The generalized hydrodynamic equations for ...

The generalized hydrodynamic equations for arbitrary collision frequency in a weakly ionized plasma E. Furkal and A. Smolyakov Citation: Phys. Plasmas 7, 122 (2000); doi: 10.1063/1.873787 View online: http://dx.doi.org/10.1063/1.873787 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v7/i1 Published by the AIP Publishing LLC.

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PHYSICS OF PLASMAS

VOLUME 7, NUMBER 1

JANUARY 2000

The generalized hydrodynamic equations for arbitrary collision frequency in a weakly ionized plasma E. Furkal and A. Smolyakov Department of Physics and Engineering Physics, University of Saskatchewan, Saskatoon, Saskatchewan S7N 5E2, Canada

共Received 17 June 1999; accepted 17 September 1999兲 Electron transport processes in a weakly ionized plasma with elastic electron-neutral collisions are studied by using the hybrid fluid/kinetic approach. The standard hierarchy of fluid moment equations is closed with expressions for higher hydrodynamic moments 共heat flux and viscosity兲 in terms of the lower moments 共temperature, density, and fluid velocity兲. The heat fluxes and viscosity moments are determined in the linear approximation from the kinetic equation in the Chapman– Enskog form. The obtained system of moment equations describe the transport processes in weakly ionized plasmas in the most general ordering, when the electron mean free path v Te / ␯ e is arbitrary with respect to the characteristic length scale k ⫺1 of the system’s inhomogeneity, and collision frequency ␯ e is arbitrary with respect to the characteristic frequency ␻ . General expressions for the nonlocal 共time and spatial dependent兲 transport coefficients are obtained. In the nonlocal limit, k v Te Ⰷ ␯ e , the derived transport coefficients describe the wave–particle 共Landau兲 interaction effects. Implications of nonlocal effects on plasma heating mechanisms are discussed. © 2000 American Institute of Physics. 关S1070-664X共00兲00601-7兴

I. INTRODUCTION

can be included into the fluid model by proper description of plasma viscosity in collisionless regimes. It has been previously shown10–12 that effects of wave– particle interaction 共Landau damping兲 can be incorporated in fluid type models. The usual hydrodynamic approach is no longer valid when k v Te ⯝ ␯ e ⯝ ␻ and the kinetic methods should be used in order to describe the relaxation processes. In previous works,10–12 collisionless effects were simply added to fluid equations such as to simply match the linear plasma response in various limits. Though such an approach is able to qualitatively describe these processes, difficulties arise as to which particular transport processes, e.g., viscosity or heat flux,10 are affected most by the collisionless interaction. Another problem occurs when the transition regimes between strongly collisional and purely collisionless have to be considered. Such situations may occur in plasma processing and light sources applications where Ramsauer gases are used and effective collision frequency may change the order of magnitude over the relatively small range of electron energies. For these cases, a rigorous Chapman–Enskog-type procedure is more suitable.13–15 This gives a uniform description of plasma transport in all collisionality regimes including kinetic effects such as Landau wave–particle resonance. An infinite system of moment equations 共equations for density n, flow velocity V, temperature T, stress tensor ⌸, etc.兲 are obtained by taking different velocity moments of the Boltzmann kinetic equation. The untruncated hierarchy of such equations is completely equivalent to the Boltzmann kinetic equation and all kinetic information is not lost in such a hierarchy. Usually, the infinite system is truncated to few equations, e.g., n, V, T, and closure relations 共expressions for stress tensor ⌸ and heat flux q in terms of the lower mo-

Electron transport processes in weakly ionized plasmas play a fundamental role in gas discharge physics and its applications, in particular, for a variety of plasma sources used in science and technology.1 In the present paper we consider the electron transport processes in a low temperature plasma (⭐10 eV兲 with a nonuniform 共in time and space兲 electric field when the main mechanism of electron scattering is elastic collisions with neutrals. We are interested in a situation when the electron mean free path is not small compared to the characteristic length scale of the system’s inhomogeneity, and the electron collisional frequency is not necessarily large compared to the frequency of the external field. These conditions become increasingly important as gas discharges for plasma processing and lighting move toward the lower pressure regimes. Such regimes, where thermal electron motion and collisionless effects are essential, are usually referred as nonlocal.2–4 One of the manifestations of the nonlocality is the modification of the electric conductivity and anomalous skin effect5 which were shown to be important in inductively coupled discharge.6,7 Similarly, other plasma transport properties such as thermal conductivity, viscosity, and diffusion will be modified in this regime. In this work we derive fluid type equations which take into account these modifications. The kinetic and hydrodynamic approach are combined such that the standard set of continuity, momentum, and energy equations8 can be used with all kinetic effects absorbed in the plasma viscosity, heat flux, and friction forces. Collisionless plasma heating due to wave–particle interaction is a main mechanism of heating in modern reactors for plasma processing.9 In this work we show that collisionless heating 1070-664X/2000/7(1)/122/13/$17.00

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© 2000 American Institute of Physics


Phys. Plasmas, Vol. 7, No. 1, January 2000

ments n, V, and T兲 are introduced. Standard closure relations are derived for the limit of the high collision frequency, ␯ Ⰷ( ⳵ / ⳵ t,k v Te ) and thus are valid only in the standard hydrodynamic 共high collisionality兲 regime. In the low collisional regimes the kinetic effects are important. The latter can be incorporated in the standard fluid moment if we do not exclude them in calculating the closure relations. The closure relations for arbitrary collisionality regimes can be obtained in closed form for the linear case, when the deviation of the distribution function from the equilibrium is not too large. Derivation of such relations constitutes the main goal of the present work. We use the Chapman–Enskog-type procedure13 to introduce the kinetic effects into the fluid equations. In this method, the total distribution function f is decomposed into two parts—a dynamical time and space dependent Maxwellian f M , and a deviation F, f ⫽ f M ⫹F. It is assumed that the spatial and temporal dependence in f M is through the hydrodynamic quantities n(r,t), T(r,t), and V(r,t). Substituting the first three fluid moment equations 关for n(r,t), T(r,t), and V(r,t)] into the Boltzmann kinetic equation, one obtains a kinetic equation for the departure part F. The deviation part, in turn, has to satisfy certain constraints which come from the fact that the first three moments n(r,t), T(r,t), and V(r,t) have to be determined by the dynamic Maxwellian distribution function f M alone. These constraints define the higher moments ⌸ and q through the lower hydrodynamic moments n, T, and V. The decomposition of the total distribution function onto the Maxwellian plus the deviation part is equivalent to the linearization of the Boltzmann kinetic equation around the equilibrium Maxwellian distribution, which is justified for the cases when the electron distribution function is close to the Maxwellian. The distribution function is not far from the Maxwellian when the rate of the electron–electron collisions is comparable to that of the electron-neutrals. These rates in turn depend on the densities and energies of the colliding species. Therefore, for a certain range of these parameters 共low electron energy and relatively high electron density兲 the electron distribution function 共EDF兲 can be close to Maxwellian distribution and linearization of the distribution function around the local Maxwellian is possible.16 The standard Chapman–Enskog procedure neglects the spatial and temporal dependencies of the perturbed distribution function in the Boltzmann equation, which is justified only in the strongly collisional limit. As plasma becomes more rarefied, the effects of the electron motion become important and one has to include the spatial and temporal derivatives of the perturbed distribution function in the kinetic equation. We take into account the spatial and temporal inhomogeneities and find the perturbed distribution function in terms of the continued fraction. As it will be shown below, in our approach, the collisional effects are characterized by a set of few effective collisional frequencies. For a given gas, these frequencies, can be calculated theoretically or deduced from experimental data. In this paper, we use the effective collisional frequencies that are calculated from a semianalytical model devel-

The generalized hydrodynamic equations for arbitrary . . .

