34
2 Phase Transitions
There exists another classification scheme of phase transitions. Landau [2] divided
phase transitions into two kinds: first kind and second kind, the latter of which is a
continuous transition between closely related phases. By expanding the difference in
thermodynamic potentials between them in terms of the so-called order parameter,
which is finite in one phase but null in the other phase, he derived some general
conclusions for phase transitions from the thermodynamic point of view [2, 3]. All
discontinuous (first-order after Ehrenfest) phase transition is classified as of the first
kind. Note that some phase transitions of the first kind are discussed within Landau’s
thermodynamic phenomenology, as will be discussed in Sect. 2.2.
Note that Fig. 2.1 assumes that two phases are “reasonably” stable at any combinations of (T, p). The true equilibrium corresponds to the state that the compound
is solely in the phase of a lower molar Gibbs energy. This phase is the (thermodynamically) stable phase. The other phase of a higher molar Gibbs energy is called a
metastable phase. It is easy to identify which of two phases are stable and metastable
if a phase transition (from one to another) occurs. This is, however, not always the
case. A textbook example is the case of elemental carbon. Around the ambient condition, its stable phase is graphite, and all of diamond, any fullerenes, and even
graphene are metastable. However, we never imagine the spontaneous transition of
the diamond to graphite! This example demonstrates the fact that we cannot know
the metastability only by simple observations.
Figure 2.1 also implies another point, which is practically essential. Since both
phases are seemingly stable around the phase boundary, overshooting such as superheating and supercooling can take place in this first-order transition. It is difficult to
construct the logic that guarantees the loss of the stability of the phase on the phase
boundary. We thus conclude that the possibility of the overshooting is intrinsic to
any first-order transitions. In turn, the observation of overshooting is regarded as the
most decisive way to assure the first-order nature of phase transitions, while assuming
that the experiment correctly watches equilibrium states. In contrast, overshooting
is impossible for higher-order transitions.
2.1.2 Entropy and Boltzmann’s Principle
Entropy plays a central role in the thermodynamic study of phase transitions, as
partly exemplified in Eq. 2.3.
In classical thermodynamics, entropy is a state function that indicates the possibility of a change of thermodynamic state of a system (process) by mechanical
means. Namely, the process is impossible if the entropy of the system is smaller after
it than that before it, whereas it is possible if the opposite holds. This description is
a form of the second law of thermodynamics.
Boltzmann constructed the statistical mechanics to reproduce the classical thermodynamics relying solely on the assumption on entropy S,
S = k B ln W,
(2.8)
2 Phase Transitions
There exists another classification scheme of phase transitions. Landau [2] divided
phase transitions into two kinds: first kind and second kind, the latter of which is a
continuous transition between closely related phases. By expanding the difference in
thermodynamic potentials between them in terms of the so-called order parameter,
which is finite in one phase but null in the other phase, he derived some general
conclusions for phase transitions from the thermodynamic point of view [2, 3]. All
discontinuous (first-order after Ehrenfest) phase transition is classified as of the first
kind. Note that some phase transitions of the first kind are discussed within Landau’s
thermodynamic phenomenology, as will be discussed in Sect. 2.2.
Note that Fig. 2.1 assumes that two phases are “reasonably” stable at any combinations of (T, p). The true equilibrium corresponds to the state that the compound
is solely in the phase of a lower molar Gibbs energy. This phase is the (thermodynamically) stable phase. The other phase of a higher molar Gibbs energy is called a
metastable phase. It is easy to identify which of two phases are stable and metastable
if a phase transition (from one to another) occurs. This is, however, not always the
case. A textbook example is the case of elemental carbon. Around the ambient condition, its stable phase is graphite, and all of diamond, any fullerenes, and even
graphene are metastable. However, we never imagine the spontaneous transition of
the diamond to graphite! This example demonstrates the fact that we cannot know
the metastability only by simple observations.
Figure 2.1 also implies another point, which is practically essential. Since both
phases are seemingly stable around the phase boundary, overshooting such as superheating and supercooling can take place in this first-order transition. It is difficult to
construct the logic that guarantees the loss of the stability of the phase on the phase
boundary. We thus conclude that the possibility of the overshooting is intrinsic to
any first-order transitions. In turn, the observation of overshooting is regarded as the
most decisive way to assure the first-order nature of phase transitions, while assuming
that the experiment correctly watches equilibrium states. In contrast, overshooting
is impossible for higher-order transitions.
2.1.2 Entropy and Boltzmann’s Principle
Entropy plays a central role in the thermodynamic study of phase transitions, as
partly exemplified in Eq. 2.3.
In classical thermodynamics, entropy is a state function that indicates the possibility of a change of thermodynamic state of a system (process) by mechanical
means. Namely, the process is impossible if the entropy of the system is smaller after
it than that before it, whereas it is possible if the opposite holds. This description is
a form of the second law of thermodynamics.
Boltzmann constructed the statistical mechanics to reproduce the classical thermodynamics relying solely on the assumption on entropy S,
S = k B ln W,
(2.8)