Entropy catastrophe

First-order solid-state amorphization occurs due to an entropy catastrophe [39] causing melting of superheated graphite and decompressed diamond below Pg when the entropy of the ordered crystal would exceed the entropy of the disordered liquid. This condition is resolved with the occurrence of a kinetic transition to a (supercooled) glass whereby the exact kinetic conditions during carbon transformation will be critically Pg-depen-dent [39]. It is important to consider the crystal to liquid transition and the effect of a superheated crystal whereof the ultimate stability is determined by the equality of crystal and liquid entropies [40]. When this condition is met, a solid below its Pg will melt to an amorphous solid, particularly... 

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. 

The theory now proceeds as developed in Sections V and VI, essentially unchanged. For example, P v) will have the same bimodal structure as shown in Fig. 14, but will now be continuous. Similar smoothing of all artificially introduced discontinuities will not affect the theory in any essential way. The loss of a sharp distinction between liquid- and solidlike cells could vitiate use of the percolation theory. The nonanalyticity in S will certainly be lost, leading to a communal entropy for which 9S/9p is always less than infinity. However, the first-order phase transition should be preserved, just as it was for most of the parameter space even when )3> 1. The discontinuity in p and v would be reduced, as would be the latent heat. One important effect of this smearing will be the appearance of a critical end point for the liquid, a temperature below which the liquid phase is no longer even metastable. The second-order transition, which is only a small region of parameter space for /8> 1, is now wiped out completely by the restoration of analyticity. Our theory thus leads to a first-order phase transition or no transition at all. However, the entropy catastrophe can be resolved within our theory only if a transition occurs. 

The free-volume model was originally derived to explain the temperature dependence of the viscosity. We have shown that it has a much broader application and can explain many of the outstanding experimental observations. This includes the existence of an entropy catastrophe at 7 and the approximate equality of Tj, and 7], first observed by Angell and coworkers.The relation between ln and 7, measured by Moynihan et al., also follows naturally and quantitatively from the notion that the liquidlike cell fraction p is the important variable that ceases to reach equilibrium when the relaxation rates become longer than the time scale for the measurement. 

Dimarzio [18] identified this state of vanishing entropy as the entropy catastrophe introduced by Kauzmann [19], and regarded T2 as the glass transition temperature Tg of the polymer. Because the temperature derivative of the entropy is discontinuous if the entropy is kept constant at A5" = 0 below T2, the glass transition on the basis of this picture is classified into a second-order transition by Ehrenfest s definition. 

The first four chapters, making up the fundamental part, contain reviews of the latest knowledge on polymer chain statistics, their reactions, their solution properties, and the elasticity of cross-linked networks. Each chapter starts from the elementary concepts and properties with a description of the theoretical methods required to study them. Then, they move to an organized description of the more advanced studies, such as coil-helix transition, hydration, the lattice theory of semifiexible polymers, entropy catastrophe, gelation with multiple reaction, cascade theory, the volume phase transition of gels, etc. Most of them are difficult to find in the presently available textbooks on polymer physics. 

An entropy catastrophe happens (left side of the curve in Figure 12.7) where 5 = 0. For any phy sical. system, the minimum number of accessible states is W = ], so S = kin IT implies that the minimum entropy is 5 = 0. At the point of the entropy catastrophe, the system has no states that are accessible below an energy = o. Compute o. 

The total heat released is the sum of the entropy contribution plus the irreversible contribution. This heat is released inside the battery at the reaction site. Heat release is not a problem for low rate appHcations however, high rate batteries must make provisions for heat dissipation. Failure to accommodate heat can lead to thermal mnaway and other catastrophic situations. 

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. 

At this point AG, AS and, consequently, AH become zero indicating an instability similar to the inflection point in P- Vspace for the van der Waals equation of state of a fluid. Such a triple point has also been predicted based on catastrophe theory by investigating the perturbations of catastrophe germs [2.155], For larger degrees of disorder, the entropy difference AS would be... 

The total number of symmetries is therefore equal to a = 12x81 = 972. The entropy correction which follows from this equals = - 8.315 x In 972 = - 57.2 J mol l K T Forgetting this correction for very symmetrical chemical species would therefore result in a catastrophic error. 

Fig. 17, we see that AS/R is equal to 0.417 when jc =l which is the Maier-Saupe prediction for monomers [63]. Then as the mole fraction of bent conformers increases the transitional entropy is also predicted to increase which is paradoxical because for a pure system of biaxial particles AS/R is predicted to decrease with increasing molecular biaxiality [65]. We shall return to this important point shortly but for the moment we note that for x of 0.5 AS/R has increased to 1.3 which is over three times the value found for monomers and so is analogous to the behavior of even liquid crystal dimers. As xf continues to decrease so AS/R increases until for a mole fraction of the linear conformer of less than 0.03 the transitional entropy falls catastrophically to a value of less than 0.03. This behavior is reminiscent of odd dimers although AS/R is somewhat less than the values usually found, as is the critical composition forx . In the limit thatx° vanishes so too does the transitional entropy, in... 

Axioms 9. The topological entropy realization while controlling the system s behaviour creates the multitude core, containing reorganization, permitting to present the entropy size H(y), entropy potential As and its adhesion catastrophe dynamic model (movement to the target attractor and the stability loss) ... 

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