According to kinetic gas theory, the lattice thermal conductivity above Debye temperature for an ideal lattice can be expressed by [12]

к = 0.33 CvvL (35)

where, Cv is the specific heat at constant volume, J/m3 K, v is the velocity of sound in solid, m/s, and L is the mean free path of scattered waves (the phonon wave­length), m.

Above the Debye temperature, thermal conductivity of electric insulators decreases with increasing temperatures. Since atomic vibrational frequency increases with temperature, an increase in wave scattering is anticipated which are shown to be due to phonon interaction. Thermal energy is transferred by the Umklapp process, in which two phonons interact to form a third [12]. According to this theory, lattice thermal conductivity is inversely proportional to absolute temperature and this becomes a minimum when phonon wavelength becomes less than the mean distance between the scattering centers. For crystalline solid, the minimum distance between the scattering centers is the interatomic distance which is the lattice parameter, ao. Therefore, the above equation becomes [12]:

(k)mn= 0.33 Cvva0 (36)

In solids, phonon-phonon scattering is due to the anharmonic components of crystal vibrations. Lattice anharmonicity increases with the mass difference between anions and cations in the ionic material and is greatest in UO2 or PuO2 [38, 91]. As a result, the thermal conductivity of the oxides of the actinide metals is considerably lower than that of most other crystalline oxides. The kinetic theory of gases shows that the collision mean free path is given by the reciprocal of the product of the collision cross-section rP and the density of scattering points (nP):

L =(1/apnp) (37)

The deviation from stoichiometry and the presence of foreign atoms or porosity result in lower values of к in actinide oxides. Further, it can be shown that the phonon mean free path should vary as 1/T. In general, phonon-phonon scattering and phonon-impurity scattering are the dominant mechanisms of the thermal conductivity in ceramics. Klemens [92] has proposed a heat conduction model in materials where the phonon-phonon (Umklapp) scattering and the phonon — impurity scattering occur simultaneously. Theoretically, the phonon component of the thermal conductivity k may be written as:

k = (A + BT)-1 (38)

where A and B are constants and T is the absolute temperature.

Thermal resistivity (R), which is the reciprocal of thermal conductivity (k), of the above oxides can be described by the following equation:

R = 1/k = A + BT (39)

The first term, A, in Eq. (39) represents the defect thermal resistivity. This results from the phonon interactions with lattice imperfections, impurities, isoto­pic, or other mass differences as well as bulk defects such as grain boundaries in the sample. The influence of substituted impurities on the thermal conductivity is described by the increase of the parameter A. The second term in Eq. (39), namely BT, represents the intrinsic lattice thermal resistivity caused by phonon-phonon scattering [1, 2]. As the temperature increases, this term becomes predominant. The parameter B remains nearly constant by substitution. The constants A and B can be obtained from the least squares fitting of the experimental data.

The thermal conductivity of nuclear ceramics is strongly influenced by the stoichiometry. Deviations from stoichiometry produce point defects, most likely oxygen vacancies or metal interstitials in hypostoichiometric compounds and oxygen interstitials or metal vacancies in hyperstoichiometric compounds. Intro­duction of point defects into the oxygen ion sublattice or substitution of Th for U on the cation sublattice provides additional centers from which phonon scattering occurs. It is reported that there is a drastic change in the uranium vacancy con­centration on varying O/M ratio around the stoichiometric composition. Many reports are available on the effect of stoichiometry on the thermal conductivity of UO2 and (U, Pu)O2 samples [1]. Thermal conductivities decrease as their hyperstoichiometry, x, increases. At low temperatures, thermal conductivity of mixed oxide can be described by a modified equation of (38) as [93, 94]:

k = 1/[A(x, q) + B(x, q)T] (40)

where, x and q denote the extent of nonstoichiometry and the Pu/Th content in the UO2 lattice, respectively. The limited amount of experimental information avail­able suggests that the coefficient A depends primarily on the O/M ratio and only very weakly on the plutonium content. The coefficient A may be written as

A = A0 + DA(x) (41)

where, A0 is very nearly equal to the A value of pure UO2. The perturbation DA arises from interactions of point defects with lattice. The magnitude of DA is proportional to the defect atom fraction and to a measure of the cross section of the defect for phonon scattering. The latter is proportional to the square of the dif­ference between the atomic radius of the defect (r{) and that of the host atom (r). The mass difference between the impurity atom and the host atom may also influence A, but this contribution is not significant in mixed oxide fuel materials. A can also be represented by the following equation as

A = [(p2Vh)/(3hv2)] X Ci, (42)

i

where V, в, h, and v denote the average atomic volume, Debye temperature, Planck’s constant, and phonon velocity, respectively. The term RC is the sum of the cross-sections of all the phonon-defect scattering centers. The analysis of the lattice defect thermal resistivity and the evaluation of phonon scattering by the various defect scattering centers in pure and mixed actinide oxides have been carried out by several authors [91-94]. Accordingly, A of Eq. (42) can be given as

A = C(C + Г0), (43)

where C = (p2Vh)/(3hv2). Ги is the scattering cross-section arising from U sub­stitution and Г0 is that from all other native defects present in the sample. The scattering cross-section Ги can be expressed in terms of the mass and size dif­ference of the substituted atom over that of the host [95]:

Ги = x(1 — x) x [(AM/M)2 + E(Ar/r)2], (44)

where, x is atomic fraction of substituted U in place of Th, AM and Ar are the mass and radius difference between U and Pu/Th atom, respectively, M and r are average mass and radius of the substituted atom, and E is an adjustable parameter which represents the magnitude of lattice strain. From the above, it is clear that scattering cross-section depends upon the mass difference between Th and U atoms, size difference between Th and U atoms, charge of U ion and microstructure.

