Flory-Fox Equation

The Flory-Fox equation(1,2) is the most popular empirical equation that relates the number-average molecular weight, Mn, to the glass transition temperature, Tg,

Tg(Mn) ≈ Tg,∞ - K / Mn

Another simple empirical formula 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.6 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.3,6
Some representative values of the Flory-Fox parameter for three common polymers are given in the Table below.

 

predicted and observed Tg-1/M relationships
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
aValues have been taken from Lit. (2).

 

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

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References & Notes
  1. T.G. Fox, and P.J. Flory, Journal of Applied Physics 21, 581–591 (1950)
  2. T. G. Fox, P. J. Flory, Journal of Polymer Science, 14, 315-319 (1954)
  3. T.J. Fox, and S.J. Loshaek, Journal of Polymer Science 15, 371 - 390 (1955)
  4. L. An, D. He, J. Jing et al., Eur. Polym. J., Vol. 33, No 9, 1523 - 1528 (1997)
  5. Joseph Bicerano, Prediction of Polymer Properties, Marcel Dekker, New York 2002

  6. The factor K is usually obtained by fitting the Flory-Fox equation to experimental data
    for a given polymer. If these data are not available, K can be estimated from Bicerano's
    equation: K = 0.002715 Tg,∞3.(5)

  • Summary

    Flory-Fox Equation

    predicts the molecular weight dependence of the glass transition temperature of amorphous polymers.

  • The glass transition temperature increases with the molecular weight and reaches a limiting value at moderate to high molecular weights.

  • The Flory-Fox equation, however, does not give accurate estimates for the entire range of molecular weights, that is, the accuracy of the prediction is much lower for very short polymers (oligomers).

  • The deviations at low molecular weights can be ascribed to the chain ends which have a greater mobility than repeat units in the middle of the chain.