Collision Integral Research Articles

Определение значений переносных характеристик в многокомпонентных средах на основе анализа микропроцессов

This paper proposes a two-level approach for mathematical modeling of processes in multicomponent gas mixtures. It involves a combination of the description of conservation laws based on macro-equations and the determination of the corresponding transfer coefficients using the analysis of micro-processes. The main purpose of this work is to develop an algorithm for determining transport characteristics in multicomponent media based on the analysis of inner micro-processes. Hence, the kinetic molecular theory of gases and liquids is used, namely, the collision part of the Boltzmann equation. It includes multiple improper integrals taking into account the effect of micro-processes in the medium on the values of transport coefficients. When integrating the Boltzmann equation by the Chapman-Enskog method, the transport coefficients are expressed through integral brackets of the Sonin polynomials, which represent the linear combinations of molecular collision integrals. Their calculations require the use of efficient mathematically sound ways including the elements of parallel computing.

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.

A weighted particle scheme for Enskog-Vlasov equation to simulate spherical nano-droplets/bubbles

The Enskog-Vlasov equation has been proven successful in capturing the complex behaviour of fluids undergoing a phase change. However, its numerical solution is computationally demanding, and this has restricted the studies to one- and two-dimensional planar flows. In this work, a weighted particle scheme is developed for the numerical solution of the Enskog–Vlasov equation in spherically symmetric geometry. It is shown how weighted schemes designed for the Boltzmann equation can be extended to cope with the non-local structure of the Enskog collision integral, and a compact expression of the mean force field is determined using the shell theorem. As an application, the growth rates of nano-droplets/bubbles in the bulk of a homogeneous metastable vapour/liquid are evaluated in a wide range of supersaturation ratios that was not possible until now due to the high computational cost required by alternate approaches. The proposed scheme significantly broadens the range of problems that can be investigated via the Enskog-Vlasov equation, and it is a stepping stone towards the simulation of general three-dimensional liquid-vapour flows.

Collision Integrals for Nitrogen and Hydrogen Ionized Gas: The Exact Values and Assessment of Approximations

Collision integrals are calculated for the H–N^{+}, N–H^{+}, N^{2+}–H, and N^{+}–H^{+} interactions in the temperature range of 1000–60,000 K. The resulting collision integrals were fitted to simple functional forms. The applicability of commonly used approximations for calculation of the collision integrals is verified. Those approximations are simplification of exact potentials (especially the repulsive ones), use of polarizability potential for atom–ion interaction, and use of Coulomb screened potential for ion–ion interaction. The reported collision integrals are based on the ab initio calculated points of the potential energy curves, so that the error of fitting the points with analytical potential energy function is not present.

Collision integrals and viscosity coefficients of argon–carbon thermal plasmas: Comparison using different interaction potentials

The calculated values of collision integrals of the majority of binary interactions that can be involved in argon/carbon (Ar/C) plasmas are presented in this work. The studied plasmas are considered to be in local thermodynamic equilibrium, and calculations are performed from 1000 to 30 000 K. Computations have been carried out first for standard potentials and second for the Improved Lennard-Jones (ILJ) potential. In this paper, a clear and well detailed report is given for each methodology adopted to calculate the omega integrals, together with the main data needed to perform these computations. The obtained values of collision integrals have been compared with some data reported in the literature, and then, they are used to estimate the viscosity of two plasma systems, namely, pure argon and a mixture of argon and carbon (Ar/C). The calculation of viscosity coefficients is made on the basis of the Chapman–Enskog method and developed to the first approximation. The important contribution of the charge exchange process and its influence on the accuracy of the diffusion-type collision integrals of neutral–parent ion systems are emphasized. Although some discrepancies are observed, comparisons of our results with those of previously published studies show an overall satisfactory agreement in most of the cases. Our investigation of the data uncertainty further confirms the suggestion that the ILJ approach is an excellent candidate to provide collision integrals with acceptable accuracy when reliable experimental data or accurate theoretical calculations are unavailable. For that, all the necessary collision integrals needed to calculate reliable transport properties of the Ar/C plasma mixtures are reported in this work.

