A popular empirical equation that relates the number-average molecular weight, Mn, to the glass transition temperature, Tg, is the Flory-Fox equation.(1,2)
Tg(Mn) ≈ Tg,∞ - K / Mn
Another relationship for the prediction of the molecular weight dependence of the Tg has been developed by Fox and Loshaek:3
Tg(Mn) ≈ Tg,∞ - K / (Mn Tg,∞2)
where Tg,∞ is the limiting value of the glass transition temperature at very high molecular weight, Mn is the number-average molecular weight and K is a constant for a given polymer that is related to the free volume present in the polymer. This dependence can be explained with the free volume theory of the glass transition;(3) due to the greater mobility of the chain ends, the free volume increases with the number of chain ends in a given volume, i.e. it increases with decreasing molecular weight. Loshaek and Fox found that the Tg,∞ is linear proportional to the free specific volume v:
(vg - vg,∞) / (Tg - Tg,∞) = const.
If this slope together with the slope of the v-T curve of the liquid monomer and the liquid polymer of infinite chain length is known, the Tg-Mn
relationship can be predicted, that is, the constant K can be estimated from thermo-physical properties that do not depend on the molecular weight.3
Some representative values of the Flory-Fox parameter for three common polymers are given in the Table below.
|Polymer||Exper. K Values||AVG Predicted K Values|
|Polystyrene||1.0 10-5||1.2 10-5|
|Polymethylmethacrylate (iso)||-||1.05 10-5|
|Polyisobutylene||0.69 10-5||0.65 10-5|
The plot below shows the dependence of the glass transition temperature on the molecular weight for polystyrene, as predicted by the Flory-Fox equation and measured by An, He and Jing (1997). The Tg's have been calculated with the software 3Ps-Tg. The predicted and measured values are in excellent agreement.
Glass Transition Temperature of Polystyrene
- T.G. Fox, and P.J. Flory, Journal of Applied Physics 21, 581–591 (1950)
- T. G. Fox, P. J. Flory, Journal of Polymer Science, 14, 315-319 (1954)
- T.J. Fox, and S.J. Loshaek, Journal of Polymer Science 15, 371 - 390 (1955)
- L. An, D. He, J. Jing et al., Eur. Polym. J., Vol. 33, No 9, 1523 - 1528 (1997)