of
TO POTENTIOWETER
WILSON
TYPE
UPPER
.-zJ-*
SEAL
TUNSSTEN
0
COIL
0 0
II r
OPTICAL PIROYETER
II J I I II
HEATER , BOTTOM ELECTRODE
TO
VACUUM
-
SAUQE
TO “ACUUM I
I INCH
TO
POTENTIOMETER
FIG. 1. Vacuum, induction heated e.m.f. cell.
oxygen of the NbO phase at the oxygen-rich composition (in equilibrium with NbO2) and at the oxygen-poor composition (in equilibrium with Nb). As electrolyte (0.8 ZrOs 0.2 CaO) is used.
Apparatus In order to carry out measurements above 1 lOO”C, the e.m.f. cell is enclosed in a vacuum and heated by induction. Fig. 1 shows the apparatus. A tungsten crucible serves as the heater and
that when the system is assembled, a & in. gap exists between the crucible and the lid. As the crucible and lid are both heated by induction, the inside of the crucible, where the e.m.f. cell is located, approximates well a black body. Both half cells and the electrolyte are in pellet form, each pellet measuring about 4 mm in diameter and 2 mm high. The top plate and bottom plate are separated by a quartz tube lt in. in diameter and 6 in. long, which also serves as a vacuum jacket. The two tungsten electrode wires are connected to
1468
M.
HOCH,
A.
S.
IYER
a potentiometer. To filter out radio-frequency currents, a filter consisting of three 0.02 PF condensers is connected across the e.m.f. leads and from each lead to ground. The temperature of the crucible is measured with an optical pyrometer. Vacuum is produced by an Eimac oil diffusion pump, and measured by a Veeco DC-2 discharge gauge. The tungsten crucible is heated by induction through power supplied by a 20 kW Thermionic generator. During the measurements, the vacuum is better than 2 x 10-5 mm. To check the system for spurious effects, the e.m.f. was measured, first using only the electrolyte, and then by making contact between the two electrodes. The e.m.f. was below 1 mV, showing that spurious voltages were negligible.
and J.
NELKEN
pressing into pellets, the material was heated to 1200-1300°C in vacuum. After regrinding, electrolyte pellets were pressed using 65,800 p.s.i. pressure, and reheated in vacuum to 1300°C. Half cells were prepared by mixing metal and metal oxide (e.g., Nb+NbsOs) to give an equimolar mixture (e.g., Nb/NbO), and the mixture heated in vacuum to 1200-1500°C until X-ray diffraction patterns indicated that the reaction was complete. The material was ground and pressed into a half cell using 65,800 p.s.i. pressure. To insure good contact, the half cells and the electrolyte pellet were placed together (one on top of the other) in the die, and pressed using 51,200 p.s.i. pressure.
Preparation of the electrolyte and the half cells
EXPERIMENTAL RESULTS Reaction Fe0 + Ni + NiO + Fe
A mixture of ZrOs and CaCOs was heated to 1000°C for eight hr in air; after grinding and
To check the equipment, the reaction FeO+ Ni + NiO + Fe was studied, using the cell FeO,
Table 1. l+ee energy change in the reaction: Fe0 + Ni + Fe + NiO; Fe, FeO/eZectrolyte (O-8 ZrOs O-2 CaO)/Ni, NiO
Temp. "K I 2 3 4 5
Table
2.
Temp.
“K
1173 * 2 1230 1250 1323 1330 1386 1440
1386 f 1399 1456 1502 1561
Vacancy
e.m.f. mV
ELLIOTT~~~ GLEIsER(g) e.m.f. mV
285 287 294 297 301
291 f 293 299 305 310
interaction energies in FeO; Cell: (0.8 ZrOs 0.2 CaO)/FeO, FesO4
e.m.f. mV 089 095 098 II2 112 122 129
3
ELLIOTT and GLEISER(g) e.m.f. mV 65 + 52 76 80 97 98 II2 123
Cell:
n1’ 0.5119 0.5118 0.5118 O-5118 0.5118 0.5118 o-5119
Fe,
6
FeOlelectrolyte --
n1 0.5296 0.5315 0.5320 0.5341 0.5343 o-5359 o-5373
EZZ
kcal/mole 2.854 & 0.064 2.600 2.620 2.782 2.729 2.805 2.803
Average 2.742 rf: 0.101 kcal/mole
INTERACTION
ENERGIES
OF
VACANCIES
Fe/electrolyte (OX&02 O*ZCaO)/NiO, Ni. Accurate thermodyn~ic data(*) are available for the above reaction. To observe variations due to half cell and electrolyte preparation, new half cells and electrolyte were used for each measurement. Table 1 contains the data, together with those of ELLIOTT and GLEISER.~~)The agreement is good. Fe0 phase The e.m.f. measurements on the cell Fe, FeO/eleetrolyte (O-8 ZrOs O-2 CaO)/FeO, FesOa are given in Table 2. From ELLIOTT and GLEISER@) the free energy change for the reaction 0.788 Fe+ F’esO* + 4 Feo.9470 can be calculated, and is also given in Table 2. This value must be larger than the value obtained by galvanic measurements, where the reaction (0.788-3y)Fe+
