Nucleation Barrier Calculator

Calculator

Select a driving-force input mode and provide material parameters. The layout adapts to your screen: three columns on large, two on medium, and one on mobile.

Driving-force mode

Choose how the thermodynamic driving force is provided.

Nucleation type

Heterogeneous uses a contact-angle reduction factor f(θ).

Contact angle θ (degrees)

Used only for surface-assisted nucleation.

Surface energy γ

Interfacial energy between nucleus and parent phase.

|ΔGv| (driving force per volume)

Magnitude of free-energy difference per unit volume.

Supersaturation S

Must be positive and not exactly 1.

Temperature T (K)

Used for Δμ = RT ln(S) and kBT scaling.

Molar volume Vm

Used to convert Δμ (J/mol) into |ΔGv| (J/m³).

Formula used

Classical nucleation free energy (spherical nucleus):

ΔG(r) = 4πr²γ − (4/3)πr³|ΔGv|

Critical radius:

r* = 2γ / |ΔGv|

Homogeneous barrier:

ΔG*_hom = 16πγ³ / (3|ΔGv|²)

Heterogeneous barrier (surface-assisted):

ΔG*_het = f(θ) · ΔG*_hom

f(θ) = [(2 + cosθ)(1 − cosθ)²] / 4

Supersaturation option: uses Δμ = RT ln(S) (J/mol) and |ΔGv| = |Δμ|/Vm (J/m³).

How to use this calculator

Select a driving-force mode: direct |ΔGv| or supersaturation inputs.
Enter γ (surface energy). Use the unit selector if needed.
If using direct mode, provide |ΔGv| (free-energy density magnitude).
If using supersaturation mode, enter S, T, and Vm.
Choose homogeneous or surface-assisted nucleation. If surface-assisted, provide θ.
Click Calculate. Results appear above the form under the header.
Use Download CSV or Download PDF to export outputs.

Example data table

Sample values for quick validation and comparison.

Case	γ (J/m²)	\|ΔGv\| (J/m³)	θ (deg)	r* (nm)	ΔG* (J)	ΔG* (eV)
Homogeneous	0.05	1.0e8	—	1.0	2.09e-19	1.31
Surface-assisted	0.05	1.0e8	60	1.0	3.27e-20	0.204

Note: Surface-assisted values use f(60°)=0.15625, reducing ΔG* while keeping r* unchanged.

Article

1) Why nucleation barriers matter

Many phase changes begin only after a “critical” cluster appears. The barrier ΔG* controls how often that cluster forms, so even modest parameter shifts can change rates by orders of magnitude. This calculator quantifies ΔG* and r* to support materials selection, process control, and experimental interpretation.

2) Driving force and units you can trust

The thermodynamic driving force is handled as a free‑energy density |ΔGv| in J/m³. If you already have ΔGv from a model, use the direct mode. If you have supersaturation S, the tool computes Δμ = RT ln(S) (J/mol) and then |ΔGv| = |Δμ|/Vm using the molar volume Vm, keeping unit conversions explicit and transparent.

3) Surface energy sensitivity (γ³ scaling)

In classical nucleation theory, the homogeneous barrier scales as ΔG* ∝ γ³, meaning a 20% increase in γ can raise ΔG* by roughly 73%. When fitting data, ensure γ is consistent with temperature, composition, and interface structure because small uncertainties strongly influence predicted kinetics.

4) Critical radius (2γ/|ΔGv|)

The critical radius r* balances surface cost and bulk gain. Larger |ΔGv| makes r* smaller and nucleation easier, while larger γ pushes r* upward. The calculator reports r* in meters, nanometers, and ångström for quick comparison with microscopy, molecular simulations, and typical defect scales.

5) Homogeneous vs surface-assisted nucleation

Surface-assisted nucleation often lowers the barrier without changing r*. The reduction is captured by the shape factor f(θ), bounded between 0 and 1. For example, θ = 60° gives f(θ)=0.15625, reducing ΔG* by about 84% relative to a bulk event under identical γ and |ΔGv|.

6) Interpreting ΔG*/(kBT)

Reporting ΔG*/(kBT) helps connect thermodynamics to probability. Values near 10–30 can yield observable events in laboratory timescales, while values far above 50 typically imply extremely rare nucleation unless strong heterogeneities or transient conditions are present. Provide T to enable this normalization.

7) Using the example table for validation

The example row (γ=0.05 J/m², |ΔGv|=1×10⁸ J/m³) produces r*≈1 nm, matching the analytical formula r*=2γ/|ΔGv|. The barrier output is consistent with 16πγ³/(3|ΔGv|²). You can change θ to confirm the f(θ) dependence.

8) Practical workflow and exporting results

Start with measured or literature γ and a realistic |ΔGv| (or S, T, Vm). Compare homogeneous and surface-assisted scenarios to bracket expected behavior, then export CSV for lab notes or simulations. Use PDF printing to attach a clean summary to reports, proposals, or quality documentation.

FAQs

1) What does the nucleation barrier represent?

It is the maximum of ΔG(r), the free‑energy cost to form a nucleus. A larger barrier means nucleation is rarer, while a smaller barrier means nuclei form more frequently under the same conditions.

2) Why does the calculator use |ΔGv| instead of ΔGv?

For nucleation, the bulk term must favor growth, so the magnitude is used. The tool takes the absolute value to avoid sign mistakes and keep r* and ΔG* physically meaningful.

3) Does heterogeneous nucleation change the critical radius?

In the standard cap‑geometry model, r* depends only on γ and |ΔGv|, so it stays the same. The surface primarily reduces the barrier through the factor f(θ).

4) What contact angles give the strongest barrier reduction?

Smaller θ gives smaller f(θ) and a lower barrier. As θ approaches 0°, f(θ) approaches 0. As θ approaches 180°, f(θ) approaches 1, resembling the bulk case.

5) When should I use supersaturation mode?

Use it when you know S, temperature, and molar volume but not ΔGv directly. The tool computes Δμ = RT ln(S) and converts it to |ΔGv| using Vm.

6) Are the results valid for non-spherical nuclei?

The default formulas assume a spherical nucleus (or spherical cap for surfaces). Strong anisotropy, faceting, or elastic strain can change the geometry and barrier, so treat outputs as a first‑order estimate.

7) How can I reduce uncertainty in my inputs?

Measure γ where possible, confirm Vm and temperature, and estimate |ΔGv| from a consistent thermodynamic model. Sensitivity is high, so small input errors can produce large changes in predicted barriers.