ORNL-2656Errata

REVIEW OF THERMAL CONDUCTIVITY AND HEAT TRANSFERIN URANIUM DIOXIDE

Page Column Line

10 2 37

11 1 10

Errata

for neutrons/cm2 read neutrons-cm^-sec" '

for and high neutron fluxes (1018 neutrons/cm2)read and an exposure of 1018 nvt,

Contract No. W-740S-eng-26

METALLURGY DIVISION

REVIEW OF THERMAL CONDUCTIVITY AND HEAT TRANSFER

IN URANIUM DIOXIDE

V. J. Tennery*

DATE ISSUED

APR 9 1959

*Summer employee, 1958, from University of Illinois

OAK RIDGE NATIONAL LABORATORYOak Ridge, Tennessee

operated byUNION CARBIDE CORPORATION

for the

U.S. ATOMIC ENERGY COMMISSION

ORNL-2656

niAmwiN i1A.hETTA ENERGY SYSTEMS LIBRARIES

3 MM5b 03bl3Mb 3

REVIEW OF THERMAL CONDUCTIVITY AND HEAT TRANSFER IN URANIUM DIOXIDE

V. J. Tennery

ABSTRACT

In an attempt to determine the possibility of increasing the thermal conductivity of U0- and thusthe efficiency of power reactors using this fuel,the theories of heat transfer in insulators and semi

conductors are reviewed and discussed. No original work was done. No theoretical basis is found,when the oxide is considered as an insulator, forexpecting small or moderate chemical additions toincrease its thermal conductivity. Additions oflarge amounts of a substance having a high thermalconductivity (BeO) have been found to increase thenet conductivity, but the systems so producedwere two-phase and sintered poorly.

It is suggested that the thermal conductivity ofUO, might be increased by treating it as a potentialoxidic semiconductor and doping it with the properelements. Study of the literature shows thatthermal conduction by intrinsic carriers has beentreated experimentally and theoretically but thatthe conductivities so found do not follow the

Wiedemann-Franz law. Conduction by extrinsiccarriers, as in a doped oxide, has not been treated

1. INTRODUCTION

The core temperatures of power reactors havesteadily been increased to the point where theiradvancement is limited primarily by the physicalproperties of the materials constituting the core.In general, reactor efficiency increases as a function of the temperature differential between thecore and its environment, and the importance ofhigh core temperatures is evident.

This need for higher operating temperatures hasled to the use of ceramic cores. One of the first

ceramics investigated for this application wassintered U02, and it is presently being used asthe fuel material in some reactors in the form of

theoretically. Recent measurements indicate thatdoping U02 with Y3+ or Nb5+ increases its thermalconductivity; but the data are sparse, and the magnitude of the increase and its dependence uponimpurity concentration and temperature have notbeen determined. Suggested experiments to testthe effects of impurity addition on the thermal conductivity are described.

The following reservations are noted:1. The effect of radiation is probably drastic

and determinable only by experiment.2. Since oxygen can suppress conductivity in

two ways, oxygen released by fissioning in U02may be a factor.

3. No detailed study has been made of the effectsof enrichment in U235.

4. Wide variations in resistivity measurementsfrom sample to sample raise the question of thestoichiometric and chemical purity of the UO.tested in the laboratories and of the effect of

impurities common to reactor-grade U0-.

canned cylindrical pellets. One of the major disadvantages of U02 as a core material is its lowthermal conductivity. Since a low conductivitylimits the rate at which heat can be removed fromthe core it is obvious that this property of theceramic is one of the major limitations on theoperating temperature and consequently on theefficiency of a reactor.

A survey of the present theories on thermal conduction in solids was made, with emphasis on U0-.Thermal conduction in insulators is discussed indetail since U02 might be classed in this group ofsolids. Thermal conduction in semiconductors is

also considered since with the proper impurityadditions U02 becomes an extrinsic semiconductor.

2. THEORY OF HEAT TRANSFER IN SOLIDS

2.1 Heat Transfer in Insulators

As noted in the introduction, a survey of theliterature of the theories of heat transfer in crystalline solids was made in an effort to determine the

possibility of increasing the thermal conductivityof an essentially pure insulator.

A theory of the thermal conductivity of insulatorswas developed in 1914 by Debye; as in his theoryof specific heat (1912), he assumed that the latticevibrations may be described by a model in whichelastic waves are propagated through a continuum.Since solids expand upon heating, these wavescannot be purely harmonic but must be anharmonic.This anharmonicity was, according to Debye, thesource of coupling between the lattice waves, sothat mutual scattering of the waves becomes possible. (He pointed out that mutual scattering isnot possible for purely harmonic waves.) As ameasure of the coupling, Debye introduced a meanfree path /, which measures the distance of travelof a wave required to attenuate its intensity by afactor e. The thermal conductivity, k, is thengiven by the equation

k= \cvl , (1)

where c is the specific heat per unit volume and vis the wave velocity.

These ideas were extended by Peierls and translated in terms of phonon-phonon interaction. InPeierls' theory a crystal is treated as a lattice ofatoms rather than a continuum. The coupling between normal modes of vibration is ascribed to

anharmonicities arising from third and higher orderterms in the potential energy of a displaced atom.The part played by these terms is analogous tothat of collisions in the theory of a perfect gas:though introducing only a small perturbation of themotion they are responsible' for the coupling between the normal modes which is essential in producing thermal equilibrium. In Peierls' theory thenormal modes of vibration are quantized and, byanalogy with the photons of radiation theory, these

R. Berman, Advances in Physics 2, 103 (1953).

P. Debye, Vortrage ilber die kinetische Theorie derMaterie und der Elektrizit'dt, 19-60, Teubner, Berlin.1914.

3R. Peierls, Ann. Physik 3, 1055 (1929).

quanta are now called phonons. For a phononassociated with an angular frequency a> and wavenumber K, ~t>a> gives the energy while UK behavesrather like a momentum.