Finite difference method for numerical solution of the initial and boundary value problem for boltzmann’s sixmoment system of equations

Boltzmann’s one-dimensional non-linear non-stationary moment system of equations in the third approximation is presented, in which the first, third and fourth equations corresponds to the laws of conservation of mass, momentum and energy, respectively. This system contains six equations and represents a nonlinear system of hyperbolic type equations. For the Boltzmann’s six- moment system of equations an initial and boundary value problem is formulated. The macroscopic boundary condition contains the moments of the incident particles distribution function on the boundary and moments of the reflected particles distribution function from the boundary. The boundary condition depends on the temperature of the wall (boundary). In this work, using the finite-difference method, an approximate solution of the mixed problem for the Boltzmann system of moment equations is constructed in the third approximation under the boundary conditions obtained by approximating the Maxwell boundary condition. For given values of the coefficients included in the moments of the nonlinear collision integral and the parameter depending on the wall temperature, as well as for fixed values of the initial conditions, a numerical experiment was carried out. As a result, the approximate values of the particle distribution function incident on the boundary and reflected from the boundary, as well as the density, temperature and average velocity of gas particles, as moments of the particle distribution function, are obtained.

Численный анализ динамики молекулярного газа при импульсной лазерной абляции

The evaporation of a molecular gas with rotational degrees of freedom caused by the action of a pulsed nanosecond laser (pulsed laser ablation) of moderate intensity is considered. Model kinetic equations that account for the exchange of translational and rotational energies using a two-temperature model are used to account for the influence of internal energy on the vapor-gas cloud dynamics. Rykov equations (R-model) are integrated when modeling the flow of a diatomic gas, and a generalization of the Bhatnagar-Gross-Krook (BGK) equation for a molecular gas is used to calculate the flow of nonlinear molecules with a number of rotational degrees of freedom equal to three. Taking into account the influence of internal energy on the gas flow is realized by relaxation terms approximating the collision integral as a sum of elastic and inelastic collisions. Since in the model kinetic equations the collision frequency does not depend on the velocities, and the influence of internal energy is taken into account only due to temperature changes, it is necessary to compare with more realistic approaches, which include, for example, the method of direct simulation Monte Carlo (DSMC). The comparison shows that the change in the average temperature over time and the vapor-gas cloud parameters (density and temperature), obtained by the solution of the kinetic equations and the DSMC method, are close. Calculations of the problem of pulsed laser ablation by direct integration of kinetic equations are carried out by the method of discrete ordinates and are a difficult computational problem, which is associated with discontinuous boundary conditions, the presence of both continuum regions and free molecular flow regions in the solution, as well as the need to calculate the vapor-gas cloud dynamics for a large time interval. To reduce computational costs, grid adaptation in physical and velocity spaces is used.

Transport coefficients in thermal QCD: A probe to the collision integral

We have studied the transport coefficients as a tool to probe the collision integral appearing in the Boltzmann transport equation. For this purpose, we have estimated the transport coefficients (momentum: ${\ensuremath{\eta},\ensuremath{\zeta}}$, heat: ${\ensuremath{\kappa}}$, and charge: ${{\ensuremath{\sigma}}_{\mathrm{el}}}$) in the kinetic theory approach with the collision integrals in Bhatnagar-Gross-Krook (BGK) and relaxation-time (RT) approximations and ask whether we can distinguish between the two collision integrals. For example, $\ensuremath{\eta}$ gets enhanced while $\ensuremath{\zeta}$ gets reduced with respect to RT. As a corollary, we then investigate the interplay among the aforesaid transport coefficients, viz. fluidity and transition point of QCD medium by evaluating the ratios, $\ensuremath{\eta}/s$ and $\ensuremath{\zeta}/s$, respectively, nature of flow (Reynolds number, RI), sound attenuation (Prandtl number, Pr), and competition between the momentum and charge diffusion ($\ensuremath{\gamma}$), etc. as further plausible tools to decipher the same. With BGK collision integral, the ratios $\ensuremath{\eta}/s$ (increase) and $\ensuremath{\zeta}/s$ (decrease) show opposite behavior, whereas Pr, RI, $\ensuremath{\gamma}$, and the ratio $\ensuremath{\zeta}/\ensuremath{\eta}$ get reduced with respect to RT. We then examine how a strong magnetic field modulates the impact of the collision integral, which, in a way, explores the dimensionality dependence of the transport phenomena, especially momentum transport because the quark dynamics is effectively restricted to 1D only and only the lowest Landau levels are populated. As a result, $\ensuremath{\eta}$ ($\ensuremath{\zeta}$) gets reduced (amplified), which will have ramifications on the ratios, viz. $\ensuremath{\eta}/s$ ($\ensuremath{\zeta}/s$) becomes smaller (larger), enhancement of Pr, $\ensuremath{\gamma}$, and $\ensuremath{\zeta}/\ensuremath{\eta}$, etc. In this study, the thermomagnetic medium effects have been incorporated by adopting a thermodynamically consistent quasiparticle model, where the medium-generated masses of the partons have been obtained from the pole of their resummed propagators calculated using perturbative thermal QCD in strong $B$.

