Glass transition theory free volume theories

We believe the difficulty is that the free-volume theory as applied to the glass-transition does not take account of the essential role of intra- and intermolecular interaction in the system and the flexibility of the polymer chains, all of which factors play an important role in the glass-transition phenomena. 

We therefore believe one of the most pressing problems in the field of free-volume theory and glass-transition theory to be the development of new concepts and obtaining of new parameters for the corresponding states. Miller37 as early as 1968 introduced the idea that the glass transition corresponds to the iso-relaxation state, which at molecular weights of polymers below the critical ones may be replaced by an iso-viscous state. 

Using the classical hypotheses of the free volume theory, the glass transition temperature for a polymer (p) and solvent (s) solution, with a volume fraction of solvent, v, is given by (Kelley and Bueche, 1960) ... 

Free Volume Theory. Free volume theory suggests that the glass transition temperature is observed for polymers when their viscosity approaches that of their liquid state. Following a derivation based on the Doolittle expression for polymer viscosity (r ) as a function of free volume (Eisenberg, 1984)... 

Similarly to Fig. 5-4 for other glassy polymer-solvent systems also the predictions of this free-volume theory are in general agreement with experimental data on the temperature dependence of D in the vicinity of Tg2. In particular, the theory predicts a step change in Ed at Tg2, and this is consistent with most experimental investigations of polymer-solvent diffusion at temperatures just above and below the glass transition temperature (6,11,15). 

The authors of the cluster theory draw the conclusion that the theory affords a sufficiently rigorous theoretical derivation of Doolittle s equation (72). Verification of the free volume theory advanced by Cohen and Grest was carried out by Hiwatari using computer simulation [97], showed that glass transition in liquids can really be described in terms of the percolation theory, the value of Pcr in this case being close to 0.2. Unlike Cohen and Grest s assumptions, however, this transition is not accompanied by a drastic change in the fluidity of the liquid near Per-... 

On the other hand, some phenomenological distributions of relaxation times, such as the well known Williams-Watts distribution (see Table 1, WW) provided a rather good description of dielectric relaxation experiments in polymer melts, but they are not of considerable help in understanding molecular phenomena since they are not associated with a molecular model. In the same way, the glass transition theories account well for macroscopic properties such as viscosity, but they are based on general thermodynamic concepts as the free volume or the configurational entropy and they completely ignore the nature of molecular motions. 

The free volume theory of glass transition is based on Doolittle s empirical assumption (29), which states that the viscosity, q, at T > Tg is related to the free volume fraction by the equation... 

The assumptions and restrictions of the free-volume theory, as well as the significance of its parameters, are discussed in detail by Vrentas and Duda (5.61. For temperatures close to the glass transition temperature, the diffusion process is free-volume dominated and the energy term can be absorbed in the pre-exponential term. Equation 36 becomes... 

While the WLF equation is based on the free-volume theory of glass transition which is concerned with the introduction of free volume as a requirement for coordinated molecular motion, the equation also serves to introduce some kinetic aspects into the quantitative theory of glass transition. 

Fig. 3.53 Viscosity data from Fig. 3.51 plotted against the volume (normalized by the ambient-pressure volume). The curves are fits obtained by the free-volume theory (eqn 3.9.5), the resulting parameters being for methanol, B = 2.22 0.05 and V = 33.8 0.1 A for 1,2-propanediol, S = 6.7 0.3 and 14 = 71.9 0.5 A. Extrapolation of these fits to T = 10 Pa s gives the indicated glass-transition pressures, P. The strong curvature of the methanol data indicates the more fragile behavior of this fluid, which is reflected in the fragility parameters (calculated from equation 5 given by Cook et a .)F For methanol mp = 289 6 and for 1,2-propanediol mp = 123 5.

Free-volume theories of the glass transition assume that, if conformational changes of the backbone are to take place, there must be space available for molecular segments to move into. The total amount of free space per unit volume of the polymer is called the fractional free volume Vf. As the temperature is lowered from a temperature well above Jg, the volume of the polymer falls because the molecules are able to rearrange locally to reduce the free volume. When the temperature approaches Tg the molecular motions become so slow (see e.g. fig. 5.27) that the molecules cannot rearrange within the time-scale of the experiment and the volume of the material then contracts like that of a solid, with a coefficient of expansion that is generally about half that observed above Tg. 

For these polymers the following equation for Tg (absolute) can be deduced from the free-volume theory of the glass transition (see section 7.5.3) ... 

The free volume theory originated some years later than the lubricity and the gel theories, when the evolution of different properties of polymers as a function of temperature, specific volume, thermal expansion coefficients, or viscosity was attempted to be explained.The relationships between these properties and some variables corresponding to polymer stracture, such as molecular weight or terminal groups content, the presence of another monomer and, of course, the presence of plasticizers, was also explained. For plasticized polymers the theory attempted to explain the diminution of the glass transition temperature with the plasticizer content. This theory is a contribution of different authors, but it was postulated by Fox and Floiy. The theory is still being used to explain some properties of plasticized polymers, i.e., viscoelastic properties. ... 

According to the glass transition theory, an important property of water is to act as an excellent plasticizer, which swells and increases the free volume of the food matrix. An increase in moisture results in increased diffusion and molecular mobility in a multi-component food system, accelerating lipid oxidation by increasing the number of catalytic sites. Metals become more effective prooxidant catalysts by their increased mobility in the mbbery state. The molecular mobility decreases in solutes of increasing molecular weight. In... 

According to the glass transition theory, lipids are dispersed in the free volume of the food matrix composed of carbohydrates and protein polymers. In the rubbery state, the lipids react readily with oxygen and become oxidized. In the glassy state, however, the lipids are stable to oxidation because they are encapsulated and there is no free volume. The glass transition temperature, which determines when the food matrix changes from one state to the other, increases with a decrease in moisture and water activity. In many foods the... 

A comprehensive analysis of the free volume theories of glass transitions of high polymers has recently been published by Kanig (121). 

The free-volume theory of the glass transition, as developed in Section 8.6.1, is concerned with the introduction of free volume as a requirement for coordinated molecular motion, leading to reptation.The WLF equation also serves to introduce some kinetic aspects. For example, if the time frame of an experiment is decreased by a factor of 10 near Tg, equations (8.47) and (8.48) indicate that the glass transition temperature should be raised by about 3°C ... 

Ramachandrarao, P. Cantor, B. Cahn, R.W. (1977). Free volume theories of the glass transition and the special case of metallic glasses. Journal of Materials Science, Vol. 12, No. 12, (December 1977), pp. 2488-2502, ISSN 0022-2461... 

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