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; the Fox equation provides a fast, single-parameter estimate that is widely used in reactor design, extrusion, and solvent-borne formulation workflows.
Methodology & Formulas
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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