Hard-sphere Collision Research Articles

Simulation of a coarse solid sphere settling in a riser with transverse vibrations

During the transportation of the deep-sea ores in lifting risers, the vibration of the lifting risers due to oceanic currents and waves could exert an effect on the movement of coarse solid ores inside, such as transversal drag, collisions between the ore particle and inner wall of the riser. This study focuses on the settling behavior of a coarse sphere with a large Reynolds number in a transversal vibration riser so as to explore the effect of riser vibration on ore movement. In the present investigation, the moving computational domain, overset mesh, and hard-sphere collision model are applied. Then, the settling characteristics of the solid sphere with different riser vibration frequencies and amplitudes are mainly analyzed and discussed. The results demonstrate that with the increase in the vibrational frequency and amplitude of the riser, the relative velocity between the single solid sphere and riser in the vibration direction and the fluctuation of the sphere's settling velocity show an increasing trend, while the average settling velocity of the sphere decreases. The lateral velocity of the sphere, which is perpendicular to the vibration direction of the riser, is influenced slightly by the vibration frequency when the riser vibrates with a low frequency. In addition, the sphere trajectory is more chaotic at low vibrational frequencies compared to lower amplitudes and higher vibrational frequencies of the riser. The lateral displacement of the sphere is found to be suppressed when the vibrational frequency or amplitude of the riser is high enough.

Dynamics of collisions and uptake of alcohol molecules with hydrated nitric acid clusters.

We investigate the collisions of different alcohol molecules with hydrated nitric acid clusters using a molecular beam experiment and molecular dynamics simulations. The uptake cross sections σp for the molecules evaluated from the experiment are in excellent agreement with the simulations. This suggests that (i) the nontrivial assumptions implemented in the evaluation procedure of the experimental data are valid, and (ii) the simulations describe correctly the major processes in the molecule-cluster collisions. We observe that σp decreases with the increasing alkyl chain length of the alcohol, and also with the branching of the molecules that have the same mass but different structures. These systematic trends can be rationalized based on the accessibility of the hydrophilic OH group, which decreases with the increasing chain length and steric hindrance. The observed trends and their interpretation differ significantly from the simple model of hard-sphere collisions. The obtained data shall be beneficial not only for the fundamental understanding of the molecule-cluster collisions, but also in the modelling of atmospheric new-particle formation and aerosol growth.

Molecular Dynamics-Based Modeling of Ion-Neutral Collisions in an Open Ion Trajectory Simulation Framework.

Ion mobility spectrometry (IMS) and ion mobility mass spectrometry (IMS-MS) methods have become increasingly popular and are important analytical techniques to determine information about the structural parameters of gas-phase analytes. The accurate description of the interaction between molecular ions and neutral background gas particles is an essential part of high-quality simulations of such modern mass- and ion-mobility-spectrometric systems. Established ion-neutral collision models (Hard-sphere collision modeling and statistical diffusion simulations) in common ion-trajectory simulation systems like SIMION use strongly simplified assumptions and are thus limited in their predictive ability. In contrast, collision cross-section (CCS) modeling programs (e.g., MOBCAL, IMoS, and Colloidoscope) allow high-quality ion mobility predictions for low-field equilibrium conditions using explicit scattering processes with a molecular dynamics-based trajectory method but cannot be used for nonequilibrium collision modeling in an ion trajectory simulation. This work presents an extension to the open-source Ion Dynamics Simulation Framework (IDSimF), which allows the simulation of ion dynamics under arbitrary and even nonequilibrium conditions. It was extended by an advanced collision model employing the molecular dynamics trajectory method for a detailed microscopic description of ion-neutral collisions within ion-trajectory simulations. We used drift tube ion mobility spectrometry (DT-IMS) to validate the predictive abilities of the model and to estimate the runtime requirements for productive simulations. Simulated high-field ion mobilities for small ion systems in a drift tube IMS are compared to experimental values from the literature and an implementation of a hard-sphere model in IDSimF for helium and argon as background gas particles. Significant improvements in ion mobility predictions using the molecular dynamics trajectory approach are obtained with deviations of only a few percent from experimental values. Therefore, the established and publicly available MD collision model will serve as foundation for nonequilibrium ion dynamics simulations and the development of improved ion dynamics modeling methods.

