8.2 Properties of Glasses
173
is often discussed as a measure of the temperature-dependent degree of the disorder.
This quantity is called the configurational entropy in the research field of glasses [29].
Here, C glass indicates both heat capacities of the glass below T g and the supercooled
liquid. By definition, S res = S conf (0) assuming the third law of thermodynamics for
the crystal.
8.2.4 Effects of Structural Inhomogeneity
Irrespective of a kind of the frozen-in degree(s) of freedom, glasses intrinsically
possess, to some extent, structural disorder, which may be difficult to characterize
quantitatively beyond the residual entropy. The structural disorder inevitably leads
to the inhomogeneity in its structure. If the degree of the disorder is weak, and only a
limited kind of disorder can exist, it can be regarded as a point defect. However, such
a situation is difficult to imagine in real glasses. On the contrary, we should consider
situations where an extent affected by each “defect” overlaps with that of others.
In this case, effects on each molecule vary from molecule to molecule, resulting in
the nearly continuous distribution of the state of surroundings. Let consider such
situations.
Although liquid-quenched glasses are plausibly more isotropic than crystalline
solids, for which the Debye T
3 –law of heat capacity had been established, the violation of the law has been reported for LQG. Heat capacities of nonmetallic
1 glasses
at low temperatures roughly obey the law, C = αT
3
+ βT [30–32]. Since the cubic
term of the heat capacity of solids originates in the dispersion relation of the sound
(acoustic) waves (see Chap. 5), the heat capacity of glasses ought to have the cubic
term. The primary issue is thus to understand the origin of the linear term. Anderson [33] and Phillips [34] independently explained the presence of the linear term
by assuming a continuous distribution of level splitting of tunneling states. The
non-vanishing density of states at the null excitation energy, n(0), leads the linear
term with β ∝ n(0). Further, their models that are mostly the same to each other
explain well the temperature dependence of the acoustic attenuation in disordered
solids. Later, the density of states excess to the Debye one has been well established
below ca. 1 meV. The peak in the density of states (or equivalently in C p T
−3 ) is
often called the boson peak. Excitations responsible for the peak in the density of
states are believed to be involved in anomalous physical properties of glasses. Some
models predict the presence of the boson peak while assuming the inhomogeneous
distribution of structural disorders [35, 36].
It is noteworthy that all of the studies on the anomalous properties of glasses
described in the previous paragraph implicitly assume the universality of the anomalies. However, there is an example that would imply the dependence of the anomalous
property on the degree and/or type of frozen-in disorder [37]. The heat capacity of
1 Conduction electrons well approximated by the Fermi gas also contribute linearly on the temperature to heat capacity, irrespective of crystalline or amorphous states.
173
is often discussed as a measure of the temperature-dependent degree of the disorder.
This quantity is called the configurational entropy in the research field of glasses [29].
Here, C glass indicates both heat capacities of the glass below T g and the supercooled
liquid. By definition, S res = S conf (0) assuming the third law of thermodynamics for
the crystal.
8.2.4 Effects of Structural Inhomogeneity
Irrespective of a kind of the frozen-in degree(s) of freedom, glasses intrinsically
possess, to some extent, structural disorder, which may be difficult to characterize
quantitatively beyond the residual entropy. The structural disorder inevitably leads
to the inhomogeneity in its structure. If the degree of the disorder is weak, and only a
limited kind of disorder can exist, it can be regarded as a point defect. However, such
a situation is difficult to imagine in real glasses. On the contrary, we should consider
situations where an extent affected by each “defect” overlaps with that of others.
In this case, effects on each molecule vary from molecule to molecule, resulting in
the nearly continuous distribution of the state of surroundings. Let consider such
situations.
Although liquid-quenched glasses are plausibly more isotropic than crystalline
solids, for which the Debye T
3 –law of heat capacity had been established, the violation of the law has been reported for LQG. Heat capacities of nonmetallic
1 glasses
at low temperatures roughly obey the law, C = αT
3
+ βT [30–32]. Since the cubic
term of the heat capacity of solids originates in the dispersion relation of the sound
(acoustic) waves (see Chap. 5), the heat capacity of glasses ought to have the cubic
term. The primary issue is thus to understand the origin of the linear term. Anderson [33] and Phillips [34] independently explained the presence of the linear term
by assuming a continuous distribution of level splitting of tunneling states. The
non-vanishing density of states at the null excitation energy, n(0), leads the linear
term with β ∝ n(0). Further, their models that are mostly the same to each other
explain well the temperature dependence of the acoustic attenuation in disordered
solids. Later, the density of states excess to the Debye one has been well established
below ca. 1 meV. The peak in the density of states (or equivalently in C p T
−3 ) is
often called the boson peak. Excitations responsible for the peak in the density of
states are believed to be involved in anomalous physical properties of glasses. Some
models predict the presence of the boson peak while assuming the inhomogeneous
distribution of structural disorders [35, 36].
It is noteworthy that all of the studies on the anomalous properties of glasses
described in the previous paragraph implicitly assume the universality of the anomalies. However, there is an example that would imply the dependence of the anomalous
property on the degree and/or type of frozen-in disorder [37]. The heat capacity of
1 Conduction electrons well approximated by the Fermi gas also contribute linearly on the temperature to heat capacity, irrespective of crystalline or amorphous states.