Entropy vanishing excess

Fig. 20. Two examples of the vanishing excess entropy paradox. The area representing the entropy of fusion on the CJ, versus T (log scale) plot has been matched to the area between the crystal and supercooled Uquid heat capacity curves (the latter being extrapolated naturally below the Tg to indicate the supercooled liquid Cg for very slow (equiUbrium) measurements. At the temperature T, S (supercooled liquid) would equal as can be...

This ows that the assumptions used in the simplified lattice model of Ch. Ill cannot be consistent. Indeed this model predicts in its zeroth approximation ( 2) a vanishing excess entropy (cf. 3.2.15), instead of an exce.ss entropy proportional to the excess free energy. This is clearly due to the assumption that the excess free energy is only related to changes in the lattice partition functions and that in spite of changes in the interaction, the excess volume is zero. 

As an extension of earlier definitions, Q may now be viewed as the excess or deficit function needed to have Q - (AE + W - S(i)/iidni) vanish identically for any process under consideration this is equivalent to the requirement imposed by the First Law that E be a function of state. For the case where only mechanical work is involved we may therefore introduce in the usual manner the total entropy function S such that... 

The second is a conjecture, based on evidence from MD simulations for simple systems, that the liquid state is terminated above as a result of the relaxation time for the equilibrium structure diverging at positive excess entropy. This would imply a finite zero-frequency shear modulus, hence indefinite mechanical stability. With the associated vanishing of... 

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... 

We may compare (17.6.2) with the corresponding expression (10.7.4) for monomer mixtures. Apart from the combinatorial entropy and factors qjqA and CA/iA, these formulae are exactly the same, when the mole fractions xa and xb are replaced by Xa and Xb- We must also notice that in our present model the configurational specific heat at constant volume cvA vanishes as a consequence of our assumption that the cell partition functions do not depend on the temperature. Therefore the detailed discussion of the effect of intermolecular forces on excess functions presented in Ch. IX-XI, applies also to polymer mixtures. For example p will again give rise to positive deviations from ideality, positive excess entropy and heat absorption. We shall not go into more detail. 

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