or other means, the equilibrium liquid is continuously losing internal equilibrium. As will be discussed here, the question arises as to whether there is any finite production of entropy and – if so – whether this quantity is of importance regarding the other terms involved in the process.
The Third Law of thermodynamics postulates that the entropy of a perfectly ordered system is zero at 0 K. In contradiction with it, however, calorimetric measurements indicate that glasses not only possess nonzero entropies at 0 K but that this residual entropy depends on thermal history as illustrated by a simple entropy cycle calculated from measured heat‐capacities and entropy of fusion (ΔSf). Beginning with a perfectly ordered crystal, whose entropy thus is 0 at 0 K, one derives the entropy of the crystal at its congruent temperature of fusion Tf, then that of the melt from this temperature down to the glass transition, and finally that of the glass down to 0 K (Figure 1). The difference S0 between this entropy and that of the crystal at 0 K is the residual entropy (Table 2), which increases with higher glass transition temperatures and, thus, with higher cooling rates, reflecting the increasingly wide distribution of configurational states obtaining with increasing temperatures.
A finite residual entropy at 0 K might seem to contradict the Third Law of thermodynamics. As justified by Jones and Simon [1], however, there is no contradiction because this law applies only to crystals and other systems in internal equilibrium, which are necessarily ordered at 0 K to minimize their Gibbs free energies. This is not the case of glasses, which do obey the Nernst theorem [2] since they cannot pass from one entropy state to another at 0 K (ΔS = 0 for two neighboring glassy states at 0 K).
Although such determinations also made for partially disordered crystals like ice Ih or CO have long been explained by simple statistical mechanical models, the very concept of residual entropy has recently been debated [3]. On the assumption that ergodicity must hold for the entropy to be defined, the proponents of a kinetic view have claimed that the configurational entropy undergoes an abrupt jump at the glass transition in order to reach the zero value of the crystal entropy at 0 K. In contrast, the proponents of the conventional view have stressed that what matters is not time averages but spatial averages of configurational microstates [3], which is the reason why the measured residual entropies do make sense physically and correlate with the specific structural features of glasses and disordered crystals.
Figure 1 Entropies of the crystal, liquid, supercooled liquid and glass phases of a substance.
By definition, equilibrium thermodynamics cannot alone account for fundamental questions raised when relaxation is too slow with respect to experimental timescales. Owing to the kinetic nature of the problem, use has been made of the formalism originally developed for the kinetics of chemical reactions by De Donder and his school [4]. With values increasing as the reaction proceeds, a new variable, the advancement of reaction, ξ(t) is defined to characterize the state of the system as a function of time, t, such that the reaction rate is simply dξ(t)/dt. This extensive variable, expressed in mol, accounts for the distribution of matter (local mass or density variation), or the molecular structure, within the system at any time. A new state function, the affinity, A, is then introduced to relate ξ(t) to the driving force of the reaction, its Gibbs free energy (at constant T and P):
(3)
The affinity A, expressed in J/mol, is the intensive conjugate variable of ξ. All time dependences are thus embedded into the time variations of the internal parameter ξ, or A, and of the other variables that are controlled experimentally (e.g. T, P).
For a relaxing system, the instantaneous entropy production was simply written by De Donder as the product of the thermodynamic force and the corresponding flux [4],
Table 1 Thermodynamic states in terms of affinity and its derivatives and in terms of rate of advancement of the process.
| Rate of advancement dξ/dt (extensive, mol/s) | dξ/dt = 0 | dξ/dt ≠ 0 |
|---|---|---|
| Affinity A (intensive, J/mol) | ||
| A = 0 and dA = 0 | True equilibrium; liquid state; σi = 0 | Unphysical |
| A = 0 and dA ≠ 0 | Isomassic state; σi = 0 | False equilibrium; nonequilibrium state; σi = 0 |
| A ≠ 0 and dA = 0 | Isomassic, isoaffine state; σi = 0 | Isoaffine state; σi ≠ 0 |
| A ≠ 0 and dA ≠ 0 | Nonequilibrium; glassy state; σi = 0 | Nonequilibrium; viscous state; σi ≠ 0 |
Liquid, glass, and relaxing liquid states are indicated by gray cells. The other cells indicate particular states that can be encountered or not during the glass transition. The value of the rate of production of entropy is indicated in each cell.
where the thermodynamic force actually is A/T, for the sake of dimensional analysis (the entropy production being in W/K).
Regarding the glass transition, the problem boils down to know A and ξ (or dξ/dt) and how they evolve with time. Depending on the values of both parameters, however, at this point several cases must be distinguished because not all of them are relevant (Table 1). The first and simplest case is that for which both A and dξ/dt are zero. It is that of the equilibrium liquid, which will thus be first considered in its metastable, supercooled extension.
3 Supercooled Liquids
Although the liquid state is generally far from simple, it can be considered as an equilibrium reference at viscosities (η) low enough that flow is easy, i.e. at high‐enough temperatures at the pressure considered. In that case, the diffusion of microscopic entities, be they molecules or atoms, obeys the Stokes‐Einstein relation, which relates the diffusivity D to the temperature and viscosity with:
(5)
where the coefficient C is a geometrical factor fixed by the boundary condition
