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Numerical computation of the parameters of vacuum systems with adsorbing surfaces
Vacuum/volume 34/number Printed in Great Britain
0042-207X/84$3.00+ .OO Pergamon Press Ltd
B/pages 509 to 511 /I 984
Numerical computation of the parameters vacuum systems with adsorbing surfaces G I Grigorov. V I Tzanev and K K Tzatzov, Lenin 72, Sofia 1184, Bulgaria received
11 August
of
Institute of Electronics, Bulgarian Academy of Sciences, Boul.
1982
The paper exposes an iterative procedure for evaluation of the distribution of incident and adsorbed gas flows in vacuum systems with adsorbing walls, based on surface co-ordination coefficients and real sticking coefficient distribution over the walls. Some nontrivial procedures connected with the evaluation and optimization ‘df such systems are discussed.
1. Introduction
The determination of molecular flow density distribution in vacuum systems with adsorbing walls is of importance for many applications. A method for evaluating the distribution of incident and adsorbed gas flow density in vacuum systems with adsorbing walls, based on the conception of surface co-ordination coefficients and taking into account the real sticking coefficient distribution over the walls was proposed in ref 1. The characteristics of a getter pumping device are determined in ref 1 from the distribution ofthe differential parameters c, s and 7 over the active surface. On this basis the effective capture coefficient as well as the efficiency criterion are evaluated. By using these quantities optimization of the geometrical configuration may be fulfilled. As far as the evaluation and optimization are connected with some nontrivial procedures, we shall utilize in this paper the main parts of the computation procedure used in ref 1.
2. Basic statements
Let us consider a vacuum system with an inner surface F on which a getter film is continuously deposited. We define the surface F as composed by N elemental areas Fti’, i= l,N, the number N determined in general by the geometrical configuration of the system and also by the getter flow distribution on the system surfaces. The main requirement in dividing this area into elements is the uniformity condition of the characteristic parameters I$,!,, vi’, v$‘, s(j),cti’ and yci’for each element. The meaning of these parameters is: v$-flow density of gas particles leaving F”‘; v$--flow density of the metal atoms deposited on F”‘; @--flow density of the gas particles desorbed from F”‘; c?‘= v$Jv$‘-relative density of the incident gas flow, where I*!:~is the flow density of the gas particles impinging on F’i’; s”’ = $‘/v~&-the sticking coefficient on F‘“‘; vi’-the flow density of the adsorbed on F(” gas particles; -*‘i’= vi’/vz’-the sorption ratio. ,
At a given distribution of the flow of deposited metal atoms vg’ all quantities above depend only on the input gas flow, i.e. on the pressure p at the input. The liaison between the elements Pi’ may be described by a set of geometrical surface co-ordination coefficients tij- ,I, which determine the mass-exchange between the jth and ith elements. From the molecular flow balance on the ith element we have:
,,(i’ = ,$’ + j$I 4j_i 0”I ,,W 1°C =
p’
. ,,(i’_ fPl -
ji,
. ,u,--p
4j-i
. $‘,
’ l’tJl.
(1)
(2)
The sorption process on continuously deposited metal films can be characterized by the relation P=f(;“‘), leading’ to: 9 (..(i’, &i)) = [7(il/~(Y(il)] _ @’ = 0. I
(3)
Analytical expressions ofl(;“‘) for different gas-metal pairs are given in ref 1. The equations (l)-(3) represent a set of 3 xN equations describing the mass-exchange between N elemental areas Ffi’ and are a basis for evaluation of vacuum systems with adsorbing walls. Solving the equation set (l)-(3) we obtain the values of vz&and ;.li’, defined above, for each element i ofthe vacuum system. In part 4 of the present paper we show the way of determining the integral characteristics of the vacuum system on the basis of these quantities. In the set of equations (l)-(3) the ranges of definition of the unknown quantities are \$+o;
oli”i’Ii’i;X,
where ;$., is determined from the maximum of the stoichiometry for a given gas-getter pair at the input gas flow density (p) considered. The set of equations (l)-(3) can be solved as follows: , N; (a) The first step is )Ji)=y$-* for all i= 1,. l We have to remember here that the index ““’ indicates that the quantity is related to the element F”‘.
