Kinetic Theory of Transport Processes in Partially Ionized Gases, Summaries of Chemistry

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INIS-mf—

FJ.F. van Odenhoven

KINETIC THEORY

OF TRANSPORT PROCESSES

IN PARTIALLY IONIZED GASES

KINETIC THEORY OF TRANSPORT PROCESSES

IN PARTIALLY IONIZED GASES

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE

TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE

HOGESCHOOL EINDHOVEN, OP GEZAG VAN DE RECTOR

MAGNIFICUS, PROF.DR. S.T.M. ACKERMANS, VOOR

EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN

DECANEN IN HET OPENBAAR TE VERDEDIGEN OP

VRIJDAG 18 FEBRUARI 1983 TE 16.00 UUR

DOOR

FERDINAND JOAN FRANCISCUS VAN ODENHOVEN

GEBOREN TE EINDHOVEN

Wees niet bang voor het langzaam voorwaarts gaan, wees slechts bevreesd voor het blijven staan

(Chinees gezegde)

Contents

page

 - References I Introduction I - II Basic equations - References - III Very weakly ionized gases - III-l The electron distribution function - III-2 The electron macroscopic equations - III-3 Form relaxation of the electron distribution - III-4 The inclusion of Coulomb collisions - References - IV Weakly ionized gases - IV-1 Heavy particle results - distribution function IV-2 Perturbation solution of the electron - IV-3 The macroscopic electron equations - IV-4 The first order isotropic correction - IV-5 Electron transport coefficients - IV-6 Modifications for a seeded plasma - References 

• V Strongly ionized gases - V-l Heavy particle results - V-2 The electron kinetic equation - V-3 The electron macroscopic equations - electron distribution V-4 The nonisotropic part of the - References
• VI Numerical results - VI-1 The isotropic correction - VI-2 Electron transport coefficients

I INTRODUCTION

One can state that the modern kinetic theory of non-equili- brium processes in dilute gases came to maturity with the works of Chapman and Enskog^The book by Chapman and Cowling^2 has never ceased to be an Indispensable textbook on this matter. Since then there have been written many new textbooks , and much has been added to the theory, especially to the kinetic theory of plasmas. More complete historical summaries can be found in the references^. The method of multiple scales is one of the important tools used in this thesis. First introduced by Sandri e.a.^4 it has developed into a valuable mathematical device^5. It has also proved to be very succesful in deriving kinetic equations from the BBGKY-hierarchy^6. The purpose of the present work is the description of transport processes and the calculation of transport coefficients of partially ionized gases. The calculations are restricted to elastic collision processes. This is certainly justified if the kinetic energy of the electrons is much smaller than the excitation energy of the first atomic energy level. There are of course, always inelastic collisions involving high energy electrons, but their influence on the values of the transport coefficients is small, because these result from integrations over the entire velocity space. In chapter II the basic equations and the multiple time scale formalism are expounded. The electrons are of special interest, since they contribute significantly to all transport processes. Because of their small mass the electrons often have a tempera- ture different than the one of the heavy particles. If there are only very few electrons the isotropic part of the electron distribution function can deviate significantly from an equili- brium Maxwelllan as a consequence of fields, gradients and temperature differences which may be present. There are two limiting cases in which the situation is relatively simple.

In the fully ionized or Spltzer limit the isotropic part of the electron distribution function is a Maxwellian and the non- isotropic part has been computed numerically by Spitzer and HSrm^. Within the framework of the Landau kinetic equation this solution is exact. In the Lorentz limit (very small degree of ionization but finite electron-atom mass ratio), on the other hand, the isotropic part is found to be a so-called Davydov distribution function^8. If the neutrals are sufficiently cold, the Druyvesteyn distribution is a special case of this distribution for the hard spheres interaction model. One can distinguish four domains for the electron density with different orderings in terms of the small parameter e which is the square root of the electron-atom mass ratio:

G = (m /m )^ h

6 Si^ (i-D

Two of these domains contain the already mentioned cases of very low respectively high degree of ionization. The definition of the different regions in terms of the ratio of the electron-electron to electron-atom collision frequency, which is proportional to the electron-atom density ratio, is now as follows: Very Weakly Ionized Gas Vee „ 2 << e V ea