Fe304 + 3 Feo.~~~-,O+Feo.~4:0
is studied (Fes.947-u0 indicates the oxygen-rich Fe0 phase). Within the large error limits of the data ( + 52 mV) this is the case. Using the values of nl and n1 (the phase boundaries of the Fe0 phase) given by DARKENand GuRRY,(~~)the interaction energy Es2 between iron vacancies is evaluated from equation (3). Eas = 2.742 + 0.101 kcaljmole and is constant; the small variations are due to the difficulty of accurately reading the phase boundaries. TiO phase The experimental data on the cell Ti, TiO/electrolyte (0.8 ZrOs 0.2 CaO)/TiO, TisOs are given in Table 3. From ELLIOTT and GLEISERCQ) the free energy change for the reaction Ti+TisOs + 3 TiO can be calculated. This value must be larger than the value given by the galvanic cell measurements, where the reaction (1 -i-2x-y)Ti+-TisOs
-+ 2 Til+,OfTil-,O
is studied, (T&+,0 is the titanium-rich TiO phase, T&O is the oxygen-rich TiO phase). Inspection of Table 3 indicates that this is not the case. The discrepancy is due to the entropy of TiO. SHOMATE(~~)upon whose entropy data Elliott and Gleiser’s values are based did not take the vacancies into account when determining the entropy of TiO. The vacancies contribute to the entropy of stoichiometric TiO at 0°K the value of S = -2R [O*SSIn 85 + O-15 In 0.151 = 1.68 e.u. Thus
IN
TRANSITION
METAL
OXIDES
1469
M. HOCH,
1470
A.
S.
IYER
the free energy of formation of TiO becomes more negative by - 168 T cal. When this correction is made, ELLIOTT and CLRrsRR’s(s) data and the galvanic cell results are in agreement. The phase boundaries of the TiO phase can be obtained from the works of BUMPS, KESSLER and HANSEN, and of SHOFIELD and BACON,~~) who agree within three per cent in the region of interest. The values of (Z.&l-Elz) for each set of data were calculated and are given in Table 3. The differences in (2En-Els) between the two sets of data, and the variation with temperature are due to the differences and very small errors in the phase boundaries as (2&l--Els) is very sensitive to composition : At 1275”K, rz; of both references”23 1s) is 0.4721.
and
J.
NELKEN
193 mV, at 1276°K 197 mV, which is somewhat higher than our results. They also studied the reaction Fes.saTO+Nb
+ NbO-tO.947
Fe
(12)
with a galvanic cell, and found AGO = -34.5 + 3.15 x10-s T f O-7 per cent kcal (11141346”K), and the reaction NbOs-t- Hs + NbO+ Hz0 with AGO = 29.49-6.10 x 10-s T & O-3 kcal (1673-1773°K). Combining these actions, and introducing the free energy mation of Fec.saTO(s) and of HsO(s), the
Nbf
NbOs -f 2 NbO
(13) per cent two reof forreaction (14)
Table 4. Yacaq~ interaction enmgies in NbO; CeE: Nb, NbO~e~ec~~o~~e(0.8 ZrOs O-2 CaO)/NbO, NbOs Temp. "EC
EMF mV
I 3
n1
(16E11- 9&a)
kcal/mole
CalZlatecl (16&1-9&a)
1160 _t 2 1217 1275 1353 1429 1503 1572
137 157 174 190 203 217 230
0.485
0*505
Changing nl from 0.5561 to 05572 (a 0.2 per cent change) changes (2En-Els) from 7.205 kcal to 7.109 kcal (a l-3 per cent change). Thus using a constant (average of values at 1275 and 1278°K) (2En--Ers) = 7.180 kcal, and 9 = 0.5567 (halfw,ay between the two values), the phase boundary nl of the TiO phase is calculated in the last column in Table 3. Taking into account the errors in the e.m.f. measurement, El1 = Ess = 2.103 + 0.025 kcal/mole and Ers = -2.974 + 0,035 kcalfmole. NbO phase The experimental data on the cell Nb, NbO/ electrolyte (08 ZrOs O-2 CaO)/NbO, NbOs are given in Table 4. GE~SIMOV et a;h(15) measured the e.m.f. of a cell similar to the one used here but with platinum electrodes and found at 1223°K
305.6 rt: 26 351.3 389.8 425.8 455.0 486.5 515.8
= 5158 kcaf/moie ?21= 0~505 0.492 0.491 0,490 O-488 0.488 0,486 0,485
