Encyclopedia of Glass Science, Technology, History, and Culture

( 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 geven though the actually observed values do vanish ( Figure 3).

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

( 8)

where T 1= 360 K is in Figure 2an 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 4showing 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 Mdefined 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 d A = 0) whereas the glass below T Mis defined by d ξ = 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.

The values are taken from the literature.

Although considering an internal parameter ξ , this approach assumes that the glass transition occurs continuously at T Mwhere 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 Por ∆α Pare 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.