Homogeneous versus grain boundary heterogeneous nucleation calculator

Nucleation at grain boundaries reduces the critical thermodynamic barrier by destroying a portion of the high-energy grain boundary interface (σ_gb) as the lenticular embryo forms. When σ_gb = 0, the boundary provides zero energetic advantage, and the heterogeneous curve becomes strictly identical to the homogeneous spherical curve (f = 1). Conversely, when σ_gb ≥ 2σ, complete wetting occurs, collapsing the barrier to zero (f = 0).

Physical parameters

Grain diameter (d) 25.0 µm

Operating temperature (T) 820 K

Equilibrium transition temperature (T₀) 1000 K

Matrix/embryo interfacial energy (σ) 0.40 J/m²

Grain boundary energy (σ_gb) 0.50 J/m²

Volumetric enthalpy of transformation (ΔH_v) 1.20 × 10⁹ J/m³

Boundary energy ratio (σ_gb / 2σ) cos θ = 0.625 (θ = 51.3°)

Geometric barrier reduction factor f 0.144 (85.6% reduction)

Critical embryo radius (r*) 3.70 nm

Homogeneous barrier (ΔG*hom) 3.24 × 10⁻¹⁸ J

Heterogeneous barrier (ΔG*het) 4.67 × 10⁻¹⁹ J

Rate ratio log₁₀ (I_het / I_hom) +15.2 (Het dominant)

Free energy vs embryo radius ΔG{r}

Lenticular embryo geometry

Thermodynamic and geometric formulations

For a spherical homogeneous embryo of radius r, the free energy of formation is:

ΔG_hom = -

4 3

π r³ ΔG_v + 4π r² σ

For a heterogeneous lenticular embryo on a grain boundary, mechanical equilibrium along the perimeter relates the boundary tension to the contact angle: cos θ = σ_gb / (2σ). The net excess interfacial energy accounts for the destruction of the grain boundary area π (r sin θ)² σ_gb. The physical barrier reduction factor is:

f{θ} =

1 2

[ 1 -

3 2

cos θ +

1 2

cos³ θ ] ΔG_het = f{θ} ΔG_hom

Physical boundary limits:
- When σ_gb = 0: cos θ = 0, giving f = 1. The grain boundary provides zero thermodynamic assistance; the heterogeneous curve is strictly identical to the homogeneous curve.
- When σ_gb ≥ 2σ: cos θ ≥ 1, giving f = 0. Complete wetting occurs along the boundary, eliminating the nucleation barrier entirely.
Critical radius (r*): The critical radius of curvature remains identical in both cases: r* = 2σ / ΔG_v. The grain boundary reduces the required volume and net surface area to form this critical curvature, but does not alter r* itself.