A particle resolved simulation approach for studying shock interactions with moving, colliding solid particles

This work applies a new combination of techniques for the fully resolved simulation of compressible, gas–particle multiphase flows. The adaptive wavelet collocation method is used to dynamically, and efficiently, adapt the computational grid to localized flow features and the particles. A characteristic-based volume penalization method that imposes arbitrary Dirichlet, Neumann, or Robin-type immersed boundary conditions, is used to enforce the no-slip condition at particle surfaces. A hard-sphere collision model is applied to capture the particle–particle collisions. Proof of concept test cases are presented, showcasing the dynamic grid adaptation and fully resolved two-way coupling between the phases that is possible with this approach. Results for a shock-driven single cylinder under viscous and inviscid conditions are presented along with a demonstration of a shock interacting with a cloud of randomly distributed cylinders and spheres.

Numerical Modeling of Spectral Hardening at a Finite-width Shock

Spectral hardening has been identified in solar flare hard X-ray observations for several decades and remains a puzzle. We examine spectral hardening under the diffusive shock acceleration mechanism using numerical simulations. The hardening is related to the finite width of the shock and is controlled by the shock Péclet number. We implement two different types of Monte Carlo simulations. The first is based on the backward stochastic differential equation method, where the Parker transport equation is solved by casting it to a set of stochastic different equations, and by following the trajectories of individual quasiparticles. In the second approach, we follow real particles and particles are assumed to move freely between scatterings from magnetic turbulence in the plasma. The scattering is modeled as either large-angle hard-sphere elastic collision, or small-angle pitch-angle scattering. We show that the results from these two approaches agree well with each other and agree with analytical results. We also use a Pan-spectrum form to fit the resulting spectra.

Covariant formulation of relativistic mechanics

Accretion disks surrounding compact objects, and other environmental factors, deviate satellites from geodetic motion. Unfortunately, setting up the equations of motion for such relativistic trajectories is not as simple as in Newtonian mechanics. The principle of general (or Lorentz) covariance and the mass-shell constraint make it difficult to parametrize physically adequate 4-forces. Here, we propose a solution to this old problem. We apply our framework to several conservative and dissipative forces. In particular, we propose covariant formulations for Hooke's law and the constant force, and compute the drag due to gravitational and hard-sphere collisions in dust, gas and radiation media. We recover and covariantly extend known forces such as Epstein drag, Chandrasekhar's dynamical friction and Poynting-Robertson drag. Variable-mass effects are also considered, namely Hoyle-Lyttleton accretion and the variable-mass rocket. We conclude with two applications: 1. The free-falling spring. We find that Hooke's law corrects the deviation equation by an effective Anti-de Sitter tidal force; 2. Black hole infall with drag. We numerically compute some trajectories on a Schwarzschild background supporting a dust-like accretion disk.

Twenty-six moment equations for the Enskog–Vlasov equation

The Enskog–Vlasov equation is a phenomenological kinetic equation that extends the Enskog equation for the dense (non-ideal) hard-sphere fluid by adding an attractive soft potential tail to the purely repulsive hard-sphere contribution. Simplifying assumptions about pair correlations lead to a Vlasov-like self-consistent force field that adds to the Enskog non-local hard-sphere collision integral. Within the limitations imposed by the underlying assumptions, the extension gives the Enskog–Vlasov equation the ability to give a unified description of ideal and non-ideal fluid flows as well as of those fluid states in which liquid and vapour regions coexist, being separated by a resolved interface. Furthermore, the Enskog–Vlasov fluid can be arbitrarily far from equilibrium. Thus the Enskog–Vlasov model equation provides an excellent, although approximate, tool for modelling processes with liquid–vapour interfaces and adjacent Knudsen layers, and allows us to look at slip, jump and evaporation coefficients from a different perspective. Here, a set of 26 moment equations is derived from the Enskog–Vlasov equation by means of the Grad moment method. The equations provide a meaningful approximation to the underlying kinetic equation, and include the description of Knudsen layers. This work focuses on the – rather involved – derivation of the moment equations, with only a few applications shown.

