Variable Hard Sphere Model Research Articles

Hermite spectral method for the inelastic Boltzmann equation

We propose a Hermite spectral method for the inelastic Boltzmann equation, which makes two-dimensional periodic problem computation affordable by the hardware nowadays. This new approach involves utilizing a Hermite expansion, whereby the expansion coefficients for the variable hard sphere model are simplified into a series of summations that can be precisely derived. Additionally, a new collision model is built with a combination of the quadratic collision operator and a simplified collision operator, which helps us to balance the computational cost and the accuracy. Various numerical experiments, including spatially two-dimensional simulations, demonstrate the accuracy and efficiency of this numerical scheme.

Mean scattering angle method for direct simulation Monte Carlo.

In order to predict the postcollision velocities in the binary collision, one needs the deflection angle given by the solution of the trajectory equation of classical mechanics. In the direct simulation Monte Carlo, it has been conventional to assume that the scattering is isotropic taking the hard sphere scattering law (the variable hard sphere model). Because the simulation carries out the ensemble average, and thus the collisional deflections should be averaged at the end, it is proposed to use the preaveraged deflection angle for each collision. This is a representative single deflection angle simulation. The present work postulates a method to estimate the mean deflection angles for arbitrary force models considering the differential cross section formula and the trajectory equation. For hard spheres, the representative single deflection simulation of normal shock waves recovers the conventional computations within 3%. For Maxwell molecules, the conventional simulation in which the isotropic scattering law is taken can exactly be reproduced by the single deflection when the representative angle is adjusted to the mean deflection angle together with the reference diameter obtained in the present method. It has been observed that the Maxwell-molecule-representative for the single angle simulation reproduces the conventional computations of any isotropic scattering models including hard spheres.

Kinetic simulations of laser-induced plume expansion from a copper target into a vacuum or argon background gas based on ab initio calculation of Cu–Cu, Ar–Ar, and Ar–Cu interactions

The kinetic simulations of plume expansion induced by pulsed laser heating of a copper target in a vacuum or low-pressure argon background gas are performed based on the direct simulation Monte Carlo (DSMC) method and ab initio quantum mechanical calculation of interactions between copper and argon atoms. The potential energy curves (PECs) for Cu–Cu, Ar–Ar, and Ar–Cu interactions are obtained in density functional theory (DFT) calculations with the van der Waals (vdW) correction. The computed Cu–Cu PEC is strikingly different from the Lennard-Jones (LJ) potentials with semi-empirical parameters, which were previously suggested for kinetic simulations of the copper vapor flows. It is found that the Lorentz–Berthelot rule cannot reliably predict the parameters of the LJ potential for cross-species Ar–Cu interaction. The DFT-vdW PECs are fitted by the Morse long-range (MLR) potentials. The MLR potentials are used to compute the outcomes of binary collisions in the DSMC method based on the solution of the classical scattering problem and to parameterize the variable hard sphere (VHS) collision model. The results of the DSMC simulations based on DFT-vdW PECs are compared with the results obtained based on various parameterizations of the VHS model. It is shown that the previously developed parameterizations of the VHS model can either over- or underestimate the plume temperature and density compared to the results obtained based on the DFT-vdW PECs. The simulations also reveal the strong effect of the cross-species collision model parameters on the flow structure in the mixing layer, which is dominated by molecular diffusion.

Numerical Investigation into the Flow Characteristics of Gas Mixtures in Knudsen Pump with Variable Soft Sphere Model.

In Knudsen pumps with geometric configuration of rectangle, gas flows are induced by temperature gradients along channel walls. In this paper, the direct simulation Monte Carlo (DSMC) method is used to investigate numerically the flow characteristics of H2–N2 mixtures in the Knudsen pump. The variable soft sphere (VSS) model is applied to depict molecular diffusion in the gas mixtures, and the results obtained are compared with those calculated from a variable hard sphere (VHS) model. It is demonstrated that pressure is crucial to affecting the variation of gas flow pattern, but the gas concentration in H2–N2 mixtures and the collision model do not change the flow pattern significantly. On the other hand, the velocity of H2 is larger than that of N2. The velocities of H2 and N2 increase if the concentration of H2 rises in the gas mixtures. The results of velocity and mass flow rate obtained from VSS and VHS models are different. Finally, a linear relation between the decrease of mass flow rate and the increase of H2 concentration is proposed to predict the mass flow rate in H2–N2 mixtures.

