Some problems of gas-solid surface interaction

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION* R G. BARANTSEV Leningrad State University, U S.S.R.

GAS-SURFACE interaction occurs in a variety of physical processes and attracts the attention of various specialists. We consider these phenomena in the aspect of rarefied gas dynamics where the surface interaction problem is one of the most interesting and complex at present. The main attention is paid to neutral monatomlc gas interaction with a simple clean surface at energies 10-2-102 eV. Quantum mechanical methods, high-temperature effects and electromagnetic phenomena are not considered here. Section 1 contains a general statement of the problem and a brief review of the physical and computational data results. Fundamental notions characterizing interaction on different levels of description and permitting the order of the phenomena of interest in a natural way are introduced in a new fashion. Each of the following sections is devoted to the central problem of a certain scale level" molecular (II), Boltzmannian (III) and that of random roughness (IV). The study is aimed at obtaining theoretical and analytical results rather than a lot of numerical data. Atomic lattice scattering is considered within the framework of the two-particle-collision theory. Hard sphere scattering is studied in detail and then attraction and softness corrections are introduced. The scattering function model account begins with a critical review and is completed by an algorithm of successive modelling incidence and re-emission velocity dependencies leading to useful formulas for the momentum and energy exchange coefficients. The roughness problem is stated for an arbitrary reflection law in the small area. A complete solution is obtained in the case of a homogeneous isotropic differentiable random surface. Asymptotic expansions are obtained for a slightly rough Gaussian surface. 1. I N T R O D U C T I O N

1.1. Fundamental Notions 1.1.1. Interaction functions Let us consider the interaction of a rarefied gas with a solid surface within the framework of the kinetic theory. The gas is specified by a set of distribution functions of atomic particles in the velocity spacef~(u), subscript i designating partmles with different both chemical and * List of mare notatmns will be found on p 76. 1

2

R, G

BARANTSEV

energy states. The distribution functions are assumed to be averaged over an elementary time-space volume the size of which ~s large as compared to that of an atom The solid body can be considered in quast-equdtbrlum m this scale This means that given atom composition and location m the lattice and the adsorption layer, their vibrational motion is completely determined by the body temperature Atom location can be described m molecular scale by space distribution functions which are determined by the body crystal structure, the surface crystallographic plane, the adsorption-layer population. In addition, random irregularities of larger size, the so-called roughness, are admissible The atomic particle colhsion with the surface can result in the following phenomena: 1 Scattering--reflection of the particle either with or without changing its internal state. 2 Sputtering--dislodging, from surface into gas, other particles belonging to the body

or trapped from gas_

3 Trapping--adsorption of the particle on the surface or its penetration into the body. Spontaneous emission of atom particles from surface into gas can also occur Let the impinging particle have a velocity ul and internal state t, the emerging one a velocity u and internal state j Now we introduce the functions via which the result of interaction by way of particle exchange can be expressed. V~ (ul, u) is the distribution density of a flux of scattered particles; its integration over all possible u results in VJ, (u0 that is the probability of t -+j-scattering for incidence velocity U 1,

W~ (ul, u) is the distribution density of a flux of sputtered particles; its integration over u yields W~ (u~) that is the average number of j-particles dislodged by an t-particle hitting with velocity Ul S~ (ul) is the trapping probability of an l-particle with incidence velocity u. It is clear that v,~ (u~) + s, (u,) = 1. R j (u) is the distribution density of a flux of emitted particles; its integration over u results in R j that msthe average number of j-particles spontaneously emitted from umt area per unit time For a joint consideration of scattering and sputtering the function T = V + W that was named in ref. 1, the boundary transform is used. Interaction functions enter the boundary conditions for distribution functions f, which satisfy the Boltzmann equations reside the gas. The boundary conditions express the fact that for e a c h j the particle flux with velocity u from the surface u.fj (u) I ..>o is made up of fluxes to the surface I Ua, If, (ul) I ~1, < o transformed by T j, (ul, u) and of emitting flux R s (u), Un f j ( u )

I~.>o

"~, I

([U,n [ft(Ul)Ts, (Ul, u)du~

+ RS(u)

(1 l)

i) Uln<0

The interaction functions can include parameters connected to f~, for example, adsorption layer population or surface temperature. In these cases the solution of the Boltzmann equations must be coupled with relations concerning adsorptton layer kinetics and heat transfer reside a body.

1.1.2. Descrlptwn levels Interaction phenomena can be studied with different degrees of detail. When the scale changes, the basic notions and laws connecting them also change. There exist description

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

3

levels on which a closed statement of rarefied gas problems is deduced for a comparatively wide range of phenomena. The molecular level ~s characterized by the Newton equations for trajectories r, (t) of an enormous number of particles. On the Boltzmannian level we have the kinetxc equations for distribution functions f~ (r, u, t), i : 1, 2 . . . . . m; m being the number of gas species. The gas-dynamic level exhibits the transfer equations for density n~, velocity U~, temperature Ti and other macroscopic quantities depending on r, t. With the reduction of the degree of detad considered, the description is simphfied owing to the reduction of the number of dependent and independent variables. This reduction may consist of the enlargement of the time-space scale on the Boltzmannian level and averaging over veloc~tles on the gas-dynamic one The simphfication is accompanied by diminishing the sphere of application. Thus, the Boltzmann equation treats &lute gases, and the conventional transfer equations the near-equlhbrium states. Connection of each level to deeper ones ~s accomplished by means of some functions which are the channels of influence of the deep processes on macroscopic phenomena. Translating effects into the language of another scale these functions make ~t possible to close the statement of the problem at a certain level. On the molecular level such a role Is played by the interaction potentials modeUmg the effect of electromc structure of atomic particles; on the Boltzmannian o n e - - t h e interaction functions introduced m Section 1.1; on the gas-dynamic o n e - - t h e accommodation coefficients. The question of a proper description level is important not only for the statement of the problem but also m the course of solution finding and m the deduction of results. The solution can be obtained more easily on a macroscopic level. It is there that the main results of interest are held. But the essentially non-equlhbrlum phenomena are connected to deep processes and for closing the statement of the problem one has to go into the more detailed descnptton levels.

1.1.3. Interaction functionals Rarefied gas dynamics mainly deals with the Boltzmannian and gas-dynamic levels. The basic notions of gas-surface interaction were introduced above on the Boltzmannlan level. Transition to the gas-dynamic one is accomphshed by averaging over velocities with the formation of average fluxes of mass, m o m e n t u m and energy for each component. The boundary transforms depend on velocities both before and after collision. So their averaging is accomphshed in two stages. Integration over u of the interaction functions weighted by 1, u, u 2 results in a set of interaction funct~onals describing the average fluxes of mass, m o m e n t u m and energy of emerging j-particles under given Interaction conditions. These functionals are completely determined by the interaction functions. Indeed, let t-particles of mass mi, number density nt, velocity ul, and internal energy El counted from a certain unexcited state impinge on a surface area with the outward normal n (F~g. I). The mass, m o m e n t u m and energy of oncoming particles to unit area per unit time are

n, [ul. I mi,

ni I Uln [mt ul,

nl [ul. [(~ ml u~ + El)

(1.2)

4

R

G_ BARANTSEV

"i

FiG. I respectively. The scattered and sputtered j-particles reduce the magnitudes of mass by: n~ J ul, I f T~ (ul, u) mj du, un>O

n, I u~. I f T ~, (u~, u) mj u du,

momentum by

(1.3)

Un> 0

energy by.

n, Iu,. If T j,(u,, u) (½ mj

u~ +

EA du,

,an> O

and the emitted j.particles by

f RJ (u) mjdu, f RJ(u) m, udu, f R'(u)(½m, u2 + Ej) du u~>O

un>O

(1.4)

Un>O

respectively. Subtracting expressions (1.3) and (1.4) from (1.2) one obtains the mass, momentum and energy of the j-component transferred to the surface under these conditions, L~(u,):n,

lua. [rn, 3o - m J T ~ ( u 0 ] - m j R

P~ (Ul)= n, Iu,.

[m,u,8,~--fT{(u.u)mludu]--fRJ(u)m,udu, un>O

Q~(ul):n,

lu,.

y,

[(½rnjuj 2-}-E,) 3o -

un>O

f T~(ul, u)(½mju 2 + E j ) du] un>O

-- f R j(u)(½m,u 2-- Ej) du u~>O

(I.5)

SOME PROBLEMS OF GAS--SOLID SURFACE INTERACTION

Summing over j produces the full values of these quantities under the same conditions, ~

L~ (ux) = L, (ul) = L r (ul) -- L a, A

~

Pf (uO = P, (uO = pr (uO - pR,

(1.6)

J

~

Ol ( u , ) = Q, (Hi)= QT ( u , ) - O". J

The emission terms may depend on 1 and ul through S~ (ul)

1.1.4. Exchange coefficients When introducing dimensionless interaction macro characteristics under the condition o f impinging/-particles o f velocity u~ it is natural to refer the quantities (1.5) and (1 6) to ni mi Ul, to n~ m~ u 2 and to ½ n~ mj u~ respectively Then the local mass, m o m e n t u m and energy exchange coefficients are obtained,

LI ( U l )

1~ (ux) = - - , n~ m~ ul

p~ (ul) --

Pl ( U l )

2' n, m~ u 1

q~ (Hi) =

QI (Ul) {

s' n~ m~ u x

(1.7)

depending on the impinging beam species and velocity. In partmular, for the scattered and sputtered particles one has

lr (u,) = cos 01 [1-- ~__j-~ T~ (u,)], J

P ~ (Ul) :

COS 01

u[~-~m~fT¢(ul, u)Udu], J

q~ (U,)= COS 01 [(l

m i

ua>

II1

0

rn, +~,)--~fT~,(ut,") j

(1.s)

+~j)' du,]

u(~12

u.> 0

where 01 =

<~ ( n , -

U l ) , El - -

2E, m t l,/12.

0.9)

In the case of a snmple gas without trapping, sputtering and emission l (u~) = O,

p(ul)=cosO,u[~--~--fV(ul, u)-udu], Ut Ilw> 0

q (u j) = cos 01

[, fv,u, u, 4 Ua> 0

(l.10)

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R.G.

BARANTSEV

If the surface is isotropic, then p(ua)

- r (ul) t - - p ( u l )

n,

(1 I1)

where the unit tangential vector t is placed on the plane (n, ul) so that < (t, - - ul) =- ~r/2 - - 01 (Fig. 1) Thus, m the simplest case the interaction functlonals are r e d u c e d to three scalarexchange coefficients o f tangential m o m e n t u m ~- (ul), n o r m a l m o m e n t u m p (ul) a n d energy

q(ul) O f c o m m o n usage are the a c c o m m o d a t i o n coefficients defined by cr1 - -

r-

- T+ +,

7--

zs

p- --p+ ~r2 - - _--7-------7, P

a-

--Ps'

q- --q+ +, q-

(1.12)

--qs

where superscript minus sign is referred to the i m p i n g i n g flow, superscript plus sign to the emerging one, a n d subscript s to the emerging flow in e q u d i b r i u m with the surface when 2h, V(u) = - - u " e x p ( - -

V=

1

hs -- 2RTs'

(1.13)

R being the gas constant, T~ the surface temperature. In the scale accepted ~-- = c o s

01 sin 01,

p - = c o s z 01 ,

q- =cos

01.

(1.14)

0~,

(1.15)

Using (1.13) a n d (1.10), (1.11) is easy to calculate ~'2 = 0 ,

p+---- ~

7r cos 01,

q+ = - - c o s

his

where hl ~ = hsu~. The signs o f emerging values in (1.12) are i m p l i e d to be such that

~=z----c

+, p = p -

+p+,

q=q---q+.

(1.16)

F r o m (1.12), (1.14)-(1.16) one can derive the f o r m u l a s connecting the exchange coefficients with those o f a c c o m m o d a t i o n 7" =

O"1 s i n

01 COS 01,

p = (2 - - ~2) cos2 01 + - ~ q=cLcos

cos 01 ,

(1 17)

O~ 1 - -

U n d e r c o n & t l o n s far f r o m e q m h b r m m the coefficients r, p, q are m o r e n a t u r a l a n d convenient for reference t h a n ~ , e2, a. In fact they are g r a d u a l l y accepted t h o u g h the t e r m " a c c o m m o d a t i o n " is still retained. It is also archaic, though, because n o n - e q u i l i b r i u m m o m e n t u m a n d energy exchange on a surface does n o t generally l o o k like " a d a p t a t i o n " o f the a t o m i c particles to surface c o n d m o n s .

1.1 5. Assoctated functionals The second stage o f averaging, the i n t e g r a t m n over incidence velocmes, is c o n n e c t e d with the d i s t r i b u t i o n functions fi. I f they are known, as for instance in the case o f a free

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

7

molecule flow past convex bodies, the mass, momentum and energy fluxes on the surface can be comparatively easily calculated on the grounds of (1.5), (1.6). The values of these quantities created by i --~j-scattering, sputtering and j-emission, L~, P~, Q~, are obtained by multiplying (1.5) by f~ (u~)/ni and integrating over Ul, < 0. The expressions (1.5) correspond to f t = r/l ~ (Ul - - E l i ) (1.18) Taking into account all possible species of emerging particles for an impinging/-component one gets L,

= --i f

f~ (ul) L, (Ul) dUl,

nl

ffln
(1.19)

P,=--

if f~(ul) P,(ul)dul,

nl

ui~
7,1 f f l (ul) Q, (uO dUl.

Q, =

Ul~
Summing, at last, over all impinging gas species one obtains the full mass, momentum and energy fluxes on the surface, L, P, Q. Let

n=

n,,

m = -

where

l

g n I = J f / (Ul)

dul,

UI=

nl mi,

n

I

Ull =-

Uli,

(1.20)

1

if

nf

f / (Ul) U 1 d u l .

(1.21)

The full exchange coefficients /----, L

p=-- P

n m U 2'

n m U~

Q q-

½ n m U~

(1.22)

can be directly expressed m terms of the partial ones (1.7) by

f

l = nl l

P

~1

utn
f m__~t A (ul) q, (111) U-~ f

m__. m .l fi (Ul) p, (ux) U---~

l

q

ux d u l ,

rn___!lm fl (uO It (ul) ~

1

~

(1.23)

uln
m

l

Vtn < 0

Under equdlbrmm conditions when UI = 0, it is natural to use the average velocity of heat motion connected with the gas temperature T1 to form the dimensionless values.

8

R . G . BARANTSEV

Coefficients ( 1.22) in contrast to (1.7) do not depend on i and u I but depend on the parameters contained m f, (u~). In the general case f, (ul) are not known a prtort and the full exchange coefficients can be calculated only after solving the problem on the Boltzmannian level. If, however, the problem has been stated on the gas-dynamic level by means of representing f~ via macroparameters, then averaging over ul depends on the representauon form and a couphng occurs between the exchange coefficients and the unknown macroparameters.

1_2. Statement o f the Problem 1.2.1. Systematization

The variety of phenomena in gas-surface interaction problems permits different classifications depending on viewpoints and criteria of ordering. Going up the scale through the description levels one ought to construct: 1. On the molecular level--the interaction potentials using electronic and nuclear ideas. 2. On the Boltzmannlan one--the interaction functions V{ (Ul, U), WJ/(Ul, u), R J (u) that may include undefined parameters connecting them wlthf~. 3. On the intermediate one--the functlonals over u weighted l, u, u 2 providing the partial mass, momentum and energy exchange coefficients 4 On the gas-dynamic o n e - - t h e full exchange coefficients closely connected w~th the macroparameters representing f~. For rarefied gas dynamics the main description level is the Boltzmannlan one. Therefore, the interaction potentials are usually assumed to be given and macro characteristics are derived from the lnteractmn functions From the physical aspect the following problems can be stated moving from simple towards complicated things: 1. 2 3. 4. 5 6. 7. 8 9

Structureless pamcle colhslons with a simple crystal lattice of motionless atoms Effects of crystal structure and surface crystallographic plane. Influence of surface temperature. Statistical roughness Adsorphon-layer kinetics Influence of internal degrees of freedom. Charged particle interaction Chemical reactions on surface Radiation

In the multidimensional space of parameters it is customary to distinguish domains of certain specific property, from the point of view of simplifying the formulations of the problem, solution methods and applications. The kinetic energy E1 of the impinging particle proves to be one of the most important factors. The energy range that is of interest for rarefied gas aerodynamics can be divided into three intervals: low energies middle energies high energies

~ 10-2-10 -1 eV, ~ 10°-101 eV, ,,~ 102-10 s eV.

9

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

The middle energy interval is: (a) the most interesting one because it corresponds to orbital velocities of artificial Earth satellites; (b) the most complicated one because these energies are comparable with those of chemlsorptmn and lattice bonds; (c) the least accessible for investigation by modern theoretical and experimental techniques.

1.2.2. Formulation of the basic problem The interaction functions hnk the Boltzmannian level with the molecular one. So, to construct them one has to begin from the molecular level, thereupon averaging over parameters which became probabilities on the Boltzmannian scale. An exact formulation of the problems w~thln the framework of classical mechamcs must contain the coupled Newton equations of interacting particles together with the initial condmons for positions and velocities. Interaction forces and lmtial condatmns are the two sources of dlfficult~es The forces are usually defined by central pair potentials Ul# ( I r~ -- r~ I ) depending on the species of interacting particles. Excitation of the internal degrees of freedom means transition to other potential curves. The initial condations are determined by atom arrangement in the lattice and the adsorption layer and by their vibrational motion in the equilibrmm state The imtlal position and velocity of the impinging particle are to be given. Let us consider a structureless particle hitting a surface block containing N atoms The motion of all N + 1 particles is described by the equations N

d 2 ro

~

dUot roi droi rol,

~

dUlj rt~ dr,~ r~j

m dt---T -d 2 r, M, dt---T --

(1.24)

dUio rio drto rio'

(1.25)

jT~l

where

r,j=r,--rj,

r l j = Iro l;

i,j=

1,2 . . . .

,N.

At the initial moment one has ro = {x, + z~o°) tan

01 COS ~01,

to = {-- ul sin 01 cos

q~l, - -

r , = r ~ °),

t=

t i : t ~ °),

y~ + zoo°) tan 0t sin 91,

Ul

sin 01 sin

1,2 . . . . .

N.

~01, - -

U1 COS

zoo°) },

(1.26)

01) ,

(1.27) (1.28)

If the surface temperature Ts = 0, coordinates r~°) are determined by the structure of the lattice and the adsorption layer and the velocities t~°~ ---- 0. The solution of the problem (1.24)-(1.28) gives quite definite values to, r~ at any moment. The transition to the Boltzmannlan level is achieved by averaging over the following parameters:

10

R G. BARANTSEV

1 The aim point coordinates x,, 3'~ uniformly distributed over all the lattice grid 2 The surface atom displacements and velootles (when T, # 0), wah a dlsmbutlon function depending on the atom arrangement and bonding springs. 3 r~,°) and ~01, in case of polycrystals Eventually one finds the distribution densities o f the fluxes o f scattered and sputtered pamcles and the trapping probablhty The interaction engagement distance Z~o°) and the number o f atoms In the block N a r e to be large enough not to affect the result. Introducing dimensionless values it is convement to scale distances, masses and energies with the lattice spacing, the sohd atom mass and the impinging particle energy, accordingly. When all the atoms in this block are the same there are only two potentials, external Uo~ = U, and internal U,j = Uo ones Let each of them contain two parameters energy and spatial ones. Then, for a given lattice structure, the problem contains eight dimensionless parameters: E, ~o ~s tz

---- external potential well depth, = internal potential well depth, = solid wbratlonal energy, -- gas a t o m mass,

a, = ao = 01 = ~/91 =

external potential range, internal potential range, incidence angle, azimuthal incidence angle

1 2.3 Surface structure and mteractlon potentials It is natural to study the atomic structure of solids in the following sequence, perfect crystals, defects, polycrystalhne and a m o r p h o u s states. Along this sequence the deterministic description gradually changes into a statistical one Let us have a look at the basic ~deas about crystal structure within the framework o f a deterministic description. (8~ The crystal lattice Is lnvarmnt under fundamental translations a, b, c taken as unit vectors o f crystallographic axes. Depending on relationships between lengths and d~rectlons o f these vectors seven crystallographic systems are distmgmshed, cubic, trlgonal, tetragonal, hexagonal, rhomblc, monoclinic and triclinic Within the cubic system the umt vectors are orthogonal and o f the same length Taking into account all other kinds of symmetry fourteen different types o f space lattices are numbered in all. The cubic system contains three types: simple, body-centred and face-centred lattices Most of the metals have the body-centred and face-centred cubic lattices. The surface form o f a crystal can vary depending on the orientation of the cleaving plane with respect to crystallographic axes. The crystal plane is characterized by the Miller indices (hkl) defined as the least integers which are proportional to the reverse values of the coordinates of points at which the plane crosses the crystallographic axes. In the cubic system these radices are proportional to the normal direction cosines Sohd atoms are in vibrational thermal motion with amplitudes that are one or two orders smaller than lnteratomic distances at normal temperatures. The distribution o f atoms m space coordinates and velocities according to classical statistical mechamcs is given by the known Boltzmann function To write it one has to know the interaction potentmls and to find the lattice spectral function. In case o f low temperatures the q u a n t u m a p p r o a c h is needed The problem is essentially slmphfied if all the atoms are assumed to be independent harmonic oscillators vibrating with the same frequency (the Einstein model). For most o f solids this

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

ll

frequency is of the order of 1013 Hz. The mean energy of the wbrational motion of a surface atom according to classical mechanics is equal to 3kTs, k being the Boltzmann constant. Modelhng interaction between atomic particles by means of central pair potentmls is an approximation with an applicabihty range not yet determined, c9-11) Empirical potentml curves illustrate the fact that at great distances attractnon dominates and at short ranges repulsion. Of widest use are' the Lennard-Jones potential

o] (1 29) and the Morse potential U(r) = D [e -2~c. . . . ) _ 2e-,C . . . . )]

(1.30)

The former describes adequately the dipole interaction between non-polar particles at great distances. Parameters e, e or D, ~, ro are found empirically. In case of dissimilar particles the potential parameters are found by using combining rules :o2.13) ,,j = ~/(,, ,j),

or,, = (or, + %)/2

or

= ~/(~, %).

