Diagram view

System parameters

Equilibrium temperature, T_e

Driving-force scale, L

Interfacial energy, σ

Mobility activation energy, Q

Avrami exponent, n

Kinetic rate scale

k(T) = k₀·exp(−Q/RT)·exp(−ΔG*(T)/k_BT). n=4 (constant nucleation rate + 3D interface-controlled growth) is the textbook default; lower n describes site-saturated nucleation and/or lower-dimensional (rod- or plate-like) growth.

What this is actually computing. k(T) = k₀·exp(−Q/RT)·exp(−ΔG*(T)/k_BT) really is evaluated from the same classical-nucleation ΔG*(T) = 16πσ³/(3ΔG_v²) used in the Spinodal, Nucleation & Coarsening Lab (ΔG_v = −L(T_e−T)/T_e), multiplied by a real Arrhenius mobility factor: the C-curve shape you see is a genuine consequence of those two competing, physically real temperature dependences (a barrier that vanishes at T_e, a mobility that vanishes at low T), not a fitted or hand-drawn shape. What's not independently measured: k₀ is a single free scale parameter (lumping atomic attachment rate and JMAK geometric constants) chosen only to put the nose at a plottable, illustrative time; a real alloy's k₀ comes from separate kinetic measurements, not from this model. The Avrami exponent n is held constant through the whole transformation; real transformations often show n drifting as the reaction proceeds (site saturation, impingement), which this constant-n model does not capture. CCT curves are built from the TTT data by the Scheil additivity rule (Σ Δt_i/t(T_i) = 1), the standard, widely-used engineering approximation for continuous cooling, not an independent re-derivation of nucleation kinetics under a moving temperature. teaching-scale model

References

Avrami, M. Kinetics of Phase Change. I. General Theory. Journal of Chemical Physics 7, no. 12 (1939): 1103–1112. doi:10.1063/1.1750380
Johnson, W.A., and R.F. Mehl. Reaction Kinetics in Processes of Nucleation and Growth. Transactions of the American Institute of Mining and Metallurgical Engineers 135 (1939): 416–442.
Kolmogorov, A.N. On the Statistical Theory of the Crystallization of Metals (in Russian). Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya 1, no. 3 (1937): 355–359.
Scheil, E. Anlaufzeit der Austenitumwandlung. Archiv für das Eisenhüttenwesen (1935). doi:10.1002/srin.193500186
Porter, D.A., K.E. Easterling, and M.Y. Sherif. Phase Transformations in Metals and Alloys, 3rd ed. CRC Press, 2009.
MIT OpenCourseWare. 3.21 Kinetic Processes in Materials, Lecture Notes. ocw.mit.edu

Growth model

Surface composition, C_s

Bulk (initial) composition, C₀

Case-depth definition, C_target

Diffusivity, D

Time, t

Growth-law check: depth² vs. time (parabolic/diffusion-controlled growth is a straight line through the origin)

What this is actually computing. Both sub-models solve Fick's second law exactly for their stated boundary conditions: the fixed-surface case via the standard erf profile, the moving-boundary case via a from-scratch Boltzmann-similarity solution whose interface position and concentration profile are cross-checked in this tool's test suite against an independent finite-difference simulation, not just algebra checked by hand. These are two different problems and this tool does not conflate them: fixed-surface has no phase change and a boundary held at constant composition (the simple carburizing textbook case); moving-boundary has a real phase change and a boundary whose own motion is solved for self-consistently (decarburization, or diffusion-controlled thickening of any product phase such as proeutectoid ferrite, same math). What's simplified in both: constant D (no composition- or temperature-dependence, though real interstitial diffusivities like carbon-in-austenite do vary), a semi-infinite solid (no back-face effects), and instantaneous local equilibrium at any interface (tie-line compositions apply immediately; no interface-attachment kinetics of its own). The moving-boundary model further treats the product phase as compositionally uniform, the standard simplification when its diffusivity or solubility range is very different from the parent phase's (valid for ferrite-into-austenite-style problems; this is not a general two-sided-diffusion solver). teaching-scale model

References

Crank, J. The Mathematics of Diffusion, 2nd ed. Clarendon Press, Oxford, 1975.
Porter, D.A., K.E. Easterling, and M.Y. Sherif. Phase Transformations in Metals and Alloys, 3rd ed. CRC Press, 2009. Chapter 5, Diffusional Transformations.
Abramowitz, M., and I.A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Equation 7.1.26 (error-function rational approximation). National Bureau of Standards, 1964.
SAE International. J423, Methods of Measuring Case Depth.

Kinetics for Real Alloys Lab · TTT/CCT diagrams and diffusion-controlled case-depth growth, from first principles

Diagram view

System parameters

Cooling path

References

Growth model

References