123

oped in Ref. 17 by using the Modified Effective Range Theory.18 We neglect the influence of the induced magnetic field on the system.19,20 We also ignore inelastic effects as well as processes which change the number density of the various species, e.g., ionization and recombination. These effects can be important, but they are beyond the goals of the present paper. II. EXPANSION OF THE DISTRIBUTION FUNCTION AND MOMENT HIERARCHY

The Boltzmann kinetic equation for the electron distribution function in a weakly ionized plasma is

⳵f ⳵ f eE„r,t) ⳵ f ⫹v• ⫺ • ⫽Cˆ 共 f 兲 . ⳵t ⳵r m ⳵v

共1兲

ˆ is the collision operator of interaction between elecHere C trons and atoms given by Ref. 21, C共 f 兲⫽

冕

v rel共 f ⬘ f 1⬘ ⫺ f f 1 兲

d␴ 3 d p 1 d⍀, d⍀

共2兲

where v rel is the relative velocity between electrons and atoms, f, f 1 and f ⬘ , f ⬘1 are the electron and neutral atom distribution functions before and after the collision, d ␴ /d⍀ is the differential cross section of electron-neutral collisions. In the present paper we neglect the electron–electron and electron–ion interactions, since the neutral atom density is considerably higher than that of the electrons and ions. We also neglect the effects of energy transfer from electrons to atoms because of the large mass difference between the two species. Thus atoms can be considered to be motionless, and their distribution function is given by f ⬘1 ⫽ f 1 ⫽N ␦ 共 p 1x 兲 ␦ 共 p 1y 兲 ␦ 共 p 1z 兲 ,

共3兲

where N is the neutral atom density. Substituting expression 共3兲 into 共2兲 we obtain the linearized collision operator, ˆ 共 f 兲 ⫽N C

冕

共 f 共 t,r⬘ ,v⬘ 兲 ⫺ f 共 t,r,v兲兲v

d␴ d⍀. d⍀

共4兲

The orthonormal eigenfunctions of the collision operator 共4兲 are 1/2 , ⌿ r,l,m ⫽L rl⫹1/2共 v 2 / v 2t 兲v l P 兩lm 兩共 cos ␪ 兲 e im ␾ /N r,l,m

共5兲

Pm l (cos ␪)

L rl⫹1/2( v 2 / v 2t )

are Laguerre polynomials, where are associated Legendre polynomials, ␪ and ␾ are the angu1/2 is the normallar variables in spherical velocity space, N r,l,m ization constant equal to N r,l,m ⫽

2 l⫹1 ⌫ 共 r⫹l⫹3/2兲共 l⫹ 兩 m 兩兲 ! . 冑␲ r! 共 2l⫹1 兲共 l⫺ 兩 m 兩兲 !

共6兲

Equation 共5兲 can also be written by using the spherical harmonics functions Y l,m ( ␪ , ␾ ), Y l,m 共 ␪ , ␾ 兲 ⫽ 共 ⫺1 兲 m

冑

共 2l⫹1 兲共 l⫺m 兲 ! m P l 共 cos ␪ 兲 e im ␾ . 4 ␲ 共 l⫹m 兲 ! 共7兲

Then we have


124


⌿ r,l,m ⫽ 共 ⫺1 兲 m

冑

E. Furkal and A. Smolyakov

4 ␲ 3/2r!

␣ 0 ⫽1⫹

2 l⫹1 ⌫ 共 r⫹l⫹3/2兲

⫻L rl⫹1/2共 v 2 / v 2t 兲v l Y l,m 共 ␪ , ␾ 兲 .

共8兲

The functions 共8兲 satisfy the relation ˆ 共 ⌿ r,l,m 兲 ⫽ ␯ l ⌿ r,l,m . C

共9兲

For elastic electron-neutral atom collisions the eigenvalues are r(2l⫹1)-fold degenerate and do not depend on the subscripts r and m. Degeneracy comes from the fact that the differential cross section of elastic electron-neutral atom interaction is independent of azimuthal angle ␾ . Since functions 共5兲 form a complete set, we can expand the unknown electron distribution function f in the ⌿ r,l,m , f⫽

兺

r,l,m

b r,l,m 共 r,t 兲 ⌿ r,l,m ,

共10兲

␦ 0⫽

b r,l,m 共 r,t 兲 ⫽

冕

e ⫺v

2/ 2 vt

By introducing the scalar a, vector b, tensor B, etc., moments of the distribution function expansion 共10兲 can be written as f ⫽ f M 0 共 a⫹v–b⫹ 共 vv⫺ v /3I兲 :B⫹••• 兲 ,

mn 0 v 4t

n⫽

冕

共11兲

⬁

兺

r⫽0

␣ r L r1/2共 x 兲 ,

共12兲

兺

␰ r L r3/2共 x 兲 ,

共13兲

兺 ␦ r L r5/2共 x 兲 , r⫽0

共14兲

r⫽0

共15兲

...,

and f M 0 is the Maxwellian distribution, x⫽ v 2 / v T2 , Note that a,b,B, . . . are related to the expansion coefficients b r,l (r,t), ⬁

a⫽

兺

r⫽0

v T2

␰ 1 ⫽⫺

,

˜ 2q , 5p 0 共18兲

冕冉冕冕

V⫽

冊

兰 vf d 3 v , n

p⫽

m 3

冕

v 2 f d 3v ,

共19兲

5 v 2⫺ f d 3v , vT 2

q⫽T

⌸⫽m

共 vv⫺ v 2 /3I兲 f d 3 v ,

共20兲

mv2 共 vv⫺ v 2 /3I兲 f d 3 v . 2

冉

f ⫽ f M 0 1⫹ 2v

⫹ f M0

v T2

⫹2 f M 0

ñ ˜T L (1/2) ⫺ L (1/2) ⫹••• 0 n0 T0 1

冉

˜ L (3/2) • V ⫺ 0

vv⫺ v 2 /3I mn 0 v T4

冊

2 ˜qL (3/2) ⫹••• 5p0 1

冉

˜ L (5/2) : ⌸ ⫺ 0

冊

2 ˜ L (5/2) ⌰ ⫹••• 1 7T 0

兺

r⫽0,m⫽⫺1

b r,1,m 共 r,t 兲 ⌿ r,1,m ,

共16兲

冊共21兲

It is convenient to separate in 共21兲 the terms related to the linearized local Maxwellian distribution f M ⫽n 共 r,t 兲

冉

冉

m 2 ␲ T 共 r,t 兲

⯝ f M 0 1⫹

冊冉 3/2

exp ⫺

m 共 v⫺V共 r,t 兲兲 2 2T 共 r,t 兲

冊

ñ ˜T 2v ˜ L (3/2) L (1/2) ⫺ L (1/2) ⫹ 2 •V . 0 0 n0 T0 1 vT

冊

共22兲

Then we can represent the total distribution function in the form f 共 r,v,t 兲 ⫽ f M 共 v,n 共 r,t 兲 ,T 共 r,t 兲 ,V共 r,t 兲兲 ⫹F 共 r,v,t 兲 .

b r,0,0 共 r,t 兲 ⌿ r,0,0 ,

r⫽⬁,m⫽1

v–b⫽

,

v2

⬁

B⫽

˜ 2V

⫹••• .

⬁

b⫽

␰ 0⫽

˜ 2⌰ ␦ 1 ⫽⫺ . 7T 0

f d 3v ,

where a⫽

˜T , T0

Using standard moments the distribution function 共11兲 is

* d 3v . f ⌿ r,l,m

2

˜ 2⌸

␣ 1 ⫽⫺

The fluid moments are given by the standard definitions

⌰⫽

where

ñ , n0

共23兲

The deviation part F consists of all higher moments 共scalar, vector, 2nd rank tensor, . . . 兲. For our subsequent analysis it is convenient to write F in the form of the spherical harmonic functions

r⫽⬁,m⫽2

共 vv⫺ v 2 /3I兲 :B⫽

兺

r⫽0,m⫽⫺2

b r,2,m 共 r,t 兲 ⌿ r,2,m .

共17兲

Expansion coefficients ␣ n , ␰ n , ␦ n are related to the standard hydrodynamics moments. For example, the first two scalar ␣ 0 , ␣ 1 , vector ␰ 0 , ␰ 1 and tensor ␦ 0 , ␦ 1 quantities are associated with the perturbed plasma density n, pressure p, mean velocity V, heat flux q, stress tensor ⌸, and energy weighted stress tensor ⌰,

F⫽

兺

f l,m 共 r, v ,t 兲 Y l,m 共 ␪ , ␾ 兲 .

共24兲

All the scalar moments are incorporated in ⬁

f 0,0⫽

兺

r⫽2

b r,0,0 L r1/2共 x 兲 ,

共25兲

the vector moments in



兺冑⌫ 共 r⫹5/2兲 b r,1,⫺1 L r3/2共 x 兲 , r⫽1 ⬁

f 1,⫺1 ⫽⫺

␲ 3/2r!

兺冑⌫ 共 r⫹5/2兲 b r,1,0 L r3/2共 x 兲 , r⫽1 ⬁

f 1,0⫽

␲ 3/2r!