509
G I Grigorov
et al: Parameters of vacuum systems with adsorbing
surfaces
(b) from (1) we obtain va:, for all i= 1,. . , N; (c) if vzi,,>O for each i we go to (d), if not-we replace yciJwith k.y”‘, O
(f) ifp=[
f
((-‘I ,(. -7’i’)/7’i’)2]o.5 se; OCE 6 1, then-end
of the
i=l
iterative procedure, if not-we replace y’“+ 1.p. (T(i’-y’i’), O
J?”
with
The iterative procedure described in (a)-(f) consists of two iteration loops. The first one, from (b) to (c), ensures that the condition v& 2 0 is fulfilled, and the second one, from (b) to (f), is the real iteration which solves the set of equations (l)-(3). 3. Procedure for solving of equation (3) It may be proven that for each gas-metal pair the functionfly”‘) is monotonic, i.e. for each c(j) [y~i”/~(r;i”)]
SP;(Y,i,)=a;
the solution of (3) is unique and may be found numerically. Let us consider a concrete case of nitrogen adsorption on titanium. By the iterative procedure the equation ~/(l-~/y,)cxP(“.8-~,‘~~)-c=0
= E< -,’5 ‘is- E= -9 Imax,
O
4. Evaluation of getter pump efficiency A getter pump may be treated as a virtual surface with an area equal to the pump inlet, and adsorbing properties equivalent to the pump action. These properties may be described by the quantities’ :
Kerr= Q./Qm, c,rr = QIQm. where
(3a) i=l
1,
is the gas flow exhausted by the pump, Q, is the inlet gas flow and
)‘,=o.s,
a=l-33.exp(-950/T,), TS--wall-surface
lim [ylf(y)]=cmax+ ..+ I rmrx at
a
i
(,,;‘.
F”‘)
i=l
In that concrete case we have:
is the getter flow deposited in the pump. An optimization of a getter pumping device requires the adoption of an adequate optimization criterion. As a possible criterion was proposed’ :
c,,, = r . yminr
lim [:/f(y)]= ;‘- ;‘m,”
[-jyf(y)]‘>O
Qm=
temperature’.
--00.
With the set of {c~} so chosen we obtain on one hand a representative set of values I$} and on the other hand we reduce considerably the time for determining the interval [c,,c,+~], involving the value c for which equation (3a) has to be solved. In that manner the iterative procedure is reduced to a calculation of the approximate cubic spline’.
has many times to be solved with Ymin
%mll,,)=
serr= Q,lQm
= C$” 2 c(i) 2 &LX= ~;~.,/~(~~;,),
~(~J,c)=[;‘/~(j’)]-c~=~
where M = int(K/Z). Then with the set of values {C&} we plot a cubic spline3 Sp,(y) with boundary conditions
at ~‘0,
&
lim [7lf(7)]‘= .,_. I rm,n
z,
where Scffmln and
lim [7lf(7)1’=+ cc. 7’ ;mal
S,ffmar
correspond
Consequently for each c in the interval Cc,,,, c,,,] equation (3) has only one solution which we find by the so called ‘method of interval halving’*. The solution we obtain from Y*=(jl~+Y~j/2. (yL and yR are the left and the right values of 7) if the conditions .F(‘i’,c).F+(j’&)So;
F(YL,C) . F’(,‘,,c) 20;
o<~,-y,~o.ooo1;
7,<0.49,
F(j.R,C)- F(;.L?C)
-
r
Of-
ooool.
