Atomistic hybrid DSMC/NEMD method for nonequilibrium multiscale simulations

https://doi.org/10.1016/j.jcp.2009.10.035Get rights and content

Abstract

A multiscale hybrid method for coupling the direct simulation Monte Carlo (DSMC) method to the nonequilibrium molecular dynamics (NEMD) method is introduced. The method addresses Knudsen layer type gas flows within a few mean free paths of an interface or about an object with dimensions of the order of a few mean free paths. It employs the NEMD method to resolve nanoscale phenomena closest to the interface along with coupled DSMC simulation of the remainder of the Knudsen layer. The hybrid DSMC/NEMD method is a particle based algorithm without a buffer zone. It incorporates a new, modified generalized soft sphere (MGSS) molecular collision model to improve the poor computational efficiency of the traditional generalized soft sphere GSS model and to achieve DSMC compatibility with Lennard-Jones NEMD molecular interactions. An equilibrium gas, a Fourier thermal flow, and an oscillatory Couette flow, are simulated to validate the method. The method shows good agreement with Maxwell–Boltzmann theory for the equilibrium system, Chapman–Enskog theory for Fourier flow, and pure DSMC simulations for oscillatory Couette flow. Speedup in CPU time of the hybrid solver is benchmarked against a pure NEMD solver baseline for different system sizes and solver domain partitions. Finally, the hybrid method is applied to investigate interaction of argon gas with solid surface molecules in a parametric study of the influence of wetting effects and solid molecular mass on energy transfer and thermal accommodation coefficients. It is determined that wetting effect strength and solid molecular mass have a significant impact on the energy transfer between gas and solid phases and thermal accommodation coefficient.

Introduction

With continued improvements in computer speed and storage capabilities, atomistic simulation methods, such as direct simulation Monte Carlo method (DSMC) and molecular dynamics (MD), have become increasingly viable options for gas dynamics research at sub-macroscopic scales. The scales of interest are distinguished by a flow Knudsen number (Kn=λ/d), the ratio of mean free path to a characteristic flow dimension, greater than 0.1.

DSMC, in particular, has been widely applied to investigate dilute gas flow phenomena at length scales on the order of a mean free path that are no longer governed by continuum fluid dynamics but are in the transition regime between continuum and free molecular flow (0.1Kn10). Knudsen layer type gas flows within a few mean free paths of an interface or about an object with dimensions of the order of a few mean free paths are in the transition regime.

When applied to flow over an interface, the relatively efficient DSMC method has significant limitations in its ability to address the important nanoscale interactions of gas molecules with solid or liquid molecules comprising the interface. The present research is aimed at removing this limitation with the development of a hybrid, multiscale simulation method that uses the MD method to resolve nanoscale phenomena closest to the interface along with DSMC simulation of the remainder of the Knudsen layer. Thus, the method spans the disparate scales of molecular diameters and molecular collisions. (For argon at standard conditions, the mean free path is 63 nm, while the effective hard sphere molecular diameter is 0.4 nm.) The nanoscale domain resolved near the interface may be a small but not insignificant portion of the Knudsen layer. We adopt the convention of referring to the overlapping scales of these flow domains and their resident phenomena as mesoscale.

The DSMC method was originally developed by Bird [1] and subsequent developments have been catalogued by Orhan et al. [2], and Prasanth and Kakkassery [3]. It has produced strikingly accurate results in a variety of dilute gas flow simulations by introducing the notion of stochastic representative particles in place of actual gas molecules. The simulation divides tracks the motions and collisions of these particles through a flow domain, which is divided into cells, by simplified molecular interaction mechanics. DSMC collisions take place with kinetic outcomes determined by modeled molecular scattering probabilities. Macroscopic flow properties are computed from sampled particle statistics over a number of independent simulation trials, represented by ensemble or time averages. A binary collision assumption limits DSMC simulation to dilute gases.

DSMC requires interface boundary conditions for the simulation particles. These are necessarily restricted to simplified mechanical models based on average surface collision dynamics or to statistically averaged particle flux distributions based on simplified molecular interactions at the interface. Since DSMC simulation is limited to dilute gases with simplified boundary interactions, it cannot resolve complex mesoscale interfacial phenomena involving interphase molecular interaction and exchange at an interface.

