Free Molecular Regime Research Articles

Ab initio modelling of transport phenomena in multi-component mixtures of rarefied gases

The direct simulation Monte Carlo method based on ab initio potentials is applied to model transport phenomena in binary, ternary and quaternary mixtures of noble gases. To this end, the Couette and Fourier problems are solved over the whole range of the rarefaction parameter spanning the near free-molecular, transitional, and slip flow regimes. Moreover, analytical solutions are obtained in the limits of the free-molecular and continuous medium regimes of flow. As a result, the shear stress, heat flux through multi-component mixtures, distributions of velocity, temperature, and mole fractions of each species are calculated as a function of the gas rarefaction. An analysis of these data points out the properties which are strongly affected by the chemical composition. It is found that the largest deviations of characteristics of mixtures from those of a single gas are observed for the binary mixture of helium and krypton, while their deviations for the ternary and quaternary mixtures are smaller. A temperature variation in a mixture leads to non-uniform distributions of chemical composition due to the thermal diffusion phenomenon. Such redistributions are analyzed in both problems and their effect on the heat and momentum transfer is evaluated. The reported results can be used to evaluate the main factors determining transport phenomena in practical applications such as vacuum systems, microfluidics, gas separation technology, space research etc.

Investigation of shock/shock interferences on the aerodynamics of a fragment in the wake of debris in a rarefied regime/at high altitude

This work presents an experimental investigation focused on the analysis of aerodynamic properties between two interacting spheres in a supersonic rarefied flow. Atmospheric re-entries of space debris, whether natural or man-made, begin at altitude 120 km, and observations of historical re-entries have shown that fragmentation occurs between 90 and 50 km. The resulting fragments interact with each other, altering their own trajectories while traversing the different flow regimes between the free molecular and continuum regimes. This study focuses on the intermediate slip regime, where viscous effects of varying magnitude can influence the nature of the interactions of the shocks and modify them from the already known behaviour in the continuum regime. Specifically, this study examines how two spheres interact with each other upon re-entry into the atmosphere, focusing particularly on the six types of shock/shock interactions identified by Edney. The experiments were performed in the MARHy wind tunnel, in a steady Mach 4 laminar flow with static pressure 2.67 Pa. To highlight the differences between the six types of interferences, a variety of set-ups and devices were used: flow-field visualization, aerodynamic forces (through two diagnoses, aerodynamic balance and the swinging sphere technique) and wall pressure measurements. Results demonstrate the identification of differences according to the type of interference observed, showing in particular the viscous effect of rarefied flows by making a comparison with the continuum regime.

Laboratory measurement of volatile ice vapor pressures with a quartz crystal microbalance

Nitrogen, carbon monoxide, and methane are key materials in the far outer Solar System where their high volatility enables them to sublimate, potentially driving activity at very low temperatures. Knowledge of their vapor pressures and latent heats of sublimation at relevant temperatures is needed to model the processes involved. We describe a method for using a quartz crystal microbalance to measure the sublimation flux of these volatile ices in the free molecular flow regime, accounting for the simultaneous sublimation from and condensation onto the quartz crystal to derive vapor pressures and latent heats of sublimation. We find vapor pressures to be somewhat lower than previous estimates in literature, with carbon monoxide being the most discrepant of the three species, almost an order of magnitude lower than had been thought. These results have important implications across a variety of astrophysical and planetary environments.

Drag on a sphere in a slow flow of a binary mixture of rarefied gases

The drag force on a sphere in an isothermal binary mixture of rarefied gases is calculated on the basis of the McCormack model for the linearized Boltzmann equation. The diffuse scattering of the gas–surface interaction law is assumed as the boundary condition. The rigid-spheres model of interatomic interaction potential is used as well as ab initio potentials for helium, argon, and krypton gases. The force is obtained in a wide range of the gas rarefaction, which covers the free molecular, transition, and slip regimes of the gas flow. In the free molecular and slip flow regimes, the problem is solved analytically, while in the transition regime, the system of kinetic equations is solved numerically via the discrete velocity method optimized to overcome the problem of discontinuity of the distribution function of molecular velocities on the convex surface. The calculations are carried out for the mixtures helium–argon and helium–krypton at 300 K. In the slip flow regime, the data available in the literature for the viscous slip coefficient of the helium–argon mixture are used, while for the mixture helium–krypton, it is calculated. The influence of the interatomic interaction potential, molar fraction, and ratio of atomic mass of species in the mixture on the drag force is analyzed.

