Tutorial: Langevin Dynamics methods for aerosol particle trajectory simulations and collision rate constant modeling

Highlights

• The methodological details of Langevin Dynamics are discussed.

• The translation Langevin equation is described for calculating particle trajectories.

• The drag, diffusion and deterministic forces acting on particles are summarized.

• Demonstrations on the usage of Langevin Dynamics are presented.

• Potential applications and limitations of Langevin Dynamics is also included.

Abstract

The Langevin Dynamics (LD) method (also known in the literature as Brownian Dynamics) is routinely used to simulate aerosol particle trajectories for transport rate constant calculations as well as to understand aerosol particle transport in internal and external fluid flows. This tutorial intends to explain the methodological details of setting up a LD simulation of a population of aerosol particles and to deduce rate constants from an ensemble of classical trajectories. We discuss the applicability and limitations of the translational Langevin equation to model the combined stochastic and deterministic motion of particles in fields of force or fluid flow. The drag force and stochastic “diffusion” force terms that appear in the Langevin equation are discussed elaborately, along with a summary of common forces relevant to aerosol systems (electrostatic, gravity, van der Waals, …); a commonly used first order and a fourth order Runge-Kutta time stepping schemes for linear stochastic ordinary differential equations are presented. A MATLAB® implementation of a LD computer code for simulating particle settling under gravity using the first order scheme is included for illustration. Scaling analysis of aerosol transport processes and the selection of timestep and domain size for trajectory simulations are demonstrated through two specific aerosol processes: particle diffusion charging and coagulation. Fortran® implementations of the first order and fourth order time-stepping schemes are included for simulating the 3D motion of a particle in a periodic domain. Potential applications and caveats to the usage of LD are included as a summary.

Introduction

The calculation of particle trajectories in the context of classical physics that permits the knowledge of both position and velocity with complete certainty by integrating Newton's Second law of motion, (i. e.) position ${\overrightarrow{r}}_p\left(t\right)$ and velocity ${\overrightarrow{v}}_p\left(t\right)$ timeseries, is of fundamental interest for visualizing aerosol particle dynamics, developing models of collision/reaction rate constants for single particle-level mass transfer processes as well as to predict the transport of populations of aerosol particles and can be accomplished at a relatively modest computational cost by integrating Langevin-type ordinary differential equations (ODEs) of motion. The term particle is broadly used here to denote nm – μm sized solid/liquid aerosol particles (spherical or arbitrary shaped), nm-sized macromolecules or molecular ions suspended in a flowing/stagnant background gas in the presence/absence of external electric/magnetic fields. In this tutorial article, we focus on the methodological details of solving the Langevin equation of motion to calculate particle trajectories and discuss two examples on the use of an ensemble of computed trajectories to infer single particle mass transfer rate constants. The interested reader is referred to the pioneering work of Uhlenbeck and Ornstein (1930) that presents an analysis of the mean values of the velocity and mean squared displacement of a free particles subject to stochastic forces from a fluid medium and establishes the connection of the Langevin equation to the Fokker-Planck partial differential equations for the phase-space (velocity and position) distribution functions (Risken & Frank, 1996). Further, in a seminal review, Chandrasekhar (1943) presents a detailed overview of the Langevin equation, the underlying probabilistic aspects and applications to colloidal and astrophysical problems. The widespread use of Langevin equations to model driven-dissipative systems is evident from its usage in diverse fields such as physics, engineering, finance and several textbooks on this topic (Coffey et al., 2004; Risken & Frank, 1996; Schuss, 2013). We restrict our scope to walking the interested reader through the steps of setting up equations of motion to calculate particle trajectories under the influence of deterministic and stochastic forces and excuse ourselves from elaborate discussions of the modeling of the physical processes themselves. Specifically, the reader is referred to prior research articles on particle charging modeling (Chahl & Gopalakrishnan, 2019; Gopalakrishnan et al., 2013a, 2013b; Gopalakrishnan & Hogan, 2012; Li et al., 2020; Li & Gopalakrishnan, 2021; Ouyang et al., 2012) and particle coagulation modeling (Gopalakrishnan et al., 2011; Gopalakrishnan & Hogan, 2011; Thajudeen et al., 2012) using Langevin Dynamics. The tutorial is organized as follows: the Methods section discusses the Langevin ODE for particle translational motion; various particle-gas, particle-particle, particle-flow and particle-field interactions relevant to aerosol systems; two numerical schemes for integrating the Langevin ODE. Subsequently, in the Demonstrations section, we discuss two case studies of using LD to model the collision rate constant for aerosol particle diffusion charging and coagulation. We conclude with a Summary of the LD methodology, potential applications and caveats to the usage of LD for modeling particle trajectories in gas-phase systems such as aerosols and dusty plasmas.