Nucleation Calculator
Copper undercooled by 200 K forms stable nuclei above a critical radius of 1.27854 nm, a cluster of several hundred atoms.
Solid–liquid copper about 0.177 J/m².
Copper 1.88 × 10⁹ J/m³.
1.27854255319149 nm
Full answer, then cut to fewer decimals (not rounded):
- 1.27854255319149 nm
- 1.2785425531914 nm
- 1.278542553191 nm
- 1.27854255319 nm
- 1.2785425531 nm
- 1.278542553 nm
- 1.27854255 nm
- 1.2785425 nm
- 1.278542 nm
- 1.27854 nm
- 1.2785 nm
- 1.278 nm
- 1.27 nm
- 1.2 nm
- Driving force ΔG_v = ΔH_f ΔT/T_m
- 276877761.4 J/m³
- Homogeneous barrier ΔG*
- 1.21197106 × 10⁻¹⁸ J
- Barrier in eV
- 7.56452837 eV
- Heterogeneous factor S(θ) = (2 + cos θ)(1 − cos θ)²/4
- 0.15625
- Heterogeneous barrier ΔG*S(θ)
- 1.893704782 × 10⁻¹⁹ J
Step-by-step solution
- ΔG_v = ΔH_f ΔT ÷ T_m = 1880000000 × 200 ÷ 1358 = 276877761.4 J/m³
- r* = 2γ ÷ ΔG_v = 1.278542553 nm
- ΔG* = 16πγ³ ÷ (3ΔG_v²) = (1.21197106 × 10⁻¹⁸) J
Nucleation Formula
r* = 2γ/ΔG_v, ΔG* = 16πγ³/(3ΔG_v²)
ΔG_v = ΔH_f ΔT / T_m; heterogeneous S(θ) = (2 + cos θ)(1 − cos θ)²/4
Worked Example
Copper solidifying with 200 K of undercooling:
- ΔG_v = ΔH_f ΔT ÷ T_m = 1880000000 × 200 ÷ 1358 = 276877761.4 J/m³
- r* = 2γ ÷ ΔG_v = 1.278542553 nm
- ΔG* = 16πγ³ ÷ (3ΔG_v²) = (1.21197106 × 10⁻¹⁸) J
Homogeneous barrier ΔG*: 1.21197106 × 10⁻¹⁸ J · Barrier in eV: 7.56452837 eV · Heterogeneous factor S(θ) = (2 + cos θ)(1 − cos θ)²/4: 0.15625 · Heterogeneous barrier ΔG*S(θ): 1.893704782 × 10⁻¹⁹ J
GATE tip
r* ∝ 1/ΔT and ΔG* ∝ 1/ΔT². Doubling the undercooling halves the critical radius and quarters the barrier, so nucleation rates rise steeply.
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Example: Mild steel (σ₀ = 70 MPa, k_y = 0.74 MPa·m^½) with 25 µm grains: Yield strength σ_y = 218 MPa.
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n_v / N = exp(−Q_v / kT)
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Example: Copper (Q_v = 0.9 eV) at 1000 °C: Vacancy fraction n_v/N = 0.000273741.
Frequently asked questions
Why is undercooling needed?
At the melting point the volume driving force is zero, so the surface energy cost makes every nucleus unstable; ΔG_v grows with ΔT, shrinking r* and ΔG*.
Why does heterogeneous nucleation dominate?
A wetting surface (mould wall, inoculant) reduces the barrier by S(θ); at θ = 60° the barrier is only 16 % of the homogeneous value.
Sources
- The International System of Units (SI), 9th edition (SI Brochure), Bureau International des Poids et Mesures .
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
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APA
Scientific Converters. (2026, October 15). Nucleation Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/nucleation-calculator/
IEEE
Scientific Converters, "Nucleation Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/nucleation-calculator/