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 / T_g,mix ≈ ∑_i ω_i / T_g,i

where T_g,mix and T_g,i are the glass transition temperature of the mixture / copolymer and of the components, respectively, and ω_i is the mass fraction of component i.
For two components A and B, the Fox equation reduces to

1 / T_g,mix ≈ ω_A / T_g,A + ω_B / T_g,B

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, ΔC_pi T_g,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 T_g values of some commercial polybutadienes are shown in the table below. The agreement is within the experimental uncertainty of T_g measurements.
 

References

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