Modeling of transport phenomena in gases based on quantum scattering

doi:10.1016/j.physa.2018.05.129

Physica A: Statistical Mechanics and its Applications

Volume 508, 15 October 2018, Pages 797-805

https://doi.org/10.1016/j.physa.2018.05.129 Get rights and content

Highlights

•
A quantum interaction is implemented into direct simulation Monte Carlo method.
•
Ab initio potential of interatomic interaction is applied.
•
Two canonical problems of fluid mechanics are solved by the proposed method.
•
The quantum scattering influence is significant in a wide range of the temperature.

Abstract

A quantum interatomic scattering is implemented in the direct simulation Monte Carlo (DSMC) method applied to transport phenomena in rarefied gases. In contrast to the traditional DSMC method based on the classical scattering, the proposed implementation allows us to model flows of gases over the whole temperature range beginning from 1 K up any high temperature when no ionization happens. To illustrate the new numerical approach, two helium isotopes $^{3}$ He and $^{4}$ He were considered in two canonical problems, namely, heat transfer between two planar surfaces and planar Couette flow. To solve these problems, the ab initio potential for helium is used, but the proposed technique can be used with any intermolecular potential. The problems were solved over the temperature range from 1 K to 3000 K and for two values of the rarefaction parameter $δ = 1$ and 10. The former corresponds to the transitional regime and the latter describes the temperature jump and velocity slip regime. No influence of the quantum effects was detected within the numerical error of 0.1% for the temperature 300 K and higher. However, the quantum approach requires less computational effort than the classical one in this temperature range. For temperatures lower than 300 K, the influence of the quantum effects exceed the numerical error and reaches 67% at the temperature of 1 K. Numerical data provided in Supplemental Material can be used to model any kind of helium flow at any temperature.

Introduction

The direct simulation Monte Carlo (DSMC) method [1] used to calculate rarefied gas flows consists of decoupling of the free motion of gaseous molecules from collisions between them. The second stage requires a physical intermolecular potential in order to obtain reliable results. Recently, a procedure to implement any potential into the DSMC method was proposed in our previous paper [2] using the phenomenological Lennard-Jones potential as an example. In contrast to phenomenological models, ab initio (AI) potentials are free from any adjustable parameter usually extracted from experimental data. Nowadays, such potentials practically for all noble gases and their mixtures are available in the open literature, see e.g. [[3], [4], [5], [6], [7], [8], [9], [10]]. Thus, the DSMC method based on AI potential [11] also becomes free from such adjustable parameters. The idea of the procedure to implement any potential into the DSMC is to generate look up tables of the deflection angle depending on the relative velocity of interacting particles and their impact parameter. The method was used to study the influence of the interatomic potential on various phenomena in rarefied gases [[12], [13], [14], [15], [16]] considering the intermolecular interaction based on the classical mechanics, that is justified at high temperatures for heavy gases. However, the quantum effects in intermolecular interactions is not negligible for light gases, e.g. helium, hydrogen, tritium, especially at moderately low temperatures [[17], [18], [19], [20], [21]]. It can be important to model helium, hydrogen and tritium flows in many technological fields such as cryogenic pumps [[22], [23]], cryogenic systems used in the huge fusion reactor ITER [[24], [25]], monochromatic beams of helium [[26], [27]], helium microscope [[28], [29]], acoustic thermometry at a low temperature [[30], [31]], experimental set-up to measure the neutrino mass [[32], [33]], etc. In spite of the high practical interest to model gases at low temperatures, the quantum scattering has not been implemented yet in the DSMC method.

The aim of the present paper is to propose a new technique to implement the quantum scattering into the DSMC method using any potential and to show the influence of quantum effects on transport phenomena in rarefied gases. For this purpose, a procedure of generation of deflection angle matrix based on quantum scattering is elaborated and a couple of classical problems of fluid mechanics is solved to evaluate the influence of quantum effects. A temperature range where the classical approach fails and the quantum theory becomes an unique alternative to simulate the transport phenomena in rarefied gases will be pointed out. It will be also shown that even at a high temperature when the classical approach works, the quantum approach reduces computational effort that makes it preferable for the whole range of the temperature. It should be emphasized that we are interested in quantum effects only in interatomic iterations. Other effects, like high densities at low temperatures when the interatomic distance is comparable to the de Broglie wavelength, are not considered here.