Thermal transport by electrical charge carriers can also contribute to thermal conduction at high temperatures. The ratio between thermal and electrical con­ductivities of metals can be expressed in terms of the ratio:

Lc = k/rT = p2k2/3e2 = 2.45 x 10-8 WX/K2, (45)

which may be called the Wiedemann-Franz ratio or the Lorenz constant. In the above equation, r is the electrical conductivity, e is the electronic charge, and k is Boltzmann’s constant. Thermal conductivity of a solid can be measured by two methods:

1. By determining the stationary heat flow through the specimen, which gives k directly,

2. By determining the variation of the temperature at a fixed plane, that is a specimen surface, due to an induced nonstationary heat flow which gives the thermal diffusivity, a.

Since the second method is more versatile and requires smaller specimen, it has become a standard method for determining k for T > 600 K. For lower temper­atures, the first method is more suited.

For the thermal diffusivity measurement, the sintered pellet was sliced into discs of about 10 mm diameter and 2 mm thickness using a low speed cut-off wheel. A pulse of laser was projected on to the front surface of the pellet and the temperature rise on the rear side of the pellet was recorded as a transient signal by using an infrared detector. The thermal diffusivity (at) was calculated from the following relationship:

at = WL2 /nt 1=2 (46)

where t1/2 is the time required in seconds to reach half of the maximum temper­ature rise at the rear surface of the sample and L is the sample thickness in millimeter. W is a dimensionless parameter which is a function of the relative heat loss from the sample during the measurement. The data have to be corrected for radiation heat losses by the method of Clark and Taylor [96].

Unlike UO2 or PuO2, ThO2 is a semitransparent material to wavelengths of the infrared region. For a laser flash experiment, all the energy of the laser pulse is not absorbed on the front face of the sample, but also in volume. Also, the temperature measurement on the rear face is skewed as the pyrometer may receive radiation produced not only at the sample surface, but also in volume. These difficulties are overcome if the faces of the samples are given a coating by graphite. A coating of graphite on both faces was used in order to make sure that the energy of the laser is absorbed on the front face and to improve the temperature recording on the rear face.

Heat transport through materials is described by two properties: thermal con­ductivity, k (under steady state conditions) and thermal diffusivity, at (under transient conditions). These two properties are related by the expression:

k(T) = at(T) • p(T) • Cp(T), (47)

Where, p the density of the material and Cp its specific heat at constant pressure. The specific heat of mixed oxide like (Th1_yPuy)O2 solid solutions was calculated from the literature values of specific heats of pure ThO2 and PuO2 and subse­quently using Neumann-Kopp’s rule. The following equations were used to calculate Cp of (Th1-yPuy)O2:

Cp(Th1_yPuy)O2 = (1 — y) • Cp(ThO2) + y • Cp(PuO2), (48)

where y is the weight fraction of PuO2.

Effect of Porosity on Thermal Conductivity

Attempts to evaluate the decrease in thermal conductivity due to porosity (P) have been made by Eucken in as early as 1932. There are many relations in the literature describing the effect of porosity on thermal conductivity. Some of them are listed below [1, 64, 97-106]:

1. Loeb км — (1 _ P) kjD (i)

2. Modfied Loeb kM — (1 — aP) кто where 2 < a < 5 (ii)

3. Kampf and Karsten kM — (1 — P2/3)кто (iii)

4. Biancharia kM — [(1 — P)/(1 — (b — 1)P)] kTD b = 1.5 for spherical pores (iv)

5. Maxwell-Eucken kM — [(1 — P)/(1 + bP)] кто (v)

6. Brand and Neuer kM — (1 — rP) kjo where r = 2.6 — 0.5 (T + 273)/1000 (vi)

7. Schultz kM — (1 — P)c кто (vii)

(kM and kTD are the thermal conductivities, respectively, in presence and absence of porosity P, 0 < P < 1).

Schultz [106] has theoretically shown that, for spherical pores distributed randomly, у of Eq. (vii) has a value of 1.5. However, in reality the above coeffi­cients for fuel pellets are larger (у > 1.5), due to the porosity being neither spherical nor uniformly distributed [1]. IAEA [40] has recommended the value of a = 2.5 ± 1.5 for the modified Loeb equation for 0 < P < 0.1. Inoue, Abe, and Sato [107] experimentally showed that у = 2.4 for 0.044 < P < 0.470 and reported that b = 2 (Eq. (v)). The IAEA recommendation (a = 2.5 ± 1.5) is in agreement with other experimenters [1, 40]. Among the above, Eq. (i) under predicts the data and Eq. (iv) accounts for the shape of the pores.