Glass transition temperature free volume fraction

Polymer Polymer type Glass transition temperature, Tg CO Fractional free volume (cm3/cm3)... 

The occupied volume also increases with temperature, but its magnitude and thermal expansion coefficient Uo. as drawn schematically in Fig. 11-6,-remain a matter of conjecture and can be estimated only indirectly. It cannot be identified with the van der Waals covolume. In Fig. 11-6, the specific volumes are normalized by the volume Og at the glass transition temperature. Thus the fractional free volume,/, a dimensionless number, is oj/og. (Alternatively,/may be defined as Vf/o, which is nearly equivalent since d varies by only a few per cent in a temperature range of interest.)... 

If we, however, consider that viscosity is inversely related to the fractional free volume, which increases from a small value at the glass transition temperature Tg linearly with temperature above this figure, then it is possible to derive an equation. 

The glass transition temperature of a dilute system, according to the free volume changes, is determined by the diluent volume fraction Vd, and changes of the thermal expansion coefficient, a, at Tg by using ... 

The viscosity dependence of intramolecular excimer formation is complex. As in the case of molecular rotors (Section 8.2), most of the experimental observations can be interpreted in terms of free volume. However, compared to molecular rotors, the free volume fraction measured by intramolecular excimers is smaller. The volume swept out during the conformational change required for excimer formation is in fact larger, and consequently these probes do not respond in frozen media or polymers below the glass transition temperature. 

Here Bd is a dimensionless constant of order unity43, q being the polymer density f ( 0.025) is the fractional free volume at the glass transition temperature Tg, with the latter taking its maximum value T at infinite molecular weight. VE denotes the... 

These authors were the first FGSE workers to make extensive use of the concept of free volume 42,44) and its effect on transport in polymer systems. That theory asserts that amorphous materials (liquids, polymers) above their glass transition temperature T contain unoccupied volume randomly distributed and in parcels of sufficient size to permit jumps of small molecules — and of polymer jumping segments — to take place. Since liquids have a fractional free volume fdil typically greater than that, f, of polymers, the diffusion rate both of diluent molecules and (uncrosslinked and unentangled) polymer molecules should increase with increasing diluent volume fraction vdi,. The Fujita-Doolittle expression 43) describes this effect quantitatively for the diluent diffusion ... 

The forming of free volume, increases the fraction a of particles participating in the fluctuation processes between several possible conformations. Intermolecular mobility for amorphous materials begins at the glass transition temperature characterized by an increase of the free volume10. ... 

Simha and Boyer22 applied the free-volume concept to describe glass transitions in amorphous polymers. By analogy with Eq. (12), it was assumed that the fractional free-volume/g at glass-transition temperature Tg is... 

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. An increase in the free volume of the macromolecule (or that volume not occupied by the molecules themselves) allows more room for the molecules to move around, and thus an accompanying reduction in the Tg. Therefore, swelling of a macromolecule by a thermodynamically compatible solute (i.e., possessing similar op values as the sorbent) will tend to increase the free volume and lower the Tg (Kelley and Bueche, 1961 Haward, 1973 Barton, 1983). Lucht et al. (1987) developed a relationship between the glass transition temperature of several coals and the weight fraction of pyridine within the coal network,... 

The rates of relaxation and retardation processes above the glass temperature are strongly dependent on the viscosity and thus on the fraction of free volume present. Because the viscosity not only depends on temperature but also on static pressure (the glass transition temperature increases approximately 1 °C per 20 bar of pressure) it is not surprising that pressure also affects the viscoelastic processes. A qualitatively relation analogous to Eq. (13.121) can be readily derived (Ferry, 1980) ... 

The influence of factors such as chemical structure, molecular weight, cross-linking and plasticizers in the glass transition of polymers can be related to the changes that they provoke on the free volume fraction, which, as we already know, reaches a critical value at the glass transition temperature. The factors affecting the glass transition can be classified into two types (1) molecular factors, i.e., those related to the chemical structure of the polymer chain, and (2) external or controllable factors. 

There is no fundamental qualitative difference in mechanisms of low molecular weight (MW) penetrant diffusion in polymers above and below glass transition temperature, Tg, of the polymers [5,6]. The difference lies only in the fact that the movement of structural units of the macromolecule that are responsible for the transfer of penetrant molecules takes place at different supermolecular levels of the polymer matrix. At T > Tg the process of diffusion takes place in a medium with equilibrium or near-equUibrium packing of chains, and the fractional free volume, P(, in the polymer is equal to the fractional free volume in the polymer determined by thermal mobUity of strucmral units of macromolecules V((T), i e., V(= vut). At r< Tg the process of diffusion comes about under nonequihbrium packing conditions, although there exists a quasi-equilibrium structural organization of the matrix, where Vf> It is assumed that in this case Vf= where is the fractional free volume... 

The average fractional free volume at the glass transition temperature fg may be expected to closely parallel B in its dependence on structural parameters. The quantity fgjB, proportional to the empirical... 

This implies that the free volume fraction at the glass transition temperature is the same for all polymers and constitutes 11.3% of the total volume in the glassy state. (Many simple organic compounds have a 10% volume increase on melting, it may be pointed out.) TTiis is the largest of the theoretical values derived, but the first. Other estimates placed the free volume at about 2%. 

Eqs 2.23-2.26 are valid for any liquid. When upon cooling the temperature reaches the glass transition region, part of the free volume fraction is no longer accessible for the molecular motion. 

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