⬁

f 1,1⫽⫺

兺

r⫽1

冑

␲ 3/2r! b L 3/2共 x 兲 , ⌫ 共 r⫹5/2兲 r,1,1 r


共27兲

creases and wave–particle interactions becomes important, the terms related with heat flux and viscosity tensor are important on the right-hand side of Eq. 共34兲. Linearizing Eq. 共34兲, we obtain for the perturbed distribution function F

共28兲

⳵F ⫹v–ⵜF⫺C 共˜f M ⫹F 兲 ⳵t

共26兲

III. THE CHAPMAN–ENSKOG–TYPE EQUATION

To describe the evolution of the dynamical Maxwell distribution function f M (v,n(r,t),T(r,t),V(r,t)) we use the hydrodynamic equations for the electron density n, flow velocity V, and temperature T,

mn

冉冉

共29兲

冊冊

⳵V ⫹V–“V ⫽⫺enE⫺“p⫺“•⌸⫹R, ⳵t

共30兲

3 ⳵T n ⫹V•“T ⫽⫺p“•V⫺⌸:“V⫺“•q⫹Q e . 共31兲 2 ⳵t The electron momentum changes are due to collisions with neutrals and given by R⫽m

冕

vC 共 f 共 r,t,v兲兲 d 3 v .

Q e⫽

冕

共33兲

where v⬘ ⫽v⫺V is the random electron velocity. Substituting Eq. 共23兲 into the Boltzmann kinetic equation 共1兲 and using the fluid moment Eqs. 共29兲–共31兲, one obtains recast Chapman–Enskog-type equation for the deviation part F,13

⳵F e ⫹v–“F⫺ E–“ v F⫺C 共 f 兲 ⳵t m ⫽⫺

冉

冊

fM m w2 I :“Vf M ⫹w• 共 “–⌸⫺R兲 ww⫺ T 3 p

冉冉

冊冊

mw 2 5 fM ⫺ w–“T . 2T 2 T

共34兲

Equation 共34兲 is an exact equation for F. No approximations have been made so far. In the strongly collisional limit, one can eliminate all but the first and the last terms on the right-hand side of Eq. 共34兲. When the role of collisions de-

冊

冉

2v2

⫺1 2

3vT

冊

冉冊

f M0 f M0 v2 5 ⵜ•q⫺ 2 ⫺ v•ⵜT , p0 T0 vT 2

共35兲

where v T2 ⬅2T 0 /m, p 0 ⫽n 0 T 0 , f M 0 is the nonshifted lowest order Maxwellian distribution function, the perturbed Maxwellian is given by Eq. 共22兲. The linearization of the distribution function is equivalent to the assumption that the energy that electron acquires in a field eE␭ where ␭ is the mean free path, is much less then the thermal electron energy kT. Since the perturbed values 共density, temperature兲 can be expressed in terms of the perturbed electric field E, the above linearization means that the perturbed density, temperature should be small in comparison with their equilibrium values. The collisional heat generation term Q e has been dropped in Eq. 共35兲, since Q e ⫽⫺R–V, and it is a nonlinear function of the flow velocity V. The collision term in 共35兲 can be written C 共˜f M ⫹F 兲 ⫽⫺

2 ␯ 1 ˜V z v f M 0 v T2

cos ␪ ⫺

2 ␯ 1 ˜V x v f M 0 v T2

sin ␪ cos ␾

⬁

⫺

兺

l⫽0,m

␯ l f l,m Y l,m 共 ␪ , ␾ 兲 ,

共36兲

where ␯ l is the lth order collision frequency defined by Ref. 23,

␯ l 共 v 兲 ⫽N v

冕

共 1⫺ P l 共 cos ␹ 兲兲

d␴共 v,␹ 兲 d⍀, d⍀

共37兲

where P l (cos ␹) are the Legendre polynomials and ␹ is the scattering angle. The collision frequencies ␯ l have to be calculated theoretically or deduced experimentally. In this work we use a semi-analytical model for ␴ ( v , ␹ ) developed in Ref. 17 for argon gas. In 共36兲 it is assumed that ˜V y ⫽0. Equation 共35兲 has to be solved for F together with the following constraints:

冕兵

mw 2 fM ⫹ ⫺1 共 ⌸:“V⫹“–q⫺Q e 兲 3T p ⫺

⫹

共32兲

The collisional heat generation in electrons as a result of collisions with neutrals is determined by mw 2 C 共 f 共 r,t,v兲兲 d 3 v , 2

冉

f M0 m v2 vv⫺ I :“Vf M 0 ⫹v• 共 ⵜ–⌸⫺R兲 T0 3 p0

⫽⫺

and the higher order tensor moments 共all ranks starting from 2nd兲 are incorporated in f 2,m , f 3,m , . . . , where m⫽⫺l,⫺l ⫹1, . . . ,l.

⳵n ⫹“•nV⫽0, ⳵t

125

1,v, v 2 其 Fd 3 v ⫽0,

共38兲

since the first three moments are determined by the local Maxwellian distribution function f M . Using Eq. 共24兲, the constraint 共38兲 can be written as

冕

⬁

0

f 0,0v 2 d v ⫽0,

冕

⬁

0

f 0,0v 4 d v ⫽0,

冕

⬁

0

v 3 f 1,0d v ⫽0. 共39兲

These equations will be used to determine the closure relations for ⌸ and q in Sec. V.


126



where v ⫽ 兩 v兩 , f l,m (k, ␻ , v ) is a function of the absolute value of electron velocity. In further analysis, for the sake of simplicity, we assume that all the inhomogeneities are in the x-direction, that is, k ⫽kxˆ, T⫽T(x), and n⫽n(x). The electric field and plasma flow have two components E⫽E x (x,t)xˆ⫹E z (x,t)zˆ, and V ⫽xˆVx (x)⫹zˆV z (x). This configuration corresponds to the simplified one-dimensional cylindrical model of the inductive plasma discharge.1 Substituting Eq. 共40兲 into Eq. 共35兲 and making the Fourier transformation, we obtain

IV. SOLUTION OF THE CHAPMAN–ENSKOG-TYPE EQUATION

To proceed further, we have to solve Eq. 共35兲 for the deviation of the distribution function F. We do it by making Fourier expansion for F 共24兲,

F 共 r,v,t 兲 ⫽

Y l,m 共 ␪ , ␾ 兲冕 f l,m 共 k, ␻ , v 兲 e i(k–rⴚ␻ t) d 3 kd ␻ , 兺 l,m

共40兲

⫺i ␻

兺 l,m

⫽⫺

⫹

⫺

f l,m 共 k, ␻ , v 兲 Y l,m 共 ␪ , ␾ 兲 ⫹ik v sin ␪ cos ␾ 2 ␯ 1 ˜V z v f M 0 v 2t

2 f M0 3 v 2t

cos ␪ ⫺

˜ x兲⫹ v 2 共 ikV

冉

2 ␯ 1 ˜V x v f M 0 v 2t

兺 l,m

f l,m 共 k, ␻ , v 兲 Y l,m 共 ␪ , ␾ 兲 ⫹

sin ␪ cos ␾ ⫺

2 f M0 v 2t

␯ l f l,m 共 k, ␻ , v 兲 Y l,m 共 ␪ , ␾ 兲兺 l,m

˜ z兲⫺ v 2 sin ␪ cos ␪ cos ␾ 共 ikV

2 f M0 v 2t

˜ x兲 v 2 sin2 ␪ cos2 ␾ 共 ikV

v f M0 vf vRx f M0 ˜ 兲 ⫹ M 0 sin ␪ cos ␾ 共 ik⌸ ˜ 兲⫺ cos ␪ 共 ik⌸ sin ␪ cos ␾ xz xx p0 p0 p0

冊

冉冊

2v2 f M0 vRz f M0 v2 5 v f M0 ˜ x兲⫺ ˜ 兲. cos ␪ ⫹ ⫺1 ikq ⫺ sin ␪ cos ␾ 共 ikT 共 p0 p0 3 v T2 v T2 2 T 0

共41兲

If the differential cross section is a function of ␪ , Eq. 共41兲 leads to coupled equations for different spherical harmonics. * ( ␪ , ␾ ) and integrating it over the solid angle we obtain Multiplying 共41兲 by Y l,m