An optimized collision-averaged variable soft sphere parameter set for air, carbon, and corresponding ionized species

A collision-averaged parameter set for air, carbon, and the corresponding ionized species for the variable soft sphere collision model is suggested which is suitable for the earth's atmosphere or mars atmosphere, for example. The parameter set is generated through collision integral fits and a number of optimization steps so that individual sub-sets can also be used for, e.g., air or without ionized species. In addition, the parameter set can be extended by further species without having to carry out the complete optimization again, which is shown in the example of argon. The limitations of the collision-average model are discussed and in which cases the collision-specific model or other models should be used. The model is compared with collision integrals from various publications.

Transport coefficients of second-order relativistic fluid dynamics in the relaxation-time approximation

We derive the transport coefficients of second-order fluid dynamics with $14$ dynamical moments using the method of moments and the Chapman-Enskog method in the relaxation-time approximation for the collision integral of the relativistic Boltzmann equation. Contrary to results previously reported in the literature, we find that the second-order transport coefficients derived using the two methods are in perfect agreement. Furthermore, we show that, unlike in the case of binary hard-sphere interactions, the diffusion-shear coupling coefficients $\ell_{V\pi}$, $\lambda_{V\pi}$, and $\tau_{V\pi}$ actually diverge in some approximations when the expansion order $N_\ell \rightarrow \infty$. Here we show how to circumvent such a problem in multiple ways, recovering the correct transport coefficients of second-order fluid dynamics with $14$ dynamical moments. We also validate our results for the diffusion-shear coupling by comparison to a numerical solution of the Boltzmann equation for the propagation of sound waves in an ultrarelativistic ideal gas.

Collisionless relaxation of a Lynden-Bell plasma

Plasmas whose Coulomb-collision rates are very small may relax on shorter timescales to non-Maxwellian quasi-equilibria, which, nevertheless, have a universal form, with dependence on initial conditions retained only via an infinite set of Casimir invariants enforcing phase-volume conservation. These are distributions derived by Lynden-Bell (Mon. Not. R. Astron. Soc., vol. 136, 1967, p. 101) via a statistical-mechanical entropy-maximisation procedure, assuming perfect mixing of phase-space elements. To show that these equilibria are reached dynamically, one must derive an effective ‘collisionless collision integral’ for which they are fixed points – unique and inevitable provided the integral has an appropriate H-theorem. We describe how such collision integrals are derived and what assumptions are required for them to have a closed form, how to prove the H-theorems for them, and why, for a system carrying sufficiently large electric-fluctuation energy, collisionless relaxation should be fast. It is suggested that collisionless dynamics may favour maximising entropy locally in phase space before converging to global maximum-entropy states. Relaxation due to interspecies interaction is examined, leading, inter alia, to spontaneous transient generation of electron currents. The formalism also allows efficient recovery of ‘true’ collision integrals for both classical and quantum plasmas.

Some Consequences of Mathematical Inaccuracies in Mechanics

The article is devoted to the study of some violations of the known laws of mathematics and classical mechanics in continuum mechanics and kinetics. The most common is open non-stationary systems. From the equations formulated earlier and some experiments, a connection traced between the gradients of physical quantities and the angular momentum (force). The use of Hamilton's formalism and the dependence of force only on the distance between particles limits the study. In the collision integral, for example, for a rarefied gas, the Lennard-Jones potential, which not related to the type under consideration, is often used. Hamilton's formalism traces the behavior of closed systems. The general form of boundary conditions and forces changes the theory proposed in the works of N.N. Bogolyubov. The results of the reformulation discussed. Even in the classical theory, after taking into account the moments, we come to the absence of symmetric stress tensor in Boltzmann theory. The symmetric tensor obtains after assumption of small influence from absence of symmetry at the condition of the forces balance. No symmetric tensor leads to the existence of two solutions. New examples of solving problems on hydromechanics, elasticity theory and kinetic theory are given. A correspondence between the terms of the Liouville equation with more general and traditional forces established for continues mechanics. Previously considered boundary layer problems, jet problems and the simplest problems of elasticity theory. The paper proposes a method for finding the second solution for no symmetric problems, if the solution of the symmetric problem we know. The mathematical inaccuracies of the theory of continuum mechanics and kinetics discussed.

