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The questions raised by the Kauzmann paradox or the PD ratio clearly illustrate the need for a more fundamental thermodynamic description of the glass transition. Following the pioneering work of Tool [12, 13] and Davies and Jones [9], different approaches and phenomenological models have been developed to deal with the glass transition range itself, many within the framework of classical nonequilibrium thermodynamics [4, 11].

The starting point has been the phenomenological concept of fictive temperature (T_f) propounded by Tool [12, 13] to characterize the state of a relaxing system at any time. This temperature is similar to an order parameter ξ. It thus overcomes the limitations of the fixed limiting temperature T_M, which characterizes only the point at which internal equilibrium is suddenly lost in a quenched state. On an analogous basis, a more detailed description is made in terms of two‐temperature thermodynamics [14] whereby the vibrational and configurational degrees of freedom are distinguished by a “classical” temperature for fast modes (phonons bath), and an effective temperature for the slow modes, respectively.

The first physical models have then relied on two different approaches. In free‐volume theories, one generally considers that the dynamics of the system is determined by the free space present around its atoms, which makes configurational rearrangements more or less easy. In entropy theories, among which that of Adam‐Gibbs is the best known [15], the same determining role is attributed to configurational entropy. In other words, these theories assign the strong increase of relaxation times with decreasing temperatures and the eventual structural freezing in to decreases of either free volume or configurational entropy. Other more recent theories of the glass transition rely on mode coupling, random first‐order transitions or energy‐landscape descriptions [e.g. 16]. These different approaches have the common goal of finding the exact expression for the structural relaxation time, or its distribution, as a function of controlling parameters such as temperature or pressure, or structural order parameter.

For the sake of simplicity, let us consider here conditions of constant pressure. If the additional parameter ξ is taken into account, the total differential of the enthalpy of a system can be written as the sum of two contributions (considering pressure, the generalization to three contributions would be obvious):

(12)

The isobaric heat capacity is written as:

(13)

The first term on the right‐hand side is the heat capacity at constant ξ, i.e. , and the second, the configurational contribution as defined by Eq. (6). To account for the kinetic nature of the glass transition, it is then necessary to rewrite Eq. (13) as:

(14)

When the rate of change of ξ becomes much smaller than the rate of change of temperature, (dξ/dt)_P ≪ (dT/dt)_P, the configurational contribution is negligible.

Hence, it is the ratio between these two rates that is controlling the relative value of the experimentally recorded configurational heat capacity. This ratio is maximum in the supercooled liquid state, and decreases throughout the glass transition range to become negligible in the glassy state (cf. Figure 3). There, only the first right‐hand side term in Eq. (14) contributes:

(15)

The next step thus consists in taking into account the time dependence of ξ at every temperature through the temperature dependence of the relaxation time τ. The simplest way to do this is to assume a simple exponential decay for ξ at fixed temperature and pressure:

(16)

where ξ_eq(P,T) is the equilibrium value of the order parameter, i.e. a variable characterizing the liquid structure that depends only on P and T. Although the relaxation time itself has been given different temperature dependences with Arrhenius, VFT, or others laws (Chapter 3.7), the important point is that they are all of an exponential nature with respect to T or P to ensure the structural freezing‐in of the system.

Interesting applications of these concepts have been made with the lattice‐hole model of liquids, which has the advantage of lending itself to an evaluation of the order parameter ξ. Schematically, this model considers a liquid as a lattice in which disorder is represented by unoccupied sites whose fraction x depends on both temperature and pressure [17]. From the equilibrium value of the order parameter, it is thus possible to solve the linear differential Eq. (16) to find its temperature dependence and, then, to calculate the variations of the heat capacity within the glass transition range under varied conditions [18]. Likewise, the configurational Gibbs free energy may also be computed analytically as a function of temperature, pressure, and order parameter. A similar approach has been followed to incorporate the effects of pressure in the expression of the structural relaxation time for determining also how the heat capacity, thermal expansion coefficient, and isothermal compressibility vary under different conditions [19].

