Calculation Of Collision Integrals Research Articles

Method and computer library for calculation of the Boltzmann collision integrals on discrete momentum lattice.

We present a general and numerically efficient method for calculation of collision integrals for interacting quantum gases on a discrete momentum lattice. Here we employ the original analytical approach based on Fourier transform covering a wide spectrum of solid-state problems with various particle statistics and arbitrary interaction models, including the case of momentum-dependent interaction. The comprehensive set of the transformation principles is given in detail and realized as a computer Fortran 90 library FLBE (Fast Library for Boltzmann Equation).

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.

Kinetic theory of polydisperse gas–solid flow: Navier–Stokes transport coefficients

The particulate phase stress and solid–solid drag force in the multifluid modeling of polydisperse gas–solid flows are usually closed using kinetic theory. This research aims to establish the hydrodynamic equations and constitutive relations of the multifluid model for polydisperse systems via species kinetic theory, in which the non-equipartition of energy and interphase slip velocity between different species are considered. Whereas previous studies have used approximations, such as Taylor series expansions, to simplify the calculation of collision integrals, the present study, for the first time, solves the collision integrals analytically without any approximations to obtain accurate constitutive relations. Explicit expressions for the constitutive laws are obtained, including the particle stress tensor, solid–solid drag force, heat flux, and energy dissipation rate up to the Navier–Stokes order. The present study offers more complete and mathematically rigorous constitutive laws for the multifluid modeling of polydisperse gas–solid flows.

A comparison of interatomic potentials for noble gases

Prediction of transport properties of noble gases requires the calculation of collision integrals, which depend on interatomic potentials as the input. However the accuracy of transport properties depends largely on the accuracy of interaction potentials. So different interatomic potentials of noble gases are compared in order to get the accurate transport properties. The forms and characteristics of Lennard-Jones, exponential repulsive, Hartree-Fock-Dispersion-B (HFD-B), and phenomenological model potentials that are used to describe the atomic interactions between noble gases are analyzed in this paper. Then the calculation method of transport properties is presented. Viscosities and thermal conductivities of noble gases based on these four potentials are obtained using Chapman-Enskog method in the temperature range for computation from 300 to 5000 K. It can be seen from the results that the interaction potentials have a great influence on the calculated results of transport properties. There are great differences between the results obtained using different interaction potentials. These differences of the calculated results can be explained according to the performance of interaction potentials. Results calculated with Lennard-Jones potential are always much lower in the high temperature range due to its overestimated repulsive part, and the exponential repulsive potential gives unreasonable results at low temperatures because there is no attractive well in this potential. Therefore, the accurate interatomic potentials for noble gases can be obtained only by comparing the calculated results with published experimental and theoretical data of other researchers. It can be found that the results obtained by HFD-B potential agree well with previously experimental and theoretical data. So it is apparent that the HFD-B potential in light of Hartree-Fock repulsion and dispersion theory can provide a realistic description of the trends and features of interatomic potentials, allowing accurate theoretical calculations to be made for transport properties of noble gases.

Computing of gas flows in micro- and nanoscale channels on the base of the Boltzmann Kinetic equation

Abstract The paper describes the methodology of computing gas flows in narrow micro- and nanoscale channels on the basis of finite-difference solution of the Boltzmann kinetic equation using the conservative projection method of collision integral calculation. Mathematical framework of the method is considered and the problem solving environment for calculation of the above mentioned flows is described. Examples of the flow calculations in the plane and 3D geometry are given.

General numerical algorithm for classical collision integral calculation

A new automatic algorithm to calculate collision integrals from isotropic interaction potentials has been developed. The proposed method overcomes the limits of existing codes that need a priori definition of potential features. The algorithm efficiency has been tested by comparison with traditional approaches. The agreement achieved in reproducing published data is good. A theoretical analysis of the asymptotic behaviors, related to the accuracy of integral evaluation with respect to interval finiteness, is also reported.

Thermophysical Properties of Nitrogen from a New Potential Energy Surface Using the Mason–Monchick Approximation

Received January 04, 2006 A recent N2–N2 potential has been used to calculate the second virial, viscosity, and diffusion coefficients. Calculations have been done up to the first quantum correction for virial coefficients and the second-order kinetic theory approximation for transport coefficients. The Mason–Monchick approximation (MMA) has been used for the calculation of collision integrals and, via a numerical analysis, a common intersection point has been found for reduced cross sections and collision integrals of different orientations. This regularity has been interpreted with the aim of the orientation dependence of the potential energy and different types of collisions between molecules. The overall agreement of the calculated second virial coefficient with experiment is reasonable but suggests that a slight re-scaling of the potential would be beneficial. In the case of transport properties, calculated and experimental results show an average deviation of about 1.6% and 0.7% for viscosity and relative diffusion coefficients, respectively.

Collision integrals in transport properties calculation of air

Chapman-Enskog method is a base of the transport properties calculation of partially ionized gases. This method represents the transport coefficients as functions of the collision integrals. These collision integrals are required for each pair of species of a partially ionized gas and different method can be used for their calculation. In the first part of the paper the calculation method of the transport properties (electrical conductivity, thermal conductivity, viscosity) is briefly described. The transport coefficients values of air (Ar, N, O, e−) for temperatures 1 000 ≤ T ≤ 20 000 K and the influence of the choice method of the collision integral calculation on results are presented in the second part of the paper.

