## Thermal expansion

The majority of matters increase their volumes with increasing T, which is called thermal expansion [Wiki]. The thermal expansivity (or thermal expansion coefficient), α, which describes the degree of thermal expansion of a matter for a given T increase, is defined by the following equation:

(1)

Although increase in P always decreases V, as Le Chatelier's principle [Wiki] suggests, there is no such necessity for thermal expansion. One of well-known phenomena against thermal expansion is water at T between 273 and 277 K, where V of water decreases with T.

Thermal expansion is caused by asymmetry of an atomic potential as shown in Fig. 1. What the blue curve means is that, if an atom approaches to a neighboring atom, it receives a strong repulsion force, whereas it receives relatively weak attractive force even if the distance between these two atoms increases. To the first approximation, a symmetric atomic potential can be approximated by a quadratic function, which is called the harmonic potential. On the other hand, an asymmetric potential is called the anharmonic potential [Wiki].

Fig. 1. Explanation of thermal expansion in view of the anharmonic atomic potential.

At nearly zero temperature (T0 in Fig. 1), an atom is located at the bottom of the atomic potential (“Zero position” in Fig. 1). With increasing T from T0 to T1, T2, …, the kinetic energy of atomic motion increases with increasing from E0 to E1, E2, … . Because of the asymmetric potential, the average atomic position is shifted to the direction of a longer atomic distance at high T, which is thermal expansion. Because of the atomic potential contain higher terms, the rate of the atomic distance increase with T increases with increasing T.

As in discussed in a separate page, lattice thermal resistance, which is inversely proportional to thermal conductivity [Wiki] or thermal diffusivity [Wiki], is also caused by anharmonicity of an atomic potential. For this reason, a matter with larger thermal expansivity has smaller lattice thermal conductivity and diffusivity.

Let us discuss P dependence of α (Fig. 2). By pressurizing, the atomic potential is distorted so that the potential minimum is located at a shorter atomic distance (blue curve). Because of the narrowing of the atomic potential, the rate of the atomic distance increase with T decreases by compression. Namely, α decreases with increasing P. The decrease in α with pressure is expressed by the Anderson Grüneisen parameter, δT. In the case of mantle minerals, α decreases in proportion to 5.5 ± 0.5 power of V.

Fig. 2. Explanation of decrease in α with P.