1 ⫺i ␻ f l,m ⫹ ik v ⌫ l,m ⫹ ␯ l f l,m ⫹ 2 ⫽⫺

⫹

⫺

冑冑冑

冑

4 ␲ 2 ␯ 1 ˜V z v f M 0 ␦ l,1␦ m,0⫹ 3 v 2t

8␲ v2 f M0 ˜ z兲⫹ 共 ␦ l,2␦ m,⫺1 ⫺ ␦ l,2␦ m,1兲共 ikV 15 v T2

16␲ v 2 f M 0 ˜ x兲⫹ ␦ l,2␦ m,0共 ikV 45 v T2

冑

冉

冑

8 ␲ ␯ 1 ˜V x v f M 0 共 ␦ l,1␦ m,⫺1 ⫺ ␦ l,1␦ m,1兲 3 v 2t

冑

4␲ v f M0 ˜ xz 兲 ⫺ ␦ ␦ 共 ik⌸ 3 p 0 l,1 m,0

冑冑

8␲ v f M0 ˜ xx 兲共 ␦ l,1␦ m,⫺1 ⫺ ␦ l,1␦ m,1兲 ⫺ 共 ik⌸ 3 2p 0

冊

4␲ v f M0 2v2 f M0 ˜ x 兲 ␦ l,0␦ m,0⫺ R z ␦ l,1␦ m,0⫹ 冑4 ␲ ⫺1 共 ikq 2 3 p0 p0 3vT

冑

8␲ v2 f M0 ˜ x兲共 ␦ l,2␦ m,⫺2 ⫹ ␦ l,2␦ m,2兲共 ikV 15 v T2

冉冊

8␲ v f M0 R 共␦ ␦ ⫺ ␦ l,1␦ m,1兲 3 2 p 0 x l,1 m,⫺1

8␲ v2 5 v f M0 ˜ 兲, ⫺ ⫺ ␦ l,1␦ m,1兲共 ikT 共␦ ␦ 3 v T2 2 2T 0 l,1 m,⫺1 共42兲

where

⌫ l,m ⫽

冑冑

冑冑

共 l⫺m⫹1 兲共 l⫺m⫹2 兲 f l⫹1,m⫺1 ⫹ 共 2l⫹1 兲共 2l⫹3 兲

⫺

共 l⫹m⫺1 兲共 l⫹m 兲 f ⫺ 共 2l⫺1 兲共 2l⫹1 兲 l⫺1,m⫺1

共 l⫺m 兲共 l⫺m⫺1 兲 f 共 2l⫹1 兲共 2l⫺1 兲 l⫺1,m⫹1

共 l⫹m⫹2 兲共 l⫹m⫹1 兲 f l⫹1,m⫹1 . 共 2l⫹1 兲共 2l⫹3 兲

共43兲

The exact solution of this infinite system of Eqs. 共42兲 can be obtained in a form of infinite continued fractions.17 The following expressions for the perturbed electron distribution functions f 0,0 and f 1,0 , and f 1,⫺1 are obtained: Downloaded 07 Oct 2013 to 155.247.166.234. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://pop.aip.org/about/rights_and_permissions


f 0,0⫽ 冑4 ␲

⫺

f M0 p 0 ik

3 v T2 ␻

f 1,⫺1 ⫽⫺

v2

⫺ 2

vT

冋册冊

5 f M0 2 T0

冉冋冉

冉冊冉

册

冊

册冊冑冊冑

1 ⫺1 ˜V z ⫹ H 2 共 k, ␻ , v 兲

H 2 共 v ,k, ␻ 兲 ⫽1⫹D 2 / 共 1⫹D 3 /1⫹D 4 /••• 兲 ,

共48兲

with coefficients

共 l 2 ⫺1 兲 k 2 v 2 共 4l 2 ⫺1 兲共 i ␻ ⫺ ␯ l 兲共 i ␻ ⫺ ␯ l⫺1 兲

冉

冊

冊

4 ␲ f M 0v ik ˜ ⌸ . 3 p 0 共 i ␻ ⫺ ␯ 1 兲 H 2 共 k, ␻ , v 兲 xz

共47兲

共 4l 2 ⫺1 兲共 i ␻ ⫺ ␯ l 兲共 i ␻ ⫺ ␯ l⫺1 兲

冉

4 ␲ 2 ␯ 1v f M 0 1 ˜V 2 3 共 i ␻ ⫺ ␯ 1 兲 H 2 共 k, ␻ , v 兲 z vt

4 ␲ f M 0v 1 R ⫺ 3 p 0 共 i ␻ ⫺ ␯ 1 兲 H 2 共 k, ␻ , v 兲 z

l 2k 2v 2

共44兲共45兲

H 1 共 v ,k, ␻ 兲 ⫽1⫹C 1 / 共 1⫹C 2 /1⫹C 3 /••• 兲 ,

D l⫽

冋

册

The effects of the higher-order spherical harmonics are included in the continued fractions H 1 and H 2 ,

C l⫽

册冊

127

1 f M 0共 i ␻ ⫺ ␯ 1 兲 1 ˜ ⫺1 ⌸ ⫺1 共 ˜V x 兲 xx ⫹4 冑␲ 2 H 1 共 k, ␻ , v 兲 H 1 共 k, ␻ , v 兲 v T ik

2 冑4 ␲ v 2 3 f M 0 1 k ˜q x , ⫺1 ˜T ⫺ ⫺ 2 H 1 共 k, ␻ , v 兲 3 v T 2 p 0 ␻ H 1 共 k, ␻ , v 兲

3 ␻ f , 2 k v 0,0

16␲ f M 0 v 3 v T2

冉冋

Vx兲 1 4 冑␲␯ 1 f M 0 共˜ ˜ x兲 ⫺ ⫺1 共 ikV 2 2 H 1 共 k, ␻ , v 兲 H k, ␻,v兲共 1 v Tk

冉冊冉冋

冑

冑冑

⫹

册冊

1 f M0 ⫺1 R x ⫺ 冑4 ␲ H 1 共 k, ␻ , v 兲 p0

4 冑␲ k v 2 f M 0

⫹ 冑4 ␲

f 1,0⫽

冉冋


,

共49兲

.

共50兲

The similar method of incorporating of the higher-order spherical harmonics was used in Refs. 15 and 24 for the problem of electron–ion collisions and in Ref. 17 for the

共46兲

calculation of the nonlocal electron conductivity in a weakly ionized plasma. To find an expression for the perturbed electron distribution functions f 0,0 , f 1,0 , f 1,⫺1 , one has to calculate the continued fractions H 1 ( v ,k, ␻ ) and H 2 ( v ,k, ␻ ) which in turn requires the knowledge of the lth order collision frequencies ␯ l . Analytical expressions for the effective collisional frequencies were found from modified effective range theory in Ref. 17. If the electron-neutral atom interaction is of the polarization type, which is assumed in the present paper, then the collision frequency ␯ l rapidly converges to a constant value17

␯ ⬁ 共 v 兲 ⫽ lim ␯ l 共 v 兲 ,

共51兲

l→⬁

so that only few low l frequencies are different from ␯ ⬁ . The first ␯ l for argon are plotted in Fig. 1. A relatively simple but accurate approximations for the continued fractions H 1 and H 2 can be developed noting the identity

冑1⫹x 2 ⫽1⫹

1 2

x2

,

x2

1 1⫹ 4 1⫹

1 4

共52兲

x2 1⫹

1 x2 4 1⫹•••

and the fact that for large l coefficients C l converge to a constant C l→ FIG. 1. The normalized effective collisional frequencies for argon gas; ␯ 1 , solid line, N is the neutral density, a 0 is the Bohr radius, v 1 is the electron temperature for 1 eV; ␯ 2 , dashed line; ␯ 3 , dotted line; ␯ ⬁ , dotted dashed line.

k 2v 2 1 1 ⫽ x 2, 4 共 i␻⫺␯⬁兲2 4

共53兲

where x⫽k v /(i ␻ ⫺ ␯ ⬁ ). Keeping a few first terms exact and replacing the rest with the approximate expression C l ⯝x 2 /4 the function H 1 can be represented asymptotically


128


H N1 ⫽1⫹C 2 /1⫹C 3 /1⫹•••⫹C N /1 ⫹

x2 4


冒冒 1⫹

x2 4

共54兲

1⫹•••.