Development of a particle collision algorithm for discontinuous Galerkin simulations of compressible multiphase flows

This paper discusses the development of algorithms for simulating compressible, four-way coupled, particle-laden flows with discontinuous Galerkin methods. Specifically, the algorithmic developments focus on the treatment of hard-sphere collisions. First, we aim to reduce the computational effort devoted to inspecting pairs of particles for collisions during the given time step. This is often one of the most expensive stages of four-way coupled particle-laden flow simulations. The proposed algorithm exploits information about relative particle positions provided by the geometric mapping of a physical element to a reference element commonly employed in discontinuous Galerkin and other finite-element-based solvers. Second, we develop an enhanced particle-wall collision methodology that better accounts for finite particle size than previously developed schemes. A particular feature of the particle-particle and particle-wall collision algorithms is compatibility with arbitrary, curved, unstructured elements. Finally, we apply the resulting framework to a number of test cases, including hypersonic dusty flow over an entry capsule and surface erosion by sandblasting.

Reevaluating the Role of Polarizability in Ion Mobility Spectrometry

With the expanding commercial availability of gas-phase separation systems that incorporate gas-phase mobility, there is a concurrent rise in efforts to cast the gas-phase mobility coefficient in terms of an ion-neutral collision cross-section (CCS). The motivating factors for this trend are varied, but many aim to complement experimental results with computationally generated CCS values from in silico structural approximations. Unfortunately, the current paradigm for relating experimental mobility results to computationally derived structures relies upon empirical approaches, including a myriad of variables that do not realistically bound the comparison. In this Critical Insight, we advocate for the development of a self-consistent experimental and computational framework that uses laboratory results to constrain the scope of the modeling effort. This paper aims to prompt discussion, challenge assumptions, and promote the development of more efficient, accurate computational techniques within the gas-phase ion measurement community. Specifically, we postulate whether experimental deviations from Langevin's polarization limit (Kpol) are suitable to estimate the relative contributions of hard-sphere collisions and long-range interactions within CCS values. Not surprisingly, different molecule classes exhibit different trends in the K/Kpol ratio when normalized for reduced mass, and the most common IMS calibrants (e.g., tune mix, polyalanine, tetraalkylammonium salts) follow different polarizability trends than many of the analytes probed in the literature. Succinctly, if gas-phase ion structure is largely invariant based upon the colliding neutral and newly developed experimental efforts can quantitatively capture ion polarizability, then modeling efforts describing a target analyte must be self-consistent as the collision neutral is changed in silico.

Direct numerical simulation of magneto-Archimedes separation of spherical particles

Abstract

An investigation of collisions of liquid coated particles

The presence of even small amounts of liquid between particles dramatically changes the dynamics of collisions. This study considers granular collisions between two particles coated with a thin layer of viscous liquid, such that the capillary number is high and viscous forces dominate. High-speed particle tracking velocimetry was used to experimentally study the collisions of two smooth spheres with and without liquid coatings. We then use these experiments to examine four theoretical models that describe the collisions. A key challenge when modelling viscous forces is that the force which is predicted as particles approach each other scales with the inverse of the distance, i.e. tends to infinity. Most existing models impose a limit to the viscous force. One recent model instead assumes a hard sphere collision. These fundamentally different approaches produce different rebound outcomes. A fair match between experiments and simulations was obtained when using the hard sphere collision model, but only if an empirically-fitted glass transition pressure model was used to describe the minimum approach distance.

Rarefied reactive gas flow in T-shape microreactors

The present study investigates the reactive gas flows in T-shape microreactors by means of a 3D direct simulation Monte-Carlo method (DSMC). The reactive process is modeled by incorporating a single step irreversible exothermic reaction of gas species A and B. The energy released in each reaction is added to the system by increasing the kinetic energy of the product gas specie C. The reactive collisions are modeled by means of a momentum preserving hard sphere collision model in 3D DSMC. We analyzed several process and gas parameters to observe their extent of influence on the characteristics of the reactive flow. The T-microreactor with thermal walls showed much higher mass carrying capacity as compared to the specular walls. The reaction front, which initially for maximum reaction rate was close to the inlet of gas specie of higher mass density, moved towards the center of the inlet part of the T-channel and eventually dissolved with decreasing reaction rates. This transition was smooth in the T-channel with thermal walls, whereas, quite abrupt in case of specular walls. Moreover, in T-microreactors with thermal walls, the amount of A and B converted to C follows an exponential decrease especially in moderate to low reaction rates. Further, with a clear separation between high and low reactive regimes, a piece-wise exponential decay in concentration of C with respect to decreasing reaction rates was also observed in the T-channel with specular walls.

