Event-driven Langevin simulations of hard spheres

A. Scala

Phys. Rev. E 86, 026709 – Published 20 August 2012

Abstract

The blossoming of interest in colloids and nanoparticles has given renewed impulse to the study of hard-body systems. In particular, hard spheres have become a real test system for theories and experiments. It is therefore necessary to study the complex dynamics of such systems in presence of a solvent; disregarding hydrodynamic interactions, the simplest model is the Langevin equation. Unfortunately, standard algorithms for the numerical integration of the Langevin equation require that interactions are slowly varying during an integration time step. This is not the case for hard-body systems, where there is no clear-cut distinction between the correlation time of the noise and the time scale of the interactions. Starting first from a splitting of the Fokker-Plank operator associated with the Langevin dynamics, and then from an approximation of the two-body Green's function, we introduce and test two algorithms for the simulation of the Langevin dynamics of hard spheres.

Received 8 March 2012

DOI:https://doi.org/10.1103/PhysRevE.86.026709

Authors & Affiliations

A. Scala

ISC-CNR Dipartimento di Fisica, Sapienza Università di Roma Piazzale Moro 5, 00185 Roma, Italy,
IMT Alti Studi Lucca, piazza S. Ponziano 6, 55100 Lucca, Italy, and London Institute of Mathematical Sciences, 22 South Audley St Mayfair, London W1K 2NY, United Kingdom

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Vol. 86, Iss. 2 — August 2012

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Images

Figure 1
Effect of the damping coefficient $γ$ on the size of the simulation step $Δ t$ (all quantities in reduced units). The diffusion coefficient $D$ from simulations is plotted versus the time-step size $Δ t$ for various $γ$ 's. As expected, the system approaches the MD value for diffusion regardless of $γ$ for $Δ t \to \infty$ . The “true” value of $D$ is obtained for $Δ t \to 0$ . We observe at small $Δ t$ 's a plateau in the $D$ vs $Δ t$ plot for $Δ t ≲ γ^{- 1}$ , signaling that the “true” value of $D$ is approached. Results are presented for packing fraction $φ = 0.30$ ; a completely analogous behavior is found at a low packing fraction $φ = 0.10$ and a high packing fraction $φ = 0.45$ .Reuse & Permissions
Figure 2
A two-body problem for hard spheres can be mapped into the problem of a point particle interacting with a larger sphere. When particles are very near, the problem further simplifies to the interaction of a Langevin particle with a reflective flat wall, the solution of which can be derived by applying the image method to the Langevin equation. In fact, the Green's function must be zero inside the wall and must satisfy the no-flux boundary conditions at the wall. Combining the free Green's function of the particle in its initial position and the free Green's function of its image (with the normal-to-the-wall component of the initial velocity reflected) satisfies both Kramers' equations and reflective boundary conditions giving the correct solution.Reuse & Permissions
Figure 3
Effects of packing fraction $φ$ (upper panel) and of the damping coefficient $γ$ (lower panel) on the time step $Δ t$ for the AGD algorithm. All quantities in reduced units; thick lines are just a guide for the eye. Diffusion $D$ is calculated averaging over 10 independent trajectories for 2000 particle systems; simulations are long at least 10 times the structural correlation time. In the upper panel, results are shown for $φ = 0.10, 0.30, 0.45$ at fixed damping $γ = 10$ . In the lower panel, results are shown for $γ = 1, 10, 100$ at fixed packing fraction $φ = 0.30$ . Notice that the estimated diffusion coefficient $D$ has a small relative variation in the wide range of dampings $γ$ 's and packing fractions $φ$ 's analyzed. As a rule of thumb, to estimate $D$ with an accuracy much smaller than $1 %$ time step of order $Δ t \sim 0.1$ is already enough.Reuse & Permissions
Figure 4
Comparison of the convergence of the measured diffusion coefficient $D$ for the two algorithms considered in the papers. The figure shows results for damping $γ = 10$ at packing fraction $φ = 0.30$ at varying time steps $Δ t$ ; within the numerical accuracy (size of the symbols is of the order of the error), both algorithms converge to the same value for $Δ t \to 0$ . An analogous behavior is observed for $φ = 0.10, 0.45$ and $γ = 1, 100$ .Reuse & Permissions

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