In the presence of a temperature gradient thephonon distribution differs from the equilibriumdistribution at uniform temperature — the Planckdistribution. Collisions between phonons tend torestore the equilibrium distribution, and the rateof the restoring process determines the thermalconductivity. Collisions are possible if the threevalues of a> and K obey the equations

(2)

and

Kl + K2 ~ K3 (3)

Equation 2 states that after a collision the resulting wave or waves still carry the same energyas before, and Eq. 3 implies that this energy isstill flowing in the same direction. Such collisionsdo not in themselves give rise to a thermal resistance. Peierls also showed that, for a discretelattice, collisions are possible in which Eq. 3does not hold but is replaced by

2rreK, + K„ — K- + - (4)

where a is the lattice constant and e is a unit

vector, the possible directions of which depend onthe crystal symmetry. Equation 4 implies that thedirection of flow of the energy is changed after acollision, and, as e can take up one of severaldirections (e.g., six directions parallel to theaxes of a cubic crystal), the result of such collisions is much the same as if the waves were

scattered at random. This type of collision, whichPeierls called an "Umklapp," or "U-process,"gives rise to thermal resistance.

It is convenient to use the concept of mean freepath, as in Debye's theory; but, since waves havebeen replaced by phonons, Eq. 1 is rewritten as

k = /jcvl . (5)

In the case just discussed / is the mean free pathfor U-processes.

Variation of Thermal Conductivity with Temperature. —As a result of the treatment developed byPeierls the theory predicts that the thermal conductivity varies as 1/T at high temperatures

(T > 9), but at low temperatures it leads to a relation of the form k = QTV exp (d/2T), where Q isa constant, v is of the order of unity, and Q is theDebye specific heat parameter. It is this lawwhich in most cases is approximated by the thermalconductivity of the common ceramic oxides. Asthe processes responsible for the form of this laware not widely understood, it will be useful todiscuss them in some detail.

In a large crystal free from impurities, cracks,and strains in its structure, the scattering of thermal waves or phonons will be caused only bycollisions between the waves themselves. In this

case the conditions described by Eqs. 2 and 4exist. Peierls named the phonon collision anUmklapp process, which can best be translated as

a "flop-over" process; at a collision the wavenumber vector "flops over" and takes up a newposition. These U-processes come about becauseof the possibility of interference effects betweenthe phonons and the lattice, by which the latticecan transfer momentum between itself and the

phonons. It is as if the phonons can suffer Braggreflections from the lattice. If a U-process is totake place at all, we see from Eq. 4 that

2neK, + K2 > (6)

This implies that phonons must have a certainminimum energy before they can undergo Umklapp,and Peierls shows that, as a rough criterion, thisminimum energy may be taken as kd/2, where k isBoltzmann's constant.

The mean free path noted in Eq. 5 can be approximated by the expression

A„= A exp (6/bT) , (7)

where A is a constant depending on the crystaland b has a value slightly greater than 2. Now thenumber of phonons with this energy is proportionalto l/exp[((V2T) - 1]. From this it can be deducedthat the temperature dependency of / or A is 1/T,and hence that of k is 1/T for T > 0, which isthe temperature range involved in the proposedapplication of U02.

There are processes involved in heat conductionother than Umklapp collisions. The phonons may

A. J. Dekker, Solid State Physics, Prentice-Hall,Englewood Cliffs, N. J., 1957.

be scattered by the crystal or grain boundaries,by impurities, and by lattice imperfections. Someyears ago, de Haas and Biermasz6 made measurements on crystals of quartz, potassium chloride,and potassium bromide at low temperatures, andtogether with Casimir7 demonstrated that in theregion where conductivity falls with temperaturethe flow of thermal energy is limited by its beingscattered at the walls of the crystal. This is theso-called "size effect," which results in themeasured thermal conductivity becoming proportional to the smallest dimension of the specimenand to the cube of the absolute temperature. Thesetwo processes, Umklapp scattering and boundaryscattering, one predominant at high and the otherat low temperatures, lead to a maximum in thethermal conductivity in the temperature regionwhere they both contribute least to the thermalresistivity 1/k, namely, at temperatures in thevicinity of (9/20.8 This maximum in the thermalconductivity exhibits values much higher thanthose possessed by any metals at room temperature. For example, a crystal of sapphire, ALO-,has a maximum value of 15.5cal-cm • »sec~ '(0C)- 'at 40°K, which is over 19 times the conductivityof copper at 25°C [approximately 0.8 cal-cm-'-sec" -C^C)- ]. Diamond, the best thermal conductor of all materials at room temperature, has amaximum value of 7.17 cal-cm-'«sec~' '(0C)~',which extends from about 40 to 100°K. However,the thermal conductivity does not drop as rapidlywith increasing temperature as it does in otherknown insulators and has a value of about 2.2

cal-cm-^sec-'.^C)-' at 100°C. This is 10times the thermal conductivity of most irons andsteels at this temperature and is over 3.5 timesthe thermal conductivity of polycrystalline BeO,which is the best oxidic thermal conductor known.'Unfortunately the position of this maximum of theconductivity in the absolute temperature scale isdictated by variables which cannot be controlled.

R. E. B. Makinson, Proc. Cambridge Phil. Soc. 34,474-97 (1938).

6W. J. de Haas and T. Biermasz, Physica 2, 673-82(1935).

7H. G. B. Casimir, Physica 5, 595 (1938); Magnetismat Very Low Temperatures, Cambridge, London, 1940.

