Diffusion processes in nuclear fuels

Journal of the Less-Common

DIFFUSION

Metals, 121 (1986)

PROCESSES IN NUCLEAR

63’7 - 564

531

FUELS

Hj. MATZKE Commission of the European Communities, ment, European Znstitute for Transuranium (F.R.G.)

Joint Research Centre, Karlsruhe Elements, Postfach 2266, D-7500

EstablishKarlsruhe

Summary

A review is given of atomic diffusion processes in nuclear fuels. Emphasis is put on the oxides UOz and (U,Pu)O,, the carbides UC and (U,Pu)C, and the nitrides UN and (U,Pu)N, but ThOz and the metals uranium, plutonium and thorium are also treated to some extent. Self-diffusion, impurity diffusion, chemical diffusion and thermal diffusion are all discussed. For ceramic fuels, the effects of deviations from stoichiometry, of build-up of fission products during reactor irradiation and of enhancement of diffusion by radiation are dealt with. Finally, a short review on fission gas behaviour is given.

1. Introduction

Diffusion phenomena in actinide metals and actinide ceramic nuclear fuels have been treated at all previous conferences of the series “Plutonium and Other Actinides”, starting with questions of self-diffusion and chemical diffusion in pure or stabilized plutonium [l, 21 and extending to ceramics including oxides [ 31 and carbides [4] of uranium, plutonium and thorium. The present review is intended to cover these three groups of materials, but it includes also the nitrides of uranium and plutonium, thus covering both aspects of today’s nuclear reactor fuels (UO, and (U,Pu)O,) and of highmetal density advanced fuels ((U,Pu)C and (U,Pu)N). Because of limitations of space, a comprehensive literature survey cannot be presented. Rather, highlights and summarizing statements are given, whenever possible treating recent developments since the last plutonium conference in Asilomar, and typical references to the original literature are given rather than a complete listing. Atomic diffusion is a basic process controlling many phenomena during fabrication and irradiation of the above nuclear fuels. Examples are sintering, creep, grain growth, fission gas bubble formation and migration, and fission gas release. All these processes are related to metal (actinide) atom diffusion, *Paper presented at Actinides 85, Aix en Provence, September 2 - 6,1985. 0022-5088/86/$3.50

OElsevier Sequoia/Printed in The Netherlands

538

since in all of the ceramic fuels treated here the metal atoms diffuse at much lower rates than the non-metal atoms (oxygen, carbon, nitrogen). The slower species is normally rate-controlling, since for true matter transport to occur all of the components of a compound have to be transported (Fig. 1).

Fig. 1. Schematic presentation of processes related to metalatom fuels.

self-diffusion in nuclear

To understand fully the diffusion processes, their dependences on temperature (including the possible effects of temperature gradients), on deviations from stoichiometry (oxygen potential, carbon content or nitrogen pressure), on impurities (chemical doping or build-up of fission products during irradiation) should all be known. In addition, at temperatures below about 0.5 Z’, (Z’, is the melting point in kelvin), direct enhancement of diffusion by radiation and by the fission events has to be considered. The available knowledge on these points for the above fuel materials is therefore summarized. Also, the existing gaps in our knowledge are pointed out.

2. Diffusion in actinide metals Many investigations exist on self- and impuritydiffusion in thorium, uranium and plutonium. These three actinide metals occur in different crystallographic forms, with phase changes at well-defined temperatures. Diffusion processes have been measured for two phases of thorium, three of uranium and five of plutonium (see Table 1). The high-temperature phases are body-centered cubic (b.c.c.), two lower temperature phases are facecentered cubic (f.c.c.); the remaining phases are more complex. Self-diffusion in f.c.c. metals is well understood: it proceeds via a vacancy mechanism, the diffusion coefficients D follow an Arrhenius-type relation

539

TABLE 1 Results of self-diffusion in actinide metals Metal

Phase [5]

Stability W)

Th

fl b.c.c. (Y f.c.c.

1360 - 1750* < 1360

U

y b.c.c. /? complex tetragonal (Y e .c . orthorhom bit

Pu

E 6’ 6 y p

b.c.c. b.c.t. f.c.c. f .c . orthorhom bit b.c. monoclinic

range

775 - 1136“ 668 - 775 < 668 480 458 315 205 125

-640a - 480 - 458 - 315 - 205

Do (cm2 s-l)

AH (kJ mol-‘)

0.5 [6] 1.7; 395 [6]

230 324; 300

1.19 x 10-a [7] 1.35 x 10-2 [7] 2.0 x 10-a [7]

112 176 192

4.5 (5.3 5.2 3.8 1.7

x x x x x

1O-3 [B] 1032)5 [B] 10-l [B] 10-l [B] 1O-2 [B]

66.9 (589)b 126.4 118.4 108

b.c.c., body-centred cubic; f.c.c., facecentred cubic; b.c.t., body-centred tetragonal; e.c., end centred. aMelting point. bDescriptive values. The T range for the &‘-phase is very narrow: 25 * 5 “C.

D = DOexp(-AH/RT) the possible exception of a slight curvature very near the melting point, Cl, owing to a contribution to diffusion of divacancies), the activation enthalpy AH is related to T,.,,.* AH(cal mol-‘) = 35T,(K) or to the latent heat of melting L m: AH = 16.5L,, and the activation entropy is positive giving a pre-exponential Do k 0.1 cm2 s-i. In b.c.c. metals, however, an anomalous diffusion behaviour is frequently observed, typical examples being zirconium and titanium: curvature is obvious when plotting log D uersus l/T in the Arrhenius diagram, and AH and D,, are lower than expected. Many sophisticated experiments have been performed to find the reasons for this. Trivial possibilities such as gram-boundary contributions were excluded, and a number of explanations were suggested. However, a definite conclusive explanation valid for all “anomalous” b.c.c. metals has still not been found. The available results for self-diffusion in different allotropic forms of actinide metals are summarized in Table 1, together with typical references, and are shown in Fig. 2. The expected behaviour is found for the f.c.c. phases of thorium and plutonium. For all b.c.c. modifications (p-Th, r-U, E-Pu) the AH values are low, and for y-U and E-Pu, the Do values are also low. The latter two phases of uranium and plutonium can therefore be regarded as belonging to the group of “anomalous” b.c.c. metals. For e-Pu, diffusion measurements under pressure [9] show a decrease in D with pressure and a negative activation volume. These facts can be taken as evidence for an interstitial mechanism for self-diffusion in E-Pu [9]. However, impurity diffusion measurements in y-U support a vacancy mechanism for diffusion (with

540

-16 1

5

10

15

20

25

lO'/T, K-' Fig. 2. Arrhenius diagram for self-diffusion in the different phases of the actinide metals thorium, uranium and plutonium [6 - 81.

in b.c.c. uranium [lo] (see below). More work is needed to define the diffusion mechanism in y-U, as well as in the remaining phases. Theoretical work might be most helpful, since the experimental techniques which have helped to define diffusion mechanisms in many other systems do not seem feasible with actinide metals: these techniques include measurements of the isotope effect of self-diffusion, quenching and Balluffi-Simmons-type experiments. Figure 2 shows also large changes in D between different phases at the temperature of the phase changes. For instance, D” drops by a factor of 150 at the r-U-&U phase change and DTh even by a factor of 200 at the /3-Tha-Th phase change if the low D Th values for cr-Th are taken as more reliable. Such changes in D are frequently encountered in other metals at phase changes as well. The data for plutonium [8] are impressive indeed in that five phases have been covered including the 6 ’ phase which exists only in a very narrow temperature region, (25 f 5) “C. Figure 3 shows available impurity diffusion data for the high temperature b.c.c. phases of 7-U and /3-Th [ 10 - 131. Data exist for many impurities, e.g. for 13 elements in /3-Th (and even for 22 elements in cx-Th). Elements generally diffusing interstitially such as oxygen, nitrogen and carbon diffuse very fast, as is also the case in many other metals. Cobalt, iron and nickel are fast diffusors as well in both P-Th and 7-U. At the temperatures of the phase changes P-U-r-U and cr-Th-&Th, their D values are higher by factors of 100 to 400 than D” and Dm. Usually, fast diffusion of impurities is related to a high polarizability and to migration of interstitial type defects [ 141, often invoking a predominantly substitutional solubility of the impu-

541

lfml6oolum

no0

s

I”‘.‘.