Classical molecular dynamics simulation or MD dates from its conception by Alder and Wainwright [4] in the late 1950s. Its history and algorithms are described in a number of texts and reference works (e.g., [5], [6]). It is essentially a simulation method that tracks the motions of modeled individual atoms, molecules, molecular clusters, or ions. MD is based on applying Newtonian mechanics to these motions with interparticle attractive and repulsive forces determined by specified potential energy functions. Most commonly, short range interactive forces are derived from a Lennard-Jones (L-J) intermolecular potential function.

The MD method is appealing for interfacial physics because it can model interaction of individual gas molecules with the liquid or solid molecules comprising a multiphase interface and can also model interphase transitions. Furthermore, it is not restricted to dilute gases and can simulate the motions of denser gases and liquid near the interface. Technically, when the MD method is applied to fluids that are not in equilibrium because they are subjected to imposed mechanical or thermal drivers, it is known as nonequilibruim molecular dynamics (NEMD).

The substantial disadvantage of MD or NEMD simulation is that it is so highly computationally intensive that its application is restricted to nanoscale systems comprised of relatively small numbers of molecules within dimensions of hundreds of nanometers. It cannot be applied to analyze mesoscale flow systems of engineering interest since it is infeasible to simultaneously apply it to the interfacial region and the larger, bulk incident-gas region. Nevertheless, it is a valuable tool for investigating interfacial physics. As in the case of DSMC, macroscopic properties can be obtained by sampling.

A multiscale method with the ability to efficiently simulate the molecular gas dynamics of bulk microscale flow over an interface, while resolving the computationally demanding nanoscale molecular interactions in a domain surrounding the interface, would be invaluable. The ability of DSMC and NEMD to handle molecular collision-scale and molecular diameter-scale types of problems, respectively, suggests that a hybrid, combing the two, could be an effective approach to development of a robust multiscale method.

Hybrid simulation schemes are abundant for microscale/continuum gas flows using DSMC coupled with Navier–Stokes, Stokes, or Euler continuum CFD methods [7], [8], [9], [10]. Likewise, many approaches have been developed for NEMD coupled with continuum Navier–Stokes CFD for nanoscale/microscale liquid flows [11], [12], [13], [14], [15], [16]. However, there has been far less work on hybridized DSMC/NEMD methods and the methods that have been developed, so far, either have substantial limitations or are untested for their ability to model certain important systems.

A combined DSMC and NEMD study of laser ablation was reported by Zeifman et al. [17] in which information was transferred from a NEMD flow domain to a DSMC flow domain but there was no two-way active coupling between the two domains. A hybrid DSMC/NEMD gas/solid simulation method for one-way DSMC to near-surface NEMD matching, which involves repositioning of incident DSMC molecules, was developed by Yamamoto et al. [18].

A simulation method was developed by Nedea et al. [19], [20], coupling NEMD solution domains to Monte Carlo (MC) solution domains. The MC method used is an extension of DSMC [1] that avoids the restriction of traditional DSMC to dilute gases. The primary emphasis of their work is on dense hard-sphere gas systems but they also demonstrate good results for thermally driven flow in a dilute gas channel bounded by isothermal walls [20]. However, their hybrid method is coupled by matching the macroscopic properties in overlapping buffer layers, and not focused on ensuring mass, momentum, and heat flux continuity across the method interface. This and the modeling artifices they employed may make it difficult for their method to completely capture the physics of systems with large gradients in the method interface region or unsteady flows.

In this article we introduce a new DSMC/NEMD hybrid method based on DSMC particle/NEMD molecule coupling with consistent particle flux dynamics across the interface between separate DSMC and NEMD solution domains. In developing our hybrid method, special attention was paid to the molecular collision model in the DSMC domain to insure compatibility near the interface with the Lennard-Jones (L-J) intermolecular potential function used in the NEMD domain. A new collision model with improved efficiency and low-temperature capability that is suitable for maintaining compatibility was also developed and implemented.

