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On the temperature dependence of the lattice thermal conductivity
Materials Science and Engineering, 60 (1983) L1-L2
L1
Letter
On the temperature dependence of the lattice thermal conductivity
S. V. TSIVINSKY All-Union Institute o f Electrothermal Equipment, Moscow 109052 (U.S.S.R.) (Accepted April 30, 1983)
Thermal conductivity is one of the main thermal characteristics of materials. It is rather difficult, however, to measure it at high temperatures. There are no theoretical or experimental dependences either which could predict with a sufficient accuracy the thermal conductivity of the most important technical materials [1]. In the present note a simple mathematical relation is suggested that allows the lattice thermal conductivity to be estimated at a given absolute temperature T in terms of the thermal conductivity K0 at some temperature To, at which it can be measured quite easily (e.g. r o o m temperature), and in terms of the special characteristic temperature 0 used in ref. 2 to describe the temperature dependences of the heat capacities of elements (and/ or inorganic compounds). The mathematical expression representing the temperature dependence of thermal conductivity is
f(O/T) -
~Co f(O/To)
-- ~ 1 + 3 1 + (O/T)a¢21
(1)
(2)
Thus, for a given material, ~ is proportional to f(O/T). The dependence f(O/T), as calculated from eqn. (2), is plotted in Fig. 1 (full line). The symbols represent the values of f(O/T) for different materials obtained from eqn. (1) using experimental values of ~0 and To and the function f(O/To) found from eqn. (2). The values of ~o, To and 0 are listed in 0025-5416/83/$3.00
3Rn
{
(OIT)~I2
t
C - { 1 + (01T)3/2} 2 1 + 3 1 + (O/T) 3/2 J
H=
3RnT
(3) (4)
{1 + (01T)3/2} 2
where n is the number of atoms in a molecule and R is the universal gas constant.
TABLE 1
where
f
Table 1. It can be seen that the values of f(O/T) calculated from eqns. (1) and (2) for various substances at different ~o and To values coincide within the experimental error of the thermal conductivity; this proves the validity of eqn. (1). Equation (1) is valid for pure dense {nonporous) dielectrics and semiconductors (with a low concentration of free charge carriers) in the high temperature range in which the thermal conductivity decreases as the temperature increases. This equation holds for both single crystals and polycrystal samples; for anisotropic single crystals the average values are taken. Equation (1) has a clear physical interpretation. As shown in ref. 2, the specific heat C and the enthalpy H (per mole or per kilomole) can be expressed in terms of O/T as
The values of 0, T Oand K0 for s o m e m a t e r i a l s Ma~rml
0(K)
To(K )
KO[3,4 ]
Ge (single crystal) Si (single crystal) MgO CaO BeO ZnO NaC1 (single crystal) SiO 2 (a-quartz, single crystal) ThO 2 UO 2 a-A1203
123 180 230 180 360 180 88 330
150 150 373 373 378 321 195 195
1.54 4.20 36 13.95 212 29.4 10.3 26.5
200 165 330
373 473 298
10.3 7.96 40
© Elsevier Sequoia/Printed in The Netherlands
L2
e
vV
~0 ~
0
0o ~ i
AO o i
2
3
T
o
i
,
¢
4
5
6
T
Fig. 1. Plot of f(O/T) vs. T/O as calculated from eqn. (2) () and eqn. (1) (o, A1203; (~, germanium (single crystal); o, silicon (single crystal); A, MgO; +, CaO;--o---, BeO; ×, ZnO; [], UO2; 0, NaC1 (single crystal) ; v, SiO 2 (s-quartz, single crystal); ~, ThO2). The experimental data on g and K0 were taken from refs. 2 and 3.
It can easily be seen from eqns. (3) and (4) that eqn. (1) follows from a more general relation K
C/Co
~o
H/Ho
{5)
where Co and H0 are the values of C and H at T = T o. Equations (1) and (5) imply a new experimental physical characteristic: at high temperatures the ratio of the lattice thermal conductivities corresponding to two temperatures is equal to the ratio of the specific heats divided by the ratio of enthalpies taken at these temperatures.
It should be mentioned in conclusion that eqns. {1)-(5) cannot be derived in terms of previous theories [1]. REFERENCES 1 R. Berman, Thermal Conductivity of Solids, Mir, Moscow, 1979, pp. 27, 70-104. 2 S. V. Tsivinsky, Elektrotekh. Prom.-st', Set. Elektroterm., (8) (1981) 1. 3 R. E. Krzizanowsky and Z. Yu. Stern, Thermo-
physical Properties of Non-metal Materials, Energia, Leningrad, 1973. 4 A. A. Bzhetanov, V. S. Bondarenko, N. V. Perelomova, F. I. Strizhevskaya, V. V. Chkalova and M. P. Shaskol'skaya, Acoustic Crystals, Nauka, Moscow, 1982.