with AGO = - 8.18 - 1.2 x 10-3 T kcal is obtained. The accuracy of this equation is hard to evaluate as Iarge extrapolations in temperature have to be made when equations (12) and (13) are combined. At 1572”K, equation (14) yields 218 mV, in agreement with 230 mV in Table 4. The phase boundaries were estimated from BRAUER’#) data as NbOs.sd and NbOl.se, and assumed independent of temperature. Using equation (lo), (l&!&1-9Els) was evaluated. The strong dependence of (16Ell- 9&s) on temperature is due to the assumed constancy in phase boundary. B~~R(~~) prepared his samples at 1950°K and cooled them to room temperature. The homogeneity range of the NbO phase will become smaller at lower temperature (similar to Fe0 and TiO}. Only small changes in phase boundary are needed to obtain constant values for the vacancy interaction
INTERACTION
ENERGIES
OF
VACANCIES
energy; this was calculated by assuming (16Eu9Els) = 515.8 kcal/mole and the oxygen-rich NbO phase boundary constant as in TiO (nl = 0.505). The Nb-rich NbO boundary n; is given in the last column of Table 4. Using again the assumption taking El2 = ‘- 1/(2)Eu and (16Eu-9E12) as 515 -8 kcal/mole (value at 1572”K), El1 = 17.9 kcal/mole and El2 = -25.3 kcal/mole is obtained. (2Ell-E1s) can also be evaluated from equation (1 l), where the vacancies were assumed to be completely random. At 1572”K, (2Eu--Els) = 55.649 kcal/mole from which, again with the assumption El1 = 16.3 kcal/mole. The El2 = -2/(2)&l, change in El1 is only 10 per cent (from 17.9 to 16.3 kcal/mole) when the vacancies are treated as random rather than completely ordered. Thus in the present method of evaluation, El1 is almost independent of the distribution of vacancies. On the other hand, El1 influences strongly the distribution (random or ordered) of the vacancies. In FeO, where Es2 = 2.7 kcal/mole, the vacancies will be somewhat ordered. But even if this is taken into account in the distribution function (using, for example, the quasi chemical method),(‘) Ess will not change.
IN
TRANSITION
METAL
OXIDES
1471
REFERENCES of Alloys, Addison1. WAGNER C., Thermodynamics Wesley Press, Reading, Mass. (1952). 2. FOWLER R. and GUGGENHEIM E. H., Statistical Thermodynamics, University Press, Cambridge (1949). J. S., Proc. Roy. Sot. A185, 69 (1946). 3. ANDERSON 4. JETTE E. R. and FOOTE F., J. them. Phys. 1, 29 (1933). 5. CO&O~G~ES R., thesis, Paris (1955). 45. 362 (1939) Z. 6. EHRLICH P.. Z. Elektrochem. anorg. Ch;m. 247, 53 (1941). Clarendon Press, 7. GUGGENHEIM E. H., Mixtures, Oxford (1952). 8. KIUKKOLAK. and WAGNER C., J. electrochem. Sot. 104, 379 (1957). 9. ELLIOTT J. F. and GLEISER M., Thermochemistry for Steelmaking, Addison-Wesley Press, Reading, Mass. (1960). 10. DARKENL. S. and GURRY R. W., J. Amer. them. Sot. 67.1398 (19451: 68.798 (1946). Amer. &em. hoc. 68,310 (1946). 11. SHOI&TE C: H.,i 12. BUMPS E. S., KESSLER 0. and HANSEN M., Trans. Amer. Sot. Metals 45, 1008 (1953). 13. SCHOFIELD T. H. and BACON A. E., J. Inst. Met. 84, 1654 (1956). 14. BRAUERG., Z. nnorg. Chem. 248.1 (1941). 15. GERASIMOVT. IA., LAVENTEVV. I., KUZNETSEVF. A. and REZUHINA T. N., paper presented at the XVIII International Congress of Pure and Applied Chemistry, Montreal (1961).
Pergamon
Chem. Solids
Press 1962. Vol. 23, pp. 1463-1471.
INTERACTION
ENERGIES
TRANSITION
METAL
M. HOCH, Materials Science Program,
OF VACANCIES
OXIDES
CONCENTRATION
Printed in Great Britain.
WITH
LARGE
OF VACANCIES* and J. NBLKBN
A. S. IYBR
Graduate School, University of Cincinnati, (Received
IN
30 April
Cincinnati,
Ohio
1962)
Abstract-Equations are derived from statistical considerations to obtain vacancy interaction energies, En, Ezz (for vacancies with same sign), and Em (with opposite sign), from the variation of oxygen partial free energy with composition for FeO, TiO, and NbO. The oxygen partial free energy was determined through electromotive force measurements with solid electrolytes. Cells of the type Nb, NbO/electrolyte (0.8 ZrOs 0.2 CaO)/NbO, NbOs were studied. The measurements were carried out in vacuum, with tungsten electrodes and induction heating. In FeO, E22 = 2.74 f 0.10 kcal/mole. In TiO, El1 = E22 = 2.10 & 0.03 kcal/mole, El2 = -2.97 + 0.04 kcal/mole. In NbO, En = E22 = 17.9 kcal/mole, El2 = -25.3 k&/mole. The constancy of the vacancy interaction energies permitted the accurate placement of the boundaries of the TiO phase. The entropy of TiO should be increased by 1.68 e.u. to account for the randomness of vacant titanium and oxygen sites.
