Deriving Reliable Nucleation Rates from Metadynamics Simulations: Application to Yukawa Fluids

TL;DR

The paper tackles predicting crystal nucleation rates from supercooled liquids by marrying metadynamics-derived free-energy barriers with classical nucleation theory. It introduces a workflow that uses local collective variables to efficiently sample nucleation pathways, then post-processes the metadynamics data through a sequence that yields CNT-consistent free-energy surfaces via $F_l(N)$ and $F_a(N)$ mappings, including finite-size corrections and Tolman curvature. The approach is validated on the Yukawa one-component plasma across screening lengths $\kappa=0,2,5$, and benchmarked against brute-force simulations, showing reliable rate predictions and a well-defined critical temperature $T_c$ for nucleation. The work demonstrates that CNT parameters fitted to metadynamics data can be extrapolated to other temperatures, enabling efficient exploration of nucleation behavior in complex systems and guiding future multi-component studies. Overall, the method provides a practical, CNT-consistent route to quantify nucleation barriers and rates from enhanced-sampling simulations with potential applicability to astrophysical and material systems alike.

Abstract

In order to solidify the usefulness of metadynamics in studying nucleation of crystals from supercooled liquids, we provide a specific procedure to calculate nucleation free energy barriers. After a pedagogical review of the important elements of classical nucleation theory and how metadynamics is used to find nucleation free energy barriers, we explain the benefits of local collective variables over more common global collective variables. We show how a metadynamics free energy barrier must be carefully postprocessed so that classical nucleation theory can be applied to calculate nucleation rates. We apply our procedure to a Yukawa plasma and show that a particular physically-motivated fit to metadynamics data reproduces low-temperature reference data, justifying the usefulness of metadynamics to predict nucleation rates and the nucleation critical temperature.

Figures (10)

• Figure 1: (Upper panel) CNT free energy surface [Equation \ref{['eq:freeEnergy']}]. (Lower panels) Differences between idealized spherical CNT clusters (upper left half of panels) and more realistic clusters (lower right half of panels)
• Figure 2: Illustration of the principles of metadynamics. (a) In blue, the system is stuck in a well of the unbiased free energy surface $F(S)$. With a compensating bias applied, the total free energy surface $F(S)+V_b(S)$ is nearly flat and the (orange) biased system can traverse anywhere. (b) The unbiased system (blue) reaches only a small range of $S$ values. When the bias is applied (orange) it can sample more $S$ values with similar probabilities. When the biased sampling is reweighted (green), the "true" sampling probability (corresponding to the result of an infinitely long unbiased simulation) is returned.
• Figure 3: A schematic of the free energy surface generated by the globally-biased metadynamics method. (A) Configurations at the bottom of the liquid free energy well contain no large crystal clusters. (B) Near the top of the free energy barrier, the simulation generates multiple nuclei or other erroneous transition states. (C) The free energy becomes much lower than the liquid free energy in highly crystallized states because the crystal is thermodynamically favored over the liquid.
• Figure 4: Cartoon of the free energy surface generated by the local biasing method. The metadynamics bias generates crystals in the light blue region, while the "outside" white region of the simulation is restrained to remain liquid. (A) There are no solid clusters at the bottom of the liquid free energy well. (B) The classical nucleation pathway is respected with only one cluster forming inside the biased region at the top of the free energy barrier. (C) The crystal grows to fill the "inside" region, but can't become larger to fill the whole simulation volume.
• Figure 5: Nucleation free energy barrier heights for Yukawa plasmas with screening parameters $\kappa=0,2,$ and $5$. Triangles show the barrier heights calculated using local biasing at moderate temperatures, and circles show barriers from global biasing at lower temperatures.