The accuracy of collision cross sections in particle modeling on copper vacuum arcs

Direct Simulation Monte Carlo and Particle-in-Cell (DSMC-PIC) method is a powerful tool in studying metal vapor vacuum arcs (VA). However, the collision cross sections (Xs) for metal atoms/ions are not evaluated yet, compared to the already developed databases for gas discharges. The Xs are subject to uncertainties which influence the accuracy and predictive ability of particle models. In this paper, the common electrode material copper (Cu) is taken as an example for survey of Xs. Various experimental data, theoretical calculation, and analytical fitting formula are reviewed. The electron-impact Cu elastic, excitation, and ionization Xs are found to have an overall uncertainty within 15%. The Cu-Cu+ resonant charge-exchange Xs are not as accurate as that of e-Cu collision. The Cu-Cu interaction Xs by variable hard sphere model requires further validation. Based on the evaluated Xs, the mean free path and collision frequency are derived in copper VA plasmas, and compared to the Debye length and plasma frequency, respectively. The influences of those spatial and time scales are discussed in developing DSMC-PIC codes for modeling of copper VA.

An efficient numerical method for solving the Boltzmann equation in multidimensions

In this paper we deal with the extension of the Fast Kinetic Scheme (FKS) (Dimarco and Loubère, 2013 [26]) originally constructed for solving the BGK equation, to the more challenging case of the Boltzmann equation. The scheme combines a robust and fast method for treating the transport part based on an innovative Lagrangian technique supplemented with conservative fast spectral schemes to treat the collisional operator by means of an operator splitting approach. This approach along with several implementation features related to the parallelization of the algorithm permits to construct an efficient simulation tool which is numerically tested against exact and reference solutions on classical problems arising in rarefied gas dynamic. We present results up to the 3D×3D case for unsteady flows for the Variable Hard Sphere model which may serve as benchmark for future comparisons between different numerical methods for solving the multidimensional Boltzmann equation. For this reason, we also provide for each problem studied details on the computational cost and memory consumption as well as comparisons with the BGK model or the limit model of compressible Euler equations.

Novel Hybrid Hard Sphere Model for Direct Simulation Monte Carlo Computations

This paper concerns the development of a novel hybrid hard sphere molecular model in direct simulation Monte Carlo technique. Usually, the variable hard sphere model is applied to simulate molecular collisions. However, this model is not accurate enough when temperatures are quite low or extremely high. The generalized hard sphere model can give the viscosity behavior of gases more accurately over a wide temperature range, but it is seldom used in practical engineering applications due to its poor computational efficiency. In this paper, a hybrid hard sphere model is presented to improve the computational efficiency of the original generalized hard sphere model. The modification is based upon the variable hard sphere and generalized hard sphere models and agrees quite well with the experimental observations. To investigate the accuracy and capability of the newly proposed model, two test cases involving thermal equilibrium and nonequilibrium are simulated. The results are compared with those of existing models and experimental data. The results show that the computational efficiency of the new model is much higher than the original generalized hard sphere model especially at low temperatures.

Plume propagation and Pt film growth during shadow-masked pulsed laser deposition in a buffer Ar gas

Shadow-masked pulsed laser deposition (SMPLD) enables the preparation of films that contain none of the droplets that are normally formed in laser irradiation of the target. The platinum (Pt) film produced by SMPLD was studied using scanning electron microscopy (SEM), atomic force microscopy (AFM), and Rutherford backscattering spectroscopy (RBS) of helium ions. The film thickness distribution across the substrate surface took the shape of a simple crater, and the film thickness on the crated “bottom” (center of the shadow area) was approximately 5 times less than that on the “mound” (edge of the shadow area). Monte Carlo collision (MCC) modeling of the laser plume movement during SMPLD was performed to clarify the role of the mask in the formation of the Pt films. The dynamics of the Pt atoms in the laser plume was studied using the vacuum deposition method through a narrow slit onto a rapidly displaced substrate, augmented by RBS measurements of the deposited film thickness along the substrate movement direction. The ionic flux was specifically measured using an ion probe. MСС simulation allowed the changes in the basic parameters of the deposited atom stream to be evaluated with the use of a mask. Comparison of the experimental and calculated distribution of the Pt film indicated that the best correlation was observed using the interpenetration model of the plume and buffer gas (argon, Ar) accompanied by elastic collisions of Pt atoms with the Ar atoms using the variable hard sphere model. Atomic flux models were utilized to imitate the growth of individual Pt crystals using the kinetic Monte Carlo method. In the SMPLD case, the root mean square roughness of the model crystal surface increased by ∼10% and the concentration of vacancies increased by ∼4% compared with the model crystal obtained by pulsed laser deposition (PLD). The surface topography of the experimental Pt films was defined by the nanocrystal nature of their structure. The use of a mask promoted the growth of relatively large nanocrystals and, thus, an increase in the surface roughness in AFM scans on the submicron scale.