-6

-10 1

lam

T,“C

800

’

. p-Th

1

5

6

1

e

1

9

10 lO’/T, K-’

Fig. 3. Arrhenius diagram for impurity diffusion in the high temperature b.c.c. phases of uranium and thorium (7-U and fl-Th) [lo - 131.

rity with slow diffusion of these atoms, but also a small interstitial solubility with fast diffusion causing the effective high D values (dissociative mechanism) in a “two-stream” model. For uranium and thorium, a vacancy mechanism with a strong impurity vacancy binding energy and a high degree of correlation between the direction of successive vacancy jumps is more probable. This is substantiated by the observed acceleration of uranium self-diffusion in 7-U caused by additions of cobalt [lo]. Also, the slow diffusor niobium causes an expected decrease in D” if alloyed to y-U. As an alternative, in recent work on diffusion of cobalt, iron and manganese in uranium, the observed fast diffusion was explained using a model assuming local melting of the crystal lattice near the impurities [15]. More rigorous statements on the details of the interaction between uranium defects and impurities must await further theoretical and experimental work. Some data also exist for phases other than the b.c.c. modifications: cobalt is also a fast diffusor in &Pu [15a] with DcO/Dpu ratios of about 104; cobalt, manganese and iron diffuse fast in a-U and P-U as well [15] and electrotransport measurements in cr-Th and &Th [ll - 131 show fast diffusion of iron, cobalt and nickel in both modifications, with relatively small discontinuities of factors of 3 to 5 at the phase boundary (T = 1350 “C). As expected, the AH values for these fast diffusors are low (37 - 80 kJ moll). 3. Self-diffusion in ceramic nuclear fuels In this section thermally activated self-diffusion in oxide fuels (UOz, (U,Pu)O,, Th02), carbide fuels (UC, (U,Fu)C) and nitride fuels (UN, (U,Pu)N) is treated. Metal atom diffusion in thermal gradients, radiation enhanced diffusion and fission product diffusion are dealt with in subsequent sections.

542

3.1. Oxides The actinide dioxides of uranium, plutonium and thorium have the fluorite structure. Their melting points are high (e.g. about 3300 “C for ThOz). UOZ is the fuel of today’s water-cooled reactors, the mixed oxide (UJVO2 9 most frequently with 20% to 25% Pu, is the standard fuel of liquid metal cooled fast breeder reactors and ThOz (or ThOz with a small addition of UOz) is considered as potential future fuel for some reactor programs, e.g. for the Canadian CANDU reactors. Both UOz and PuOz, and therefore also (U,Pu)O,, can show large deviations from the stoichiometric composition with the O/M ratio of 2.00 (M = metal, hence uranium, plutonium and/or thorium). UOz, ThOz and PuOz can all be substoichiometric as MOz-% with oxygen vacancies being formed as dominant defects, charge compensation being achieved by reducing some of the metal atoms, e.g. to Pu3+. UOz can also be overstoichiometric as UOz+X, containing oxygen interstitials together with U5+ and U6+ ions. In fact, UOz +-X is the oxide phase showing the largest width of a single-phase field in an extensive list of non-stoichiometric oxides [16] : the single phase UOz +x extends from uole65 to UOz.z5 (at about 2500 “C) with a very wide A(O/M) of 0.60. The excess oxygen in UOz +X does not occupy the normal ($, i, $) interstitial sites of the fluorite structure. It rather forms larger clusters by displacing some (often two) normal oxygen atoms from their lattice sites, thus creating oxygen vacancies and more oxygen interstitials (Willis clusters). In substoichiometric oxides, sg. PuOz -%, the oxygen vacancies can also form clusters if x Z. 0.05. The simplest cluster consists of an oxygen vacancy and two Pu3+ ions. At larger deviations from stoichiometry, larger clusters can form. As mentioned above, oxygen is the mobile species in the actinide dioxides of the fluorite structure: at 1400 “C and for UOs, D”/Du > lo5 (depending on x). Since for true mass transport to occur all atomic species have to be transported, the slowest is usually the rate-controlling species (see Fig. 1). In the following, most enphasis is therefore always put on metal atom diffusion, even when discussing carbides and nitrides. 3.1.1. Oxygen diffusion Oxygen diffusion, however, is also of interest and importance, e.g. for any reduction or oxidation treatment. For oxidation and reduction, chemical diffusion of oxygen is ratedetermining. The corresponding diffusion coefficients 0” are deduced from measurements of the rate at which thermodynamic equilibrium conditions are achieved by diffusion of point defects (oxygen vacancies for MO*-, , oxygen interstitials for MO,.,) within the chemical gradient set-up when the oxygen pressure of the gas atmosphere or the temperature are suddenly changed. A prerequisite is that oxygen transport across the gas-solid interface is fast and not-rate-limiting. Most frequently, weight changes are measured to determine D. If equilibrium is achieved, self-diffusion coefficients D describe oxygen transport. Since no suitable radioactive tracers exist for oxygen (as for nitrogen, and in contrast to all other diffusing species considered in this

543

article), either mass spectrometry with “0, or different suitable nuclear techniques must be used to measure diffusion profiles of the naturally occurring inactive “0 isotope (see ref. 16 for a summary). This complicates the measurements. A somewhat simplified relation [16 - 181 between fi and D for oxygen diffusion in e.g. U02 +X is given by @=FD=DO_

(2 + xl d{A~(O,)l 2RT

dx

As shown in ref. 17, the thermodynamic factor F (and hence essentially the oxygen potential A q02)), varies equally strongly with x as does Do in uo 2+x (see Fig. 4), but has the opposite trend. The product of Do and F, and hence so, are nearly constant and thus fairly independent of x. This is also shown in Fig. 4 as a band. The fro values are very high indeed, thus explaining the high rates of oxidation and reduction of the oxides MO2 fx . T,“C

-7 wl

2omlEa

12oolaJl

KKlforlJ0, ior IhO,

Fig. 4. Normalized Arrhenius diagram, plotted as T,/T, of selfdiffusion processes in ‘IhOz, UO2 fx and (Ue.spUe.~)O~+, (MO2,,) (T,, melting point in kelvin). Shown are results for oxygen (0) diffusion in ThO2, UO2 and PuO2, and for metal atom (thorium, uranium, plutonium) diffusion in ThO 2, U02 and MO?. The effect of deviation from stoichiometry is also shown: in the upper half for oxygen diffusion in UO2 and U02+X for three values of x (10e5, lop3 and lo-‘), and in the lower half for plutonium diffusion in MOzkX at 1600°C [17 - 201. Included are recommended diffusion data for the fiiion rare gas xenon diffusing either unperturbed as single gas atom (low concentration), or affected by trapping, i.e. retarded by gas-gas or gas-defect and gas-damage interactions (high concentration). Some precursors of the rare gases which are also volatile, diffuse faster; the example shown is tellurium [Zl, 221.

544

In contrast, Do is strongly dependent on X, as mentioned above and as shown for four compositions (UOZ, U02_oooo1, UOz.ool,and UOZ.i) in Fig. 4. These lines are calculated [ 18, 191, but they are also well supported by experimental results. Theoretically, a maximum in Do is expected at x = 0.1 because of defect clustering. However, oxygen diffusion measurements have so far not clearly revealed the presence ot these clusters. Conditions are similar for substoichiometric oxides MOz_% [e.g. ref. 231 where fast oxygen diffusion is due to the large concentration of oxygen vacancies, and the small slopes in the Arrhenius diagrams are caused by the the migration enthalpy of oxygen vacancies. Very low value of A=., limited data are a&able for UOZ ._ where the formation of shear-structures have also been reported (see, e.g., ref. 3). More data exist for (U,Pu)02_X and oxides such as (U,Ce)02 _-x where charge compensation is easily achieved by reducing the added metal, e.g. Pu4+ to Pu3+. Again, no indication of the above-mentioned type of clusters (oxygen vacancy cations) is seen in oxygen diffusion, neither for Do nor for o”O. As with MOz +%, further work should look in more detail at the possible (and expected) influence of the existence and dissociation or change of size and shape of these clusters on oxygen diffusion. 3.1.2. Metal atom diffusion Metal atom diffusion in fluorite type dioxides is much slower than oxygen diffusion. Many of the early results were untypical because of experimental difficulties, largely caused by surface effects, as described in detail in refs. 3 and 20. Reliable results are, however, available for ThOl, +x . UOZ%X and UJ,WO, Because of the low mobilities of the metal atoms thorium, uranium and plutonium, metal atom selfdiffusion, e.g. uranium diffusion in U02, plutonium in Pu02 etc. are ratedetermining for many interesting high temperature processes (creep, sintering, densification, swelling etc.). Table 2 contains some of the necessary information and Fig. 4 includes the general trend of metal diffusion processes in fluorite-type oxides. A reduced temperature scale is used, hence instead of 104/T, T,,,/T is plotted on the abscissa to normalize results for materials of different melting points, T, in kelvin. Figure 4 clearly shows the fact that metal atom diffusion is much slower than oxygen diffusion. A melting point relation is also implied for the different oxides of the fluorite structure, and, as a matter of fact, also for the related fluorites CaF, and BaF2, since results for U02, Th02 (and even CaF, with more than a factor of 2 in T,) nearly coincide [17]. The earlier data that were affected by surface effects and experimental difficulties are not considered anymore. The true metal volume diffusion shows high activation enthalpies (e.g. 5.6 eV for plutonium in U02, see Table 2, or about 6.5 eV for thorium in Th02 [16]). Calculated values are higher still (see below). The point defect model for U02 and other fluorite-type dioxides is summarized in Table 2. As for oxygen diffusion, significant changes in AH and D between

