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The relationship between oxygen thermomigration and
defect structure in some non-stoichiometric oxides
D. Norris
To cite this version:
D. Norris. The relationship between oxygen thermomigration and defect structure in some
non-stoichiometric oxides. Journal de Physique Colloques, 1980, 41 (C6), pp.C6-331-C6-334.
JOURNAL DE PHYSIQUE Colloque C6, supplemen.nt au tz0 7 , Tome 41, Juillet 1980, puge C6-331
The relationship between oxygen thermomigration and defect structure
in some non-stoichiometric oxides
D. I. R. Norris
Berkeley Nuclear Laborator~es, Berkeley, Gloucestershire, England
Resume. - La thermomigration de l'oxygtne h 1'Ctat solide dans les oxydes non staechiometriques depend de
la maniere dont le deftaut ou l'excirs d'oxygkne se rapporte au rCseau cristallin. On a utilise le concept d'une d'oxygkne like a deux cations reduits dans un modkle cluster decrivant la thermomigration de l'oxygtne dans
(U, Pu)02-,. Des relations ont CtC etablies pour la variation stcechiomCtrique a 1'Ctat solide produite par un recuit suffisamment long dans un gradient de temperature et pour le temps requis pour I'approche de l'etat sta- tionnaire. On discute la capacitk du modtle de tenir compte des donnkes publikes sur la redistribution de l'oxygtne. On presente de nouveaux resultats experimentaux pour (U, Ce)02-, et I'on compare cette matikre a (U, Pu)02-,.
Abstract. - Solid state oxygen thermomigration in non-stoichiometric oxides depends upon the manner in which oxygen deficiency or excess is accommodated in the crystal lattice. The concept of one oxygen vacancy bonded to two reduced cations has been employed in a cluster model describing oxygen thermomigration in (U, Pu)O,
-,.
Equations have been derived for the steady state stoichiometry variation produced by a sufficiently long anneal in a temperature gradient and for the time to approach the steady state. The ability of the model to account for published data on oxygen redistribution is discussed. New experimental results for (U, Ce)02-, are presented and comparisons made between this material and (U, Pu)02-,.1. Introduction. - A temperature gradient can pro- 2. The cluster model of thermomigration. - The
duce redistribution of the oxygen in a non-stoichio- thermomigration of any component within a solid is metric oxide : this is a well-known phenomenon described by non-equilibrium thermodynamics [e.g. 21. which has considerable practical importance for In steady state, the concentration C j of component j
oxide fuel in nuclear reactors. However, it has only varies according to :
recently been recognised [I] that the redistribution
depends upon the manner in which the oxygen defi- In C j = In A - (Hj
-
Q,*)
ciencv (or excess) is accommodated in the crystal - . R T ( l
+
2 In yj/2In
C j ) (2) lattice. The present paper reviews progress made inthe understanding of oxygen thermomigration, parti- cularly in (U, Pu)02-,, by taking into account defect structures and energetics.
On annealing a non-stoichiometric oxide in a temperature gradient, the oxygen distribution changes until it reaches
a
steady state in which composition varies with temperature along the gradient. It is customary to describe this situation by the equation :where x is the stoichiometry deviation, A a constant, R the gas constant and T the absolute temperature.
The energy parameter Q , usually found to be positive, is called here the Arrhenius slope ; alternatively - Q
can be regarded as an apparent heat of transport. There are alternative mechanisms for transport of oxygen, including gas phase routes, but our interest here is in the interpretation of measurements of the Arrhenius slope when solid state diffusion dominates other mechanisms.
where Hj is the effective formation enthalpy, y j the
activity coefficient and A is a constant determined by a boundary or conservation condition. The heat of transport Qf is the energy flux consequent upon unit flux of component j in an isothermal system. For the treatment of oxygen thermomigration in (U, Pu)02-, or a similar material, it suffices to deal merely with the oxygen sub-lattice, since the diffusion coefficient of oxygen is several orders of magnitude higher than those of the cations. The migrating component is the oxygen vacancy.
