JMAK (Avrami) Transformed Fraction

Hold a steel at a temperature where austenite is unstable and it will transform — but not all at once. Very little happens at first, then the transformation accelerates, then it slows again and approaches completion asymptotically. Plot fraction transformed against the logarithm of time and you get a sigmoid, and that sigmoid is the same shape for pearlite forming in steel, for recrystallisation of a cold-worked metal, for precipitation in an aluminium alloy, and for crystallisation of a polymer. The equation that describes it was derived independently by Johnson and Mehl and by Avrami in 1939, with Kolmogorov's earlier work behind both, and is universally called the JMAK or Avrami equation.

The derivation is worth understanding because it explains the exponent. Suppose nuclei of the new phase appear at random points and grow outward at a constant rate. If they never ran into each other, the transformed volume would grow as a simple power of time — as t³ for spheres from a fixed set of nuclei, as t⁴ if new nuclei keep appearing throughout. Call that hypothetical volume the extended volume, which can exceed the sample. The trick is then to account for impingement: the probability that a randomly chosen point has not yet been swallowed is exp(−V_extended), by exactly the same Poisson argument that gives the exponential in radioactive decay. So X = 1 − exp(−ktⁿ), and the exponential is not an assumption but the impingement correction.

That is why n is not a free fitting parameter — it encodes the geometry of nucleation and growth. Three-dimensional growth from a fixed number of sites all present at the start gives n = 3; if nucleation continues at a constant rate throughout, n = 4. Two-dimensional or plate-like growth gives n = 2, and growth along a line or on grain edges gives n = 1. And if growth is diffusion-controlled — the particle limited by how fast solute can be carried away, so its radius grows as √t rather than linearly — every one of those values is halved in its time dependence, giving the classic half-integers at 1.5 and 2.5. Fitting your own data and finding n near 4 or near 1.5 is telling you something real about what is nucleating and how it is growing.

k is unit-bound in a way that catches people out. Its dimensions are time^(−n), so its numerical value depends on the exponent AND the time unit together. This site works in seconds. The same transformation fitted with time in minutes gives a k that is 60ⁿ times smaller — a factor of 3600 at n = 2, and 12.96 million at n = 4. A k lifted from a paper without both its exponent and its time unit is unusable. All of the temperature dependence hides in k as well: it follows an Arrhenius form, and in a diffusional transformation it passes through a maximum somewhere between the equilibrium temperature, where the driving force is zero, and low temperature, where diffusion has effectively stopped. That competition between driving force and mobility is what puts the nose on a TTT curve, and the nose is the whole reason a quench has to be fast.

The assumptions are the honest part. Nucleation is random in space — which is exactly what it is not in a real alloy, where grain boundaries, grain edges, grain corners and inclusions are all strongly preferred sites. A site-saturated transformation that nucleates only on boundaries pulls the measured n down, and a heavily heterogeneous one can give values that no geometric interpretation fits. Growth rate is constant, which fails once solute has to diffuse ahead of the interface. And the equation is isothermal, which no real furnace cycle is.

That last point is what separates a TTT diagram from a CCT diagram. A TTT (time–temperature–transformation) diagram is this equation solved for time at every temperature, plotted as a set of constant-fraction curves. A real component cools continuously, and to predict what happens you cut the cooling curve into small isothermal steps and add the transformation from each — the additivity rule, due to Scheil, the same Scheil as the segregation equation. The result is the CCT diagram, which is shifted down and to the right of the TTT diagram it was built from. Reading a continuous cool off a TTT diagram is a standard error and it always predicts a harder result than you get.