’
.
yL 2 0.49 are satisfied. The iterative solution of (3a) needs considerable computing time. That is why we first solve equation (3a) for a set of values {c~},k= 1,N (k is an odd number) c~=c,,,,,+[c(Y=~.~~)-cmin3 C&= c(y -0.49) 510
. (k-l),
+ cc,,, - c(7=0.49)].
k= 1, (M(k-l),
1)
k=M,K;
Figure 1. r as a function of sCll.
to the limiting
working
G I Grigorov et al: Parameters of vacuum systems with adsorbing
surfaces
~..-***.****++*************. Figure 2. Segment of a Fortran-program
r*(s,ff)
-*
*************.
l V
for computing a set of data [T&r,), s~,~@~)],suitable for determination of the pump device efficiency.
pressures p,., and P,,,~,at the pump inlet; the functions are determined
.4
r and r*
as
= 7s. Serf.
Such criterion takes into account the pumping speed as well as the rate of getter utilization. The character of the dependence r=f(s,rl) is shown in Figure 1 and we have the following features: (a) r(s,,,M.5. serfif ~c~r+~cf~(Po), (b) UsefIb - CCif ~.ff-+ 1, (c) the function r(s,,,) shows a marked maximum which it changes quickly.
around
The main inaccuracy in the evaluation of E results from the inaccurate determination of r(s,,,) around its maximum due to the numerical technique used. To avoid this we compose an algorithm which allows us to obtain sufficient information of this maximum. The algorithm, in the form of a segment of a Fortranprogram is shown on Figure 2. The quantities used are as follows : p. --the allowed maximum value of the gas pressure at the
pump inlet, prnsx= the maximum work-pressure, pmin= the minimum work-pressure, p,,,,,= the limit pressure at which the condition r’(seIT)= - COmay be treated as satisfied.
‘Call Calc’ statement implies the computation of the system parameters at the current value (p,) of the pressure. The main feature of the algorithm shown on Figure 2 is that first it finds out the region in which the function has its maximum and then it calculates in more detail its value in this area. With the set of values [r(s,,,),s,,,(p,)] obtained in such a way we determine the efficiency E by means of a cubic spline’ Sp3(s) with boundary conditions
~~Xs,&d ~0.5 andSPiCse~~(P,im)l= - 00. We have used the calculation procedure described above for evaluation and optimization of different vacuum systems with adsorbing surfaces. An example of its application to the case of cylindrical body systems is given in ref 1.
References
’ G Grigorov and K Tzatzov, Vacuum, 33, 139 (1983). ’ R W Hamming, Numerical Merhods /or Scienrisrs and Engineers, McGraw-Hill, New York (1973). ’ A A Kirilenko and V P Loginov, Zarubezhnaja Radioelekrronika. 2, 3 (1978).
511
0042-207X/84$3.00+ .OO Pergamon Press Ltd
B/pages 509 to 511 /I 984
Numerical computation of the parameters vacuum systems with adsorbing surfaces G I Grigorov. V I Tzanev and K K Tzatzov, Lenin 72, Sofia 1184, Bulgaria received
11 August
of
Institute of Electronics, Bulgarian Academy of Sciences, Boul.
1982
The paper exposes an iterative procedure for evaluation of the distribution of incident and adsorbed gas flows in vacuum systems with adsorbing walls, based on surface co-ordination coefficients and real sticking coefficient distribution over the walls. Some nontrivial procedures connected with the evaluation and optimization ‘df such systems are discussed.
1. Introduction
The determination of molecular flow density distribution in vacuum systems with adsorbing walls is of importance for many applications. A method for evaluating the distribution of incident and adsorbed gas flow density in vacuum systems with adsorbing walls, based on the conception of surface co-ordination coefficients and taking into account the real sticking coefficient distribution over the walls was proposed in ref 1. The characteristics of a getter pumping device are determined in ref 1 from the distribution ofthe differential parameters c, s and 7 over the active surface. On this basis the effective capture coefficient as well as the efficiency criterion are evaluated. By using these quantities optimization of the geometrical configuration may be fulfilled. As far as the evaluation and optimization are connected with some nontrivial procedures, we shall utilize in this paper the main parts of the computation procedure used in ref 1.
