Kauzmann temperature, glass transition

Let us now turn to a discussion of the relation of the temperature dependence of the polymer melt s configurational entropy with its glass transition and address the famous paradox of the Kauzmann temperature of glass-forming systems.90 It had been found experimentally that the excess entropy of super-cooled liquids, compared with the crystalline state, seemed... 

Here, Cv h(T) and Svlh(T) are the vibrational contributions to the heat capacity and the entropy, respectively. Note that the slope of the replica symmetry-breaking parameter with respect to temperature is not unity as predicted by one-step replica symmetry breaking. Rather, the slope is governed by three factors the Narayanaswamy-Moynihan nonlinearity parameter x, the Kauzmann temperature, and the ratio of the Kauzmann temperature to the glass transition temperature. 

FIGURE 6 Schematic representation of changes in specific volume and enthalpy with temperature. The effect of annealing of a glass, and the associated molecular reorientation is also shown in addition to the melting (7 ) and Kauzmann (Tk) temperatures, the two glass transition temperatures (7gi and arc pointed out. Source From Ref. 16. 

The entropy crisis described in the preceding paragraph is the result of an extrapolation. With the exception of He and " He (Wilks, 1967)," there is no known substance for which a Kauzmann temperature is actually reached. Nevertheless, the extrapolation needed to provoke a conflict with the Third Law is indeed modest for many substances (Angell, 1997), and what intervenes to thwart the imminent crisis is a kinetic phenomenon, the laboratory glass transition. This suggests a connection between the kinetics and... 

Figure 2.5 shows also the concentration dependence of the inverse Kauzmann temperature T (entropy catastrophe temperature). For the pure metal, T is much higher than the temperature T0 as discussed. The 77-line should also decrease with increasing concentration and end in the triple point(C, 7 )[2.21] as follows from its definition (AS = 0). It is interesting to note that at this point the real Kauzmann temperature and the inverse Kauzmann temperature meet. But in real systems, the amorphous phase has an excess entropy (small fraction of the entropy of fusion) when compared to the corresponding crystal, the exact amount determined from the kinetics and timescale of the glass transformation. Therefore, another glass transition temperature line with finite excess entropy must be considered, which will be parallel to the Tg-line (above it) and cross the T0- and 77-lines not exactly in the triple point. 

In the absence of any interceding phenomena, it seems necessary only to allow the supercooled liquid longer and longer periods of equilibration to ensure that its entropy will fall below that of the crystal at the temperature Jk (denoted Fj and in Refs. 58 and 106, respectively). Such an occurrence, though not actually in violation of the third law at finite temperatures, would imply a contradiction of the Nemst heat theorem on the approach to 0°K. This seems unlikely and raises the question of the existence of a thermodynamic singularity underlying the glass transition at or above the Kauzmann temperature T. ... 

Since the homogeneous nucleation probability cannot be measured for glass-forming liquids, it has not been possible to either prove or disprove this denial of an in-principle ground state for the hquid state of simple substances. The plausibility of Kauzmann s resolution, however, has suffered from the identity of behavior of crystallizable and atactic (noncrystalliz-able) polymers, and by the experimental contrasts in the composition dependencies of homogeneous nucleation temperatures and glass transition temperatures (7 and T ) observed in binary solutions. 

Fig. 21. Entropy versus log-temperature diagram for the hard-sphere model. The solid curves give the computer simulation values for the supercooled fluid, glass, and crystal. The dashed curves have the following bases (a) a calculation from the virial equation using the known first seven coefficients and higher coefficients obtained from the conjectured closure (the plot corresponds quite closely with that calculated from the Camahan-Starling equation ) and (i>) an extrapolation of higher temperature behavior such as that used by Gordon et al., which implies a maximum in the series of virial coefficients. The entropy is defined in excess of that for the ideal gas at the same temperature and pressure. Some characteristic temperatures are identified 7, fusion point 7 , upper glass transition temperature T/, Kauzmann isoentropic point according to closure virial equation.

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