(1.31)

When the interaction is collective the potentials are added together. As a result the potential well depth for each atom inside a metal lattice, identified w~th the bond energy Eo, reaches the value of several electron volts, ca) On the surface it is naturally somewhat less. The dipole attraction energy for a gas particle summed over the semi-infinite lattice decays like r - a , that is much slower than r - 6 . Besides (1.29) and (1.30) other potential models are also used In the hmlting cases the s~mphfied versions are suitable For instance, when kinetic energy is so high that attraction can be neglected the simplest model used is a hard sphere having a single space parameter. And for slight deviations from equlhbrmm a part of the potential near the well is approximated by a parabola, having only one energy parameter specifying the bonding spring. 1.2.4. Modelling on intermediate levels The gas-surface lnteracUon problem as stated on the molecular level includes a number of microparameters (e ,, co, %/z, a., ao, 01, etc.), too numerous to make a snmllarlty with respect to them practically feasible in aerodynamics. On the other hand, the avadable aerodynamic similarity macroparameters are not usually sufficient for proper correlation of experimental data m a rarefied gas. Fixing the Mach and Knudsen numbers and a temperature factor does not determine the aerodynamic functions exactly enough. Straggling proves to be unacceptably large even for Co. It means then that there exist factors not taken into account, which ought to be introduced into aerodynamic considerations and controlled. Thus, there is a gap between the level on which snmllarlty parameters are introduced and the level where they become manifest in aerodynamics. To bridge the gap one ought to consider intermediate levels of description and try to begin modelling there. The parameters that appear in these models wdl be closer to the aerodynamic ones. Three intermediate levels can be distinguished between the molecular and aerodynamic ones: Boltzmannlan, moment and local gas-dynamic ones. On the first level one has to model the interaction functions instead of solwng problems of molecular collisions for some models of interaction potentials; on the second o n e - - s o m e of their moments; on the third

1_3

R. G

BARANTSEV

one--the local gas-dynamic quantities. In other words, we have to deal with an a priori approximation of dependences on velocities after collision, those before collision and coordinates, m consecutive order Thus, the functions of many variables are gradually reduced to those of a smaller number of arguments, in the long r u n - - t o parameters With approachlng the gas-dynamic level of modelling one gains in simplicity but accuracy is lost. The problem is to find the level of modelhng on which an acceptable compromise is achieved Experience in the use of different models will enable us to judge them and make a choice. Of great importance in doing so is testing models by means of the regression analysis of the data of physical and computational experiments. Further on, the factorial analysis will be needed to minimize the number of parameters and to isolate the most essential ones

1_3 Experimental Results 1 3 l. Low and middle energies

The main techniques in experimental studies of gas-surface interaction on the Boltzmannlan level are the ones involving the molecular beam. A number of experiments with beams determining the scattering function V (ul, u) for various gases and surfaces has been carried out. The scattering lndicatnx, that is the directional distribution of the scattered particle flux, was mainly measured, V,o (Ul, 01 " 0, ~) = ~ V (111, u) u 2 du. i) o

(1.32)

Most attention was paid to V,o in the incidence plane. Experimental evidence in low energy beam scattering is already rich enough to state certain general trends: 1. At thermal velocities atomic particle scattering from solid surfaces tends to be diffuse and depends weakly on surface and gas species. Under these circumstances trapping with following re-emission takes place, rather than reflection, the surface being covered by a considerable adsorption layer. 2. Deviation from the diffuse scattering at thermal velocities can happen with increase of the incidence angle 01, decrease of the gas particle mass m, surface cleanliness, decrease of the adsorption energy Eo, increase of the surface temperature Ts, and decrease of roughness. 3 As the incidence velocity rises the scattering ceases being diffuse In most cases this occurs at energies as low as ~ 10 -1 eV. The scattering function deviation from the diffuse one is not generally a linear combination of diffuse and specular ones. 4. The non-diffuse scattering indlcatrix V,~ is characterized by a more or less sharp peak. The maximum of V,., for inert gases on comparatively clean surfaces is nearer to the specular direction than to the normal ( I 0,, 01 I < Or,) and more often lies above the specular ray The difference 01 -- 0,, increases when 01 approaches 90 ° and decreases with rising E1 and diminishing m and T~, the maximum becoming more prominent. Adsorption energy E~ makes the peak smoother and diversely affects its position. Three trends of research can be distinguished in recent years: 1. Expansion of the range of investigated objects, phenomena, parameters at low energies observing high purity standards

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

13

2. Measurement of scattering particle velocity. 3. Research m the middle energy range. Information of the research on metal surfaces up to 1971 is gathered and systematized in ref. 7 (see also ref. 14). On cleaned surfaces at moderate energies the scattering function V (u, 0, q~) has as a rule one maximum with respect to each of the arguments and is quahtatively specified by the maximum position urn, 0,, and by its width ~u, %, %. The parameters u,,, au are considered e~ther with a fixed direction or as averaged over all the &rections; the parameters 0,,, ~0, % are usually averaged over the velocities. The dependence of these scattering characteristics on 01, El, Ts,/z, Eo, etc., is the subject of research. At low energies, with an increase in 01 and El, the values of 0,, and u,, increase and the dispersions decrease; as a rule 0m < 01, % < % < %. With an increase in Ts the value of 0,, decreases, u,, increases (for Ts < TO and %, a u can behave diversely. The function um (0) decreases more often, the relation of u,, to u~ and u~ varies In ref. 15 the scattering function was resolved into diffuse and d~rected components, the latter preserving the tangential momentum. H2- and D2-scatterlng with dissocmtion, recombination and desorption was observed in ref 16, the angle-dependence of adsorbed, desorbed and reflected particles was found to be interconnected by a law simdar to the Kirchhoff law in optics. With approaching the middle energies the character of the E~-dependence changes. The difference ~ = 0,, -- 01 becomes positive, achieves maximum at a certain E* and then decreases to zero. The quantity E* depends on 01 so that E* cos 01 is nearly constant. The value of this parameter for argon on silver equals ~ 0.2 eV tl 7~ The function a n (El) changes into an increasing instead of a decreasing one. The same transformation happens to u,, (0). The character of the dependence of 0,,, %, u,. on 01 holds. The existence of the two qualitatively dafferent scattering regimes, thermal and structural, was antiopated by Oman t2°) and observed in refs. 18, 17. A comparison between experimental and numerical results carried out in the second regime in refs. 17-19 may be recognized as hopeful, In the first regime a certain agreement between experimental data and the so-called cube theory results was observed (better for soft cubes t22~ than for hard ones t21)) but only a quahtative one and far from being complete. 1.3.2. High energtes Applying to the aerodynamically tugh velocities we find ourselves in the field of energies, which in many other branches of physics are considered low The bulk of data on ionic bombardment of surfaces accumulated in physics until now is concerned with energies of the order of 102-104 eV. t2a-26) As distinct from the range of near-thermal velocities the set of phenomena is essentially expanded here, and the Interaction structure becomes more complicated. Let us list the main interaction functions which are relevant in collisions of positive ions with a solid surface: V++, V~, V °, S+, scattering in the form of positive ions (reflection), negative ions, neutral atoms and trapping In any form; W++, W+, W g_, W_~, W~_, W~_, sputtering of positive and negative ions, neutral atoms, electrons, foreign adsorbed particles and surface atoms The most Investigated phenomena are: V + + W + ---- T + (Ul, u), secondary ion-ion emission,

14

R G

BARANTSEV

W~ (ul, u), secondary ion-electron emission, W~ (uj, u), surface sputtering by ions_ Uniting scattering and sputtering ions into one group is due to the difficulties of their experimental separation. For high surface temperatures It is also difficult to separate secondary Ions from the auto-emitting ones Ion-neutral emission has been studied less because of the difficulties of neutral particle detection The investigation of new phenomena naturally begins with the determination of energy thresholds: E~, E5 and others, at which these phenomena become appreciable. At first the total characteristics are found, for example, T+ (ul) - K, secondary ion emission yield, W.~ (u 1) ~ 7, secondary electron emission yield, W~ (ul) - N, surface sputtering yield Then the angular and energy distributions of emerging particles are delved into, l.e the interaction functions proper. All the quantities are studied as dependent on the impinging beam energy E~, particle mass m, incidence angle 01, lomzatlon potential V,, surface temperature Ts, surface composition and state. The V,-dependence resolves the phenomena into two essentially different groups. bombardment by alkaline ions, the ionization potential of which is of the order of exit work (,,~ 4-5 eV), and bombardment by inert gas ions, the ionization potential of which (12-25 eV) exceeds the exit work ~vmore than twice Thus, the secondary ion emission T +, which is the main subject of investigation with alkaline ions, in the case of inert gases is almost absent because neutrahzatlon occurs with high probability. The secondary electron emission mechanism of these groups is also absolutely different. When V~ > 29 there can be potential electron escape owing to the energy of Ion neutralization even for low velocities (of thermal order). When V, < 2~0 the escape occurs owing to the kinetic energy of the ion. The kinetic electron emission thresholds are numbered by hundreds of electron volts Nitrogen and oxygen, by the Ionization potential, are close to argon and xenon When a surface is bombarded by positive gas ions of the energies of 10~-103 eV there can occur neutralization, secondary emission of electrons, atoms and negative ions, trapping of the particles and sputtering of the surface It is established that almost all the impinging positive Ions are neutrahzed. For example, with He +, Ne +, Ar + on clean W, Mo, Si (100) and contaminated W, Hf, Ge(III), Hagstrom ~27) has obtained T2 ( u l ) = 4 10 . 4 - 2 ", 10 .3 irrespective of El. With overwhelming probability the Auger neutralization occurs, m which two surface electrons take part. One of them descends to the ground level of the atom, the other is excited (in particular, emerges). Most of the neutralization energy is spent on surface heating. For velocities like u~ ~ 10 km/sec this energy can exceed the kinetic one. The secondary electron emission yield 7 with all tested targets and singly charged ions for E1 < 1 keV is almost independent of E~, which is indicative of the potential mechanism of electron escape. The value of 7 does not practically depend on m, 01 (up to E1 ~ 102 eV), T, (on clean surfaces) Near-linear dependence on V~ ~ is observed With q, decreasing from 5 to 4eV, 7 changes in the interval of 0 20-0.25 for He + and 0.06~0 12 for Ar + For the nitrogen and oxygen ions, 7 ~ 0.1 in case of metals and 0.03~3.06 in case of semiconductors.

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

15

When E1 > 1 keV, ~, begins to rise owing to the kinetic emission. Velocity distribution of the electrons ~s near equilibrium with the mean energy of the order of a few electron volts. Energy thresholds of sputtering metal surfaces by inert gas ions for normal incidence are within the interval of E1 = 20°50 eV. In this range the sputtering yield is N ~ 10-s-10 -4. The behavlour of energy thresholds for metals depending on the atomic weights has a periodic character. The values of E s are roughly equal to the quadrupled heat of sublimation. The minimum sputtering thresholds are those of such metals as Pb, Au, Ag, Cu, the maximum ones are those of Ta, Zr, W. The dependence on the gas atom mass and the surface temperature is weak. The sputtering yield with an increase in the beam energy rises up to saturation at E1 104-10 s eV. For instance, with Ar ÷ on Ag N ~ 1 at E~ = 200eV and N ~ 10 at E1 > 20 keV. Mass dependence of N in near-threshold range is not monotonous. With an increase in the incidence angle a maximum is reached at 01 ~ 60°-80 ° where N is 5-20 times more than for the normal incidence. The angular dismbutlon of sputtered atoms for the normal incidence with increasing E1 changes from a subcosine to a supracosine one. A predominant forward sputtering for the oblique incidence has been observed only in the near-threshold range. Mean energies of sputtered particles are within the interval from a few to 102 eV. The energy distribution is almost unlnvestigated. There have been data only for charged particle sputtering, the thresholds of which are higher An appreciable penetration only begins at E1 > 102 eV. On tungsten the trapping probability of Ne ÷, Ar ÷, Kr ÷ equals 10 -2 at E1 ~ 70, 150, 250 eV respectively. (28) The formation of negative ions can happen for a strong affinity of the gas atom with an electron, for example, in the case of oxygen. The neutral component of secondary emission is almost unlnvestigated. There are some reasons to believe (see ref. 23, §8.2) that T~_ (Ul, u) by its properties is much like T + (Ul, Ill) of alkaline ions, which has been studied in sufficient detail. In particular, one ought to expect multlmodel angular and energy distributions. But direct observations are just only beginning. A series of experiments has been carried out by Devlenne and his collaborators (29. 30.3) on the interaction of a neutralized argon beam with metal and glass surfaces at E1 = 102-104 eV. In case of aluminium the average relative energies E,,/E1 of particles in the incidence plane for 01 ---- 60 ° and E1 = 2 5 keV (u~ ~ I00 km/sec) equal 0 25, 0.28, 0.40, 0.46 at exit angles 0 = 0 °, 30 °, 60 °, 80 °. Out of the incidence plane for the latitude 0 = 60 ° at azimuthal angles 25 °, 50 °, 75 ° E,,/Ea = 0.29, 0.28, 0.24 respectively. The velocity distribution for all 0 is stook-shaped with a slower decay towards larger velocities. For 0 = 60 ° the velocities are within the interval of 40-100 km/sec with u,, = 63.4 km/sec. The dependence of exit energy on E1 is given in ref. 30 In the specular direction for 0~ = 60 °, E,,/E1 diminishes from 0.6 to 0.23 within the interval E1 = 0.1-0.6 keV, then it gradually increases and at E1 > 2 keV keeps on the level of 0.44. Of some Interest is the indlcatrix evolution in the incidence plane with increasing E~ For 100 eV there is one sharp specular peak; for 200 eV a near-normal maximum appears, which for 400 eV already overgrows the specular one; at about 600 eV the specular maximum vamshes but a subspecular one ( ~ 70 °) appears; this one becomes predominant by 2 keV. So the transform structure at high energies is rather complicated. At energies E~ ~ 100102 eV the angular and energy distributions have only been obtained for the alkaline ions so far. (26'31) For gas atoms and ions the force characteristics of interaction have been measured (2,32) The tangential momentum transfer ~- rises with

16

R G. BARANTSEV

increasing/~ and E j, the normal momentum transfer p dlmimshes with increasing tz and can depend differently on El Accommodation coefficients or1, cr2 diminish as 01 rises.

1.4. Computattonal Results 1 4.1 Evolutton o f computational expertments The basic problem stated in Section 1.2.2 contains a lot of functions and parameters: ri (t); r (°), i"(°), i = 0, 1, .. , N , xa, Ya, q~l, e,, %, ~, ix, a,, ao, 01 Without additional assumptions it is not amenable to an analytic solution. The formulation of analytic methods is preceded by studying this multidimensional problem by means of high-speed computers. First of all, one-dimensional versions of the problem were considered. Those reduce the number of functions, equations and initial conditions three-fold, rule out the need of averaging over aim point coordinates xa, >'~ and azimuthal angle ~01 and rule out the incidence angle Ol. True, the problem degenerates somewhat the scattering lndicatrix and the tangential momentum transfer disappear But the velocity distribution and the normal momentum and energy transfer remain. There remains a possibility of studying these variables dependlng on depths ~,, Co and ranges a,, ao of external and internal potentials, on vibrational energy es and on relative mass/~. There remains a posslblhty of considering the important question of the number N of atoms in the block of effective interaction Moreover, the one-dimensional models enable us to make progress in studying such factors as thermal lattice vibrations, surface impurities, internal degrees of freedom of gas particles. The methodological importance of one-dimensional models is quite large Transition to spatial lattices was accomphshed by a gradual expansion of the statement of the problem and inclusion of the parameters which have been absent in the one-dimensional models. The appearance of models which to some extent take into account spatial effects, remaining In their essence still one-dimensional, was characteristic of the first stage. A general feature of such works was that the collisions were considered to happen along the centre hne and a gas particle was beheved to interact with a single lattice atom only. With all their insufficiency these models enabled us to make clear a number of important regularities and to develop the calculation technique. An essential point of the method ensuring success in solution of the problem with a semiinfinite harmonic atom chain was constructing the response function to an elementary perturbation of the end atom_ F O. Goodman (aaa) constructed the response functions of multidimensional semi-infinite simple cube lattices to the perturbation applied to one of the surface atoms along the normal which made it possible to introduce a new important parameter into the problem, the lattice dimension True, the very statement of the problem basically still retained one-dimensional character: a gas particle directly interacted with a single surface atom which it hit along the centre line Parameters Xa, Y~, ~1, 01 and functions V,o, ~- were absent as before The dependence of S and ~ on/~, Co, E,, a,, ~s was mainly studied An approximate way of taking into account the inchned and off-centre collisions was proposed in ref. 33b. The solution of the problem of atom scattering from crystals in the multidimensional formulation was started five years ago (A. I. Erofeev, J A. Ryjov, D S. Strijenov, R. A_ Oman et al.). Most of the authors used the model of independent harmonic oscillators (the Einstein lattice) for which in eqs (1.25) only one term of the sum over j remained and

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

17

was hnear. The first series of numerical results enabled us to draw the following qualitative conclusions for middle energies: 1. The dependence on target parameters xo, yo is strong and for averaging over the lattice spacing a significant number ( ~ 102) of trajectories is required, especially for evaluating V ( ~ 103). 2. The scattering indicatnx V,, has a near-specular maximum that can be either above or below the specular ray. The indicatrix width dimmtshes with increasing 0~ and ~. 3. The lattice structure type, the surface crystallographic plane, parameters ~1, E~, ~o, ao, a . (in real intervals) affect the m o m e n t u m and energy transfer comparatively weakly. The main parameters are t~, 01, ~ ,. 4. W~th an increase in t~ the normal m o m e n t u m transfer p decreases and the energy transfer q increases; with an increase in 01 coefficients p and q decrease. Curve q (El) has a characteristic minimum. 5. With a decrease in tz the number N of atoms in the block of effective interaction approaches unity. This happens the more rapidly, the smaller E. is and the larger 01 IS,

Numerical calculations of scattermg on crystals were carried out also for high energies. For instance, the scattering functions were evaluated for copper and argon atoms on the side (100) of a copper monocrystal in ref. 34 using a scheme of successive pair collisions wtth the potential U(r) = A exp ( - - r/b). For Cu ~ Cu El = 3 keV, A = 22.6 keV, b = 0.196 A; for Ar ~ Cu E1 = 2.2 keV, A = 16.3 keV, b = 0.196/~. For incidence in the plane (110) at angles of 01 = 0 °, 45 °, 70 ° the reflection y~eld for copper equals 0.02, 0 08, 0.63; for argon, 0.04, 0.15, 0.70 respectively. The angular distribution, roughly speaking, ~s near-specular; the fine structure is complicated, dependent on 01, assocmted w~th the side form. The energy d~stribut~on has several maxima, two main ones at about E = 0.8 E~.

1.4.2. Recent computational results Now we shall review briefly fundamental papers published during the last three years. F O. G o o d m a n ~33b) pointed out two possible reasons for disagreement between numerical results obtained in 1967 and the experimental ones. Firstly, the interaction potential depth taken in the calculaUons seems to be considerably overestimated. Secondly, the particles wluch have not left the surface after the first collisions must not be considered as trapped. Many of them leave after some hopping. In ref. 20 the numerical calculations of scattering on the Einstein lattice are continued. The interaction between gas atoms and surface atoms is described by the Lennard-Jones potential. Parameters ~,, a,, ~o,/~ are taken accordingly to Ne, Ar, Xe on Ag (111); E1 = 10- 2-10 eV, 01 = 50 °, Ts = 560°K. Decreasing E, brings the results nearer to the experimental ones. Two qualitatively d~fferent interaction regimes have been found: thermal scattering, at low energies, and structural scattering, at middle ones. With a transition from one regime to the other the trends m El-dependence of 0mand ~0 are reverted; energy transfer regularities still remain. In the first regime with an increase in E1 the trapping probabdity S decreases, 0m increases, % decreases. The maximum re-emission velocity um ts gained at 0 ~ 0 for T~ < Ts, at 0 ~ 01 for T1 ~ Ts and at 0 ~ 90 ° when the structure scattering dominates. For Ne the structural scattering starts at El ~ 0.2 eV, for Ar it does at E1 ~ 1.7 eV. In general the transition to the second regime is encouraged by dlmimshing parameters c,, E~, a,.

]8

R G. BARANTSEV

In the series ~3s~ the scattering on the three-dimensional Einstein lattice is also considered but with the Morse external potential this time The p a r a m e t e r s are taken for He on NI (111) a n d (100), E1 = 10-2-1 eV, 01 ---- 0 - 7 0 ' , T~ ~ 0-1200°K The scattering function m ref. 35a has two lobes which s m o o t h d o w n with increasing E1 a n d Ts a n d decreasing 01. When T, rises, the lndlcatrlx on the whole moves t o w a r d s the normal. The q~-dlsperslon is small, especially for high El a n d 01 ~- 90 c In ref 35b an a d s o r p t i o n layer o f a r g o n or oxygen a t o m s b o n d e d with the lattice a t o m s by the L e n n a r d - J o n e s potential is added_ The energy transfer q increases with decreasing a d a t o m mass a n d a d a t o m - l a t t l c e b o n d i n g energy. The m a x i m u m value of q is g a m e d when the layer ~s half-packed, back- a n d off-plane-scattering being appreciable. The existence o f the thermal a n d structural scattering regimes has been confirmed in ref 35c In the first regime Vo, depends weakly on the lmplnglng a t o m d l s t r J b u t l o n f ( u ~ ) W i t h an increase in E~ the value o f 0,, becomes m o r e than 01 but then a p p r o a c h e s 0~ again with dispersion diminishing steadily. F o r T~ > 600°K the scattering function weakly depends on T~ The velocity distribution is b l m o d a l for high E1 a n d more sensitive to f ( u l ) . The influence o f a n h a r m o m o t y o f a three-dimensional lattice on the exchange coefficients is studted in ref. 36. The M o r s e or L e n n a r d - J o n e s potentials are used with p a r a m e t e r s for N2 on Fe, M o , W; E~ 5-40 eV, T~ ~ 0 Interaction t~me ~-, rises with increasing t~ a n d decreasing El A n h a r m o n l c l t y s o m e w h a t increases r . a n d the a c c o m m o d a t i o n coefficients along with it, however, the effective interaction block size does not increase, Dependence on the external potential is stronger In ref 37 the L e n n a r d - J o n e s potentials are used with the p a r a m e t e r s for A r on A r a n d A g (111) a n d the scattering lndicatrix Is calculated, EL = 0_2-13 eV, 0~ ~ 30-60 ~, T~ = 0 F o r 01 ~ 30-45 ° back-scattering is observed W i t h an increase in E1 o r 0~ the value o f 0,, a p p r o a c h e s 01 from above, ~ (0~) diminishes, % ~ 60 ° at 0~ 30 ° and % -~ 20-30 ° at 0~ = 60 °. Scattering function a n d a c c o m m o d a t i o n coefficients calculations are a c c o m p l i s h e d m ref. 38 on a h a r d sphere square a r r a y with a s m o o t h e d attractive field E , P a r a m e t e r s E,, a , , t~, 01, ~1 are v a n e d The particles whose energy after colhslon is larger than E , but the energy o f m o t i o n along the n o r m a l is less than E , are referred to as s e m i - t r a p p e d ones M o s t o f the semit r a p p e d particles eventually escape Their c o n t r i b u t i o n to the a c c o m m o d a t i o n coeffioents turns o u t to be noticeable. The energy a n d n o r m a l m o m e n t u m a c c o m m o d a t i o n coefficients increase with increasing ~ a n d ~., diminish with increasing 01 a n d a , The tangential m o m e n t u m a c c o m m o d a t i o n coefficient increases slightly with t~, increases with ~,, increases strongly with a , a n d depends weakly on 01. The scattering function is structural Non-single scattered particles m a k e a lobe which is rather p r o m i n e n t in some interval o f a , The s e m i - t r a p p e d particles affect httle the d~smbution f o r m as they escape nearly diffusely. W h e n 0~ grows the dispersion diminishes and the p e a k a p p r o a c h e s the surface. W h e n c, decreases the peaks become higher and move t o w a r d s the specular ray, in the range o f 0 ~< ~. ~< 0.5, the d i s p l a c e m e n t being a p p r o x i m a t e l y 15 ~o/, F o r E, = 0 there is also a n o t h e r p e a k at 80 ° which d i s a p p e a r s by s e m i - t r a p p i n g already for E, ~ 0.05. W~th an increase i n / ~ the peak leans t o w a r d s the surface a n d the ind~catrlx becomes sharper a n d higher. The a . - d e p e n d e n c e is n o t m o n o t o n i c . W h e n a , increases three regimes can be distinguished A t first the multiple colhsions are n u m e r o u s b u t the scattering is nearly a diffuse one a n d so the structure is almost invisible. Then the n u m b e r o f multiple colhslons decreases

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

19

and they are strongly focused in the incidence plane. With a. further rising both multiplicity and structure disappear, and the distribution narrows. The velocity spectrum generally has two characteristic bands corresponding to single and multiple colhslons, the latter being higher than the former The average velocity grows monotonically with the increase of the reflection angle. A comparison with experiment enables us to find the best-fitting values of E°. These turn out to be one order smaller than those obtained by summing the Lennard-Jones potentials. This is in accordance with the conclusions of papers 33c and 20. For low energies (E~ = 0 065 eV) a good agreement between computational t3s~ and experimental ~ag) data has been achieved in ref. 35d

