Grain Growth ⚙️

Metal microstructure evolution, Hall-Petch strengthening, and Arrhenius kinetics

Annealing temperature T
900 °C

Initial grain count N₀
80

Grain boundary energy γ
0.50 J/m²

Mean Grain Size d̄

— µm

No. of Grains N

—

Growth Rate K(T)

— µm²/h

Yield Strength σ_y

— MPa

Annealing Time

0 h

Grain Growth Exponent

n = 2

Physics & equations

Normal grain growth follows the parabolic law (Beck's law):

d̄² − d₀² = K(T) · t, K(T) = K₀ · exp(−Q/RT)

where Q is the activation energy for grain boundary migration, R = 8.314 J/mol·K, and K₀ is the pre-exponential factor. Grain boundaries migrate toward their centre of curvature, reducing total boundary area and surface energy.

Hall-Petch relation connects grain size to yield strength:

σ_y = σ₀ + k_HP / √d̄

where σ₀ is the lattice friction stress and k_HP is the Hall-Petch coefficient. Finer grains mean more grain boundary area — stronger barriers to dislocation motion — hence higher yield strength.

The simulation uses a Voronoi tessellation to define initial grain structure, then applies curvature-driven boundary smoothing via iterative Monte Carlo Potts-model steps.