Q

R. Berman, F. E. Simon, and J. Wolks, Nature 168,277 (1951).

9T. W. Powell, Research 7, 492-550 (1954).

Thus there is essentially no possibility of displacing this maximum by some means to the temperature region in the vicinity of the Debye temperature (870°K for UOJ. The low-temperatureconductivity of U02 has not been measured; hencethe value at the maximum is not known. However,it is safe to say that if such a displacement couldbe accomplished the thermal conductivity problemof U02 would be solved.

Most of the common oxides do not obey Debye'sor Peierls' equations exactly, but the generaltrend is for the thermal conductivity to decreaseas 1/T at high temperatures. Uranium dioxideexhibits this behavior between 600 and 1400°C.

From considerations of U-processes, impurity scat

tering, and the isotopic scattering predicted byKlemens'0 in 1955 and found experimentally byBerman in 1956, it is evident that anything disturbing the periodicity of the lattice will lower thethermal conductivity from the theoretical maximumin a manner which cannot at present be calculated.From Klemens' work on isotopic scattering wemight expect the enrichment of U02 with U todecrease the thermal conductivity from the valuesfound by Kingery and Hedge . This isotopicscattering might explain the anomalous behaviorof enriched fuel pellets noted by Eichenberg et al.It was found that fuel pellets of 19.5%-enrichedU02 had to have abnormally low thermal conductivities to explain the center melting which occurred in the pellets during in-pile operation. Nodetailed study has been made on the effect ofenrichment on the thermal conductivity of U02.This anomaly might also have been due in part tothe greater rate of bond disruption and generationof lattice distortion caused by the higher population of U and its resulting fission products inthe enriched specimens.

Therefore, treating U02 as an insulator, there isapparently no way to improve the thermal conductivity of very dense U02 ceramics without introducing a considerable quantity of another phase

P. G. Klemens, Proc. Phys. Soc. (London) A68,1113 (1955).

R. Berman and E. L. Foster, Proc. Roy. Soc.(London) A237, 344-54 (1956).

12R. C. Westphal, WAPD-MM-538 (June 1954).1 3

J. C. Hedge and I. B. Fieldhouse, Measurement ofThermal Conductivity of Uranium Oxide, AECU-3381(Sept. 20, 1956).

J. D. Eichenberg, Effects of Irradiation on BulkU02, WAPD-183 (Oct. 1957).

which has a high thermal conductivity. Thisapproach suggests the use of U02 cermets or ofmetal grids, with the resulting problem of dilution.A great deal of work has already been done in thisdirection in the AEC program.

2.2 Heat Transfer in Semiconductors

However, U02 is not what might be considereda good insulator, since less than 1 ev separatesthe conduction band from the valence band. The

electrical conductivities of the oxides of uranium

have been studied by many investigators, and theirwork has been summarized by Meyer and by Katzand Rabinowitch. The oxides UO.. and U,0.are reported to be metal-excess semiconductors.The conductivity of UO, at room temperature isvery low; the reported values range from 10 toless than 10 (ohm-cm)- , hence this oxidemight be thought of as an insulator. Part of theconductivity of U,0o is ionic in nature. The re

's o

ported values, which include both the ionic andelectronic contributions to the conductivity, varyfrom 2 x 10-4 to 1x 10-7 (ohm-cm)-1. In U02,conductivity values ranging from 3x 10- to 4x 10(ohm-cm)- have been measured at 27°C. Thevariation of conductivities is, of course, causedby variations in the composition, purity, apparentdensity, and crystalline perfection of the specimens studied. Prigent studied the electricalresistivity of U02 pellets and found they wereunacceptable for thermistor applications due tothe wide variations in resistance from sample to

sample. However, he did find that the conductancewas an exponential function of temperature. Eventhough polycrystalline U02 exhibits relativelyhigh electrical resistivities it can properly beclassed an oxidic semiconductor, an energy ofabout 0.4 ev being required to free a positivecarrier from the interstitial oxygen in U02+ .

The theory of heat conduction in semiconductorshas largely originated in the past six years. Asemiconductor is unique in that in a thermal sense

it is a combination of an insulator and a metal. In

15

16,W. Meyer, Z. Elektrochem. 50, 274 (1944).

J. J. Katz and E. Rabinowitch, Natl. Nuclear EnergySer. Div. VIII 5 (1951).

17R. K. Willardson, J. W. Moody, and H. L. Goering,J. Inorg. & Nuclear Chem. 6, 19-33 (1958).

,8J. Prigent, /. phys. radium 10, 58-64 (1949).

elemental metals the lattice conductivity is usuallynegligible compared with the thermal conductivitycomponent due to the conduction electrons, whereasthe opposite is the case in insulators.

Most of the work in this area has been done on

the elemental semiconductors germanium and silicon, although some work has been publishedrecently on a few intermetallic compounds. Innearly all these cases only intrinsic conductionmechanisms are considered; that is, the chargecarriers are raised from the valence band to the

conduction band solely by thermal excitation, andthe forbidden band is devoid of occupied acceptoror donor levels. Price derived an expression forthe electronic component of thermal conductivityin such a model. He proposed that a temperaturegradient in a semiconductor leads to gradients ofthe carrier concentrations, and hence to carrierdiffusion currents N:

N = -°\°2 / ^

a, +o2\e2J

r aX

~k~T 3+y, +y. grad T , (8)

where A is the energy gap between the bands, a iselectrical conductivity, and the thermal diffusionfactors yj and y. are each L for pure latticescattering. Price assumed nondegeneracy in theelectron and hole distributions and neglected anyimpurity band transport effects. Each electron-hole pair transports annihilation energy A, as wellas band energy of order kT, and hence the contribution of carrier flow to the thermal conduction is

a, ala2 / K

Oy+02 \e2

A2+ aA + BkT

kT(9)

where a and B depend on the scattering processes.For silicon and germanium the term A in Eq. 9could attain about 0.3 watt units at their meltingpoints. Thus the possibility of a thermal conductivity minimum at high temperatures is demonstratedin this type of solid.