Mutual Intersection Points of Reduced Collision Integrals for Lennard–Jones (n-m), Hulburt–Hirschfelder, and Tang–Toennies Potential Energy Functions

We investigate the reduced collision integrals Omega ^{(ell ,s)*}(T^*) for 1 le ell le 4, ell le s le 7 and ell + s le 8 for several isotropic potential energy functions: the Lennard–Jones (n-m), the Hulburt–Hirschfelder, and Tang–Toennies potential. It is observed that for a given ell and s, Omega ^{(ell ,s)*}(T^*) shows a mutual intersection region at a reduced temperature 0.39< T^*=T^*_{ell s} < 2.22 which is nearly independent of the potential energy function used.

The effect of chirp parameter on laser propagation in collisional quantum plasma

This manuscript investigates the effect of the chirp factor on the relativistic self-focusing of a Gaussian chirped pulse laser in collisional quantum plasmas. Nonlinear equations are derived to describe the propagation of a linear chirped laser frequency; afterward, by considering dielectric function obtained by quantum kinetic equation and BGK collision integral in coordinate space, evolution differential equations of laser beam width are found; and numerically solved by the fourth-order Runge-Kutta method. It studies oscillations of the beam width along the propagation direction for different values of collision frequency, chirp parameter, plasma density, and laser intensity. Numerical analyses show that a laser beam is defocused after focusing on a definite value; nonetheless, the positive chirp parameter could reduce the defocusing and strengthen the self-focusing in collisional quantum plasma. Also, the higher chirp value leads to a smaller oscillation amplitude and a greater oscillation rate. Furthermore, the self-focusing effect becomes stronger by increasing plasma density and laser intensity. These consequences could be beneficial in the improvement of fast ignition experiments.

Transport properties for neutral C, H, N, O, and Si-containing species and mixtures from the Gordon and McBride thermodynamic database

Accurate transport properties of non-ionized gas mixtures of C, H, O, N, and Si-containing species at temperatures up to 4000 K are essential in many scientific fields. Mixture transport properties are computed through the solution of linear transport systems, requiring collision integrals as functions of temperature for each binary collision pair in the mixture. Due to the dimensionality of the problem, no such database exists for all the 180 hydrocarbons and silicon species detailed in the nine-coefficient polynomial thermodynamic database of Gordon and McBride, widely used in many applications. This constraint was overcome by using a phenomenological inter-molecular potential energy surface suitable for transport properties, which describes the pair interaction approximated with two fundamental species physical properties, namely the dipole electric polarizability and the number of effective electrons participating in the interaction. These two parameters were calculated with ab initio quantum chemistry calculations, since they were not always available in literature. The studied methodology was verified and validated against other approaches at a species and collision integral level. Transport properties for a variety of equilibrium mixtures, including planetary atmospheres and chemical compositions of thermal protection materials relevant to aerospace applications, were calculated, assessing the predictive capabilities of this new database.

Quasilinear theory for inhomogeneous plasma

This paper presents quasilinear theory (QLT) for a classical plasma interacting with inhomogeneous turbulence. The particle Hamiltonian is kept general; for example, relativistic, electromagnetic and gravitational effects are subsumed. A Fokker–Planck equation for the dressed ‘oscillation-centre’ distribution is derived from the Klimontovich equation and captures quasilinear diffusion, interaction with the background fields and ponderomotive effects simultaneously. The local diffusion coefficient is manifestly positive-semidefinite. Waves are allowed to be off-shell (i.e. not constrained by a dispersion relation), and a collision integral of the Balescu–Lenard type emerges in a form that is not restricted to any particular Hamiltonian. This operator conserves particles, momentum and energy, and it also satisfies the$\smash {H}$-theorem, as usual. As a spin-off, a general expression for the spectrum of microscopic fluctuations is derived. For on-shell waves, which satisfy a quasilinear wave-kinetic equation, the theory conserves the momentum and energy of the wave–plasma system. The action of non-resonant waves is also conserved, unlike in the standard version of QLT. Dewar's oscillation-centre QLT of electrostatic turbulence (Phys. Fluids, vol. 16, 1973, p. 1102) is proven formally as a particular case and given a concise formulation. Also discussed as examples are relativistic electromagnetic and gravitational interactions, and QLT for gravitational waves is proposed.