INTRODUCTION
SOMEof the transition metals of Groups IV, V, VII, and VIII form stable monoxides of the NaCl-type structure. Closer examination, however, shows that in all cases the structure is stable only in the presence of a large concentration of vacancies. The aim of the present work is to obtain some information about these vacancies. THEORY
The theory of compounds with small deviations
by WAGNER(~)and by FOWLER and GUGGENHEIM.(~) The theory was expanded by ANDERSON(3)to include larger deviations. However, he considered only the binary compounds AB where the defect is present solely in atom A, either as a vacancy or as an interstitial. In the compounds under present investigation, the defects occur only as vacancies, but both components can show these vacancies. The grand partition function, GPF, for a crystal of ideal composition AB with only vacancies as defects can be written as follows:t
(1)
N! X
from stoichiometric
cva vs!(N-
v2)!
composition
[a2K2(T)1(N-Y2) exp-(v&2+
has been given
* This research was supported in part by the Aeronautical Research Laboratory, Office of Aerospace Research, United States Air Force, under Contract AF 33(616)6299.
&sE2s+~&sE12)/kT
where N is the number of lattice sites of component 1 and also of component 2; VI and I’s are the vacancies of components 1 and 2; El and Es
1463
t Expansion of Fowler equation (1304).
and Guggenheim,
p.
552,
1464
M.
HOCH,
A. S.
IYER
are the energies necessary to create a vacancy of component 1 and component 2, respectively; Eli, E22, and El2 are the interaction energies per pair between 1 and 1,2 and 2, and 1 and 2 component vacancies; the S’s are the number of those pairs; al and LZ~are the absolute activities of components 1 and 2 in the crystal phase; and El(T) and Es(T) are contributions of an added atom of component 1 and 2, respectively, to the partition function for the normal vibrational modes of the crystal; K is the Boltzmann constant; T is the absolute temperature. Furthermore, if the crystal shows symmetrical behavior around the stoichiometric composition, then El1 = Eea is assumed. In the present investigation, the subscript 1 refers always to oxygen, 2 to metal. The equilibrium conditions are obtained by replacing the sum by its largest term,@) and differentiating the log of the partition function with respect to the number of vacancies, and equating to zero. During the differentiation, the number of metal atoms, i.e., (N- Vs) in equztion (l), is kept constant. Fe0 phase It is well known that the Fe0 phase exists only in the presence of excess oxygen, and that a large concentration of vacancies is present in the lattice. According to the measurements of JETTE and FOOTE(~) and of COLLONGUES,(~) the iron lattice contains vacancies whereas the oxygen lattice is, within experimental results (+ 1 per cent), completely filled. Su = &a = 0, &a = 2/Z (Vi/N), assuming that the interaction energy does not influence the random distribution of vacancies. The error in the evaluation of Eas introduced by this random assumption is small, as is shown in the NbO case below. z (the number of nearest neighbor iron sites around each iron site) = 12. Replacing the S’s in equation (1) and carrying out the differentiation gives :
COLLONGUES(~)the mole fraction of oxygen, no is
N no =
lzl =
ZN-V2
Writing equation (2) for a second composition, (denoted by ‘), subtracting the two equations and replacing Vs and Vi in terms of nl and n;, yields = RTlnz
2?21- 1 , nl -----+RTlnal-RTlna;
-RTln
(3)
2n;-1
Measurement of the oxygen activity as a function of composition permits the evaluation of Ess from equation (3). TiO phase EHRLICH@) studied the structure and density of TiO. He concluded that at the stoichiometric composition 15 per cent of the titanium and oxygen lattice sites are empty and the vacancies are distributed randomly. In a titanium;rich material, more titanium sites are occupied and more oxygen sites are empty. The total number of vacancies is constant over the whole composition range of the TiO phase. Carrying out the differentiation, again assuming random distribution of vacancies, with z [the number of nearest neighbor titanium (oxygen) sites around each titanium (oxygen) site] = 12 and z’ (the number of nearest neighbor titanium sites around each oxygen site) = 6, gives RT,n
(N-
V1J2. V2
N2. Vl
-EI+&;
-2
RT In CZ#CIT]
X
N2 results
1--=
Vz”
_z,E12
Es
kT
of JETTE and
0
(2)
N2
FOOTE(~) and
(V2--Vl)N+KV2 [
1 1 (4)
V,2-2V2N+
Eu
[a1(KlT)1-:$
ZVsNa-
the
J. NELKEN
ZVlN+
N- V2 In y+ln
From
and
N2
Vz”
Writing equation (4) for a second composition (denoted by ‘), subtracting the two equations, and
INTERACTION
ENERGIES
OF
VACANCIES
replacing the V’s in terms of nr, gives (12-&l--6&a) x [nl-n;]
al-RTln
TRANSITION
u;]
[a+(1 +RTln
?2~(1*7nr- ‘7) l-l.7
nr
-RTln
METAL
OXIDES
1465
the number of atoms on the x1 sublattice is [a+ (1 + K)Nl- KVr], and the number of vacancies on the yl sublattices [r~-(l-K)Nl+kVr]. The mass action law requires
[2.89(n1+n;)-6.291 = -2[RTln
IN
n;a(1*7?2;-*7)
-K)Nr-KVr]
[a-_Nlf(l
-k)Vr]
= a2,
from which
l-1*7n;
Measurement of the oxygen activity as a function of composition permits the evaluation of (12&6Els) from equation (5).