Effect of intermolecular potential on compressible Couette flow in slip and transitional regimes

The effect of intermolecular potentials on compressible, planar flow in slip and transitional regimes is investigated using the direct simulation Monte Carlo method. Two intermolecular interaction models, the variable hard sphere (VHS) and the Lennard-Jones (LJ) models, are first compared for subsonic and supersonic Couette flows of argon at temperatures of 40, 273, and 1,000 K, and then for Couette flows in the transitional regime ranging from Knudsen numbers (Kn) of 0.0051 to 1. The binary scattering model for elastic scattering using the Lennard-Jones (LJ) intermolecular potential proposed recently [A. Venkattraman and A. Alexeenko, “Binary scattering model for Lennard-Jones potential: Transport coefficients and collision integrals for non-equilibrium gas flow simulations,” Phys. Fluids 24, 027101 (2012)] is shown to accurately reproduce both the theoretical collision frequency in an equilibrium gas as well as the theoretical viscosity variation with temperature. The use of a repulsive-attractive instead of a purely repulsive potential is found to be most important in the continuum and slip regimes as well as in flows with large temperature variations. Differences in shear stress of up to 28% between the VHS and LJ models is observed at Kn=0.0051 and is attributed to differences in collision frequencies, ultimately affecting velocity gradients at the wall. For Kn=1 where the Knudsen layer expands the entire domain, the effect of the larger collision frequency in the LJ model relative to VHS diminishes, and a 7% difference in shear stress is observed.

DSMC Simulation and Experimental Validation of Shock Interaction in Hypersonic Low Density Flow

Direct simulation Monte Carlo (DSMC) of shock interaction in hypersonic low density flow is developed. Three collision molecular models, including hard sphere (HS), variable hard sphere (VHS), and variable soft sphere (VSS), are employed in the DSMC study. The simulations of double-cone and Edney's type IV hypersonic shock interactions in low density flow are performed. Comparisons between DSMC and experimental data are conducted. Investigation of the double-cone hypersonic flow shows that three collision molecular models can predict the trend of pressure coefficient and the Stanton number. HS model shows the best agreement between DSMC simulation and experiment among three collision molecular models. Also, it shows that the agreement between DSMC and experiment is generally good for HS and VHS models in Edney's type IV shock interaction. However, it fails in the VSS model. Both double-cone and Edney's type IV shock interaction simulations show that the DSMC errors depend on the Knudsen number and the models employed for intermolecular interaction. With the increase in the Knudsen number, the DSMC error is decreased. The error is the smallest in HS compared with those in the VHS and VSS models. When the Knudsen number is in the level of 10−4, the DSMC errors, for pressure coefficient, the Stanton number, and the scale of interaction region, are controlled within 10%.

Simulation of gas diffusion in highly porous nanostructures by direct simulation Monte Carlo

A Direct Simulation Monte Carlo (DSMC) method is utilized to simulate gas diffusion in nanoscaled highly porous layers. An open source solver has been extended with the variable soft sphere (VSS) binary collision model and the inflow boundary model was adjusted for small numbers of DSMC particle initialization. Comparison with the analytical diffusion equation illustrate the improvement of the VSS model compared to the variable hard sphere model (VHS). Subsequently, several highly porous particle layers (gas sensors synthesized by flame spray pyrolysis and isotropic layers) build up by 10nm particles have been investigated. Results for DSMC gas diffusion in the porous structures are in agreement with the well established dusty gas model (DGM). However, while DGM requires measurements or estimations of pore sizes, porosity, and tortuosity and furthermore is limited to homogenous layers, the present contribution shows significant advantages of DSMC in describing gas diffusion in non-isotropic porous structures.

