## Flory-Fox Equation

A popular empirical equation
that relates the number-average molecular weight, *M _{n}*, to the glass
transition temperature,

*T*, is the

_{g}*Flory-Fox*equation.

^{(1,2)}

*T _{g}*(

*M*) ≈

_{n}*T*-

_{g,∞}*K*/

*M*

_{n}Another relationship for the prediction of the molecular weight dependence of the *T _{g}* has been developed by Fox and Loshaek:

^{3}

*T _{g}*(

*M*) ≈

_{n}*T*-

_{g,∞}*K*/ (

*M*

_{n}*T*

_{g,∞}^{2})

where *T _{g,∞}* is the limiting value of the glass transition temperature at very high molecular weight,

*M*is the number-average molecular weight and

_{n}*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

*T*is linear proportional to the free specific volume

_{g,∞}*v*:

(v_{g} - v_{g,∞}) / (*T _{g}* -

*T*) = const.

_{g,∞}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 *T _{g}*-

*M*relationship can be predicted, that is, the constant

_{n}*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 *T _{g}*'s have been calculated with the software 3Ps-Tg. The predicted and measured values are in
excellent agreement.

### Glass Transition Temperature of Polystyrene

##### References

- 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)