545 TABLE 2 Predicted and measured activation energies for the actinide dioxides MOzkx of the fluorite structure and experimental and calculated energies of formation and migration of oxygen and uranium defects in LlOz together with Arrhenius activation enthalpies Predicted activation energies based on a point [ 31 and neglecting entropy effects

MO2

+x

MO2 MO2

--x

Oxygen

Metal

AH&

A@

++o

AGs - AGFO + AH&

+ A@&

Experimental 3.0 -4.0 about 9.5 6-7

Migration energies (eV) O-vacancy, AHE,, O-interstitial, AHg,i U-vacancy, AHC v U-interstitial, AEitl;,i

0.5 - 0.7 0.8 - 1 .O about 2.4 possibly low

+x

--x

uo2

+x

a

Calculatedb 4.3 19.4 7.3 0.5 0.6 6 .O 8.8

energies (eV)c

uo2 uo2

- 2AGFo + AH&

+ AH& (x 6 0.02) FM - AGS + AHi,i(x Z 0.02) with contributions of a cluster mechanism and of increased mobilities of M3+ ions

Formation energies (eV) 0-F’renkel pair, AGFo U-Frenkel pair, AGFU Schottky trio, AGS

uo2

model

“acG,

AH&

Arrhenius activation oxygen

defect

0.8 - 1.0 2.6 0.5 - 0.7

0.6 2.8 0.5

Uranium uo2 uo2

--x

about about about about

2.6 5.6 7.8 (x 5 0.02) 5 (x Z 0.02)

3.8 8.5 13.3 18.1

AT, migration enthalpy; O,v, oxygen vacancies; Oj, oxygen interstitials; M,v, metal vacancies; Mj, metal interstitials. aFrom refs. 3,16 - 20,23 and 33a. bFrom refs. 24 - 26. c The suggested Arrhenius equations for stoichiometric UO2 are [ 20 ] Do = 0.26 exp(-2..G(eV)/kT)

cm2 6-l

DM = 0.65 exp(-5.6(eV)/kT)

cm2 s-r

where M = F’u since the most recent data were obtained for plutonium-tracer diffusion. The theoretical values are obtained from the relations given above. Note that recent calculations considering four different mechanisms for cation migration (a) direct vacancy jump, (b) oxygen vacancy assisted cation vacancy jump, (c) interstitial cation jump and (d) cation vacancy jump assisted by two oxygen vacancies always yield high values. Mechanism (a) is favoured with 7.8 eV for U02 + x, and mechanism (b) with 11.1 eV for U02 and with 13.5 for U02_x [26].

546

stoichiometric, hyperstoichiometric and hypostoichiometric oxides are predicted. Figure 5 shows calculated oxygen and metal defect concentrations and a schematic presentation of activation enthalpies as a function of x: in MO* fX [ 31 and defect concentrations in UOz as a function of temperature [ 271. As expected, metal vacancies dominate in MOz +X. Figure 6 shows that a temperaturedependent minimum in DM exists, the example being plutonium diffusion in (U0.sPu0.2)02 +X. The minimum occurs at O/M = 1.98 at 1500 “C. An interesting fact can be seen: at constant p(Oz), diffusion at 1600 “C can be slower than at 1500 “C. For MO* fx, DM increases approximately with x2, as postulated in Table 2. Similar data for D” in U02+% are further discussed below.

1

I

I

2mo

2m

ml temwmture.

(a)

K

O/M -ratio

(b)

Fig. 5. predicted concentrations of oxygen and uranium defects in UOz (a) as a function of temperature in UOs and (b) as a function of deviation from stoichiometry at 1600 “C [3, 271. Also shown (b) is the dependence of thegctivation enthalpy AH for metal atom diffusion on x in MO2 fX.

1.95 1 %GpuQ.I) z c3

O/M-mtio 2.00 2.02

1.98 ‘?tx

16OO’C

rrgle uystols bocpr diflurm

-13 _

fa 15Nl’C

B

-11

-18

-

,

,

,

,

,

-22 -20 -16 46 -11

,

,

,

-12 -10

,

-8

-6

, -1

, -2

,_ 0

Lo9 piO,l.(atm)

Fig. 6. Dependence of the diffusion of plutonium in (U&‘u~.~)Os~;, on oxygen Partial pressure at 1500 and 1600 “C. The range of deviation from stolchiometry covered by the data extends from overstoichiometric MOs_e, to understoichiometric Mol.92 [ 20 I.

The concentrations for metal interstitials shown in Fig. 5 are obtained with the assumption that the increase in DM on the left-hand side of the minimum (increasing sub~oichiome~) is due to ~~~titi~s taking over mass transport. There is, however, another phenomenon that might contribute to this increase in DM. Most of the available results of actinide diffusion in non-stoichiometric MOz +5 suffer from the fact that the tracer atgms change their valence together with the matrix atoms, whenever the AG(OZ) of the annealing atmosphere is changed: though, for example, diffusion of U4’ and Pu4+ in UOZ and (U,Pu)O, was measured, the measurements of the diffusion of uranium in uo Z+r, or of plutonium in (U,Pu)02_, etc. were performed with at least some of the tracer atoms at the changed valence of Us+ (or possible U6+) or of Pu3+ paralleling the oxidation of UO? and reduction of (U,Pu)02. The only exception in the publ~hed data are the results of Fig. 6 for (U~)O* +X where no valence change is expected for plutonium, and the increase in Dp” with x is a strong argument in favour of the point defect model of Table 2. However, valence effects could still affect fast diffusion of uranium in uo 2 +y and diffusion of plutonium in (U,Pu)O, -%. Recently, some experiments were made to fill this remaining gap [Zl]. ThOZ was chosen as a matrix of practically constant composition, and the annealing atmospheres were chosen in a way such that most of the tracer was either trivalent, quadrivalent or pentavalent. No significant difference was seen between the diffusion rates of Th4+ (ionic radius, 1.02 A), U4+ (ionic radius, 0.97 A) and Pu4+ (ionic radius, 0.93 a) indicating that ionic size is of little importance in actinide diffusion. Us’ was found to diffuse much more slowly than U4’, despite its smaller ionic size, and Pu3+ was found to diffuse faster than Pu4+, despite its larger ionic size (1.08 Ii). These results clearly demonstrate the validity of the model stating that changes in point defect concentrations caused by changes in oxygen-to-metal ratio are decisive for the observed fast metal atom diffusion in MO2 +%. In contrast, the increased mobility of trivalent ions may contribute to the increase in DPu for O/M < 1.98 in Fig. 6, in addition to the previously postulated contribution of metal interstitials to mass transport. More experiments are needed to determine this contribution quantitatively. Recent calculations [25, 261 are in qualitative agreement with these results. The migration energies of U4+ and Pu4+ in UOZ are predicted to be identical, whereas U5* is calculated to show a AHm larger by about 18% than that of U4’, and Pu3* to show a AH” smaller by about 30% than that of Pu4+. This confirms that ionic size is not an important parameter in metal atom diffusion, and that the effects of large ionic size are overcompensated by those of the charge state. The formation of clusters of oxygen defects in MO*+, is not reflected in the metal atom diffusion data. Possibly, the clusters dissociate at the diffusion ~mperatures. In contrast, the lack of a further increase in DM below O/M = 1.95 (see Fig. 6, ~(0~) < lo-i6 atm) is compatible with the formation of the Pu3+-Vo-Pu3+ clusters which are indicated as being stable up to T 2 1600 “C. We note that the dependence