In general the thermal conductivity k of a semiconductor may be written as the sum of two components: k the lattice conduction and k the

S eelectronic conduction; thus

k = k + Kg e

19 P. J. Price, Phys. Rev. 95, 596 (1952).

(10)

For sufficiently high temperatures k decreasesas 1/T, as was noted in the discussion on insulators. At the same time k is expected to increasevery rapidly with temperature owing to the increasing number of free charge carriers in thesemiconductor. The total heat conductivity wouldtherefore be expected to show a minimum at intermediate temperatures. A calculation of k basedon the assumptions currently made in calculatingthermoelectric forces leads to the well-known

Wiedemann-Franz law,

". =2(-J To , (ID

where

k - Boltzmann constant,

e = electronic charge,

a = electrical conductivity.

The introduction of experimental values of a intoEq. 11 for a given semiconductor gives values ofk far too small to be observed. Measurements of

the thermal conductivity of the intermetallic compound Mg2Sn show that there is no indication of aminimum and that the total heat conductivity decreases as 1/T approximately. Busch andSchneider reported an anomalous behavior in thethermal conductivity of InSb. Accurate measurements revealed a pronounced minimum in k at100°C, and just below the melting point of thecompound (400°C) k attained a value of one-fourththat of copper. A plot of k as a function of 1/Tyielded a straight line at low temperatures; anextrapolation of this line gave k for higher valuesof T. It was found that the electronic conduction

k determined in this manner could be representedby the expression

k£ = const, exp (-AE^^T) . (12)

From these data it was concluded that the experimentally determined k of InSb was larger thanwhat one would expect according to Eq. 11 byapproximately a factor of 100.

Frohlich and Kittel point out that in the theoryof the electronic part of the thermal conductivityof semiconductors in the intrinsic range it is important to take into account the transport of ionization energy E. In the case where the electron and

20G. Busch and M. Schneider, Physica 20, 1084-86

(1954).

21H. Frohlich and C. Kittel, Physica 20, 1086 (1954).

hole mobilities are equal the electronic contribution can be understood in terms of the expression

^e = CeuJ , (13)

where k is the electronic thermal conductivity,C is the carrier heat capacity per unit volume, jiis the carrier velocity, and / is the mean free path.The number of carriers, N, excited for E » kT isporportional to exp (-E/2kT), so that very roughly

C = Nk(EAT)" (14)

which is larger than the usual translational heatcapacity by a factor of the order of (E/kT) . Thisfactor at room temperature may be of the order of100 or more in representative semiconductors, sothat it is important to take it into account. However, since equal electron and hole mobilities in asemiconductor are unusual, a more general theoryis still required.

J. M. Thullier presented a comprehensivetheory for thermal conduction in intrinsic semiconductors in 1955 and specifically attempted toexplain the anomaly observed in InSb. Thullier'scalculations involve an attempted modification ofthe Wiedemann-Franz ratio by taking into accountboth the kinetic and ionization energies transportedby the charge carriers. However, his results differmarkedly from Eq. 11 only when rifj. = Piib, wheren and p are the electron and hole concentrationsand \in, p. their respective mobilities. This condition would be unique and would not be expectedto exist in an oxide; it certainly would not apply

in the extrinsic case of a doped oxide.

Joffe23 has summarized certain rules on thermalconductivity:

1. For substances with similar structuresdecreases as A increases, where A is the

phononmean atomic weight of the elements comprisingthe sample.

2. The phonon conductivity decreases with increase in the proportion of the ionic part of thechemical bond; in fact k decreases along

phononthe following sequence: the elements of group IV,compounds of elements of groups III and V, thenII with VI, and finally I with VII.

3. Ionic crystals with ions of equal masses havehigher conductivities than crystals with ions of

22

23

J. M. Thullier, Compt. Rend. 241, 1121-22 (1955).

A. F. Joffe, Can. J. Phys. 34, 1342-53 (1956).

different masses. This rule was stated by Euckensome years ago.

From rule 2 we see that U02 would be expectedto possess a low thermal conductivity, since itscation is of group VI and its cation and aniondiffer greatly in mass.

Joffe also investigated the effect of impurityadditions on the thermal conductivity of a semiconductor. All the impurities investigated hadthe same coordination number as the host theywere replacing; so the data illustrate the effecton the lattice conductivity only, the number ofcharge carriers remaining essentially the same.The considerations here are the same as those

enumerated for impurity scattering in insulators.The effect of substituting Se for Te in PbTe isillustrated in Fig. 1.

400

r 300

E 200

100

UNCLASSIFIEO

ORNL-LR-DWG 32559A

PbSe

30%^15%23

PbTe + PbSe

&*

Sr^'5% & V**r°*^8-^

PbTe a1̂

100 200 300

TEMPERATURE (°K)

400

Fig. 1. Thermal Resistivity vs Temperature for Se

Additions In PbTe.

Denoting by X the free path of phonons in anundisturbed lattice and by a the lattice constant,the ratio of the undisturbed lattice's conductivityto that of the disturbed lattice is

Ko— = 1

K

N(15)

where S - S Ja stands for the cross section of an

atom expressed in lattice units. This expressiontakes into account both the quantity of impuritiesand the kind of distortion they produce. Experiments on various semiconductors revealed some

cases of interstitial impurities with 5 > 1 up toS = 10. The introduction of Te atoms in the lattice

of GaSb led to S = 7, while in the same substanceSe gave S = 3. Apparently in some cases S mustbe taken as less than 1. For instance, a clustering

• 1/3in groups of n atoms leads to S For

n = 10, S = 0.46. For most of the semiconductorsinvestigated, Eq. 15 holds good on the conditionthat S = 1 for substitutional impurities; for instance,Si in Ge, Sn in Si, Se in PbTe.