Numerical study of microdevice with surface acoustic waves for separation of gas mixtures

Recently, it was shown that traveling surface acoustic waves (SAWs) can affect gas flow in microchannels. The effect of SAWs was studied in free-molecular flow regime, and it was shown that SAWs can induce separation of gas mixtures. In the present work, this effect is studied for denser flow regimes, which are more interesting from a practical point of view. The problem is studied numerically using own modification of the direct simulation Monte Carlo method on the example of a neon–argon mixture. The main finding is that SAWs still enhance separation of gas mixtures outflowing into vacuum through a microchannel under all studied rates of gas rarefaction up to Kn≈0.1. Another important practical result is that effect is present for wave speeds typical for existing SAWs (≈1000 m/s) and in a wide range of SAW amplitude to channel height ratios. Influence of other practical aspects, such as channel length, masses of species, and available magnitudes of material surface speed, are also briefly discussed.

Numerical investigation of transmission probability characteristics in the first low-density region of a laser wakefield accelerator

Transmission probability is an important parameter in vacuum science and technology that needs to be accurately characterised for system design. Typically, this is computed using the test particle Monte Carlo method. However, this approach is valid only in the free molecular regime. In this work, we propose a methodology to compute transmission probability using the direct simulation Monte Carlo method, which makes possible the characterisation of vacuum devices for a wider range of operating pressure conditions from viscous to molecular flow regimes. This is applied to study the gas expansion characteristics in the first low-density drift region of a laser wakefield accelerator. Validations are first made by comparing the results against the test particle Monte Carlo method in the free molecular regime. The transmission probability is then characterised for a wide range of operating conditions, revealing interesting insights helpful towards not only a fundamental understanding but also in making design considerations.

Modeling of Brownian Force on a Spherical Particle Trajectory in Free Molecular Gas Flow Regime

Modeling of Brownian Force on a Spherical Particle Trajectory in Free Molecular Gas Flow Regime

Inversion in binary gas mixtures in rarefied flow conditions: Direct simulation Monte Carlo solution and comparison with the analytical solutions at free molecular regime

This paper analyzes the mixing of gases in a plane channel at rarefied conditions. The direct simulation Monte Carlo method is employed to simulate gas mixing in parallel mixers working at different Knudsen numbers and having different values of wall accommodation coefficient. Results show that the normal-to-wall component of the mole fraction gradient may have the same sign as the corresponding component of the diffusive mass flux vector near the diffuse solid walls in contrast to the predictions of Fick's law for continuum conditions. This non-continuum behavior, which is called “inversion” in the present study, will become more pronounced at higher Knudsen numbers, whereas it will become less evident for smaller wall accommodation coefficients. To confirm that the observed phenomenon is consistent with the basic physical laws governing the rarefied gas dynamics and it is not an artifact of the numerical method, a new analytical model based on the kinetic theory of gases is developed for the parallel mixers that have diffuse walls and are working in the free-molecular regime. Excellent agreement is observed between the analytical and direct simulation Monte Carlo results in the free molecular flow regime. Both methods predict the occurrence of inversion near the diffuse walls at highly rarefied flow conditions.

Comparison of different Gaussian quadrature rules for lattice Boltzmann simulations of noncontinuum Couette flows: From the slip to free molecular flow regimes

The lattice Boltzmann (LB) method can be formulated directly from the Boltzmann equation with the Bhatnagar–Gross–Krook assumption. This kinetic origin stimulates wide interest in applying it to simulate flow problems beyond the continuum limit. In this article, such a thought is examined by simulating Couette flows from the slip to free molecular flow regimes using the LB models equipped with different discrete velocity spaces, derived from the half-range Gauss Hermite (HGH), Gauss Legendre (GL), Gauss Kronrod (GK), and Gauss Chebyshev first and second quadrature rules. It is found that the conventional HGH-based LB models well describe noncontinuum Couette flows in the slip and weak transition flow regimes. Nonetheless, they suffer from significant errors with the further increasing Knudsen number, even if a large number of discrete velocities have been employed. Their results contrast with those by the LB models derived from the other Gaussian quadrature rules, which have far better accuracy at large Knudsen numbers. In particular, the GL- and GK-based LB models well capture the velocity fields of Couette flows in the strong transition and free molecular flow regimes. These numerical simulations in this article highlight the importance of velocity discretization for the LB simulations at different Knudsen numbers. They reveal that the LB models based on the Gauss Hermite (GH) quadrature rule are not always the best choice for simulating low-speed bounded flows at moderate and large Knudsen numbers; under strong noncontinuum conditions, those non-GH-based LB models proposed in this article have yielded more accurate results.

