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Fox Equation (Copolymer & Blend Tg)

Predicts the glass transition temperature (Tg in Kelvin) of miscible polymer blends or random copolymers based on mass fractions (w1, w2) and homopolymer Tgs.

Mathematical Canonical Relation
1Tg=w1Tg1+w2Tg2\frac{1}{T_g} = \frac{w_1}{T_{g1}} + \frac{w_2}{T_{g2}}
Plain Formula: 1/Tg = (w1/Tg1) + (w2/Tg2)

Live Engineering Parameter Solver

Input your production or test values below for instant, verified output.

Verified Engine
w1
Range: 0 – 1 fraction
Tg1_C
Range: -100 – 350 °C
Tg2_C
Range: -150 – 350 °C
Calculated Engineering Result:
11.9 °C (285 K)
Engineering Verification:Flexible rubbery behavior at room temperature (25°C).

When & Where to Apply

Use when compounding miscible polymer blends or formulating plasticized PVC to calculate resulting Tg.

Common Engineering Pitfalls

  • Entering temperatures in Celsius instead of Kelvin
  • Applying to immiscible phase-separated polymer blends

Physical Assumptions & Validity Limits

  • Complete thermodynamic miscibility / single amorphous phase
  • Additivity of free volume
  • Temperatures expressed in Kelvin (K)

Variable Symbols, SI Units & Dimensional Specifications

SymbolVariable MeaningSI UnitTypical Industrial Range
T_gGlass transition temperature of blend/copolymerK (°C)—
w_1Weight fraction of Component 1fraction0.70
T_{g1}Glass transition of Component 1 in KelvinK358.15
w_2Weight fraction of Component 2fraction0.30
T_{g2}Glass transition of Component 2 in KelvinK193.15

Worked Industrial Numerical Example: Plasticized PVC Tg Reduction with DOP Plasticizer

Problem Statement:

Unplasticized PVC has Tg1 = 85°C (358.15 K). If 30 wt% DOP plasticizer (Tg2 = -80°C = 193.15 K) is compounded into PVC (w1 = 0.70, w2 = 0.30), calculate resulting flexible PVC Tg.

Step-by-Step Numerical Solution:
  1. Apply Fox Equation: 1/Tg = (0.70 / 358.15) + (0.30 / 193.15)
  2. Compute term 1: 0.70 / 358.15 = 0.0019545 K^-1
  3. Compute term 2: 0.30 / 193.15 = 0.0015532 K^-1
  4. Sum inverses: 1/Tg = 0.0035077 K^-1
  5. Invert to get Tg: Tg = 285.08 K = 11.93°C
Final Answer: Tg = 11.9°C (285.1 K)
💡 Engineering Interpretation: Adding 30% DOP plasticizer depresses PVC Tg from 85°C down to 11.9°C, converting rigid pipe resin into flexible room-temperature sheet packaging.

Authoritative Technical Reference & Source Provenance

Second-Order Transition Temperatures and Related Properties of Polymers by T. G. Fox (Bull. Am. Phys. Soc., 1956), 1, Page 123.
Connected Curriculum Lesson

Glass Transition Temperature (Tg) & Free Volume Theory

Study Full Lesson