Polymer Glass Transition: Free Volume & WLF Equation

Material

Controls

Temperature T 300 K

Free volume at T_g, f_g 0.025

WLF constant C₂ 51.6 K

Live State

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Free volume f(T)

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Segmental mobility

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WLF shift log₁₀(a_T)

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Specific volume V(T)

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How it works

Above the glass transition temperature T_g, empty space between chain segments — free volume — grows linearly with temperature (Doolittle / free-volume theory):

f(T) = f_g + α_f (T − T_g),   T ≥ T_g
f(T) ≈ f_g,                   T < T_g  (frozen in)

Below T_g the free volume needed for a segment to hop into a neighboring gap can no longer be created fast enough, so it freezes at f_g and long-range chain motion effectively stops — the material is glassy. The temperature dependence of segmental relaxation time / viscosity above T_g follows the Williams–Landel–Ferry (WLF) equation:

log₁₀ a_T = −C₁ (T − T_g) / [C₂ + (T − T_g)]
C₁ = 1 / (2.303 f_g),   C₂ = f_g / α_f
η(T) = η(T_g) · a_T

a_T is the ratio of relaxation time (or viscosity) at T to its value at T_g. The "universal" constants C₁ ≈ 17.4 and C₂ ≈ 51.6 K fit most amorphous polymers within ~100 K of T_g.

Temperature slider — sweeps T through T_g; chains vibrate weakly (glass) below it and wiggle diffusively (rubber/melt) above it, driven by the mobility computed from a_T.
Material dropdown — sets a realistic T_g for five common polymers.
f_g and C₂ sliders — change how abruptly the free volume opens up and how sharply mobility rises above T_g.
Cool ramp — animates T sliding downward in real time so the specific-volume curve traces its characteristic kink at T_g live.

Real-world relevance: this is why a rubber band goes brittle in a freezer (below its T_g) but stays flexible at room temperature, and why processing engineers pick molding temperatures well above a polymer's T_g to keep it flowable.

Bead-spring chain segments Bounding cell

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