Entropy crisis

The truncation of the high temperature series in Eq. (4) at order is a vahd concern when applying the theory at low temperatures. This concern also extends to GD theory, which implicitly involves a truncation at order p. At some point, these perturbative treatments must simply fail, but we expect the lattice theories to identify faithfully the location of the entropy crisis at low temperatures, based on numerous previous comparisons between measurements and GD theory. Experience [96] with the LCT in describing equation of state [97] and miscibility [98] data indicates that this approach gives sensible and often accurate estimates of thermodynamic properties over wide ranges of temperatures and pressures. In light of these limitations, we focus on the temperature range above Tg, where the theory is more reliable. 

There has been a wealth of activity based on the idea that glassy dynamics is due to some underlying thermodynamic transition [1-25], If a glass former shows a jump in some an appropriate order parameter without the evolution of latent heat, then such a system is said to exhibit a random first-order transition [94,95]. Models of this kind, which include the p-spin glasses [110], and the random energy model [111], do not have symmetry between states but do have quenched random long-range interactions and exhibit the so-called Kauzmann entropy crisis. 

Keenan, J. H., Gyftopoulos, E. P. and Hatsopoulos, G. N., "The Fuel Shortage and Thermodynamics The Entropy Crisis," Proceedings of MIT Energy Conference, M. Macrakis, Editor, MIT Press, Cambridge, MA (1972). 

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... 

In many examples, particularly in fragile glass-forming liquids discussed in detail in chapter 3, the entropy lost by the time the system arrives at Tg is such a large fraction of AS m that the entropy crisis is imminent in just under 20 K below Tg. Therefore, any suggestion that the remaining entropy may be lost so slowly that Tk becomes equal to 0 K appears impossible. Kauzmann s own resolution of this paradox has been... 

Our main aim was to clarify the role of clusters in the process of liquid solidification and to show that formation of the polycluster glass structure is a typical result of this process. On the basis of the developed approach, such important problems as entropy crisis and relaxation phenomena in supercooled liquids can be considered in detail, but this is far from the topic of this paper and will be considered elsewhere. 

To develop our approach, we borrow ideas from both approaches. We then derive the consequences of our gap model schematically presented in Figure 10.2. We prove that assuming the existence of the gap in (10.3) and the validity of (10.12), the IG transition must occur at a positive temperature Tk (see Section 10.9). This then assures a rapid entropy drop near Tk as a precursor of the impending entropy crisis, which is discussed in Section 10.3.6. 

Gibbs and Di Marzio [18] used the above spH,dis (g) to demonstrate the entropy crisis in polymers (see Ref [52] for details). This calculation was the first one of its kind to demonstrate the entropy crisis. Despite its limitation, to be discussed below, the work by Gibbs and Di Marzio has played a pivotal role in elevating the Kauzmann entropy crisis from a mere curious observation to probably the most important mechanism behind the glass transition, even though the demonstration was only for long molecules. 

Communal entropy being less than a positive value, no matter how small, cannot be as fundamental an entropy crisis as the requirement Scomm = 0 to argue for an ideal glass transition. For example, liquid helium shows no glass transition when its entropy becomes equal to such a small positive value. Thus, we will adhere to Scomm = 0 as the most fundamental requirement forthe entropy crisis. This also rules out using the excess entropy AS x (T) in (10.16) used by Kauzmann and various other authors, to be used as a signal of an entropy crisis when it vanishes at a positive temperature, since thermodynamics itself does not rule... 

It should be emphasized that there is no violation of thermodynamics just because ASex(T, V) has become negative. Although it is not very common, it is possible for SCL entropy to be less than that of the crystal. On the other hand, a negative entropy is impossible. Thus, it appears more natural to identify the Kauzmann paradox with a component of the entropy becoming negative. Accordingly, as discussed in Section 10.4.1, we interpret the Kauzmann paradox as the following entropy crisis ... 

Under extrapolation, a negative component of S(T, V), and not a negative Sex(T, V), signals the entropy crisis. Its implication is simply that such states cannot occur in Nature, and the onset of the crisis is gradually the underlying thermodynamic driving force for GT in molecules of all sizes. 

An alternative thermodynamic theory for the impending entropy crisis based on spin-glass ideas has also been developed in which proximity to an underlying first-order transition is used to explain the glass transition [83]. 

Figure 10.11 The free energies and entropies from the two FP solutions. The entropy crisis occurs at Tk, below which Sn (dashed cyan) becomes negative, but the corresponding free energy F) (cyan) remains concave despite an unphysical communal entropy. This explains their labeling as unphysical in the figure. The...

A number of attempts to solve the entropy crisis and to develop further the theory of an ideal glass transition have been made on the basis self-consistent mode-coupling theory, (99-101) statistical mechanics (102), cluster-hole theory (103), and computer simulation (104). 

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