Engineering & Science
Fox Equation & Gordon-Taylor Copolymer Glass Transition Temperature (Tg)
Predict the glass transition temperature Tg (in °C via absolute Kelvin!) of random copolymers, plasticized PVC, and miscible amorphous blends using the Fox harmonic rule and Gordon-Taylor equation.
Never plug Celsius directly into the Fox equation—always convert to Kelvin!
Formulating acrylic latex paints, SBR tire rubbers, or amorphous pharmaceutical solid dispersions requires tuning copolymer glass transition temperature Tg by blending a glassy monomer (Component 1, e.g., styrene Tg1 = +100 °C = 373.15 K) with a rubbery comonomer or plasticizer (Component 2, e.g., butadiene Tg2 = -70 °C = 203.15 K)! Plugging Celsius into 1/Tg = w1/Tg1 + w2/Tg2 crashes at 0 °C or flips sign for sub-zero rubbers—ALL glass transition mixing laws MUST use absolute Kelvin (T_K = T_C + 273.15)!
The parameter-free Fox equation takes the weight-fraction harmonic mean in Kelvin: Tg_K = 1 / ((w1 / Tg1_K) + ((1 - w1) / Tg2_K)), then subtracts 273.15 to return °C! When thermal expansion increments Delta_alpha differ strongly between components, the Gordon-Taylor equation weights Component 2 by k_GT = (rho1 * Delta_alpha2) / (rho2 * Delta_alpha1): Tg_K = (w1 * Tg1_K + k_GT * (1 - w1) * Tg2_K) / (w1 + k_GT * (1 - w1)). At w1 = 0.70 styrene / 0.30 butadiene, Fox predicts +25.12 °C, while Gordon-Taylor (k_GT = 0.50) predicts +70.00 °C!
| Copolymer Composition (Tg1 = 100 °C, Tg2 = -70 °C) | Fox Equation Tg (°C) | Gordon-Taylor Tg (k = 0.50) | Physical Behavior at 25 °C |
|---|---|---|---|
| 70 wt% Styrene / 30 wt% Butadiene | +25.12 °C (298.27 K) | +70.00 °C (343.15 K) | Leathery / Glassy Resin |
| 50 wt% Styrene / 50 wt% Butadiene | -10.07 °C (263.08 K) | +43.52 °C (316.67 K) | Fox: Flexible Elastomer at RT |
| Invalid Weight Fraction w1 > 1 | Guarded Input Error | Out of [0, 1] Bound | "Invalid Blend Inputs" |
Fox Equation Copolymer Glass Transition Temperature Tg (°C)
Converts B2 and B3 from °C to Kelvin (+273.15), evaluates harmonic mean 1 / (w1/Tg1_K + (1-w1)/Tg2_K), and converts back to °C.
=IF(OR(B2 <= -273.15, B3 <= -273.15, B4 < 0, B4 > 1), "Invalid Blend Inputs", ROUND((1 / ((B4 / (B2 + 273.15)) + ((1 - B4) / (B3 + 273.15)))) - 273.15, 2))Gordon-Taylor Blend Glass Transition Temperature Tg (°C)
Evaluates (w1*Tg1_K + k*(1-w1)*Tg2_K) / (w1 + k*(1-w1)) - 273.15 using Gordon-Taylor constant k in B5.
=IF(OR(B2 <= -273.15, B3 <= -273.15, B4 < 0, B4 > 1, B5 <= 0), "Invalid Blend Inputs", ROUND((((B4 * (B2 + 273.15)) + (B5 * (1 - B4) * (B3 + 273.15))) / (B4 + (B5 * (1 - B4)))) - 273.15, 2))Try the example
Use the original values or change them to test your understanding.