the Avrami equation
/ AV-rah-mee /
This is the tidy formula that draws the S-curve. The Avrami equation is a simple expression for the fraction y of material transformed after time t, and it reproduces the slow-fast-slow sigmoidal shape seen in real transformations.
The formula is y = 1 - exp(-k t^n), where y runs from 0 to 1, t is time, and k and n are constants for a given transformation and temperature. The exponent n reflects the nucleation and growth geometry (often between 1 and 4), while k sets the overall speed and packs in the strong temperature dependence. At small t, y is tiny; as t grows the exponential drives y toward 1. For example with k = 0.01 per second^n and n = 2, at t = 10 s the exponent is -0.01 times 100 = -1, so y = 1 - exp(-1) = 0.63, meaning 63 percent transformed.
It gives engineers a compact way to describe and interpolate transformation data, and to define a rate. But it assumes random nucleation and constant conditions, so it is an empirical fit, not a law from first principles. The values n and k are extracted from experiment, and it works best for isothermal (constant-temperature) holds, which is exactly the setting of a TTT diagram.
Fitting fraction-transformed data to y = 1 - exp(-k t^n) and reading off k lets you compare how fast the same reaction runs at two different temperatures.
One equation for the whole S-shaped history.
The Avrami constants k and n are empirical fits valid for a specific temperature and condition; do not treat them as universal material constants.