Section snippets

Numerical method

The DSMC method consists of a decoupling the free-motion of molecules from intermolecular collisions during each time steps $Δ t$ . Here, the free-motion of particles is considered to be classical that is valid under the condition [[34], [35]] $\frac{n h^{3}}{{(2 π m k_{B} T)}^{3 ∕ 2}} ≪ 1,$ where $n$ is the gas number density, $h$ is the Planck constant, $m$ is the atomic mass of the gas, $k_{B}$ is the Boltzmann constant and $T$ is the gas temperature. This condition is well satisfied at the atmospheric pressure and at any temperature above

Differential cross section

According to the quantum theory of scattering [[17], [18], [19], [20], [21]], the DCS of undistinguishable particles with a spin $s$ consists of two terms and reads $σ^{(B)} (g, χ) = \frac{s}{2 s + 1} σ^{'} (g, χ) + \frac{s + 1}{2 s + 1} σ^{''} (g, χ),$ $σ^{(F)} (g, χ) = \frac{s + 1}{2 s + 1} σ^{'} (g, χ) + \frac{s}{2 s + 1} σ^{''} (g, χ),$ for boson and fermions, respectively. Both $σ^{'}$ and $σ^{''}$ are expressed via the speed $g$ and deflection angle $χ$ as $σ^{'} (g, χ) = \frac{2}{k^{2}} {| \sum_{l = 1, 3, 5, \dots}^{\infty} f_{l} (g, χ) |}^{2},$ $σ^{''} (g, χ) = \frac{2}{k^{2}} {| \sum_{l = 0, 2, 4, \dots}^{\infty} f_{l} (g, χ) |}^{2},$ $f_{l} (g, χ) = (2 l + 1) exp (i δ_{l}) sin δ_{l} P_{l} (cos χ),$

Matrix of deflection angle

The vector $σ_{T j}$ and the matrix $ξ_{i j}$ were calculated for helium-3 and helium-4 having the atomic masses [39] 3.01605 u and 4.00260 u, respectively. The AI potential for these two species is the same and can be found in Refs. [[3], [4], [6]]. For our purpose, the potential proposed in the work [3] was chosen as the most complete and exact at the moment. The authors of the paper [40] used this potential to calculated the viscosity and thermal conductivity of both helium-3 and helium-4. The

Examples

In order to illustrate the technique and to estimate the influence of the quantum effects, two classical problems of fluid mechanics related to transport phenomena through helium were solved.

The first problem is a heat transfer between two parallel plates fixed at $x = \pm H ∕ 2$ . The plate at $x = - H ∕ 2$ is kept at a temperature $T_{0} + Δ T ∕ 2$ , while the other plate has a lower temperature $T_{0} - Δ T ∕ 2$ . We are interested in the heat flux $q_{x}$ as a function of the gas rarefaction $δ$ and of the equilibrium temperature $T_{0}$ .

Conclusions

An interatomic interaction based on quantum scattering was implemented into the direct simulation Monte Carlo method applied to transport phenomena through rarefied gases. Such an implementation allows us to model flows of light gases like helium over the whole temperature range beginning from 1 K up any temperature when no ionization happens. As an example, two helium isotopes $^{3}$ He and $^{4}$ He have been considered in two classical problems of fluid mechanics, namely, heat transfer between two

Acknowledgment

The author acknowledges the Brazilian Agency CNPq, Brazil for the support of his research, grant 303697/2014-8.