Derivation and validation of a hard-body particle-wall collision model for non-spherical particles of arbitrary shape

In this work an extended three-dimensional model to describe the process of wall collisions of arbitrary shaped non-spherical particles is developed. This model is derived as the generalized solution of the hard-sphere wall collision model described in Crowe et al. (2012) [1]. The validation of the new model is performed versus experimental results of the collision between cylindrical particles and a flat plate ((Sommerfeld et al., 2015 [2]), Sommerfeld and Lain (2018) [3]). The agreement between the measurements of linear normal and tangential velocities and the model predictions was found to be very good while the rotational velocities forecasted by the model showed also a satisfactory agreement with the experiments.

New Aspects of the Airglow Problem and Reactivity of the Dioxygen Quintet O2(5Πg) State in the MLT Region as Predicted by DFT Calculations

Dioxygen in the quintet O2(5Πg) state is a weakly bound species near the entrance of the O(3P) + O(3P) recombination channel. It was predicted by ab initio calculations in 1977 and detected experimentally in 1999. Meantime, the O2(5Πg) species was tentatively assumed as intermediate in transport properties calculations for the rarefied gases of the Earth's upper atmosphere, though its potential energy curve is still debated. Besides six other strongly bound low-lying states of dioxygen, the O2(5Πg) state is an important potential candidate for modeling energy transfer and airglow of the upper atmosphere. A number of photochemical kinetic schemes designed to simulate energy flow upon atomic and molecular oxygen collisions in the rarefied mesosphere take into account a participation of the O2(5Πg) state in energy relaxation processes responsible for terrestrial nightglow. All mechanisms of energy redistribution are based on the hard-sphere collision models. The possibility of chemical interactions between the quintet excited state of dioxygen and other atmospheric components has not been considered so far in photochemistry of the upper atmosphere. In the present paper, the chemical reactivity of the quintet O2(5Πg) species is calculated for the first time in the framework of the density functional theory. Definitely, O2(5Πg) is the most reactive species among all other metastable dioxygen states below 5.1 eV. Quintet products of the O2(5Πg) state association with heavy inert gases, H2O, N2, and CO2 are predicted to be chemically significant, while the complexes with abundant H2 and He species are rather weak and not important even in the mesopause low-temperature region. The complex with N2 molecule is unexpectedly stable with dissociation energy 4 kJ/mol, which can strongly influence the abundant termolecular association O + O + N2 → O2 + N2 process. Reaction with meteoritic ablated Mg atom produces metastable 5A1 excited state of MgO2 being more strongly bound than the ground 3A2 state of magnesium peroxide.

On the meaning of “collision rate constants” for ion-molecule reactions: Association of hydrogen atoms with C6H5+ and small alkyl radicals with C7H7+ ions

Limiting high pressure rate constants for the association of H with C6H5+ and of CH3, C2H5, and n-C3H7 radicals with C7H7+ molecular ions are analyzed in terms of “rigidity factors” arising from the anisotropy of the interaction potentials and “collision rate constants” in the absence of anisotropy (the latter corresponding to phase space theory, PST). Model calculations based on ab initio potential energy surfaces show that the PST rate constants kPST(T) exceed collision rate constants from conventional ion-molecule capture theory (in this case given by Langevin rate constants kL). They can be represented by kPST(T) = kL + kh.sph. where kh.sph. denotes hard-sphere collision numbers with collision radii r0. The r0 derived from the modelled kPST(T) are related to properties of the interaction potentials. Applications to other ion-molecule reactions are proposed.

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.

