β Scribed by S.O. Gladkov
No coin nor oath required. For personal study only.
It is shown that in a certain class of amorphous materials the isobaric behavior of the heat capacity at high temperatures follows l/ T' which then changes to the smoother 1 /fi.
The Vogel-Fulcher empirical law for the temperature dependence of polymer viscosity is explained from general thermodynamic principles.
For one reason or another the interest for investigation of disordered materials did not wake up until twenty or so years ago. As usually happens during the period of pioneering studies, there have been many reports of what seemed to be the abnormal behavior of some parameters of disordered systems including such as heat capacity, viscosity, susceptibility, etc.
As for the theory, the pioneering works apparently belonged to Frenkel [l] who was the first to put forward the idea of a free volume. On the basis of this concept it was already possible at that time to explain, at least semiquantitatively, a number of experimental facts in the field of metal melting. In the later works by Cohen, Turnbull and Grest [2-81 the free volume concept was applied to polymers and other glass-like systems. The authors, for example, explained the originally empirical Vogel-Fulcher law [9, lo] (see below) in terms of the coefficient of volume expansion of liquid.
The appropriate formulas may be conveniently derived using the model of quantum tunneling of a heavy particle across the potential barrier. At absolute zero temperature the problem was solved by Livshits and Kagan in ref. [ll]. The authors set themselves the task of calculating the probability of formation of the nucleus of a new phase, generated as a particle tunnels through the anharmonic barrier.
Of the more recent work in the field we would note a series of articles by Patashinsky et al. [12-151 where the theory of melting of solids was developed on the basis of the short crystalline order model. In our further treatment we will need to refer to ref. [15] in more concrete detail and we will dwell at length on the problem of application of the local crystal order model to the problem at hand.
Our discussion of the theory will be split up in two parts. In the first we will describe the heat capacity of some polymers as a function of temperature. The treatment relies on the free volume concept with regard to molecular tunneling to the inside of the free volume, whereby it is possible to adequately describe the equilibrium properties of certain disordered systems.
In the second part we will consider the quasi-equilibrium state of the liquid substrate and derive the Vogel-Fulcher law in a consistent and rigorous way.
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