Читать книгу Materials for Biomedical Engineering - Mohamed N. Rahaman - Страница 155

Whether a material can conduct heat or not is quantified by its thermal conductivity κ, defined by the equation

(4.45)

where, q is the heat flux, that is, the amount of heat flowing across unit area per unit time, and dT/dx is the temperature gradient down the material. Equation (4.45) is valid for the steady‐state condition in which the heat flux is independent of time and the minus sign signifies that the heat flow is down the temperature gradient, that is, from high to low T. As q has units of W/m², the units of κ are W/m/K.

Transport of heat in solids occurs by flow of electrons and by vibration of atoms (or ions) in the crystal lattice. Thus, we can write

(4.46)

where, κ_e and κ_l are the contributions to the thermal conductivity due to the electrons and lattice vibrations, respectively. Depending on the material, one or the other conduction mechanism often dominates. In pure metals with a high degree of metallic bonding, such as copper, thermal conduction occurs predominantly by the sea of nearly free electrons that surround the metal ions. Because of the large number of electrons and ease with which they flow, metals are good thermal conductors, that is, they have a high thermal conductivity. As the flow of electrons is responsible for both thermal and electrical conduction, metals are both good electrical conductors and good thermal conductors, a relationship quantified by the Weideman–Franz law given by

(4.47)

where, σ ′ is the electrical conductivity, T is the temperature and L is a constant equal to 2.44 × 10⁻⁸ WΩ/K².

Thermal conduction by lattice vibrations dominates in ceramics as these materials contain few free electrons. An atom vibrating about its equilibrium position due to its thermal energy can be thought of as a sphere supported by springs (Figure 4.22a). However, the atoms in a crystal lattice are bonded together and, thus, they do not vibrate independently of one another. Instead, the oscillations are connected which leads to a wave behavior (Figure 4.22b). Using the wave‐particle duality of quantum mechanics, the waves associated with atomic oscillations are quantized and the quantum of energy or “particle” associated with these lattice vibrations is called a phonon. A phonon represents a quantum of atomic lattice vibration, analogous to a photon that is a quantum of electromagnetic radiation (light). Phonons can show properties such as scattering, for example, that are typical of particles.

Figure 4.22 (a) Vibration of a sphere connected by springs. (b) The atomic vibrations in a crystal lattice are connected which leads to a wave behavior.

As phonons are not as efficient as electrons in conducting heat energy due to scattering by lattice imperfections, ceramics typically have a lower thermal conductivity than metals (Figure 4.23). Glasses typically have a lower thermal conductivity than ceramics because phonon scattering becomes more effective with a more disordered atomic structure. Ceramics and glasses have a low thermal conductivity and a low electrical conductivity. On the other hand, diamond has a low electrical conductivity and a high thermal conductivity, higher than that of metals. The low electrical conductivity of diamond is due to its strong covalent bonding (no free electrons). In comparison, the high thermal conductivity is due to two factors, the highly ordered crystal structure and the strong atomic bonding that strongly limits the incorporation of impurities into its structure. These two factors severely limit phonon scattering in the crystal lattice.

Figure 4.23 Chart showing the thermal conductivity values for a variety of materials.

In polymers, heat conduction occurs by vibration and rotation of the long chain molecules. Polymers have a low thermal conductivity, although, for a semicrystalline polymer, the conductivity increases with increasing volume fraction of crystalline regions because phonon scattering in the crystalline region is lower than that in an amorphous region of the same composition.