The seeding method: A test case for classical nucleation theory in small systems

Molecular dynamics simulations are widely used to investigate nucleation in first-order phase transitions. Brute-force simulations, though popular, are limited to conditions of high metastability, where the critical cluster and the nucleation barrier are small. The seeding method has recently emerged as a powerful alternative for exploring lower supersaturation regimes by initiating simulations with a pre-formed nucleus. In confined systems (NVT ensemble), the seeded simulations are particularly effective for determining stable cluster properties and provide a stringent test case for classical nucleation theory (CNT). In this work, we perform NVT seeded simulations of Lennard-Jones condensation in small systems and compare them with CNT predictions based on several thermodynamic models, including equations of state, perturbation theory, and ideal gas approximation. We find that CNT accurately predicts stable cluster radii across a wide range of conditions. Notably, even the very simple ideal gas approximation proves useful for initializing seeded simulations. Furthermore, seeded simulation results correspond to the critical cluster radii of infinite systems: CNT predictions with good equations of state show very good agreement with simulations, while the perturbation theory and the ideal gas approximation perform well at low temperatures but deviate significantly at high temperatures.

💡 Research Summary

This paper presents a systematic validation of classical nucleation theory (CNT) for condensation processes in small, canonical (NVT) systems using Lennard‑Jones (LJ) particles and the seeding technique. Traditional brute‑force molecular dynamics (MD) simulations can only access high supersaturation regimes where critical clusters are tiny and nucleation barriers are low. The seeding method, which initializes a simulation with a pre‑formed liquid droplet, enables the study of lower supersaturation conditions. When applied in the NVT ensemble, mass conservation together with chemical‑potential and mechanical equilibrium leads to two possible stationary states: an unstable critical droplet (the traditional CNT critical nucleus) and a larger, stable droplet that represents the equilibrium configuration of the confined system.

The authors first derive the thermodynamic framework for a spherical liquid droplet of radius R embedded in a vapor of density ρ_v, using the capillary approximation (sharp interface, constant surface tension γ). The Helmholtz free‑energy variation (Eq. 3) yields the equilibrium conditions μ_l = μ_v and the Laplace relation p_l − p_v = 2γ/R. Because the total number of particles N is fixed, the vapor density is not prescribed but depends on the droplet size via the volume fraction Φ = V_l/V. Solving the coupled equations (4)–(6) gives two solutions for a given box size L and overall density ρ: a small, unstable cluster and a larger, stable one. In the limit of an ideal vapor and incompressible liquid the familiar CNT expression R* = 2γ/(ρ_l T ln S) is recovered, but quantitative predictions require accurate chemical potentials and pressures.

To obtain these thermodynamic quantities, the authors employ the Johnson‑Zollweg‑Gubbins (JZG) equation of state (EOS) for the LJ fluid, including a mean‑field correction for the truncated potential (cutoff 6.78 σ). The JZG EOS reproduces the LJ phase diagram obtained from independent MD simulations. Surface tension γ(T) is measured directly from MD using the Kirkwood pressure‑tensor method and follows a power law γ = γ₀(1 − T/T_c)^a with γ₀ = 2.838, a = 1.252, and critical temperature T_c = 1.305 (LJ units).

Simulation details: LAMMPS is used with a timestep of 0.005 τ, Nose‑Hoover thermostat, and periodic boundaries. A liquid seed of density (\bar\rho_l) and radius R is cut from an equilibrated bulk liquid and placed in a cubic box of side L. The remaining volume is filled with vapor particles to achieve the desired overall density ρ. Various seed radii (R = 2–9.7 σ) are tested in a box of L = 50 σ at ρ = 0.025 and T = 0.8. For seeds larger than the critical size (R > 3 σ) the system rapidly relaxes to a stable droplet of radius ≈ 8.5 σ, in excellent agreement with the CNT free‑energy minimum. The smallest seed (R = 2 σ) dissolves, confirming the CNT prediction of an unstable nucleus. The critical radius is thus bracketed between 2 and 3 σ.

The authors then explore a temperature range 0.7 ≤ T ≤ 1.1. Using the JZG EOS and the measured γ, they solve Eqs. (4)–(6) for several box sizes (L = 30, 50, 70) and plot the two solution branches as functions of overall density ρ. The lower branch corresponds to the unstable critical droplet, the upper branch to the stable droplet. As L decreases, the two branches merge at a density where supersaturation is insufficient to overcome the “super‑stabilization” effect; nucleation is then suppressed. For comparison, the authors also compute the branches using the ideal‑gas approximation for the vapor. The ideal‑gas results match the JZG EOS predictions very well for the stable branch, especially at low temperatures, but deviate noticeably at higher temperatures where vapor compressibility becomes important.

Seeding simulations are performed for each box size by initializing a seed close to the theoretically predicted stable radius for the highest density considered. The simulations run for 5 000–10 000 τ, and the droplet quickly reaches equilibrium, confirming that a good theoretical estimate of the target radius dramatically reduces equilibration time.

Key conclusions:

1. CNT, when supplied with accurate thermodynamic input (EOS and surface tension), predicts the size of the stable droplet in confined NVT systems across a wide range of temperatures and densities.
2. The stable droplet in a finite system has the same radius as the critical nucleus of an infinite system at the corresponding supersaturation, providing a stringent test of CNT.
3. Even the crude ideal‑gas approximation is sufficient to generate a reasonable initial seed, especially at low temperatures, making it a practical tool for setting up seeded simulations.
4. The super‑stabilization effect in very small boxes eliminates nucleation, a phenomenon correctly captured by the two‑solution CNT framework.
5. The seeding method, guided by CNT, enables efficient exploration of low‑supersaturation regimes that are inaccessible to brute‑force MD.

Overall, the work demonstrates that the seeding approach, combined with a rigorous CNT analysis, offers a powerful and computationally efficient pathway to study nucleation in small systems and to benchmark thermodynamic models of simple fluids. Future extensions could address more complex molecular fluids, anisotropic interfaces, and dynamic growth kinetics beyond the static equilibrium analysis presented here.