Further cautionary tales on thermostatting in molecular dynamics: Energy equipartitioning and non-equilibrium processes in gas-phase simulations.

Molecular dynamics (MD) simulations of gas-phase chemical reactions are typically carried out on a small number of molecules near thermal equilibrium by means of various thermostatting algorithms. Correct equipartitioning of kinetic energy among translations, rotations, and vibrations of the simulated reactants is critical for many processes occurring in the gas phase. As thermalizing collisions are infrequent in gas-phase simulations, the thermostat has to efficiently reach equipartitioning in the system during equilibration and maintain it throughout the actual simulation. Furthermore, in non-equilibrium simulations where heat is released locally, the action of the thermostat should not lead to unphysical changes in the overall dynamics of the system. Here, we explore issues related to both obtaining and maintaining thermal equilibrium in MD simulations of an exemplary ion-molecule dimerization reaction. We first compare the efficiency of global (Nosé-Hoover and Canonical Sampling through Velocity Rescaling) and local (Langevin) thermostats for equilibrating a system of flexible compounds and find that of these three only the Langevin thermostat achieves equipartition in a reasonable simulation time. We then study the effect of the unphysical removal of latent heat released during simulations involving multiple dimerization events. As the Langevin thermostat does not produce the correct dynamics in the free molecular regime, we only consider the commonly used Nosé-Hoover thermostat, which is shown to effectively cool down the reactants, leading to an overestimation of the dimerization rate. Our findings underscore the importance of thermostatting for the proper thermal initialization of gas-phase systems and the consequences of global thermostatting in non-equilibrium simulations.

Conductive heat transfer through nanoconfined argon gas: From continuum to free-molecular regime

Heat transfer across nanogaps between solid surfaces has received an increased attention driven by both recent experimental investigations and technological applications as well. Most of the studies concerned, however, vacuum conditions and relatively few attention has been paid to the situation where a gas occupied the gap. This latter situation is difficult to handle theoretically because a continuum description of heat transfer across a nanoconfined gas medium completely breaks down. This breakdown maybe related not only to the partially ballistic nature of heat transfer between the surfaces but also to the existence of an adsorbed gas layer which affects energy transfer across the gap. In this work, to better understand the interplay of these effects, we present a series of molecular dynamics simulations of argon gas confined between either metallic or silicon walls held at different temperatures. The gas density is tuned so that the gas experiences a wide range of Knudsen numbers ranging from the continuum to the free molecular regime. It is shown that the Debye temperature of the solid controls the amount of adsorbed gas close to the walls, which in turn controls heat conduction in the middle of the channel. While the nanochannel walls significantly impact the density and temperature distributions of the rarefied gas, the pressure and the heat flux across the gas domain converge toward a plateau as the gas becomes denser. Finally, we also derive new analytical formulas to calculate the gas pressure, induced heat flux, and effective thermal conductivity of the nanoconfined gas. These formulae, which take into account the effects of an adsorbed gas layer, are shown to hold over a broad range of Knudsen numbers in the range 0.05–20. This framework allows one to predict nanoscale heat transfer across a gas confined by a variety of solid walls.

Computational study of rarefied gas flow and heat transfer in lid-driven cylindrical cavities

The gas flow characteristics in lid-driven cavities are influenced by several factors, such as the cavity geometry, gas properties, and boundary conditions. In this study, the physics of heat and gas flow in cylindrical lid-driven cavities with various cross sections, including fully or partially rounded edges, is investigated through numerical simulations using the direct simulation Monte Carlo (DSMC) and the discrete unified gas kinetic scheme (DUGKS) methods. The thermal and fluid flow fields are systematically studied for both constant and oscillatory lid velocities, for various degrees of gas rarefaction ranging from the slip to the free-molecular regimes. The impact of expansion cooling and viscous dissipation on the thermal and flow fields, as well as the occurrence of counter-gradient heat transfer (also known as anti-Fourier heat transfer) under non-equilibrium conditions, is explained based on the results obtained from numerical simulations. Furthermore, the influence of the incomplete tangential accommodation coefficient on the thermal and fluid flow fields is discussed. A comparison is made between the thermal and fluid flow fields predicted in cylindrical cavities and those in square-shaped cavities. The present work contributes to the advancement of micro-/nano-electromechanical systems by providing valuable insight into rarefied gas flow and heat transfer in lid-driven cavities.