2. Basic statements
Let us consider a vacuum system with an inner surface F on which a getter film is continuously deposited. We define the surface F as composed by N elemental areas Fti’, i= l,N, the number N determined in general by the geometrical configuration of the system and also by the getter flow distribution on the system surfaces. The main requirement in dividing this area into elements is the uniformity condition of the characteristic parameters I$,!,, vi’, v$‘, s(j),cti’ and yci’for each element. The meaning of these parameters is: v$-flow density of gas particles leaving F”‘; v$--flow density of the metal atoms deposited on F”‘; @--flow density of the gas particles desorbed from F”‘; c?‘= v$Jv$‘-relative density of the incident gas flow, where I*!:~is the flow density of the gas particles impinging on F’i’; s”’ = $‘/v~&-the sticking coefficient on F‘“‘; vi’-the flow density of the adsorbed on F(” gas particles; -*‘i’= vi’/vz’-the sorption ratio. ,
At a given distribution of the flow of deposited metal atoms vg’ all quantities above depend only on the input gas flow, i.e. on the pressure p at the input. The liaison between the elements Pi’ may be described by a set of geometrical surface co-ordination coefficients tij- ,I, which determine the mass-exchange between the jth and ith elements. From the molecular flow balance on the ith element we have:
,,(i’ = ,$’ + j$I 4j_i 0”I ,,W 1°C =
p’
. ,,(i’_ fPl -
ji,
. ,u,--p
4j-i
. $‘,
’ l’tJl.
(1)
(2)
The sorption process on continuously deposited metal films can be characterized by the relation P=f(;“‘), leading’ to: 9 (..(i’, &i)) = [7(il/~(Y(il)] _ @’ = 0. I
(3)
Analytical expressions ofl(;“‘) for different gas-metal pairs are given in ref 1. The equations (l)-(3) represent a set of 3 xN equations describing the mass-exchange between N elemental areas Ffi’ and are a basis for evaluation of vacuum systems with adsorbing walls. Solving the equation set (l)-(3) we obtain the values of vz&and ;.li’, defined above, for each element i ofthe vacuum system. In part 4 of the present paper we show the way of determining the integral characteristics of the vacuum system on the basis of these quantities. In the set of equations (l)-(3) the ranges of definition of the unknown quantities are \$+o;
oli”i’Ii’i;X,
where ;$., is determined from the maximum of the stoichiometry for a given gas-getter pair at the input gas flow density (p) considered. The set of equations (l)-(3) can be solved as follows: , N; (a) The first step is )Ji)=y$-* for all i= 1,. l We have to remember here that the index ““’ indicates that the quantity is related to the element F”‘.
509
G I Grigorov
et al: Parameters of vacuum systems with adsorbing
surfaces
(b) from (1) we obtain va:, for all i= 1,. . , N; (c) if vzi,,>O for each i we go to (d), if not-we replace yciJwith k.y”‘, O
(f) ifp=[
f
((-‘I ,(. -7’i’)/7’i’)2]o.5 se; OCE 6 1, then-end
of the
i=l
iterative procedure, if not-we replace y’“+ 1.p. (T(i’-y’i’), O
J?”
with
The iterative procedure described in (a)-(f) consists of two iteration loops. The first one, from (b) to (c), ensures that the condition v& 2 0 is fulfilled, and the second one, from (b) to (f), is the real iteration which solves the set of equations (l)-(3). 3. Procedure for solving of equation (3) It may be proven that for each gas-metal pair the functionfly”‘) is monotonic, i.e. for each c(j) [y~i”/~(r;i”)]
SP;(Y,i,)=a;
the solution of (3) is unique and may be found numerically. Let us consider a concrete case of nitrogen adsorption on titanium. By the iterative procedure the equation ~/(l-~/y,)cxP(“.8-~,‘~~)-c=0
= E< -,’5 ‘is- E= -9 Imax,
O
4. Evaluation of getter pump efficiency A getter pump may be treated as a virtual surface with an area equal to the pump inlet, and adsorbing properties equivalent to the pump action. These properties may be described by the quantities’ :
Kerr= Q./Qm, c,rr = QIQm. where
(3a) i=l
1,
is the gas flow exhausted by the pump, Q, is the inlet gas flow and
)‘,=o.s,
a=l-33.exp(-950/T,), TS--wall-surface
lim [ylf(y)]=cmax+ ..+ I rmrx at
a
i
(,,;‘.