548

of AH on x in MO2 f x given in the upper curve of Fig. 5(b) is clearly revealed in creep measurements on UO?, UO? +X, and (U,Pu)02 fX, and is a minimum in mobility. The minimum in DM is of practical importance since often the corresponding O/M of about 1.98 is the specified composition of (U,Pu)02 fuels. For instance, when mechanically mixed powders of UOz and PuOz are homogenized, metal diffusion will be very slow or starting interdiffusion might even build-up a diffusion barrier if the anneal is performed such that the O/M ratio is close to that of the minimum in D. Chemical or interdiffusion of uranium and plutonium in the system U02-Pr.10~ shows a similar minimum in a plot of DM uersus ~(0,) as do the self-diffusion coefficients D [28]. A change in ~(0,) can overcome this situation (see also below). An interesting feature not encountered in most other interdiffusion studies arises from the fact that at most oxygen potentials, the valence of plutonium is different from that of uranium. This causes the O/M ratio in the two parts of the interdiffusion couple to be different as well and to vary locally along the interdiffusion profile. For instance, in substoichiometric (U, -rPu,)02_, , the average valence of plutonium at constant Ad(0,) is up, = 4 -2x/y, or x = 0.5y(4 - upu). The deviation x from stoichiometry is therefore a function of y at constant up,. This gives rise to an [O]/[M] gradient along the interdiffusion profile. Therefore, somewhere along the profile, the conditions for the minimum in DM and BM may be met thus setting up a local diffusion barrier. The consequences of the different valences of uranium and plutonium and of the O/M gradient that forms along the interdiffusion profile in MOz_ is that p = and that different types of Arrhenius behaviour are ~“cc~Puo, 9 Aa( expected depending on the parameters which are kept constant, i.e.

Obviously, the situation is less severe in the hyperstoichiometric region since both U&+x and (U, -$‘uY)Oa+x contain U4+ atoms that can be oxidized to U5+ and U6+ whereas plutonium remains quadrivalent. Though an Arrhenius-type presentation of the data is a convenient, but’ strictly speaking inadequate presentation, the data yield a slope corresponding to 5.6 eV, identical with that of metal tracer diffusion in UOz [28]. 3.1.3. The X transition A last point to be mentioned here is that of the effect of a possible high temperature phase change (X transition) in oxide nuclear fuels. Such A transitions, evidenced by an anomaly (a peak) in the specific heat, are known to occur at about OST, in halides (CaF2, PbP2, SrClz etc.) and were postulated for fluorite-type oxides (ThOz, possibly U02 etc.) [29]. The transition is connected with a dynamic disorder in the non-metal sublattice

549

(fluorine, or oxygen) and with a rise in anion electrical conductivity to a value close to that of the ionic melt. For some time, a quasi-molten anion sublattice was postulated. A more recent picture [30] favours a concentration of about 0.5 to 2 mol.% of anion defects with an unusually large mobility. An effect, via the Schottky defect equilibrium, on the otherwise very small diffusion rates of cations might exist and would be of technological importance. For instance, UOz as the fuel of today’s water-cooled nuclear power stations, and in particular (U,Pu)O, may operate with central temperatures above the transition temperature. A recent analysis [31] has indeed revealed a peak in the high temperature specific heat. Also, anion diffusion rates at about 0.82’, are even higher in UOz than they are in the fluorides (D Z 10e6 cm’ s-l) and recent experiments at very high temperatures using neutron diffraction, quasi-elastic diffuse and inelastic neutron scattering [32] have unambiguously shown that significant Frenkel disorder does occur in UOz (and ThOz) above 2000 K. The existence of a X transition is certainly also consistent with other recent observations: (a) creep of UOz , (U,Pu)O, and ThOz shows a behaviour equivalent to an anomalous increase in creep rate not predicted from an extrapolation of low temperature data [33] ; (b) high-temperature thermal transient testing of irradiated mixed oxide fuel indicates the existence of a temperature, again at around 2670 K, above which massive plastic swelling of the fission gas containing fuel occurs [34]. All this evidence points to a non-electronic mechanism involving both anion and cation atomic defects at about 2400 “C in UO? (and about 2700 “C! in ThOz etc.), probably both as point defects and as clusters. The clusters, however, are expected to have short lifetimes, e.g. 3 ps at the highest temperatures. Therefore, cation diffusion (including fission product diffusion) should also be affected; this phenomenon has not been considered yet in any of the many existing fuel performance codes. Direct cation (uranium) diffusion measurements [35] exist for CaFz where the known onset of the dynamic disorder in the anion sublattice at Th is connected to a sudden increase in D”. Obviously, for T > Th metal atom mobilities, metal atom defect concentrations or, quite probably, cluster configurations change and cause this significant increase in D. A recent theoretical treatment [27] using the free energies of formation of defects suggested by the present author confirms that metal atom defects have to be considered as well whenever T > Th in e.g. UOz (see Fig. 5). Quite evidently, the question of atomic transport at very high temperatures in the fluorite-type oxides deserves increased attention and more experimental work should be devoted to help to answer the question on the importance of the Bredig (or X) transition on high temperature kinetic processes. 3.2. Carbides Monocarbides of uranium and plutonium (as well as nitrides, see Section 3.3) are considered as advanced nuclear fuels. Because of their high metal atom density, they offer smaller doubling times in breeder reactors.

550

UC was the fuel of the WR-1 reactor in Canada, and (U,Pu)C is the fuel of the fast reactor FBTR in India. A recent monograph [36] describes the science and technology of these advanced fuels in detail. The monocarbides have the rocksalt structure. At high temperatures, they show a large single phase field which extends from about UC&s to UCiB9for UC1 +X at 2300 “C. In pratical application, since reactor operating temperatures using carbides are much lower than those using oxides, because of the higher thermal conductivity of the carbides, substoichiometric carbides are two-phased consisting of the monocarbide and metal (U,pU) inclusions whereas hyperstoichiometric carbides consist of the monocarbide and precipitates of the higher carbides (M&, M&). Substoichiometric carbides cannot be used as nuclear fuels because of compatibility problems with the cladding and since they show a very high swelling rate. Diffusion in such materials was nevertheless studied because of the basic scientific interest and since the results help to understand the diffusion mechanisms. 3.2.1. Carbon diffusion Carbon diffusion has only been studied in UC1 +X since the weak p radiation of the only available tracer, 14C cannot be measured reliably in the presence of plutonium. Many authois have reported experimental data (see refs. 36, 37 and 38). Unfortunately, much of the early work disregarded the U-C phase diagram. The reported activation enthalpies AHc showed a wide scatter, since frequently phase boundaries were passed within the experimental temperature range without correction for the enhanced diffusion in the two-phased material (UC+U, UC+U& or UC+UC& Diffusion mechanisms via carbon vacancies, interstitials, Cz pairs and an interstitialcy mechanism were all suggested, and even an open controversial discussion on the mechanism of carbon diffusion was carried out [39, 401. The most recent data [ 381 are shown in Fig. 7. DC varies only slightly with x The AHc values are 4.0, 3.7 and again 4.0 eV atom-’ for these in uci,,. three sets of specimens of either clearly hypostoichiometric, fully stoichiometric or clearly hyperstoichiometric composition. Similarly high AHc values have also been observed in other experiments performed in the single-phase high temperature field or, at least, if the experiments were extended to above the phase boundary, and if a discontinuity in diffusion was allowed for. Previous lower AHc values (about 2.5 eV) represent two-phased materials and cannot be accepted as reliable anymore. The most probable diffusion mechanism [37] consists in dissociation of Cz pairs with jumps of single carbon atoms to other single carbon atoms to form a new pair. According to theoretical calculations [44] such pairs exist not only in UC2 and UC1 +X , but also in stoichiometric UC (e.g. 2% at 2400 K) and even in the hypostoichiometric UC0.92 of Fig. 7 (about 0.1% at 2000 K). 3.2.2. Uranium and plutonium diffusion Uranium diffusion in UC1 fX and plutonium diffusion in (U,Pu)CI +% have also been measured extensively. As shown in Fig. 7, D” is much smaller

