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Avrami (JMAK) Isothermal Crystallization Fraction & Half-Time Formula
Calculate Johnson-Mehl-Avrami-Kolmogorov (JMAK) transformed phase fraction X(t) = 1 - EXP(-k * t^n) and crystallization half-time t_1/2 = (LN(2) / k)^(1 / n) for polymers, steels, and glasses.
Model sigmoidal S-curve nucleation, growth, and impingement in isothermal annealing!
During isothermal polymer spherulite crystallization, austenite-to-pearlite steel transformation, or metallic-glass devitrification, transformation starts slowly during nucleation, accelerates rapidly as 3D grains grow, and levels off asymptotically as growing grains impinge on one another! The Johnson-Mehl-Avrami-Kolmogorov (JMAK) Equation captures this exact sigmoidal S-curve after correcting for phantom grain overlap: X(t) = 1 - EXP(-k * (t ^ n))!
Here Avrami exponent n (typically 1 to 4) encodes nucleation mode + growth dimensionality (e.g., n = 3 for instantaneous site-saturated 3D spheres, n = 4 for constant-rate sporadic 3D nucleation), and setting X(t_1/2) = 0.50 isolates the Crystallization Half-Time: t_1/2 = (LN(2) / k) ^ (1 / n)!
| Annealing Time t (Polymer Spherulites: k = 0.02 min^-3, Avrami n = 3) | Extended Volume k * t^n | Transformed Fraction X(t) | Microstructure Stage |
|---|---|---|---|
| Early Induction (t = 2.0 min) | 0.160 | 0.1479 (14.8%) | Isolated nuclei growing |
| Half-Time t_1/2 = 3.260 min | 0.6931 (= LN(2)) | 0.5000 (50.0%) | Peak transformation rate |
| Late Impingement (t = 4.0 min) | 1.280 | 0.7220 (72.2%) | Spherulite impingement |
Avrami-JMAK Transformed Volume Fraction X(t) = 1 - EXP(-k * t^n)
Evaluates transformed fraction in [0, 1] from rate constant B2 (> 0), Avrami exponent B3 (> 0), and hold time B4 (>= 0).
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0), "Invalid Avrami Parameters", ROUND(1 - EXP(-B2 * (B4 ^ B3)), 4))Isothermal Crystallization Half-Time t_1/2 = (LN(2) / k)^(1 / n)
Computes the exact hold time needed to reach 50% phase transformation.
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Avrami Parameters", ROUND((LN(2) / B2) ^ (1 / B3), 3))Try the example
Use the original values or change them to test your understanding.