Vogel-Fulcher-Tammann Viscosity

Glass viscosity spans about fifteen orders of magnitude between a runny melt and an annealed solid, and it does not obey Arrhenius. Plot $\log\eta$ against $1/T$ for a soda-lime glass and the line curves — steeply upward as the glass cools toward its transition — so a single activation energy cannot describe it. Vogel in 1921, Fulcher in 1925 and Tammann and Hesse in 1926 each arrived independently at the same repair: put a divergence temperature in the denominator. $\log_{10}\eta = A + B/(T - T_0)$. It works, over a wide range, for almost every glass-forming liquid.

The mind the units warning comes first, because it is the mistake that actually gets made. Classical glass technology quotes viscosity in POISE, and the reference points every glassmaker knows by heart are all in poise: working point $10^4$, softening point $10^{7.6}$, annealing point $10^{13}$, strain point $10^{14.5}$. This site works in pascal-seconds, and 1 Pa·s = 10 poise, so every one of those exponents drops by exactly 1 — working point $10^3$ Pa·s, annealing point $10^{12}$ Pa·s. That single unit shift moves $A$ by 1 and changes nothing else. A set of VFT constants copied from a poise-based table into a pascal-second calculation is wrong by a factor of ten at every temperature, and it looks entirely plausible while being wrong.

Those reference points are what the equation is for. Every glass process is specified by a viscosity and run by a temperature. The gob feeder wants the glass at a particular stiffness; the forming machine wants another; the annealing lehr is built around the annealing point, where internal stress relaxes in about fifteen minutes, and cools slowly through to the strain point, where it does not relax at all any more. Fibre drawing, container forming and the float bath each live at their own point on this curve. Given a fit, this equation converts every one of those viscosities into a furnace setpoint, which is exactly why the constants are worth measuring.

VFT is a fit, not a law, and it extrapolates badly. Three constants tuned over a viscosity window — typically the few orders of magnitude someone actually measured — and outside that window the fit has no authority. It is worst going downward. As the temperature approaches $T_0$ the equation predicts genuinely infinite viscosity, and no glass does that: real glasses go on flowing below $T_0$, immeasurably slowly but not at zero rate. Constants fitted around the working point should not be trusted at the annealing point, and a single set claiming to span all fifteen decades is generally a compromise that is a little wrong everywhere.

$T_0$ itself invites over-reading and deserves care. It is a fitting parameter that usually lands 50 to 150 K below the measured glass transition temperature. There are theories — Adam-Gibbs, and the ideal-glass picture behind it — that give it a physical reading as the temperature where configurational entropy would vanish, and whether that is a real thing or a coincidence of functional form remains one of the genuinely unsettled arguments in condensed matter. What is not in doubt is that no glass has ever been observed to stop.

The constant $B$ plays the role an activation energy plays in Arrhenius, and the analogy is worth handling carefully. Because the denominator is $(T - T_0)$ rather than $T$, the effective activation energy is not constant — it climbs steeply as the glass cools. That climb is what "fragility" names in Angell's classification. A strong liquid like fused silica stays close to Arrhenius all the way down and has a small $T_0$; a fragile one turns up sharply near the transition. Soda-lime is moderately fragile, and adding network modifiers makes a glass more fragile, which shortens the working range — the temperature window in which the glass can be formed. That is a real production consequence of a composition change, and it is visible in these three constants before it is visible on the shop floor.