Williams-Landel-Ferry (WLF) Equation: Viscoelastic Shift Factor log10(aT)

Predict 6+ decades of polymer melt viscosity drop within Tg to Tg + 100 °C!

Between the glass transition temperature Tg and Tg + 100 °C, free-volume expansion causes amorphous polymer relaxation times and zero-shear viscosities to plummet far steeper than simple Arrhenius kinetics! Malcolm Williams, Robert Landel, and John Ferry (1955) discovered that shifting dynamic mechanical (DMA) frequency sweeps horizontally by factor aT = eta(T) / eta(Tr) collapses decades of data onto one universal master curve: log10(aT) = (-C1 * (T - Tr)) / (C2 + (T - Tr))!

When the reference temperature Tr is chosen directly at Tg (e.g., Tr = Tg = 100 °C for polystyrene), universal WLF constants are C1 = 17.44 and C2 = 51.6 K (or °C difference)! Heating just +30 °C above Tg to T = 130 °C yields log10(aT) = (-17.44 * 30) / (51.6 + 30) = -6.4118—meaning relaxation time and viscosity drop by over SIX ORDERS OF MAGNITUDE (aT = 10^-6.4118 = 3.9e-7)!

WLF Logarithmic Time-Temperature Shift Factor log10(aT)

Guards against the Vogel-Fulcher-Tammann denominator singularity C2 + (T - Tr) <= 0, then computes (-C1 * (T - Tr)) / (C2 + (T - Tr)).

=IF(OR(B2 <= 0, B3 <= 0, (B3 + (B5 - B4)) <= 0), "Outside WLF Validity (T - Tr <= -C2)", ROUND((-B2 * (B5 - B4)) / (B3 + (B5 - B4)), 4))

Linear Viscosity / Relaxation Shift Multiplier aT = 10 ^ log10(aT)

Exponentiates 10^log10(aT) to give the direct ratio eta(T) / eta(Tr).

=IF(OR(B2 <= 0, B3 <= 0, (B3 + (B5 - B4)) <= 0), "Outside WLF Validity (T - Tr <= -C2)", ROUND(10 ^ ((-B2 * (B5 - B4)) / (B3 + (B5 - B4))), 8))

Further reading

• Microsoft Support — POWER function