2. S P H E R E A T O M L A T T I C E R E F L E C T I O N 2.1. Surface Model 2.1. l Pair interactmn The nature of interaction between an atomic particle and a solid surface essentially depends on the ratio of the effective interaction time ~ ° the the average period of the lattice vibrations ~'o ~ 10 -la see If ~-. ~ ro the lattice affects the gas particle as a whole. On the contrary for ~-. ~ ro the particle really interacts with one or two lattice atoms only. In the first case a continuum model of solid is appropriate, in the second one a model of free or bound atom is vahd. These observations and corresponding estimates were made as far back as 1938/42) The effective interaction time mainly depends on the incidence energy E1 and the gas-to-surface atom mass ratio/~. For low energies the estimates of ~. lead to continuum models. The scattering and sputtering theories for energies E1 > 102 eV rest upon the inequality E1 >> Eo where Eo is the lattice bond energy ( ~ 1-10 eV) Under these conditions the lattice atoms can be treated as free and the interaction can be considered as a sequence of pa~r collisions. ~23-26~ For ~ < 1 such a scheme seems to be statable up to E~ ~ Eo because when approaching Eo penetration does not happen practacally and single reflection dominates. Quantitative compansons carried out on the basis of experimental (31~ and computational t2~ data confirm well this assumptton at El ~ 10--102 eV. For ~ >t 1 collisions are mainly multiple and with each of them the gas particle energy ~s lost, so that the pairwise character of interaction breaks down at higher energies ExFor the middle energies, E~ ~ 1-10 eV, m the case of/~ < 1 the free atom model is still rather more valid than a contmuum model. Indeed, let us estimate the effective lnteractmn time for a single head-on collision by formula ~-° = 2b/u~, b being the coefficient in the exponentml repulsion potential U(r) ~ exp (-- r/b). C42~Supposing b ----0.25 × 10 -B cm, for ux = 10 km/sec we have r . / % = 0.05 In fact this value proves to be overestimated. According to the computational experiment data (2°~ ~-,/% is small enough even for E~ ~ 1 eV. In ref. 44b the free atom model is justified for/z = 0.3 at all the incidence angles and for/~ = 0 5 at 0~ > 30 °. It may be stated that at the middle energies for/z and E, of the order of 10 -~ reflection practically takes place really from the free surface atoms. Diminishing one of these parameters allows the increase of admissible values of the other. For example, at ~. ~ 10- 2 the values of/~ may be raised up to 0.5 approximately. More accurate estimates of the limits of the pair interaction range can be found by means of a systematic asymptotic analysis of the problem for small z,/ro An attempt of this kind

20

R. G. BARANTSEV

was made in ref. 44c with a slmphfied model. The normal head-on collision of a gas particle with a simple cubic lattice of elastically coupled atoms is considered The external interaction is described by the exponential repulsion potential. An asymptotic formula for the energy accommodation coefficient and an asymptotic estimatxon for the validity of the single collision assumption have been obtained In ref. 45 the asymptotic study is accomplished with two parameters Besides T,/~o the tangential-to-normal momentum exchange coefficient ratio r/p is also assumed as small The scattering from the Einstein lattice with the Morse external potential at E1 -~ 1-10 eV is considered Explicit asymptotic formulas for the scattering lndlcatNx and the tangential momentum exchange have been obtained 2.1.2 The form of the interactmn potential The choice of the interaction potential is of primary significance in the palr-colhslon theory When energies are very high the interaction is reduced to the Coulomb repulsion of nuclei. At comparatively low energies when electronic shells are deformed insignificantly the hard sphere model is usually used, the radii of which are assumed to depend on E1 In the intermediate interval electronic shells are already crushed but still exert a shielding influence. The interval limits for ions of various gases bombarding Sliver and copper targets are evaluated in ref 24 For example, in the case of O + - C u the range of weak shielding is 29keV < El < 5.39 , 103keVOn condition that significant overlapping of the electronic shell occurs, Flrsov (461 has obtained the following comparatively simple form of the shielding potential for the intermediate interval of Ej

U(r)

z, z2 e" r

X

(zl/2 4 z,/2)2. 3

(2 1)

where a = 0 468 /~,, z~e and z2e are the nucleus charges, x is the Thomas-Fermi funcuon (tabulated). The hard sphere model with a constant cross-section produces a scattering yield that does not depend on E1 which is not consistent with experimental data. For E1 ,,~ 1-30 keV it is shown (2L26'47) that the pair interaction theory using the Flrsov potential makes it possible to interpret all the main regularities of atom or ion reflection from solid surfaces including the fine structure of energy and angular scattering spectrums on monocrystals. For E1 "~ 102 eV within the framework of the pair collision theory various ways have been proposed to interpret experimental data (a) An approximate account of the bond energy by introducing the increased mass of target atoms, the so-called "effective" mass. (b) Taking into account the El-dependence of the collision cross-section. (c) Extending the Flrsov potential to energies below 1 keV Each of these ways is not flawless: hard spheres with any masses and radii provide a contradiction to experiment; lsotropic scattering m the centre-of-mass system and the Firsov potential for Ei < 1 keV is without theoretical foundation. We do not have yet any sufficiently rehable and simple potential model for E~ ~ 102 In the pair interaction theory one has to use potentmls of sufficiently short range because the distances between consecutive collisions are small. At the middle energies, first of all,

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

21

the potentmls of zero range, Le. barriers, are tested. Various b a m e r forms chosen are. sphere, t48~ cube, t2~) trapezium, t2'3) The improvements were also carried out in d~fferent ways. However, nowhere has the search for a proper potentml led to a generally acceptable model so far. Conformably to rarefied gas aerodynamics conditions a hard sphere lattice reflection theory has been proposed and developed in ref 49b. In ref 50 the momentum accommodatnon coefficients are calculated m the two-dimensional case for ~ = 0.

2.1.3. Model description The interaction potential between gas and surface atoms, for a first approximation, will be assumed to be the impenetrable sphere, the effective radius of whsch generally may depend on the encounter velocity and the incidence angle. For E1 < 102 eV th~s radius must be large enough not to permit an apprecmble penetration into the surface. Under such conditions the single reflection of a gas particle from a free surface atom is most probable, the second collisions with nelghbourmg surface atoms are possible and the higher multiplicity collisions are hardly probable. As for the small tz, the energy loss is not large, the second collisions may be assumed as pairwise too, even if at other effective radfi. The influence of the ftrst colhsion on the second surface atom state before the impact will be neglected. Then the gas particle future does not depend on the relaxation process of energy transfer Into the bulk. The separation of the reflection problem from the more complicated remaining part of interaction is a significant asset of the model considered. This step permits the non-consideration of the potential of solid atom interaction with each other and the structure of tuner lattice layers in solwng the problem There remains the external layer structure and five dimensionless parameters:/~, % a,, 0~, ~pl Let us consider the triangular array of atoms in the surface plane correspondang to a (111)-face of a face-centred cubic lattice or a (100)-face of a hexagonal lattice. In the first case the triangle side is l = d/x~2 where d is the lattice spacing I f gas and solid atoms are regarded as hard spheres of a and A radn respectively and / (but not d) xs chosen as the unit of length, then

aq-A

a , -- ~

lq-cr

-- ao " - - - ~

(2.2)

where 2A ao = -f-,

a cr = ~ .

(2.3)

In case of a dense sphere packing in the surface layer ao=

1,

a,----(1 + a ) / 2 .

(2.4)

However, one can do w~th one space parameter a , without assuming the dense packing. To rule out penetration a. > l/v/3 (2.5) must be satisfied. The dependence of a , on the velocity and the incidence angle is assumed to be weak, the solid atom vibration energy is small as compared to El. Reserving the minor factors for subsequent corrections, as the main variant, the single reflection at E, = 0, a o = const, will

22

R G BARANTSEV

be considered The shadowing effect pertaining to grazing angles will be treated from the very beginning. Simultaneously simplified results corresponding to the shadowing neglected will be written. Effect o f E, has been evaluated m refs 49c and 7

2_2. Scattermg Function 2.2 1. Single reflection Let (b, t, n) be the Cartesmn system centred on a target surface atom. The ax~s n is directed along the outward surface normal to the area ds. The impinging particle velocity ua is parallel to the plane (t, n), u~, ~< 0. The centres of six netghbourlng surface atoms and six next neighbours are arranged at the points o f coordinates bk tk = ½[(x/(3) + 1)

cos( kw) 1)k (~/(3) -- l)]sm ~pt +--~- ,

(

k = 1, 2,

., 12.

(2.6)

The angle ~ E [0, 7r/3] specifies the latnce o n e n t a t m n with respect to the plane (u, ua). Even values of k correspond to the nearest atoms, odd ones to the next to nearest Drawing spheres of radius a . and centres at (&, tk, 0) one has a surface on whmh gas particle centres occur at encounter moments. Under condition (2.5) this surface has no hole and consists of spherical caps whose lines of intersection project on the plane (b, t) like a h o n e y c o m b o f regular hexagons. As the leaving velocity u on ds can be attained as a result o f a number o f collismns the scattering function V (u~, u) can be written in the form

V=

V~ + Vz + . .

+ V,,,

(2 7)

V,. being the scattering function o f m-fold reflected parncles Integration over all u, > 0 results in 1 = N1 -? N2 + . . ÷- Nm, being the fraction o f m-fold reflected particles When N~ is near one, one can neglect the difference between V,,/N,, (m > 1) and V~/N~ supposing V =

(2.8)

V 1 / N 1.

In the remaining part of the section I/1 will be considered in detail Taking that during the pair interaction o f a gas particle with a surface a t o m there ~s no ambmnt a t o m influence, it is natural to proceed as in the two-body problem Let us introduce, in the velocity space u, a spherical coordinate system (R, a, ~b) centred at uc = u~ t'/ (I + tz), axis a = 0 directed opposite Ul. For an elastic impact the velocity vector end remains on the sphere R = ua/(1 + tz) Hence (49~)

1

V~ = ~

8 (R -- Ra) V1,~c

(2 9)

K7 where

R

=

I U --

U c

I,

R 1 = Ul/(l + t0,

a -- < ( u -- uc,

u~),

(2 10)

Vlo,o being the scattering function in directions from the point uc- It is clear that one ought to put down I"1,oc = 0 for all the directions in which the single scattering is geometrically impossible. A n d for all possible directions of single scattering, (a, ~b) ~ oJc, in the case o f hard sphere potential one has Vl,oc = const (49a) The value o f this constant is to be determined

SOME P R O B L E M S OF GAS---SOLID S U R F A C E I N T E R A C T I O N

23

from the normahzatlon condition: integration of VI~,o over all possible direcUons of scattering by the central atom must result in one. But the integration over the solid angle oJc results in N1. Further dealing with sohd angles centred at uc becomes inconvement. Therefore we introduce the spherical coordinates centred at the atom: (r, a, 8) for r = {b, t, n } and (u, 0, 9) for u = {ub, u . u. }, so that 8, r sin a sin 8, r cos a},

r :-

{ r s i n cz c o s

u=

{ u s l n 0 c o s 9, u sin 0 sin tp, u cos 0}.

(2.11) (2.12)

The impinging particle velocity now can be written in the form ul = {0, --u, sin 01, --ul cos 01}

(2 13)

where 01 = < (-- ul, n) is the incidence angle Let us link the scattering direction (8, 4') with the impact point r , (a., a, fl)- For that it is convenient to introduce an auxlhary coordinate system by rotation of (b, t, n) through the angle 01 about the axis b. In the rotated system the spherical angles (al,/31) can be simply expressed in terms of (o, 4'), 4'=fll,

~ = 2a 1.

(2.14)

Thus,

fa

c=/sin

dOa4':4/cos%sln,

: 4 f cos

sm

,d,

la31

as : a f cos

(2.15)

where cos al = cos 01 cos ~ + s m

01 sin ~ sin 3-

(2.16)

The solid angle coc in the system (0, 4') corresponds to a solid angle f~ m the system (a, 3) cutting out a single collision domain on the spherical cap. This domain boundary will be referred to as the single reflection contour. Impacts can happen all over the cap part, that is to say, visible from the impinging particle side. Integral (2.15) over the corresponding solid angle ~)~ can be simply connected with the visible cap part projection onto the plane orthogonal to ul. The same projection area falls on every cell's lot and it can be evaluated in a simpler way by projecting onto the mentioned plane the hexagonal cells base from the plane (b, t). As this hexagon area equals x/(3)/2 then 4 f cos al df~ -- a,42 x/3 2 cos 01. flvis

Hence a2 Vlo,o -- 2 ~/3 cos 01'

(2.17)

The solid angle w filled by vectors u (u, 0, ~0) m the velocity space for the single reflection will

24

R

G

BARANTSEV

be referred to as the single reflection cone. A full expression o f Vt c a n n o w be w r i t t e n m the form 1/1=

u22~/3.cos01

1 +t~ul,

--1

,u~,

\

(2 18)

O, u ~ o J

It r e m a i n s to find the b o u n d a r y o f o~,

(2.19)

~, (0, ~ ) = 0, c o n n e c t e d , n a t u r a l l y , to the single reflection c o n t o u r ,

r (~, ¢~)

=

0

(2.20)

These e q u a t i o n s i n c l u d e f o u r p a r a m e t e r s : tz, a , , 01, ~ol. 2.2.2.

Single reflection boundary

I n o r d e r to d e t e r m i n e the f u n c t m n s _P a n d ), let us first find a r e l a t t o n b e t w e e n the d i r e c t i o n (0, 9) o f reflected partmles a n d the i m p a c t p o i n t (~,/3). Let u,, u l , a n d u,, u l , be the c o m p o n e n t s o f velocities u, ul a l o n g a n d across the c e n t r e line. F o r a n elastic i m p a c t Ur =

U l r , Ur -~- U l r (~tL - -

1)/(/~ + 1). H e n c e

~--1 U ~ - Ur +

Ur - -

~+l

~--I Ulr +

Ulr =

AS

then

Ulr --

--

~+l

2 Utr +

at

--ulr

=

ul

- - - -

I+/~

Ulr

U 1 COS (Z1 r , / a t ,

u = ul + ~

2

r,

ul cos al a,--'

(2 21)

T h e p r o j e c t i o n s o n the axes b, t, n are 2 ' [ s m 0 cos ~ - u~ l +~ //_

sin ~ cos ~ cos a l ,

-u- sin 0 sin ~p - - - -2 sm ~ sin ~ cos ~1 - - sin 01, ux 1 +/z u 2 - - cos 0 - - - uI 1 +/z

(2.22)

COS ~x COS ~z1 - - COS 01,

and u~

(1 + t~) 2 c°s2 ~1

(2 23)

cos el is expressed t h r o u g h a,/3, 01 by f o r m u l a (2.11). T h u s , COB 0

cos ~ cos a l - - (1 + t~)/2 cos 01 ~/{[(1 + ~-)/2] 2 - - ~z cos 2 ~t } ' sin ~ sm ~ cos ~l - - (1 + t~)/2 sin 01

t a n cp =

s i n ~ COS ~ COS ~1

(2.24)

(2.25)

25

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

Now we have to construct the contour P. The single reflection domain is first of all within the spherical cap f2o, the boundary equation of which is -Po (a, fl; ~x, a.) -- 2a, sm a cos (fl -- ~t) -- 1 = 0; I/~ -

~1 I ~< g,

~

/~ +

= ~ (~).

(2.26)

A part f21 of this cap can be shaded by nelghbouring atoms. The falling shadow boundary will be denoted by _r'~. A locus in the visible part of the cap after impact when the velocity vector u is tangent to one of the neighbourmg caps will be referred to as the second impact boundary and denoted by/'2- This part of the contour separates the single reflection domain from the multiple reflection domain O2. Separating also the own shadow domain f2' 1 with the boundary P'l we have f~ = f~o -- f~l -- f2'1 -- f~2-

(2.27)

But because/"1 is always below /'2 the contour P in fact consists of pieces of Po,/'1 and /'2.

Let us now find r'j and _r'2 taking into account only twelve neighbourmg atoms with centres at the points rk = {b~, tk, 0}, k = 1, 2 . . . . . 12. The falling shadow boundary corresponds to the vector ut to be tangent at one of the neighbouring caps. IfBk is a tangent point on the atom of centre O, and A is an impact point on the central atom then the triangle ABkOk is rectangular and Irk -- r , ]2 = al + [(rk -- r , ) . ul/ul] 2.

(2.28)

From that the part F1 ~k~ of the shadow boundary cast by the kth atom is described by the equaUon /'~k~(~,fl;

~o1,01 , a . ) - - -

a. c o s a l - - r k s m O 1sin

-- [r2 -- 2 rk ak s,n a cos (fl -- q~1-- k-~)]

~1 +

=0.

(2.29)

Similar arguments with the vector u result m an equation like (2.28) for the second impact boundary. Expressing there u/u by means of (2 22)-(2,23) we obtain the equation of the part /'2 ~k) of the second impact boundary for the kth atom

/'~k~(a, fl; q~l, 01, a.,t~)= {COSa~ [--rksin~cos (fl--q~l - - 6 ) + ½(l --/~)a.] +~(l+t~)rksln01sin(~o+~)}2--

[(1~-------~)2

-- /z cosZ al] [r2 -- 2rk a, sin a cos (fl -- q,l -- k~)] = O.

(2.30)

Given the parameters (~01, 01,/z, a,), the domain f2 is formed as a minimum domain cut out by the lines _r't
(2.31)

26

R

G

BARANTSEV

The b o u n d a r y ~, of the single reflection cone w is formed by vectors u reflected from points of the contour P Substituting expressions (2.31) into the right of (2 24), (2 25) we obtain parametric equatmns o f ~,, O:O~(fl,

~ , 0 1 , ~ 1 , a.),

q~ = ~v~ (fi;

qvl, 01, t~l, a.).

(2 32)

Without taking account o f shadowing and second colhs~ons the reflection contour can be s~mply determined f r o m (2 26). The function cLr appearing there changes periodically between ~m,. ---- arc sin [1/(2a.)] and a,.~. = arc sin [1/(v'(3)a.)], the ratio sin am../sm am,. = 2/~/3 ~ 1, 15 being independent o f a.. The averaging over ~01 turns the hexagon into a circle with constant at- Determlmng its radius so as to conserve the projection area on the plane (b, t) we have ~r = ~* =-- arc sin

.

(2.33)

Averaging over ~1 in (2.31), (2.32) is proper for polycrystalline surfaces. 2.2.3.

Scattering indicatrtx

Let us find the direction distribution density o f singly reflected particles

Zlo,= f vl u2au. 0

Inserting here the expression (2.13) for Vt witlun the cone o~ we obtain

(1 + ~ ) 2 a ~ 2v/(3) cos 01

f 8[X(v,$,~)]v 2 d r 0

where X (v, ~b,/~)

1 ul

~

U

/~ 1+~

-

v=u/ul, 4 = ~ (0, 9, 01)=

I11

I ,

1 1+~

<(u, ul).

The function X (v, 4,/~) vanishes when v = Vo (¢,, ~) -_= ~

1

(~ cos ¢ + v/(l -- t~2 sin' 4)}.

l+g

With

8v

= V=L o

v/(1 _ p2 sin 2 ~b)

(2.34)

SOME PROBLEMS OF G A S - S O L I D SURFACE I N T E R A C T I O N

27

we finally have Vl,o =

a,2 {tz cos ~b + a/(l - - / j , 2 sin 2 ~b)}2 2~/(3) cos 01 ~/(1 --/~2 sin 2 ~b)

(2.35)

Each reflection direction has its own velocity ratio U/Ul = vo (4, Iz) by (2.34). Both Vo and Vl,o depend on (0, ~0) through the declination angle only. When ~ diminishes, these quantities increase and the larger ~, the steeper the curves. At/~ ---- 0 a .2

vo = 1,

1:1,o= 2x/(3)cos 01"

(2.36)

It must be noted that eq. (2.34) follows only from the momentum and energy conservation laws in pair collision and so it is valid for any interaction potential, but eq. (2.35) is due to the hard sphere potential.

2.3. Momentum and Energy Exchange Let us evaluate the average over ~01 momentum and energy exchange coefficients using the formulas (2.8) and (2.18) for V (Ul, u). At first we find the partial exchange coefficients then the full ones for the Maxwelhan &stnbution of impinging particles.

2.3.1. Partial exchange coefficients The dimensionless momentum and energy exchange coefficients defined by (1.7) are expressed with no trapping, sputtering and emission by the scattering function in the form (1.10). After averaging over ~x eq. (1.11) also holds. Taking t- and n-projections of p and using besides (2.12), (2.13) the eqs. (2.22), (2.23) together with the normalization of V we obtain ~'(uO

-

2 cos 01 f V(ul, u) sin 0` sin/3 cos 0-1 du, 1+~

-

-

f

?(uO = 2 c o s 01

1+~

V(III, n ) c o s 0` c o s 0`1 dli,

(2.37)

uj>O

q(nl) = 4~ cos 01 f (I + ~)~

v<.. u)

cos

u~>O

The angles a,/3, 0`1 are related to 0, w by (2.16), (2.24), (2.25). Let V = VI/N1, I:1 determined by (2.18) and N1

= f V1 du.

(2.38)

~a> 0

For the 8-function in (2.18) let us turn, in the velocity space u, to the spherical coordinates (R, o, 4) centred at Ul/z/(1 + tL) again. Then the integrals over R in (2.37), (2.38) become

28

R

G. BARANTSEV

trlv]al The solid angle dwc sin odod6 as shown in (2.15) is connected to the sohd angle d ~ = sin adadfl by a simple relationship dcoc 4 cos ~ df2] All that results in N I ( O ] , tx, a . ) - -

V'(3)2aZ" cos 01

f

cos ~t d~),

(2.39)

f~

7"(01, ]/,, a.)

=

N1 ~/(3) (1 + tz)

4a.

4a.

p(01, tz, a.) = N1 ~/(3) (1 + / ~ )

f

sin a sin/3 cos 2 al dr2,

(2 40)

f

cos a cos z al df~,

(2.41)

f~

8/z (1 a,~ -r/z) z f c°sS ctl dO q(01, tz, a.) : NI a/(3)

(2 42)

f~

where cos at = cos 01 cos ~ + sin 01 sm a sin/3, ~ = {0 ~< ct ~< ar (/3; 01,/z, a . ) , 0 ~< /3 < 2 zr}. The integration over a can be fulfilled analytically, but over/3 generally numerically. In case o f n o r m a l incidence w h e n 01 : 0, ~r : const, we have N 1 :

2qT 2 -~ a, sin 2 ctr,

1- :

0,

(2.43) p--

1 ( 2 - - s i n 2czr), 1 +/~

2/~ - (2 -- sin 2 a r ) . (1 + t z ) 2

q=

o.:~ I0

o9

1

~

~

07 NI

06

:l

/

~

i

05 04

,

03

Oj °

rO

08

Ni

/,~

__ _.....~

~

07 06 ~ 05

15

30

45

01* FIG

2

60

70

80

29

SOME PROBLEMS OF G A S - S O L I D SURFACE I N T E R A C T I O N

,u.=o 03

O2 =0

a,:l

b L : ~~

0

Y

~

\

\ 50

6O

90 0

50

0o

60

9O

0o FIG

3

Some results of numerical integration in (2.39-42) using data of Section 2.2 for a r are given in Figs. 2-5. With an increase in ~ for all the incidence angles, p increases and q decreases. Decreasing a , mainly affects increasing ~. I f neglecting the shadowing and the second colhsions we take a r = a , by (2.33) the integration in (2.39-42) can be carried through resulting in the following simple formulas: NI=

1,

sin 2 ~.

.

~--- - - s m O l c o s 0 1 , l+tz 1

P = 2(1 + ~)

[sin 2 a* + cos 2 G (4 -- 3 sin 2 a,)],

/~cos01 [3sin 2 a , + c o s 2 0I ( 4 q -- (1 + ~)-------"~

(2.44)

5sin 2ct,)].

A comparison between these approximate formulas and the more exact ones (2.39-42) will be given in Section 2.4.1. 2.3.2. Full exchange coefficients If the distribution function of impinging particlesf(ul) is known the full m o m e n t u m and energy exchange coefficients are expressed according to (1.23) m terms of the partial ones as follows

1

P = n

;

f(ul) p(u,)

a

aul,

(2.45)

i dux,

(2.46)

U1

Utn
I

q=n

f(ul)

LIra< 0

30

R.G. BARANTSEV

15

// _ I

=.=3

"F=0 I

°.= I

I0

I/ P

,%

x. k

05

0

60

30

90

0

3O

o7

60

90

8=*

FIG 4

n and U1 being the number density and the average velocity o f the impmglng flow. In the coordinate system connected to n and U1 we have UI = {0, -- U1 sin 0o, -- UI cos 0o} Let

f(ul) = n

(2.47)

exp [ - - h (ul -- UI)2].

(2.48)

O6 =I

a,=l

I q

q 04

,

02

0

.

%.