High-order polynomial approximations for solving non-inertial particle size density in flames

A novel numerical framework is discussed to simulate the time evolution of non-inertial particle size distributions (or number density functions) in flames. The generic form of the population balance equation is considered featuring nucleation, surface growth/loss and agglomeration/coagulation. This balance equation is first recast in a form that is prone to minimize spurious numerical errors in the simulation of surface growth/loss and collision integrals. Formally, this is achieved classifying the terms of the equation into: (i) Lagrangian transport in size-space (surface growth/loss), (ii) relaxation rates of the particle density at a given size (non-uniform growth/loss and negative contribution of collision integrals) and (iii) sources (nucleation and positive contribution of collision integrals). To secure accuracy, a high-order modal decomposition of the particle size distribution is introduced within every section of size considered. A Legendre polynomials basis is used with Gauss-Lobatto quadrature points. By construction, the method performs very well for dealing with particle surface growth/loss and it is also highly accurate for the estimation of the collision integrals thanks to the high-order quadrature. This is confirmed simulating canonical test cases of the literature to compare the numerical results against exact and analytical solutions. With a discretisation based on about 40 sections of size and with Legendre interpolation at the 5th-order, very good accuracy is obtained up to the third moment of the distributions for particle size ranging over up to 8 orders of magnitude. The method is cast to minimize computing cost. Strategies to couple this novel numerical framework with the simulation of carbon particles dynamics in flames are discussed.

Singular Behavior of the Macroscopic Quantity Near the Boundary for a Lorentz-Gas Model with the Infinite-Range Potential

Possibility of the diverging gradient of the macroscopic quantity near the boundary is investigated by a mono-speed Lorentz-gas model, with a special attention to the regularizing effect of the grazing collision for the infinite-range potential on the velocity distribution function (VDF) and its influence on the macroscopic quantity. By careful numerical analyses of the steady one-dimensional boundary-value problem, it is confirmed that the grazing collision suppresses the occurrence of a jump discontinuity of the VDF on the boundary. However, as the price for that regularization, the collision integral becomes no longer finite in the direction of the molecular velocity parallel to the boundary. Consequently, the gradient of the macroscopic quantity diverges, even stronger than the case of the finite-range potential. A conjecture about the diverging rate in approaching the boundary is made as well for a wide range of the infinite-range potentials, accompanied by the numerical evidence.

Detailed Description of the Collision Frequency in the Solar Atmosphere

This work aims to provide an accurate description and calculations of collision frequencies in conditions relevant to the solar atmosphere. To do so, we focus on the detailed description of the collision frequency in the solar atmosphere based on a classical formalism with Chapman–Cowling collision integrals, as described by Zhdanov. These collision integrals allow linking the macroscopic transport fluxes of multifluid models to the kinetic scales involved in the Boltzmann equations. In this context, the collision frequencies are computed accurately while being consistent at the kinetic level. We calculate the collision frequencies based on this formalism and compare them with approaches commonly used in the literature for conditions typical of the solar atmosphere. To calculate the collision frequencies, we focus on the collision integral data provided by Bruno et al., which is based on a multicomponent hydrogen–helium mixture used for conditions typical for the atmosphere of Jupiter. We perform a comparison with the classical formalism of Vranjes & Krstic and Leake & Linton. We highlight the differences obtained in the distribution of the cross sections as functions of the temperature. Then, we quantify the disparities obtained in numerical simulations of a 2.5D solar atmosphere by calculating collision frequencies and ambipolar diffusion. This strategy allows us to validate and assess the accuracy of these collision frequencies for conditions typical of the solar atmosphere.

The impact of ion mobility coefficients on plasma discharge characteristics

In this paper, the high-accuracy ion mobility coefficient based on the Chapman–Enskog approximation to the solution of the Boltzmann equation for low pressure radio frequency plasma discharges is presented. We employ two-dimensional fluid simulations of the argon filled axisymmetric reactor, where the effect of new ion-kinetics-based fluid closure is compared to theoretical expressions and experimental data. The spatial profiles of plasma composition in the low pressure radio frequency capacitively coupled plasma are presented, which includes the metastable reactions in the simulation. Moreover, inelastic collision integrals terms, due to charge exchange inelastic collisions between ions and neutral species, have been also considered. A Monte Carlo simulation of kinetic ion energy distribution of impinging on the radio frequency powered electrode provides a measure of accuracy of the new transport model. From our simulation, the results that mirror the influence of ion mobility coefficient obtained by the Chapman–Enskog method on plasma physical quantities under different pressures, frequencies, and electrode gaps is in good agreement with experimental measurement results and theoretical expressions.