Thermal Conduction

(or heat conduction), one of the ways in which heat, that is, the energy of thermal motion of mi-croparticles, is transferred from hotter regions of a substance to less hot regions so as to equalize the temperatures of the regions.

Thermal conduction involves the transfer of energy directly from particles—such as molecules, atoms, or electrons—with high energies to particles with lower energies. If the relative change in temperature T at the distance of the mean free path Ī of a particle is small, then the Fourier law holds: the heat flux q is proportional to the temperature gradient grad T —that is,

(1) q = –λ gred T

Here, λ is the coefficient of thermal conductivity, or simply the thermal, or heat, conductivity. It is independent of grad T and depends on such factors as the state of aggregation of the substance (see Table 1), the atomic and molecular structure of the substance, the temperature, the pressure, and, in the case of a mixture or solution, the composition of the substance.

Deviations from the Fourier law may occur in the following cases: at high values of grad T, for example, in intense shock waves; at low temperature (for liquid helium He II); and at high temperatures of the order of tens or hundreds of thousands of degrees. In gases at such high temperatures, energy is transferred predominantly not through collisions between atoms but through radiation. In dilute gases in which Ī is comparable with the distance L between the walls bounding the volume of gas, the molecules collide more frequently with the walls than with each other. The conditions for the applicability of the Fourier law are violated here, and the very concept of a local gas temperature becomes meaningless. In this case, heat transfer between bodies in the gaseous medium is considered, rather than the process of thermal conduction in the gas. Conductive heat transfer in a continuous medium is described by the heat equation.

For an ideal gas consisting of rigid spherical molecules of diameter d, the following expression holds, according to the kinetic theory of gases, for λ when d « l « L:

where ρ is the density of the gas, c_v is the specific heat of the gas at constant volume V, and v is the mean velocity of the molecules. Since Ī is proportional to 1/p and ρ ~ p, where ρ is the gas pressure, the thermal conductivity of such a gas does not depend on the pressure. Moreover, λ and the viscosity η are related by the following equation:

In the case of a gas consisting of polyatomic molecules, an important contribution to λ is made by the internal degrees of freedom of the molecules. This fact is taken into account by the equation

λ = ηc_v[(9γ – 5)/4]

where γ = c_p/c_v (c _p is the specific heat at constant pressure). In the case of real gases, the thermal conductivity is a rather complicated function of temperature and pressure and increases with increasing T and p. The value of λ for a gaseous mixture may be larger or smaller than that for the components of the mixture—that is, the thermal conductivity is a nonlinear function of composition.

In dense gases and liquids, the mean distance between the molecules is comparable with the dimensions of the molecules, and the kinetic energy of the molecules is of the same order as the intermolecular potential. Thus, the transfer of energy through collisions is much more intense than in dilute gases, and the rate of transfer of molecular energy from hot isothermal fluid layers to cooler layers is close to the rate of propagation of small pressure disturbances. Since this rate of propagation is equal to the speed of sound, we have λ = ρc_pu_s L̄ where u_s is the speed of sound in the fluid and L̄ is the mean distance between the molecules. This equation is best satisfied by monatomic liquids. The value of λ for liquids generally decreases with increasing T and slowly increases with increasing p.

The nature of thermal conduction in solids depends on the type of solid. In dielectrics, which lack free electric charges, the energy of thermal motion is transported by the quasiparticles known as phonons, which are quanta of the elastic vibrations of the atoms of a crystal (seeLATTICE_VIBRATIONS and QUASIPARTICLES). In solid dielectrics, λ ≈ cv̄l̄. Here, c is the specific heat of the dielectric, which is the same as the specific heat of the phonon gas; v̄ is the mean velocity of the phonons, which is approximately equal to the speed of sound, and l̄ is the mean free path for the phonons. The existence of a definite finite value of l̄ is a consequence of the scattering of phonons by phonons and by lattice imperfections, particularly grain boundaries and the boundary of the specimen. The temperature dependence of λ is determined by the temperature dependence of c and l̄. At high temperatures (T » θ_D, where θ_D is the Debye temperature), the phonon-phonon scattering associated with the anharmonicity of the vibrations of the atoms in the crystal is the principal mechanism limiting l̄. The phonon-phonon mechanism of thermal resistance (1/λ is known as the thermal resistivity) is accounted for by Umklapp processes, as a result of which the phonon flux is slowed down. The greater the value of T, the lower the value of l̄ and the greater the probability of the occurrence of Umklapp processes. When T » θ_D, Ī ~ 1/T, and, consequently, λ ~ 1/T, since under these conditions c depends weakly on T. As T decreases (T « θ_D), the mean free path determined by phonon-phonon scattering markedly increases [l̄ ~ exp (θ_D /T)] and is generally limited by the size R of the specimen. When T « θ_D, the specific heat decreases approximately as T³. Consequently, λ passes through a maximum with decreasing temperature. The temperature at which maximum λ occurs is determined by the approximate equality l̄(T) ≈ R.

In metals, thermal conduction is a result primarily of the motion and interaction of current carriers, that is, conduction electrons. In the general case, the thermal conductivity of a metal is equal to the sum of a lattice phonon component λ_lat, and an electronic component λ_elec: λ = λ_lat + λ_elec. Moreover, at ordinary temperatures generally λ_elec » λ_lat. In the thermal conduction process, each electron transports, when a temperature gradient is present, the quantity of energy kT, where k is the Boltzmann constant. Consequently, the Wiedemann-Franz law holds over a broad range of temperatures. It states that the ratio of the electronic component λ_elec of the thermal conductivity to the electrical conductivity a is proportional to the temperature:

where e is the electronic charge. Since in most metals λ_lat « λ_elec, λ_elec can be replaced by λ in the Wiedemann-Franz law with a high degree of accuracy. Observed deviations from (3) have been explained by the inelasticity of electron collisions. In the case of the semimetals Bi and Sb, λ_lat is comparable with λ_dec because of the small number of free electrons.

The phenomenon of heat transfer in semiconductors is more complicated than in dielectrics and metals for two reasons: first, both components of the thermal conductivity, λ_elec and λ_lat, are important in semiconductors; second, the thermal conductivity of semiconductors is greatly affected by impurities, bipolar diffusion processes, exciton transfer, and other factors.

The influence of pressure on the λ of solids is expressed with good accuracy by a linear dependence of λ on p. In many metals and minerals, λ increases with increasing. p