In accordance with Eq. 15, the influence of impurities is more pronounced the larger the freepath AQ/a. A contamination of Si (with An/« = 40)or Ge (with An/a = 30) is already noticeable atN/NQ = 10-2 (1% of impurities), while in PbTe(with A./a = 3) 10% of impurities are required toproduce an equal effect.

For semiconductors with a large phonon freepath, 1/k rises quickly with the introduction ofimpurities. For Ge with XQ/a = 35, 1% of impuritiesincreases the thermal resistivity by 35%, whilefor PbTe with \Q/a = 3 the resistivity would increase by only 3%. Unfortunately, the phonon freepath as a function of temperature is not knownfor U02.

Another important consideration in semiconductors is the way phonons and electrons combinein the heat transport. The assumption is madethat they act independently, and this is reasonableup to a concentration of electrons of 1020 per cubiccentimeter. A check on this assumption was madeby Joffe by introducing impurity atoms and in thisway changing the electrical conductivity to a largedegree without affecting the phonon conductivitymarkedly.

Figure 2 illustrates the dependence of the thermalconductivity on a, the electrical conductivity, inPbTe. Between a = 1000 and a = 2500 (ohm-cm)-1the electrons must be considered as nondegenerate,that is, they obey the ideal gas law. From Fig. 2we see that in this case an increase in the electrical conductivity definitely increases the thermalconductivity of PbTe.

Another important mechanism of thermal transferis that of radiation through the solid. This additional mechanism of energy transport would attainsignificance only at high temperatures. Genzelhas computed the corresponding Ak:

16 ae ,Ak = T3

3 k(16)

where e is the electronic part of the dielectricconstant and a is the factor in Stefan's law. This

—- 10

o«•-

X

k 8*—*

>t-

> 6h-C)=>n

z 4oo

1

< '<L2rrijjI 0

UNCLASSIFIEDORNL-LR-DWG 36517

EA'

y /

yy a

^w*

B'

B

IC c1

0 1000 2000 3000 4000

°\ ELECTRICAL CONDUCTIVITY (ohm-1 cm-1)

Fig. 2. Dependence of Thermal Conductivity of PbTeon the Electrical Conductivity, a. AB and A'B'= K ;D, beginning of degeneracy.

effect has been observed by McQuarrie in MgO,BeO, and ALO,. Apparently an oxide must betranslucent before this effect is appreciable.Hence it was not observed by Kingery in U02 upto 1500°C or by Hedge and Fieldhouse13 at1680°C.

Possibility of Increasing the Conductivity ofU0_. — From this discussion we can draw some

conclusions about the feasibility of improving thethermal conductivity of U02:

1. The oxide must be as close to theoretical

density as possible, as* gas-filled voids in thesolid act as thermal insulators.

2. As was noted previously, the effect of theintroduction of any foreign ions into the U02lattice will be to decrease the phonon conductivity.The magnitude of this depression cannot be computed, because the phonon free path as a functionof temperature is not known for U0o. W. D.Kingery ° has pointed out that in single-phaseceramics, lattice thermal conductivity may, undersuitable conditions of elevated temperature orlattice disturbance by irradiation, be reduced to

24M. McQuarrie, ;. Am. Ceram. Soc, 37, 91-95 (1954).25.Effects of Irradiation on Bulk U00, WAPD-183

(Oct. 1957). 2

a minimum value set by phonon scattering at meanfree paths of the order of the lattice spacing; inthe case of U02, such a minimum value would beapproximately 0.0173 wcm-1'(0C)-'. Using thisvalue for the phonon free path at elevated temperatures, the value of S in Eq. 15 could be determinedby experimentally measuring k and kq at hightemperatures for samples with controlled impurityconcentrations.

In conjunction with the theoretical considerations just discussed there is a possibility ofincreasing the thermal conductivity of polycrystal-line U02. This might be accomplished by introducing an ion (as the oxide) into the lattice whosesize would permit it to isomorphously replaceU4+ and whose valence was preferably either 5+or 3+. In the case of a 3+ ion the lattice wouldgain one-half a positive hole per substituted ion,whereas a 5+ ion would result in the release ofa free electron to the lattice. These chargecarriers may then be able to transport enough heatin the presence of a thermal gradient to result ina sizable electronic conduction component. SinceU4+ is easily oxidized to U6+ (this occurs inU02 at room temperature with the absorption ofoxygen) the advantage of adding a 5+ ion may becounterbalanced by the ability of its uranium nearneighbors to alter their valence so as to reducethe lattice strain. The substitution of a 3+ ionfor U4+, on the other hand, has been found tostabilize the oxidizing tendency of U02 at highconcentrations. The atomic species selected forsubstitution into the U02 must be so chosen thatits size will not drastically distort the lattice.Ions whose masses differ markedly from that ofU4+ will act as scattering centers and may reducethe already low phonon conductivity. The Gold-schmidt rule, which states that an ion whoseradius is within ±15% of that of the host ion can

be substituted into a lattice without severe lattice

distortion, would limit the ionic radius selection tothe range 0.8245 to 1.1155 A, using 0.9700 A asthe radius of the U ion. The species chosenshould also have only one stable valence state.The 3+ ions of the rare earths all fall within this

range. Unfortunately the high neutron crosssections of some would prohibit their use in a fuelelement. If the required impurity concentrationwas low, then some of the higher-cross-sectionmembers of the family may be acceptable. The

rare earths having the lowest cross sections andtheir ionic radii are listed:

3 +La

.3+

3+Pr

Tb

Ho

Er

3+

3 +

3+

.3+

3+Yb

Lu3+

Cross Section Ionic Radius

(barns) (angstroms)

8.8 1.061

0.7 1.034

11.0 1.013

44.0 0.923

64.0 0.894

166.0 0.881

120.0 0.869

36.0 0.858

108.0 0.848

Samarium has an ionic radius very close to thatof U4+, but its cross section of 10,000 barns excludes its consideration as a possibility.

