166
8 Molecular Glasses
However, there exists its limit on the low-frequency side, the existence of which is
practically unavoidable. Namely, any measurements in science on materials are to
be done by a human. Her/his daily time scale practically limits the time to wait for
thermal equilibrium. If the τ becomes sufficiently more prolonged than the daily
time scale (typically, 100 s – 1 h), the system looks macroscopically stable in time,
resembling a system in thermal equilibrium, but undergoing, in reality, a prolonged
approach to equilibrium. Thus, we cannot observe, at sufficiently low temperatures,
the equilibrium state, which is the subject of the third law of thermodynamics. If the
change from the equilibrium state to a non-equilibrium state on cooling accompanies
detectable anomalies in properties of the system, the change is often called a glass
transition. The glass transition has been regarded as one of the most challenging
issues in condensed matter science with a long history [1–3].
The above description of glass transitions based on the comparison of the relaxation time τ of the system and the daily time scale implies that any systems undergo
a glass transition. This expectation is because τ goes to infinity on approaching the
absolute zero according to the Arrhenius law (Eq. 8.21). This naïve outcome is, however, not the case. Essential is a distinction between relevance and irrelevance of the
slowest mode for the system’s property. For example, the reorientation of a benzene
molecule around its sixfold axis is a classic example of molecular motion in the crystal [4]. Its rate is well described by the Arrhenius equation, resulting in excessive
length than the daily time scale around 50 K (τ ≈ 10
3 s). However, the reorientation
is well described as hopping between six equivalent states. Even if the reorientation
ceases, the averaged configuration inside the crystal remains the same. Noticeable
anomalies are not expected in the properties of the crystal, accordingly. Indeed, there
has not been known the presence of glass transitions for crystalline benzene [5]. On
the other hand, if the slowest mode involves the transition between nonequivalent
configurations of the system, the change between equilibrium and non-equilibrium
states would be a glass transition.
8.1.1.4 Thermodynamic Symptoms of Glass Transitions
According to the above definition of glass transitions, they differ from phase transitions in the usual sense of thermodynamics. The glassy state is out of equilibrium
and may differ from state to state, depending on how the relevant degrees of freedom
became frozen-in. The occurrence of a glass transition on cooling depends on the time
scale used in observation, or, more practically, on a cooling rate. For a glass transition
on heating, i.e., for a transition from a non-equilibrium state to the equilibrium state,
not only the time-scale but also the given state of the glass affects behaviors. Since, at
least, a motional mode is practically prohibited below the glass transition, degree(s)
of freedom involved in the mode cannot change their states below the glass transition. This immobility means that the thermodynamic functions related to the first
law of thermodynamics (enthalpy in usual experiments) no longer change their magnitudes even with cooling. Thus, the temperature dependence of enthalpy becomes
weaker, as shown in Fig. 8.3. Namely, the contribution of the degrees of freedom to
8 Molecular Glasses
However, there exists its limit on the low-frequency side, the existence of which is
practically unavoidable. Namely, any measurements in science on materials are to
be done by a human. Her/his daily time scale practically limits the time to wait for
thermal equilibrium. If the τ becomes sufficiently more prolonged than the daily
time scale (typically, 100 s – 1 h), the system looks macroscopically stable in time,
resembling a system in thermal equilibrium, but undergoing, in reality, a prolonged
approach to equilibrium. Thus, we cannot observe, at sufficiently low temperatures,
the equilibrium state, which is the subject of the third law of thermodynamics. If the
change from the equilibrium state to a non-equilibrium state on cooling accompanies
detectable anomalies in properties of the system, the change is often called a glass
transition. The glass transition has been regarded as one of the most challenging
issues in condensed matter science with a long history [1–3].
The above description of glass transitions based on the comparison of the relaxation time τ of the system and the daily time scale implies that any systems undergo
a glass transition. This expectation is because τ goes to infinity on approaching the
absolute zero according to the Arrhenius law (Eq. 8.21). This naïve outcome is, however, not the case. Essential is a distinction between relevance and irrelevance of the
slowest mode for the system’s property. For example, the reorientation of a benzene
molecule around its sixfold axis is a classic example of molecular motion in the crystal [4]. Its rate is well described by the Arrhenius equation, resulting in excessive
length than the daily time scale around 50 K (τ ≈ 10
3 s). However, the reorientation
is well described as hopping between six equivalent states. Even if the reorientation
ceases, the averaged configuration inside the crystal remains the same. Noticeable
anomalies are not expected in the properties of the crystal, accordingly. Indeed, there
has not been known the presence of glass transitions for crystalline benzene [5]. On
the other hand, if the slowest mode involves the transition between nonequivalent
configurations of the system, the change between equilibrium and non-equilibrium
states would be a glass transition.
8.1.1.4 Thermodynamic Symptoms of Glass Transitions
According to the above definition of glass transitions, they differ from phase transitions in the usual sense of thermodynamics. The glassy state is out of equilibrium
and may differ from state to state, depending on how the relevant degrees of freedom
became frozen-in. The occurrence of a glass transition on cooling depends on the time
scale used in observation, or, more practically, on a cooling rate. For a glass transition
on heating, i.e., for a transition from a non-equilibrium state to the equilibrium state,
not only the time-scale but also the given state of the glass affects behaviors. Since, at
least, a motional mode is practically prohibited below the glass transition, degree(s)
of freedom involved in the mode cannot change their states below the glass transition. This immobility means that the thermodynamic functions related to the first
law of thermodynamics (enthalpy in usual experiments) no longer change their magnitudes even with cooling. Thus, the temperature dependence of enthalpy becomes
weaker, as shown in Fig. 8.3. Namely, the contribution of the degrees of freedom to