Application of eq. (2) to oxygen vacancies depends
upon knowledge of Q,*, of H, and of the consequences
of deviations from dilute solution : detailed discussion is given elsewhere [3]. The literature contains various theories for heats of transport, but disagreements remain unresolved. An empirical rule suggests that
Q,* is likely to be small relative to Hv. Consideration
of the manner in which Hv should be defined when there is migration only within the solid state suggests that it is the local lattice strain enthalpy associated
C6-332 D. I. R. N O R R I S
with an oxygen vacancy - any definition involving
removal of an oxygen atom out of the solid state into a gaseous phase cannot be relevant. In general, the local enthalpy involved in deviating from a perfect crystal lattice is positive, so that both Hv and also the equivalent quantity for an oxygen interstitial are positive. From eq. (2) then the direction of oxygen thermomigration is always such as to produce the highest defect concentration at the highest tempera- ture. When vacancies are the dominant defects, as in (U, Pu)Oz-,, oxygen redistributes down the tempe- rature gradient. When interstitials are the dominant defects, as in (U, Pu)02+,, oxygen redistributes up the temperature gradient.
The cluster model defines species j as the fully mobile defects only and considers these to be in dilute solution, so that the activity term may be omitted from eq. (2). In a non-stoichiometric oxide such as (U, Pu)02-,, most of the oxygen deficiency is accommodated in some kind of ordered structure o r cluster and the treatment requires equations to be devised describing the equilibrium between the mobile state and each bound state. Direct experimental information on ordered vacancy structures is insuf- ficient and recourse must therefore be had to a model. We choose to follow that of Manes and Manes- Pozzi [4], which successfully accounts for both the thermodynamic functions of (U, P u p z - , and also the phase diagram of the system. The basic idea is that an oxygen vacancy has a tendency to associate with a pair of Pu cations which are reduced to the trivalent state, i.e.
The symbol V signifies a vacancy not associated with a pair of Pu cations : in terms of the chemical balance of the equation, the addition o f a vacancy is equivalent to the removal of an oxygen atom. The subscript i refers to the fact that not all Pu sites are equivalent, depending on the proximity of other Pu2O3 complexes. Thus complexes may be classified as types 1, 2, 3, etc., and calculations based upon the geometry and sym- metry of the crystal lattice distinguish between them. The quasi-chemical equation governing the equili- brium of eq. (3) is :
where xi is the vacancy concentration accommodated in clusters of type i, yi is the fraction of cations that are Pu ions able to participate in the formation of complex i and the assumption C, 4 x, is made. The binding enthalpy E , is the enthalpy decrease when the reaction of eq. (3) goes from left to right. Consi-
dering the first three cluster types only, values of y , are available [4], while y 2 and y 3 depend upon x,
and x z respectively. The final result obtained by putting previous equations together is :
These equations describe the steady state. The characteristic time t for approach to the steady state is given [5] by :
where c is
QP
D VT/(RT~), z is the diffusion distance and D is the diffusion coefficient for mobile oxygen vacancies.3. Comparison with experiment. - On the basis of the foregoing model, theoretical curves have been calculated to fit experimental data obtained by Evans and Aitken [6]. Their results are shown in figure 1
with the fitted curve, which assumes increments of 4 kcal/mole in the binding energies of successive complexes and then requires a Ql of 28 kcal/mole. This is consistent with Hv being approximately half the Frenkel energy and both EBl and Q,* being rela-
tively small.
Temperature (OC)
Rec~procal Temperature ( K-' )
f i g . I . - T h e clusrcr model fitted to oxygen t h e r m o m t g r a t ~ o n data of Evan\ . ~ n d A ~ t k c n 151 for ,I 2 000 h a n n r ; ~ l of lony pellets of U, ,,Puo z s 0 2 - ,. F r o m [[I.
Experimental data collected by Sari and Schu- macher [7] are plotted in figure 2 as measured Arrhe- nius slope versus plutonium valence. The latter is a convenient representation of stoichiometry deviation by assuming that 'the uranium valence is 4 and that the oxygen deficiency corresponds to a plutonium valence of less than 4. The experiments are difficult and the scatter on the points is large, but the theore- tical curves are at least consistent with them. Unfor- tunately, the accuracy is insufficient to test the pre- dicted variation with plutonium concentration y.