30

60

90~

30

60

s,°

s, FIG 5

90

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

31

We calculate the full exchange coefficients with no account of the shadowing and the second collisions. Substituting (2.48) and (2.44) into (2 45), (2.46) results in s i n 2 tz,

~"

(1 + /z) s 2 Io11,

1

P -- 2(1 + tz) S2 [sin2 eL, (/2oo + lo2o) + 2(2 -- sin 2 a,) Ioo2], q --

(2.49)

--/z [3 sin 2 a , (12ol + lo21) + 2 (2 -- sm 2 a.) Ioo3], (1 + / z ) 2 s 3

where

if

I~jk = -~.3/2 -

v~ v / ~ exp [-- (v -- s) 2] dr,

(2.50)

Vn
v = ~v/hul, s = x/hUl, s = J ~ M ,

(2.51)

M being the M a c h number. It is useful to have a table of the integrals (2.50). F o r i,./, k = O, 1, 2, 3, i + j + k <<. 3 they are --1 Iooo = ½(1 + erfz), Ii~k = laoo = 0, lool ' - - 2V'~r X(z), s sin 0 0 Iolo = -- ½ s sin 0 0 (1 + erfz), loll = - X(z),

2~/~

12oo = J:(1 + erfz), lo2o = ¼(1 + e r f z ) (1 + 2 s 2 sin 2 0o), z loo2=2~rr

[

.

]

X(z)+-~z

(1 + e r f z )

1

,/21o=

--I ~ s s i n 0 o(1 + e r f z ) ,

--1

I2ol = -- - X(z),/o21 -X(Z) (1 + 2s 2 sin 2 0o), 4x/~r 4X/~r

io12_zssinOo[

2X/~r

X~rr

X(z) + ~

]

(1 + e r f z ) ,

Io3o = -- ¼ s s i n 0o (1 + e r f z ) (3 + 2 s 2 sin 2 0o),

(2.52)

--1 Ioo3 = ~ [(1 + z 2) x(z) + ½ zx/rr (1 + erfz)]. Here and henceforth X (z) = exp ( - - z 2) + V'~rz (1 + erf z),

(2.53) exp ( - - x 2) dx, z = s cos 0o.

erf z = - ~ 0

32

R G BARANTSEV

Making use o f the above we obtain ~-(8o, s:/~, a,)

sin g a . sin 0o 1 - - / z 2sx/rr X(z),

1 [sin 2 a, p(Oo, s,/~, a,) -- 2(1 ÷ / z ) ~ [ ~ (l ÷ erf z) (1 ÷ s 2 sm 2 0o)

(2

c°s0° [

- - s i n 2 ct.) ~

~J~(1

X(Z) q- 2z

q(Oo, s, ~, a,) = (1 + i~)z s3x/~r

-k e r f z )

]},

sm 2 a , X(z) (1 + s 2 sin 2 0o)

+ (2 -- sin E a,) [(1 ÷ z 2) x(z) + l z ~ / r r (1 + erfz)]}.

12.54)

F o r s --~ oo these formulas are of course reduced to (2.44) with 0 l = 0o. 2 4. Esttmation and Refinement 2.4.1 Shadowing and multiple colhstons The main results m Section 2 3 have been obtained for the single reflection with due regard for shadowing; the second and next collisions are approximately lmphed by (2.8). In the simplified variant (2.33), (2.44) shadowing and multiple collisions are completely left out of account. The difference between these two variants is due to both mentioned factors, the shadowing becoming apparent at grazing angles only. Let us estimate the influence o f each o f these correcting effects on the exchange coefficients. At first we qualitatively examine the shadowing role leawng multiple collisions out of account. Consider a vertical section of gas particle centre locus at impact m o m e n t s (Fig. 6). Neglecting shadowing we in fact include in the integration over f~ the part A B of its own shadow and the part C D o f the falling shadow. In the former case cos al and sin/3 are negative; m the latter they are positive. As can be seen from (2 40-42) the inclusion o f A B raises p and reduces • and q but the inclusion of C D raises all three coefficients. The own shadow is larger than the falling one but m it Icos aa[ Is smaller Hence the correctmns to ~- and q f r o m A B and C D are to some extent mutually compensated and neglecting shadowing must reveal itself mainly m increasing p F o r instance, in the case of 01 = 90 ° f r o m (2.44) for ~ ---- 0 25 we have p = 0 11 at a , -- 1 and p = 0.20 at a . = 0.75 instead o f p=0, Thus the shadowing affects ¢ and q insignificantly and also p comparatively slightly. For taking it into account it is reasonable to apply a simpler model than in Section 2.2. However,

FIG. 6

33

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

one o u g h t to d o it after the multiple colllsmns whose effect is stronger have been a d e q u a t e l y t a k e n into account. TABLE 1 MULTIPLE COLLISIONEFFECT

(/~ = 0 25, a, = 0.75) 01° q-

0

by (2.44) by (2.40)

15

30

45

0 098 0 167

0 170 0 268

0 196 0.268

by (2.44) by (2.41)

121 1.36

1 14 1.28

0.96 1 08

0.70 0.80

by (2.44) by (2 42)

0.483 0.540

0 451 0 506

0.365 0.420

0 254 0.304

As can be seen f r o m Table 1, the a p p r o x i m a t e a c c o u n t o f multiple collisions b y (2.8) increases significantly the exchange coefficients, p a r t i c u l a r l y r. F o r a m o r e a c c u r a t e estimate one o u g h t to distinguish distributions o f particles after one o r a few collisions. L e t us calculate, for example, the velocity gained with ~b-deflection after two collisions. Let 61 e [0, 6] be the angle o f deflection after the first collision. Similar to (2.34) we o b t a i n u - - - - v0(¢, 61, ~') ul

1 b ' c o s ¢1 (1 + ~)2 -

+ ~/(1 - - t~2 sin 2 4'i)} [~ cos (6 - - 61) + w/( I - - t~2 s m 2 (6 - - ~bl)}].

(2.55)

The d e p e n d e n c e Vo (61) l o o k s like a symmetric arch with the t o p at the p o i n t 61 = 6[2. So it m a k e s sense to consider

Vo(6, ~b/2,~) Vo,, =

Vo(6, 0,/~)

~ cos 6/2 + x/(1 - - tz2 sin 2 ~b/2)]2 -----(1 ÷ t~) ~ cos ~b + x/(1 --/~2 san 2 ~b)]"

(2.56)

TABLE 2. MAXIMUM VELOCITY RATIO AFTER TWO COLLISIONS

~o

30

60

90

120

150

/~ = 0.25

1 02

1.06

1.11

1.14

1.I1

t~=05

1.04

I 16

I 27

1.35

1.26

180

VOm

A s c a n be seen f r o m T a b l e 2 the velocity o f reflected particles in a given direction increases slightly with collision n u m b e r . Some results at 01 ---- 0 have been o b t a i n e d in ref. 53 for V2o.

2.4.2.

Numerical calculations

The degree o f accuracy o f the a p p r o x i m a t e analytic t h e o r y o f single reflection can be e s t i m a t e d b y m e a n s o f c o m p u t e r calculations o f multiple reflection f r o m hard sphere

34

R G BARANTSEV

lattices Such calculations have been carried out by F. O. G o o d m a n C51) wxth square a n d t r m n g u l a r latUces for F < 1 a n d a n u m b e r o f values o f 01, ~Pl, a,In the m a r e calculation series o f the scattering f u n c u o n the lattice ~s square, ~01 - - 20 r, the p a r a m e t e r s / z a n d a , are t a k e n for inert gases on tungsten a n d mckel. The u-space ~s divided into 3600 cells and in every case 22,500 trajectories are calculated for aiming p o i n t averaging. The results represented m ref 51a are for hehum, argon a n d xenon on tungsten wtth c o r r e s p o n d i n g values o f (/z, a , ) = (00218, 1 0), (0217, 1.1), (0.714, 1.2) at 0~ = 0, 45 °, 67.5 °. Some selected secUons ~0 = const a n d 0 = const o f the scattering indtcatrix a n d

0 28

I

~q~

~=0 a,=O 9 sq IQthce

0 20 --

14 12 I0 8

012/ 008 O04 0

15

30

45

60

7,5

90

81° FIG. 7

also dispersion plots o f velocity m a g m t u d e d i s t r i b u t i o n are shown In addatlon to the conclusions t h a t have been included into Section 1.4.1 we note that all m a i n properties o f the scattering function are c o n t a i n e d in (2 18). In other words, the a p p r o x i m a t e theory developed a b o v e proves to be quahtatively a d e q u a t e even in case o f X e - W . Calculations o f the a c c o m m o d a t i o n coefficients cry, cr2, ~ have been carried o u t for ---- 0.001, 0.1 (0.2) 0.9, a , = 0.9 (0.1) 1.3, 01 = 0 (22.5 °) 67 5 °, ~ = 0 (15 °) 45 ° with the square lattice and ~ol = 0 (10 °) 30 ° with the triangular one. A full set o f the results has been p u b l i s h e d m ref 51b on thirty-six figures. F o u r plots gwen m ref. 51b illustrate the dependence o f or1, ~r2, ~ on each o f the p a r a m e t e r s /~, a , , 01, cpa at the fixed values: a . ~ 0.9, 01 = 67.5 °, ~0~ = 0, t~ = 0.1 for a square lattice. The plots o f T (01), p (01), q (0a) on Fig. 7

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

35

are re-evaluated from fig. 21 o f ref. 51b by the formulas (1.17) at his = oo. F o r comparing them to those obtained in Section 2.3 the slight influence o f lattice type and ~1 can be taken into account t h r o u g h an effective a ,. On a square lattice under condation a , > 1V'2

(2.57)

the gas particle target at impact consists o f spherical caps with quadrangle bases. Averaging over q~1 conserving the projection area on a horizontal plane changes the quadrangle c o n t o u r /" into a circle with constant polar angle a r = a , - arc sin

.

(2.58)

Triangular and square lattices under condition (2 57) will be referred to as equivalent if the values of a , for them are the same. It follows from (2.33) and (2.58) that the parameters a , of such lattices are related by a, =

a7 ~ 0 93 a,D

(2.59)

so that for a ,a = 0.9, a ,~ ~ 0.84. Correlation between a ° and ~1 is more complicated because it depends essentially on 01. On a square lattice increasing ~1 f r o m 0 to 45 ° at an oblique incidence corresponds to decreasing a°. Hence at ~1 = 0 the effective a , must be slightly above the average. With a° = 0.875 the approximate values of the exchange coefficients (2.40)-(2.42) at ~ = 0.1 prove to be quite close to the exact ones. The plots o f p (01) and q (01) practically coincide with those on Fig. 7 and the values o f ~- at all 01 prove to be overestimated by not more than one unit o f the second figure. Thus, the approximate account o f multiple colhslons by (2.8) at ~ = 0.1 is justified with a high accuracy. The range o f the external potential decreases with an increase in El, the c o n d m o n (1.5) or (2.57) is violated and a penetration o f gas atoms into the lattice bulk occurs. Some computer calculations at E1 ---- 1 keY, ~ = 1 with a , depending on E1 have been carried out in ref. 52. Scattering together with penetration for a , = const has been also considered in ref. 44a.

2.4.3.

Soft-sphere lattice

A main deficiency o f the hard sphere model consists in the potential steepness which does not provide a proper direction distribution o f scattered particles. The scattering indicatrix is very susceptible to even slight changes o f the potential barrier inclination and form. One o f the p r i m a r y tasks is finding the simplest adequate potential model within the framework o f pair interaction theory. Keeping the spherical symmetry we now try a slightly inclined rather than a vertical barrier that makes spheres somewhat pliant, or soft. Such use o f the term "soft", in our opinion, is more natural than in ref. 22 with respect to cubes with an a t t r a c u o n well. The smallness o f the inchnation v makes it possible to stay within the framework o f pairinteraction theory. We shall find an asymptotic solution o f the problem for small v and/L. Here we limit ourselves to the case o f normal incidence (01 = 0) and averaging over azimuthal angle ~01 reduces the problem to a two-dimensional one in a radial section o f the

36

R. G. BARANTSEV

spherical cap a n d the s u r r o u n d i n g tore (Fig_ 8). T h e h o r i z o n t a l area per a t o m is e q u a l to •rp~. F o r a t r i a n g u l a r lattice p . - - (~'(3)/2zr) ~, for a s q u a r e o n e p . --~ (1/~) ÷. T h e a i m i n g d i s t a n c e p = a . sin ~ c h a n g e s w i t h i n the i n t e r v a l [0, p . = a . sin a.] F r o m the p o t e n t i a l U (r) the a s y m p t o t i c s o l u t i o n c o n t a i n s o n l y o n e p a r a m e t e r

,.={ <, n

(2.60)

Ul U

O.

B

p

0

b

2p. FIG 8

S o l v i n g the first c o l l i s i o n p r o b l e m u n d e r the initial conchtlons rl=

{a. s l n a , a . c o s a ) ,

r2=

{0,0),

ui =

{0, -

u2 =

{0, 0 } .

1},

(2.61)

w i t h i n 0 (t~) a n d 0 (v) we o b t a i n the i n t e r a c t i o n t i m e tk = 4v a . cos a,

(2.62)

the exit p o i n t c o o r d i n a t e s rk = a . , sin ek = sin a (1 + 4v cos 2 a),

(2.63)

the exit veloc]ty u = 1 - - 2 ff cos 2 ~,

(2.64)

sin 0 ---- sin 2a [1 + (t~ + 2v) cos 2a]. T h e single reflection d o m a i n is l i m i t e d b y p = pi = a . sin ~1 for w h i c h the exit r a y either j u s t leaves the p o i n t A or is t a n g e n t to the tore (Fig. 8). Let us find t h a t v a l u e a . = ak. for w h i c h ak = ~ . a n d 0 = rr/2 - - ~ , s i m u l t a n e o u s l y . F r o m (2.63) a n d (2.64) we o b t a i n sm ~k* = ½ (1 + v - - / # 2 ) ,

(2.65)

SOME PROBLEMS OF G A S - S O L I D SURFACE I N T E R A C T I O N

37

and correspondingly sin a I . :

½ (I --

2v --/~/2).

(2.66)

F o r a , ~< ak° the value o f ~1 can be f o u n d f r o m (2.63) in the f o r m sm aa = sin a . (1 - - 4v cos 2 a.), 0 ~< a . ~< a~.

(2.67)

I n o r d e r to d e t e r m i n e ax for a . >t a k . we a p p l y the relationships B C = a . = B D + D C , B D = 2 p . cos 0, D C = a . sin (0 - - ak) leading to the e q u a t i o n

a , = 2 p . cos 0 + a . sin (0 - - a~).

(2.68)

Inserting here a, (a~) f r o m (2.63) and 0 (a~) f r o m (2.64) we have the following a s y m p t o t i c solution sin a 1 = Ao (1 - - A,, ~ - - A~,v), ak. ~< a . ~< ~r/2,

(2.69)

where Ao :

1 + (1 - - 16 sin a . + 32 sin 2 ~,)* 8 sin a . 2[Ao(l

A.

=

A~ =

+2sina.)--

(8Aosma,--

,

(2.70)

I] 1)

,

(2.71)

16Ao (1 - - Ao2) sin a ,

(8 Ao s m

~,

(2.72)

-- 1)

The single reflection cone b o u n d a r y for all cz. ~ [0, ,r/2] IS d e t e r m i n e d t h r o u g h aa b y sm Ok = sin 2 a l [1 + (/L + 2 0 cos 2a~]. The single scattering function for 0 ~< 0 ~< 0~ is VI=(1

+/~)2V1,~c 8

u--1

(2.73)

i}

+/zul

- - ~ 1+/~

(2.74)

'

where unlike (2.17) a ,2 V~o, - 4rr p2 (1 + 4v u COS 0).

(2.75)

F o l l o w i n g (2.8) we p u t V = V 1 / N I where N1 = sin 2 a l / s i n 2 a , .

(2.76)

Then V=

(1 + 2~) (1--4vucosO) 4~r sin 2 a t

Vu2 du =

3

u

1 - - 2 (/~ + 2v) cos 8 4 r r Sift 2 (xI

-~---~--ux/ 1 +~

0~<0~<0~.

l} 1+/~'

(2.77)

(2.78)

38

R.G.

BARANTSEV

Table 3 gxves al, 01, Nx for a number of ~, at t~ ~ 0.1, v = 0 and 0.1 Figure 9 shows the influence of v on the scattering lnd,catrlx. TABLE 3 (tL -- 0 1 )

lff

20 °

30'

40 °

50 °

60 ~

GI

v~0 v~0.1

10 ° 6o10 •

20 ° 12o50 ,

30 c 20030 •

34°10 , 26~30 '

35"50' 28o50 ,

36~50 ' 30 °

0~

v=0 v=O1

22' 15°50 ,

43°50 • 33°15 '

65o30 , 53 °30'

74"30" 70°20 '

78 ~20' 78 °30'

80' 4 0 ' 85 °

Nx

v=0 v=01

1

1

1

0.418

0.762 0 480

0 586

0 375

0.479 0.334

0 490

0 396

As can be seen from the table the potential barrier mchnatlon slgmficantly reduces the single reflection fraction N1. So we need to calculate V2. That has been done in ref. 53. 2 4.4.

Attraction and adsorption

Modelling of atoms by hard spheres splits a many-parUcle interaction problem into a sequence of binary collisions. Within the framework of this theory the coupling of interaction can be taken into account only indirectly.

p.=O I o

o

o

OI

0

I0

20

30

40

,50

O* FIG 9

60

70

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

39

One possiblhty is to Introduce an effective surface atom mass M o f f > 1 designed to take into account atom bonds in the lattice. This approach was not successful. The reason is that other atoms affect attraction rather than repulsion In other words coupling is first and foremost by attraction. H a r d sphere lattice reflection taking account of many-particle attraction has been considered in ref. 54. When a particle enters the attraction field of relative depth ~ = Eo/(rnu~/2) its normal velocity rises up to ~/(ul,2 + ~). Then the lattice colhslon occurs which results in obtaining a velocity u, with a probability density V (ul~, u,). If u, 2 > ~ the particle leaves the attraction field, its normal velocity being a/(uo 2 -- E,), otherwise trapping (or diffuse scattering) takes place As duo, = u, (u 2 + ~o)- ~ du, one can express the scattering function on a lattice with an attraction field directly through that of the lattice without the field by Un

V~ (ul, u) = ~/(u2, 4- ~) V{ul~, ul,, ~/(u~2, 4- ~o); u~, u,, ~/(u 2 -t- E,)}.

(2.79)

In ref. 54 on the basis of (2.79) the calculations of p and q have been accomplished for a nitrogen atom scattering from tungsten at ul = 5-15 km/sec, 01 = 30 ° and'45 °, E, = 1 eV and 2 eV. The existence of adsorption and desorptlon considerably complicates the interaction problem. The scattenng and sputtering functions are connected to the distribution functions in the ambient gas through the filhng of the adsorption layer. The kinetic equation describing this coupling has been obtained and studied by B. V. Fillppov (1", b) Including some adsorption, desorptlon and dislodging models (also see P. J Pagnl and J. C. Keck(2)). Macromodels of molecular beam interaction with a relaxing adsorption layer have been considered in ref. ld, pp. 30~0. An analytic study of hard sphere lattice scattering m the presence of an adsorption layer has been accomplished in ref. 55. Gas atoms and adatoms are attracted to the surface by the square-well potential whose depth has a periodic variation along the surface. Single and double collisions, scattering and dislodging are considered; E1----1-10eV, T~ = 0. Analytic formulas for V~, W,o, u,, have been obtained On a clean surface 0m increases and % decreases as tz and 0~ increase and ~o decreases. The function u,, (0) rises monotonically. An increase in ~o moves the plots higher but does not change their qualities. I f t~ decreases, um increases. W~th a decrease in 01 the value of u,, becomes smaller and more uniform in 0. On a contaminated surface the periodicity of the potential results in a multllobal sputtering function. The results of calculations made for argon on silver covered by water molecules are in qualitative agreement with the experimental data obtained at the middle range of energies on surfaces not cleaned quite well (ref. 2a, pp. 253-68). 3. S C A T T E R I N G F U N C T I O N

MODELLING

3.1. Review of Available Models

3.1.1. Specular-diffuse reflectton For the purely specular reflection V = 8 (u - - ul + 2 ul,). The diffuse reflection with full accommodation is defined by the scattering function (1.13).

40

R. G. BARANTSEV

N o w let cr particles be reflected diffusely and I cr specularly We also assume that the &ffusely emerging particle temperature Te m a y differ f r o m the surface temperature (not full accommodation). Then V ( u l , u) :

2hnz (I -- o) 8 (u -- ul + 2ul.) + ~ - - u. exp ( - - ha U 2)

(3.1)

7T

where DI h e --

2k Td'

Evaluating the m o m e n t u m and energy exchange coefficients (1.10) we obtain ~- :

cr sin 01 cos 01,

p=(2--cr)

cos 201 + ~

q=crcos01(1--h-~a

cos0a,

(3 2)

).

where

hie = ha u~

(3 3)

A comparison o f (3.2) with (1.17) for ha = h~ gwes Crl = o2 = ~ = ~, that is for full temperature a c c o m m o d a t i o n every a c c o m m o d a t i o n coefficient equals the fraction of diffusely reflected particles. If hi :~ hs according to (1.12) we have p - _ p+ ~1 : o r , or2 : o r - P P~

q - _ q~~ : ~ - q -- q+

(3.4)

where p+ = ½

cos 017

q+ = ~

cos 01 ;

(3.5)

the remaining terms are explained in (1.14-15). The specular--daffuse reflection scheme contains two dimensionless parameters, ~ and hie, that m a y depend on ul and surface properties. The &ffuse coefficient cr has been defined so that it m a y change within the interval [0.1] only; the limiting values correspond to the purely specular and purely diffuse reflection cases. The p a r a m e t e r hid m a y generally take any positive value. When his :~ 2 instead of hla one can use the energy a c c o m m o d a t i o n coeffioent = o

1 -- 2/hla

(3.6)

1 -- 2/h1~"

F o r hla = 2 the energy flux of the particles reflected &ffusely equals the impinging energy flux. I f 2 ~ hie <<,his then 0 ~< a ~< ~. The specular--diffuse reflection scheme appeared during the first years o f molecular physics. The daffuse coefficient g was introduced by Maxwell, the a c c o m m o d a t i o n coefficient by Smoluchowski and Knudsen. These parameters served to take account of

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

41

velocity slip and temperature j u m p effects under near equlhbrium conditions. At small deviations f r o m t h e r m o d y n a m i c eqmlibrium cr and a are close to 1. Under essentially non-equilibrium condltlons of hypersonic rarefied gas aerodynamics the function (3 1) obviously cannot express all the variety o f possiblhtles in distribution of reflected particles. Even for description o f m o m e n t u m and energy exchange the n u m b e r of parameters in this model is not sufficient. However, due to lack of knowledge of real reflection laws the specular-d~ffuse scheme dominated in aerodynamics for a long time thus exaggerating its role. It has been to a certain extent an obstacle in the way o f studies on gas-surface interaction physics and led to some disorienting conclusions. Lately the situation has been corrected but bad consequences of the habit of using the specular-diffuse model linger on.

3.1.2. Generahzed versions One of the first competitors with the specular-diffuse scheme was a reflection model p r o p o s e d in ref. 57. According to that model the scattering occurs into a cone o f semi-angle flo, the axis being forward-directed at an angle 0,, connected with the incidence angle 01 by sin 0,, = (sin 01)v. Inside the cone the scattering indicatnx is in proportion to cos (rr/2 fl//3o),/3 is the angle between a reflected particle velocity vector and the cone axis. The relative velocity magnitude is connected to the energy a c c o m m o d a t i o n coefficient, a, and the parameters v and flo to the m o m e n t u m ones, or1 and a2. F o r v = 1,/30 = 0, ~ = 0 we have the specular reflection, for v = ~ , / 3 0 = ~r/2, a = 1 the diffuse one. Although that model has one p a r a m e t e r more than the specular-diffuse one the complication is formal and the f o r m chosen is not very convenient to be used. In other words, the task of compromising simphclty and accuracy has not been optimally solved. Therefore the Schamberg model has not been widely adopted. A n o t h e r generahzation of the specular-diffuse scheme undertaken in ref. 58 as directed to making the diffuse part o f the scattering function (3.1) more accurate an the case when = o (uO. Let 1 -- cr particles reflect specularly and cr by a law independent o f U1, that is

V (Ul,

U)

=

[1 -- cr (ul)]8 (u -- u I

Jr- 2 ul,)

+ ~ (ul) V. (u).

(3.7)

The function V. (u) can be determined assuming (3.7) to expand to an equihbrium state for which the b o u n d a r y condition (1.1) is as follows: u, exp ( - - hs u z) = -- ~ ul, V(ul, u) exp ( - - hs u21) dUl.