551

-11.*wcmst -16I

3

1

5

6 lOLlI, K-'

Fig. 7. Arrhenius diagram for the diffusion of 14C in UC of three different carbon contents, and for the diffusion of 233U in stoichiometric singlecrystalline or arccast UC [ 38,411. The corresponding Arrheniusequations are given in Table 3.

than DC. A straight line in the Arrhenius diagram is obtained for pure zonerefined single crystals (metallic impurities, less than 30 ppm) whereas curvature is found in less-pure arccast UC with metallic impurities of less than about 120 ppm. The most probable cause of this curvature and for the lower AHU below about 2100 “C is the influence of these impurities. Extrinsic diffusion is thus operative with vacancy-impurity interactions enhancing diffusion. Intrinsic diffusion is only operative above 2100 “C in the arc-cast UC, whereas it is still observed at 1500 “C in the zone-refined single crystals. Since technological grade carbides are less pure than the arc-cast UC, and since fission products grow in at a high rate during reactor operation (up to 500 ppm per day in a liquid metal fast breeder reactor)), extrinsic diffusion and impurity effects will control the in-pile kinetics of carbide fuels. Di’” in (U,F’u)C is larger than D” in UC at any given temperature. The reason is a combination of faster diffusion of the smaller plutonium atom (Dp”/Du is about 3 to 5 in UC) and the somewhat bigger lattice spacing in MC as compared with UC. We note that (UgU)C usually also has lower densities, smaller grain sizes and higher impurity contents than UC, all features that tend to accelerate overall diffusion rates. The Arrhenius equation in Table 3 is for a typical technological carbide (composition (“0.~0.2)c0.9800.0~0.02%

Deviations from the stoichiometric composition strongly influence metal atom selfdiffusion rates [42]. DM decreases and AHM increases with x in UCi+_, andMCi+,, as is obvious from Fig. 8. It is suggested that carbon atoms donate electrons to the metal-metal bond thus increasing the binding energy, and hence AH”.

552 TABLE 3 Results of self-diffusion in nuclear carbides and nitrides [41 - 43 ] System

Temperature

C in UCl.0 U in UCr_e U in UCi_e

-

Sintered

dependence

Pu in

W0.&0.2)c

D = 24 exp(-354200/RT) D = 11.7 exp(-594100/RT) D = 6.9 exp(-590400/RT) + 3.6 x 10e5 exp(-351700/RT) D = 0.013 exp(-402000/RT)

Pu in

WO.&O.~W

D = 0.25 exp(-496600/RT)

Remarks

Single crystals Arc cast, T > 2100 “C Arc cast, T < 2100 “C Sintered

As elsewhere in this review, D is in cm2 s-l and AH in kJ mol-‘.

09

10

11 C/M-ratio

Fig. 8. Dependence of D” in UC and of AH~ and AH~ in UC and (U,Pu)C on C/M ratio (M = metal) [36, 421. It should be noted that (UPu)Ci_, does not exist in practice. D, as always in this paper, is given in cm2 s-l. AH is given in kilocalories per mole and in electron volts (1 kcal mol-’ = 4.18 kJ mol-‘).

As mentioned above, impurities also have an important effect on metal atom self-diffusion in UC and MC. For instance, vanadium, tungsten, tantalum, iron and nickel have an enhancing effect on uranium self-diffusion. However, zirconium and, to a smaller extent, large impurities such as cerium and lanthanum can also decrease D” in the doped carbides. There is no obvious relation between the enhancement or the diminution factor of the impurities and the lattice parameter of the doped carbide or the atomic size of the impurity atom. The integrated effect of all fission products, however, is such as to enhance metal atom diffusion significantly, in particular at lower temperatures. As expected, the effect of adding impurities decreases with increasing temperature.

553

Finally, uranium diffusion in the higher carbides U& and UC? is faster than that in UC at constant temperature [45]. Grain boundary diffusion in pure and doped UC1 fX is essentially faster (about lo3 to lo5 times) than lattice diffusion with a AH of about 55% that of bulk diffusion. In the monocarbide region, uranium mobility at grain boundaries increases as C/U ratio decreases (as does lattice diffusion) but it is impeded by the addition of impurities [ 461. 3.3. Nitrides In contrast with carbides, the mononitrides UN and (U,Pu)N have a narrow single phase field (also NaCl structure), even at high temperatures. Higher nitrides (only the sesquinitride I&N3 exists) are of less practical importance than the higher carbides [ 361. Diffusion data exist for nitrogen in UN, for uranium in UN and for plutonium in (U,Pu)N. 3.3.1. Nitrogen diffusion As with oxygen, no suitable radioactive tracer exists for nitrogen. Therefore, either mass spectrometric measurements with i5N or a nuclear reaction induced in the diffusion specimen following the diffusion anneal are needed to measure DN values. DN varies with the square root of nitrogen pressure, p(Np), of the annealing atmosphere, indicating an interstitial diffusion mechanism [47]. The reported activation enthalpies vary between 230 and 400 kJ mol-’ (2.4 - 4.2 eV atom-‘). However, since UN is nonstoichiometric and since the Arrhenius plots were based on constant p(N,) rather than on constant composition, no fundamental significance should be attached to these AH values. 3.3.2. Metal diffusion The available data for uranium diffusion in UN are contradictory: an early study [48] yielded the implausible result that at given p(N,), D” is independent of temperature. Simultaneously, an approximately linear relation between D” and p(N,) was reported. However, later work [49] yielded much lower D” values (40 to 100 times lower), indicating that the early results were not necessarily representative of volume diffusion. Reliable data on the dependence of D” on T and p(Nz) have still to be measured. In contrast with UN, such data exist for plutonium diffusion in (U,Pu)N [43]. An Arrhenius relation is shown in Table 3, and the dependence on p(N,) for three temperatures is shown in Fig. 9. These results obtained at constant p(N,) do not therefore represent constant composition. The U-N, IQ-N and (U,FQ)-N systems are characterized by a very large change in metal and nitrogen potential as the single-phase mononitride field is traversed [36]. The diffusion data were obtained within this range of p(N,). The slope in the plot of Dp” versus p(NJ at 1650 “C is in agreement with a vacancy mechanism for plutonium diffusion in a single phase field with the nitride being in equilibrium with the surrounding nitrogen through the incorporation of singly charged nitrogen atoms into interstitial lattice

1 0

1

2

3 log pvlorr

Fig. 9. Dependence of Dh m . (UO#uO_P)N on nitrogen partial pressure (D in centimetres squared per second) [ 43 1.

positions, whereas the independence of HP” on p(N,) at the higher temperature of 1810 “C could imply that diffusion occurs in a matrix of constant composition. Alternatively, the effect of varying nitrogen content could have been compensated by changes in the carbon or oxygen content of the technological grade sinters used. Obviously, more work on the phase relations and on the p(Nz) dependence of metal atom diffusion is desirable. Arrhenius diagrams constructed for different N/M ratios show a trend in D, and AH which is different from that shown for carbides in Fig. 8. Whereas in carbides, DM increases and AHM decreases with decreasing C/M ratio, the opposite is indicated for nitrides: DM decreases and AHM increases with decreasing N/M ratio. This can be taken as an indication that the bonding conditions are different in NM and MC. 3.3.3. Carbonitrides Diffusion experiments on carbonitrides MCI-,N, show that substituting carbon by nitrogen reduces diffusion rates [50]. This has been observed for carbon and uranium diffusion in the series UC-U(C,N)-UN and for plutonium diffusion in the series MC-M(C,N)-MN. Typically, this decrease amounts to one to two orders of magnitude in D at temperatures of 1450 to 1800 “C. The most pronounced decrease in atomic mobilities occurs at around MCo.sNo_s. Figure 10 shows this behaviour for self-diffusion and the related processes of grain growth and restructuring in the thermal gradient existing during irradiation. The decrease in kinetic rates when passing from the carbides to the nitrides corresponds to temperatures higher by about 200 to 300 “C which are needed to achieve rates in MN that are comparable with those in MC. Another example of the same trend is in-pile fission gas release [ 521. [The actual diffusion coefficients for a given (technological grade) carbonitride depend also on the content of metallic impurities (which generally increase diffusion rates) and on the total X/M ratio (X = C+N+O) where values greater than 1 tend to decrease D. These two dependences may overcompensate the effects of variations in the nitrogen content.]