Molecular Dynamics Simulations of Shock Waves in Mixtures of Noble Gases

We study the structure of a normal shock wave in noble gas mixtures (Xe-He and Ar-He) of various compositions using molecular dynamics and direct simulation Monte Carlo. The molecular dynamics simulations are first validated against experimental data. Good agreement is found between the molecular dynamics solutions and the experimental measurements, with the exception of the parallel temperature profile in the 24.7% Ar-He mixture, despite the satisfactory agreement between the parallel velocity profiles. Secondly, a validation against direct simulation Monte Carlo solutions obtained with the accurate generalized hard sphere model is presented. As expected, the generalized hard sphere direct simulation Monte Carlo and molecular dynamics solutions are in near-perfect agreement. Finally, molecular dynamics results are compared to those obtained with the lower fidelity variable hard sphere model, which, if inappropriately parametrized, fails to describe the shock wave structure. This work exemplifies how full molecular dynamics solutions could be used to precisely discern differences between phenomenological models of various accuracy.

Consistent treatment of transport properties for five-species air direct simulation Monte Carlo/Navier-Stokes applications

A general approach for achieving consistency in the transport properties between direct simulation Monte Carlo (DSMC) and Navier-Stokes (CFD) solvers is presented for five-species air. Coefficients of species diffusion, viscosity, and thermal conductivities are considered. The transport coefficients that are modeled in CFD solvers are often obtained by expressions involving sets of collision integrals, which are obtained from more realistic intermolecular potentials (i.e., ab initio calculations). In this work, the self-consistent effective binary diffusion and Gupta et al.–Yos tranport models are considered. The DSMC transport coefficients are approximated from Chapman-Enskog theory in which the collision integrals are computed using either the variable hard sphere (VHS) and variable soft sphere (VSS) (phenomenological) collision cross section models. The VHS and VSS parameters are then used to adjust the DSMC transport coefficients in order to achieve a best-fit to the coefficients computed from more realistic intermolecular potentials over a range of temperatures. The best-fit collision model parameters are determined for both collision-averaged and collision-specific pairing approaches using the Nelder-Mead simplex algorithm. A consistent treatment of the diffusion, viscosity, and thermal conductivities is presented, and recommended sets of best-fit VHS and VSS collision model parameters are provided for a five-species air mixture.

Direct simulation Monte Carlo modeling of metal vapor flows in application to thin film deposition

Thin films of metal for electronics, nano/microelectromechanical systems and optical coatings are often prepared by various vacuum deposition techniques. Modeling such metal vapor flows using methods such as the direct simulation Monte Carlo (DSMC) can aid in the design and analysis of deposition systems and accelerate development of films with desired properties. The determination of suitable variable hard sphere (VHS) molecular model parameters for DSMC simulations using measured growth rate distribution is demonstrated with aluminum vapor as an example. Axisymmetric DSMC simulations using a VHS model corresponding to a reference diameter of 0.8 nm and a viscosity-temperature exponent of 1 are shown to agree well with available experimental data. The model is then used in two-dimensional DSMC simulations to study the interaction of plumes from multiple sources. An expression for substrate mass flux assuming no interaction between sources agrees well with DSMC simulations for a mass flow rate of 0.1 g/min corresponding to a Knudsen number (Kn) of about 0.1. The non-additive interaction of plumes at a higher flow rate of 1 g/min corresponding to a Kn of about 0.01 results in a higher mass flux non-uniformity in the DSMC simulations which is not captured by the simplified analytical expression.

On a modified Monte-Carlo method and variable soft sphere model for rarefied binary gas mixture flow simulation

The effect of new terms in the improved algorithm, the modified direct simulation Monte-Carlo (MDSMC) method, is investigated by simulating a rarefied binary gas mixture flow inside a rotating cylinder. Dalton law for the partial pressures contributed by each species of the binary gas mixture is incorporated into our simulation using the MDSMC method and the direct simulation Monte-Carlo (DSMC) method. Moreover, the effect of the exponent of the cosine of deflection angle (α) in the inter-molecular collision models, the variable soft sphere (VSS) and the variable hard sphere (VHS), is investigated in our simulation. The improvement of the results of simulation is pronounced using the MDSMC method when compared with the results of the DSMC method. The results of simulation using the VSS model show some improvements on the result of simulation for the mixture temperature at radial distances close to the cylinder wall where the temperature reaches the maximum value when compared with the results using the VHS model.