THE RELATIONSHIP BETWEEN OXYGEN THERMOMIGRATION AND DEFECT STRUCTURE C6-333 3 0 Y y = 0.25 y
-
0.6 a 'LO 3'9 :8 :7 3'6 5: ; L 3'3' 3.2 Pluton~um ValenceR g . 2. -Measured Arrhenrus sloprs fbr steady slate oxygen thermomigration in U , - , . P U ~ O ~ - ~ [7] w ~ t h the predictions of the cluster model.
Calculation of t for the conditions of the experiment yields an approximate value of 2 100 h. The agree- ment is sufficiently close to give confidence in the met hod.
4. Comparison between (U, Ce)02-x and (U,
Pu)02-,. - Table I shows some preliminary experi- mental results for (U, Ce)02
-,.
Oxygen-to-metalTable I. - Measured Arrhenius slopes for steady
state oxygen thermomigration in U -,Ce,.02 -.r.
Cerium Arrhenius valency z;
-
slopeQ
Y x = 4 - 2 T / y kcal/mole - -- - - 0.282 0.003 3.979 40 0.402 0.012 3.940 65 0.402 0.019 3.905 8 0.402 0.027 3.866 2(*I
0.148 0.020 3.730 0(*) For reasons given in the text, this result should be regarded as a lower limit
ratios were determined by X-ray lattice parameter measurement, a technique successful as long as the material remained single-phase [8]. However, at the larger deviations from stoichiometry, extra features appeared in the X-ray powder photographs. Measure- ment of the dominant lines then gave the value for the major phase only, the true O/M being probably lower.
The work shows that oxygen vacancies in this case also redistribute from low to high temperatures, as expected from eq. (2) with a positive H,. The Arrhe-
nius slope Q decreases with increasing x as in (U, Pu)02-, but more rapidly by a factor of about 2. Thermomigration becomes negligible, i.e. Q appro- aches zero, at a cerium valence of 3.7 compared to a plutonium valence of 3.3. This is too large a difference to be explained by the measurement technique and, in terms of the cluster model, suggests that the for- mation of immobile complexes is easier in (U, Ce)02 -.r
than in (U, P U ) O ~ - ~ .
5. Discussion and conclusions. - The vacancy complexes that have been assumed almost certainly over-simplify the actual defect structures. From a fundamental point of view, the cluster model should not be regarded as more than a step towards under- standing of the effect. It does, however, provide a successful interpretation of oxygen thermomigration data, explaining in a natural way the important trends that have been observed. Its use to make predictions of fast reactor fuel behaviour [5] is there- fore well justified.
As a method of obtaining information on defect structures, thermomigration measurements suffer from the disadvantages of being difficult to make and of limited accuracy. However, the comparison between (U, Pu)O,
-,
and (U, Ce)02-, does suggest that immobile complexes incorporating some oxygen defi- ciency are more easily formed in the latter material.Further investigation will be worthwhile.
Acknowledgment. - This paper is published by permission of the Central Electricity Generating Board.
DISCUSSION
Question. - M . J . GILLAN. Reply. - D. I. R. NORRIS.
The value of the heat of transport is not easy to There have been theoretical arguments relating estimate. However there is some information on the heat of transport to the migration energy, but I
other ionic crystals, for example potassium chloride. find these unconvincing. Results for potassium chlo- There the reduced heat of transport is comparable ride are consistent with the more empirical approach with the migration energy. Is this consistent with used in my paper.
References
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, <, r, 292.
1 0 1 .
[6] EVANS, S. K. and A I T K ~ , E. A,, General Electrlc Report [3] NOKRIS, D. I. R , J. NUCI. M u t e r . 79 (1979) 118. GEAP- 14036 (1975).
[4] MANES, L and MANES-Pozzr, B., Pluronium 1975 and Other 171 SARI. C. and SCHUMACHFR, G., J . Nucl M u t e r . 61 (1976) 192. Actinides, eds. Blank, H . and Lindner, R. (North-Holland, (81 NORKIS, D. 1. R., COLEMAR, S . C. and KAY, P., CEGB Report
