Above the glass transition temperature Tg, 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 Tg 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 Tg 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
This 2D version renders the same theory as a lattice-gas polymer model: each chain is a snake of beads occupying cells on a grid, surrounded mostly by empty lattice sites (free volume). Instead of continuum springs, chains move by reptation — a bead at one end vanishes while a new bead appears at the opposite end in a random empty neighboring cell — the classic discrete-time algorithm for simulating polymer diffusion on a lattice. The probability that a reptation move is attempted and accepted each frame is set directly by the WLF mobility a_T computed above, so the same number that predicts viscosity in the lab also throttles how often chains actually slither through the lattice here.
- Temperature slider — sweeps T through Tg; below it chains only jitter in place (glassy), above it they reptate through the lattice at a rate set by a_T (rubbery/melt).
- Material dropdown — sets a realistic Tg for five common polymers.
- fg and C₂ sliders — change how abruptly the free volume opens up (density of highlighted void cells) and how sharply mobility — and hop rate — rises above Tg.
- Cool ramp — animates T sliding downward in real time so the specific-volume curve traces its characteristic kink at Tg live, and hop rate visibly collapses.
- Drag / scroll on the lattice — pan and zoom the grid; it does not change the physics, only the view.
Real-world relevance: this is why a rubber band goes brittle in a freezer (below its Tg) but stays flexible at room temperature, and why processing engineers pick molding temperatures well above a polymer's Tg to keep it flowable.