A similar form is valid for H 2 . The final the second-order approximate expressions for H 1 and H 2 in the algebraic form are17 1 k 2v 2 共冑1⫹k 2 v 2 / 共 i ␻ ⫺ ␯ ⬁ 共 v 兲兲 2 ⫹1 兲 H 1 共 ␻ ,k v , ␯ 兲 ⫽1⫹ , 3 i ␻ 共 i ␻ ⫺ ␯ 1 共 v 兲兲关共冑1⫹k 2 v 2 / 共 i ␻ ⫺ ␯ ⬁ 共 v 兲兲 2 ⫹1 兲 ⫹ 共 8/15兲 k 2 v 2 / 共共 i ␻ ⫺ ␯ 1 共 v 兲兲共 i ␻ ⫺ ␯ 2 共 v 兲兲兴 H 2 共 ␻ ,k v , ␯ 兲 ⫽1⫹

1 k 2v 2 共冑1⫹k 2 v 2 / 共 i ␻ ⫺ ␯ ⬁ 共 v 兲兲 2 ⫹1 兲 . 5 共 i ␻ ⫺ ␯ 2 共 v 兲兲共 i ␻ ⫺ ␯ 1 共 v 兲兲关共冑1⫹k 2 v 2 / 共 i ␻ ⫺ ␯ ⬁ 共 v 兲兲 2 ⫹1 兲 ⫹16/35k 2 v 2 / 共共 i ␻ ⫺ ␯ 2 共 v 兲兲共 i ␻ ⫺ ␯ 3 共 v 兲兲兴共56兲

As shown in Ref. 17 these expressions provide very good fit to the exact continued fractions for a wide range of the parameter ␻ /k v . In particular, collisionless effects of the Landau damping are recovered with high accuracy.

V. THE CLOSURE RELATIONS AND GENERALIZED TRANSPORT COEFFICIENTS

The closure relations for the components of the plasma ˜ , ⌸ ˜ , and the heat flux ˜q are found viscosity tensor ⌸ xz xx x from the constraints 共38兲 which can be written as

冕冕冕

⬁

0 ⬁

0 ⬁

0

v 2 f 0,0d v ⫽0,

共57兲

v 3 f 1,0d v ⫽0,

共58兲

v 4 f 0,0d v ⫽0.

共59兲

Equations 共44兲–共46兲 contain also the electron momentum change terms R x and R z . These can be determined from their definitions R z ⫽m ⫽⫺

R x ⫽m ⫽⫺

冕

v zC共 f 兲d 3v

8 ␲ mV z 3 v T2

冕

冕␯ ⬁

0

1v

4

f M 0d v ⫺

冑冕 4␲ m 3

⬁

0

␯ 1 v f 1,0d v , 3

共60兲

3 v T2

冕␯ ⬁

0

1v

4

f M 0d v ⫺

冑冕 8␲ m 3

⬁

0

Substituting expressions 共44兲–共46兲 into 共57兲–共61兲 and resolving the obtained system one obtains closure relations for ⌸ xz , ⌸ xx , and q x , ˜ 兲, R x ⫽⫺l x mn 0 ˜V x ⫺tn 0 共 ikT

共62兲

R z ⫽⫺l z mn 0 ˜V z ,

共63兲

˜ xx ⫽⫺ ␩ xx mn 0 共 ikV ˜ x 兲 ⫺␵n 0˜T , ⌸

共64兲

˜ xz ⫽⫺ ␩ xz mn 0 共 ikV ˜ z兲, ⌸

共65兲

˜q x ⫽rp 0 ˜V x ⫺ ␹ 共 ikT ˜ 兲.

共66兲

Exactly the same closure equations 共62兲–共66兲 could be obtained using the definitions 共19兲–共20兲 for heat flux and viscosity. Equations 共62兲–共65兲 define nonlocal transport coefficients such as the thermal force coefficient t, the stress force coefficient ␵ induced by the temperature gradient, the heat pinch coefficient r, the thermal conductivity ␹ , and viscosity coefficients ␩ x,x , ␩ x,z . Exact expressions for these coefficients are given in the Appendix. Closure relations 共62兲–共65兲 are used together with the system of moment equations, ⫺i ␻ñ ⫹ikn 0 ˜V x ⫽0,

共67兲

˜ ⫹R , ⫺i ␻ mn 0 ˜V z ⫽⫺enE z ⫺ik⌸ xz z

共68兲

˜ ⫹R , ⫺i ␻ mn 0 ˜V x ⫽⫺enE x ⫺ik ˜p ⫺ik⌸ xx x ˜ x ⫹ikq ˜ x ⫽0. ⫺i ␻ 23˜p ⫹ 25 p 0 ikV

共69兲

Substituting expressions 共62兲–共65兲 into Eqs. 共68兲 and 共69兲 we obtain the x- and z-components of the electron hydrodynamic velocity given by

v xC共 f 兲d 3v

8 ␲ mV x

共55兲

␯ 1 v 3 f 1,⫺1 d v .

˜V z ⫽⫺

␴z E , en 0 z

共70兲

˜V x ⫽⫺

ñ ˜T ␴x E z ⫺d n ik ⫺d T ik , en 0 n0 T0

共71兲

共61兲 The first term in 共60兲–共61兲 is the standard friction force due to the finite velocity of the electron component with respect to the neutral component, and the second is the thermal force related to the heat flux and higher order vector moments. The thermal force arises by virtues of a gradient in the electron temperature and dependence of the collision frequency on electron velocity.

where

␴ z⫽

e 2n 0 z m ␯ eff

,

共72兲




z FIG. 2. The normalized effective dissipation frequency ␯ eff as a function of the wave vector.

␴ x⫽ d n⫽

d T⫽

e 2n 0 x m ␯ eff

T x m ␯ eff

,

共73兲

,

共74兲

T 共 1⫹t⫺ ␨ 兲 x m ␯ eff

.

x FIG. 3. The normalized effective dissipation frequency ␯ eff as a function of the wave vector.

ⰇkvT , that represent a local limit case, the coefficients can be simplified. Then, case the local thermal force t and heat pinch coefficient r are t⫽r⫽

冋冕冋冕

⬁v4

共共 v 2 / v T2 兲 ⫺ 共 5/2兲兲 f M 0 d v i␻⫺␯1

0

•

共75兲

⬁

0

册

册

v4 f M0 dv . i␻⫺␯1

共79兲

The stress force coefficient is

The effective dissipation frequencies are x ␯ eff ⫽⫺i ␻ ⫹k 2 ␩ xx ⫹l x ,

共76兲

z ␯ eff ⫽⫺i ␻ ⫹k 2 ␩ xz ⫹l z .

共77兲

Relations 共72兲, 共73兲, 共75兲, and 共74兲 define the nonlocal electric conductivity ␴ z and ␴ x , and diffusion coefficients of the considered system. Converted back to space–time variables the transport coefficients become the integrodifferential operators acting on temperature, density, flow velocity, and electric field. Expression 共72兲 for the nonlocal transverse electrical conductivity coincides with the one found in the previous paper.17 As an illustration we show the nonlocal z x ␯ eff , ␯ eff , and t for argon gas in Figs. 2, 3, and 4. It is necessary to note that there exists a simple symmetry relation15 between the heat pinch coefficient r, thermal force coefficient t, and stress force coefficient % given by r⫽t⫺ ␨ .

129

␵⫽0.

共80兲

The coefficient of thermal conductivity is

␹⫽

再

2␲m ⫺ 3T 0 ⫹t

冉冕

⬁

0

冕

⬁v6

共共 v 2 / v T2 兲 ⫺ 共 5/2兲兲 f M 0 d v

0

v6 f M0 dv i␻⫺␯1

冊冎

i␻⫺␯1

,

共81兲

where viscosity coefficient ␩ ⫽ ␩ xx ⫽ ␩ xz is given by

共78兲

It is interesting to note that Eq. 共78兲 holds true not only for strongly collisional case, but for any values of the nonlocality parameter k v T /(i ␻ ⫺ ␯ ) including collisionless case. The existence of the relation 共78兲 is a direct consequence of the generalized Onsager symmetry principle.22 VI. THE TRANSPORT COEFFICIENTS IN THE FLUID „COLLISIONAL… LIMIT, ␯ ⯝ ␻ Ⰷ kv T

The transport coefficients obtained in the previous section are complicated functions of the electron temperature T, frequency ␻ , and wave number k. In the fluid limit, ␯ ⯝ ␻

FIG. 4. The nonlocal thermal force coefficient ␶ as a function of the wave vector.


130



electron energy 共the Ramsauer effect兲 in argon gas modifies the thermal force coefficient t, so that it can be positive 共the thermal force is in the opposite direction to the temperature gradient兲 or negative 共the thermal force is codirectional with the temperature gradient兲 depending on the value of electron temperature.26 In the case when ␻ Ⰷk v T , ␯ e , the continued fractions H 1 and H 2 are given by H 1 ⫽1⫺

1 k 2v 2 , 3 ␻2

共85兲

H 2 ⫽1⫺

1 k 2v 2 , 5 ␻2

共86兲

and the transport coefficients are z x FIG. 5. The local effective dissipation frequency ␯ eff⫽ ␯ eff ⫽ ␯ eff as a function of the electron temperature, ␻ ⫽0.