We present the results of validating the particle flux continuity at the DSMC/NEMD interfaces in both equilibrium and nonequilibrium systems by comparisons with Maxwellian and Chapman Enskog theory, respectively. To verify the method’s accuracy in unsteady systems, an oscillatory Couette flow simulation was compared with a pure DSMC solution. In addition, the hybrid method’s computational efficiency was compared with that of the pure NEMD method for different DSMC/NEMD solver partition size ratios and system particle numbers.

Finally, the hybrid method was applied to investigate interaction of argon gas with solid surface molecules in a mesoscale parametric study of the influence of wetting effects and solid molecular mass on energy transfer and thermal accommodation coefficients. The results are compared with results from DSMC simulations using the thermal accommodation coefficients determined from the hybrid simulations.

Section snippets

DSMC molecular collision modeling

Various molecular collision cross section models have been developed for use in DSMC through consideration of the intermolecular force or potential. For DSMC applications, the goal generally is to reproduce the dependence of viscosity and self diffusion on temperature of actual gases. The traditional and most widely used collision model is the variable hard sphere (VHS) model [1]. Some newer molecular collision models, such as the variable soft sphere (VSS) [21] and the generalized hard sphere

Description of DSMC/MD hybrid method

In most hybrid methods that divide the physical domain to two or more parts and assign each one to different method solver, matching is achieved by establishing overlapping or “buffer” regions along the interfaces between the methods extending into their respective domains, wherein both methods are applied. The mean properties and momentum or heat fluxes from each domain are “copied” to the buffer, to generate the mean properties or flux continuities between the methods.

Wang and He [11]

Validation of the method

The newly developed DSMC/NEMD method, which has no buffer zone, raises the question of whether the method interface is continuous in a particle sense and physically realistic. To validate the method and its boundary treatment, we chose three benchmark problems, an equilibrium gas system between isothermal diffusive walls, a Fourier thermal system and an oscillatory Couette flow. In the NEMD solver, molecular motions are driven by intermolecular forces derived from Eq. (1) with the force cut-off

Assessment of computational performance

To assess the computational performance of the hybrid method, we compared the hybrid CPU times in the simulation of the equilibrium gas system for three different partitions of the region between the isothermal diffusive walls into NEMD and DSMC domains. Systems were simulated containing 8000, 16,000, and 32,000 gas molecules. For each size of the system, we chose three different values for the NEMD/DSMC solver partition ratio, rMD: 25%, 50%, and 100% . To facilitate the comparison, the hybrid

Application of method to investigate thermal accommodation at a surface

One of the advantages for using the NEMD method at a surface is its capability of studying the physics of interface phenomena. One use of this capability is the determination of the thermal accommodation coefficient often used as an approximate boundary condition for molecular interaction surface physics in kinetic theory-based computations such as DSMC.

A thermal accommodation coefficient for total energy of molecules at the gas–solid surface, aE can be defined as cf. [18]aE=Ei¯-Er¯Ei¯-Ew¯where

Conclusions

The atomistic hybrid DSMC/NEMD simulation method developed shows great promise as a versatile tool for investigating multiscale interfacial systems involving Knudsen layer gas flow over a surface with nanoscale surface interaction phenomena. It offers a distinct performance advantage over the use of NEMD alone for investigating such systems and does not depend on physically questionable artifices for its implementation. The generalized soft sphere DSMC collision model developed in conjunction

Acknowledgments

This work was supported by NSF Cooperative Agreement No. HRD-0206162, to the CREST Center for Mesoscopic Modeling and Simulation and NSF Award number 0934206 to the PREM, City College-Chicago MRSEC Partnership on the Dynamics of Heterogeneous and Particulate Materials.

References (27)

  • D.B. Hash et al.

    Assessment of schemes for coupling Monte Carlo and Navier–Stokes solution methods

    J. Thermophys. Heat Transfer

    (1996)
  • T. Werder et al.

    Hybrid atomistic-continuum method for the simulation of dense fluid flows

    J. Comput. Phys.

    (2006)
  • X.B. Nie et al.

    A continuum and molecular dynamics hybrid method for micro- and nano-fluid flow

    J. Fluid Mech.

    (2004)
  • Cited by (0)

    View full text