Phase Transitions
Since states of ensembles of a vast number of molecules are specified as thermodynamic phases, phase transitions are central in issues concerning the change in materials properties. This chapter summarizes the thermodynamic aspects of phase transitions, i
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Phase Transitions
2.1 Background 2.1.1 Thermodynamic Aspects The fusion of a crystal is, together with boiling/condensation, a representative of phenomena known as phase transitions. It is useful to summarize the terminology and properties of phase transitions in (macroscopic) thermodynamics. Note that the following descriptions assume thermodynamic equilibrium.1 A phase is not only a thermodynamic concept but also a reality. A phase must be uniform chemically and physically if viewed macroscopically, i.e., averaged over a volume that contains a sufficiently large number of molecules. All intensive variables (not only temperature, pressure, and density but also composition) are uniform in a single phase. A pure substance is assumed for a while. Generally, a substance has plural phases like gas, liquid, and solid, which are representative three phases of matter. Although these three phases are easily distinguished from each other under normal conditions, this is not generally true, as indicated by the gas-liquid critical point, where gas and liquid merge to a single state. Thermodynamics says that a phase is characterized by the Gibbs energy (G), a thermodynamic potential relevant to the conditions where temperature (T ) and pressure ( p) are independent variables. Now, we write the molar Gibbs energy of phase H as μH (T, p). Then, a surface in a (T, p, μ) space can be identified as phase H. When a substance has two phases, H and L (Fig. 2.1), two surfaces μH (T, p) and μL (T, p) generally cross with a crossing line. Since the Gibbs energy must be minimum at thermodynamic equilibrium, the stable phase is interchanged upon crossing the crossing line. Thus, the substance must change its form between phase H
1 We
here assume a usual meaning of “equilibrium.” It is seemingly stable and steady against time of our daily time scale. Some class of non-equilibrium and “relaxing” states is treated in Chap. 8.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 K. Saito, Chemical Physics of Molecular Condensed Matter, Lecture Notes in Chemistry 104, https://doi.org/10.1007/978-981-15-9023-8_2
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2 Phase Transitions
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H L p T Fig. 2.1 Phase relation of two phases H (pink surface) and L (green surface) in (T, p, μ) space with Δtrs s > 0 (positive molar entropy of transition) and Δtrs v < 0 (negative molar volume of transition). In this case, H phase is a high-temperature phase and a high-pressure phase with respect to L phase because a phase having lower molar Gibbs energy, μ, is more stable at high temperatures and high pressures
and phase L upon the crossing. This change is a phase transition. On the crossing line, two phases have the same molar Gibbs energy. Both phases have equivalent stability. Namely, on the crossing line, two phases can coexist. The crossing line projected on the (T, p)-plane is identified as a coexistence line of the two phases, accordingly. The projected curve also serves as a phase boundary on the plane. Interestingly, when
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