␩ ⫽⫺

1 5

冋冕

⬁

0

v6 f M0 dv 共 i ␻ ⫺ ␯ 1 兲共 i ␻ ⫺ ␯ 2 兲

册冒冋冕

⬁

0

册

v4 f M0 dv . i␻⫺␯1

共82兲 The friction coefficients in this regime are l x ⫽l z ⫽

兰 ⬁0 共 ␯ 1 v 4 f M 0 d v / 共 i ␻ ⫺ ␯ 1 兲兲兰 ⬁0 共 v 4 f M 0 d v / 共 i ␻ ⫺ ␯ 1 兲兲

.

共83兲

In the local limit, the contribution of the viscosity ␩ to the effective dissipation frequencies ␯ x and ␯ z is small as 2 / ␯ 2 Ⰶ1 and can be neglected. Then, the local electric k 2 v Te conductivity coefficients are

␴ x⫽ ␴ z⫽

4␲e2 3T 0

冕

⬁

0

v4 f M0 dv⫽␴. i␻⫺␯1

共84兲

The similar expression for the local electric conductivity was obtained in Ref. 25. The behavior of the local dissipax z ⫽ ␯ eff is shown in Fig. 5 for ␻ ⫽0. tion frequency ␯ eff⫽ ␯ eff Figure 6 shows the local thermal force t. As one can see the nonmonotonous behavior of the transport cross section with

4␲m 3 p 0i ␻

冕␯

4␲m ␵⫽ 3 p 0i ␻

冕␯

t⫽

1v

4

1v

冉冊冉冊 v2

v T2

4

⫺

5 f dv, 2 M0

v2

5 ⫺ f M 0d v , 2 2 vT

5 v T2 ␩ xx ⫽ ␩ xz ⫽⫺ , 6 i␻ ne 2 ␴ x ⫽ ␴ z ⫽⫺ , mi ␻

5 n v T2 ␹ ⫽⫺ , 4 i␻ ⫽⫺

v T2

2i ␻

r⫽O

冉冊 k 2 v T2

␻2

, 共87兲

.

As we can see, in this limit the transport coefficients are purely imaginary, so that there is no dissipative 共irreversible兲 processes in a system, which is not an unexpected result since there are no collisions ( ␻ Ⰷ ␯ ) as well as there is no wave–particle interaction ( ␻ /kⰇ v T ). In the case when ␯ is independent of electron velocity, the local transport coefficients can be calculated analytically to give the known results26

␶ ⫽0,

␩⫽

v T2

2␯

r⫽0,

,

␴⫽

␵⫽0, ne 2 , m␯

␹⫽ d n⫽

5nT , 2m ␯ v T2

2␯

共88兲 .

As was mentioned earlier, the above coefficients can be used to describe the transport processes in a weakly ionized plasma only in a case of very low energies of the electrons 共low electron temperature兲 and high density of neutrals, so that the condition ␯ Ⰷ ␻ Ⰷk v T still holds. VII. NONLOCAL LIMIT; ␯ , ␻ Ⰶ kv T

In strongly nonlocal limit, ␯ , ␻ Ⰶk v T , effects of electron collision is unimportant. The transport coefficients 共A1兲– 共A6兲 can be simplified by using the asymptotic form for the continued fractions H 1 and H 2 . Taking the limit ␯ , ␻ Ⰶk v T in the expressions for the exact continued fractions 共47兲 and 共48兲 we have FIG. 6. The local thermal force coefficient as a function of the electron temperature, ␻ ⫽0.

H 1 ⫽⫺

2 kv , ␲ i␻

共89兲



H 2 ⫽⫺

␶ ⫽⫺

kv 4 , 3␲ i␻⫺␯1

48冑␲ 5n 0 k v T

2 r⫽⫺ , 5 l x⫽

␹⫽

冕 ␲ 冕

8␲ 5n 0

2 3/2 l z⫽ n 0v T

␩ xx ⫽

冕

⬁

0

⬁

0

0

v T2

n v T2

⫺

3 f d v ⫽0, 2 M0

2 ␵⫽ , 5 共91兲

,

冉冊

共92兲

v 3␯ 1 f M 0d v ,

共93兲

2 冑␲ v T2 5k v T

冉冊 v2

5 冑␲ k v T

v ␯1 2

⬁

共90兲

␯ 1v 2 9


,

ne 2 冑␲ ␴ z⫽ , m kvT

v2

v T2

⫹1 f M 0 d v ,

␩ xz ⫽ d n⫽

v T2

冑␲ k v T 5

,

v T2

2 冑␲ k v T

␴ x⫽ , d T⫽

5ne 2

1 , 2 冑␲ m k v T 3

v T2

2 冑␲ k v T

共94兲 .

The calculated transport coefficients coincide with those found in Refs. 13 and 15 for the adiabatic limit in the electron–ion plasma. The coincidence of the transport coefficients in the adiabatic limit for electron-neutral and electron–ion plasma is not surprising, since this limit is collisionless indeed and the main mechanism of the transport processes in both systems is the wave–particle interaction 共Landau damping兲. It is worth noting that in the present case, the higher moments 共viscosity, heat flux兲 as well as the diffusive electron flux are proportional to the perturbed lower moments 共temperature, density, mean velocity兲共in the local case the higher moments are proportional to the gradients of the corresponding lower moments兲. VIII. CONCLUSIONS

We have considered nonlocal electron transport processes in weakly ionized plasmas when the main mechanism of electron scattering is the electron-neutral atom interaction. We have used the moment approach similar to the Chapman–Enskog expansion method, which determines the higher moments 共stress tensor, heat flux vector兲 in terms of the lower moments 共density, temperature, and mean velocity兲. Because the standard Chapman–Enskog method is essentially an expansion of the total distribution function in a small parameter ␧ which is the ratio of the mean free path to the length scale, it cannot be used to describe the system for the most general collisionality regime. An increase in the value of ␧ leads to a larger number of terms in the expansion and in the collisionless limit one deals with an infinite number of equations. The modification to the Chapman–Enskog procedure developed in this paper is based on the expansion of the total distribution function in the complete set of eigenfunctions of the collision operator.23 The obtained infinite system of equations for the expansion coefficients was solved in terms of the continued fractions and the total distribution function is found as function of ␻ and k. Obtained

131

transport coefficients are complicated functions of the electron temperature T, wave number k, and frequency ␻ in Fourier space. Converted back to the space–time variables they become the integrodifferential operators. In the local limit k v Te Ⰶ ␯ e , ␻ Ⰶ ␯ e we obtain the local transport coefficients which are the nonmonotonous functions of the electron energy due to the nonmonotonous energy dependence of the differential cross section of electronneutral atom interactions. Thus results in the possibility of a negative thermal force, which may be directed against the temperature gradient, and thus the modification of the total heat flux in the system. In the strongly nonlocal limit k v Te Ⰷ ␯ e , the transport processes are due to the wave–particle interaction 共Landau mechanism兲. Unlike to the local case where the higher moments 共heat flux, viscosity兲 are expressed in terms of the gradients of the corresponding lower hydrodynamic moments 共temperature, density, mean velocity兲, it is shown that in the nonlocal case ( ␯ , ␻ Ⰶk v T ) the higher moments are proportional to the perturbed lower moments 关see Eqs. 共91兲–共94兲兴. The method of solving the kinetic equation developed in this work is particularly important for higher electron energies, ⭓0.1 eV, when the differential cross section depends on the scattering angle. If ␯ l ⫽const 共the differential cross section is independent of the scattering angles兲 for all l, than one can solve the Boltzmann kinetic equation with a simple collision term C(F)⫽⫺ ␯ ( v )F. 32 Such collisional term is valid only in the limit of very small electron energies. In this limit there is a weak dependence of the differential cross section on the scattering angle and electron energy that can be neglected.17 One of the most important results of this paper is a demonstrated possibility to describe and interpret a number of kinetic effects within the hydrodynamic approach. Such effects are important for transport processes and heating in material processing reactors.1 For instance, in low pressure regimes, when particle motion is important, the character of electric conductivity is modified leading to the anomalous skin effect.6,17 In fact, these effects corresponds to the contribution of the plasma viscosity. It can easily be seen from 共76兲 and 共77兲 that in the nonlocal regime effective plasma resistivity is determined by viscosity effects (k 2 ␩ xz and k 2 ␩ xx terms are dominant兲. Similarly, in the nonlocal regime, viscosity becomes dominant in the plasma heating14 where it exceeds collisional heating in the factor of Q vis /Q coll 2 ⯝k 2 v Te / ␯ 2 Ⰷ1, where Q coll⫽⫺R–V⫽⫺R z V z ⫺R x V x , Q vis⫽⫺ ␲ :“V⫽⫺ ␲ zx

