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Time‐dependent effects appearing at the glass transition are clearly observed in the heat capacities measured for PVAc (Figure 2), which is a model polymeric system extensively studied because of its excellent glass‐forming ability and standard T_g close to room temperature. The observed hysteresis loop between cooling and heating demonstrates that the heat capacity does not only depend on T and P but also on time. Moreover, upon heating, the heat capacity shows a typical overshoot, i.e. an endothermic event, named structural recovery process. To come back to the initial liquid state, the system needs to recover the amount of internal enthalpy that has previously been lost. From such measurements, it is possible to determine the configurational contribution to the heat capacity . Here, it is defined by the difference at every temperature between the heat capacities actually measured and estimated for the glass phase:

(6)

This type of definition also applies to other thermodynamic variables such as the thermal expansion coefficient α_P, or the isothermal compressibility κ_T. A configurational contribution consequently represents the thermodynamic contribution that originates in configurational changes in the liquid.

The glassy state then is defined as that for which the configurational movements have been frozen‐in, i.e. . In this state, only the vibrational motions, i.e. the fast degrees of freedom (faster than the experimental timescale), contribute. To define this contribution over the entire temperature interval of interest, an extrapolation of the glass heat capacity from low to high temperatures is needed (Figure 2). The heat capacity of the supercooled liquid can also be extrapolated toward low temperatures (Figure 2). The difference between these values for the supercooled liquid and the glass,

Figure 2 Heat capacity of PVAc measured across the glass transition range by differential scanning calorimetry at the same rate of 1.2 °C/min first upon cooling (solid circle) and then upon heating (empty circle). Dashed lines: fits made from the heat capacities measured for the glass and supercooled liquid.

(7)

then yields the equilibrium configurational contribution, which keeps increasing below T_g even though the actually observed values do vanish (Figure 3).

From the equilibrium and actual configurational contributions, the variation of the configurational enthalpy ∆H ^conf and entropy ∆S ^conf, taken between two temperatures, are calculated with:

(8)

where T₁ = 360 K is in Figure 2 an arbitrarily selected reference temperature.

Absolute values of both state functions could be obtained from the enthalpy and entropy of an isochemical crystalline compound through the crystallization values of these functions (see Figure 1). For lack of such a compound for PVAc, only relative values are thus presented (Figure 4) in such a way that both the actual and equilibrium values are equal from 360 K to the temperature of about 315 K at which internal equilibrium is lost. Since these variations are similar for the configurational enthalpy and entropy, only the former is shown in Figure 4.

Figure 3 Configurational heat capacity of PVAc across the glass transition range upon cooling: configurational contribution (solid circle) and equilibrium configurational contribution (empty circle).

Figure 4 Difference between the configurational enthalpy of PVAc and a zero reference‐value taken at 360 K. Actual value (solid circle) and equilibrium value (empty circle). Inset: magnification of Figure 4 showing extrapolated values of the glass and supercooled liquid of this differential configurational enthalpy intersecting at the point M, which defines the limiting fictive temperature T_M = T_f ′.

Contrary to their equilibrium counterparts, which continue to decrease upon cooling, both the actual configurational enthalpy and configurational entropy level off in the amorphous state (Figure 4). Owing to the large width of the glass transition range, the heat capacity variations at the glass transition are much too smooth to be interpreted as reflecting the discontinuity of a second‐order phase transition. Such a discontinuity can nonetheless be identified at a temperature T_M defined by the intersection of the extrapolated glass and supercooled liquid (Figure 4, inset). Both configurational enthalpy and entropy are thus continuous at that temperature, which separates the glass from the supercooled liquid. The same applies to other properties such as volume. Because entropy and volume are the first derivatives of the Gibbs free energy with respect to temperature and pressure, respectively, the following relations initially derived by Ehrenfest should hold when second‐order derivatives of the free energy vary discontinuously at this point M:

(9a)

(9b)

To express these equations in terms of discontinuities of equilibrium configurational contributions at T_M, e.g. of Eq. (7), Prigogine and Defay [7] assumed that the supercooled liquid is in internal equilibrium down to T_M (i.e. A = 0 and dA = 0) whereas the glass below T_M is defined by dξ = 0. These two equalities can then be grouped to yield the so‐called Prigogine–Defay (PD) ratio [7]:

(10)

Table 2 Thermodynamic parameters measured from five different glass‐formers.

Material	Tg (K)	ΔS0 (J/K/mol)	PD ratio	TK (K)	T0 (K)
SiO2	1480	5.1	>103	1150	NA Arrhenius relaxation
CaAl2Si2O8	1109	36.2	1.5–22	815	805
Glucose	305	1.7	3.7	241	242
PVAc	301	NA No crystal	2.2	239	250
Glycerol	183	19.4	3.7	134	123
Se	295	3.6	2.4	207	226

The values are taken from the literature.

Although considering an internal parameter ξ, this approach assumes that the glass transition occurs continuously at T_M where ξ_g = ξ_l. If so, it would follow from Eq. (9) that the PD ratio should be unity. As indicated by the values listed for widely different glass‐forming liquids (Table 2), however, calculated PD ratios are higher or even much higher than unity. One can explain such values by taking into account the kinetic nature of the glass transition [8]. Physically, it is making sense to assume that isobaric temperature derivatives such as ∆C_P or ∆α_P are not measured under the same kinetic conditions as an isothermal pressure derivative like ∆κ_T. Whereas this inconsistency may be removed if more than one internal order parameters ξ are involved in the thermodynamics of the glass transition [9], the problem may in contrast be compounded by the uncertainties arising from the extrapolation procedures used for deriving the relevant parameters at the temperature T_M.

Another puzzling fact has been long ago pointed out by Kauzmann [10] who wondered what would happen if the entropy of a supercooled liquid were extrapolated down to temperatures much lower than the experimentally observed T_g. The conclusion was that it would become lower than that of the isochemical crystal at a temperature T_K, thus termed the Kauzmann temperature (Table 2), which could suggest that the liquid undergoes a continuous phase transition toward the crystalline phase at T_K analogous to the critical point of fluids.

One way out of the paradox implies kinetic arguments and assumes that the viscosity of the supercooled liquid diverges at a temperature close to T_K. This assumption may be represented by the Vogel–Fulcher–Tammann (VFT) equation (Chapter 4.1):

(11)

where the temperature T₀ of the viscosity divergence is actually close to the Kauzmann temperature (Table 2) even though they may depend on the specific sample and the method of measurement.

Another way out is to take with great caution the extrapolations of the heat capacity and other thermodynamic functions of the supercooled liquid. As long pointed out [e.g. 11], there is no current theory for these properties in liquid state analogous to the Einstein or Debye models that provide functional forms at all temperatures for heat capacities of crystals.

As derived from strikingly old questions in glass science, these counterintuitive features indicate that glasses cannot be described by equilibrium thermodynamic states only. Nonequilibrium thermodynamics is, therefore, likely to be useful to characterize glasses and the glass transition.