Time-Temperature Superposition and the WLF Equation
The TTS Principle
Time-temperature superposition (TTS) is the most powerful concept in polymer viscoelasticity for structural engineering. It states that changing the temperature of an amorphous polymer has the same effect on its viscoelastic properties as shifting the logarithmic time (or frequency) scale. Heating the material is equivalent to extending the observation time; cooling is equivalent to shortening it.
This principle applies to thermorheologically simple materials — those where all molecular relaxation mechanisms share the same temperature dependence. A single shift factor aT then describes the horizontal shift required at each temperature. Most glass interlayers (PVB, SentryGlas, EVA, TPU) satisfy this condition in the transition and terminal zones (Ferry, 1980, Ch. 11).
Practical consequence: a single isothermal relaxation test covers only 3–4 decades of log(t). By testing at 9–12 temperatures and applying TTS, the resulting master curve spans 15–20+ decades — from nanoseconds to centuries — covering the full structural design range from wind gusts (3 s) to permanent loads (50 years).
When TTS fails
TTS breaks down when:
- Multiple relaxation mechanisms with different temperature dependences are active simultaneously (e.g., near secondary transitions)
- Structural changes occur with temperature: crystallisation, melting, or chemical reactions
- The material is deep in the glassy zone, where sub-Tg motions have different activation energies (Ferry, 1980, Sec. F)
For EVA interlayers, Centelles et al. (2021) observed a secondary crystallisation near 44°C that complicates TTS in that temperature region. The master curve was still constructable outside this anomalous range.
The Shift Factor aT
The shift factor aT quantifies how far an isothermal curve at temperature T must be shifted horizontally on the log(t) axis to superpose with the curve at the reference temperature T0:
treduced = t / aT (time domain)
ωreduced = ω · aT (frequency domain)
- aT > 1 (log aT > 0): T < T0 — curve shifts to longer reduced times. The material is stiffer; relaxation is slower.
- aT = 1 (log aT = 0): T = T0 — no shift. Reference temperature.
- aT < 1 (log aT < 0): T > T0 — curve shifts to shorter reduced times. The material is softer; relaxation is faster.
Master Curve Construction
The step-by-step procedure:
- Measure isothermal curves at 9–12 temperatures spanning from below Tg to well above Tg. Each curve covers 3–4 decades of log(t).
- Choose a reference temperature T0 (typically 20°C for engineering applications or Tg for polymer science).
- Shift each curve horizontally on the log(t) axis until it overlaps with the T0 curve. The shift amount is log(aT) for that temperature. Centelles et al. (2021) used the Closed-Form Shifting (CFS) algorithm for automated, objective shifting.
- Verify superposition quality: shifted curves must overlap smoothly. If G′ and G″ were both measured, both must shift by the same aT.
- Fit the WLF equation to the log(aT) vs T data to obtain C1 and C2.
- Fit a Prony series to the master curve (typically 12–20 terms).
The WLF Equation
The Williams-Landel-Ferry equation is the empirical expression that describes the temperature dependence of aT for amorphous polymers above Tg:
log aT = −C10(T − T0) / (C20 + T − T0)
Where C10 and C20 are empirical constants specific to the material and the choice of reference temperature T0.
Converting between reference temperatures
If WLF constants c10, c20 are known at T0, constants at any other reference T1 are (Ferry, 1980, Eq. 24–25):
c11 = c10 · c20 / (c20 + T0 − T1)
c21 = c20 + T0 − T1
The Vogel temperature T∞
The WLF equation implies a temperature T∞ = T0 − C20 at which aT → ∞ (relaxation times become infinite). Below T∞, the material is effectively frozen. T∞ is typically about 50°C below Tg.
Arrhenius alternative below Tg
Below Tg, the WLF equation becomes less accurate. The Arrhenius form provides a better description (Ferry, 1980, Eq. 44–46):
log aT = ΔHa / (2.303 R) · (1/T − 1/T0)
Where ΔHa is the apparent activation energy and R = 8.314 J/(mol·K) is the gas constant.
