Fast event-driven simulations for soft spheres: from dynamics to Laves phase nucleation

TL;DR

This work introduces an event-driven Monte Carlo (EDMC) framework as a rejection-free substitute for molecular dynamics when simulating steeply repulsive potentials, demonstrated on Weeks-Chandler-Andersen (WCA) particles. EDMC samples the canonical ensemble exactly by treating pair interactions as stochastic collision events, yielding dynamics that closely resemble MD while delivering large speedups at low temperatures. The authors validate EDMC against MD for both monodisperse and binary WCA systems, showing accurate thermodynamics, diffusion, and relaxation, and use it to reveal spontaneous Laves-phase nucleation at temperatures previously inaccessible to MD. In particular, EDMC enables long-time exploration of glassy dynamics and crystal nucleation in a near-hard-sphere regime, including direct observation of MgZn$_2$- and MgCu$_2$-like nucleation trajectories, with potential broad applicability to systems with short-range steep interactions. The method offers a practical, efficient tool for studying dynamics and phase behavior in soft-mphere models where finite-time-step MD struggles, while highlighting caveats related to potential invertibility, long-range interactions, and parallelization.

Abstract

Conventional molecular dynamics (MD) simulations struggle when simulating particles with steeply varying interaction potentials, due to the need to use a very short time step. Here, we demonstrate that an event-driven Monte Carlo (EDMC) approach first introduced by Peters and de With [Phys. Rev. E 85, 026703 (2012)] represents an excellent substitute for MD in the canonical ensemble. In addition to correctly reproducing the static thermodynamic properties of the system, the EDMC method closely mimics the dynamics of systems of particles interacting via the steeply repulsive Weeks-Chandler-Andersen (WCA) potential. In comparison to time-driven MD simulations, EDMC runs faster by over an order of magnitude at sufficiently low temperatures. Moreover, the lack of a finite time step in EDMC circumvents the need to trade accuracy against simulation speed associated with the choice of time step in MD. We showcase the usefulness of this model to explore the phase behavior of the WCA model at extremely low temperatures, and to demonstrate that spontaneous nucleation and growth of the Laves phases is possible at temperatures significantly lower than previously reported.

Figures (9)

• Figure 1: Schematic view of the collision process between two particles as processed by the conventional time-driven MD approach (dashed line), and the present EDMC approach (solid lines). The stochastic nature of the EDMC simulation method can yield a range of different trajectories. Note that for simplicity, the blue particle is considered to be fixed. The background shading and dotted contour lines illustrate the interaction potential $U(r)$, with darker shading indicating higher energy.
• Figure 2: Comparison between dynamics of EDMC (marked by solid dots ($\bullet$) and lines) and LAMMPS (marked by empty squares ($\Box$) and dashed lines) for monodisperse WCA at density $\rho \sigma^3 = 0.65$, for a range of temperatures. a) Mean squared displacement as a function of time. b) Intermediate scattering function $F(q,t)$ ($q = 2\pi / \sigma$) as a function of time. c) Relaxation time $\tau_\alpha$ as a function of the wave-vector $q$. Note that error bars (determined as twice the standard error) are shown for all points but are typically smaller than the point size. All lines are guides to the eye.
• Figure 3: a) Typical snapshot of a direct coexistence simulation. b) WCA phase diagram at very low temperatures. Empty circles represent recent estimates also obtained from interface pinning simulationsattia2022comparing. The estimate for $\beta P_\mathrm{coex}^\mathrm{HS} \sigma_\mathrm{HS}^3= 11.55668$ was obtained using the same methodology for a pure hard-sphere system of the same size smallenburg2024simple. Error bars are determined as twice the standard error on five independent measurements, and are mostly smaller than the points. The dashed line shows the expected low-temperature scaling (see Ref. attia2022comparing).
• Figure 4: MD measurements of the average energy of the WCA model at density $\rho \sigma^3 = 0.68$ as a function of the MD time step, for different temperatures as indicated. The solid orange lines are cubic fits to the data at $\delta t/\tau_\epsilon < 0.03$, and the dashed green lines indicate the extrapolation of that fit to $\delta t = 0$. For these results, we used $N = 8000$.
• Figure 5: Relative performance of the EDMC and MD methods for monodisperse WCA systems of several sizes, expressed as the ratio of the number of time units simulated per second in each simulation method, a) at fixed density $\rho \sigma^3 = 0.68$ and b) at fixed temperature $k_B T / \epsilon = 0.001$. Performance evaluation was obtained from successive short simulations of duration $100 \tau_{kT}$ started from an equilibrated fluid system. Standard deviations are typically found to be much less than $1 \%$.