Dense gas flow simulations in ultra-tight confinement

Modeling dense gas flows inside channels with sections comparable to the diameter of gas molecules is essential in porous medium applications, such as in non-conventional shale reservoir management and nanofluidic separation membranes. In this paper, we perform the first verification study of the Enskog equation by using particle simulation methods based on the same hard-sphere collisions dynamics. Our in-house Event-Driven Molecular Dynamics (EDMD) code and a pseudo-hard-sphere Molecular Dynamics (PHS-MD) solver are used to study force-driven Poiseuille flows in the limit of high gas densities and high confinements. Our results showed (a) very good agreement between EDMD, PHS-MD, and Enskog solutions across density, velocity, and temperature profiles for all the simulation conditions and (b) numerical evidence that deviations exist in the normalized mass flow rate vs Knudsen number curve compared to the standard curve without confinement. While we observe slight deviations in the Enskog density and velocity profiles from the MD when the reduced density is greater than 0.2, this limit is well above practical engineering applications, such as in shale gas. The key advantages of promoting the Enskog equation for upscaling flows in porous media lie in its ability to capture the non-equilibrium physics of tightly confined fluids while being computationally more efficient than fundamental simulation approaches, such as molecular dynamics and derivative solvers.

Sodium ion self-diffusion in molten NaBr probed over different length scales.

The single-particle dynamics of sodium ions in molten sodium bromide has been investigated with quasielastic neutron scattering. A detailed and rather extensive data analysis procedure allowed determination of the pure sodium ion dynamics with increasing wave vector. Two different evaluation procedures agree perfectly on the resulting diffusion coefficient of sodium ions on long distances. A simple kinetic theory based on binary collisions of hard spheres is not able to reproduce the sodium diffusion coefficient. The derived reduced linewidth from modeling with a Lorentzian spectral function decreases with increasing wave vector towards the first structure factor maximum. That deviation from the hydrodynamic behavior signals the hindrance of the microscopic diffusion process due to the so-called cage effect when microscopic length scales are probed in a dense fluid. The observed quadratic wave-number-dependent decrease might be evidence for a coupling to density fluctuations as the source of the changes in the diffusion process. The results indicate that in the molten salt NaBr near the melting point the self-diffusion process might be governed by similar processes as already observed in dense metallic liquids.

Spectral gaps for reversible Markov processes with chaotic invariant measures: The Kac process with hard sphere collisions in three dimensions

We develop a method for producing estimates on the spectral gaps of reversible Markov jump processes with chaotic invariant measures, that is effective in the case of degenerate jump rates, and we apply it to prove the Kac conjecture for hard sphere collision in three dimensions.

Silica nanocluster binding rate coefficients from molecular dynamics trajectory calculations

Oxide nanoparticle growth from vapor phase precursors occurs in high temperature aerosol reactors first via the formation of nanoclusters, which are nanometer-scale condensed-phase species composed of 101-102 atoms. The binding rate for nanoclusters, defined as the rate at which two nanoclusters collide with and stick to one another, is hence of interest in predicting nanoparticle size distribution functions in an aerosol. We have utilized molecular dynamics (MD) trajectory calculations to determine the homogeneous (equal-sized) and heterogeneous (disparate-sized) binding rate coefficients of SiO2 (silica) nanoclusters composed of 18, 144, and 333 atoms. MD trajectory calculations incorporated all-atom models of nanoclusters using a combined Born–Huggins–Mayer-Lennard-Jones potential model, which accounts for both short range interactions and electrostatic interactions. MD calculations were utilized to determine the binding probability as a function of both initial relative velocity and impact parameter; integration of the binding probability across impact parameter and relative velocity space yields the binding rate coefficient. MD trajectory calculations reveal that the most common type of encounters between nanoclusters leading to binding are grazing collisions, i.e. instances where collision would not occur without induced dipole potential influences. The resulting binding rate coefficients are found to be extremely weakly dependent on system temperature, which is in contrast to the use of rate coefficient models which are the product of a hard-sphere collision rate coefficient and a constant enhancement factor (leading to a rate coefficient proportional to the square root of temperature). Enhancement factors defined from MD trajectory calculations fall in the range of 3–9 as system temperature decreases from 1500 K to 300 K. Such large values suggest that potential interactions need to be considered when modeling oxide nanocluster growth in the gas phase.