555

UK.N)

WW

C-difhJSiWl (0 -10-gcm2

300-

0

s-‘l

IO -w)%nz

PIJ-diisial ~‘1

IU,pu)IC.NI zone IV-III

IUPJNLN~ IU.k)(C.N)

U-diffusim

10=lO~ucmz

gun ~‘1

(d’-di

growth -l&nun])

(burn-up-I,

ok”/.]

iI

0.5

1.0 0

0.5

0 1.0

05

1.0 0

0.5

0 1.0

ix

10

N/GN)

Fig. 10. Diffusion and diffusion-related processes in the series carbide-carbonitridesnitride as function of nitrogen content. The temperature increase to obtain given rates as compared with monocarbides is shown [ 42,50,51].

As with the monocarbides (see above), impurities and, very importantly, the accumulation of fission products during reactor operation affect uranium and plutonium diffusion. Work on MC, M(C,N) and MN with added (inactive) fission products to simulate burn-up values of 3, 10 and 16 at.% showed that the pronounced increase in Dp” with burn-up followed by a saturation found in MC and in the carbon-rich carbonitride MC,sN,, is not observed in MN: MN shows a maximum in Dp” at intermediate burn-up followed by a decrease towards the Dpu values for MN without fission products [43]. The explanation of this different behaviour is the retarding action of the rare earths. These are soluble in nitrogen-rich fuels and compensate the enhancing effect, of the other fission products. They are not soluble in the carbide. With data for only three bum-ups available, only a trend can be indicated. More data would be welcome.

4. Radiation enhanced diffusion in ceramic fuels In addition to the chemical effects of fission products, fission causes matter transport to be accelerated as a result of radiation enhanced diffusion (coefficients D*). Data exist for uranium and plutonium diffusion in the oxides UOz and MOz as well as in the series UC-U(C,N)-UN and MCM(C,N)-MN [53, 541. The results shown in Fig. 11 prove that D* is completely temperature independent and athermal below about 1000 “C. The enhancement factors D*/D can therefore be very high indeed. In the cold outer part of a fuel they can easily be more than lOlo, as is obvious by comparison with the results for thermally activated diffusion which are schematically represented in Fig. 11. A possible temperature dependence of D* is indicated for oxides between about 1000 and 1200 “C. The reason for the significant enhancement is the radiation damage and nuclear collision phe-

556

nomena together with thermal and pressure effects that occur during the slowing down of the fission products. The latter effects are more pronounced in the poorly conducting oxides than in the more metallic carbides where the distribution of energy originating from electronic stopping of fission fragments over large volumes is much easier than in the oxides. Similar results with largely athermal radiation enhancement below about 1000 “C have also been observed for the related phenomena of in-pile creep and fission gas diffusion, e.g. refs. 55 and 56. The ratios of radiationenhanced creep rates in the series MO,-MC-MN was very similar to that for D* shown in Fig. 11.

-20’

i

’

6

.

’

8

.

‘1 -L--J--J 10 14 15 21 23 25

10L/T. K-’ Fig. 11. Radiation enhanced d_iffusion of uranium and plutonium in ceramic nuclear fuels (normalized to a fission rate F = 5 X 101* fissions cme3 6-l [ 53, 541. (It should be noted that D* is proportional to 6’)

5. Thermodiffusion in temperature gradients Pronounced temperature gradients VT exist in operating nuclear fuels, in particular in oxide fuels. An LMFBR oxide pin typically has a diameter of about 5 mm, a central temperature of 2400 “C and a surface temperature of 600 “C and therefore a V 2 of 1800 “C in about 2.5 mm. Local VT values can be lo4 K cm-‘. In carbide, nitride or metal fuels with their higher thermal conductivity, typical VT values are 2000 - 3000 K cm-‘. These gradients exert a driving force on moving atoms, i.e. in a homogeneous two (or more)component phase, an unmixing occurs, if one component diffuses preferentially to the hot (or the cold) end. This effect is called thermotransport, thermal diffusion or Soret effect. The (simplified) equation describing the flux J of atoms of type i Ji=-D

aCi -+ ax

QiCi

_ RT2

557

shows that this flux is proportional to D (or d in the case of metal atoms in (U,Pu)02, for example) and to the heat of transport Q*. 5.1. Metal atoms No thermodiffusion of uranium or plutonium has been observed in nuclear carbides or nitrides. In contrast, an important redistribution of uranium and plutonium frequently occurs in the mixed oxide (U,Pu)O* [e.g. see refs. 51, 57 - 591. Fig. 12 shows a radial electron microprobe scan of a (U,Pu)O, fuel after irradiation [ 591. Obviously, there is an enrichment of plutonium in the hot centre. There are two possible competing processes: pore migration, causing the formation of the central void, can transport plutonium (for O/M L 1.98) or uranium (for O/M Z. 1.97) into the centre. Thermal diffusion (with Q* = -35 kcal mol-‘) can contribute to plutonium enrichment as well (see, e.g. ref. 59). However, there is still a dispute on the relative contribution of thermal diffusion [60]. If it exists, it is a slow process which is effective for long irradiations. Initially, plutonium transport by migrating pores is dominant. More fuel pins, and in particular pins with low O/M ratio, should be analyzed to obtain a definite answer. 5.2. Carbon In carbide fuels, carbon transport towards the cold metal sheath is often observed. Carbon movement in the radial thermal gradient certainly exists. The problem is that technological (U,Pu)C usually contains some oxygen impurities. Therefore, CO is formed which can transport carbon down the temperature gradient as well. Quantitative data 1611 exist for UC? where carbon also migrates to the cold side (with an apparent activation enthalpy of 76 kcal mol-’ and a high heat of transport Q* = 92 + 27 kcal mall’ at 1600 “C, Q* is possibly T dependent).

fuel radius, mm

Fig. 12. Measured plutonium enrichment in the hot centre of an LMFBR fuel pin (band) with prediction for plutonium transport by thermal diffusion and evaporation-condensation via pore migration [ 59 1.

558

5.3. Oxygen Thermal diffusion of oxygen in UOZ f X, (U,Pu)O* fX and PuOZ__ is very pronounced indeed. Both in-pile and out-of-pile experiments (see, e.g. ref. 62) show that in hypostoichiometric fuels, oxygen migrates to the cold side (towards the cladding in a fuel element), whereas in hyperstoichiometric fuels, it migrates in the opposite direction (hot side, or centre in a fuel element). Therefore, Q* is negative for both oxygen vacancies and oxygen interstitials. The effect of the plutonium content is not important [62]. It has been reported that Q* decreases from about -25 kcal mol-’ for small x values towards zero for higher x values (0.06 to 0.16), and hence depends on the valence of uranium and plutonium. These results were used to predict radial oxygen profiles in operating fuel elements. Despite these considerable efforts and the significant success achieved, some questions of details are still being discussed, e.g. the relative impor-tance of electrotransport phenomena (Seebeck effect) due to metals used to encapsulate the fuel [63], and indicated (but unexpected) very low values of O/M ratio in the centre of quenched fuel elements [64].