Nonlinear Region

" 6 e^ = # ( e 2 ) ea

Weakly Ionized Gas

ea

Strongly Ionized Gas V ea

Adjacent to the region of the very weakly ionized gases lies a region where the equation for the electron distribution function in zeroth order of e is non-linear and the form of the distribution function varies with the electron density between a Davydov and a Maxwell distribution. In chapter III the first two regions are considered. An order- ing different from the work of van de Water^10 is assumed. Some results additional to his are obtained.

conversion by means of an MHD-generator'^1. Therefore some attention is also paid in this thesis to new transport processes and higher order corrections to transport coeffi- cients in alkali seeded noble gas plasmas. This attention is rewarding, because for these plasmas a better comparison with experiments appears to be possible. All results of the calculations and the comparisons with experiments are collected in chapter VI. The method used in this thesis consists of an expansion of the unknown quantities into an asymptotic series in the small parameter e supplemented by the method of multiple time scales. The general form of the solution f of the relevant kinetic equation in each order is found in terms of an expansion into harmonic tensors (see appendix C ) : f(c) = f( 0 )^ (c) +

+e^2 (f( 2 )^ (c) + f(2)(c).c + f(2)(c):) +

• (1-2) where c is the peculiar velocity, is the harmonic tensor of second rank and f denotes an isotropic correction of order n. Nonisotropic parts give rise to expressions for the transport coefficients, isotropic parts appear in the contribu- tions of the nonisotropic part.s in higher order. The expansion generally used in the litterature is a two-term expansion of the form: f(c) = f( 0 )^ (c) + f(1)(c).c , (1-3)

which is sufficient for the calculation of transport coeffi- cients in lowest order. The method applied in this thesis gives results up to second order in e and describes both fast and slow transport phenomena by means of the multiple time scales formalism.

References

1. S .Chapman, Phil.Trans.R.Soc,216(1916)279,217 (1917) 118, Proc.Roy.Soc.,A98(1916)1. Q.Enskog,Inaugural dissertation,Uppsala 1917.
2. S.Chapman and T.G.Cowling:"The mathematical theory of non- uniform gases",Camgridge University Press 1970.
3. J.O.Hirschfelder,C.F.Curtiss and R.B.Bird:"Molecular theory of gases and liquids",.J.Wiley 1954. L.Waldmann:"Transporterscheinungen in Casen von mittleren Druck",in:iïandbuch der Physik, Springer 1958. CCercignani:"Mathematical methods in kinetic theory", Plenum press 1969. J.H.Ferziger and H.G.Kaper:"Mathematical theory of trans- port processes in gases", North Holland Publ. Comp. 1972.
4. G.Sandri,Ann.Physics,24(1963)332,380. E.A.Frieman.J.Math.Phys. ,4^1963)410. J.E.McCune,T.F.Morse and G.Sandri:Rar.Gas Dynam.J_(1963)115.
5. A.H.Nayfeh:"Perturbation methods", J.Wiley 1973.
6. P.P.J.M.Schram,:"Kinetic equations for plasmas", Ph.D.thesis Utrecht 1964.
7. L.Spitzer and R.Harm, Phys.Rev. ,89_( 1953)977.
8. B.Davydov,Phys.Zeits.der Sowjetunion,8Q935).
9. M.J.Druyvesteyn,Physica,1£( 1930)61, j_( 1934)1003.
10. W.van de Water,Physica,85C(1977)377.
11. M.Mitchner and C.H.Kruger:"Partially ionized gases",.J.Wiley

f n k T ( r , t ) = Am c^f ( r , v , t ) d J^ v , (2-2) i S S ™ S S S " ~ where the peculiar velocity c = v - w. —s — —s If equation (2-1) is multiplied by appropriate functions of velocity and integrations over the entire velocity space are performed one obtains so-called moment equations. Choosing these functions as: 1, m v, and ^m v^2 the moment equations are s - s the conservation equations for the particle number density, momentum and energy respectively: 3n j^-+ V-(ngWg) = 0, (2-3a) 3w s s 3t -s -s -s s-s s^ ~ (^) t*s S t^ S^ C^ (2-3b)