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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.
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A technique for the separation of rarefied gas mixtures based on microelectromechanical systems (MEMS) has recently attracted the attention of researchers. The mechanism of gas separation is the thermally induced flow caused by inhomogeneous temperature gradients within the microchannel, which distinguishes the velocity of different gas species. In this paper, the effects of Knudsen number $K n$ and equilibrium temperature $T_{0}$ on the flow properties of each gas species within the gas mixture, such as velocity, molar fraction, streamlines, temperature, and pressure, are investigated by the Direct simulation Monte Carlo (DSMC) method. The flow of the gas mixture is numerically simulated using a quantum scattering-based ab initio (AI) potential and compared with the results of numerical simulations based on the variable soft sphere (VSS) model. The results show that the gas velocity accelerates with increasing temperature, but decreases with increasing Knudsen number. The streamlines are very sensitive to the Knudsen number, yet they are only affected by the temperature in the transitional flow. When the conditions are the same, He flows faster than Ne and the streamlines are smoother. The gas separation factor increases with equilibrium temperature at $T_{0} < 100 K$ and decreases slightly with equilibrium temperature at $T_{0} \geq 100 K$ . The gas separation efficiency is best in the transitional flow. The temperature discrepancy for the different gas species is less affected by the equilibrium temperature and Knudsen number, with the maximum discrepancy rate being only 2.727% for He and Ne. The pressure discrepancy is more affected by the equilibrium temperature and Knudsen number, with the maximum rate of discrepancy reaching 39.228% for He and Ne. At $T_{0} = 10 K$ and $K n = 1$ , the discrepancy rates of the gas separation factor, temperature, and pressure obtained based on the AI potential and the VSS model reach maximum values of 44.802%, 4.675%, and 52.716%, respectively. Compared to the VSS model, the AI potential based on quantum scattering is applicable to a wider range of temperatures and is more accurate for both low and high temperatures. Furthermore, the AI potential is more accurate than the VSS model in the transitional flow. The results obtained from the AI potential based on quantum scattering will be more desirable when performing gas separation studies based on thermally induced flows in rarefied gases.
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Citation Excerpt :
Zhu et al. [11] and Sharipov [12] studied transport phenomena in gases at low temperatures based on quantum scattering, including the problem of heat transfer between two parallel plates. No influence of the quantum effects was detected for the temperature 300K and higher, but the behaviors of helium isotopes 3He and 4He are qualitatively different at a temperature below 20K [12]. If quantum effects are not taken into account, the error in the calculated heat flux can be about 20% for the helium-neon mixture at temperatures of the order of 10K [11].
The heat transfer in a rarefied gas between two solid surfaces moving relative to each other was studied. The problem was solved for nitrogen using the event-driven molecular dynamics (EDMD) method under the following conditions: (1) а wide range of flow regimes ( $K n = 0.05 - 20$ ), (2) high relative velocity of surfaces, comparable to the thermal velocity of gas molecules, (3) significant difference in surface temperatures, (4) rotational degrees of freedom in gas molecules and (5) complex geometry of solid surfaces. Special attention was paid to the heat removal from the hot surface, namely the quantitative assessment of the reduction of heat removal due to viscous friction and effect of the transverse profiling of surfaces on the heat transfer between them.
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View full text

Modeling of transport phenomena in gases based on quantum scattering

Highlights

Abstract

Introduction

Section snippets

Numerical method

Differential cross section

Matrix of deflection angle

Examples

Conclusions

Acknowledgment

Int. J. Heat Mass Transfer

Comput. & Fluids

Int. J. Heat Mass Transfer

Vacuum

Vacuum

Vacuum

Vacuum

Cryogenics

Vacuum

Int. J. Heat Mass Transfer

The DSMC Method

Direct simulation Monte Carlo method for an arbitrary intermolecular potential

Phys. Fluids

Relativistic and quantum electrodynamics effects in the helium pair potential

Phys. Rev. Lett.

Phys. Rev. Lett.

Ground state potential energy curves for He2, Ne2, Ar2, He-Ne, He-Ar, and Ne-Ar: A coupled-cluster study

J. Chem. Phys.

New basis sets for the evaluation of interaction energies: an ab initio study of the He-He, Ne-Ne, Ar-Ar, He-Ne, He-Ar and Ne-Ar van der Waals complex internuclear potentials and ro-vibrational spectra

Phys. Chem. Chem. Phys.

Ab initio potential energy curve for the helium atom pair and thermophysical properties of dilute helium gas. I. Helium-helium interatomic potential

Mol. Phys.

Ab initio potential energy curve for the neon atom pair and thermophysical properties of the dilute neon gas. I. Neon-neon interatomic potential and rovibrational spectra

Mol. Phys.

Ab initio pair potential energy curve for the argon atom pair and thermophysical properties of the dilute argon gas. I. Argon-argon interatomic potential and rovibrational spectra

Mol. Phys.

State-of-the-art ab initio potential energy curve for the krypton atom pair and thermophysical properties of dilute krypton gas

J. Chem. Phys.

Coupled cluster calculations of the ground state potential and interaction induced electric properties of the mixed dimers of helium, neon and argon

Mol. Phys.

Ab initio simulation of transport phenomena in rarefied gases

Phys. Rev. E

Benchmark problems for mixtures of rarefied gases. I. Couette flow

Phys. Fluids

Quantum Collision Theory

Ground state potential energy curves for He $_{2}$ , Ne $_{2}$ , Ar $_{2}$ , He-Ne, He-Ar, and Ne-Ar: A coupled-cluster study