Other ions which should fit into the U02 latticeare:

IonCross Sect!

(barns)

on Ionic Radius

(angstroms)

Ca2+ 0.42 0.99

Y3+ 1.4 0.93

Surprisingly, Sylvania-Corning reports thatNb5+, 1.1 barns, 0.70 A, increases the thermalconductivity of U02, even though its radius isbelow the 0.8245-A limit.

The substitution of a 3+ ion into the lattice

would markedly alter the electronic structure ofthe solid, since the substitution of an X ion forU4+ would result in the following:

U4+ + 202- is replaced by

X3++ 1.5O2- + 0.5a . (17)

We see that the addition of the X3+ results in thecreation of half a hole per substituted ion; the

97hole in turn can act as a charge carrier.

Since all the theories discussed have been con

cerned primarily with elemental and intermetallic

Personal conversations, Sylvania-Corning, Bayside,N. Y., August 1958.

J. S. Anderson, Bull. soc. chem. France [5]20,781 (1953).

semiconductors, we cannot extend them directlyto oxides. However, the principles cited shouldcertainly be valid. If the number of charge carriersin the U02 could be increased sufficiently andif their mobility was great enough, then the thermalconductivity of the oxide would be expected toincrease with temperature. The effect of an in-pi le environment on these properties may be drasticand would have to be measured.

Hartmann pointed out that the increase of electrical conductivity with the addition of oxygenmakes U02 a positive hole conductor. He madeHall-coefficient measurements and obtained a

mobility value of 10 cm «v «sec for the chargecarriers, which is a rather low mobility. No datawere found on measurements of these propertiesfor U02 which contained 3+ impurities. The addition of 3+ ions to the U02 lattice will result inunsatisfied oxygen bonds and p-type conduction.The effect on the thermal conductivity of non-stoichiometry in the oxide is unknown. Since theexcess oxygen fits into interstices in the fluoritelattice of the U02 and subsequently oxidizes someof the U to U6+, the oxygen may suppress theconductivity by two processes. By occupying anonpermitted lattice site, the oxygen acts as animpurity scattering center. The U6+ resultingfrom the oxidation distorts the crystalline electricfield and hence also acts as an impurity scatteringcenter. This suppression of the phonon conductivity would probably mask any increase inthe electronic thermal conductivity component dueto the decreased resistivity which accompaniesoxidation.

From theoretical considerations it follows thatthe logarithm of the density of carriers plotted vsthe reciprocal temperature should yield a straightline of slope -&E/2k, where AE is the donorionization energy. As the temperature is increasedto such values that the intrinsic excitation be

comes important, the slope changes gradually to—Eqap/2^. The heat carried by the intrinsicallyexcited carriers is apparently small in UO,, as isevidenced by the horizontal nature of the curvesin the high-temperature region of Fig. 3. Therefore, if the impurity carriers are able to transmita significant amount of thermal energy in thiscase, one would expect that at high temperatures

28 W. Hartmann, Z. Physik 102, 709 (1936).

0.08

0.02

UNCLASSIFIEDORNL-LR-DWG 35713

\\ ^-KINGERY 95% THEORETICAL/;

L

• ^~~ —c —

HEDGE AND FIELD'hOUSE, ~^~75% THEORETICAL pS-^

|i

0 200 400 600 800 1000 1200 1400 1600 1800

TEMPERATURE (°C)

Fig. 3. Thermal Conductivity vs Temperature of U0-.

the electronic component of the thermal conductivity would approach a constant value as the condition of complete ionization of the impuritycenters is approached. Since the phonon conductivity would continue to decrease with temperatureuntil the mean free path approached the latticeconstant, the thermal conductivity may again decrease at extremely high temperatures, providedno significant radiation component appeared.

3. MEASUREMENTS OF THERMAL

CONDUCTIVITY

3.1 Review of Thermal ConductivityMeasurements on UO,

Snyder and Kamm at Princeton determined thethermal conductivity of uranium dioxide powders,but these values are low because of the low

densities of the samples that were tested. Weeksat Argonne reported a thermal conductivity of0.023 ca|.cm-,.sec-1.(0C)~' at 70°C for a samplewith a density of 10.23 g/cm as compared to

0.00034 cal.cm-'.sec-'.CC)-' at 100°C for thePrinceton value. Weeks's value is approximately68 times as large as the Princeton value, clearlyillustrating the importance of high density for

29T. M. Snyder and R. L. Kamm, CP-76 (May), CP-92(May), CP-96 (May), and CP-124 (June) (1942).

