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Particle charging in the afterglows of non-thermal plasmas typically take place in a non-neutral space charge environment. We model the same by incorporating particle-ion collision rate constant models, developed in prior work by analyzing particle-ion trajectories calculated using Langevin Dynamics simulations, into species transport equations for ions, electrons and charged particles in the afterglow. A scaling analysis of particle charging and additional Langevin Dynamics calculations of the particle-ion collision rate constant are presented to extend the range of applicability to ion electrostatic to thermal energy ratios of 300 and diffusive Knudsen number (that scales inversely with gas pressure) up to 2000. The developed collision rate constant models are first validated by comparing predictions of particle charge against measured values in a stationary, non-thermal DC plasma from past PK-4 campaigns published in Phys. Rev. Lett. 93(8): 085001 and Phys. Rev. E 72(1): 016406). The comparisons reveal excellent agreement within ±35% for particles of radius 0.6,1.0,1.3 μm in the gas pressure range of ~20-150 Pa. The experiments to probe particle charge distributions by Sharma et al. (J. Physics D: Appl. Phys. 53(24): 245204) are modeled using the validated particle-ion collision rate constant models and the calculated charge fractions are compared with measurements. The comparisons reveal that the ion/electron concentration and gas temperature in the afterglow critically influence the particle charge and the predictions are generally in qualitative agreement with the measurements. Along with critical assessment of the modeling assumptions, several recommendations are presented for future experimental design to probe charging in afterglows. 

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 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. 

Particle shape strongly influences the diffusion charging of aerosol particles exposed to bipolar/unipolar ions and accurate modeling is needed to predict the charge distribution of non-spherical particles. A prior particle-ion collision kernel β_i model including Coulombic and image potential interactions for spherical particles is generalized for arbitrary shapes following a scaling approach that uses a continuum and free molecular particle length scale and Langevin dynamics simulations of non-spherical particle-ion collisions for attractive Coulomb-image potential interactions. This extended β_i model for collisions between unlike charged particle-ion (bipolar charging) and like charged particle-ion (unipolar charging) is validated by comparing against published experimental data of bipolar charge distributions for diverse shapes. Comparison to the bipolar charging data for spherical particles shows good agreement in air, argon, and nitrogen, while also demonstrating high accuracy in predicting charge states up to ±6. Comparisons to the data for fractal aggregates reveal that the LD-based β_i model predicts within overall ±30% without any systematic bias. The mean charge on linear chain aggregates and charge fractions on cylindrical particles is found to be in good agreement with the measurements (~±20% overall). The comparison with experimental results supports the use of LD-based diffusion charging models to predict the bipolar and unipolar charge distribution of arbitrary shaped aerosol particles for a wide range of particle size, and gas temperature, pressure. The presented β_i model is valid for perfectly conducting particles and in the absence of external electric fields; these simplifications need to be addressed in future work on particle charging. 

Based on the prior work of Chahl and Gopalakrishnan (2019) to infer particle-ion collision time distributions using a Langevin Dynamics (LD) approach, we develop a model for the non-dimensional diffusion charging collision kernel β_i or H that is applicable for 0≤Ψ_E≤60,0≤Ψ_I/Ψ_E ≤1,Kn_D≤2000 (defined in the main text). The developed model for β_i for attractive Coulomb and image potential interactions, along with the model for β_i for repulsive Coulomb and image potential interactions from Gopalakrishnan et al. (2013b), is tested against published diffusion charging experimental data. Current state of the art charging models, Fuchs (1963) and Wiedensohler (1988) regression for bipolar charging, are also evaluated and discussed. Comparisons reveal that the LD-based model accurately describes unipolar fractions for 10 – 100 nm particles measured in air (Adachi et al., 1985), nitrogen and argon but not in helium (Adachi et al., 1987). Fuchs model and the LD-based model yield similar predictions in the experimental conditions considered, except in helium. In the case of bipolar charging, the LD-based model captures the experimental trends quantitatively (within ±20%) across the entire size range of 4 – 40 nm producing superior agreement than Wiedensohler’s regression. The latter systematically underpredicts charge fraction below ~20 nm in air (by up to 40%) for the data presented in Adachi et al. (1985). Comparison with the data of Gopalakrishnan et al. (2015), obtained in UHP air along with measurements of the entire ion mass-mobility distribution, shows excellent agreement with the predictions of the LD-based model. This demonstrates the capability to accommodate arbitrary ion populations in any background gas, when such data is available. Wiedensohler’s regression, derived for bipolar charging in air using average ion mass-mobility, also describes the data reasonably well in the conditions examined. However, both models failed to capture the fraction of singly and doubly charged particles in carbon dioxide warranting further investigation.