⳵Vz ⳵Vx ⫺ ␲ xx . ⳵x ⳵x

共95兲共96兲

The importance of the plasma compressibility 共pressure and viscosity兲 on the heating in collisionless regimes has been noted in Ref. 27. In our approach we use the kinetic equation which was linearized for small deviations from the equilibrium Maxwellian distribution. Thus the derived transport coefficients are strictly speaking valid only for the linear case. This is one of the most significant limitations of our theory. Never-


132



共98兲

heating terms.28,29 The Fourier components in this model are calculated numerically. Such a hybrid model was proven to be effective in investigating of collective transport processes in a tokamak plasma.28,29 Such a fluid-kinetic model can also be helpful in fluid modeling of plasma discharges as an alternative to exact kinetic calculations.30–34 It contains all kinetic effects associated with the wave–particle interaction which cannot be described by the local collisional approximation and describes the transition between collisional and collisionless regimes. Nonlocal effects associated with thermal particle motion are also important for laser heated plasmas where a similar approach24,35,36 can be useful.

共99兲

ACKNOWLEDGMENTS

theless, our nonlocal transport coefficients represent a significant improvement over the standard 共local兲 transport coefficients which are independent of the wave vector and the wave frequency26 contrary to the nonlocal case. The obtained transport coefficients can be used in hybrid fluid-kinetic model for numerical simulations. This model can be represented as follows:

⳵ n⫹“• 共 nV兲 ⫽0, ⳵t

共97兲

d V⫽⫺enE⫺ⵜ p⫺ⵜ•⌸⫹R, dt

mn

5 3 d p⫹ p“•V⫹“•q⫽Q. 2 dt 2

Note that such model would retain the main hydrodynamic 共convective兲 nonlinearites. The plasma viscosity and heat flux in this model are calculated from the lower moments by using the linearized closures developed in this paper. The plasma heating term Q 关Eqs. 共95兲 and 共96兲兴 is calculated in the quasilinear approximation. The summation over all Fourier harmonics is assumed in the heat flux, viscosity, and

t⫽

This work was supported by Natural Sciences and Engineering Research Council of Canada. Helpful discussions with J. D. Callen on the Chapman–Enskog closures are acknowledged with gratitude. APPENDIX: THE CLOSURE RELATIONS FOR CALCULATIONS OF THE TRANSPORT COEFFICIENTS

4 ␲ mi ␻ 关 E 1 共 A 3 B 2 ⫺A 2 B 3 兲 ⫹E 2 共 A 1 B 3 ⫺A 3 B 1 兲 ⫹E 3 共 A 2 B 1 ⫺A 1 B 2 兲兴

␩ xx ⫽⫺

冉

␵⫽ ⫺

r⫽

4 ␲ mi ␻ p 0k 4

A 1B 1

关 E 1 共 B 3 /B 1 ⌬⫺A 3 /A 1 ⌼ 兲 ⫹E 3 共 ⌼⫺⌬ 兲兴

关 A 3 B 2 ⫺A 2 B 3 兴关 A 3 B 1 ⫺A 1 B 3 兴

3i ␻

1

v T2 k 2

A B

1

关 A 3 B 1 ⫺A 1 B 3 兴

冊

关 A 3 B 1 ⫺A 1 B 3 兴

␩ xz ⫽

2

k D

2

⫺

共A5兲

共A6兲

,

k 2 关 A 3 B 1 ⫺A 1 B 3 兴

4 ␲ mi ␻ E 1 共 A 3 B 1 ⌼⫺B 3 A 1 ⌬ 兲 ⫹E 3 共 ⌬⫺⌼ 兲

⌬⫽⫺i ␻ ⫹

ik 2 F 2 , 3␻ A1

共A9兲

⌼⫽⫺i ␻ ⫹

ik 2 G 2 , 3␻ B1

共A10兲

⌿⫽

8␲m 3 v T2

G 3⫺

A 3 B 1 ⫺A 1 B 3

A 1⫽

,

冕冉 ⬁

v2

0

共A7兲 l z⫽

C1 D

, 2

共A2兲

,

8␲m␻2 v T2 k 2

共A11兲

E 1.