30J. L. Weeks and R. L. Seifert, Rev. Sci. Instr. 24

1054-57 (1953).

good thermal conduction. Englander in Francereports an average value of 0.01 cal«cm~ »sec- •(°C) at approximately 70°C for material havinga density of 9.2 g/cm . Kingery at MIT has mademore recent measurements on U02. Kingery andVasilos reported data on U02 having 73% oftheoretical density. The measurement was madeutilizing a linear flow comparison method.Kingery has also measured the thermal conductivity of U02 having 95% of theoretical densityby this method, and this is the only measurementthat has been made on high-density material overan extended temperature range. He gives anempirical equation for the thermal conductivity asa function of temperature:

K= 0.143e-'-207Txl°-3

wcm- '.<°t(°C)-1 ,

where T=°K, T= 1473°K.Calculation of conductivities at various tempera

tures gives the following values:

(°C) [wcm-1.(°C)-1] [cal-cm-1.sec-1-(°C)-1]

100 0.0912

300 0.0715

500 0.0563

600 0.0500

800 0.0392

1000 0.0307

0.0218

0.0171

0.0135

0.0120

0.0094

0.0073

McCreight reported thermal conductivity valuesfor several compositions including U02-MgO,U02-BeO, and U02-Zr mixtures. He found that a29 wt % addition of BeO to U02 increased thethermal conductivity by nearly a factor of 4 overthat of U02 at 450°C. This large addition probably resulted in the presence of two phases (nox-ray analyses were made of the resulting bodies)and the BeO phase conducted the additional heat.

Bowers et al. reported that BeO additions of

up to 15 wt % to U02 improved the thermal shockresistance of the body, whereas Zr02 and Ce02did not affect this property.

31

32,M. Englander, CEA-79 (1 951).

10

L. R. McCreight, Thermal Conductivity Data forSome Nuclear Fuels, KAPL-822 (Oct. 1952).

33D. J. Bowers et al.. Effect of Ceramic or Metal

Additives in High U02 Bodies, BMI-1117 (July 24, 1956).

A recent measurement of the thermal conductivity

of U02 was made by Hedge and Fieldhouse atArmour. Using a stacked disk method which willbe described in some detail later, they obtainedvalues of the thermal conductivity to 1670°C.Unfortunately their U02 samples had a density of8.17 g/cc, 75% of theoretical density; Thisprobably explains why their values are considerably lower than Kingery's. Due to severe crackingof the samples the measurements were not extended to higher temperatures. A comparison ofHedge and Fieldhouse's values with those ofKingery for material with 95% of theoretical densityis shown in Fig. 3. The dashed section of thecurve is extrapolated data.

The only measurements on U02 with small impurity additions that were encountered are thosepresently being made by Sylvania-Corning. Theirequipment is a modification of the linear flow comparison method of Kingery. The maximum temperature limit of the apparatus is 550°C. Yttrium andother 3+ ions are being substituted for U inU02, and the additions examined thus far appearto improve the thermal conductivity of the oxide,although only a small amount of data has beenobtained thus far. If these measurements are

accurate, then the data are conclusive proof thatan electronic thermal conduction component can beinduced in polycrystalline U02 with a subsequentincrease in the thermal conductivity.

Since the theory of heat transfer in semiconductors will not presently permit the calculationof the magnitude of the electronic component, k ,its presence and magnitude can be deduced only byphysical measurements.

Flinta has recently reported the effect of »ulatively low fluxes*!.8 x 10 neutrons»cm2§ onre

the in-pile thermal conductivity of UO~.measurements, which were made in Sweden, wereperformed on annular U02 pellets having 91% oftheoretical density. The thermal conductivity wasdetermined with data obtained from a thermocouplenear the center of the U02 together with a thermocouple located near the outside surface of the

pellets. The method was essentially a stackeddisk method, the samples being 15 mm in diameterand 20 mm in length. Five disks were stackedtogether, with a tungsten rod passing through the

The

34J. E. Flinta, Thermal Conductivity of U07, TID-7546, p 516-25 (1957).

central hole and serving as a heat source inaddition to that caused by the fissioning. Withthe outer surface of the pellets cooled to 50°C bywater, thermal cycling of the center to 1800°C,out of pile, resulted in a marked decrease in themeasured /c, presumably due to shock failure ofthe pellets. The thermal conductivity was foundto be erratic with neutron irradiation; it dropped to0.007 wcm- •(°C)~1 after three heating cycles.At about 350°C and high neutron fluxes' (l&ILrj£iJ.-trnn'./rm \_ r increased from a minimum of 0.005to 0.007 wcm-1.(0C)-1. This behavior could notbe explained.

The latest measurement of k for U02 was madeby Eichenberg in the Materials Testing Reactor.The samples consisted of PWR pellets, 0.3535 in.in diameter, and they were stacked in a stainlesssteel capsule. The central temperature of the U02was measured with a thermocouple in the centralhole of the pellets. Thermocouples located in theinterior of the capsule wall provided data fromwhich the heat generation in the U02 was calculated. The pellets had an estimated exposure of0.7 x 10 nvt thermal. The thermal conductivitywas determined up to 480°C, and the values wereabout 50% of the Armour data shown in Fig. 3.

3.2 Methods of Measuring Thermal Conductivityof Insulators and Semiconductors

In general the methods employed for measuringthe thermal conductivity of semiconductors andinsulators are the same. Some methods which

utilize certain unique properties of metals, forexample, high electrical conductivity, are applicable only to good conductors, while others canbe used for either metals or nonmetals.

Thermal conductivity may be measured by eitherstatic or dynamic methods. In dynamic methodsthe temperature is Varied suddenly or periodicallyfor one portion of the sample and the temperaturechange with time is measured to determine thethermal diffusivity, n/cp. In static methods, thesample is allowed to come to a steady state andthe temperature distribution is measured to determine the thermal conductivity k by an integrated

35J. D. Eichenberg, An In-Pile Measurement of theEffective Thermal Conductivity of UO-., WAPD-T-813(March 28, 1958). 2

•k .ii^ .jlziJj^D

form of

dT

dxQ = kA (18)

where Q is the amount of heat flowing per unittime through an area A and the temperature gradientin a direction perpendicular to A is dT/dx. Thisexpression also serves to define k, the thermalconductivity.

