A hybrid computational method coupling the lattice‐Boltzmann (LB) method and a Langevin‐dynamics (LD) method is developed to simulate nanoscale particle and polymer (NPP) suspensions in the presence of both thermal fluctuation and long‐range many‐body hydrodynamic interactions (HIs). Brownian motion of the NPP is explicitly captured by a stochastic forcing term in the LD method. The LD method is two‐way coupled to the nonfluctuating LB fluid through a discrete LB forcing source distribution to capture the long‐range HI. To ensure intrinsically linear scalability with respect to the number of particles, a Eulerian‐host algorithm for short‐distance particle neighbor search and interaction is developed and embedded to LB‐LD framework. The validity and accuracy of the LB‐LD approach are demonstrated through several sample problems. The simulation results show good agreements with theory and experiment. The LB‐LD approach can be favorably incorporated into complex multiscale computational frameworks for efficiently simulating multiscale multicomponent particulate suspension systems such as complex blood suspensions.
Tutorial: Langevin Dynamics methods for aerosol particle trajectory simulations and collision rate constant modeling
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.
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- Award ID(s):
- 1903432
- NSF-PAR ID:
- 10221107
- Date Published:
- Journal Name:
- Journal of aerosol science
- Volume:
- 155
- Issue:
- June
- ISSN:
- 0021-8502
- Page Range / eLocation ID:
- 105746
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Summary -
The relative velocities and positions of monodisperse high-inertia particle pairs in isotropic turbulence are studied using direct numerical simulations (DNS), as well as Langevin simulations (LS) based on a probability density function (PDF) kinetic model for pair relative motion. In a prior study (Rani et al. , J. Fluid Mech. , vol. 756, 2014, pp. 870–902), the authors developed a stochastic theory that involved deriving closures in the limit of high Stokes number for the diffusivity tensor in the PDF equation for monodisperse particle pairs. The diffusivity contained the time integral of the Eulerian two-time correlation of fluid relative velocities seen by pairs that are nearly stationary. The two-time correlation was analytically resolved through the approximation that the temporal change in the fluid relative velocities seen by a pair occurs principally due to the advection of smaller eddies past the pair by large-scale eddies. Accordingly, two diffusivity expressions were obtained based on whether the pair centre of mass remained fixed during flow time scales, or moved in response to integral-scale eddies. In the current study, a quantitative analysis of the (Rani et al. 2014) stochastic theory is performed through a comparison of the pair statistics obtained using LS with those from DNS. LS consist of evolving the Langevin equations for pair separation and relative velocity, which is statistically equivalent to solving the classical Fokker–Planck form of the pair PDF equation. Langevin simulations of particle-pair dispersion were performed using three closure forms of the diffusivity – i.e. the one containing the time integral of the Eulerian two-time correlation of the seen fluid relative velocities and the two analytical diffusivity expressions. In the first closure form, the two-time correlation was computed using DNS of forced isotropic turbulence laden with stationary particles. The two analytical closure forms have the advantage that they can be evaluated using a model for the turbulence energy spectrum that closely matched the DNS spectrum. The three diffusivities are analysed to quantify the effects of the approximations made in deriving them. Pair relative-motion statistics obtained from the three sets of Langevin simulations are compared with the results from the DNS of (moving) particle-laden forced isotropic turbulence for $St_{\unicode[STIX]{x1D702}}=10,20,40,80$ and $Re_{\unicode[STIX]{x1D706}}=76,131$ . Here, $St_{\unicode[STIX]{x1D702}}$ is the particle Stokes number based on the Kolmogorov time scale and $Re_{\unicode[STIX]{x1D706}}$ is the Taylor micro-scale Reynolds number. Statistics such as the radial distribution function (RDF), the variance and kurtosis of particle-pair relative velocities and the particle collision kernel were computed using both Langevin and DNS runs, and compared. The RDFs from the stochastic runs were in good agreement with those from the DNS. Also computed were the PDFs $\unicode[STIX]{x1D6FA}(U|r)$ and $\unicode[STIX]{x1D6FA}(U_{r}|r)$ of relative velocity $U$ and of the radial component of relative velocity $U_{r}$ respectively, both PDFs conditioned on separation $r$ . The first closure form, involving the Eulerian two-time correlation of fluid relative velocities, showed the best agreement with the DNS results for the PDFs.more » « less
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Meier-Schellersheim, Martin (Ed.)We introduce a Stochastic Reaction-Diffusion-Dynamics Model (SRDDM) for simulations of cellular mechanochemical processes with high spatial and temporal resolution. The SRDDM is mapped into the CellDynaMo package, which couples the spatially inhomogeneous reaction-diffusion master equation to account for biochemical reactions and molecular transport within the Langevin Dynamics (LD) framework to describe dynamic mechanical processes. This computational infrastructure allows the simulation of hours of molecular machine dynamics in reasonable wall-clock time. We apply SRDDM to test performance of the Search-and-Capture of mitotic spindle assembly by simulating, in three spatial dimensions, dynamic instability of elastic microtubules anchored in two centrosomes, movement and deformations of geometrically realistic centromeres with flexible kinetochores and chromosome arms. Furthermore, the SRDDM describes the mechanics and kinetics of Ndc80 linkers mediating transient attachments of microtubules to the chromosomal kinetochores. The rates of these attachments and detachments depend upon phosphorylation states of the Ndc80 linkers, which are regulated in the model by explicitly accounting for the reactions of Aurora A and B kinase enzymes undergoing restricted diffusion. We find that there is an optimal rate of microtubule-kinetochore detachments which maximizes the accuracy of the chromosome connections, that adding chromosome arms to kinetochores improve the accuracy by slowing down chromosome movements, that Aurora A and kinetochore deformations have a small positive effect on the attachment accuracy, and that thermal fluctuations of the microtubules increase the rates of kinetochore capture and also improve the accuracy of spindle assembly.more » « less
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Abstract Precipitation efficiency and optical properties of clouds, both central to determining Earth's weather and climate, depend on the size distribution of cloud particles. In this work theoretical expressions for cloud droplet size distribution shape are evaluated using measurements from controlled experiments in a convective‐cloud chamber. The experiments are a unique opportunity to constrain theory because they are in steady‐state and because the initial and boundary conditions are well characterized compared to typical atmospheric measurements. Three theoretical distributions obtained from a Langevin drift‐diffusion approach to cloud formation via stochastic condensation are tested: (a) stochastic condensation with a constant removal time‐scale; (b) stochastic condensation with a size‐dependent removal time‐scale; (c) droplet growth in a fixed supersaturation condition and with size‐dependent removal. In addition, a similar Weibull distribution that can be obtained from the drift‐diffusion approach, as well as from mechanism‐independent probabilistic arguments (e.g., maximum entropy), is tested as a fourth hypothesis. Statistical techniques such as the
χ 2test, sum of squared errors of prediction, and residual analysis are employed to judge relative success or failure of the theoretical distributions to describe the experimental data. An extensive set of cloud droplet size distributions are measured under different aerosol injection rates. Five different aerosol injection rates are run both for size‐selected aerosol particles, and six aerosol injection rates are run for broad‐distribution, polydisperse aerosol particles. In relative comparison, the most favourable comparison to the measurements is the expression for stochastic condensation with size‐dependent droplet removal rate. However, even this optimal distribution breaks down for broad aerosol size distributions, primarily due to deviations from the measured large‐droplet tail. A possible explanation for the deviation is the Ostwald ripening effect coupled with deactivation/activation in polluted cloud conditions. -
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.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/https://doi.org/10.1016/j.jaerosci.2021.105746
