Modeling of the Glass Transition

The simplest model to represent the glass transition is based on the hole theory which was developed by Frenkel and Eyring some 60 years ago and is described in more detail in Sect. 6.1.3 (see also Sect. 4.4.6). The equilibrium number of holes at T is N and each contributes an energy e, to the enthalpy. As given on Fig. 6.5, the hole contribution to the vibrational heat capacity Cp and its kinetics is represented by  

N represents the instantaneous number of holes and x is the relaxation time towards the hole equilibrium. During a TMDSC experiment, as outlined in Sect. 4.4.6, N and x are temperature and, thus, time dependent. First, a solution of Eq. (2)is attempted for quasi-isothermal temperature-modulated DSC, to be followed by a numerical solution for standard TMDSC. The basic solution of Eq. (2) is  

The curve is elevated relative to zero by a constant amount (0.125) and has a contribution of 2o), double the modulation frequency (+, second harmonic). Both these contributions are not included in the experimental reversing heat flow which contains only the contribution to the first harmonic ( , see Sect. 4.4.3). Accepting the present analysis, it is possible to determine y and x from the reversing heat capacity by matching the last term of the equation in Figs. 6.118 and 4.131, and then use the paramters describing the match to compute (A), the actual response of the TMDSC to the quasi-isothermal temperature modulation. 

This modeling reveals a glass-transition-like behavior in systems describable with the hole model. It also shows that the reversing heat capacity by TMDSC is only approximately the apparent reversible heat capacity. In addition, it was shown in Sect. 

3 that an asymmetry exists, not treated in the simple hole model, causing a change of the relaxation time z with N (see also Fig. 4.126). Additional observations point to deviations from the exponential response of the first-order kinetics of Eq. (2), above. AU these complications point to the fact that the glass transition is cooperative and needs a more detailed model for fuU representation. 

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To keep the liquid at metastable equilibrium while cooling it to T2 Tq, the cooling rate would have to be infinitely slow. It has been argued that in this hypothetical limit a thermodynamic transition of some kind, possibly second order, intervenes to prevent the excess entropy from becoming catastrophically negative. However, another possibility is that the true dependence of excess entropy on temperature deviates from the linear extrapolation to zero, and the excess entropy varies much more slowly with temperature near Tb than it does at higher temperatures. This latter possibility is found in some simple models of the glass transition discussed below. 

Accordingly, since the dispersion and xa are obtained independently as separate and unrelated predictions, in such models the dispersion (or the time/frequency dependence) of the structural relaxation bears no relation to the structural relaxation time. This means it cannot govern the dynamic properties. As have been shown before [2], and will be further discussed in this chapter, several general properties of the dynamics are well known to be governed by or correlated with the dispersion. Therefore, neglect of the dispersion means a model of the glass transition cannot be consistent with the important and general properties of the phenomenon. The present situation makes clear the need to develop a theory that connects in a fundamental way the dispersion of relaxation times to xa and the various experimental properties. 

We now discuss the impact of this general property on theories and models of the glass transition. The primary concern of most theories is to explain the temperature and pressure dependences of the structural relaxation time ra. The dispersion (n or >kww) °f the structural relaxation is either not addressed or... 

Jenkins and Hay [205] discussed kinetic modeling of the glass transition and of enthalpic relaxation resulting from physical aging, as observed by differential scanning calorimetry. 

Jackie, J., Models of the glass transition. Rep. Prog. Phys.,49,171-231 (1986). 

The hole model of the glass transition. To get a better feeling for the nature of the glass transition to polymers, the simple, 70-year old hole-model of the liquid [8,9] was used for its description [10,11]. The model is shown schematically on the upper right of Fig. 6.5. The main distinction between liquid and solid is the low viscosity of the liquid (see Sect. 5.6). This low viscosity is linked to the long-range mobility of the holes through their ability to diffuse or, after collapse, to reform elsewhere. 

C. Ligand Field Model of the Glass Transition in Macromolecule-Metal Complexes... 

Fig. 5.15. Storage shear moduli measured for a series of fractions of PS with different molecular weights in the range M = 8.9 10 to M = 5.81 10. The dashed line in the upper right corner indicates the slope corresponding to the power law Eq. (6.81) derived for the Rouse-model of the glass-transition. Data from Onogi et...

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