2.13.4.1 Thermal Conductivity

It is appropriate to discuss thermal and electrical conductivity as coupled phenomena. Thermal con­ductivity is considered a sum of phonon and electron contributions to conductivity. The phonon contribu­tion to thermal conductivity should decrease with temperature, as atomic vibrations inhibit phonon
transport. The contribution to thermal conductivity due to electrons is calculated by the Wiedemann — Franz law,41 according to

LT

P

where ke is the electronic thermal conductivity, L is the Lorentz constant (2.44 x 10~8W O K~2), T is absolute temperature, and P is electrical resistivity. Generally, electrical resistivity of metals increases with temperature; in transition metal carbides, elec­tron thermal conductivity increases with tempera­ture. At low temperatures heat is mainly conducted by phonons, which are scattered strongly by con­duction electrons.42-44 At intermediate tempera­tures, both electrons and phonons contribute to thermal conductivity, but in the transition metal carbides the electronic component is dominant. Phonon scattering by carbon vacancies becomes important above about 50 K, contributing to a decrease in thermal conductivity with increasing temperature. At high temperatures, thermal conduc­tivity increases approximately linearly with temper­ature. The temperature dependence of electronic thermal conductivity is plotted in Figure 14; this was computed from the Wiedemann-Franz law and

a linear fit to the electrical resistivity measurements of Taylor45 and Grossman46: p = 0.79T + 36.3.

Experimental measurements of thermal conduc­tivity of ZrCx as a function of temperature between

1.8 and 3400 K are also plotted in Figure 14. The overall trend is a steep increase of thermal con­ductivity with temperature up to 50 K, followed by a slight decrease in an intermediate temperature range

F~ 70 60	T	T	T
50
40
30
20
10
0
0

Figure 14 Thermal conductivity of ZrCx as a function of temperature.

(up to 100-1000 K) and then a more gradual increase up to the melting temperature. Room-temperature thermal conductivity has been reported between 20 and 40 W m-1 K-1, meeting or exceeding that of Zr metal.47 A source of experimental scatter in thermal conductivity is sample porosity, which is not always reported by authors.

Room temperature thermal conductivity is also a strong function of C/Zr ratio (Figure 15). Storms and Wagner35 measured thermal diffusivity of hot — pressed ZrC064-1 (0.01-0.1 wt% O) by the laser flash method,48 computing thermal conductivity from sample density and heat capacity according to

k = a dCp [6]

where k is thermal conductivity (W m-1 K-1), a is thermal diffusivity (m2 s — ), d is density of the sample (kgm — ), and Cp is heat capacity (Jkg-1K- ). As described in Section 2.13.3.4, Cp was available for ZrC096 but not for other compositions and Cp versus x was estimated by assuming that it was parallel to that of NbCx. A maximum room temperature ther­mal conductivity of 45 W m-1 K-1 occurs at near­stoichiometric compositions, with a steep drop-off as carbon atoms are removed from the lattice. Further reduction of the C/Zr ratio below approximately

ZrC09 has little effect on thermal conductivity, which approaches a constant value of 10 W m-1 K — .

From a fit to literature electrical resistivity measure­ments and the Wiedemann-Franz law, Storms and Wagner calculated the composition dependence of the electronic component of thermal conductivity as

ke = 1.05 x 103 0.00382 + — [7]

e 55 + 950(1 — x)

where x is the C/Zr ratio, and a Lorenz number of 3.5 x 10-8 V2 K-2 was used (by assuming that the ther­mal conductivity in the low-carbon region was entirely electronic). By taking the difference between their experimentally measured thermal conductivities and their calculated electronic thermal conductivities, Storms and Wagner expressed the phonon thermal con­ductivity as a function of composition by the equation

0.007

(1 — x)2 where x is the C/Zr ratio. As plotted in Figure 15, electronic thermal conductivity is dominant for highly nonstoichiometric ZrCx, while lattice or phonon conductivity makes a larger contribution in near­stoichiometric ZrCx. The effect of a decrease in C/Zr ratio is proposed by Avgustinik et al.49 to reduce the

50
E ‘Й >. >  о га TO c о о га E ф
0.6	0.7		0.9	1.0

connectivity of the lattice while introducing vacancies and increasing the concentration of nonlocalized elec­trons. The net effect is an increase in phonon scattering and a decrease in conductivity with deviation from stoichiometry.

Storms and Wagner also studied the effect on thermal conductivity of tripling the oxygen content in ZrCo.64-o.682 from 0.042 to 0.125-0.13 wt%. They found that thermal conductivity was affected little by varying oxygen content in the low-carbon region but asserted that 0.6 wt% O in ZrC0.92450 produced a more noticeable effect. They suggested that impuri­ties which substitute for carbon (i. e., O or N) reduce the vacancy concentration and have the same effect on thermal conductivity as an increase in C/Zr ratio. The effect of impurities on thermal conductivity is correspondingly more pronounced for ZrC0.9-1.0. Too few measurements of well-characterized near­stoichiometric samples are available to assess this phenomenon more conclusively.

Neshpor eta/.1 measured room-temperature ther­mal conductivity of 85-95% dense sintered ZrC06-0 9 containing 1.4 wt% nitrogen by a steady-state heat — flow method, repeating this study with Avgustinik et a/.49 after decreasing N content to 0.05 wt%. Other room-temperature measurements by heat

flow or thermal diffusivity measurements42,49-55 are
consistent with the trend established by Storms and Wagner, but by covering only one composition, or compositions only below the drop-off at ZrC09, the individual studies fail to capture the true trend.