Structure-sensitive diffusion and grain boundary transport calculator

In polycrystalline solids, grain boundaries act as high-diffusivity short-circuits with a lower activation energy (Q_gb ≈ 0.5 Q_L) [E. W. Hart: Acta Metallurgica 5 (1957)597, A. J. Mortlock: Acta Metallurgica 8 (1962)132]. The measured effective diffusivity (D_measured) exhibits a distinct knee: at high temperatures, bulk lattice transport dominates and the slope reflects Q_L; at low temperatures, grain boundaries dominate and the slope bends to match Q_gb. Refining the grain size increases the boundary area fraction f, driving the transition knee to higher temperatures.

Microstructural parameters

Grain size (d) 25.0 µm

Operating temperature (T) 650 °C (923 K)

Grain boundary structural width (δ) 0.50 nm

Lattice activation energy (Q_L) 200.0 kJ/mol

Boundary activation energy (Q_gb) 105.0 kJ/mol

Crossover temperature (T*) 628 °C (901 K)

Grain boundary volume fraction (f) 6.00 × 10⁻⁵

Effective diffusivity (D_measured) 1.45 × 10⁻¹⁵ m²/s

Lattice diffusivity (D_L) 9.54 × 10⁻¹⁶ m²/s

Dominant transport mechanism Lattice dominant (65.8%)

Arrhenius plot (log₁₀ D vs 1000/T)

Polycrystalline microstructure cross-section

Thermodynamic and geometric formulations

In a polycrystalline metal characterised by average equiaxed grain size d and grain boundary thickness δ, the volume fraction occupied by boundaries is:

ratio of areas f ≈

3 δ d

The net effective diffusion coefficient follows the Hart–Mortlock relationship:

D_measured = (1 - f) D_L + f D_gb

Equating both terms gives the exact crossover temperature T* where lattice and boundary flux contributions are equal:

T* =

Q_L - Q_gb R ln[(1 - f)D_0,L / (f D_0,gb)]

Above T* (high temperatures): Lattice jumps are frequent enough to dominate the bulk flux: (1 - f)D_L ≫ f D_gb. The slope of the effective curve aligns with the steep lattice line (Q_L).
Below T* (low temperatures): Lattice diffusion freezes out exponentially due to the larger barrier Q_L. Fast diffusion through the boundary pathways dominates the net flux (f D_gb ≫ D_L). The effective curve bends and runs parallel to the shallow grain boundary line (Q_gb).
Effect of grain size: As grains coarsen (d increases), f decreases, pushing the transition knee T* to lower temperatures and widening the lattice-dominated regime. Grain refinement pushes T* upwards, expanding boundary dominance into higher temperatures.