a2 - VrNr
&a_
(5)
N+lr,
(N+'v;)2
Using only the first term in the expansion, and keeping in mind that the same conditions hold for the Nb lattice,
NbO phase
a &Gc=(6) the structure of NbO. By Nl+ Vl N2+V2 taking a NaCl-type cell and removing the eight corner atoms (e.g., niobium) and the atom at the The number of vacancies on each sublattice (the cube center (e.g., oxygen), the NbO structure is vacancies ,a;e distributed evenly among the yl, obtained. At the stoichiometric composition y1 , and y1 sublattices) is then given as : twenty five per cent of the niobium and oxygen on x1 sublattice: [N--N11 vacancies; on xa subsites are empty. lattice : [N- Na] vacancies In this structure the vacancies are ordered. To treat the vacancies statistically, the two interon y; sublattice : VI/3 vacancies ; on yb sublattice : penetrating f.c.c. lattices (containing the niobium [WI vacancies atoms and the oxygen atoms) are treated separately, in a fashion similar to the treatment of the AuCua on y;’ sublattice : VI/3 vacancies; on yg’ sublattice : lattice by GUGGENHEIM.(7) The oxygen sublattice [ v2/3] vacancies is divided into fou; interpenetrating simple cubic 1,, lattices, x1, y;, yr , y;“. The total number of on ~1 sublattice: Vr/3 vacancies; on y2“I suboxygen lattice sites is 4N (N on each sublattice). lattice: [v2/3] VaCanCieS. At the stoichiometric composition at 0°K the x1 The evaluation of $1, &a, and &a is done lattice contains all the vacancies, the yl sublattices according to GUGGENHEIM.(~) For Sir and for Saa all the atoms. At a finite temperature, however, a atoms are found on x1 and, correspondingly M each atom has x = 12 neighbors, of which onethird (i.e., f our ) are on each of the other three vacancies on the yr sublattices. To deviate from stoichiometric composition, Nl atoms and VI sublattices. For Sla each atom has x’ = 6 neighbors, of which one-third (i.e., two) are on each of vacancies can be added. Of the Nl atoms (1 -K) the other three sublatti:es. JTlr~s for $2 tk Nl will go on the xl sublattice, kNr on the yl subneighbors of x1 are,,on y2, y2 , y2 ; those of y2 lattices ; of the vacancies VI, (1 - K)Vl will go on are on x1, y1 and y1 .) the yr sublattices, KVl on the x1 sublattice. Thus BRAIJER(~~)studied
SIP = 3z[N-Nr]$+i
S22
[
T.$
1
3 = !$N_N1]V1+i
=;
[N-N2]V2+;
1” [1 k
[$I
1466
M. HOCH, s12=
A. S.
IYER
3,;
[;I
g,-,,;.
and J. NELKEN
[Iv-Nz].
3,;
[T]
[q]
.9
(7)
In the calculation of S1l, &a, and S1s it is assumed that the vacancies are distributed randomly on each sublattice (e.g., V1/3 vacancies distributed randomly on the y;’ sublattice). To carry out the evaluation of equation (1) relations between N1, Na, VI and Va are required. From equation (6) V1+N1
=
v2+N2
Furthermore, it is assumed number of atoms is constant, 3N+N1-
(64
that:
V1+3N+Ns-
(1) the total
Va = 3N;
(8)
and, (2) when a change of composition takes place, half of the change is done by introducing atoms (N1 or Na), the other half by introducing vacancies (sE11-z’E12)(2.25(fl1+n;)-5*25)(~1-n;)
+&ln(l+2n1)3(3-2n1)(6nl-l)5i3 4
(5 -
(1+2n;)s(3_2n;)(6+1)513
Again, the measurement of oxygen activity as a function of composition permits the evaluation of (16&l- 9-&s) from equation (10). To obtain an estimate of the error introduced by the assumption of random distribution of vacancies (in Fe0 and TiO, and on each sublattice in NbO), in the evaluation of E1l and E12, the NbO phase can be treated as the TiO phase, i.e., all vacancies uniformly distributed over all sublattices. In this case, VI+ Vs = 0.50 N and the final equation corresponding to equation (5) is -2[RTln
al-RTln
+ RTln
-
(9)
N2 = VI
In writing the partition function, the distribution of vacancies on each sublattice must be considered, and an equation similar to equation 7.21.5 of GUGGENHEIM(7) is obtained. Carrying out the differentiation, combining the resulting equation with that at another composition (designated by ‘) and replacing the V’s by the mole fraction of oxygen no = 121= (3N+N1V1)/6N results in: (16-k-
g.%) (nl= -(RTIn
n;)(nl+
a;)
(l-
s-.5) 1.5n1)
(l-
(111 \- -I
1*5n;>
As in the NbO phase the vacancies are almost completely ordered, the error in the evaluation of El1 and E1s by equation (11) will indicate a much larger error (perhaps tenfold) than is introduced by the random assumption in Fe0 and TiO and on each sublattice in NbO As the interaction energy of vacancies is coulombic in nature, and is inversely proportional to the distance, and in both TiO and NbO the closest distance for similar charges is d2 times that for opposite charges, it will be assumed that EIZ = -1/(2)-C. APPARATUS
