System parameters

Ω interaction energy

Temperature, T

Click anywhere on the diagram to test a composition/temperature point. The consolute point marks T_c = Ω/2R, where the miscibility gap closes to a point.

Binodal (equilibrium miscibility gap) Spinodal (d²G/dx²=0) Stable Metastable Unstable

Click inside the shaded region to classify a point.

Schematic mechanism (illustrative, looping, not a real time axis)

What this is actually computing. Reuses the exact same regular-solution G(x,T) = Ω·x(1−x) + RT[x ln x + (1−x) ln(1−x)] as the Regular Solution & Scheil-Gulliver Lab. The binodal is the same common-tangent / lower-convex-hull construction as that Lab's Mode 1, applied to a single solid-solution phase. The spinodal is the exact analytical curve d²G/dx² = 0, solved in closed form: x = ½ ± ½√(1−2RT/Ω). Both are real, tested calculations. The small animation below the diagram is not a solution of the Cahn-Hilliard equation (that needs an additional gradient-energy parameter κ this model does not include); it is a labeled, schematic sketch contrasting the two mechanisms qualitatively. teaching-scale model

References

Cahn, J.W. On Spinodal Decomposition. Acta Metallurgica 9 (1961): 795–801. doi:10.1016/0001-6160(61)90182-1
Cahn, J.W., and J.E. Hilliard. Free Energy of a Nonuniform System. I. Interfacial Free Energy. Journal of Chemical Physics 28, no. 2 (1958): 258–267. doi:10.1063/1.1744102
MIT OpenCourseWare. 3.21 Kinetic Processes in Materials, Lecture Notes. ocw.mit.edu
MIT OpenCourseWare. 3.205 Thermodynamics and Kinetics of Materials, Lecture Notes. ocw.mit.edu

Nucleation parameters

Undercooling, ΔT

Interfacial energy, σ

Advanced (T_m, L held fixed above)

Transformation temperature, T_m

Latent heat (volumetric), L

ΔG_v is linearized in undercooling: ΔG_v = −L·ΔT/T_m.

Volume term (−) Surface term (+) ΔG(r), homogeneous

Relative nucleation rate vs. undercooling (attachment kinetics held fixed)

What this is actually computing. ΔG(r) = (4/3)πr³ΔG_v + 4πr²σ exactly, with r* = −2σ/ΔG_v and ΔG* = 16πσ³/(3ΔG_v²) both cross-checked numerically against the turning point of that curve. Heterogeneous nucleation scales the entire curve by the wetting-angle shape factor S(θ) = (2+cosθ)(1−cosθ)²/4, which is why r* is identical in both cases and only the barrier height changes, an important and easy-to-miss subtlety. The rate panel shows exp(−ΔG*/k_BT) only, i.e. the thermodynamic barrier's effect alone; it does not include the diffusivity fall-off at large undercooling that produces a TTT-diagram "nose", that is a separate tool. teaching-scale model

References

Volmer, M., and A. Weber. Keimbildung in übersättigten Gebilden. Zeitschrift für Physikalische Chemie 119 (1926): 227. doi:10.1515/zpch-1926-11927
Becker, R., and W. Döring. Kinetische Behandlung der Keimbildung in übersättigten Dämpfen. Annalen der Physik 416, no. 8 (1935): 719–752. doi:10.1002/andp.19354160806
Porter, D.A., K.E. Easterling, and M.Y. Sherif. Phase Transformations in Metals and Alloys, 3rd ed. CRC Press, 2009. Chapter 4, Diffusional Transformations.
Kelton, K.F., and A.L. Greer. Nucleation in Condensed Matter: Applications in Materials and Biology. Pergamon, 2010.
MIT OpenCourseWare. 3.21 Kinetic Processes in Materials, Lecture Notes. ocw.mit.edu

Coarsening parameters

Capillary length, ℓ

Diffusion-solubility prefactor, A

Initial mean radius, r̄₀

N = 50 particles, shown for visual clarity. Model time is in arbitrary units (D and C_e are lumped into A, not tied to a specific alloy). The r̄³ ∝ t scaling law itself was verified separately against a 1,000+ particle ensemble.

Gibbs-Thomson effect: local solubility vs. particle radius

Particle population (small particles shrink and feed the large ones)

Mean radius cubed vs. time (watch the asymptotic straight line emerge)

What this is actually computing. Gibbs-Thomson is linearized: C(r)/C_e = 1 + ℓ/r, with capillary length ℓ = 2σV_m/RT (V_m fixed at a representative metallic value and folded into the ℓ slider). Each particle obeys the mean-field growth law dr/dt = (K/r)(1/r̄ − 1/r), K = A·ℓ, integrated with a small Euler step. Rather than asserting the disputed closed-form LSW rate-constant prefactor (textbooks differ on the numerical constant by convention), this runs the population forward and lets r̄³ ∝ t emerge, matching a mean-field volume-conservation law derived and checked directly (d(Σr³)/dt = 0 exactly between particle-death events, since Σ(r/r̄) = N by definition of the mean). A 50-particle animation is visibly noisier than that law, real physics of a finite population, not a bug; the R² > 0.9 scaling check in the verification notes uses 1,000 particles for that reason. teaching-scale model

References

Lifshitz, I.M., and V.V. Slyozov. The Kinetics of Precipitation from Supersaturated Solid Solutions. Journal of Physics and Chemistry of Solids 19 (1961): 35–50. doi:10.1016/0022-3697(61)90054-3
Wagner, C. Theorie der Alterung von Niederschlägen durch Umlösen (Ostwald-Reifung). Zeitschrift für Elektrochemie 65, no. 7–8 (1961): 581–591. doi:10.1002/bbpc.19610650704
Zang, L. Lecture 9: Particle Coarsening, Ostwald Ripening. University of Utah course notes. my.eng.utah.edu
MIT OpenCourseWare. 3.21 Kinetic Processes in Materials, Lecture Notes. ocw.mit.edu