Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов

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.

      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

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