Present experimental evidence indicates thatthe conductivity is not a function of the heatflowing, and Eq. 18 may be said to be verified byexperimental evidence.

The dynamic or non-steady-state methods arenot usually utilized for the determination of thethermal conductivity at high temperatures sincethey require a periodic heat source and knowledgeof the heat capacity at the temperature of measurement.

A steady-state method often utilized for ceramicsis one using a long bar specimen. One end isheated, while the other is connected thermally toa heat sink. Thermocouples are inserted at selected distances along the bar, and these temperatures plus the known amount of heat flowingalong the bar permit the calculation of k. Theoretically this is a good method, but practically ithas serious disadvantages. One of the main disadvantages, which is the one that plagues allthermal conductivity determinations, is that theradial heat flow from the bar must be either neglected or decreased by means of thermal guardrings. Any radial heat flow is a violation of theboundary conditions assumed in Eq. 18, and theaccuracy of the measurement is seriously affected.

This loss can be reduced by decreasing thelength of the bar so that the longitudinal surfacearea is small with respect to the cross sectionalarea. These conditions require the sample toassume the shape of a thin plate. When dealingwith samples of low conductivity the plate thickness can be quite small and reliable values ofdT/dx still be obtained. However, where ceramicsare concerned the forming of large flat platesamples is often not possible. Guard rings arestill required and they are inevitably imperfect.

A method of ensuring correct heat flow withoutthe use of heat guards is to employ a specimenwhich completely surrounds the heat source. Thismay consist of an infinite cylinder or slab, surrounding an infinite heat source, or it may consist

11

of a hollow sphere or spheroid. Shapes approximating an infinite cylinder or slab are satisfactoryif only the center section is employed (this isequivalent to using heat guards), but they are difficult to form from ceramic materials, particularlyfrom dense oxides. A spherical shape is satisfactory, but the highly curved isothermals present adifficulty in temperature measurement.

Loeb has developed an expression for the caseof a hollow ellipsoid (prolate spheroid). The expression makes possible the use of an envelopesample of relatively small size which can be formedfrom pure materials and which has nearly flatisothermals in the central section. Kingeryet al.37 describe this method and the other envelope methods in considerable detail. Theellipsoids are approximately 11 cm long (i.e.,2a = 11 cm) and 4.2 cm thick (2b = 4.2 cm). Theyare heated internally either by a conductor woundon a shaped core or by a susceptor shaped to fitthe internal cavity of the ellipsoid.

The ellipsoids were cast of selected refractoryoxides, hence any oxide whose structure is susceptible to change in casting could not easily beformed to a suitable shape. In the case of MgOand CaO special casting procedures were necessary. These envelope methods can be used atthe high temperatures in the range of interest andcould be utilized if ellipsoids of dense U02 canbe formed.

Another method improved upon by Kingery is thecomparative flow method, wherein samples ofknown conductivity are placed in series with theunknown samples. This method is the one presentlybeing used by Sylvania-Corning for U02, and theyare encountering great difficulty in eliminatinglateral heat flow. The maximum temperatureattained by Kingery in this measurement has been900°C, and it is not likely that guards would besatisfactory at higher temperatures.

As was noted previously, one of the latest measurements of the thermal conductivity of U02 wasmade by Hedge and Fieldhouse at Armour. Theradial heat flow method of Powell was used

rather than the linear flow method of Kingery.This method consists in measuring, under steady-

36

37u

A. L. Loeb, ;. Appl. Phys. 22, 282 (1951).

12

W. D. Kingery et al.. The Measurement of ThermalConductivity of Refractory Materials, NYO-3643 (March15, 1953).

38R. W. Powell, Proc. Phys. Soc. 50, 407 (1938).

state conditions, the radial heat flow and radialtemperature drop in a vertical stack of disks.The disks were in the form of annular rings, andthe heat which flowed through the disks wassupplied by an electric heater centered in theaxial hole of the stacked disks. In order to attain

high temperatures the entire column of disks wasplaced in an electrically heated furnace in whicha helium atmosphere was maintained.

In comparison with the prolate spheroid or otherenvelope methods and the linear flow methods,this method has some decided advantages whichdeserve enumeration. First, the problem of guarding is lessened, since the direction in which flowis desired to be decreased is now the longitudinal

rather than the radial. The interfaces between

the disks act as natural thermal barriers and hence

reduce this longitudinal flow. The interfacialthermal resistance is one of the most serious

disadvantages of the comparative method describedpreviously. Thermal guards placed on the top andbottom of the column reduced the longitudinal flowto the point where it was negligible. Second, thedisk samples, which had an inside diameter of0.50 in. and an outside diameter of 3.0 in., couldbe formed by dry process techniques rather thanby casting. Hydrostatic pressing of the greensamples may permit the forming of high-densitysamples.

If high-density U02 samples of this shape couldbe formed, this method appears to be the simplestfor giving reliable data. It was checked by Hedge,using Armco iron as a standard, and the resultsagreed with Powell's within experimental error upto 790°C, the maximum temperature of the test.

A recent report by Paine and Stonehouse involves the measurement of the thermal conductivityof intermetallic compounds. Their method resembles that of Hedge and Fieldhouse, the maindifference being that the radial heat flow is inwardrather than outward. Also, all the heat is suppliedby a molybdenum winding around the outside ofthe cylindrical specimen. The heat is conductedaway by water flowing through a tube in the centerof the column of disks. For high temperatures thistype of heat sink would create severe thermalstresses in a sample and would probably lead tocracking in a brittle material having a low conductivity, such as U02.

39 R. M. Paine and A. J. Stonehouse, An Investigationof Intermetallic Compounds for Very High TemperatureApplications, WADC-TR-57-240 (May 1958).

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