Soot Formation in Methane Pyrolysis Reactor: Modeling Soot Growth and Particle Characterization.

Methane pyrolysis is a very attractive and climate-friendly process for hydrogen production and the sequestration of carbon as solid material. The formation of soot particles in methane pyrolysis reactors needs to be understood for technology scale-up calling for appropriate soot growth models. A monodisperse model is coupled with a plug flow reactor model and elementary-step reaction mechanisms to numerically simulate processes in methane pyrolysis reactors, namely, the chemical conversion of methane to hydrogen, formation of C-C coupling products and polycyclic aromatic hydrocarbons, and growth of soot particles. The soot growth model accounts for the effective structure of the aggregates by calculating the coagulation frequency from the free-molecular regime to the continuum regime. It predicts the soot mass, particle number, area, and volume concentration, along with the particle size distribution. For comparison, experiments on methane pyrolysis are carried out at different temperatures and collected soot samples are characterized using Raman spectroscopy, transmission electron microscopy (TEM), and dynamic light scattering (DLS).

Reynolds Analogy Coefficient in the Longitudinal Cylindrical Couette Problem: from the Continuous Medium to Free Molecular Flow

The gas Couette flow for a cylindrical geometry of bounding surfaces that move in the longitudinal direction relative to their symmetry axis is considered. For a monoatomic gas, the relation between the shear stress and the energy flux transferred to the longitudinal-flow surface (Reynolds analogy) is studied. In the case of continuous medium and free molecular flow regimes, simple explicit analytical expressions for the Reynolds analogy coefficient are obtained, which depend only on the Eckert number and are independent of the ratio of the cylinder radii. The transitional regime for various values of the Knudsen number is studied using the direct simulation Monte Carlo (DSMC) method. It is shown that, in this case, the Reynolds analogy coefficient at a fixed ratio of the radii and the Knudsen number depends on the relative velocity and temperatures of the surfaces mainly through the Eckert number. A relationship between the linear energy fluxes transferred to the cylindrical surfaces is found.

The ocean fine spray

An important fraction of the atmospheric aerosols comes from the ocean spray originated by the bursting of surface bubbles. A theoretical framework that incorporates the latest knowledge on film and jet droplets from bubble bursting is here proposed, suggesting that the ejected droplet size in the fine and ultrafine (nanometric) spectrum constitutes the ultimate origin of primary and secondary sea aerosols through a diversity of physicochemical routes. In contrast to the latest proposals on the mechanistic origin of that droplet size range, when bubbles of about 10–100 μm burst, they produce an extreme energy focusing and the ejection of a fast liquid spout whose size reaches the free molecular regime of the air. Simulations show that this spout yields a jet of sub-micrometer and nanometric scale droplets whose number and speed can be far beyond any previous estimation, overcoming by orders of magnitude other mechanisms recently proposed. The model proposed can be ultimately reduced to a single controlling parameter to predict the global probability density function of the ocean spray. The model fits remarkably well most published experimental measurements along five orders of magnitude of spray size, from about 5 nm to about 0.5 mm. According to this proposal, the majority of ocean aerosols would have their extremely elusive birth in the collapsing uterus-like shape of small bursting bubbles on the ocean surface.

Clustering and collision of Brownian particles in homogeneous and isotropic turbulence

The collision rate between primary nanoparticles in the turbulent flow is of significance for accurately predicting the growth rate of agglomerates during the flame synthesis process. In this work, the clustering and the collision of Brownian particles in the free-molecular regime in homogeneous isotropic turbulence are investigated using the direct numerical simulation and the Langevin dynamics. It is found that the Brownian motion can distribute the particles more uniformly, leading to the formation of a plateau on the curve of the radial distribution function (RDF). This anti-clustering effect is significant only at a small separation distance of particle pairs and cannot be observed when the separation is larger than a critical value. The radial relative velocity (RRV) is significantly enhanced, especially at the small separation distance, when the Brownian motion is taken into account. A velocity superposition analysis shows that the statistics of RRV of Brownian particles can be obtained by adding a random variable onto the RRV of non-Brownian particles with the same Stokes number. The increased radial relative velocity counteracts the anti-clustering effect of the Brownian motion and leads to the increase of the geometric collision kernel. Finally, the collision kernel of Brownian particles is formulated as a function of asymptotic values of RDF and RRV at a vanishing separation. The proposed collision kernel enables us to estimate the collision rate between nanoparticles that are much smaller than the Kolmogorov length scale.

