Thermal expansion of solids: review on theories
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Thermal expansion of solids: review on theories V. A. Drebushchak1,2 Received: 31 August 2019 / Accepted: 20 January 2020 © Akadémiai Kiadó, Budapest, Hungary 2020
Abstract The coefficient of thermal expansion of a solid can be derived from (1) anharmonicity of atomic vibrations; (2) lattice dynamics; (3) equation of state by G. Mie; (4) equation of state by E. Grüneisen; and (6) potential of interatomic interaction. Only the last theory in this list provides us with the equation describing correctly all features in the thermal expansion: (1) proportionality between thermal expansion and heat capacity; (2) various values of “plateau” for the coefficient of thermal expansion at temperatures close to Debye temperature; and (3) acceleration of the thermal expansion in the vicinity of melting point. Keywords Anharmonicity · Gruneisen · Heat capacity · Interatomic potential · Thermal expansion
Empirical thermal expansion of solids as a function of temperature Here, we remind basic definitions and information on the thermal expansion/expansivity of solids. Linear thermal expansion coefficient is
𝛼=
d ln l 1 dl = , l dT dT
(1)
where l is a linear size of a sample measured and T is temperature. Volume thermal expansion coefficient is
𝛽=
d ln V 1 dV = , V dT dT
(2)
where V is the volume. For isotropic solids, β = 3α. Experimental data on thermal expansion of solids provide us with very wide diversity in functional relations β(T) for different samples in various temperature ranges: increasing, decreasing, positive and negative values of β, smooth and sharp (peak) changes. Here, we will discuss only those features of thermal expansion that are typical of most solids, or, in other words, the regular temperature function.
* V. A. Drebushchak [email protected] 1
V.S. Sobolev Institute of Geology and Mineralogy, SB RAS, Pr. Ak. Koptyuga, 3, Novosibirsk, Russia 630090
Novosibirsk State University, Pirogova, 2, Novosibirsk, Russia 630090
2
Suitable example of such an introductory picture is shown in Fig. 1, borrowed from Ref. [1]. Thermal expansion behaves very similar to heat capacity. It starts from zero at zero temperature and then increases according to the temperature function of heat capacity, often aT3 or γT + aT3. Then, the fast increase ceases and finally tends to increase slowly near a level (“plateau”) different for different samples. In Fig. 1, the level is about 3·10−5 K−1 for titanium, 7·10−5 K−1 for aluminum, 13·10−5 K−1 for sodium chloride, and 21·10−5 K−1 for sodium. The heat capacities of the substances in Fig. 1 are shown in Fig. 2 (data from [2, 3] for Na, [4] for NaCl, [5] for Al, [6] for Ti). Comparison between Figs. 1 and 2 makes it clear very important difference between thermal expansion and heat capacity. Experimental heat capacities in Fig. 2 finish their growth somewhere between 25 and 30 J g-at.−1 K−1. Theories predict that the isochoric heat capacity of solids goes asymptotically to the upper limit of 3R (≈ 25 J g-at.−1 K−1), if without phase transitions and/or additional contribu
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