Critical Nucleus Radius Calculator

This calculator uses classical nucleation theory for a spherical nucleus. The competition is between surface energy and bulk free energy.

• Critical radius: r* = 2γ / ΔGv
• Barrier height: ΔG* = 16πγ³ / (3ΔGv²)
• Size estimate: N* ≈ (4/3)πr*³ / v (requires molecular volume v)

If you choose supersaturation mode, the bulk driving force is estimated as Δμ = kBT ln(S), then ΔGv = Δμ/v.

1. Select a mode: direct ΔGv or supersaturation S.
2. Enter surface tension γ and choose its unit.
3. For ΔGv mode, enter a positive magnitude for ΔGv.
4. For S mode, enter S > 1, temperature, and a volume value.
5. Press Calculate. Results appear above the form.
6. Use the CSV or PDF buttons to export your results.

In classical nucleation theory, a forming nucleus pays a surface-energy cost and gains a bulk free-energy benefit. The critical radius r* is the turning point: clusters smaller than r* tend to shrink, while clusters larger than r* tend to grow. This calculator converts your inputs into consistent SI values and reports r* in meters, nanometers, and angstroms for quick comparison with microstructural length scales.

You enter surface tension γ (N/m or J/m²) and a driving-force term for nucleation. In ΔGv mode the driving force is the free-energy density difference ΔGv (J/m³). In supersaturation mode the tool estimates ΔGv from kBT ln(S) and a molecular or molar volume. Unit conversion is critical because r* scales directly with γ and inversely with ΔGv.

Use ΔGv mode when you already have a thermodynamic free-energy density from a model, database, or fitted experiment. Use supersaturation mode when you can measure or estimate the supersaturation ratio S and temperature T. The calculator converts volume inputs given per molecule or per mole, then computes ΔGv so that both modes produce the same core outputs: r*, ΔG*, and optional N*.

Many condensed-matter systems yield r* on the order of 1–10 nm, but values can be much smaller or larger depending on γ and ΔGv. For example, doubling γ doubles r*, while increasing ΔGv by 10× reduces r* by 10×. The example table shows how different input pairs can coincidentally produce similar r* yet very different barriers, highlighting why both outputs should be reported together.

Because ΔG* scales as γ³ and as 1/ΔGv², barrier height can change dramatically with modest parameter updates. A 20% increase in γ raises ΔG* by about 73%, while a 20% increase in ΔGv lowers ΔG* by about 31%. If you are fitting experimental nucleation rates, use this calculator to explore parameter sensitivity and identify which measurement uncertainty dominates your predictions.

ΔG* is reported in joules and in electron-volts for convenience. Barriers near a few tenths of an eV can correspond to rapid nucleation under favorable conditions, while barriers of several eV often imply slow or rare events unless aided by defects or surfaces. Use the eV value for intuitive thermal comparisons (kBT is about 0.026 eV at 300 K), and use joules for rigorous thermodynamic bookkeeping.

If you provide a molecular or molar volume, the calculator estimates N* from the critical volume (4/3)πr*³ divided by the per-particle volume. This helps connect continuum theory to atomistic simulation sizes or to approximate particle counts in clusters. If you are unsure, start with a known molar volume at the relevant temperature and convert it using the built-in unit options.

This tool assumes a spherical nucleus and a constant γ, which is a good first model but may deviate for anisotropic crystals or very small clusters. When documenting results, report γ, ΔGv (or S, T, and volume), and the chosen units. If heterogeneous nucleation dominates, treat the outputs as an upper bound for the homogeneous case and adjust using an appropriate geometric factor.

1) What does r* mean physically?

It is the threshold radius where surface cost and bulk driving force balance. Clusters smaller than r* are unstable and tend to dissolve, while larger clusters tend to grow into a new phase.

2) Why must ΔGv be positive in this calculator?

The formulas use the magnitude of the bulk driving force. A negative sign is handled by convention; you enter a positive value so r* and ΔG* remain meaningful and nonnegative.

3) When should I use supersaturation mode instead of ΔGv mode?

Use supersaturation mode when you can estimate S and temperature but do not have ΔGv directly. The calculator converts kBT ln(S) and volume into an equivalent ΔGv.

4) What volume should I enter for N*?

Use the per-molecule volume if available, or a molar volume if that is what you have. The tool converts molar volume into per-molecule volume using Avogadro’s number automatically.

5) Why can two cases have the same r* but different ΔG*?

r* depends on γ/ΔGv, while ΔG* depends on γ³/ΔGv². Different combinations can match the ratio but still change the cubic and squared scaling in ΔG*.

6) Are the results valid for heterogeneous nucleation?

The calculator reports homogeneous nucleation values for a spherical nucleus. For heterogeneous nucleation on surfaces, the effective barrier is typically reduced by a geometric wetting factor.

7) How do I export results for a report?

After calculation, use “Download CSV” for spreadsheets or “Download PDF” for a quick printable summary. Exports include the converted SI values and the displayed radius and barrier.