Direct simulation Monte Carlo modeling of e-beam metal deposition

Three-dimensional direct simulation Monte Carlo (DSMC) method is applied here to model the electron-beam physical vapor deposition of copper thin films. Various molecular models for copper-copper interactions have been considered and a suitable molecular model has been determined based on comparisons of dimensional mass fluxes obtained from simulations and previous experiments. The variable hard sphere model that is determined for atomic copper vapor can be used in DSMC simulations for design and analysis of vacuum deposition systems, allowing for accurate prediction of growth rates, uniformity, and microstructure.

Large-scale molecular dynamics simulations of normal shock waves in dilute argon

Large-scale molecular dynamics (MD) simulations using the Lennard-Jones potential are performed to study the structure of normal shock waves in dilute argon. Nonperiodic boundary conditions in the flow direction are applied by coupling the MD domain with a two-dimensional finite-volume computational fluid dynamics (CFD) solver to correctly generate the inflow and outflow particle reservoirs. Detailed comparisons are made with direct simulation Monte Carlo (DSMC) solutions using the variable-hard-sphere (VHS) collision model. By performing realistic MD simulations of full shock waves, this article presents a more sensitive evaluation of the VHS model parameters (via temperature and velocity distribution functions) than is possible using available experimental density measurements. In the high temperature range (300–8000 K), where the Chapman–Enskog theory supports the VHS model assumptions, near-perfect agreement between MD and DSMC solutions is demonstrated and inverse shock thickness predictions reproduce experimental measurements. In the low temperature range (16–300 K), theory predicts and MD simulation confirms that the VHS collision model becomes less valid.

Elastic molecular collision models for quantum mechanical scattering in the Monte Carlo simulation of rarefied gas flow at low temperatures

Elastic molecular collision models for Monte Carlo simulation of rarefied gas flow, such as the variable hard sphere (VHS), variable soft sphere (VSS), and variable sphere (VS) molecular models, were extended to quantum mechanical scattering, and then applied to the simulation of spherical expansion of He4. The profiles of number density, flow velocity, and parallel and perpendicular kinetic temperatures obtained with the VHS, VSS, and VS models were in good agreement with those yielded by direct calculation of quantum mechanical scattering by Koura [Phys. Fluids 11, 3174 (1999)].

열천이 현상을 이용한 마이크로 펌프내의 희박기체유동 해석

Rarefied gas flows through two-dimensional micro channels are studied numerically for the performance optimization of a nanomembrane-based Knudsen compressor. The effects of the wall temperature distributions on the thermal transpiration flow patterns are examined. The flow has a pumping effect, and the mass flow rates through the channel are calculated. The results show that a steady one-way flow is induced for a wide range of the Knudsen number. The DSMC(direct simulation Monte Carlo) method with VHS(variable hard sphere) model and NTC(no time counter) techniques has been applied in this work to obtain numerical solutions. A critical element that drives Knudsen compressor Is the thermal transpiration membrane. The membranes are based on aerosol or machined aerogel. The aerogel is modeled as a single micro flow channel.

Simulation of heat transfer in high speed microflows

The dynamic flow behavior and heat transfer characteristics of high speed gas flows through two-dimensional microchannels are investigated by using direct simulation Monte Carlo (DSMC) method. In order to study the effects of the Knudsen number ( Kn) on fluid flow and heat transfer behaviors, flows in both slip flow regime and transition flow regime have been investigated. The collision process has been treated in a statistical way using the NTC (no-time-counter) scheme. The VHS (variable hard sphere) model was chosen to simulate the collision from the kinetic point of view. The different viscous effects produced from different Kn significantly influenced the development of the flow and also the development of the thermal boundary layers. The bulk flow has a high increase in temperature due to the wall friction and the deceleration of the supersonic flow as it enters the narrow channels, and this dominates the wall heat flux phenomenon. The effect of gas species on the flow properties was studied using helium and nitrogen. It has been found that the wall temperature jump, the wall heat flux, and the centerline temperature for helium are much higher compared to nitrogen. No significant effects of the variable wall temperature distribution were observed on the total amount of heat transferred between the bulk flow and the channel walls.