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 Tg 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 TM = Tf ′.
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 TM 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)
To express these equations in terms of discontinuities of equilibrium configurational contributions at TM, e.g.
(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 |
|