(3.8)

Uln
( -- f ul, uln
e -n'":

O'(Ul)du I )-1 ,

(3.9)

Thus, in the case of a variable cr the non-correlated part of the scattering function (3.7) differs f r o m the diffuse one by the factor a (u -- 2 u,). Denotingf~ (u) = const exp (--h,u 2) f r o m the b o u n d a r y condition (1.1) by (3.7), (3.9) we obtain the relation f(u) --f(u -- 2ux.) = o(u -- 2u.) [f~(u) --f(u -- 2u.)],

u. > O,

(3.10)

42

R. G . B A R A N F S E V

which Is similar to the K r o o k model of the Boltzmann equation. The coefficient cr (ut) Is a measure of surface a c c o m m o d a t i o n for a gas particle of velocity u~ In ref 58a cr (ul) is approximated by e(ul) = exp ( - - h I /,/2) _+_ ~ c [1

exp (-- h2 UlZ)]

(3. I 1)

corresponding to a typical dependence of the energy a c c o m m o d a t i o n coefficient on the incidence velocity. It has been shown that the constants hi, cry, h2 are sufficient for a good agreement with the experimental data by Thomas. (za~ For a further refinement o f this scheme the Maxwelllan distribution (see Section 3.2) has been introduced into (3.7) instead of the specular c o m p o n e n t Th~s results in V -- ~(1 -- ,~2) exp

{h

(1 -- ~2------~[u~ + (u,

)

~ ut,) 21 [(1 -- e) 8(u~ + ut,)

+ cr 2h, u. exp ( - - h~ u,2)] The parameters cr and ,c are connected to the tangentml and normal m o m e n t u m a c c o m m o dation coefficients. Representations o f V as sums of the &ffuse and Maxwelhan distributions have been also considered in refs. 59 and 60; different dispersions in every velocity c o m p o n e n t are assumed. In ref. 60 the scattering lndlcatrix has nine parameters which are estimated by means o f regression analysis on the basis of available experimental data.

3.1 3. Ray model Experimental and computational data (Section 1.3 4) show that in m a n y cases the reflected particle distribution is distinctly concentrated The scattering mdicatrix m a x i m u m gets particularly sharp for grazing incidence angles. The velocity magnitude distribution in the principal direction is also peaked With increasing E1 there occur several lobes of the lndicatrix with a characteristic velocity for each. To model such a reflection it is reasonable to apply the ray method approximating the scattering function by means of a single or several delta functions. In case of a single maxim u m this model has been suggested in ref. 61 as a hmitlng case of the Maxwelhan scattering function. It gives the main asymptotic term for a small dispersion of the distribution Changing the reflection velocities by their mean value we put (3 12)

V ( u l , u) - - 8[u - - u ~ ( u l ) ] ,

which provides the reflection with a certain velocity Um (Ul)- The value o f u~ can be identified with the most probable velocity o f the real peaked distribution or connected to macroscopic interaction characteristics. The m o m e n t u m and energy exchange coefficients in the case of (3.12) have the form p = cos 0, (ul - u,.),

(3 13)

q=cos01(1

(3.14)

- u ~ 2)

where, in the local coordinate system connected to the normal (Fig. 1), U1 =

{0, --

sm 01, --

cos

01 }.

(3 15)

SOME PROBLEMS OF GAS--SOLID SURFACE INTERACTION

43

F o r an lsotropic surface p = {0, -- ~, - - p }

(3.16)

and the vector Um being in the plane (n, ul) can be represented as U m = An ÷ Bul.

(3.17)

Assuming u,~ forward-directed we have u~ ---- {0, -- Um sm Ore, Um COS Ore},

(3.18)

Thus, the ray model o f reflection f r o m an lsotroplc surface involves two dimensionless parameters 0 , and u,. They can be related, for instance, to the tangential and normal m o m e n t u m fluxes (~-, p) by 0 m : arc tan

Um

:

--

~

1

s i n 01 COS 01 - -

p -- cos 2 01 [(~- --

sm

01 COS

T

,

01) 2

(3.19)

÷ (p -- COS2

01)2] 1/2.

(3.20)

Cos /71

In (3.17) then we have A --

p - -

cos 0:

T

sin 01'

B--

'7" 1 -

-

sin 0: cos 01"

(3.21)

The third macroscopic parameter o f interaction is, for the ray approximation, a consequence o f the two others. Having determined 0m and u,~ in terms o f real (~-,p), the energy flux q follows from (3.14). But if um is determined through a real q by um = [(cos 01 -- q)/cos 01]+

(3.22)

then the m o m e n t u m flux magnitude is constrained by (3.13) to the f o r m (r 2 ÷ p2), = cos 01 [(sin 01 -- u~ sin 0,) 2 + (cos 01 + U m COS 0m)2]t.

(3.23)

In general, in the ray model, the q u a n t m e s ~-, p, q are interconnected by the relation (7 -- sin 01 cos 01) 2 ÷ (p -- cos 2 01) 2 = cos 01 (cos 01 -- q),

(3.24)

which can serve as an admissibility criterion o f this model on the macroscopic level o f description. W h e n the scattering function is suffic:ently concentrated, different relations between (0,~, urn) and the real characteristics o f interaction will gwe the same result within the frames o f accuracy o f the model Smgularmes of the ray model and its apphcatlons m aerodynamic problems have been investigated m ref. 62 It extends considerably the posslblhtles o f analytic study o f h~ghly rarefied gas flows

3.1.4. Reciproctty law The definition o f the scattering function calls for some necessary requirements which must also be met by models o f V within their accuracy. Consideration o f these restrictions in the process o f modelling makes the search more definitely directed. First o f all, the function V must be non-negatwe. Then, for a balance or in the absence o f trapping, sputtering and

44

R.

G.

BARANTSEV

emission the function V (ul, u) must be normahzed over the half-space u, > 0. Moreover, a so-called reciprocity relationship ~63-65~ takes place that follows from the time-reversibility of the fundamental equations. It is similar to the known detailed balance principle In the problem of collision between two gas particles Under the conditions of equlhbrium this relationship looks obvious enough V(Ul, U) l u l , l e x p ( - - h ~ u 2 ) = V(

u,-ul)

Pu, l e x p ( - - h ~ u ' ) .

(3.25)

However, while the scattering function is defined independently of f i t must be fulfilled in a non-equilibrium gas as well. A more fundamental proof of (3.25) taking into account the dynamics of the wall atoms is given in refs. 64 and 65. A number of papers in ref. 2c (C. Cercignam, I. K u ~ r et al ) are devoted to constructing and studying models of V satisfying eq. (3 25). A comparison with the experimental data taken from ref 2a, b has rather well stood by the model

V(ul, u) -7r

a,(2

2u, a, a, (2

--

~" 1 a,) exp t - - -~-n [u"z + (1 -- a,) u~,]

1 ~t)[u'--(1

}(

- - a , ) u , , ] 2 Io 2 V ( 1 - - a , ) u , u , .

--

)

(3.26)

(1 n

where Io is the Bessel function, velocities have been scaled with ~/(2RTs); ~, and a. are two parameters, the former being the tangential momentum accommodation coefficient, the latter the normal motion energy one. It is clear that the modelling of V taking account of (3.25) is accomplished with respect to u and u~ simultaneously. It should be emphasized that extension of (3.25) to include non-equilibrium conditions imphcates the target surface to be invariable. But the state of the surface really depends on the ambient gas and this is just mainly responsible for the essential u~-dependence of V. The relationship (3.25) is valid in a velocity range where the state of the surface is not noticeably different from equilibrium. Assuming V to depend on f, the tlme-reversibihty will be revealed in a more complicated way. A direct experimental verification of the reciprocity principle has been made (77)

3.2. Maxwellian Dtstrtbution of Reflected Particles Now we give a detailed account of the scattering function model proposed in ref. 67 and thoroughly investigated in ref. 61. 3 2.1. Scattering function and exchange coefficients For a fixed incidence velocity ul = U1 the scattering function IS connected with the distribution density of emerging particles by a simple relationship V(U1, u ) - -

u, 01f(u) [..>o

COS

(3 27)

that follows from (1.1) for the 8-distribution of the impinging flow, f ( u l ) I.t.
(3.28)

U1 = {0, -- sin 01, -- cos 01 }, 01 • [0, ~r/2].

(3.29)

Instead of V(Ua, u) one can m o d e l f ( u ) I ..>o.

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

45

S. Nocllla (6T) has suggested the a p p r o x i m a t i o n o f f ( u ) 1..> o by the Maxwellian distribution function f ( u ) I,.>o = n

exp [-- h(u -- U)2],

(3.30)

0, U c o s

(3.31)

n, U, h depending on U1. On an ]sotropic surface U =

{0, =[= U s l n

0},

the upper sign corresponding to a forward-directed U, the lower sign to a back-directed one. The normalization condition for the scattering function y]elds

f u, f d u :

COS 01.

(3.32)

U.> 0

The exchange coefficients according to (1.I0) take the f o r m P = UI cos 01

(3.33)

-- f u . f u d u , iI uj>O

q ---- cos 01 -- f u, fu 2 du.

(3.34)

ha> 0

Introducing v = v'hu,

s = v'hU,

(3.35)

the integration in (3.32)-(3.34) can be earned out using (2.52). Taking account o f (3.16) we obtain

nU

cos 01 -- 2s ~/-'-----~X(Z),

(3.36)

n U 2 sm 0 ~- -- sin 01 cos 01 = -4- 2s a/~r X(Z),

p __ cos2 01

--

nU2 cos O [ 2s CTr

(3.37)

X/lr

X(Z) + ~

nU 3

]

(1 + erf z) ,

(3.38)

a/~r

cos 0, -- q = 2s 3 a/r~ [(2 + s z) X(Z) + T

z(1 + erfz)]

(3.39)

where Z

=

S COS 0,

(3.40)

and X (z), erf z ]s shown by (2.53). 3.2.2. Model parameters through exchange coefficients The function (3.30) being defined in the half-space u, > 0 only, the p a r a m e t e r s n,

U, T = m/(2 kh) have not the usual meaning o f density, velocity and temperature. Let us

46

R, G BARANTSEV

r

/

-2 o

,o !/ /

31

A J

__-J7 J

-5

-2

-I

I

0

2

3

FIG 10

express t h e m t h r o u g h the m o m e n t u m a n d e n e r g y e x c h a n g e coefficients. E l i m i n a t i n g n, O, U, s f r o m (3.36)-(3.40) results in one t r a n s c e n d e n t a l e q u a t i o n for z ~b(z) ~:(z) = 2 + z ~b(z)

( p - - cos z 01) 2 cos 01 (cos 01 - - q) - - ( r - - sin 01 cos 02) 2

(3.41)

where ~(z) = z

--2~(1

+erfz).

(3.42)

T h e b e h a v i o u r o f x (z), ~ (z) 5 ( z ) i s s h o w n on Fig. 10. T h e i r a s y m p t o t i c e x p a n s i o n s are for z ----~- - oo

l

(

X(Z) = 2z----i e -z2

¢(z)

= ---

1 ~(z) -

z~

1 z

3

o(1)]

1 - - 2z---2 + --~--],

+--

3

2z ~

9 2z ~ +

0(1)

+--,

zs

0(0 Z6 •

(3.43)

47

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

for z -~- -]- oo

X(z) = 2X/~rz ÷ ~

e -~2

i ¢(z)=

z +2z

2V,----~e -'2 3

1 --

,

(

o(i) l -- -~],

(3.44)

0(1) + - z- z -

~(z) = i - 2 - ~ The c o n d i t i o n

(p

- - c o s 2 01) 2 Jr- (T - - s i n 01 c o s 01) 2 ~

c o s 01 ( c o s 01 - -

q)

(3.45)

is necessary and sufficient for u m q u e n e s s o f the s o l u t m n o f (3.41). O n solving eq. (3.41) we find successively h

2Vrr¢(z) c o s 2

~b2(z) c°s2 01 (p - c o s 2 01) 2,

s 2 = z 2 + CZ(z)

U =

s(p

-

-

x(z) (p -

(7 - sin 01 cos 01) 2 , ( p -- cos 2 01) 2

cos 2 0 0 )

¢(z) cos 01

cos 0

----

01

c o s 2 01)' (3.46)

z -. s

O n the plane (~-, p) the solvabihty c o n d i t i o n (3.45) picks out a circle centred at p o i n t (sin 01 cos 01, cos 2 01) w h o s e r a d m s m a y take values f r o m 0 to cos 01, if q / > 0. The inequalities ~- ~< sin 0, cos 01, p / > cos z 0t, the f o r m e r c o r r e s p o n d i n g to the upper sign o f

Qgl

spec

-_Fo75

I S qz=O

IC

c5 /z=O 5\

/ 05-

/

/

/

/ I

/

/

/

//

i

I O5 T

Fro. II

48

R.G. BARANTSEV

(3.31), are in the left upper quadrant o f this orcle. For q < cos 3 01 a part o f it is cut off by the line r 0 Centres o f all the circles (3 45) he on the semlclrcumference (p__ 1)2 ~ r2

~,

r /> 0

(3 47)

A typical domain of permissible values o f r, p is represented on Fig 11 (for 01 = 40°). The specular reflection point is at the left upper corner o f the domain. F o r the diffuse reflection r=sln0

lcos01,

p=cos

201 + ~

cos 01,

q=

1 --~

cos 01,

and, therefore, a lower part o f the right b o u n d a r y of ~/(~r/8) ~ 0 63 o f the radius IS only filled. The values o f the exchange coefficients obtained in Section 2 by the hard sphere lattice reflection theory at ~ = 0, 0.25, 0.5, a , = 0.75, fall Into the permissible domain for all 0x. Table 4 contains the Maxwellian distribution parameters for the hard sphere lattice reflection in case o f ~ = 0.25, a , = 0.75 for 0~ = 0 (15 °) 75 ° A direct physical interpretation of these parameters can be provided, for instance, by means o f a formal continuation o f (3.30) over the whole velocity space u. Then n, U, T = m/(2kh) become the density, velocity and temperature o f this distribution Table 4 shows their 0~-dependence. In particular, for 01 up to about 50 °, U, < 0 and U, h change little; with a further increase in 0~ the average velocity increases and h also increases quickly, which indicates the forming o f a sharp peak. TABLE4. 01° n

U 0o h

0

15

30

0 277 24.2 0 350 165 5 3 40

0 270 29.8 0 353 180 3.66

45 0.373 6.12 0 323 98.7 4 14

0 303 15.0 0.316 142 7 3.47

60 0.451 2 58 0 485 75.5 5.17

75 0 721 0 602 0.757 55 5 13.9

3.2.3. Scattermg indicatrtx O f particular interest is the direction distribution o f the emerging particle flux, 1

Vo,

-

i v u, f u 2 du.

(3.48)

cos

o

Substituting the function (3 30) here and integrating white taking account o f (3.31), (3.35), (3.36) and (3.42) we obtain

Iio --

COS

exp ( - - s 2 sin 2 y) X(S cos y) [1 --k s cos y~ (s cos 7,)]

(3.49)

cos >, = cos 0 cos ~ •

(3.50)

×(s cos 0)

where

oa, ~ are the spherical angles o f the vector u.

sin 0 sin ~ sin %

49

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

~"~/

OI O*

Fro 12

Thus, the scattering indlcatrzx of the model (3.30) involves only two parameters: 0 and s. Typical middle sections of (3.49) for the upper sign in (3.31) are shown on Fig. 12. In the case s = 0 the distribution is diffuse. When s -+ oo the distribution concentrates in the direction 0 = 0 if 0 < ~r/2 or m ~ = rr/2 if 0 /> rr/2. M a x i m u m V,o is reached at 0 ---- o,,; whose 0-dependence for several s is shown on Fig. 13.

90

jf

60

o

o~ 30

30

60

90

120

150

180 e,.-

0o

FIG 13

3.3. Scattering Function Expansions The scattering function models considered above approximate u-dependence o f V and contain two or three parameters each depending on Ul. Such representations of a two-vector function by means of a few one-vector functions are inherently approximate. It is useful to have some algorithms to m a k e those models more accurate. An exact representation can be reached by expanding V (ul, u) in some complete system o f functions of u366, 6s, 691 The coefficients o f such an expansion depending on ul are

50

R. G

BARANTSEV

substituted for the scattering function m their infinite totality only Finite sums of the series give approximate representations having diverse accuracy If V is expanded in orthogonal polynomials their form is completely determined by the weight function which must secure the existence of all moments It Is suitable to choose the weight function so that ~t should contain the cardinal properties of V. Then one can expect the expansion to be effective, i.e. a g o o d accuracy will be reached with not too many terms The function V (ul, u) defined in the half-space u, > 0 may generally be continued over the whole space u. Moreover, u~ being fixed, it can be changed b y f a c c o r d m g to (3.27) Each o f these operations leads to a different expansion even for a fixed weight We consider two variants: the whole velocity space expansion with the Maxwellian weight and the half-space expansion o f V with the diffuse weight

3 3 1 Expansion in the whole space Let us continue analytically the scattering function V (Ul, u) over the half-space u, < 0. The velocity Ul being fixed, instead o f Vwe can deal with the distribution density o f emerging particles cOS 0 a

f ( u ) ---- - -

V(u, u).

(3.51)

/dn

As a weight we take the Maxwellian function

(3.52)

fo(u) = n(2h) 3/2 co(v) where v = x/(2 h)(u -- U),

(3.53)

w(v) = (27r) -3/2 exp(-- v2/2).

(3.54)

The orthogonal polynomials in u-space with the weight (3 52) are the spatial Hermite ones defined by HO,~ l~,

(v) ---- ( - - 1)m co-l(v) ,~m

6q/)~t ,.

.,~Vt,~

~(v).

(3 55)

These are the tensors of rank m and of order 3. In particular, H ( ° ) ( v ) = 1,

H~ l ) ( v ) =

v. . . . u~2) Ij (v) = v~ v,

3t j,

(3.56)

H(3) ijk (v) = v, vj vk -- (vl 3jk + vj 3tk + vk 3tj). The expansion o f f in H (') is

f ( u ) = fo(U)

/_.. --1 m! m=O

a,,,) l...... I t,

-~,m) i~, _,t. (v),

(3.57)

-,Ira= 1

the coefficients ,,~,,) -I . . . . .

n1 f f ( u ) H ~m q, ,l. (v) du

(3.58)

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

51

are the tensors of rank m containing (m + 1) (m ÷ 2)/2 different components. For m ~< N we have (N + 1) (N + 2) (N + 3)/6 scalar coefficients in all. As the system of a (-) is complete the weight function parameters n, U, h can be arbitrarily chosen, for example, n = 1, U = 0, h = ½. But if they are considered as unknowns, m the same meaning as m the case o f (3.30), then the coefficients a (-) must satisfy the following equations: a (°) = 1, aJ *) = 0, a ~ ) = 0. (3.59) The system of w ~ (v) H (~") (v) as closed and orthogonal. So the expansion (3.57) with the coefficients (3.58) converges in the mean for a n y f ( u ) for which the product f o : ~ IS squareintegrable over u-space. The finite sums of (3.57) provide the refinements of (3.30) with an increasing n u m b e r of parameters.

3.3.2. Expansion in half-space The analytic continuation of V (u,, u) over the whole u-space is not always possible and convenient. Let us construct an expansion o f this function m polynomials which are orthogonal over the half-space u, > 0 with the weight

Vo(u) --

2-----7-v. e×p

--

.

(3.60)

These polynomials can be written in the form H i m,,k )

(v) : H, (Vb) Hj (v,) H* (v,,),

i,j,k:O,

1,..

,m;

tq-jq-k=m,

where H~ (x) are the Hermite polynomials o r t h o n o r m a h z e d in - - ~ weight (2~r)- ~ exp (--xZ/2), Ho(x) ---- I,

H i ( x ) = x,

(3.61)

< x < oo with the

Hz(x) : (x 2 -- i ~ ) / ~ / 2 , . . .

and H~, (x) are the polynomials orthonormahzed (--x2/2),

(3.62)

in 0 ~< x < ~ with the weight x exp

(x-21), (3.63)

4-9

(

H*2(x) = V'(5~r 2 -- ~-~r q-- 64)

x2

_V'(2~r) _ x ÷ 3~r -- 81, . . . 4 --~" 4 --9/

The expansion of V m r_,¢,~)

V(ul, u) = Vo(u) ~

B,,~ (ul) H, (vb) Hj (v,) H~, (v.)

(3.64)

I,J.k=O

where B,jk(Ul) :

f V(ul, u) H, (vb) H j (v,) H~, (v,) du. un>O

(3.65)

52

R G. BARANTSEV

On an Isotroplc surface the coefficients (3.65) with an odd first index vanish so that V :

Vo {1 _ Bo,o H1 (t,~) + BooJ H : (v.)

+ B o l l Hj (vt) HI (v,) + B2oo //2 (v0) ~- Bo2o//2 (v,) + Booz H I (v,) ÷

(3.66)

.}

With a finite sum of (3.66) instead of the scattering function one has a certain number of macroparameters (3.65). By means of them one can combine, in particular, the momentum and energy exchange coefficients for a given incidence velocity p -- cOS 0a {U, q:cos01

x/(2h)l IBoxot + ( J i

{ '[ 2(') 1 --~

46-

~

+B°°'J(--~

))n]},

(3.67)

BooJ 6- ~/2(B2oo 6-Bo2o)

6- ~/(5r~ 2 -- 36 ~- + 64) B ] ]4 -- rr 0021 j .

(3.68)

3.4 Successive Modelhng of V (u,, u) 3.4.1

Statement of the problem

The scattering function V results from solution of the problem on gas particle collision with an atomic lattice of a solid. This problem contains a lot of parameters and proves to be greatly complicated From the aerodynamics viewpoint, even if it is successfully solved, the similarity with respect to all those microparameters is practically hardly realizable. Therefore it makes sense to look for statements and solutions of gas-surface interaction problems on intermediate description levels between the molecular and aerodynamic ones (see Section 1.2.4). Three such levels can be distinguished: Boltzmannian, moment and local gas-dynamic. We are within the first one when modelling in some way the dependence of the scattering function on the leaving velocity u. We are within the second one when approximating the dependence of moments of the scattering function, in particular p and q, on the incidence velocity Ux_ Within the third one the question IS to approximate the dependence of local gas-dynamic variables on the coordinates. Each of these steps reduces the number of arguments and by that simplifies the statement of the problem, at the expense of ~ts "coarsening". Actions like these can seem too arbitrary, superficial and transient In order to appraise their methodological role at their true value it is useful to compare the Boltzmannian level with the molecular one where a quite profound experience has been stored by physicists. The molecular physics problems could be solved starting from the nucleus-electron level. But, practically, interaction potential models are used as a rule. A set of standard models with a small number of parameters prove to be sufficient. On the Boltzmannlan level the scattering function models promise to play a similar role. Such a situation is possible at other description levels as well. There is already a fair amount of scattering function models. Accumulating the variety of separate models brings up the question of introducing a certain systematization in modelling. First of all, we ought to choose, in the most convenient way, the sequence of

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

53

independent variables considered in the multidimensional space of arguments u, ul and interaction microparameters. Then we ought to separate those typical qualitative properties which must be contained in the models. Then the notions must be denved from experimental data, which are being formed irrespective of the form of the model, in order to be able to introduce on their basis universal macroparameters.

3.4.2.

u-dependenceapproximation

Interaction modelling on the Boltzmannian level of description consists in representing the scattering function in the form

V(ul, u)

=

Vmod [a, (ul),

az (ul), • • , au (a,); u].