n; - 2)
al-RTln
n$5
a;]
n;s(1*5 .;--5)
RTln
--
N1 = vs
(lo)
(5-64513
=
(Vs or VI). Thus,
6n1)5/3
The
evaluation
AND EXPERIMENTAL PROCJXWRE
of
the
vacancy
interaction
INTERACTION
ENERGIES
OF
VACANCIES
IN
TRANSITION
METAL
OXIDES
1467
bottom electrode. The crucible measures Q in. in diameter, 4 in. wall thickness, and 2 in. high. It rests on a tungsten tripod. One leg of the tungsten tripod is connected to a thin tungsten wire which passes through the bottom plate of the vacuum system. The second electrode is a h in. tungsten rod, which enters the vacuum system through a Wilson type seal in the top plate. The lid of the crucible, which has a &in. hole in the center, is mounted on the tungsten rod in such a fashion
energies by equations (3), (5), (10) and (11) requires the difference in partial free energy of oxygen at two compositions. This is obtained by e.m.f. (electromotive force) measurements using solid electrolytes. The galvanic cell technique using solid electrolytes has been developed by KIUKKOLA and WAGNER.@) Cells of the type Nb, NbO/electrolyte/NbO, NbOs are used. This cell measures the difference between the partial free energies of
TO POTENTIOWETER
WILSON
TYPE
UPPER
.-zJ-*
SEAL
TUNSSTEN
0
COIL
0 0
II r
OPTICAL PIROYETER
II J I I II
HEATER , BOTTOM ELECTRODE
TO
VACUUM
-
SAUQE
TO “ACUUM I
I INCH
TO
POTENTIOMETER
FIG. 1. Vacuum, induction heated e.m.f. cell.
oxygen of the NbO phase at the oxygen-rich composition (in equilibrium with NbO2) and at the oxygen-poor composition (in equilibrium with Nb). As electrolyte (0.8 ZrOs 0.2 CaO) is used.
Apparatus In order to carry out measurements above 1 lOO”C, the e.m.f. cell is enclosed in a vacuum and heated by induction. Fig. 1 shows the apparatus. A tungsten crucible serves as the heater and
that when the system is assembled, a & in. gap exists between the crucible and the lid. As the crucible and lid are both heated by induction, the inside of the crucible, where the e.m.f. cell is located, approximates well a black body. Both half cells and the electrolyte are in pellet form, each pellet measuring about 4 mm in diameter and 2 mm high. The top plate and bottom plate are separated by a quartz tube lt in. in diameter and 6 in. long, which also serves as a vacuum jacket. The two tungsten electrode wires are connected to
1468
M.
HOCH,
A.
S.
IYER
a potentiometer. To filter out radio-frequency currents, a filter consisting of three 0.02 PF condensers is connected across the e.m.f. leads and from each lead to ground. The temperature of the crucible is measured with an optical pyrometer. Vacuum is produced by an Eimac oil diffusion pump, and measured by a Veeco DC-2 discharge gauge. The tungsten crucible is heated by induction through power supplied by a 20 kW Thermionic generator. During the measurements, the vacuum is better than 2 x 10-5 mm. To check the system for spurious effects, the e.m.f. was measured, first using only the electrolyte, and then by making contact between the two electrodes. The e.m.f. was below 1 mV, showing that spurious voltages were negligible.
and J.
NELKEN
pressing into pellets, the material was heated to 1200-1300°C in vacuum. After regrinding, electrolyte pellets were pressed using 65,800 p.s.i. pressure, and reheated in vacuum to 1300°C. Half cells were prepared by mixing metal and metal oxide (e.g., Nb+NbsOs) to give an equimolar mixture (e.g., Nb/NbO), and the mixture heated in vacuum to 1200-1500°C until X-ray diffraction patterns indicated that the reaction was complete. The material was ground and pressed into a half cell using 65,800 p.s.i. pressure. To insure good contact, the half cells and the electrolyte pellet were placed together (one on top of the other) in the die, and pressed using 51,200 p.s.i. pressure.
Preparation of the electrolyte and the half cells
EXPERIMENTAL RESULTS Reaction Fe0 + Ni + NiO + Fe
A mixture of ZrOs and CaCOs was heated to 1000°C for eight hr in air; after grinding and
To check the equipment, the reaction FeO+ Ni + NiO + Fe was studied, using the cell FeO,
Table 1. l+ee energy change in the reaction: Fe0 + Ni + Fe + NiO; Fe, FeO/eZectrolyte (O-8 ZrOs O-2 CaO)/Ni, NiO
Temp. "K I 2 3 4 5
Table
2.
Temp.
