Hong Life CalculatorsUnderstanding the copolymer glass transition calculator (Fox equation) — concept and principle
The glass transition temperature Tg is where an amorphous polymer changes from hard and glassy to soft and rubbery. Paint, adhesive and coating chemists combine a "hard" monomer with a high Tg and a "soft" one with a low Tg to hit a target in between. The Fox equation gives a quick estimate of the Tg of a random copolymer or a fully miscible blend from the weight fractions and Tg values of its two components.
How it is calculated
- 1/Tg = w₁/Tg₁ + w₂/Tg₂, with all temperatures in kelvin.
- w₂ = 1 − w₁.
The calculator converts your °C inputs to kelvin (adding 273.15), applies the equation, and converts the result back to °C.
The principle
Thomas Fox proposed in 1956 that each component contributes to the reciprocal of Tg in proportion to its weight fraction. The idea rests on free volume: a polymer becomes rubbery when its free volume reaches a critical level, and each component adds its share. Because it averages reciprocals, the Fox equation is a weighted harmonic mean, so the result always lies below the simple weighted average and leans toward the lower Tg. Kelvin must be used because the equation involves ratios; using °C, especially with negative values, gives nonsense.
Worked example
- Defaults: w₁ = 0.6, Tg₁ = 105 °C, Tg₂ = −50 °C, so w₂ = 0.4.
- Convert: Tg₁ = 378.15 K, Tg₂ = 223.15 K.
- 0.6 ÷ 378.15 = 0.0015867 and 0.4 ÷ 223.15 = 0.0017925; sum = 0.0033792 K⁻¹.
- Tg = 1 ÷ 0.0033792 = 295.93 K.
- Tg = 295.93 − 273.15 = 22.78 °C.
A simple linear average would give 0.6 × 105 + 0.4 × (−50) = 43 °C, about 20 °C higher.
| Weight fraction polymer 1 | Tg (°C) |
|---|---|
| 0.15 | -35.38112849 |
| 0.3 | -18.7125735 |
| 0.45 | 0.4692363813 |
| 0.6 | 22.77906365 |
| 0.75 | 49.04997136 |
| 0.9 | 80.43966059 |
| 1.2 | 166.0078064 |
How to read the result
A Tg near room temperature, as here, means the material is at the boundary between glassy and rubbery at ambient conditions. For latex paints this relates to film formation: a binder must be soft enough (often with help from coalescing agents) to form a continuous film at the application temperature. To find the composition for a target Tg, adjust w₁ until the result matches, or solve w₁ = (1/Tg − 1/Tg₂) ÷ (1/Tg₁ − 1/Tg₂).
Common mistakes
- Using mole fractions instead of weight fractions.
- Entering the Tg of a homopolymer measured by a different method or heating rate than your target data.
- Applying the equation to immiscible blends, which show two separate Tg values instead of one.
Frequently asked questions
How does the Gordon–Taylor equation differ?
It adds an adjustable constant k: Tg = (w₁Tg₁ + k w₂Tg₂) ÷ (w₁ + k w₂), which fits curved data better when it is available.
Can I use it for more than two monomers?
Yes, the general form is 1/Tg = Σwᵢ/Tgᵢ; this calculator handles the two-component case.
Does it work for plasticised polymers?
It is often used as a first estimate with the plasticiser's own Tg, but accuracy is limited.
Sources: Fox, T. G. (1956). Influence of diluent and of copolymer composition on the glass temperature of a polymer system. Bulletin of the American Physical Society, 1, 123. · Gordon, M. & Taylor, J. S. (1952). Ideal copolymers and the second-order transitions of synthetic rubbers. Journal of Applied Chemistry, 2, 493–500.