L1
Letter
On the temperature dependence of the lattice thermal conductivity
S. V. TSIVINSKY All-Union Institute o f Electrothermal Equipment, Moscow 109052 (U.S.S.R.) (Accepted April 30, 1983)
Thermal conductivity is one of the main thermal characteristics of materials. It is rather difficult, however, to measure it at high temperatures. There are no theoretical or experimental dependences either which could predict with a sufficient accuracy the thermal conductivity of the most important technical materials [1]. In the present note a simple mathematical relation is suggested that allows the lattice thermal conductivity to be estimated at a given absolute temperature T in terms of the thermal conductivity K0 at some temperature To, at which it can be measured quite easily (e.g. r o o m temperature), and in terms of the special characteristic temperature 0 used in ref. 2 to describe the temperature dependences of the heat capacities of elements (and/ or inorganic compounds). The mathematical expression representing the temperature dependence of thermal conductivity is
f(O/T) -
~Co f(O/To)
-- ~ 1 + 3 1 + (O/T)a¢21
(1)
(2)
Thus, for a given material, ~ is proportional to f(O/T). The dependence f(O/T), as calculated from eqn. (2), is plotted in Fig. 1 (full line). The symbols represent the values of f(O/T) for different materials obtained from eqn. (1) using experimental values of ~0 and To and the function f(O/To) found from eqn. (2). The values of ~o, To and 0 are listed in 0025-5416/83/$3.00
3Rn
{
(OIT)~I2
t
C - { 1 + (01T)3/2} 2 1 + 3 1 + (O/T) 3/2 J
H=
3RnT
(3) (4)
{1 + (01T)3/2} 2
where n is the number of atoms in a molecule and R is the universal gas constant.
TABLE 1
where
f
Table 1. It can be seen that the values of f(O/T) calculated from eqns. (1) and (2) for various substances at different ~o and To values coincide within the experimental error of the thermal conductivity; this proves the validity of eqn. (1). Equation (1) is valid for pure dense {nonporous) dielectrics and semiconductors (with a low concentration of free charge carriers) in the high temperature range in which the thermal conductivity decreases as the temperature increases. This equation holds for both single crystals and polycrystal samples; for anisotropic single crystals the average values are taken. Equation (1) has a clear physical interpretation. As shown in ref. 2, the specific heat C and the enthalpy H (per mole or per kilomole) can be expressed in terms of O/T as
The values of 0, T Oand K0 for s o m e m a t e r i a l s Ma~rml
0(K)
To(K )
KO[3,4 ]
Ge (single crystal) Si (single crystal) MgO CaO BeO ZnO NaC1 (single crystal) SiO 2 (a-quartz, single crystal) ThO 2 UO 2 a-A1203
123 180 230 180 360 180 88 330
150 150 373 373 378 321 195 195
1.54 4.20 36 13.95 212 29.4 10.3 26.5
200 165 330
373 473 298
10.3 7.96 40
© Elsevier Sequoia/Printed in The Netherlands
L2
e
vV
~0 ~
0
0o ~ i
AO o i
2
3
T
o
i
,
¢
4
5
6
T
Fig. 1. Plot of f(O/T) vs. T/O as calculated from eqn. (2) () and eqn. (1) (o, A1203; (~, germanium (single crystal); o, silicon (single crystal); A, MgO; +, CaO;--o---, BeO; ×, ZnO; [], UO2; 0, NaC1 (single crystal) ; v, SiO 2 (s-quartz, single crystal); ~, ThO2). The experimental data on g and K0 were taken from refs. 2 and 3.
It can easily be seen from eqns. (3) and (4) that eqn. (1) follows from a more general relation K
C/Co
~o
H/Ho
{5)
where Co and H0 are the values of C and H at T = T o. Equations (1) and (5) imply a new experimental physical characteristic: at high temperatures the ratio of the lattice thermal conductivities corresponding to two temperatures is equal to the ratio of the specific heats divided by the ratio of enthalpies taken at these temperatures.
It should be mentioned in conclusion that eqns. {1)-(5) cannot be derived in terms of previous theories [1]. REFERENCES 1 R. Berman, Thermal Conductivity of Solids, Mir, Moscow, 1979, pp. 27, 70-104. 2 S. V. Tsivinsky, Elektrotekh. Prom.-st', Set. Elektroterm., (8) (1981) 1. 3 R. E. Krzizanowsky and Z. Yu. Stern, Thermo-
physical Properties of Non-metal Materials, Energia, Leningrad, 1973. 4 A. A. Bzhetanov, V. S. Bondarenko, N. V. Perelomova, F. I. Strizhevskaya, V. V. Chkalova and M. P. Shaskol'skaya, Acoustic Crystals, Nauka, Moscow, 1982.