(3.69)

The representation method permits various approaches. For instance, one can first choose a form of u-dependence of Vmod and then look for parameters; or first introduce parameters and then look for a representation form. F r o m the viewpoint of universalizing, the second approach ss more preferable now. Indeed, the experience of physical and computational experiment has not yet given us any general prescription how to construct the u-dependence of V but it has already given a universal set of parameters in terms of whtch the scattermg functions are characterized independently of their forms. On clean surfaces for incidence energies ~ 10-2-101 eV the function V (u) considered in the spherical variables u, 0, ~o has, as a rule, one clear maximum in each of the arguments and is qualitatively characterized by ItS position and width. These mean (u,,, 0,, ~0,,) and dispersion (%, or0, %) values become the conventional scattering characteristics. The parameters urn, ~ are considered either for a fixed direction or averaged over directions; the parameters Om, or0, % are usually averaged over the velocity magmtude. While describing a single-peaked distribution qualitatively it is not necessary to distinguish between the mean and most probable values. For a quantitative certainty let

u. (u,)= f v (u. a) u u, o. O

(3 70)

Un>0

Arranging models in a certain sequence from simple to complicated ones we shall gradually enlarge the set of parameters a, (ul) choosing the simplest form of Vmoa at each stage, c76) At the first stage we hmit ourselves to the mean velocity of reflection u,,, at the second one we add %, or0, %, then the correlation parameters, then higher moments. A general criterion of maximum simplicity to choose Vmo~has not yet been formulated. At the first stage of approximation the ray model of the scattering function (3.12) seems most suitable. On an Isotropic surface the vector am has two non-zero components m the incidence plane simply connected with u,, and 0m. The mean value of azimuthal angle is ~m = 3~r/2 for forward-reflection or ~Or~= zr/2 for back-reflection. The function (3 12) is valid for the approximate description of a keenly directed scattering or of average interaction characteristics c75) under condmons when am-similarity is sufficient. I f am does not depend on 01 or for small um when the influence of Or, is not essentml It is natural to assume 0m ---- 0, i.e. am = u,~n. Then (3.12) may be regarded as an average version of the diffuse scattering function, the parameter um/ul being connected to the temperature f a c t o r . (76~

54

R G BARANTSEV

According to (l.10) m case of (3 12) for %, -- 3~/2 one has

r

:--

COS01 (sin OJ

ltm Sill 0 m) , II1

P = COS O1 (COS O1 ~- -um - c o s 0 m) ,

(3 71)

Ul

q = cos 01 (1 --

Urn2t

u~J'

The parameters u,,, 0= can be expressed by the tangential and normal m o m e n t u m fluxes in the f o r m of (3.19), (3 20). At the second stage of approximation some straggling around mean values u,,, 0,., 9,. is permitted. In this case making use of (3.12), each delta-function of a scalar argument m 8(u -- u.,) --

8(u -- u=) 8(0 -- 0,,) u2 sin 0 3(~o -- ~o.,)

(3.72)

is replaced by a certain peak-shaped function A having a corresponding dispersion. The n o r m a l distribution has a fairly simple form and provides m a x i m u m information, the zeroth, first and second m o m e n t s being fixed. F o r x E ( - - ~ , + ~ ) 1 exp [ ~/(2~r) cr----'~

A(x -- x,,)

( x - - x = ) 2] 2-e'~ J

(3 73)

In case of a finite interval 0 ~< x ~< a the parameters and the normalization are determined otherwise. However, assuming the dispersion small, one can use (3.73) asymptotically if x., ~ (0, a) Then V

=

0.3

A(u - - u..) A ( 0 - i,/2

sln 0

A(~ -- Cpm)

(3.74)

Another version of the approximation in spherical variables has been proposed in ref. 70 Inserting (3.74) into (1.10) and evaluating the integrals by a

j g(x) ±(x - x,.) dx = g(x,.) + "~ g"(x,) o

+

(3 75)

we obtain

"r c°s 01{ sin Ol -- -u= - sin

O,n [1 -- ½(ee2 + cry)]} ,

(3.76)

//1

Mm lll

~]

p ~- cos 01 Icon 01--- - - cos 0,, (1 -- ½ try) ,

(3.77)

q = cos 01

(3.78)

l

uS u~

eu "

These formulas evidently show some important properUes concerning the ways in which the dispersions affect the exchange coefficients. Thus, for example, ~- increases with ~ + %,2 p dlmimshes with increasing e02, q diminishes with increasing %,2" the velocity

SOME PROBLEMS OF G A S - S O L I D SURFACE INTERACTION

55

straggling affects the energy exchange only; the azimuthal angle one effects the tangential m o m e n t u m exchange only. I f a small fraction a of diffuse scattering is included so that V : a Vj ÷ (1 -- ~) Va then for h~ ~ 1 the following expressions will be respectively added to eqs. (3.76)-(3.78). Um

Urn

M1

I11

U2

~r ~ cos 01 sin 0,,, -- o - - cos Oj cos 0.. ~r u~ cos 01.

(3.79)

With the correlation parameters ~,0, a,¢, ~0¢ brought into (3.74), ~- a n d p are respectwely added by

-- cos 0x cos 0,, a~0, -- cos 0a sin 0m %0,

(3.80)

and q remains invariable. So a,¢ and cr0¢ are negligible within the first-order effects.

3.4.3.

u~-dependenceapproximatwn

In the local coordinates connected to the incidence plane we have ul = {0~ - - u l sin 01, --u~ cos 01 }. Let us first consider the dependence on the incidence angle 0~. First of all the exchange coefficients data obtained at the m o m e n t level of research can be used. They show that with an increase in 0j from 0 ° to 90 °, the values o f p and q decrease monotonically to zero, r varying from zero to zero through a m a x i m u m . Such dependencies are conveniently a p p r o x i m a t e d by trigonometric functions. In ref. 71, for example, the formulas ~- -- sm 01 0"1 cos 01 ÷ ~'2 cos2 01), P =Pl

cos 01 q-p2 co s2 0 ,

(3.81)

q = ql cos 01 -+-q2 cos 2 0~ have been used. F o r some simple reflection models, m particular for the specular-diffuse one, they are exact. The approximate formulas emerging f r o m (3.81) for rx = ql = 0, q2 = 3.6 tz (1 q- tz) -2 have been proposed in ref. 51, t~ being the gas-to-surface a t o m mass ratio. They approximate fairly well computational results for a hard sphere lattice reflection at t~ ~ 0.5, a . ~ 1. In ref. 72 0 = 0 , o0 ~ 01-I, q ~ (rr/2 -- 01) cos 01 are taken. A more general approach ~s to expand r, p, q in trigonometric ser~es. M i n i m u m systems o f functions are advantageous. In the interval [0, rr/2] the systems {cos (2 k -- 1) 01 }, {cos 2 (k -- 1) 01 }, {sin (2 k -- 1) 0~ }, {sin 2 k 01 ), k = 1,2 . . . . . are complete and minimum. The most effective expansions are those in the functions satisfying the agreement conditions at end points. As we have ~-(0) = ~- (~r/2) = 0, p (0) :~ 0, p (¢r/2) = 0, q (0) ¢ 0, q (rt/2) = 0 it is expedient for p and q to take the first system and for ~---the fourth one which after extracting sin 01 also turns into the former, i.e. ~- = sin 01 ~ .

r k cos (2k -- I) 01,

k---I

P = ~,

Pk cos (2k -- 1) 0~,

k=l

q=

~.qkcos(2k--1)

01 .

,alma,mini

k=l

Coefficients ~k, Pk, qk and the upper limits o f the sums are to be determined by fitting.

(3.82)

56

R

G. BARANTSLV

If the n u m b e r o f parameters in a scattering functmn model does not exceed three, ~t is generally possible to express them by the exchange coefficients and to refine the approxImations o f r, p, q using V-data. For instance, for ray reflection the parameters u,,, 0,, are expressed by ~-, p, according to (3.19), (3.20) In ref 76 an attempt is made to select on the bas~s of (3.81) such approximations of-r and p which would properly describe the behavJour o f u,, (01) and 0nt (01). Two t w o - p a r a m e m c variants have been considered r2 -- 0, P2 ~ r~ and r~ = 0, P2 1 For a more detailed modelhng o f V when the number o f parameters exceeds three, the information about the exchange coeffioents ts not sufficient to approximate the u j-dependence o f V. One has to approximate the parameters unt, 0,,, %, Go, %, directly. Experimental and computational results (Sectmns l 3-4) show the following general regularities in 01-dependencies o f the mean and d~spersmn characterlsncs of reflecuon. W~th an increase in 0 a from 0 ° to 90 ° parameters o,, G0, % decrease to nearly zero, unt increases to a value of the order o f ux, 0,~ increases being near 01 Choosing the simplest trigonometric functions describing such a behaviour we assume =

tYu2 =

2 COS 2 0 1 , O"02 O'uO

Unt =

U 1 --

(b/1 - -

= ~2oo cos 2 0t ' %2

=:

2 c°s2 01: %0

(3.83) (3.84)

UntO) C O S 2 0 1 ;

or, = 0~ -+- ~ (00, ]~ } ,~ 1.

(3 85)

It is difficult now to establish the form o f r/ (01). We assume ~7 (0) = ~/ (~r/2) = 0 and propose for the present the following two varmnts: = ~/o sin 20l,

(3.86)

~7 = ~o sin 40j

(3 87)

In the first case ~ is o f constant sign, in the second one ,/changes stgn at 0~ = 45 °. Values o f '7o can be both positive and negative. Having a small fraction o f the diffuse c o m p o n e n t it ~s natural to assume = cro cos 2 01

(3.88)

Then let a~o = R o s m

20~ ~,~0-

(3.89)

Subsntutmg eqs. (3.85)-(3,86) and (3.88), (3.89) into (3 76)-(3 78) we obtain

% --

(4o +

O'O0 ) COS 2 01

-

O"0 C O S 2 0 1 ]

-

2Ro %0 Cr0ocos* 0~

t'

-- ½ e2o cos 2 0~ -- eo cos2 0a) -- 2Ro a,o aoo sm z 01 cos 2 0~t , q=cos01

1 --

1 --

1 --unt°

cos z 0~

(3 90)

(3.91)

(1 --~roCOS2 0~)

/gl !

~ o2

C O S 2 01

"

(3.92)

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

57

The ul-dependence has been studied to a smaller extent, There are the approximate formulas for the energy accommodation coefficient a.c73,ss) It is known that when the incident energy E1 increases within 10-2-10-1 eV the diffuse fraction and the dispersions generally decrease, 0,, and u,, increase. A further increase of E1 results in changing some properties. The transition from one, thermal, to another, structural, scattering regime was found in ref. 20, confirmed in ref. 35 and observed in refs. 17 and 18. According to the computational results t2°) with the transition to the second regime the trends in the Eldependence of 0,, and or0 invert while the energy transfer regularities remain. By the experimental data (17) in the transition region the difference ~7 = Om -- 0i becomes positive, gains a maximum at a certain E* and then decreases to zero. The function a0 (E~) changes into increasing Instead of decreasing. The same transformation happens to u,, (0) The character o f the dependence o f 0,,, or0, u , on 01 remains unchanged These qualitative regularities can be taken as a basis of modeling the ul-dependence, but for a reliable choice of universal parameters further research is needed. The approximate formulas proposed above must be subjected to a regression analysis (6°.~4) on the basis of the quantitative data of physical and computatmnal experiments to determine the numerical values of parameters and also the exactness and the applicabihty range of the models. One ought to begin with the simplest variants because the inclusion o f many parameters at once is fraught with losing stability, which is connected to the limited amount of experimental data. This approach wall help to elicit those macroscopic similarity parameters which are so needed in the aerodynamics of highly rarefied gases. (76) 4. R O U G H S U R F A C E S C A T T E R I N G 4.1. Problem Statement 4. I. 1. Area size

The scattering function V (ul, u) is defined as the distribution density of particle flux in the half-space of u-velocity after reflecting from the surface at the point r, at the instant ts on the conditmn that the incidence velocity was ul. It is implied that the reflection occurs on an elementary area d S containing the point rs. The question of this area size ought to be considered m more detail Reflection laws can essentially depend on the scale of the investigation. For instance, with increasing the area size from the molecular one, on a corrugated surface the multiple collisions o f molecular scale become possible, which results in changing the leaving velocity from dS. Quantitative changes lead to some qualitative stages with new regularities and the form of V is different at each level. In the rarefied gas dynamics problems the size of d S must correspond to the size of the space element over which the dlstnbutmn function in the Boltzmann equation is averaged. The gap between the molecular and gas-dynamic sizes can be rather large (from 10 -7 to l01 cm) Let us assume, for definiteness, that the elementary length is of order 10-2-10 -1 cm For an impinging atom the area d S of such a size is a vast, generally non-uniform, field w~th the roughness of random character and comphcated structure, o°o) The conventxonal standards specify technical surface roughness by the mean deviations within 10-6-10-2 cm. Near the surface the distribution function suffers the microfluctuations due to the surface roughness. They are taken into account if the size of the elementary volume and, respectively, the area d S are chosen small enough. But then the boundary equation contains the random P .A S _ - - C

58

R. G. BARANTSEV

small-scale roughness component. On the gas-dynamic level such fluctuations by themselves are not usually interesting and are to be averaged over larger elementary volumes and respective areas dS. Exclusions are the problems hke the shock structure near a boundary where the small size of the element ( 1 0 - 7 - 1 0 - 6 cm) is due to the large gradients of other origin. With changing the size of dS from the molecular to a gas-dynamic one the large-scale roughness part entering boundary equations diminishes and the small-scale part which is subject to averaging increases. Dealing with the scattering function we begin with the second component of roughness. So we shall assume that the size of dS exceeds the molecular one ( ~ 10-7 cm) and can extend up to the gas-dynamic one. Near the upper limit the normal N to the mean level of dS becomes determined.

4 1.2. Roughness operator After impinging upon dS with velocity ul an atom obtains the leaving velocity u from the area as a result of one or several collisions within dS with atom clusters appertaining to small ds of molecular size (,,~ 10 -7 cm) Hence, in order to construct the scattering function V one has to solve two problems: (1) collision with a small area ds; (2) statistical description of migrations and encounters within dS. The former is formulated as constructing the scattering function in the small area. The latter can be solved independently if the successive collisions with small areas are separated from each other clearly enough. The solution of it must provide the scattering function on dS in the form

V ~ SVo

(4.1)

where S is the roughness operator c78~) that is determined by surface geometric properties unless the atoms migrating in dS collide with each other. Such collision probability grows with decreasing the mean free path of the gas and increasing the roughness size. This case of coupling the scattering function with the distribution function is not considered here. Thus, in order to find S we need the geometric description of the random surface. Let us assume that the rough surface on dS is described by a homogeneous isotropic random function z = ~ (x, y) having continuous derivatives with respect to x and y. The homogeneity and ~sotropy imply that the mean surface level is constant and the correlation function depends on the distance between points only. This assumption ought to be considered as a priori averaging the statistical properties of the surface on dS. One possible generalization is to introduce several layers of homogeneous roughness (79~) The continuous &fferentiabdity of g (x, y) implies that the normal n to the mean level of a small area ds changes continuously. As we shall see soon the operator S is built up from the multidimensional distribution densities of g, ~ , ~ at different points (x, y).

4.2. The Complete Solution 4.2.1. Structure of S Let the mean level of the surface be taken as the plane (x, y). An atom impinges upon dS (Fig. 14) with the velocity ul = {0, -- ul sin 01, -- ul cos 01 }, 01 E [0, 7r/2].

(4.2)

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

t

59

Z

b ~

~

ds

,, Y

FIG 14

We have to find the probability V (ul, u) for the atom to leave dS with the velocity u = {u sin 0 cos ~0, u sin 0 sin % u cos 0}

(4.3)

belonging to the element du of the half-space uz > 0. It can occur after one, two and more colhsions within the large area so that it is natural to represent V by the series

V:

VI + V2 + . . .

÷ V,,

(4.4)

where V,, is the function of m-fold reflection. The integration over uz > 0 yields 1 : N1 + N2 + • • • + Nr,, Arm being the m-fold reflection probability. Let us consider V1 in detail. In this case the only collision can occur with the small area ds on the height Zo within dzo with the normal n directed in (o, if) within do, = sin od Odff (Fig. 14). The order of the events happening to the singly reflected atom are : (a) Free motion along the incidence ray a01 without crossing the surface up to the point O1. (b) Collision in the neighbourhood dzo of the point O1 with the small area ds oriented in dw. (c) Reflection from d~ with the velocity u in du. (d) Free motion along the reflected ray 01b without further crossing the surface. Integrating the product probability of these events P {ABCD } over all the possible levels Zo and orientations (o, if) of the encounter we obtain the probability of the single reflection into du

Vlau=f f f P(AOCDL oJ

z o

I f the surface does not change during interaction we can write down

P{ABCD) : P(B} P(CIB} P{ADIBC} where P{CIB} = Vo (ul, u) du, P{B} = p (0~, Zo, n) dzo dw is the encounter probabihty,

60

R. G BARANTSEV

P { A D I B C } = H (0~, 0, 4', Zo, n) Is the c o n d m o n a l p r o b a b d l t y o f the free m o t i o n along the rays a01 a n d Ojb Thus, ,Jr

Vj =

iil

Vo

um
n

p 1-I dzo dco

(4 5)

--~

The o t h e r terms in (4.4) are constructed slmdarly. The expression for I'm contains the p r o b a b d i t l e s o f rn successive encounters, m functions Vo, the c o n d i t i o n a l p r o b a b d l t y o f the free m o t i o n a l o n g the straight parts o f the b r o k e n line with m corners, a n d 6 m - - 3 integrations over all the possible c o o r d i n a t e s a n d orientations o f each e n c o u n t e r a n d over all the interm e d m t e v e l o o t i e s The p r o b l e m is eventually r e d u c e d to constructing the f u n c h o n s p a n d / 7 . 4.2.2. Encounter probabihty We have to construct the function p (01, Zo, n) m (4.5). Figure 15 shows the e n l a r g e m e n t o f the ne~ghbourhood o f the p o i n t O1 Let the atom-surface e n c o u n t e r occur w~thln the infinitesimal interval 01A o f the ray a01 Then dzo -- OaB = O~A cos 0a. Let us d r a w the

.

Or FIG. 15

p l a n e A C D p e r p e n d i c u l a r l y to n Crossing 01A with the area ds oriented by n is equivalent, within the small quantities o f higher order, to that o f this area with the element Oj C - dzo* o f the z-axis The p r o b a b i l i t y o f the latter is

f(zo, Zox, Zoo) dz o 3(Zo=, Zoy)/O(o, 4) do dq~ w h e r e f ( z o , Zoo, Zor) is the j o i n t dastrlbution density o f functions ~, ~=, ~ at the same p o i n t (0, 0). W i t h Zo~ = - - t a n ~ cos 4', Zor ---- - - tan ~ sm q~ we have 3(Zo=, Zo~)/30, q~) = sin ~/cos a ~ The relation between dzo a n d dzo* remains to be f o u n d As can be seen f r o m Fig. 15, OlD = 01A cos al = O 1 C c o s ~. Hence dzo* = cos al dzo/(COS ~ cos 01) Thus, cos al

p(01, Zo, n) = f(zo, Zo=, Zor) cos 01 cos 4

(4.6)

where cos al = cos 01 cos ~ + sin 01 sm ~ sin q~

(4.7)

The next e n c o u n t e r probabilities are o f the same form, except for the c o n d i t i o n a l d l s m b u t l o n densities

61

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

4.2.3. Free path probability Constructing H is the most c o m p h c a t e d part of the problem. We shall analyse the function H (0 I, O, % Zo, n) in (4.5). In order to m a k e clear the geometric meaning of H

b~ Z

/

fo

FIG 16 let us consider the cross-section o f the surface g by the dihedral angle with the edge 01z and the sides lying along the rays a01 and 01b (Fig. 16). In this section we have the r a n d o m function

f

~(0, - - t ) ,

t ~< 0,

(4.8)

~(t) = [~(t cos % t sin ~v), t / > 0. the stationary state of which is interrupted with the transition through the point t ---- 0, except for the case ~ = 3 ~r/2. The equation o f the broken line aOlb is -- t cot 01, t ~< O,

z ~- h(t) - Zo 4-

(4.9) t c o t 0, t 1> 0.

The cases 01 = 0 and 0 ---- 0 are trivial because the conditional probability of the free m o t i o n along the corresponding ray then equals one. By definition, the r a n d o m function ~:(t) intersects the function h (t) at point t upwards (downwards) if ~:(t) = h(t) and there exists ~ > 0 such that ~: (t 4- 8) >1 h (t 4- 8), ~ (t -- 8) ~< h (t -- 8), [~ (t 4- 8) ~< h (t 4- 8), ~: (t -- 8) >t h (t -- 8)] for all 8 ~ (0, e). Intersections downwards for t < 0 and those upwards for t > 0 under conditions

Zo <~ ~ (0, O) ~ Zo 4- dzo, Zo,, <~ ~,, (0, O) <<.Zoo, 4- dzo,,,

(4.10)

zoy ~< ~y (0, O) <~ Zo~ 4- dzor will be called by the overshoots of the r a n d o m function ~:(t) over the broken line h(t).

R. G BARANTSEV

62

So H is the probablhty of no overshoot The r a n d o m process overshoot problems have been studied m many fields of physics, especially m radlophyslcs t8o-82.92) The problem solved in ref. 83 is closest to our situation. In order to construct H we shall find the probability o f the opposite event, i e that o f all possible overshoots. Let us first obtain the probability o f a single overshoot d(t) A within a small interval (t -- A, t) for t > 0 (Fig 17). Crossing the ray 01b upwards can occur for all ~'(t) - ~t >~ cot 0. Consider the overshoot through the Interval A A ' with an Inclination ~t E [z, z, + dz,]. Together with this overshoot the function ~, within small quantities o f

o

\

C

f B

t

+A

o

,-/x

FIG 17

higher order, intersects, with the same inclination, the element dz ---- A ' C = B C -- A ' B = A B z t -- A B cot 0 = (zt -- cot 0)A. The probability o f the latter event is f ( z o + t cot 0, z,]Zo, zoo,, zo,) (z, -- cot O) Adzt w h e r e f ( z , z, IZo, Zoo, z0y) is the joint distribution density o f the function ~ and its derivative with respect to t at the point (t cos % t sin ~0) under condition (4.10). By integrating over all the possible overshoot inchnations we obtain oo

d~(t) = f f ( z o + t cot O, zt I Zo, Zoo, Zo,) (z, -- cot O) dz,, t > O.

(4.11)

cotO

F o r t < 0 an overshoot will be considered within the interval (t, t + A). Operating with y ---- -- t > 0, instead o f t, we obtain similarly oo

dl(t) = f f ( z o

+ y cot 0~, zylzo, Zox, Zoo) (z~

-

-

cot 0~) dz~, t < O.

(4 12)

COl 01

N o w we can find the probablhty o f s overshoots in m intervals (ti, ti + At), i = 1, 2 , . . . , m, for t < 0 and in n intervals (t~ -- Aj, Q ) , j - ~ 1, 2, . . . , n, for t > 0, i.e. ds (q, t2, • , L)

SOME PROBLEMS

OF GAS-SOLID

SURFACE INTERACTION

63

A t A 2 . . . As, s = m + n. The f o r m u l a for the j o i n t d i s t r i b u t i o n density o f s overshoots, ds, is o b t a i n e d b y way o f the n a t u r a l g e n e r a h z a t i o n o f (4.11) and (4.12). It is d~(tl, t2 . . . . .

ts) = dm.(yl . . . .

. . . . COt 0t

f(z,

y,. ; t~ . . . . .

zj; z~v, zj,

t.)

Zo, Zox, Zo,) (Zl, - - cot 0 1 ) . . . (z,, - - cot 0) d z l , . . ,

dz,,;

COt 0

zl = Zo + Yt cot 01, zj = zo + tj cot 0; i---- 1 , 2 . . . .

,m;j=

1,2 ....

(4.13)

,n;m+n----s.

Let us t a k e n o w the interval t ~ [ - - T, T] a n d divide it into 2 N small intervals o f length A = T I N . The o v e r s h o o t in the interval i will be referred to as the event 31 -----1, no o v e r s h o o t 8~ = 0. By the definition o f d,, w~thm the small quantities o f higher order, P{3,1 = 1. . . . .

3,, ~- 1} :

ds ( t q . . . . .

tq) A , 1 . . . A , .

5_ 1 = 0 , 5 1

=0 .... ,SN:0}.

W e are interested in P{8_N=O

.....

AS the events 8~ ---- 1 a n d 8~ = 0 are m u t u a l l y exclusive we have P ( 8 _ N = 0 .....

St_ 1 = 0, 8t = 0, 31+t = 1, . . . , 8N ----- 1 }

= 0, 8~+1 = 1, . . . , 8N : =1 .....

=

P{8_ N

1} - - P { 8 _ N = 0 . . . . , 8~_x :

=

0,

. ..,

8i_

1

0, 8l ---- 1, 81+1

8 N = 1}.

The a p p h c a t i o n o f this recurrent f o r m u l a 2 N times expresses the p r o b a b i l i t y o f no o v e r s h o o t in I - - T , 7] b y the p r o b a b i l i t i e s o f all the possible overshoots, P{8_ N=O .....