A numerical tool to calculate ion mobility at arbitrary fields from all-atom models

In an effort to calculate electrical mobilities in the free molecular regime using electrical fields that are non-negligible, an algorithm based on the calculation of collision integrals and the two-temperature theory has been created and tested. The algorithm calculates the mobility based on the effective temperature of the ion using a 4-6-12 potential interaction with or without an ion-quadrupole potential (for the case of nitrogen gas). The ion's energy is also approximated so that a relation between the effective temperature of the ion and the field over concentration may be given. The algorithm is parallelized and tested against experimental results for small ions in light gases successfully. The algorithm has also been tested in nitrogen for mid-sized analytes at room temperature and at 100K despite the fact that the inelasticity of collisions has not yet been considered. At 100K, the reduction of the capture radius with the increase of the electric field is apparent for many of the analytes producing “humps” in the reduced mobility vs. the field over concentration curve. For the smallest ions, two humps are observed. One pertaining to the dispersion forces and a second one due to the ion-induced dipole interaction.

Squeeze-Film Damping of Microbeam and Microplate Resonators in the Free Molecular Regime

Background: Correct modeling of squeeze-film damping (SFD) is an important consideration in the design of high-Q microresonators. In 2002, using the molecular dynamics (MD) approach, Bao et al. developed an analytical MD model for the evaluation of the SFD of a parallel-plate device in the free molecular regime. Their model was based on the energy exchange between the oscillating plate and gas molecules. Bao’s model is now widely used in microsystem design. However, Bao’s model cannot reduce to the air damping model in free space and is unsuitable for flexible microbeams. Objective: This paper first presents a more accurate analytical model for the evaluation of the SFD of the parallel plate. Then the present analytical model is extended to model the SFD of flexible microbeams with deformed shapes. Methods: This paper is based on the momentum transfer between the vibration plate and gas molecules. Results: The analytical results of the present model have shown a good agreement with the available experimental results. Conclusion: The limitations in the previous model are overcome.

A Review of Research on the Vacuum Plume

Chemical and electrical thrusters are generally utilized to control the attitude and orbit of spacecraft in aerospace. When they are firing, the exhaust expands into the vacuum environment, known as the vacuum plume. The plume flow can collide with spacecraft surfaces due to sufficient expansion, exerting adverse effects on the spacecraft, such as heating load, force/torque, contamination, and sputtering. Therefore, it is vital to investigate the vacuum plume to ensure the function and safety of the spacecraft. This review introduces the ground test and numerical simulation methods of the vacuum plume for chemical and electrical thrusters. The vacuum environment, invasive, and non-invasive (optical) measurements of the ground test are concluded. Numerical simulation of plume flow and its effects is exampled. The hybrid CFD-DSMC (computational fluid dynamics and direct simulation Monte Carlo) algorithm is employed to simulate the gas plume flow spanning continuum and transitional and free molecular flow regimes for chemical thrusters. By contrast, the PIC-DSMC (particle-in-cell plus direct simulation Monte Carlo) algorithm is used for the plasma plume flow containing charged particles exhausted by electrical thrusters. Moreover, the topics of fast prediction of the vacuum plume, plume–surface interaction, and plume–Lunar/Mars regolith interaction are proposed for future research.

Differences in water and vapor transport through angstrom-scale pores in atomically thin membranes

The transport of water through nanoscale capillaries/pores plays a prominent role in biology, ionic/molecular separations, water treatment and protective applications. However, the mechanisms of water and vapor transport through nanoscale confinements remain to be fully understood. Angstrom-scale pores (~2.8–6.6 Å) introduced into the atomically thin graphene lattice represent ideal model systems to probe water transport at the molecular-length scale with short pores (aspect ratio ~1–1.9) i.e., pore diameters approach the pore length (~3.4 Å) at the theoretical limit of material thickness. Here, we report on orders of magnitude differences (~80×) between transport of water vapor (~44.2–52.4 g m−2 day−1 Pa−1) and liquid water (0.6–2 g m−2 day−1 Pa−1) through nanopores (~2.8–6.6 Å in diameter) in monolayer graphene and rationalize this difference via a flow resistance model in which liquid water permeation occurs near the continuum regime whereas water vapor transport occurs in the free molecular flow regime. We demonstrate centimeter-scale atomically thin graphene membranes with up to an order of magnitude higher water vapor transport rate (~5.4–6.1 × 104 g m−2 day−1) than most commercially available ultra-breathable protective materials while effectively blocking even sub-nanometer (>0.66 nm) model ions/molecules.