Talk:Kauzmann paradox

Possible Error?

This article seems like it could be a slight misinterpretation of Kauzmann's paradox.

RE: In thermodynamics, the Kauzmann Paradox is the apparent result that it is possible to obtain a supercooled liquid with an entropy lower than that of its corresponding crystal.

The latter scenario, although rare, is not impossible in nature and therefore cannot be defined as being paradoxical. For example, consider systems of hard spheres, which are known to crystallize at high pressures/low temperatures. This phenomenon implies that the free energy of the crystal is lower and hence the entropy of the crystal is higher than that of the liquid since there is no internal energy contribution for hard spheres. The crystal has more entropy because the additional free volume per particle gained by crystallizing increases the total number of states accessible to the system.

The author was correct to point that extrapolating the liquid entropy to zero temperature poses a problem, since the entropy becomes negative, which is impossible by definition. Kauzmann theorized that to avoid this scenario the rate of change of entropy with respect to temperature must suddenly change, which is precisely what happens at the glass transition, hence the paradox.

68.41.6.53 08:38, 2 September 2006 (UTC)A

The article is wrong, but in order to explain it you need a different definition of configurational entropy. In glass physics the term configurational entropy usually describe the part of entropy that doesn't come from phonons. In other words you can calculate a phonon entropy (or vibrational entropy) from the phonon spectrum, and subtract it from the total entropy to get the configurational entropy. The Kauzmann paradox states that the configurational entropy of the liquid seems to become smaller than the configurational entropy of the crystal.

This doesn't mean that the liquid becomes an equilibrum state, because the crystal may still have a much larger vibrational entropy than the liquid. However it would still be somewhat contra-intuitive. The problem is that the configurational entropy is a meassure of the number of local energy minima available to the system. We expect that a crystal without defects coresponds to a single energy minimum, so thus it would have a configurational entropy of zero. On the other hand a liquid is disordered, so we would expect it to have a configurational entropy larger than zero.

The conventional way to meaure all this is to perform a Differential_scanning_calorimetry experiment in which a sample is heated through the glass transition. You can use the graph to estimate the total heat capacity and the phonon heat capacity. If you combine these you get an estimate of the configurational entropy.

For a definition I think the best source that I know of is Davis Wales's book on "Energy Landscapes". you you are interested in expermental details, then have a look at Enthalpy relaxation and recovery in amorphous materials, J. Non-Cryst. Solids 169, 211 (1994).

I think that the best fix is to rewrite the article on configuratinal entropy or perhaps to introduce a new article named configuratinal entropy (glass). I don't know which approach is better. I also mentioned the problem on the Talk:Configuration_entropy user: nielsle —Preceding comment was added at 15:24, 3 February 2008 (UTC)

Kauzmanns own explanation

If I remember correctly Kauzmann speculated that the free energy barrier of crystallisation could vanish at low temperatures. This means that the supercooled liquid would become unstable. The original article also mentions a Kauzmann paradox for volumes.

Where is the paradox

It is also regarded paradoxial that the liquid should have a lower configurational entropy than the crystal. At sufficiently low temperature a finite system will always be vibrating around a minimum of the potential energy function. We can introduce a socalled configurational entropy to describe the number of populated energy minima. This entropy is related to the heat capacities.

A perfect crystal has a configurational entropy of zero. This means that the ensemble only populates a single potential energy minimum.

Kauzmann predicted a temperature at which the configurational entropy liquid vanishes as well. This means that the supercooled liquid is trapped in the global energy minium (of the luquid subset of phase space). Many people regard this prediction as paradoxial or even wrong. See also

http://arxiv.org/abs/cond-mat/0212487

Please make a redirect

Please make a redirect of ideal glass transition temperature to this page. Then we can make a link to here from Glass (disambiguation)