Kauzmann paradox

Stillinger, E H., Supercooled liquids, glass transitions, and the Kauzmann paradox. J. Chem. Phys 88,7818(1988). 

Interestingly enough, if a glass-forming liquid were cooled slowly enough (at several times the age of the universe ) such that it follows the dotted line shown in Fig. 9.8Z) at a temperature T kau the entropy of the supercooled liquid would become lower than that of the crystal — a clearly untenable situation first pointed out by Kauzmann and referred to since as the Kauzmann paradox. This paradox is discussed in greater detail in Sec. 9.4.2. 

Finally, it is worth noting that the values of Tq or needed to fit the viscosity data are close to the temperature at which the Kauzmann temperature, Tkau is estimated from extrapolations of other properties such as those shown in Fig. 9.8, lending credence to the model. This model also provides a natural way out of the Kauzmann paradox, since not only do the relaxation times go to infinity as T approaches 7)., but also the configuration entropy vanishes since in glass at T = T only one configuration is possible. 

In Fig. 22, the two dashed curves have a similar significance. Curve a, as in Fig. 21, requires the existence of a thermodynamic transition associated with a divergence of the relaxation time at some temperature T> the most natural choice being the temperature (marked TJ, "), which corresponds to the Batchinski-Hildebrand Fq " for the system (see Fig. 4). Curve b represents an alternative resolution of the Kauzmann paradox for this system and implies that the heat capacity must have a maximum, under... 

Hunt, A., An explanation for the Kauzmann paradox and its relation to relaxation-times, J. Non-Cryst Solids, 175, 129-136 (1994). 

In equation 2 Tr denotes the freezing temperature at an infinitely slow cooling rate. For this limit the abbrevation Tr was chosen to remind of the Kauzmann paradox (6,33), which originates from a similar extrapolation procedure. Tr will... 

Melting is normally driven by an entropy gain, then AS > 0. With the decrease of temperatures from T, the integral at the right-hand side of (6.53) decays gradually from zero to —AS, as demonstrated in Fig. 6.16b. However, a linear extrapolation to AS = 0 reaches a finite temperature rather than zero absolute temperature, which can be defined as T. This result implies that below Ts, Si < Sc-Apparently, the amorphous liquid state could not be more ordered than the crystalline solid state, which is against the third law of thermodynamics. Early in 1931, Simon pointed out this problem (Simon 1931). In 1948, Kauzmann gave a detailed description, and proposed that there should exist a phase transition such as crystallization before extrapolation to to avoid this disaster (Kauzmann 1948). Therefore, this scenario is also called the Kauzmann paradox. 

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