= Am S v2{ [J (f f)^v,

t*s S t^ S^ C^ (2-3c) In the energy equation the following notation was introduced:

C = I kT + \m w2. (2-4)

The conservation equation for the particle number density is called the equation of continuity. Equations (2-3b) and (2-3c) are also frequently called equation of motion and of energy respectively. In the right-hand side of these equations the term corresponding to t=s disappears because it represents collisions between identical particles for which the above functions of velocity are collisional invariants^1 "^2. Physically this means that there is no net exchange of momentum and of energy between like particles. One could have simplified equation (2-3c) further with the aid of equation (2-3b) and have arrived at the following form of the energy equation: , 3T 42 n k{-r-£ + w «7T } + 7«qs l (^) 3t - s s' a (^) s + P :- s - s Vw » J ftii (^) s s c (^2) LI Jst d^3 v,' (2-5) t*s

a result that can also be obtained directly from equation (2-1) with the velocity function \m c^2. Another quantity of impor- s s tance is the mass-velocity or plasma-velocity defined as: £ m n w s s s It is possible to define diffusion velocities U with respect to this plasma-velocity:

U := w - w. (2-7) -s -s - in In a weakly ionized gas (WIG), however, the density numbers of the charged particles are small. It follows that the mass velocity almost equals the hydrodynamic velocity of the neutral component. For later use diffusion velocities u are defined: u := w - w. (2-8) -s -s -a Now return to equation (2-1) and consider the right-hand side of this equation. It consists of a sum of collision integrals describing the variation in time of the distribution function f due to elastic encounters only. One can distinguish two different types of interaction: one based on a short-range intermolecular potential and one of the Coulomb type, which varies as l/r, r being the distance between two interacting particles. The first of these applies to all collisions between charged particles and neutral particles and between neutral particles mutually, and will be described by the well known Boltzraann collision integral:

J s t ( f s , f t ) = 2/d3M 38 6<l2« 1 .*>{f 8 Cr ^ )ft (v+ S + ^ ) +

t s t s

-fs(v)ft(v+g)}. (2-9)

Here g = v - v is equal to the relative velocity just before a collision. The validity of the Boltzmann collision integral is based on the stnallness of the number of particles in a sphere

inA is the so-called Coulomb logarithm. Herein A is the inverse of the plasmaparameter e , and is proportional to the number of electrons in a sphere with radius equal to the Debye lenght r :

A = 7 - ner3. (2-14) P In a plasma one distinguishes three characteristic lenghts: the Debye length A^, which is a measure of the distance over which the potential of a charged particle is shielded by the surroun- ding charged particles, the mean interpartlcle distance r and the Landau lenght r , which is the distance of closest approach Li between two like charged particle? with thermal velocities. These lenghts are defined as:

r

L = 7Tx>

r

o=*-

' ' D = £ T A

C 2

~

One can verify that the plasma parameter is proportinal to the ratio of the Landau- to Debye lenght, but also that the plasma parameter connects all three characteristic lenghts in (2-15). The condition for these lenghts to be well separated is that the plasma parameter should be very small. The plasma is then called ideal. The Landau collision integral results after making two cutt- off's: in the derivation of this expression there appears an integral over the interaction distance diverging at zero and infinity. The approximation made is that one introduces the lenghts r and r as integration boundaries. This leads to the factor lnA. This factor has to be much greater than unity. Speaking in more physical terras one could say that the Landau lenght is so small that there are relatively very few short range collisions. Because of the effect of screening the upper boundary can be replaced by the Debye lenght: collisions with larger impact parameter contribute little to the collision integral. Next the electron Boltzraann equation will be considered in more

detail. To solve this complicated equation an expansion into a

small parameter e will be used, e being the square root of the

electron-atom mass ratio:

e = <me /ma A (2-16)