From the configurational Gibbs free energy calculated for the lattice‐hole model, one readily simulates with the definition (3) of the affinity its variations upon vitrification (cooling) and structural recovery (heating) [19]. Thermodynamic data measured on o‐terphenyl may be used to simulate the corresponding affinities during temperature ramps (Figure 5): cooling at 0.3 K/min from an initial temperature of 255 K is followed by heating at the same rate, and then by further cooling at 0.5 K/min preceding final heating at 20 K/min. That the supercooled liquid begins to lose internal equilibrium from 248 K is indicated by the departure at this temperature of the affinity curve from the zero line, which represents the maximum (equilibrium) value of the affinity during cooling. The affinity then linearly decreases with temperature below 240 K in the glassy state, with higher values for slower cooling as a result of lower glass transition temperatures. Upon heating, the affinity begins to increase linearly according to the same line pathway before crossing the equilibrium line. It then exhibits a peak whose position shifts toward higher temperatures and whose magnitude and width increase with the heating rate in ways such that the configurational heat capacity and the other thermodynamic coefficients can be simulated [19].

Figure 5 Simulated affinities of o‐terphenyl in the glass transition range upon cooling and heating as calculated from the lattice‐hole model. Solid circle for −0.3 K/min and solid square for −0.5 K/min; empty circle for +0.3 K/min and empty square for +20 K/min. The horizontal line represents equilibrium (A = 0). Inset: entropy production rates calculated from the previous affinities. Solid circle upon cooling and empty circle upon heating.

The entropy production can also be calculated from Eq. (4) (inset in Figure 5). In agreement with previous results [18], it shows a single peak upon cooling but two peaks upon heating. With respect to the experimental data, the advantage of the calculation is thus to distinguish clearly two contributions to the entropy produced when heat is brought to the material. The first peak is associated with a decrease of the configurational energy of the system, which is taking place because of the delay introduced by the relaxation time, even though heat is being supplied. As to the second peak, it is in contrast associated with the configurational energy necessary to recover internal equilibrium in the supercooled liquid state.

Here the wording “is associated” instead of “represents” is necessary because the entropy produced and configurational entropy changes necessarily differ as a result of the irreversible nature of the glass transition. Whereas the entropy production is the product of the thermodynamic force and flux (see Eq. (4)), the variation of the configurational entropy is written as, see Eqs. (8) and (13):

(17)

The rate of entropy production thus reflects the spontaneous or irreversible microscopic processes that take place within the system during relaxation. As dictated by the Second Law of thermodynamics, it is always positive whether upon cooling or heating (Figure 5, inset). Physically, it can be thought of the heat irreversibly generated by friction at a microscopic scale. The resulting thermal power P_i = Tσ_i, where σ_i is the entropy creation in Eq. (4), is produced much too quickly to be compensated instantaneously by an exchange of heat with the surrounding heat bath. Under this circumstance, this is why an effective or fictive temperature can be defined. This surrounding heat bath is sometimes called the phonon bath since it is characterized by fast or vibrational modes.

On the contrary, the change in configurational entropy is a reversible process related to the heat exchanged with the surrounding heat bath whose relevant thermal power is:

(18)

Because the configurational entropy becomes constant upon vitrification, its variations have vanished (i.e. the configurational heat capacity) below the glass transition range. Above this range, in the supercooled liquid state, they of course differ from zero as indicated by

(19)

In the transition range, the variations of the configurational entropy of the system are consequently positive or negative upon heating and cooling, respectively. As already evaluated long ago either theoretically [9] or experimentally [20, 21], the entropy produced is generally negligible with respect to the configurational entropy changes. The integration of the heat capacity curves measured by calorimetry is thus a pertinent way to access to the absolute value of the residual entropy at 0 K [3]. As seen from a direct comparison of Eqs. (4) and (17), arriving at this conclusion is tantamount to neglecting the affinity with respect to the enthalpy of advancement of the configurational change at every temperature:

(20)