6. Fission product diffusion in ceramic fuels Some data exist on the diffusion of soluble metallic fission products in ceramic nuclear fuels. The limited available data point to a substitutional migration on metal atom sites. In contrast, the diffusion of volatile fission products, and in particular of the inert gases xenon and krypton has been studied very extensively, in order to understand better their release, for safety reasons, and their precipitation into gas bubbles, to understand fuel swelling. A recent review summarizes many of the important aspects of gas diffusion in UOZ [65] and fission gas mobility in carbides and nitrides is treated in ref. 36. More recently, emphasis was put on short-lived volatile fission products and on transient behaviour during power and temperature increases [66 - 681. Also, hardly any fission inert gas is formed directly by fission. Rather, most gas is formed via an (also volatile) precursor. The precursors for xenon are tellurium and iodine which decay into xenon by /3 decay. The diffusional behaviour of tellurium and iodine has therefore recently also received attention. These frequently diffuse faster than the inert gas (see Fig. 4) and their mobility should be taken into consideration in order to explain the observed location or release of the fission gases. Two important features of inert gas diffusion were recognized fairly early: unperturbed diffusion as single xenon atoms occurs only at low concentrations of gas (less than about lop5 at.%), of damage and of defects. At higher gas or damage (defect) concentrations, gas-gas or gas-defect interactions cause “trapping” of the gas evidenced by retarded release. At still higher gas concentrations, fission gas bubbles form. Therefore, more than the question of the diffusion of single gas atoms, the problem of an effective diffusion coefficient is important for technological purposes. Such

559

a Deff should aIlow for trapping phenomena, precipitation of gas into bubbles and preexisting pores with subsequent re-solution by fission spikes (or by thermal re-solution), precipitation of the gas at grain boundaries and final transport out of the fuel into the gap or to the plenum. As explained in an earlier review [65], a total of five different diffusion coefficients has to be defined. Also, the transport of gas to the grain boundaries can occur in a number of ways: as single gas atoms, via the mobility of gas-filled bubbles or pores (either directed or indirected bubble diffusion, or sweeping processes by moving grain boundaries, dislocations etc.). Release from the grain boundaries occurs subsequently via network and tunnel formation or via cracks that preferentially form during power changes, shut-down or start-up of the reactor. Diffusion of single gas atoms can be described by the Arrhenius relation D = 0.5 exp(-3.9

(eV)/12T) cm2 s-i

Trapping increases this AH value by about 0.5 to 0.8 eV [ 651. This is shown in Fig. 4 by two chain lines. Theoretical calculations [69] have helped to understand fission gas diffusion and fission gas bubble behaviour better. The early deduction of the author that a tri-vacancy (Schottky-trio) is the equilibrium site for xenon atoms was confirmed by these calculations. The interesting problem of gas bubble diffusion cannot be treated here in detail for reasons of space. We rather conclude this section on diffusion with a word of caution. To measure reliable diffusion coefficients requires either depth profiles or time dependences of the diffusion process to be obtained; preferably, both should be obtained. Trivially, the temperature must be known. But knowledge that the distribution (hopefully properly determined) of the diffusing atom has followed diffusion kinetics is equally important. The author has shown before that many of the previously published results on “self-diffusion” and “fission gas release” in reality were artifacts not typical of diffusion [16, 651. With this past experience, care should be taken to measure properly the parameters before postulating quite dramatic changes in diffusional behaviour of nuclear fuels as is indicated in recent gas release work [68] where temperatures are calculated rather than measured and the time dependence is not followed.

7. Example of an application of diffusion results in technology An example of direct application of diffusion results in technology concludes this paper. As mentioned above, the metal atom diffusion coefficient, e.g. D” in U02, is ratecontrolling for many high-temperature kinetic processes, and D” increases by some orders of magnitude if UOz is oxidized. For instance, if UOz.i is used rather than U02, D” increases by about a factor of lo4 [70, 711. In reactor technology, stoichiometric UOz is needed, and therefore sintering is usually done under a reducing atmosphere. This requires high sintering temperatures.

560

Application of the diffusion results (see Fig. 13 and Fig. 14(a)) led to a changed process with oxidative sintering in CO2 followed by a reduction step to stoichiometric U02 [73,74]. As shown in Fig. 14(b), sintering under oxidizing conditions with the associated fast diffusion and matter transport can be performed at temperatures at least 400 “C lower than those in reducing conditions. This procedure involves (i) important savings in energy by using 1100 “C rather than 1750 “C, (ii) savings in time, (iii) savings in capital investments (cheaper furnaces, sintering boats etc.). Since 1977, pellets for fuel elements have been produced by this method. The necessary subsequent reduction step can be performed at low temperatures as well, because of the high values off)’ governing reduction (see above and Fig. 4).

Fig. 13. Schematic presentation of parameters and kinetic processes affecting sintering and uranium diffusion coefficients D” in UOa +% (p, density).

-‘6 2.0 (a)

2.05

210

215 220 O/U - ratio lb)

Fig. 14. (a) Dependence of uranium self-diffusion at 1500 “C and UOa and UOa+, as a function of excess oxygen content [ 70 - 721; UOa (ex AUC powder) for reducing and oxidizing conditions are technological Nikusi process (low temperature-short time sintering

of creep at 1250 “C in (b) sintering curves of shown, as used in the [ 73,74 I).

561

8. Conclusions The above review has shown that much information exists on diffusion processes in nuclear fuels. For the metals thorium, uranium and plutonium, the effect of phase changes is important. For the ceramic nuclear fuels, Fig. 15 shows schematically how different parameters affect metal atom (thorium, uranium, plutonium) selfdiffusion which is the rate-limiting step for high temperature diffusion-controlled kinetic processes. The nonmetal atoms (oxygen, carbon, nitrogen) diffuse much faster than the metal atoms. Much is known about atomic diffusion in MOz, MC, M(C,N) and MN (M = U and/or Pu), reliable correlations exist between measured D values and other interesting high temperature processes (sintering, gas release, creep etc.). For the oxides, a point defect model explains the observed features. The influence of temperature gradients and irradiation (fission) on diffusion has also been determined. However, there are still gaps which ought to be filled by future work. Examples of such gaps are the diffusion of fission products other than the inert gases, of other impurities, the influence of cluster formation and dissociation, the thermal diffusion contrienhancement

factor

for metal

Fig. 15. Schematic presentation of factors nuclear fuels: oxides, carbides and nitrides.

affecting metal self-diffusion in ceramic

bution to actinide redistribution and the formation of oxygen gradients in operating fuel elements, final values of the formation energies and migration energies of point defects, possible effects of the valence of the actinides on their diffusion and reliable data for the diffusion of gas-filled bubbles.

References 1 D. Calais, M. Dupuy, M. Mouchnino, A. Y. Portnoff and A. Van Craeynest, in A. E. Kay and M. B. Waldron (eds.), Plutonium 1965, Chapman & Hall, London, 1967, p. 358.