The closure coefficients for calculations of the transport coefficients are

⌿ mn 0

p 0k 2

1 B 3 A 1 ⌬⫺A 3 B 1 ⌼

共A4兲

,

where l x ⫽⫺

mn 0 k

⫹ 2

共A3兲

3n i ␻ 关 A 2 B 1 ⫺A 1 B 2 兴 ␹⫽ , 2 k 2 关 A 3 B 1 ⫺A 1 B 3 兴 D1

⌿

where

⫹t ,

共 ⌬⫺⌼ 兲

⫺

共A1兲

,

关 A 3 B 1 ⫺A 1 B 3 兴

p 0k 2

共A8兲

A 2⫽

冕

⬁

0

v2

冊

1 ⫺1 f M 0 d v , H 1 共 ␻ ,k v , ␯ 兲

冉冊冉 v2

v T2

⫺

5 2

共A12兲

冊

1 ⫺1 f M 0 d v , H 1 共 ␻ ,k v , ␯ 兲



A 3⫽ B ⫽ 1

B ⫽ 2

B 3⫽ E 1⫽ E ⫽ 2

E ⫽ 3

C 1⫽ D ⫽ 1

D 2⫽ F ⫽ 1

F ⫽ 2

G ⫽ 1

G ⫽ 2

G 3⫽

冕冉冕冉冕冉 ⬁

v2

0

⬁

v

4

0

⬁

v

4

0

v2

⫺ 2

vT

v4

0

冊冉冉冊

⬁

1v

0

⬁

1v

0

f M0 3 dv, 2 H 1 共 ␻ ,k v , ␯ 兲

冊

B 3⫽ 共A13兲

1 ⫺1 f M 0 d v , H 1 共 ␻ ,k v , ␯ 兲 v2

5 ⫺ 2 vT 2

冕冕␯ 冉冕␯ 冉 ⬁

冊


v

冊

1 ⫺1 f M 0 d v , H 1 共 ␻ ,k v , ␯ 兲

f M0 3 ⫺ dv, 2 v T 2 H 1 共 ␻ ,k v , ␯ 兲

2

2

冊

冊冉冉冊冉 v2

5 ⫺ 2 vT 2

1v

0

⬁

v

4

0

⬁

0

⬁

0

3 ⫺ 2 vT 2

冊冊

1 ⫺1 f M 0 d v , H 1 共 ␻ ,k v , ␯ 兲

共A15兲

共A16兲

4

k2 3i ␻

E 2 ⫽⫺

k2 3i

E 3 ⫽⫺

k2 3i

共A17兲

4

v f M0 dv, 共 i ⫺ 1 兲 H 2 共 ,k v , 兲

1 ⫺1 f M 0 d v , v 共i ⫺ 1兲 H 1 共 ,k v , 兲

⬁

0

v4 f dv, H 1 共 ,k v , 兲 M 0

⬁

0

F 1 ⫽⫺

0

⬁

0

⬁

共A19兲

6

v f dv, H 1 共 ,k v , 兲 M 0 v4

0

1 ⫺1 f M 0 d v . H 1 共 ,k v , 兲

1

共A20兲

冕冕 ␻ 冕 ␻ 冕 ␻

k2 A ⫽⫺ 3i 2

k2 B 1 ⫽⫺ 3i k2 B ⫽⫺ 3i 2

0

0 ⬁

0

0

⫺ 共 3/2兲 …f M 0 dv, i␻⫺␯1

0

1v

4

f M0 dv, i ⫺ 1 ⬁

0

v6 f M0 dv, 共 i ⫺ ␯ 1 兲共 i ␻ ⫺ ␯ 2 兲

共A24兲

v4 f M0 dv, i ⫺ 1

p 0k 2 , 4 ␲ mi ␻

F 2⫽

G 3⫽

,

3 p0 , 4␲ m

15nk 2 v T4 48␲ i ␻

G 1 ⫽⫺

15nk 2 v T4 48␲ i ␻

⫺ 共 5/2兲 …f M 0

共A21兲 dv,

A ⫽0, 3

v6 f M0 dv, i␻⫺␯1

A 1 ⫽⫺

n , 4␲

B 1 ⫽⫺

3 p0 , 4␲ m

E 2 ⫽⫺ E 3 ⫽⫺

F 1 ⫽⫺ G ⫽⫺ 1

⫺ 共 5/2兲 …f M 0

i␻⫺␯1

dv,

共A22兲

,

共A25兲共A26兲

,

The closure coefficients in this limit are A 2⫽

⬁

0

1v

2

1v

2

⬁

0

⬁

2 1v

0

v2 v T2 v2 v T2

1v

0

A 3⫽ i ␻ n v T2

8 冑␲ k v T

i␻n 8 冑␲ k v T

,

B 3 ⫽⫺

,

i ␻ n v T2 8 冑␲ k v T

3

冊冊

⫺

5 f dv, 2 M0

⫺

3 f dv, 2 M0 D 1 ⫽⫺

f M 0d v ,

共A27兲

3 nT 0 , 4␲ m 共A28兲

3

G ⫽⫺ 2

冕冕

⬁

0

v 2共 i ␻ ⫺ ␯ 1 兲 f M 0d v ,

⬁

0

,

f M 0d v ,

⬁

3 4k

n , 4␲

B 2⫽

冕␯ 冕␯ 冉冕␯ 冉 ␲ 冕␯

2

i␻⫺␯1

⬁ v 6 „共 v 2 / v 2 兲 T

⬁ 1 v 4 „共 v 2 / v 2 兲 T

nvT , D ⫽⫺ 8 冑␲ k

v4 f M0 dv, i␻⫺␯1

⬁ v 4 „共 v 2 / v 2 兲 T

共A23兲

dv,

2. Nonlocal limit, ␯ , ␻ Ⰶ kv T

1. Fluid limit, ␯ ⯝ ␻ Ⰷ kv T ⬁

⫺ 共 5/2兲 …f M 0

i␻⫺␯1

0

16␲

C 1 ⫽⫺ k2 A 1 ⫽⫺ 3i ␻

4

f M0 dv, i ⫺␯1

0

15n v T4

E 1 ⫽⫺

1 ⫺1 f M 0 d v , v 4共 i ⫺ 1 兲 H 1 共 ,k v , 兲

1v

⬁ 1 v 4 „共 v 2 / v 2 兲 T

k2 5

⬁

2

共A18兲

⬁␯

冕 ␯␻ ␯ 冕 ␻ 冕␻␯

D 1 ⫽⫺

G 2⫽

1 ⫺1 f M 0 d v , H 2 共 ,k v , 兲

冕 ␻ ␯ 冕 ␻ 冕␯ ␻

E 1 ⫽⫺

D 2⫽

1 ⫺1 f M 0 d v , H 1 共 ␻ ,k v , ␯ 兲

f M0 dv, 共 i ⫺ 1 兲 H 2 共 ,k v , 兲

⬁

0

v2

1v

0

⬁

2

3p0 , 4␲m

C 1⫽

1 ⫺1 f M 0 d v , H 1 共 ␻ ,k v , ␯ 兲

冕␯ 冕 ␻ ␯␯ ␻ ␯ 冕冉 ␻ ␯ 冊冕 ␻␯ ␻ ␯ 冕 ␻␯冉 ␻ ␯ 冊冕 ␻ ␯ 冕 ␻␯冉 ␻ ␯ 冊冕 ␻ ␯ 冕 ␯冉 ␻ ␯ 冊 ⬁

共A14兲

2

133

F 2 ⫽⫺

i␻nvT 4 冑␲ k

v 共 i ␻ ⫺ ␯ 1 兲 f M 0d v ,

, 共A29兲

4

i ␻ n v T3 2 冑␲ k

,

G 3 ⫽⫺

冕

⬁

0

v 4␯ 1 f M 0d v .

共A30兲


134 1


M. A. Lieberman and A. J. Lichtenberg, Principles of Plasma Discharges and Materials Processing 共Wiley–Interscience, New York, 1994兲. 2 I. D. Kaganovich, V. I. Kolobov, and L. D. Tsendin, Appl. Phys. Lett. 69, 3818 共1996兲. 3 I. D. Kaganovich, Phys. Rev. Lett. 82, 327 共1999兲. 4 I. D. Kaganovich and L. D. Tsendin, IEEE Trans. Plasma Sci. 20, 66 共1992兲; 20, 86 共1992兲. 5 E. S. Weibel, Phys. Fluids 10, 741 共1970兲. 6 V. A. Godyak and V. I. Kolobov, Phys. Rev. Lett. 79, 4589 共1997兲. 7 V. A. Godyak, R. B. Piejak, B. M. Alexandrovich, and V. I. Kolobov, Phys. Rev. Lett. 80, 3264 共1998兲. 8 S. I. Braginski, in Review of Plasma Physics, edited by M. A. Leontovich 共Consultants Bureau, New York, 1965兲, Vol. 1, p. 214. 9 M. A. Lieberman and V. A. Godyak, IEEE Trans. Plasma Sci. 26, 1 共1998兲. 10 G. Hammet and F. W. Perkins, Phys. Rev. Lett. 64, 3019 共1990兲. 11 M. M. Turner, Phys. Rev. Lett. 75, 1312 共1995兲. 12 M. M. Turner, Plasma Sources Sci. Technol. 5, 159 共1996兲. 13 Z. Chang and J. D. Callen, Phys. Fluids B 4, 1167 共1992兲. 14 A. I. Smolyakov and E. Furkal, Bull. Am. Phys. Soc. 42, 1717 共1997兲. 15 A. V. Brantov, V. Y. Bychenkov, V. T. Tikhonchuk, and W. Rozmus, Phys. Plasmas 5, 2742 共1998兲. 16 V. A. Godyak and V. Kolobov, Phys. Rev. Lett. 81, 369 共1998兲. 17 E. Furkal, A. Smolyakov, and A. Hirose, Phys. Rev. E 58, 965 共1998兲. 18 T. O’Malley, Phys. Rev. 130, 1020 共1962兲. 19 M. Tuszewski, Phys. Plasmas 5, 1198 共1998兲; Phys. Rev. Lett. 77, 1286 共1996兲.

E. Furkal and A. Smolyakov 20

A. I. Smolyakov and I. Khabibrakhmanov, Phys. Rev. Lett. 81, 4871 共1998兲. 21 E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics 共translated from Russian by T. B. Sykes and R. V. Franklin兲共Pergamon, New York, 1981兲. 22 J. W. Dufty and J. M. Rub, Phys. Rev. A 36, 222 共1987兲. 23 I. P. Shkarofsky, T. W. Johnston, and M. P. Bachinski, The Particle Kinetics of Plasmas 共Addison–Wesley, Reading, MA, 1966兲. 24 E. M. Epperlein, R. W. Short, and A. Simon, Phys. Rev. Lett. 69, 1765 共1992兲. 25 G. G. Lister, Y.-M. Li, and V. A. Godyak, Appl. Phys. 79, 8993 共1996兲. 26 V. E. Golant, A. P. Zhilinsky, and I. E. Sakharov, Fundamentals of Plasma Physics 共Wiley, New York, 1977兲. 27 M. M. Turner, Phys. Rev. Lett. 71, 1844 共1993兲. 28 G. W. Hammet, W. Dorland, and F. W. Perkins, Phys. Fluids B 4, 2052 共1992兲. 29 W. Dorland and G. W. Hammet, Phys. Fluids B 5, 812 共1993兲. 30 N. S. Yoon, S. S. Kim, C. S. Chang, and D. I. Choi, Phys. Rev. E 54, 757 共1996兲. 31 N. S. Yoon, S. M. Hwang, and D. I. Choi, Phys. Rev. E 55, 7536 共1997兲. 32 K. C. Shaing and A. Y. Aydemir, Phys. Plasmas 4, 3166 共1997兲. 33 Yu. M. Aliev, I. D. Kaganovich, and H. Shlueter, Phys. Plasmas 4, 2413 共1997兲. 34 V. I. Kolobov, D. P. Lymberopolous, and D. J. Economou, Phys. Rev. E 55, 3408 共1997兲. 35 V. Y. Bychenkov, W. Rozmus, V. T. Tikhonchuk, and A. V. Brantov, Phys. Rev. Lett. 75, 4405 共1995兲. 36 W. Rozmus and V. T. Tikhonchuk, Phys. Rev. A 42, 7401 共1990兲.
