represents the energy differences between the minima of the potential energy wells that are explored as temperature increases (Figure 18).
The glass transition can thus be viewed as the point from which atoms begin to explore positions characterized by higher potential energies. Regardless of the complexity of this process at a microscopic level, this spreading of configurations over states of higher and higher potential energy is the main feature of atomic mobility. As a consequence, configurational heat capacities are positive. This feature, in turn, is consistent with the fact that any configurational change must cause an entropy rise when the temperature increases as required by Le Chatelier principle. As for relaxation times, they decrease with rising temperatures because large thermal energies allow potential energy barriers to be overcome more easily.
Another general feature of interatomic potentials is their anharmonic nature: displacements of the vibrating atoms from their equilibrium positions are not strictly proportional to the forces exerted on them. Because increasing vibrational amplitudes result in increasing interatomic distances (Figure 18), the thermal expansion coefficient is generally positive for glasses. In the liquid, it increases markedly when even greater interatomic distances result from configurational changes.
2.4.4 Compressibility and Permanent Compaction
An important difference between crystals and liquids concerns the effects of pressure on their structures. The former are stable as long as the variations in their bond angles and distances induced remain consistent with their long‐range symmetry. A transition to a new phase takes place when this constraint is no longer respected. In contrast, the lack of long‐range order makes a wide diversity of densification mechanisms possible in a liquid, whose structure thus keeps constantly adjusting to varying pressures through changes in short‐range order characterized by shorter equilibrium distances and steeper slopes around the minima pictured in Figure 18. The compressibility is thus greater for a liquid than for its isochemical crystal. It is also made up of vibrational and configurational contributions. Because the shape of interatomic potentials determines the vibrational energy levels, compression is termed vibrational for the elastic part of the deformation. As for the configurational contribution, it is related to the aforementioned changes in the potential energy wells.
If a liquid is quenched as a glass at high pressure, the final glass recovered after decompression will be denser than its counterpart formed at room pressure because only the vibrational part of the compression is eventually recovered (Figure 19). But permanent densification can also be achieved at room temperature through compression of a glass at a few tens of kbar (Chapter 10.11). The effects of pressure and temperature on the properties of glasses are thus of a different nature since the kinetics of pressure‐ and temperature‐induced configurational modifications are markedly different for given frequencies or experimental timescales. This dissimilarity mainly originates in the fact that the shape of potential energy wells varies little with temperature, but significantly with pressure. If a high kinetic energy is needed to overcome potential barriers at constant pressure, the changes in these barriers with pressure can lead by themselves to new configurational states, at low temperatures, if the pressure is high enough.
Figure 19 Permanent compaction of polyvinyl acetate after compression at 800 bar (80 MPa) in the liquid state.
Source: Data from [44].
2.4.5 Kauzmann Paradox
When viscous liquids escape crystallization, why do they eventually vitrify instead of remaining in the supercooled liquid state? One answer to this question is purely kinetic and relies only on increasingly long relaxation times on cooling. If experiments could last forever, any glass would eventually relax to the equilibrium state. Then, the glass transition would result only from the limited timescale of feasible measurements. A simple thermodynamic argument known as Kauzmann's paradox [45] indicates that this answer is incorrect. At its basis is the existence of a configurational contribution that causes the heat capacity of a supercooled liquid to be generally higher than that of an isochemical crystal and its entropy to decrease faster than that of a crystal when the temperature is lowered (Figure 20). If the entropy is extrapolated to temperatures below the glass transition range, it becomes lower than that of the crystal at a temperature TK, which is high enough for such an extrapolation to remain reasonable. Although this situation is not thermodynamically forbidden, it seems unlikely that an amorphous phase could have a lower entropy than an isochemical crystal.
Figure 20 Kauzmann catastrophe for amorphous selenium and ortho‐terphenyl (C8H14). Differences between the glass transition and Kauzmann temperatures indicating the smallness of the Cp extrapolations performed.
Source: Data from [46, 47].
The conclusion is that an amorphous phase cannot exist below TK. The temperature of such an entropy catastrophe constitutes the lower bound to the metastability limit of the supercooled liquid. As internal equilibrium cannot be reached below TK, the liquid must undergo a phase transition before reaching it. This is, of course, the glass transition. In its original form, Kauzmann's paradox implicitly neglects possible differences in vibrational entropy between the amorphous and crystalline phases. This simplification is actually incorrect but it does not detract from the gist of the argument, for taking into account such differences would only shift TK slightly. A more rigorous statement of the paradox is that the catastrophe would occur when the configurational entropy of the supercooled liquid vanishes.
2.4.6 Potential Energy Landscape: Ideal Glass and Fragility
Among the great many statistical mechanical models that have attempted to account for the glass transition and solve Kauzmann's paradox, the early one proposed by Gibbs and Di Marzio [48] is of special interest. It predicts that the supercooled liquid would transform to an ideal glass through a second‐order transition at the temperature T0 at which its configurational entropy would vanish. Since then, the existence and the nature of such a transformation have been much debated. This debate notwithstanding, the important point for our discussion is the result subsequently derived by Adam and Gibbs [49] on the basis of a lattice model of polymers. This result is a very simple relationship between relaxation times and the configurational entropy of the melt, viz.
(10)
where Ae is a pre‐exponential term and Be is approximately a constant proportional to the Gibbs free energy barriers hindering the cooperative rearrangements of the structure.
Qualitatively, this theory assumes that