Physical Basis: Free Volume
The WLF equation is not merely empirical — it has a physical foundation in free volume theory (Ferry, 1980, Ch. 11, Sec. C).
The key assumptions:
- The fractional free volume f (the ratio of “empty” space to total volume) controls molecular mobility: more free volume → faster relaxation.
- Free volume increases linearly with temperature above Tg: f = f0 + αf(T − T0)
- The viscosity (and all relaxation times) depends exponentially on 1/f: ln η ∝ B/f
Combining these gives the WLF equation with:
C10 = B / (2.303 · f0) C20 = f0 / αf
Where f0 is the fractional free volume at T0, αf is the thermal expansion coefficient of free volume, and B ≈ 1.
At Tg, the fractional free volume is approximately constant across polymers: fg ≈ 0.025 ± 0.005 (about 2.5%). This universality explains why “universal” WLF constants exist as rough approximations (C1g ≈ 17.44, C2g ≈ 51.6 K when T0 = Tg). However, Ferry cautions that the variation between polymers is too large for universal values to be reliable:
Do not rely on universal WLF constants. Ferry (1980, p. 289) explicitly states: “the actual variation from one polymer to another is too great to permit use of these universal values except as a last resort in the absence of other specific data.” Always use material-specific constants from experimental characterisation.
WLF Constants for Glass Interlayers
Centelles et al. (2021) determined WLF constants from stress relaxation data at T0 = 20°C for seven commercial interlayers:
| Material | C1 | C2 (°C) | T∞ = T0−C2 (°C) |
|---|---|---|---|
| EVALAM (EVA) | 8.99 | 92.8 | −73 |
| EVASAFE (EVA) | 13.2 | 133.8 | −114 |
| PVB BG-R20 | 14.7 | 88.0 | −68 |
| Saflex DG-41 | 17.5 | 110.2 | −90 |
| PVB ES | 20.2 | 125.1 | −105 |
| SentryGlas | 102.3 | 604.5 | −585 |
| TPU | 15.8 | 196.7 | −177 |
SentryGlas has exceptionally large C1 and C2 because its Tg (≈ 55°C) is far above the reference temperature T0 = 20°C. At 20°C, SentryGlas is still in the glassy zone, where the WLF curve is extremely steep. The PVB materials show C1 ≈ 15–20 and C2 ≈ 88–125°C, which are typical polymer values when T0 is near but above Tg.
Impact on Design Accuracy
The TTS/WLF approach transforms structural glass design accuracy. Galic et al. (2022) measured laminated glass beam deflections and compared with different calculation methods:
| Method | Interlayer input | Error vs experiment |
|---|---|---|
| EN 16612 simplified | Stiffness families (lookup ω) | ≈ 61% |
| SCIA Engineer (glass addon) | Stiffness families | ≈ 64% |
| Wolfel-Bennison + TTS | Actual G(t,T) from master curve | ≈ 3% |
| Abaqus with Prony + WLF | Full viscoelastic (TRS model) | < 1% |
The accuracy bottleneck is the interlayer data, not the structural solver. A free analytical method with TTS-derived G data (3% error) beats expensive FEM with stiffness families (64% error). The master curve + WLF equation replaces the entire EN 16613 stiffness family system.
From master curve to design G value
For any design scenario defined by load duration td and temperature Td:
- Compute aT(Td) from the WLF equation
- Compute the reduced time: treduced = td / aT
- Evaluate the Prony series at treduced to get G at the design condition
This single procedure replaces the entire EN 16613 stiffness family lookup table with a continuous, physics-based calculation.
See TTS in Action
Our Prony Calculator uses TTS-derived master curves for every interlayer in the database. Select a material, choose a temperature, and see the full relaxation curve — spanning decades that no single test could cover.
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