Fox Equation

Several approaches have been proposed for estimating the glass transition temperature of mixtures and random copolymers from knowledge of the properties of the pure components. Although different in detail, the proposed relationships are all based on the additivity of basic thermophysical properties. One of the most widely used equation for predicting glass transition temperatures of amorphous mixtures and random copolymers is the Fox equation:(1)

1 / Tg,mixi ωi / Tg,i

where Tg,mix and Tg,i are the glass transition temperature of the mixture and of the components, and ωi is the mass fraction of component i.

The Fox equation has been derived from experimental findings. However, it can be directly derived from the Gordon-Taylor equation, if we assume that the product of the glass transition temperature and the change in specific heat, ΔCpi Tg,i, is identical for all compounds.

The Fox equation should only be applied to components with similar structure and / or solubility parameter (cohesive energy density), that is, to mixtures of components with very weak or no specific intermolecular interaction. A prominent example are (random) copolymers of 1,2-polybutadiene, and cis- and trans-1,4-polybutadiene. The experimental and predicted Tg values of some commercial polybutadienes are shown in the table below. The agreement is within the experimental uncertainty of Tg measurements.

Glass Transition Temperature of Polybutadienes

Polybutadiene (%cis / %trans / %vinyl) Exper. Tg (K) Predicted Tg (K)
Lanex Buna (38 / 52 / 10) 180 184
Lanex Buna (92 / 4 / 4) 168 168
Aldrich (10 / 0 / 90) 243 246.5
Firestone 645 (96 / 4 / 0) 164 166
The glass transition temperatures have been calculated with the software 3Ps-Tg from Triton Road using the values Tg,cis = 165 K, Tg,trans K = 190 K and Tg,vinyl = 261 K

References
  1. T.G. Fox, Bull. Am. Phys. Soc. 1, 123 (1956)
  • Summary

    Fox Equation

     should only be applied to blends and random copolymers if the components have similar solubility parameter.

  • The Fox equation predicts a negative deviation of the Tg composition curve from the additive rule of mixtures.

  • The Fox equation should not be applied to nonrandom mixtures.

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