Glass Transition Temperature Approximation (Fox Equation)
Reference ID: MET-3B2B | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The glass-transition temperature Tg marks the onset of large-scale segmental motion in an amorphous polymer. In process engineering it governs:
Minimum film-formation temperature for coatings and adhesives.
Maximum service temperature for thermoplastic parts.
Storage conditions to avoid cold-flow or embrittlement.
When two amorphous polymers are blended, the resulting Tg is not a linear average; while the Fox equation provides a fast, single‑parameter estimate, the more detailed Gordon‑Taylor approach can predict the glass transition temperature of mixtures. Gordon‑Taylor method for polymer blends is widely referenced in reactor design, extrusion, and solvent‑borne formulation workflows.
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Convert laboratory data to absolute temperature
\[ T_{g,i}\;[\mathrm{K}] = T_{g,i}\;[^\circ\mathrm{C}] + 273.15 \]
Apply the Fox equation
The additivity rule for the inverse glass-transition temperature of the blend is
\[ \frac{1}{T_{g,\mathrm{blend}}} = \frac{w_1}{T_{g,1}} + \frac{w_2}{T_{g,2}} \]
where wi are the mass fractions satisfying w1 + w2 = 1.
Recover the blend temperature in Celsius
\[ T_{g,\mathrm{blend}}\;[^\circ\mathrm{C}] = T_{g,\mathrm{blend}}\;[\mathrm{K}] - 273.15 \]
Regime
Mass-fraction constraint
Typical accuracy
Compatible amorphous blends
0 < w1 < 1
±3–5 K
Highly immiscible systems
Same
±10–20 K
The Fox equation, 1/Tg = w1/Tg1 + w2/Tg2 + …, is a fast, solvent-free way to predict the Tg of a miscible polymer blend or copolymer from the weight fractions (w) and known Tg values of the pure components. Use it during early-stage extrusion or compounding trials when you need a quick target temperature for dryer or melt-zone set-points and DSC data are not yet available.
Typical deviation is ±3–7 °C for fully miscible amorphous systems; errors grow when:
Components are partially miscible or phase-separate during cooling
Strong hydrogen bonding or specific interactions occur (e.g., PMMA/PVPh blends)
One component crystallizes above its Tg (e.g., PET in a blend)
Always run a DSC check on the final compounded pellet before releasing process parameters to production.
Fox ignores diluent mobility; for plasticized resins treat the plasticizer as an additional low-Tg component (Tg ≈ −80 °C for common phthalates) and reduce weight fractions of all species to a 100 % polymer-plus-plasticizer basis. For mineral-filled systems Fox is not valid—use empirical correlations or DSC instead.
Worked Example – Estimating the Glass Transition of a PMMA/Plasticizer Blend
A small compounding line needs to predict the glass-transition temperature of a 60/40 (wt %) poly(methyl methacrylate)/plasticizer mixture before extrusion. The plant engineer uses the Fox equation to avoid trial-and-error runs.
Knowns
Tg,1 = 100 °C (PMMA)
Tg,2 = –10 °C (plasticizer)
w1 = 0.600 (mass fraction PMMA)
w2 = 0.400 (mass fraction plasticizer)
Step-by-Step Calculation
Convert individual Tg values to kelvin:
Tg,1 = 100 + 273.15 = 373.15 K
Tg,2 = –10 + 273.15 = 263.15 K
Apply the Fox equation in reciprocal form:
\[
\frac{1}{T_{g,\text{blend}}} = \frac{w_1}{T_{g,1}} + \frac{w_2}{T_{g,2}}
\]
\[
\frac{1}{T_{g,\text{blend}}} = \frac{0.600}{373.15} + \frac{0.400}{263.15} = 0.001608 + 0.001520 = 0.003128\ \text{K}^{-1}
\]
Invert to obtain the blend Tg in kelvin:
Tg,blend = 1 / 0.003128 = 319.7 K
Convert back to °C for plant use:
Tg,blend = 319.7 – 273.15 = 46.5 °C
Final Answer
The predicted glass-transition temperature of the 60/40 PMMA/plasticizer blend is 46.5 °C.
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