Free-energy REconstruction from Stable Clusters (FRESC): A new method to evaluate nucleation barriers from simulation

TL;DR

The paper introduces FRESC, a method to reconstruct the free energy of formation of the critical nucleation cluster by stabilizing a small cluster in the canonical ($NVT$) ensemble and mapping its excess thermodynamics to the Gibbs free energy in the relevant ensemble. The key idea is to compute $\Delta \Omega^*(N,V,T)$ from excess quantities via $\Delta \Omega^* = \int_{N_{min}}^N \mu^{ex}(N') dN' + p^{ex}(N) V - \mu^{ex}(N) N + F^{ex}(N_{min},V,T)$, avoiding Classical Nucleation Theory and any explicit cluster coordinate. The method is demonstrated for condensation in a Lennard-Jones truncated–shifted fluid, producing results in excellent agreement with Umbrella Sampling while using far smaller systems, and showing a clean volume scaling that collapses onto a universal curve. This approach enables efficient evaluation of nucleation barriers across a wide range of supersaturations and holds promise for complex molecular systems where traditional techniques are computationally prohibitive.

Abstract

We present a simulation technique to evaluate the most important quantity for nucleation processes: the nucleation barrier, i.e. the free energy of formation of the critical cluster. The method is based on stabilizing a small cluster by simulating it in the NVT ensemble and using the thermodynamics of small systems to convert the properties of this stable cluster into the Gibbs free energy of formation of the critical cluster. We demonstrate this approach using condensation in a Lennard-Jones truncated and shifted fluid as an example, showing an excellent agreement with previous Umbrella Sampling simulations. The method is straightforward to implement, computationally inexpensive, requires only a small number of particles comparable to the critical cluster size, does not rely on the use of Classical Nucleation Theory, and does not require any cluster definition or reaction coordinate. All of these advantages hold the promise of opening the door to simulate nucleation processes in complex molecules of atmospheric, chemical or pharmaceutical interest that cannot be easily simulated with current techniques.

Figures (9)

• Figure 1: Comparison between the free energy landscape of formation of a cluster of size $n$ in the $\mu VT$ or $NPT$ ensembles, and the canonical $NVT$ ensemble. In the $\mu VT$ or $NPT$ ensembles there is only one extremum, a maximum in the associated free energy $\Delta \Omega$ or $\Delta G$, corresponding to the critical nucleus. Contrarily, in the $NVT$ ensemble at similar conditions, there may be two extrema, a maximum and a minimum in $\Delta F$, corresponding to the critical cluster and a stable cluster, respectively.
• Figure 2: Average values of the (a) pressure $p$ and (b) excess chemical potential $\mu_{ex}$ as a function of the number of particles $N$ in the simulation box, for $V=2000 \sigma^3$ and $T=0.625$. The dots are the results of the simulations, and the lines represent the values of the EoS for the homogeneous fluid by Heier et al heier2018equation. The shaded area depicts the standard deviation and the dashed blue line in (a) represents the value of the pressure at coexistence.
• Figure 3: Helmholtz free energy of formation of the stable cluster as a function of the total number of particles in the simulation box $N$, for $V=2000 \sigma^3$ and $T=0.625$. The estimated error bars are smaller than the symbol size.
• Figure 4: Grand free energy of formation of the critical cluster, $\Delta \Omega^*$, as a function of the supersaturation $\Delta \mu_s$, for $V=2000 \sigma^3$ and $T=0.625$. The empty blue symbols correspond to all simulated values of $N$ in the range from $N=1$ to $N=600$. The filled dark blue symbols indicate the range of values that are expected to be more accurate. The black empty circles represent the value of the Gibbs free energy of formation as a function of the supersaturation obtained using Umbrella Sampling simulation data from Ref. Aasen2023. The shaded area represents the estimated standard deviation in $\Delta \Omega^{*}$.
• Figure 5: Free energies of formation of the critical cluster, $\Delta \Omega^*$ as a function of the supersaturation $\Delta \mu_s$, at $T=0.625$ for different simulations with $V=250~\sigma^3$, $V=500~\sigma^3$, $V=1000~\sigma^3$, $V=2000~\sigma^3$, $V=4000~\sigma^3$, and $V=8000~\sigma^3$. The filled circles are the results of the simulations, and the open circles represent the values of previous US simulations of Ref. Aasen2023. The shaded area represents the estimated standard deviation in $\Delta \Omega^{*}$.