562 2 D. Caiais, J. A. Cornet, J. F. Cornet, A. Languihe and C. Remy, in Plutonium 1970 and Other Actinides, Nucl. Metall., 17 (1970) 135. 3 Hj. Matzke, in Plutonium 1975 and Other Actinides, North-Holland, Amsterdam, 1976, p. 801. 4 C. H. de Novion, in Proc. Plutonium 1981 and Other Actinides: Actinides in Perspective, Pergamon, Oxford, 1982,p. 175. 5 S. Ahrland et al. (eds.), The Chemistry of Actinides, Pergamon Texts in Inorganic Chemistry, Vol. 10, Pergamon, Oxford, 1975. F. L. Oetting, M. H. Rand and R. J. Ackermann, The Chemical Thermodynamics of Actinide Elements and Compounds, Part 1: Actinide Elements, International Atomic Energy Agency, Vienna, 1976. 6 F. Schmitz and M. Fock, J. Nucl. Mater., 21 (1967) 317. M. Himmel and E. D. Aibrecht, cited in J. F. Smith, 0. N. Carlson, D. T. Peterson and T. E. Scott, Thorium-Preparation and Properties, Iowa State University Press, 1975. 7 S. J. Rothman, L. T. Lloyd and A. L. Harkness, Trans. Metall. Sot. AZME, 218 (i960) 605. S. J. Rothman and N. L. Peterson, in Diffusion in Body-Centered Cubic Metals, American Society of Metals, 1964,~. 183. Y. Adda and A. Kirianenko, J. Nucl. Mater., 2 (i959) 120; 6 (1962) 130. Y. Adda, A. Kirianenko and C. Mairy, J. Nucl. Mater., 3 (1959) 300. G. B. Federov, E. A. Smirnov and S. S. Moiseenko, Metall. Metalloved. Chist. Met., 7 (1968) 124. 8 W. Z. Wade, D. W. Short, J. C. Walden and J. W. Magana, Metall. Trans. A, 9 (1978) 965. M. Dupuy and D. CaIais, Trans. Metall. Sot. AZME, 242 (1968) 1679. R. Tate and E. Cramer, Trans. Metall. Sot. AZME, 230 (1964) 639. J. Cornet, J. Phys. Chem. Solids, 32 (1971) 1489. 9 D. Caiais and J. A. Cornet, Me’m. Sci. Rev. Met., 69 (1972) 493. 10 N. L. Peterson and S. J. Rothman, Phys. Rev. A, 136 (1964) 842. 11 F. A. Schmidt, R. J. Conzemius and 0. N. Carlson, J. Less-Common Met., 59 (1978) 53. 12 W. N. Weins and 0. N. Carlson, J. Less-Common Met., 66 (1979) 99. 13 F. A. Schmidt, M. S. Beck, D. K. Rehbein, R. J. Conzemius and 0. N. Carlson, J. Electrochem. Sot., 131 (1984) 2170. 14 W. K. Warburton and D. Turnbull, in A. S. Nowick and J. J. Burton (eds.), Diffusion in Solids, Academic Press, New York, 1975,~. 171. 15 L. V.Pavlinov,Zzu. Akad. Nauk. SSR Met., 6 (1984) 59. 15a C. Charissoux. D. Caiais and G. Gallet, J. Phys. Chem. Sol., 36 (1975) 981. 16 Hj. Matzke, Diffusion in non-stoichiometric oxides. In T. S. wrensen (ed.), NonStoichiometric Oxides, Academic Press, New York, 1981,~. 155. 17 Hj. Matzke, Diffusion in ceramic oxide systems, In Adu. Ceram., American Ceramics Society, Proc. ACS Meeting, Nucl. Div., Cincinnati (1985), in print. 18 W. Breitung, J. Nucl. Mater., 74 (1978) 10. 19 G. E. Murch, Diff. Defect Data, 32 (1983) 9. 20 Hj. Matzke, J. Nucl. Mater., 114 (1983) 121. 21 Hj. Matzke, unpublished results, 1986. 22 Hj. Matzke, Radiat. Eff., 53 (1980) 219. 23 A. S. Bayoglu and R. Lorenzelli, Solid State Zonics, 12 (1984 j 53. 24 C. R. A. Catlow and W. C. Mackrodt (eds.), Computer simulation of solids, Lect. Notes Phys., 166 (1982). 25 C. R. A. Catlow,Proc. R. Sot. (London) Ser. A, 353 (1977) 533. 26 R. A. Jackson, A. D. Murray, J. H. Harding and C. R. A. Catlow,AERE Harwell Rep. TP. 1109,1985. 27 S. W. Tam, J. K. Fink and L. Leibowitz, J. Nucl. Mater., 130 (1985) 199. 28 D. Glasser-Leme and Hj. Matzke,Solid State Zonics, 12 (1984) 21i.

563 A. S. Dworkin and M. A. Bredig, J. Phys. Chem., 72 (1968) 1277. J. Schoomnan,Solid State Ionics, 5 (1981) 71. J. Ralph and G. J. Hyland, J. Nuci. Mater., in print. K. Clausen, W. Hayes, M. T. Hutchings, J. E. Maedonald, R. Osborn and P. Schnabei, Rev. Phys. Appl., 19 (1984) 719. 33 L. Leibowitz, J. K. Fink and 0. D. Slagle, J. Nucl. Mater,, 116 (1983) 324. 33a Hj. Matzke,J. Phys. Colloq. 34, CS (1973) 317. 34 E. H. Randklev and C. A. Hinsman, in E. C. Norman M. G. Adamson, A. Bottax, C. M. Cox, W. W. Little, Jr., E. T. Weber, J. D. B. Lambert and J. T. A. Roberts (eds.), Fast Breeder Reactor Fuel Performance, American Nuclear Society U.S. Department of Energy, Washington, DC, 1979, p. 405. 35 D. Glasser-Leme and Hj. Matzke, Solid State Chem., 3 (1982) 201. 36 Hj. Matzke, Science and Technology of Advanced LMFBR Fuels, North Holland, Amsterdam, in print. 37 Hj. Matzke,Solid State Ion& 12 (1984) 25. 38 H. Matsui and Hj. Matzke,J. Nucl. Mater., 89 (1980) 41. 39 G. E. Murch,J. Nuel. Mater., 55 (1975) 355; 57 (1975) 231; 58 (1975) 244. 40 S. Sarian,J. Nucl. Mater., 54 (1974) 151; 57 (1975) 237. 41 Hj. Matzke, J. L. Routbort and H. A. Tasman,J. Appl. Phys., 45 (1974) 5187. 42 Hj. Matzke and M. H. Bradbury, Euratom Rep. EUR-5905 EN, 1978. 42a T. Inoue and Hj. Matzke, J. Nucl. Mater., 91 (1980) 1. 43 M. H. Bradbury and Hj. Matzke,J. Nucl. Mater., 75 (1978) 68; 91 (1980) 13. 44 F. Jeanne, PhI3 Thesis, Univ. Grenoble, France (1972). 45 Hj. Matzke and C. Politis, Solid State Commun., 12 (1973) 401; Reaktorta~ng BerIin 1974, German Atomforum, Bonn, 1974,~. 269. 46 J. L. Routbort and Hj. Matzke, J. Am. Ceram. Sot., 58 (1975) 81, 47 J. W. Droege and C. A. Alexander, U.S. Rep. BMI-1882, 1983, 1970; 1918,197l. 48 D. K. Reimann, D. M. Kroeger and T. S. Lundy,J. Nucl. Mater., 38 (1971) 191. 49 J. L. Routbort and Hj. Matzke, unpublished results, 1975. 50 H. Matsui, M. H. Bradbury and Hj. Matzke, Nuc~. Sci. I&g., 66 (1978) 406. 51 C. Sari, Euratom Rep. EUR-5812 EN% 1978. 52 M. Coquerelle and C. T. WaIker, Nucl. Technot., 48 (1980) 43. 53 Hj. Matzke, Radiat. Eff., 75 (1983) 317. 54 Hj. Matzke, in G. Simkovitch and V. S. Stubican (eds.), Transport in Nonstoichiometric Compounds, Plenum, New York, 1985,~. 331. 55 D. Brucklacher and W. Rienst, J. Nucl. Mater.. 4.2 (1972) 285. 56 J. A. Turnbull, C. A. Friskney, J. R. Findlay, F. H. Johnson and A. J. Walter, J. NucI. Mater., 107 (1982) 168. 57 M. Bober and G. Schumacher, Adv. Nucl. Sci. TechnoE., 7 (1973) 121. 58 M. Bober, C. Sari and G. Schumacher, J. Nucl. Mater., 39 (1971) 265; 40 (1971) 341. 59 W. Breitung, Rep. Kernforschungszentrum Karlsruhe KfK-2240, 1976. 60 P. Jean Baptist,J. Nucl. Mater., 120 (1984) 316. 61 T. D. Gulden,J. Am. Ceram. Sot., 55 (1972) 14. 62 C. Sari and G. Schumacher,J. Nucl. Mater., 61 (1976) 192. 63 F. Miflot and P. Gerdanian,J. Nucl. Mater., 9.2 (1980) 257. 64 F. Ewart, K. Lassmann, Hj. Matzke, L. Manes and A. Saunders, J. Nucl. Mater., 124 (1984) 44. 65 Hj. Matzke, Radiat. Eff., 53 (1,980) 219. 66 I. J. Hastings, C. E. L. Hunt, J. J. Lipsett and R. D. MacDonald, Res. Mechanica, 6 (1983) 167. 67 J. J. Lipsett, C. E. L. Hunt and I. J. Hastings, Canada Rep. AECL-8069,1983. 68 Ib. Misfeldt, J. Nucl. Mater., 135 (1985) 1. 69 R. A. Jackson and C. R. A. Catlow, J. NucE. Mater., 127 (1985) 161,167. 70 J. F. Marin and P. Contamin, J. Nucl. Mater., 30 (1969) 16. 71 Hj. Matzke,J. Nut?. Mater., 30 (1969) 26. 29 30 31 32

564 72 B. Burton, Diffusional Creep of Polycrystalline Materials, Diffusion and Defect Series, Vol. 5, Trans. Tech. Publ., Switzerland, 1977. 73 W. D&-r and H. Aasmann, Proc. 4th Znt. Meet. on Modern Ceramic Technologies, CZMTEC, St. Vincent, Italy, 1979, Elsevier, Amsterdam, 1980. 74 G. Maier, H. Assmann, W. D&r, R. Manzel and M. Peehs, Jahrestagung Kerntechnik, German Atomforum, Bonn, 1982,p. 485.