8_a = 0 , 5 1

=0,..

N

= 1 --

Y,'

P{8,=

I)+

, 8N = 0 )

£

P{Sq = 1,8, 2 =

1}--...

|1
I=--N

2N

-N
s=0

2N

~(--1)s

~

s=O

--N<_tl<..

ds(tq . . . .
T h e limit o f that as N becomes infinite (A ~ O) is oO

s=O

T

--T

ts

--T

t2

--T

, tq) A q . . . Aq, (do = 1).

64

R

G

BARANTSEV

At last, by tending T to mfimty and using the symmetry of the functions ds (arising from the symmetry of the densities f ) we obtain ,z,

.o

s! s=0

3c

l>,f f -,13

d~(tl, . . ., ts) dt~ . .

dt~.

(4.14)

-oo

Let us write down some first terms of the series using the notaUons of (4.13). :20

/ - / = 1 -- f [do1 (r) + dto (r)] d r + 0 oo

:~

+'ff

(4.15)

0 0

The physical meaning of each term can be clearly seen here. Constructing the conditional probabdlty of the free motion along the broken line with m corners differs from the above only by unessential formal details and by the distribution d e n s l t i e s f c o n t a m i n g 3 m conditions.

4.3 A p p r o x t m a t e

Formulas

As can be seen from the construction, the roughness operator in the general case proves to be highly complicated F r o m the applied viewpoint it makes sense to consider some narrower classes of problems and to find for them the corresponding simplified representations of S. The specified condition provichng simplifications can be made by way of either choosing the particular models of rough surface or approximating some parts of the operator. The two-dimensional models of a strongly rough surface formed by rectilinear elements were studied in refs 93 and 94. The numerical calculations with models were also carried out in ref. 95 In ref 96 the experimental investigations were made on the special surfaces of coarse-grained roughness. Multiple reflections were studied in refs. 98 and 99. For approximating S the principal difficulty is connected with the cumbersome expression ( 4 . 1 4 ) o f H which is in fact a continual integral expansion. Let us consider the possibilities for the simplified representation of the conditional probabihty of the free motion along one ray, 0 < t < oo, emphasizing the two limiting cases of correlations. The numerical calculations o f / / f o r the lnterme&ate case have been carried out in ref. 84.

4.3.1. S t r o n g c o r r e l a t t o n s When the dependence between the values of ~: at different points is strong enough the multiple overshoots are hardly probable and m the expansion hke (4.14) only a few first terms are essential In the limiting case H ~ 1 It is not necessary to bring in corrections to this limiting value by the way of adding the next cut-off terms of the series. Other expansions are possible which have simpler expressions

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

65

of the first terms. It is fairly effective, for instance, to approximate/7 by the conditional probability that ~ < h at certain luckily chosen points tl, t2, •.., t,, n ~ ~.~ss) If one evaluates the conditional probability for ~: to be above the ray z = zo q- t cot 0 it turns out that in a wide range of zo, zo < cot 0, 0 < ~r/2, the maximum is at the distance of the order of the correlation radius p. For small t this probability is small because of the correlation and for large t the ray goes out of reach of fluctuations. Therefore in the first order (n = 1) it is expedient to assume ta = p. Let us calculate the conditional probabdity of the inequahty ~:(p) < h(p) for the normal process with the correlation function B(r) : ,2 exp (-- r2/pz). Writing down the correlation matrix of random variables ~:(0), ~:'(0), ~:(p) i

{R,s } : 1/e

0

1/e

1

x/2/e

)

~¢/2/e 1

it is not difficult to find the conditional distribution density

f(zalZo, Zo)--

1

{--1

exp

( z.

Z0

e

~- 0 ) 2}

where D = 1 -- 3/e 2,

The probability sought for is obtained by zo + p cot 0. Using the notation

lntegratmgf(zplZo, z o) over zp from -- oo to

1; V(2-)_

¢~(x) -- - -

exp (-- t2/2)

dt

(4.16)

we have H~O{-~D[

(go 1--!)

- - z o' -p + o cOt 0]

(4.17)

This formula provides a good approximation when alp is small and 0 is not close to rr/2. The result will be more accurate if the maximum point is found as a function of parameters zo, z o, 0. For n > 1 it is reasonable to choose the basic points in the same way. Strong correlations mean that the surface inclination fluctuation al is small. The asymptotic analysis o f / 7 for a~ --~ 0 has been carried out in ref. 79 b, d. 4.3.2.

Weak correlations

When the dependence between the values of ~ at different points rapidly decreases with the distance, the role of correlations diminishes. In the limiting case all the overshoots are independent (the Poisson system), ds (h . . . . . ts) = d~ 0 1 ) . . • da (t~), and the series for H is reduced to the simple form oo

/7=exp{--f

dl(t) dt}. 0

(4.18)

66

R

G. BARANTSEV

The corrections to this limiting expression can be constructed by means of different kinds of developments based on the weak relation factor It is convement, for example, to take

//

exp { ~

( - -s! l)S f • - f gs (ti . . . .

~=1

0

t) dt t . . . d6 }

(4 19)

0

where g~ are the correlation functions the advantage of which over de consists in that for the ergodic systems g~ ( t i , . . . , t~) - + 0 if It, -- 61 -+ oo for at least a single pair of (i, j). The functions d, and g~ are simply interconnected by the relations, gi (tl) ~ d~ (tO, g2 (t~, t 2 ) = d2 (t~, t2) -- dl (q) dl (t2), etc In the limiting case g~ = 0 for s >i 2 The first stage in the way of making it more exact ~s using the system of pair-correlated overshoots (the Gaussian one) for which gs = 0 for s /> 3 and

dt}. 0

(4.20)

O0

Worth noticing also is the system of non-meeting points (s~) that permits summing the series in (4.19). Another algorithm of refinement can be constructed on the basis of the Markov processes of increasing order (am.a6) Having the continuous derivative ~:' (t) it is natural to begin with the process defined by the fluctuation equation of the second order and the lninal conditions for ~: and ~:'. Its conditional distribution density w (z, z', t ]Zo, Zo) satisfies the Fokker-Planck equation ~3w ,gw ~ A ~2w + ~z' [(23z' + ,~2z)w] + --2 -~z #t = -- z' ~ - 'z (4.21) where/3, ~, A are the process parameters. At an initial point

w(z, z', 0 I Zo, z o) = 3(z -- Zo) 3 (z' -- SoL

(4.22)

z o < cot 0 The boundary conditions are (see refs. 87, 86) w--->0 w=0

for

for

z--->-- ~ ,

Z=Zo +tcot

0,

z'-->:[:oo;

(4.23)

z' <~ cot 0, t > 0

(4 24)

The solution of the problem (4 21)-(4.24) integrated in the limits -- ~ < z < Zo + t cot 0, -- oo < z' < + ~ provides the probability for the trajectory to be below the ray in the interval [0, t]. By integrating (4 21) while taking account of (4 22)-(4.24) we obtain

H=l_;;(z,_cotO) 0

w(zo+tcotO,

z,,tlZo, Z'o) d z ' d t .

(425)

cot0

Thus, in the case of the Markov process of the second order the evaluation of the continual integral/7 is reduced, in essence, to the solution of the boundary problem for eq. (4.21) with the absorbing boundary (4.24). But to find this solution is not simple either. There are, besides, the difficulties with the formulation of Markov process applicability conditions for

67

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

the problem considered. (s6) An additional comphcation is the violation of the stationary state of ~ (t) with the tranmion through the corner. The asymptotic evaluations o f / 7 for a~ -+ oo have been given in ref. 79 b, d. 4.4. Gaussian Surface 4.4. I. Gaussian distributions Both the full solution of the problem and its slmphfied variants contain the multidimensional distribution densities f that become determined by specification of surface properties. In many cases the roughness resulted from the action of a lot of independent random factors, the influence of each being uniformly small. Then, according to the central limit theorem, the distributmns of ~ at different points (x, y) may be assumed as close to the normal (Gaussian) ones. (aS) Let

f(Zl,

g2,

".-,

Zn) --

D,j z, zj ,

V'{(2zr)"D} exp

(4.26)

l,J= 1

where D is the determinant of the dispersion matrix {M,~}, Dij, the algebraic adjunct of the (]-element, M u = B(rij), rij = v/{ (x~ -- xj) 2 + (y~ -- yj)2 ), B(ro), the correlation function defined as the mathematical expectation of the product ~ (x, yi)~ (xj, yj). The normality being lnvariant under differentiation, the joint distribution densities of and its derivatives also have the form (4.26), except the dispersion matrix is more complicated. It contains the correlation functions of ~, ~, ~y and the mutual ones. They are all expressed through B(r~j) by means of differentiation. ~Taa) Let us write them down in the following table"

¢(xj, y j)

¢x(xj, yj)

¢,(xj, y~)

OB

OB

~xj

Oyj

~B

02B

~2B

cgxi

3x, Oxj

Ox, Oyj

OB

02 B

~92B

OYi Ox,

OYt OYj"

~(xl, Y,)

B

~(x,, y,) ~,(xi, y,)

For example, the dispersion matrix of ~(0, y), ~y(0,y), ~(0, 0), ~x(0, 0), ~y(0, 0) taking part in (4.12) and (4.6) is

[~

(o)

o --B"(O)

S(y) B'(y)

" \-s

(y)

o -s"(o)

o o

o 0

-S'(y) \ B"(y)l

o

o

--S'(O) o

--S"(O)/

!

(4.27)

Thus, in case of the normal surface all the distribution densities entering the roughness operator are expressed through the correlation function B(r) only.

68 4.4 2.

R G BARANTSEV

Single reflection

It is useful to separate the simplest form of S reached on a slightly rough surface. The stronger the correlations, the smaller the number of overshots and multiple reflections. Therefore, on a sufficxently gentle surface, we can simultaneously let / / - 1 in (4 5) and hmlt ourselves only to the first term in (4.4)_ Then the expressmn of V~ is somewhat simplified because lntegratlngf(zo, Zoo, Zoo) over Zo results l n f (Zoo, Zo~).Keepmg Nt for normahzatlon and control we have

V(uj, u)

=

,ff

Vo(Ul, u)f(zo~, Zo,) cos 01 cos 4 tg"

~11

(4.28)

utn
On a normal surface, according to (4.26) and (4 27), f ( Z o x ~ Zoy) - -

1

27r In"(0) I

exp ~z2~ + z2y~ [ 2B-~ J

1 { - - t a n 2 a} 2fro2 exp 2cr~

(4.29)

where

(r~ = -- B"(O)

(4.30)

Thus, the single reflection function on a normal gentle surface is

,

V -- 2mr~N------~

f f

Vo exp

(

tan 2 o~ dw 2~----~ ! (1 + tan 01 tan a sin 4') cos s a

(4.31,

/dln < 0 "CI/n

Here the only parameter of roughness is the inclination fluctuation al From the physical viewpoint within the framework of approximation (4.28) it is natural to enlarge the range of integration up to all a ~ [0, ~r/2), ~ ~ [0, 2r0. This results m the simplest approximate form of the roughness operator on a normal gentle surface, 2r~ n / 2

tan 2 o~ tan o do d~ (4.32) ! (1 + tan 01 tan v~ sin ~b) cos2------~

1

V_ _ _

0

0

The simplification leading to (4.31) and (4.32) for small ol can be regarded as justified enough when the incidence and reflectmn angles are not close to rr/2. With 0j and 0 approaching ~/2 the formulas obtained become rough. The non-uniformity of the approximation for 0 -+ zr/2 becomes particularly apparent when Vo has a grazing reflection spectrum. For 01 --* ~r/2 the shadowing effect increases. The "own-shadow" probability

27r~

exp

2a~

cos O1 cos 4 a

al>nl2

can be estimated comparatively easily. Its values are gwen in Table 5.

(4.33)

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

69

TABLE 5.

Ol °

o~

60

75

80

1/5 1/3 1/2

0 0.005 0.028

0.015 0.071 0.173

0.062 0 182 0.353

Judging by this table the single reflection formulas can be used for 01 ~ 75 °, ol ~ 1/3.

4.4.3. Information o f t 1 The practical use of the above results to real rough surfaces needs the knowledge of the mchnatlon fluctuation o 1 of these surfaces. Among the roughness criterm named in conventional standards o~ is not given. At the same time, to conclude something about ol by the smoothness class is rather difficult. The devices to measure the frequency characteristics of rough surfaces directly exist now as test patterns only. At present crl is determined by means of the profilograph treatment wnth the help of the Rice formula (aa.92) or other considerations. (ag~ The values of ~rl obtained on the surface patterns of smoothness classes 4-13 proved to be of order 10-2-10 -1. For such c,1 the accuracy of formula (4.32)is quite acceptable, but the roughness effect proves to be rather weak. However, the profilographs used are essentially coarse owing to the lack of resolution.C9°) Therefore, for instance, the surface area value measured proves to be significantly smaller than the real one. (91) If we assume the profilographs treated to have the small-scale roughness unresolved then, as it is shown m ref. 79a, the dispersions of the two levels are in a first approximation added and the roughness effect Increases accordingly. Thus, the existing data allow some hope that the formulas (4.31), (4.32) are of a direct practical interest, besides the theoretical one. The parameter ol should be included m the certificate characteristics of aerodynamic coatings, simple and rehable methods to measure it should be worked out and the production of the necessary exact measuring devices started. "°1-2)

4 5. Asymptoncsfor Small cr1 The simplified representations of S for a gentle surface were obtained in Section 4.4 on the basis of considerations which were not mathematically rigorous. It is desirable to estimate the orders of the terms omitted with respect to crI and to retain only the lower orders in the expressions retained.

4.5.1. Scattering function Let us find, first of all, the asymptotic expansmn of S in the simplest representation (4.32).

70

R. G. BARANTSEV

The function Vo(ul, u) is given m the coordinate system (b, t, n) connected to the normal in the micro-scale n = {sin o cos 4', sin o sin 4', cos a}.

(4.34)

In accordance with (4.2) the axes are oriented so that in (b, t, n) I11 =

{0, -- U 1

U =

sin al,

{U s i n 0. c o s / 3 ,

- - Ul c o s

u sin a

(4.35)

0.1 },

sin/3,

u cos

(4.36)

0.}.

Thus, in (4.32) V = V(ul, 01; u, 0, 9; ch), Vo = Vo(ul, 0`1 ; //, O., /3), and according to (4.2), (4 3), (4.34)-(4.36) the variables al, a, fl are expressed through o, 4', 0 1 , 0 , t~0 by COS 0`1 =

COS 01 COS ~ AV S l n

01 sin

4',

a sin

cos a = cos 0 cos a + sin 0 sin t9 COS

(4.37)

(4' --

cp),

(4.38)

cos 01 cos 0 + sin 01 sin 0 sin,p -- cos al cos a tan/3

=

sin 01 sin 0 cos rp cos a -- [cos 01 sin 0 sin (4' -- ~o) + sin 01 cos 0 cos 4'] sin o (4 39)

The last formula has been obtained with the help o f the relations b --

-n l - )< n

,t sinai +ncosa,

--

ul sin a I

I11

.

(4.40)

/d 1

Let us separate the normalized peak-shaped function 1 ( h(o, °"1) = 0"-~exp

tan 2 a ~ t a n a ~-~2 ] cos: o

(4.41)

and write down (4.32) in the f o r m n/2

N1 V = | h(o, o,1) A(o; up O, ; u, O, q~) do

(4.42)

t/ 0

where

if

A = ~

2r~

(1 + tan 01 tan a sin 4') Vo(ul, al ; u, a, 3)d4'

(4.43)

0

As h(~,el) ~

8(0), in order to obtain the asymptotic expansion o f V we have to expand

the funcUon A = A(o) near a = 0. F o r a sufficiently smooth Vo

A(a) = A(O) + A'(O) a + ½ A"(O) a 2

- ~ O(b~2)

(4.44)

where

if

A(0) = ~

0

2/t

Vo I~=o d~,

(4 45)

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

1;{

71

2~

A'(0) = ~

tan 01 sin ~ Vo

+ ~

(4.46)

~=o

0 2/t

A"(O) =

1

~

fl

OVo

2 tan 01 sin ~ ~

a=o

2vo

+ ~

447,

~=o

0 Then (4.48)

N 1 V = A(0) -]- A'(0) &/2 Crl -]- A"(0)o~ -~- o (cr12)

The function Vo d e p e n d s on o t h r o u g h a l , a a n d ft. Using (4.37)-(4 39) we find that for ~ --->0 o,1-.-->01, o..--+O, fl'-'->rp, O~ I -~ - ÷ - - sin

O~ ~,-~

~ - - cos (~ - - ~o),

- - -~- - - cot Ot cos $ - - c o t 0 sin ($ - - 9),

~2~ 1

--

002

~2~ (4.49)

-~- cot 01 cos 2 ~, ~-~ ~ cot 0 sin 2 ($ - - ~o), (1 + 2 cot 2 0) sin ~o cos ~o cos 2~b

~02

+ [(1 + 2 cot 2 0) sin 2 cp - - (1 - - cot 2 al + cot z 0)] s i n 2 q~. Hence, Vo I~q= o = Vo (ul, 01 ; /-/, 0, ~p), ~Vo Oo ~=o - -

(4 50)

~Vo ~Vo ~Vo 001 sin ~ - - ~ cos (~ - - qo) - - ~ [cot 01 cos ~ + cot 0 sin (~b + ~)] (4.51)

f r o m which, b y (4.45) a n d (4.46), A(0) = Vo, A'(0) = 0.

(4.52)

By calculating in the same way c92V°~02 ~=o a n d A"(0) we eventually o b t a i n

f

~2Vo

NI V ~--- Vo + Ol2 ½ ~ c~2Vo

-[- ~

02V 0

cot 01 COS ~0 -~- ½ - - ~

~Vo (½ cot

+ -~

~V 0

+ " ~ 1 (½ cot 01 - - t a n

. 02Vo

-1- ~ ~

~Vo t a n

0 - - t a n 01 sin ~o) - - - ~

~2V 0

01) -~ ~

,

£~2V0

sin ~ + ~ 0 ~

cot 0 cos ~0

[c0t 2 01 - - 2 cot 01 c0t 0 sin ~o + cot 2 O] 01 c o t 0 cos ~0 + o (cry).

(4.53)

I f some derivatives d o n o t exist the f o r m u l a s change accordingly beginning with (4.44).

72

R

G. BARANTSEV

The estimates of the influence of the integration limits in (4.31) ~Tsc~ and of multiple collisions ~79b) show that the asymptotic expansion (4 53) does not depend on the simplifications leading to (4.32) and is rigorous for fixed 01 < ~r/2, 0 < ~/2 However, the expansion (4.53) becomes invalid when the Incidence and reflection angles approach ~r/2 The influence of multiple reflections on the asymptotics of V in this region remains unknown so far. The exact integration limits in n for the single reflection have been considered in ref 78c where it has been found, in particular, that for 0 r < ~/2 and 0 ~r/2 ~1 [~Vo c~Vo ~/(2~r) [_c~0, sin q, + ---~-

N~ V = ½ Vo + ~

cot 01cos ~ -- Vo tan 01sin ~

]

÷o(a~).

The expansions (4 53) and (4.54) within the framework of 4.31 remain vahd ff 0

(4.54) ~ ~r/2

slowly enough in the first case and rapidly enough in the second one. The transitmn from one set of asymptotics into the other occurs for cos 0 = o (e~). The asymptoUcs of V1 in this layer is burdened by the complicated integration limits appearing in (4.43)

4.5.2.

Exchange coefficients

The summarized m o m e n t u m and energy fluxes onto the surface for a fixed incidence velocity ul introduced in Section 1 1.4 are calculated by the formulas - :fffv ,,-u

q----

cos 01 du,

(4.55)

Ul

itz ~. 0

f f f(u ) V 1 --

cos 01 du.

(4.56)

uz>O

Substituting here the expressions of V through 11o obtained above we shall seek the s~mplifled p and q representations on a gentle normal surface for an arbitrary 1Io. The principal difficulty consists m that the asymptotic development of S for or1 -+ 0 is not known all over the integration range uz > 0. Besides (4.53), the integral formulas of the single reflection (4.32) and (4.31) also lose validity as 0 ~ ~/2 However, the dangerous zone is small and we can hope that m a wide class of Vo the specifications for p and q near 0 = w/2 wall require only a small correction. As a first stage it makes sense to extend the single reflection formulas all over the integration range in (4.55), (4.56). The simplest result is reached by taking V in the form (4.32), inverting the integrations and changing uz > 0 into u, > 0. Then the normalization yields N1 = 1, and from (4.55), (4.56) we obtain 2n ~t/2

h(o, ,,1) po cos-----~'

p = ~ O

0 2n n / 2

q = ~

h(o~, o.1) 0

O

qo cos

(4.58)

SOME PROBLEMS OF GAS--SOLID SURFACE I N T E R A C T I O N

73

where h(o, al) has been defined by (4.41) and

o--fff

cos al du,

, u~>O

fff

qo=

(4.59)

U1

Vo 1 - -

cosaldu.

(4.60)

un>O

Assuming the functions Po (ul, t : l ' l ) , qo (ul, al) to be known we can obtain the asymptotic expansions of the integrals (4.57), (4.58) in the same way as for (4.42). Let us first project (4.57) onto the axes y and z, 2~ ~/2

h(O, al) (,or, -}- pon,) "~os~"

1 2~

"r =

--

0

(4.61)

0

2n

hi2

,ff h(o, a,)

p = ~

(zot= q- pon=) do dff. cOS v~

(4.62)

0 0

Separating the integrals 2z Ar

~

m

1

27r cos o

f (zoty + pon~) d~, O

2x

1 f( ot= q-pon=)d~,

Ap = m 2~ cos o

(4.63)

0

2~

,f

Aq =

2~r cos ~

qo d~ 0

we have 2~

• ,p,q = f h(o, al) A,,p,q (~; ul, 01) do.

(4.64)

0

If the functions A,,p,~ (~) permit the expansion (4.44) near zero the asymptotics of the integrals (4.64) is like (4.48). Using the formulas (4.37), (4.34), (4.40) for calculating the deri-

74

R

G

BARANTSEV

vatlves we eventually obtain

{

r(ul, 0j,ox)= ro(u,,O~) 4- o~ ½ [702 4 - ~

c o t 0 , 4- %(1 - - c o t 20,)

p(ul, O,,ol) = po(u,, 01) 4- o~ ½ \ ~

cot 01 4 - ~

4- ~

l

-- ?0~J

4 - r o c o t el

k o(o;),

÷o(oZ~), (4.65)

q(ua, 01, Ol)

=

qo(ul, 01) 4- o2 ½ \ - ~

4- ~

cot 0~

4- q0

4- o(o~).

In particular, for 0~ = 0 we have r = 0, and

p = po + Ol ~--g-~- +

~ 0 # + oto~), (4.66)

2 ( ~2q° 4- qo)

q = qo 4-% \ - ~

4- o(o~)-

The asymptotlcs taking account of the exact integration limits in (4.31) has been f o u n d in ref. 78c It contains the functionals o f 1Io whmh are not reduced to to, Po, qo The multiple reflection effect is not yet taken into account.

4.5.3. Results for particular Vo F o r illustration, we examine the roughness effect for two particular cases o f the reflectmn in the small area, writing down the scattering function and the m o m e n t u m and energy exchange coefficients for them on a gentle normal surface (a) Diffuse Vo F o r the diffuse reflection on ds (see (1.13)) 2h 2

Vo = Vo(u, ~; h) = - - u cos ct exp ( - - hu 2)

(4.67)

7T

where h - * is the most probable velocity o f emerging atoms. According to (4 53) for 01 < ~r/2 -- ~,0 < r r / 2 - - ~ w e h a v e

V1 = Vo(u, O; h) (1 -- %2 + o~ tan 01 tan 0 san ~o) ÷ o(o~)

(4 68)

The extension o f this to include all uz > 0 after n o r m a h z a t l o n gives

V = Vo(u, O; h) (1 + o 21 tan 01 tan 0 san ~o) + o(o~)

(4.69)

F r o m (4.59), (4 60) and (4 67) we find %(%) = sin % cos %,Po(%, ha) = 2h---7 cos % + cos 2 %, (4 70)

qo(at, Al) = COS al ( 1 - - ~ ) , A, = ~hul.

