Thermodynamic Predictions of Thermal Expansivity and Elastic Compliances at High Temperatures and Pressures Applied to P
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QUILIBRIUM thermodynamics of elastic solids is a very old subject that has had a scientific basis even well before the work of Gibbs.[1] The work presented in this paper includes some of the advanced thermodynamic concepts of stressed solids from J. C. M. Li’s, Institute of Metals, Mehl Lecture[2] when he was the Hopeman Chaired Professor at the University of Rochester. This paper will use the concept of strain-volume as the extensive, tensorial state variable in the thermodynamic analysis presented. Strain-volume thermodynamics is applied to an abundant, important Earth mineral (Mg, Fe)SiO3 with the perovskite crystal structure.[3] The connected changes in physical, mechanical, elastic properties at high temperatures and pressures are compared to the available crystallographic experimental data. Thermodynamic assumptions based on energy per unit volume and properties found from density functional theory or simple internal vibrational energies are also used for comparison to the strain-volume thermodynamics that is developed here. The internal energy, u, in thermodynamic systems was first found in solids by Einstein[4] in his celebrated work
S.J. BURNS, Professor, is with the Materials Science Program, Department of Mechanical Engineering, University of Rochester, Rochester, NY 14627. Contact e-mail: [email protected] Manuscript submitted November 1, 2015. METALLURGICAL AND MATERIALS TRANSACTIONS A
of 1907 and he further developed this concept in Reference 5. He was able to show that quantized, harmonic oscillators described the constant volume internal energy of solids. Thus, the internal vibrations of a solid are intimately related to the heat capacity of crystalline materials. He used the measured temperature dependence of the constant pressure heat capacity of diamond to compare to his theoretical expressions with excellent comparative results; a characteristic temperature, hE was specified in the comparison of theory to experimental values. Figure 1 is the heat capacity data from Reference 4. Debye[6,7] changed the conditions of the quantum ground state so all oscillators were not all at the same frequency and found a constant volume heat capacity expression with a characteristic, low temperature T3 signature. Debye[6] compared his theoretical expression to the measured temperature dependence of Cu and Al, he also calculated the constant volume heat capacity. The comparison to experimental data was also excellent provided a characteristic temperature, hD is correctly chosen. Both Einstein and Debye used the internal energy at constant volume to find their heat capacities; their measurements are from data collected on an atmospheric isobar, even though both their theories are for isometric conditions. It is straight forward thermodynamically to convert from constant pressure to constant volume heat capacities but since these differences are often typically small no corrections are applied. The state variables are not the same when the temperature is increased. Thus, the solids measured
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