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T4 · Rheology / FlowPolymer Rheology & Melt FlowGATE 2027 Syllabus (XE5)
Williams-Landel-Ferry (WLF) Shift Factor Equation
Calculates the horizontal shift factor aT for constructing viscoelastic master curves across temperatures from Tg up to Tg + 100°C based on fractional free volume expansion.
Mathematical Canonical Relation
Plain Formula: log(aT) = -C1 * (T - Tg) / (C2 + (T - Tg))
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T_exp
Range: -20 – 250 °C
T_glass
Range: -80 – 200 °C
C1_val
Range: 10 – 25 dimensionless
C2_val
Range: 30 – 80 K
Calculated Engineering Result:
log(aT) = -6.41 (aT = 3.87e-7)
Engineering Verification:Temperature shift of 30 K above Tg.
When & Where to Apply
Use when applying Time-Temperature Superposition (TTS) to predict long-term polymer creep, relaxation modulus, and dynamic shear storage/loss moduli.
Common Engineering Pitfalls
- Applying the WLF equation at temperatures far above Tg + 100°C (Arrhenius relation applies at T > Tg + 100°C)
- Mixing up Celsius and Kelvin temperature differences
Physical Assumptions & Validity Limits
- Temperature regime is between Tg and Tg + 100°C
- Universal constants C1 = 17.44 and C2 = 51.6 K for reference state at Tg
- Free volume increases linearly above Tg
Variable Symbols, SI Units & Dimensional Specifications
| Symbol | Variable Meaning | SI Unit | Typical Industrial Range |
|---|---|---|---|
| \log a_T | Logarithm of temperature shift factor | dimensionless | — |
| T | Experimental test temperature | °C (or K) | 130 |
| T_g | Polymer glass transition temperature | °C (or K) | 100 |
| C_1 | Universal WLF constant C1 | dimensionless | 17.44 |
| C_2 | Universal WLF constant C2 | K | 51.6 |
Worked Industrial Numerical Example: Polystyrene Rheological Time Shift at 30°C Above Tg
Problem Statement:
For Polystyrene with Tg = 100°C, calculate the shift factor log(aT) and shift multiplier aT at test temperature T = 130°C using standard WLF constants C1 = 17.44 and C2 = 51.6 K.
Step-by-Step Numerical Solution:
- Calculate temperature difference: T - Tg = 130 - 100 = 30 K
- Evaluate numerator: -17.44 * 30 = -523.2
- Evaluate denominator: 51.6 + 30 = 81.6
- Calculate log(aT): -523.2 / 81.6 = -6.41
- Calculate aT: 10^(-6.41) = 3.89 × 10^-7
Final Answer: log(aT) = -6.41 (aT = 3.89 × 10^-7)
💡 Engineering Interpretation: At 130°C, molecular relaxation processes occur approximately 2.5 million times faster than at Tg, compressing long-term creep timescales into laboratory testing windows.
Authoritative Technical Reference & Source Provenance
Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers by M.L. Williams, R.F. Landel, J.D. Ferry (J. Am. Chem. Soc., 1955), Vol 77, Page 3701-3707.
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