This choice seems obvious and the next step is that all

dimensionless numbers, obtainable from the dimensionless

electron Boltzraann equation, are expressed as powers of e. The

equations will, however, not be made dimensionless. All terras

will be multiplied by the appropriate power of e. The distribu-

tion functions will be expanded into e and in the end e is put

equal to unity, so that e merely plays a bookkeeping role. For

a weakly ionized gas the electron Boltzraann equation reads as

follows:

3f eE

-r-^- +9t e v V f- e - e —m «V fv e - u)ce - - (vxb) «V fv e = eJ ee + J ea+ eJ ei.,

e

wherein b is a unit vector in the direction of a constant

external magnetic field B. The electron cyclotron frequency:

uice = —m , has been taken of the order of the electron-atom

e

collision frequency:

w T = 0 ( 1 ). (2-18)

ce ea

Here Tea is the mean collision time between two successive

collisions of an electron with a neutral atom:

v

—T = v ea = n v _ C r ' =a Te^ea r—£X. v(2-19) '

ea ea

Thermal velocities are defined as v =(kT /ra ) and Q is the

AS S o S C

elastic collision cross section for momentum transfer of

particles s with particles t defined as follows:

... IT

Q s t (g) = J2ita(g,x)(l-cosx)sinxdx, (2-20)

o

where g is the relative speed of the colliding particles.

Next the heavy particle Boltzraann equations have to be considered. For a weakly ionized gas one obtains:

— « -.^ 3f 3 t

- E^2 V • Vf = a ae h J aa

£^2 J , , (2-24)

ai 3f eS -r-^1 + E^2 v7f. + E^2 (— + ID ,VXb)'V f, = e \ j. + E J , + E^2 J... 3t - i m c i - - v i ie ia ii (2-25) Some extra assumptions have been made in these equations. The time variable has been scaled with T , so that in these equa- tions the choice v = v = v / E has been made. The heavyea^ ^ ea aa ia particle electron collision terms receive an additional factor e^2 because of the fact that momentum transfer in these colli- sions is rather inefficient. From these assumptions it follows that Q x *- E Q - Q. , which is reasonable provided that ea x^ aa 4 ia ' r charge transfer is not taken into account. At the same time it is assumed that the temperatures of the different components are of the same order of magnitude, so that v = v ~ E V. In the chapters to follow solutions of Ti 13 Te kinetic equations will be found by means of a perturbation expansion: f = (E.Y.O^ = fo0)(E»Y,t) + e f ^ r. v. t )^ +....^ (2-26) s s s It is known that such an expansion nay often lead to secular behaviour, i.e. it contains terras f ,. and f such that the ratio f ,/f goes to infinity withs,n+i increasings,n time, so s,n+l s,n that the expansion fails. One possibility to avoid these secularities is to make use of the multiple time scale forma- lism^1 *"^7. For that purpose it is observed that there are different time scales to be distinguished: tg is called the fastest time scale which is connected with the mean free time between two successive collisions of an electron with an atom; tn 0 = Tea. Then successive time scales are defined in the

following manner: t 1 = tQ/e, t 2 = tQ /e^2 etc. The t 2 time scale will appear to be the timescale on which energy relaxation

between electrons and atoms takes place. In the multiple time scale formalism new time variables T are defined as follows: n rn := ent, (2-27)

so that the time derivative transforms as:

IF * l? 0

'h^^h^ <

~

> Thus the formalism consists of a transformation from one time variable to a certain number of time variables T which are n treated as independent. In this way extra freedom is created, that will be used to eliminate the secularities which may occur. This is the essence of the multiple time scales forma- lism. The expansion (2-26) then transforms as: f (r,v,t) +^ f (r,v,x ,T ,..) + f (r,v,x ,T ,..) +... (2-29)

The procedure is then as follows: the collision integrals are also expanded in powers of £ and the expansion (2-29) is substituted into the Boltzmann equation. Terms of equal power of e are collected and equated to zero. The resulting equations are then solved for the functions f. The conservation equa- tions will be treated in a similar manner, and will serve to find solutions to the kinetic equations. Substituting the resulting solutions into the general expressions (2-2) trans- port coefficients are obtained, mostly as integrals over the electron-atom cross scetions. For realistic cross sections numerical integration schemes have to be resorted to.