Crystallization of hard spheres revisited. I. Extracting kinetics and free energy landscape from forward flux sampling

Abstract

We investigate the kinetics and the free energy landscape of the crystallization of hard spheres from a supersaturated metastable liquid though direct simulations and forward flux sampling. In this first paper, we describe and test two different ways to reconstruct the free energy barriers from the sampled steady state probability distribution of cluster sizes without sampling the equilibrium distribution. The first method is based on mean first passage times, and the second method is based on splitting probabilities. We verify both methods for a single particle moving in a double-well potential. For the nucleation of hard spheres, these methods allow us to probe a wide range of supersaturations and to reconstruct the kinetics and the free energy landscape from the same simulation. Results are consistent with the scaling predicted by classical nucleation theory although a quantitative fit requires a rather large effective interfacial tension.

The effect of hydrodynamics on the crystal nucleation of nearly hard spheres

“…Several biasing techniques have been used to access volume fractions lower than 0.535, like umbrella sampling and forward flux sampling. 27,30,31 Recently, a new simulation technique has provided quite accurate results on both the thermodynamic and kinetic aspects of nucleation. This technique is known as the seeding approach, 42,43 and it has been applied to study the nucleation barrier of nearly hard spheres with βε = 40.…”

“…28 Therefore, it has been questioned whether the softness of the interaction potentially might be the missing ingredient in numerical models to recover the experimental nucleation rates. A numerical investigation initiated by Kawasaki and Tanaka 29 and followed by Filion et al 30 and Richard and Speck, 31,32 has looked into this issue by simulating nearly hard particles interacting through the Weeks-Chandler-Andersen (WCA) potential. 33 The nucleation rate of this softer system is in good agreement with the earlier numerical results for hard spheres, 25,27 thus proving that introducing a small amount of softness does not influence the nucleation rate.…”

Classical nucleation theory for the crystallization kinetics in sheared liquids

“…Having collected a set of trajectories, we can now compute the stationary probability distribution P + (n) to observe a configuration with a droplet of size n. As shown in Refs. [40,48], we can reconstruct the actual distribution P (n) = P + (n)/[1 − P B (n)]. In our simulations, we also compute the average number of clusters of size n, which allows to correct P (n) for small clusters [40].…”

“…[40,48], we can reconstruct the actual distribution P (n) = P + (n)/[1 − P B (n)]. In our simulations, we also compute the average number of clusters of size n, which allows to correct P (n) for small clusters [40]. We interpret F (n) ∼ ln P (n) as an effective free energy governing the nucleation kinetics, from which we extract the nucleation work W = ∆F as the height of the barrier.…”

Crystallization of hard spheres revisited. II. Thermodynamic modeling, nucleation work, and the surface of tension

“…We find a quasi-linear behavior, which indicates that the coefficient c n plays only a minor role. From the fit we obtain a n 687, which corresponds to the interfacial tension γ n,∞ = [3a n (∆ρ coex ) 2 /(32π)] 1/3 = 0.574 (7) for a flat interface. This value is in good agreement with the previous estimate γ ∞ 0.56.…”