Influence of collision integrals on the electrical conductivity of a gas system

The calculation method of the electrical conductivity of the partially ionised gases is based on the Chapman-Enskog method of the kinetic theory of a multicomponent gas mixture. The electrical conductivity depends not only on the concentrations of the components but also on the collision integral values. In this paper the electrical conductivity of the gas system of dissociation and ionisation products of SF6 were calculated for some methods of the collision integral calculation that are mentioned in the literature. The hard sphere model, the Lennard-Jones potential, the polarizability method, the Coulomb interaction and the electron-neutral interactions are used.

Accurate Calculation of Collision Integrals Using Adaptive Methods

In this work, we have developed accurate and efficient methods for calculating the scattering angle, time integral and electronic energy loss for binary collisions in matter. In contrast to the conventional integration and the hard sphere approximations, our new time integral is obtained by calculating the difference between the distance to the target atom and the ion path length. To compute the scattering angle and time integral accurately and efficiently, an adaptive Simpson quadrature using a smoother integrand is adopted. The energy loss arising from the local electron-concentration-dependent electronic stopping power is obtained by tracing the ion trajectory. To reduce the computation time, a second-order differential equation for the ion motion using an adaptive Runge-Kutta-Fehlberg algorithm is developed.

Fluid equations for temperature anisotropy based on kinetic drifts

Moments of the Boltzmann equations describing transport of plasma and anisotropic forms of energy are derived for ions and/or electrons. Formal closure of the moment equations is obtained by a standard reduction technique applied to the fourth moment. It is shown how the heat-flow tensor can be decomposed into four separate tensors describing the flows of parallel and perpendicular energy in the directions parallel and perpendicular to the magnetic field; the solutions for these tensors are obtained from ten coupled equations. The transport of plasma density n and perpendicular and parallel temperatures Tperpendicular to and T/sub /// can be described by three strongly coupled nonlinear differential equations characterized by an advective velocity which is the fluid analogue of the guiding centre drift velocity. The transport equations show how specific kinetic drift mechanisms contribute to transport driven by gradients in n, Tperpendicular to and T/sub /// and by gradients in the electric and magnetic fields; collisional effects are included and involve separate calculations of collision integrals. Finite-Larmor-radius, "gyro-fluid" or micro-turbulence effects cannot be described by the equations derived.

Accurate collision integrals for the attractive static screened Coulomb potential with application to electrical conductivity

The results of accurate calculations of collision integrals for the attractive static screened Coulomb potential are presented. To obtain high accuracy with minimal computational cost, the integrals are evaluated by a quadrature method based on the Whittaker cardinal function. The collision integrals for the attractive potential are needed for calculation of the electrical conductivity of a dense fully or partially ionized plasma, and the results presented here are appropriate for the conditions in the nondegenerate envelopes of white dwarf stars. 25 refs.

Transport properties of open-shell systems: Fine structure effects on collision integrals for oxygen and fluorine atoms with rare gases

Decoupling schemes for the efficient calculation of collision integrals for open-shell atoms when fine structure effects cannot be neglected are developed. Test computations of collision integrals, diffusion coefficients and interaction viscosities as a function of temperature are presented for the representative cases of ground-state oxygen and fluorine atoms interacting with rare gases, exploiting recent accurate information on these interactions.

Analytic parametrization of transport coefficients of Lennard–Jones (n,6) fluids

Simple, easy to use analytic formulas for the calculation of neutral-gas transport properties such as viscosity, thermal conductivity, and diffusion coefficients are presented. The formulas are based on least-squares parametrizations of accurate numerical calculations of collision integrals for Lennard–Jones (n,6) fluids with n=9, 12, and 18. The accuracy of these formulas is shown to be of the order of 1%–2% for a wide range of fluid conditions. Analytic formulas for reduced transport coefficients are also proposed. The simplicity and accuracy of these results allow the inclusion of reliable neutral-gas transport properties in the computer modeling of high-pressure discharges, e.g., arc lamps, without the penalty of large increases in execution times.

An essentially exact evaluation of transport cross-sections for a model of the helium-nitrogen interaction

Essentially exact calculations of the transport collision integrals for a realistic model of the intermolecular pair potential of helium and nitrogen are reported. The collision dynamics for the interaction of the atom and molecule have been treated within the framework of the Arthurs and Dalgarno closecoupled formalism and the calculations are free from all approximation except inevitable numerical round-off. The direct calculation of collision integrals covers the temperature range 70 K to 300 K. The range has been extended upwards to 500 K with the aid of complementary classical calculations without loss of accuracy. The results serve as the first set of ‘benchmark’ calculations of transport collision integrals for a heavy molecular system with a realistic anisotropic pair potential. They are used to assess the accuracy of various approximate schemes for the evaluation of the same quantities. It is found that classical trajectory methods are in good agreement with the close-coupled calculations above 2...

Fick's First Law Correction by an exact solution of the boltzmann equation

AbstractIt is shown that there exists a dominant solution of the Boltzmann equation for an artificial model gas in terms of a relation for the diffusion flux differing from the regular formulation of Fick's first law. Its mathematical structure corresponds to the Cattaneo‐formulation based on heuristics of the Fourier heat flux vector. This dominant solution allows a realistic estimation of a diffusion relaxation time and serves as the basis for valuable information concerning two actual problems of hypersonic gas dynamics: first, a useful relation for the calculation of collision integrals for diffusion coefficients, especially at high temperatures; second, a new type of a chemical reaction equation, which is particularly relevant in very fast relaxation processes encountered in the chemistry of fuels.

Numerical methods for the direct solution of the kinetic Boltzmann equation

Methods of solving the classical non-linear Boltzmann equation, based on the discrete approximation of the distribution function in a phase space, the finite-difference approximation of a transfer operator and the calculation of collision integrals using random quadrature formulas, are considered. The solution of the set of finite-difference equations is obtained by one of the iteration methods.