“K
1173 * 2 1230 1250 1323 1330 1386 1440
1386 f 1399 1456 1502 1561
Vacancy
e.m.f. mV
ELLIOTT~~~ GLEIsER(g) e.m.f. mV
285 287 294 297 301
291 f 293 299 305 310
interaction energies in FeO; Cell: (0.8 ZrOs 0.2 CaO)/FeO, FesO4
e.m.f. mV 089 095 098 II2 112 122 129
3
ELLIOTT and GLEISER(g) e.m.f. mV 65 + 52 76 80 97 98 II2 123
Cell:
n1’ 0.5119 0.5118 0.5118 O-5118 0.5118 0.5118 o-5119
Fe,
6
FeOlelectrolyte --
n1 0.5296 0.5315 0.5320 0.5341 0.5343 o-5359 o-5373
EZZ
kcal/mole 2.854 & 0.064 2.600 2.620 2.782 2.729 2.805 2.803
Average 2.742 rf: 0.101 kcal/mole
INTERACTION
ENERGIES
OF
VACANCIES
Fe/electrolyte (OX&02 O*ZCaO)/NiO, Ni. Accurate thermodyn~ic data(*) are available for the above reaction. To observe variations due to half cell and electrolyte preparation, new half cells and electrolyte were used for each measurement. Table 1 contains the data, together with those of ELLIOTT and GLEISER.~~)The agreement is good. Fe0 phase The e.m.f. measurements on the cell Fe, FeO/eleetrolyte (O-8 ZrOs O-2 CaO)/FeO, FesOa are given in Table 2. From ELLIOTT and GLEISER@) the free energy change for the reaction 0.788 Fe+ F’esO* + 4 Feo.9470 can be calculated, and is also given in Table 2. This value must be larger than the value obtained by galvanic measurements, where the reaction (0.788-3y)Fe+
Fe304 + 3 Feo.~~~-,O+Feo.~4:0
is studied (Fes.947-u0 indicates the oxygen-rich Fe0 phase). Within the large error limits of the data ( + 52 mV) this is the case. Using the values of nl and n1 (the phase boundaries of the Fe0 phase) given by DARKENand GuRRY,(~~)the interaction energy Es2 between iron vacancies is evaluated from equation (3). Eas = 2.742 + 0.101 kcaljmole and is constant; the small variations are due to the difficulty of accurately reading the phase boundaries. TiO phase The experimental data on the cell Ti, TiO/electrolyte (0.8 ZrOs 0.2 CaO)/TiO, TisOs are given in Table 3. From ELLIOTT and GLEISERCQ) the free energy change for the reaction Ti+TisOs + 3 TiO can be calculated. This value must be larger than the value given by the galvanic cell measurements, where the reaction (1 -i-2x-y)Ti+-TisOs
-+ 2 Til+,OfTil-,O
is studied, (T&+,0 is the titanium-rich TiO phase, T&O is the oxygen-rich TiO phase). Inspection of Table 3 indicates that this is not the case. The discrepancy is due to the entropy of TiO. SHOMATE(~~)upon whose entropy data Elliott and Gleiser’s values are based did not take the vacancies into account when determining the entropy of TiO. The vacancies contribute to the entropy of stoichiometric TiO at 0°K the value of S = -2R [O*SSIn 85 + O-15 In 0.151 = 1.68 e.u. Thus
IN
TRANSITION
METAL
OXIDES
1469
M. HOCH,
1470
A.
S.
IYER
the free energy of formation of TiO becomes more negative by - 168 T cal. When this correction is made, ELLIOTT and CLRrsRR’s(s) data and the galvanic cell results are in agreement. The phase boundaries of the TiO phase can be obtained from the works of BUMPS, KESSLER and HANSEN, and of SHOFIELD and BACON,~~) who agree within three per cent in the region of interest. The values of (Z.&l-Elz) for each set of data were calculated and are given in Table 3. The differences in (2En-Els) between the two sets of data, and the variation with temperature are due to the differences and very small errors in the phase boundaries as (2&l--Els) is very sensitive to composition : At 1275”K, rz; of both references”23 1s) is 0.4721.
and
J.