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

75

N o w (4.65) provides

= ~o(01) + ~2 ~

sin 01 + o ~ ) ,

p = po(01) _ o2 ~

cos 01 + o(cr2),

(4.71)

q = qo (0~) + 0(o2). It is noticable that the procedure concerning the range of Integration over u used when deriving the formulas (4 57), (4.58) affect the first terms of the N1 and p expansions. I f the integration is carried out in u, > 0 the coefficients of Ol 2 in these expansions are equal to 0 and -- x/rr/(2A1) cos 01 respectively; if it xs made in u~ > 0, then they are -- l and 0 (see ref. 78a). A n d with the exact hmits of integration (u~ > 0 and u~ > 0 together) we have obtained, in ref. 78c, the values --1/2 and --~/rr/(4 A1) cos 01. This fact shows that for the diffuse Vo the single reflection approximation (4 31) is insufficient, first o f all, for N~ and p. (b) Specular Vo. For the specular reflection on ds Vo =

(4.72)

8 ( u - - u I q- 2Uln ).

This function does not have the properties permitting to apply the asymptotic expansion (4.53). G o i n g back to (4.3 I) let us integrate it exactly using the rule of integrating the delta function of a complicated argument. In the spherical coordinates we have u = u~, (0, ~) and (#, ~) interconnected by the relations sin 0 cos ~0 = 2 cos al sin 0 cos ~, sin 0 sin

W =

--

sin 01 -- 2 cos a~ sin 0 sin if,

cos0=--cos01+2cosa

(4.73)

lcoso,

and c o s 2 or.1 =

c o s (u, - - U l ) =

c o s 01 c o s 0 ~- s i n

01 s i n

0 s i n ~.

(4.74)

F r o m here sin OdOdqo= 4 cos al sin od od~,

(4.75)

and the integration of (4.31) now results in V = 8(u -- ut) exp [ - - tan 2 ~/2 ~ ] 8 ~r o2 u~ N1 cos 01 cos 4 0

(4.76)

where ~ is expressed through 01, 0, 9~ by (4.73), (4.74). The n o r m a h z a t i o n shows that for 01 < rr/2 - c N1 differs f r o m I by an exponentially small value (see ref. 78a). Then, with ~'o = 0, po(~l) = 2 cos 2 ~1, qo : 0

(4.77)

we find from (4.65) • = 2 , 1 2 sin 2 01 P = P o (01) q=0.

-

-

+o

(o12),

4 ~i 2 (1--~ sin 20,) + o (012),

(4.78)

76

R G. BARANTSEV

In ref. 78a these f o r m u l a s have been o b t a i n e d directly f r o m (4 76) and given in the tables o f Nt, T , p for 0~ = 0 (15 ° ) 75 ° , ~rx = 1/10, 1/5, 1/3. The multiple reflectmn effect for the specular Vo seems rather w e a k a s y m p t o t i c a l l y The two-dimensional scattering p r o b l e m on a gentle n o r m a l surface with specular Vo m the single reflection a p p r o x i m a t i o n was considered in ref. 97. L I S T OF M A I N N O T A T I O N S

!

d l s t r i b u t m n function o f gas particles impinging particle velocity U emerging particle velocity V,W,R scattering, sputtering, a n d emlssmn functions S trapping probabdlty v,~ scattering indicatrlx E , , E , , E o incidence, a d s o r p t i o n , a n d b o n d energies U., Uo gas-surface a n d lathce a t o m interaction potentmls T~ surface t e m p e r a t u r e E,, EO, E s dimensionless energy p a r a m e t e r s dimensionless spatml p a r a m e t e r s a , , O"0 p gas-to-surface a t o m mass ratio p o t e n t i a l b a r r i e r inclmatlon Y b, t, n Cartesian c o o r d i n a t e s related to the n o r m a l 0,6 p o l a r a n d a z i m u t h a l angles o f u ~j mcMence angle z,p,q tangential a n d n o r m a l m o m e n t a a n d energy exchange coefficients m e a n velocity a n d direction o f scattering Urn, Om dispersion p a r a m e t e r s o f scattering function r a n d o m surface S roughness o p e r a t o r B c o r r e l a t i o n function 171 i n c h n a t l o n fluctuation o f the r o u g h surface Ul

REFERENCES 1. Rarefied Gas Aerodynamws, Ed. by S. V. VALLANDER(a) I--t963, (b) II--1965, (c) III--1967, (d) IV-1969, (e) V--1970, (f) VI--1972. Lemngrad Umverslty, Leningrad. (In Russian ) 2 Rarefied Gas Dynamtcs, Proc_ of the Intern. Syrups (a) V--1967, (b) VI--1969, (c) VII--1972. Academm

Press, N Y.-L 3 High and lntermedtate Energy Molecular Reams, Proc of the Intern Symps_ (a) I--Entropm, No 18,

1967, (b) II--Entropie, No. 30, 1969, (c) III--Entropie, No 42, 1971_ 4. Fundamentals of Gas-Surface lnteractwns, Proc. of the Symp, San Diego, 1967, Academic Press,

N_Y -L 5 Oddzlatywante gazu ze gclankamt statymi, Praca zblorowa, IPPT PAN, Warszawa, 6/1969

6 R. G BARANTSEV,A I. EROFEEV,YU. D NACORNYKH,D. S STR1J~NOVand A. A PYARNPUU,Rarefied gas Interaction with solid surfaces Proc of the IIlrd All-Umon Conference on Rarefied Gas DynamtcL Novoslbirsk, 1972 (In Russian.) 7 R.G. BARANTSEV,Recent research results on gas-surface interaction. Survey lecture at the Xth Symp on Advanced Problems and Methods in Fluid Mechamcs, Fluid Dynamic Transacnons, Vol. Vl, Part l, 17-75, Warszawa, 1971. 8 CH. KFI'rEL, Introduction to Sohd State Physics, John Wiley & Sons, Inc., New York, 1956. 9. J. O. HIRSCHFELDER,CH F. CORTISSand R B BraD, Molecular Theory of Gases and Liquids, John Wiley & Sons, Inc., New York, 1954_

SOME PROBLEMS OF GAS-SOLID SURFACE INTERACTION

77

10. F. O. GOODMAN,Interaction potentials of gas atoms w~th cubLc latUces on the 6-12 pairwise model. Phys. Rev 164, No. 3, 1113-23 (1967). 11. D. R. WILLIAMS, L J SCHAADand J. N. MURRELL, Deviations from pa~rwise addlttvlty m lntermolecular potentials J. Chem. Phys. 47, No. 12, 4916-22 (1967). 12 P T. SIKORA, Combining rules for spherically symmetric intermolecular potentials J. Phys B 3, No. 11, 1475-82 (1970) 13 R J. G o o d and C n J. HOPE, New combining rule for mtermolecular distances in lntermolecular potential functions. J Chem. Phys. 53, No. 2, 540-3 (1970) See also 55, No. 1, 111-16 (1971). 14. A. G. STOLE, D L. SMITH and R P. MERRILL, Scattering of the rare gases (He, Ne, Ar, Kr, and Xe) from platinum (III) surfaces. J. Chem Phys 54, No 1, 163-9 (1971) 15. M. N. BISI-IARAand S. S. FISHER, Observed intensity and speed distributions of thermal-energy argon atoms scattering from the (iII) face of silver. J Chem Phys. 52, No 11, 5661-75 (1970) 16. R. L. PALMER, J N SMITH, JR, H. SALTSeURG and D. R. O'KEEFE, Measurements of the reflection, adsorption and desorption of gases from smooth metal surfaces J Chem Phys 53, No. 5, 1666-76 (1970). 17 D . R . MILLER and R B. SUBBAR-~O,Scattering of 0 06-2.5 eV neon and argon atoms from a silver (III) crystal J. Chem. Phys. 52, No. 1,425-31 (1970), see also ref. 2c 18 M.J. ROMNEY and J. B. ANDERSON, Scattenng of 0.05-5 eV argon from the (ill) plane of silver. J. Chem. Phys. 51, No. 6, 2490-6 (1969) 19. 'V S. CALIA and R. A. OMAN, Scattering cross-section measurements for eplthermal A r on Ag (III) surfaces J. Chem. Phys. 52, No. 12, 6184-8 (1970). 20. R. A. OMAN, The effects of interaction energy m numerical experiments on gas-surface scattering [2b], 1331--44 21. R M. LOGAN and J. C. KECK, Classical theory for the interaction of gas atoms with solid surfaces. J. Chem. Phys 49, No 2, 860-76 (1968). 22. R. M. LOGAN, Calculation of the energy accommodation coefficient using the soft-cube mode Surface Sci 15, No. 3, 387-402 (1969). 23. U. A ARIFOV, Atomic Particle Interaction with SohdSurface, Nauka, Moscow, 1968 (In Russian.) 24. M. KAMI~SKY, Atomic and Ionic Impact Phenomena on Metal Surfaces, Springer-Verlag, BerlinHeidelberg-New York, 1965. 25. G. CARTER and J S COLLIGON,lon Bombardment of Sohds, Heinneman Educ. Books L t d , London, 1968. 26_ Materials of the All-Union Conferences on Emission Electronics, Proc Acad SCL USSR, Phys. Ser. (a) XIII--33, Nos. 3, 5, 1969; (b) XIV---35, Nos 2, 3, 1971 (In Russian.) 27 H . D . HAGSTROt,I, Reflection of noble gas ions at sohd surfaces. Phys. Rev. 123, No 3, 758-65 (1961). 28 K. J CLOSE and J YARWOOD,The trapping of low-energy noble gas ions at a tungsten surface Brit J Appl. Phys 18, No II, 1593-8 (1967). 29. F. M DEVIENNE,J. C ROUSTANand M RIVES,Vitesses et rrparUuon spatlale des mol6cules "rrflechies" aprrs impact sur une surface d'alummium d'un jet moleculalre d'argon de haute 6nergie J de Phys 27, No. 1/2, 83-93 (1966) 30 J C ROUSrAN, Reflection des jets moirculaires d'argon d'rnergle 6gale ou suprrleure h 100 ev aprrs Impact sur une surface d'alumlnmm Compt. rend. Acad Sct. 265, No. 20, Bl104-6 (1967). 31 C . A . VISSER,J. WOLLESWINKEL and J. Los, On the interaction of hyperthermal potassium atoms with a tungsten surface. [3b], 61-64. 32 J. W BORrNG and R. R HUMPI~RlS,Drag coefficients for free molecule flow in the velocity range 7-37 km/sec. AIAA J 8, No 9, 1658-62 (1970); see also [2b], 1303-10 33 F O. GOODMAN, (a) Classical perturbatmn theory of the thermal accommodation coefficient in n dimensions Surface ScI. 11, No. 2, 283-316 (1968); (b) Classical theory of small energy accommodation coefficients. 3"_ Chem. Phys. 50, No. 9, 3855-63 (1969), (c) On the trapping process in gas-surface mteracUons, [2b], 1105-18. 34 V. E JtJRASOVAand D S Kagpuzov, Ion reflection from monocrystals for obhque incidence. Sohd State Physics (USSR) 9, No. 9, 2058-13 (1967). (In Russian ) 35 J. LORENZEN and L. M. RAFF, Theoretical investigations ofgas-sohd interaction phenomena J Chem. Phys_ (a) 11--49, No. 3, 1165-77 (1968), (b) 111--52, No. 3, 1133-42 (1970); (c) IV--52, No. 12, 613440 (1970), (d) Quantitative comparison of gas-surface theory with molecular beam data, 54, No. 2, 674-9 (1971) 36 A A PYAR~PUU, Interaction calculation of monoenergeUc gas atom beam with a three-dimensional crystal. AppL Mech. and Techn. Physics, No. 2, 162-6 (1970). (In Russian.) 37. F. C. HURLaUr, Gas-surface interaction studies employing a three-dimensional coupled lattice model, [3b], 107-12. 38 D . P . JACKSON and J. B. FRENCH,High energy scattering of inert gases from well-characterized surfaces. II Theoretical, [2b], 1119-34.

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39 S_ YAMAMOTOand R E STICKNEY, Molecular beam study of the scattering of rare gases from the (110) face of a tungsten crystal J Chem Phys 53, No. 4, 1594-1604 (1970). 40. V I VEKSLER, Secondary emission of atomic particles from metals bombarded by positive tons of low and moderate energies Fan, Tashkent, 1970 (In Russian ) 41. F O_ GOODMAN, Review of the theory of the scattering of gas atoms by solid surfaces Surface Science 26, No. 1, 327-62 (1971). 42 JA. I. FRENKEL, On the theory of accommodation and condensation Prog. m Phys Sct 20, No 1, 84-120 (1938) (In Russian ) 43 D S. KARPUZOV and V E JURASOVA,Ion scattering by a monocrystal of Cu at low energies (50-500 eV), [26], 393-7 (In Russian ) 44. A I. EROFEEV, (a) On interaction of atoms with a solid surface. EngngJ (USSR) 4, No. 1, 36-44 (1964); (b) On energy and momentum exchange between gas atoms and molecules and a solid surface. Appk Mech. and Techn Physics, No. 2, 135-40 (1967), (c) On fast particle interaction with a sohd surface. CAGI Trans 1, No. 4, 52452 (1970) (In Russian ) 45. J FALCOVITZ,L TRILLING, H Y WACHMANand J L KECK, Theory of atomic scattering from crystal surfaces, [2c] 46. O. B FIRSOV, Atom interaction potential calculation, J. Exp Theor Phys. 33, No 3, 696-9 0957) (In Russian ) 47 E. S PARILIS, Some Problems of Atom Colhston Theory in Gases and on Solid Surface, Tashkent, 1970. (In Russian.) 48. B BAULE, Theoretlsche Behandlung der Erschelnungen In verdtinnten Gasen Ann. d. Phys 44, 145-76 (1914) 49. R. G BARANTSEV,(a) On the impact transforms of the kinetic equation of rarefied gas aerodynamics. [la], 80-91, (b) The model of isolated reflection of gas atoms from sohd surfaces, [lb], 253-71 ; (c) Atom reflection by a hot lattice of hard spheres, [26b], 424-6 50 R. H NOTTER, Y. J KAKU and N_ F SATHER, A molecular model for tangential momentum accommodation, AIAA J , 8, No_ 11, 2064-6 (1970); see also 9, No 5, 965-6 (1971) 5l F O_ GOODMAN, (a) Three-dimensional hard spheres theory of scattering of gas atoms from a solid surface I Limit of large incident speed Surface Sci 7, No 3, 391-421 (1967); see also ref 2a; (b) NASA Report CR-933, 1967 52 V. E. JURASOVA, Anlsotropy of reflection of bombarding ions by a monocrystal surface. Proc Acad. Sct USSR, Phys Ser, 28, No 9, 1470-3 (1964) (In Russian.) 53 R. G BARANTSEVand N_ I MERKULOVA, Soft sphere lattice scattering I First approximation for normal incidence. Leningrad University l/estnik, No 7, 82-88, 1972. (In Russian.) 54. E N. MURZOVA,B. V FILIPPOV and I. M TSITELOV,An improved model of molecular beam interaction with a clean surface, [ld], 41-45 55. K C. CHIANG and E L KNtYrH, Interactions ofhyperthermal gas particles with contaminated surfaces. J. Chem. Phys. 53, No 6, 2133-42 (1970); see also Rep No. 69-53, Unlv of Calif., 1969 56 M R. BusHy, J D HAYGOODand C. H. LINK, JR, Classlcalmodel for gas-surface Interaction. J. Chem. Phys 54, No 11, 4642-7 (1971). 57 R. SCnAMnERG, A new analytic representation of surface interaction for hyperthermal free-molecule flow with application to satelhte drag Heat Transfer and Fluid Mech Inst. Papers, Stanford, 14 p p , 1959. 58. M_ EPSTEIN, (a) A model of the wall boundary condition in kinetic theory. A1AA J 5, No 10, 17971800 (1967), (b) Application of an improved model of the wall boundary condition to molecular beam scattering from a solid surface, [3b], 73-78 59 G. DAURY, Contribution b. l'6tude de la r6flexlon mol6culaire Ann. de Phys. 4, No 3, 327459 (1969). 60 T. MARSHALL JR., Modified maxwelhan models for surface re-emission In free-molecule flow, [2b], 1079-86. 61. R G BARANTSEVand V G LANDMAN, Maxwelhan representation of reflected atom distribution by mass, momentum and energy fluxes, Leningrad University Vestnik, No 19, 58-63 (1966) (In Russmn.) 62 R . G . BARANTSEVand V M FEDOROVA, (a) Ray model for atom reflection from surface Arch. Mech. Stos. 21, No. 3,383-8 (1969): Application of ray reflection model for calculation of near-free-molecular gas flows, [le], 91-106. 63. E P_ WENAAS,Equilibrium cosine law and scattering of the gas-surface interface J Chem Phys 54, No. 1, 376-88 (1971) 64. J KUg~ER, Reciprocity in scattering of gas molecules by surfaces. Surface Sct, 25, No 2, 225-37 (1971) 65. C CERCIGNANIand M LAMPIS, Kinetic models for gas-surface interactions, Transp. Theory andStat Phys., 1, No. 2, 101-114 (1971)

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66. C. CERCIGNANI,(a) Boundary value problems in linearized kinetic theory. Proc. Syrup. Transport Theory 1, 249-68, SIAM-AMS (1969), (b) Mathematical Methods m Kinetic Theory, Plenum Press, N Y., 1969; (c) Models for gas-surface interactions: comparison between theory and experiment, [2c], 1971 67. S. NOCrLLA, The surface re-emission law m free molecule flow, Proc of the 3rd Int. Syrup on Rarefied Gas Dynamics, Academic Press, N.Y.-L., 1, 327--46 (1963). 68 R. G BARANTSEVand V P. PgOVOTOROV, Boundary transform expansion, [lc], 67-74 69. S F SHEN,Parametric representation of gas-surface interaction data and the problem of slip-flow boundary conditions with arbitrary accommodation coefficients, [3a], 138-45. 70 F. O GOODMAN,Empirical representation of velocity distribution density function of gas molecules scattered from a solid surface. The Structure and Chemistry of Sohd Surfaces, Proc. of the 4th Int. Materials Symp., 47 pp. Wiley, N.Y.-L.-S -T., 1969 71 R G. BARANTSEV, Aerodynamics of non-convex bodies in a stationary free-molecule flow, [ld], 46-55. 72. E C HURLBUT and F S SHERMAN, Application of the Nocilla wall reflectionmodel to free-molecule kinetic theory Phys Fluids11, N o 3, 486-96 (1968) 73. F. O. G O O D M A N and H Y WACHMAN, Formula for thermal accommodation coefficients.J Chem Phys 46, N o 6, 2376-86 (1967) 74 R. N. MIROSHIN,Linear regression analysis of experimental data in a rarefied gas, [le], 14-38. 75. G R. KARR and S. M. YEN, Aerodynamic properties of spinning convex bodies in a free molecule flow,

[2c] 76 R . G . BARANTSEV,Modelling of gas-surface interaction and the problem of highly rarefied gas flow past bodies, [2c] 77. D R MILLER and R B SUaaARAO, Direct experimental verification of the reciprocity principle for scattering at the gas-surface interface J Chem. Phys 55, No 3, 1478-9 (1971). 78 R. G BARA~TSEV, (a) Gas molecule reflection on rough surfaces, [la], 107-51; (b) Asymptotic formulas for momentum and energy exchange coefficients on body surface In rarefied gas flow. Lemngrad University Vestnik, No. 13, 69-76 (1963), (c) Single reflection asymptotics on a slightly rough surface. IbM No. 7, 126-33 (1967). (In Russian.) 79 R N_ MIROSHIN, (a) Gas atom scattering on a rough wall formed by two-scale irregularities, Leningrad Umversity Vestnik, No. 19, 154-6 (1963), (b) Boundary transform asymptotics with respect to the roughness parameter, I--[lc], 124-51, II--ibtd, No 13, 96-101 (1968), (c) On the Rice series in the random function theory, ~btd, No 7, 55-65 (1968), (d) Asymptotics of the second factorial moment of the number of intersections upwards of the straight line kt + a by a Gausslan stationary process tbtd, No. 19, 109-20 (1968). (In Russian.) 80 S. O. RICE, Mathematical analysis of random norse, Bell System Tech J 23, No 3, 282-233 (1944), 24, No 1, 46-156 (1945). 81 R. L STRATONOVICH,Selected problems of the fluctuation theory m radtotechnics, Sov. Radio, Moscow, 1961 (In Russian.) 82 V I. TIKHONOV, (a) Statistical radlotechmcs. Sov. Radio, Moscow, 1966, (b) Random process overshoots Nauka, Moscow, 1970 (In Russian.) 83. P I. KUZNETSOV,R. L STRATONOVICH and V I TIKHONOV,On the duration of random function overshoots J Techn. Phys 24, No. 1. 103-12 (1954) (In Russian.) 84 M V ANOLIK and R. N. MIROSI-nN, (a) Evaluation of continual integrals in the problem of gas atom reflection from a rough surface Leningrad University Vestmk, No 7, 52-55 (1971), (b) Single reflection of gas atoms from a rough surface, Calculation Methods, No. 7, Leningrad University, 1971. (In Russian.) 85. M. S LONGUET-HIGGINS,On the intervals between successive zeros of a random function Proc Roy. Soc A 2.46, No 1244, 99-118 (1958). 86. R. G. BARANrSEVand R N. MIROSHIN,On the approximate representations of the roughness operator. [la], 152-61 87 C M HELSTROM,The distribution of the number of crossing of a gaussian stochastic process IRE Trons, IT-3, No 4, 232-7 (1957) 88 Ju. V LINNIK and A. P. Husu, Mathematical-statistical description of grinded surface profile irregularities Engng. Collect 20, 154-9 (1954) (In Russian ) 89. E V. RYJOV, Geometric characteristics of surface roughness and waviness. Coll News in the Friction Theory, 19-34, Nauka, 1966. (In Russian.) 90 B. S. DAVYDOV, Foundations of the Probe Method for Determining Surface Roughness, Standard Publ House, Moscow, 1959 (In Russian.) 91. I. L. ROIH,V. V ORDYNSKAYAand I. P. BOLOTICH,On mechanical treatment influence on metal surface area size, Dokl Akad. Sci USSR 146, No. 6, 1316-17 (1962). (In Russian.) 92. H. CRAMER and M R. LEADBE't'rER, Stationary and Related Stochastic Processes, John Wiley, 1967, N Y.-L -S.

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93. A. I EROFEEV, (a) Wedge-shaped cavity m a free molecule gas flow, Engng. J 5, No. 5, 862-7 (1965); (b) On roughness influence on gas flow-sohd surface interaction Mech of Ltqutd and Gas, No. 6, 82-89 (1967), (c) On roughness form influence on gas flow-sohdsurfacelnteractlon lbid, 124-7 (1968). (In Russian ) 94. M. C SMITH, Computer study of gas molecule reflecttons from rough surfaces, [2b], 1217-20. 95. D C_ LOOK, JR. and T. J LovE, Investigation of the effects of surface roughness upon reflectance, A1AA Paper No_ 70-820, 1970 96. P. BARK and A. NIKURADSE, (a) Recherches exp6rimentales sur la distribution de l'mtensit6 des jets mol6culaires r6fl6chls en fonction du degr6 de rugoslt6 sur les surfaces de lalton, [3a], 111-14, (b) Etudes sur rlnfluence de la rugoslt6 d'une surface sohde sur la r6flexion de jets mol6culalres, [3b], 93-97 97 T . J . HEALY, The scattering of particles from rough surfaces, [4], 435-47. 98. J J_ MARTIN, Multlpath reflections from a random surface, JASA, 47, No. 5, 1303-9 (1970). 99_ P J. LvrqcI-I and R J WAGNER,Rough-surface scattering" shadowing, multiple scatter, and energy conservation. J Math. Phys 11, No. 10, 3032-42 (1970) 100. R. VAN HARDEVELD and F. HARTOG, The statistics of surface atoms and surface sites on metal crystals. Surface Sci 15, No 2, 189-230 (1969) 101. Apphcatton of Probabdity Theory and Mathemattcal Statlsttcs Methods for Studymg Surface Roughness. Abstracts to the Seminar, May 1971, Nauka, Leningrad. (In Russian.) 102. J. S. BENDAT and A. G PIERSOL, Measurement and Analysis of Random Data, John Wiley & Sons, 1967 N Y - L -S.