NELKEN
193 mV, at 1276°K 197 mV, which is somewhat higher than our results. They also studied the reaction Fes.saTO+Nb
+ NbO-tO.947
Fe
(12)
with a galvanic cell, and found AGO = -34.5 + 3.15 x10-s T f O-7 per cent kcal (11141346”K), and the reaction NbOs-t- Hs + NbO+ Hz0 with AGO = 29.49-6.10 x 10-s T & O-3 kcal (1673-1773°K). Combining these actions, and introducing the free energy mation of Fec.saTO(s) and of HsO(s), the
Nbf
NbOs -f 2 NbO
(13) per cent two reof forreaction (14)
Table 4. Yacaq~ interaction enmgies in NbO; CeE: Nb, NbO~e~ec~~o~~e(0.8 ZrOs O-2 CaO)/NbO, NbOs Temp. "EC
EMF mV
I 3
n1
(16E11- 9&a)
kcal/mole
CalZlatecl (16&1-9&a)
1160 _t 2 1217 1275 1353 1429 1503 1572
137 157 174 190 203 217 230
0.485
0*505
Changing nl from 0.5561 to 05572 (a 0.2 per cent change) changes (2En-Els) from 7.205 kcal to 7.109 kcal (a l-3 per cent change). Thus using a constant (average of values at 1275 and 1278°K) (2En--Ers) = 7.180 kcal, and 9 = 0.5567 (halfw,ay between the two values), the phase boundary nl of the TiO phase is calculated in the last column in Table 3. Taking into account the errors in the e.m.f. measurement, El1 = Ess = 2.103 + 0.025 kcal/mole and Ers = -2.974 + 0,035 kcalfmole. NbO phase The experimental data on the cell Nb, NbO/ electrolyte (08 ZrOs O-2 CaO)/NbO, NbOs are given in Table 4. GE~SIMOV et a;h(15) measured the e.m.f. of a cell similar to the one used here but with platinum electrodes and found at 1223°K
305.6 rt: 26 351.3 389.8 425.8 455.0 486.5 515.8
= 5158 kcaf/moie ?21= 0~505 0.492 0.491 0,490 O-488 0.488 0,486 0,485
with AGO = - 8.18 - 1.2 x 10-3 T kcal is obtained. The accuracy of this equation is hard to evaluate as Iarge extrapolations in temperature have to be made when equations (12) and (13) are combined. At 1572”K, equation (14) yields 218 mV, in agreement with 230 mV in Table 4. The phase boundaries were estimated from BRAUER’#) data as NbOs.sd and NbOl.se, and assumed independent of temperature. Using equation (lo), (l&!&1-9Els) was evaluated. The strong dependence of (16Ell- 9&s) on temperature is due to the assumed constancy in phase boundary. B~~R(~~) prepared his samples at 1950°K and cooled them to room temperature. The homogeneity range of the NbO phase will become smaller at lower temperature (similar to Fe0 and TiO}. Only small changes in phase boundary are needed to obtain constant values for the vacancy interaction
INTERACTION
ENERGIES
OF
VACANCIES
energy; this was calculated by assuming (16Eu9Els) = 515.8 kcal/mole and the oxygen-rich NbO phase boundary constant as in TiO (nl = 0.505). The Nb-rich NbO boundary n; is given in the last column of Table 4. Using again the assumption taking El2 = ‘- 1/(2)Eu and (16Eu-9E12) as 515 -8 kcal/mole (value at 1572”K), El1 = 17.9 kcal/mole and El2 = -25.3 kcal/mole is obtained. (2Ell-E1s) can also be evaluated from equation (1 l), where the vacancies were assumed to be completely random. At 1572”K, (2Eu--Els) = 55.649 kcal/mole from which, again with the assumption El1 = 16.3 kcal/mole. The El2 = -2/(2)&l, change in El1 is only 10 per cent (from 17.9 to 16.3 kcal/mole) when the vacancies are treated as random rather than completely ordered. Thus in the present method of evaluation, El1 is almost independent of the distribution of vacancies. On the other hand, El1 influences strongly the distribution (random or ordered) of the vacancies. In FeO, where Es2 = 2.7 kcal/mole, the vacancies will be somewhat ordered. But even if this is taken into account in the distribution function (using, for example, the quasi chemical method),(‘) Ess will not change.
IN
TRANSITION
METAL
OXIDES
1471
REFERENCES of Alloys, Addison1. WAGNER C., Thermodynamics Wesley Press, Reading, Mass. (1952). 2. FOWLER R. and GUGGENHEIM E. H., Statistical Thermodynamics, University Press, Cambridge (1949). J. S., Proc. Roy. Sot. A185, 69 (1946). 3. ANDERSON 4. JETTE E. R. and FOOTE F., J. them. Phys. 1, 29 (1933). 5. CO&O~G~ES R., thesis, Paris (1955). 45. 362 (1939) Z. 6. EHRLICH P.. Z. Elektrochem. anorg. Ch;m. 247, 53 (1941). Clarendon Press, 7. GUGGENHEIM E. H., Mixtures, Oxford (1952). 8. KIUKKOLAK. and WAGNER C., J. electrochem. Sot. 104, 379 (1957). 9. ELLIOTT J. F. and GLEISER M., Thermochemistry for Steelmaking, Addison-Wesley Press, Reading, Mass. (1960). 10. DARKENL. S. and GURRY R. W., J. Amer. them. Sot. 67.1398 (19451: 68.798 (1946). Amer. &em. hoc. 68,310 (1946). 11. SHOI&TE C: H.,i 12. BUMPS E. S., KESSLER 0. and HANSEN M., Trans. Amer. Sot. Metals 45, 1008 (1953). 13. SCHOFIELD T. H. and BACON A. E., J. Inst. Met. 84, 1654 (1956). 14. BRAUERG., Z. nnorg. Chem. 248.1 (1941). 15. GERASIMOVT. IA., LAVENTEVV. I., KUZNETSEVF. A. and REZUHINA T. N., paper presented at the XVIII International Congress of Pure and Applied Chemistry, Montreal (1961).