F”‘)
i=l
In that concrete case we have:
is the getter flow deposited in the pump. An optimization of a getter pumping device requires the adoption of an adequate optimization criterion. As a possible criterion was proposed’ :
c,,, = r . yminr
lim [:/f(y)]= ;‘- ;‘m,”
[-jyf(y)]‘>O
Qm=
temperature’.
--00.
With the set of {c~} so chosen we obtain on one hand a representative set of values I$} and on the other hand we reduce considerably the time for determining the interval [c,,c,+~], involving the value c for which equation (3a) has to be solved. In that manner the iterative procedure is reduced to a calculation of the approximate cubic spline’.
has many times to be solved with Ymin
%mll,,)=
serr= Q,lQm
= C$” 2 c(i) 2 &LX= ~;~.,/~(~~;,),
~(~J,c)=[;‘/~(j’)]-c~=~
where M = int(K/Z). Then with the set of values {C&} we plot a cubic spline3 Sp,(y) with boundary conditions
at ~‘0,
&
lim [7lf(7)]‘= .,_. I rm,n
z,
where Scffmln and
lim [7lf(7)1’=+ cc. 7’ ;mal
S,ffmar
correspond
Consequently for each c in the interval Cc,,,, c,,,] equation (3) has only one solution which we find by the so called ‘method of interval halving’*. The solution we obtain from Y*=(jl~+Y~j/2. (yL and yR are the left and the right values of 7) if the conditions .F(‘i’,c).F+(j’&)So;
F(YL,C) . F’(,‘,,c) 20;
o<~,-y,~o.ooo1;
7,<0.49,
F(j.R,C)- F(;.L?C)
-
r
Of-
ooool.
’
.
yL 2 0.49 are satisfied. The iterative solution of (3a) needs considerable computing time. That is why we first solve equation (3a) for a set of values {c~},k= 1,N (k is an odd number) c~=c,,,,,+[c(Y=~.~~)-cmin3 C&= c(y -0.49) 510
. (k-l),
+ cc,,, - c(7=0.49)].
k= 1, (M(k-l),
1)
k=M,K;
Figure 1. r as a function of sCll.
to the limiting
working
G I Grigorov et al: Parameters of vacuum systems with adsorbing
surfaces
~..-***.****++*************. Figure 2. Segment of a Fortran-program
r*(s,ff)
-*
*************.
l V
for computing a set of data [T&r,), s~,~@~)],suitable for determination of the pump device efficiency.
pressures p,., and P,,,~,at the pump inlet; the functions are determined
.4
r and r*
as
= 7s. Serf.
Such criterion takes into account the pumping speed as well as the rate of getter utilization. The character of the dependence r=f(s,rl) is shown in Figure 1 and we have the following features: (a) r(s,,,M.5. serfif ~c~r+~cf~(Po), (b) UsefIb - CCif ~.ff-+ 1, (c) the function r(s,,,) shows a marked maximum which it changes quickly.
around
The main inaccuracy in the evaluation of E results from the inaccurate determination of r(s,,,) around its maximum due to the numerical technique used. To avoid this we compose an algorithm which allows us to obtain sufficient information of this maximum. The algorithm, in the form of a segment of a Fortranprogram is shown on Figure 2. The quantities used are as follows : p. --the allowed maximum value of the gas pressure at the
pump inlet, prnsx= the maximum work-pressure, pmin= the minimum work-pressure, p,,,,,= the limit pressure at which the condition r’(seIT)= - COmay be treated as satisfied.
‘Call Calc’ statement implies the computation of the system parameters at the current value (p,) of the pressure. The main feature of the algorithm shown on Figure 2 is that first it finds out the region in which the function has its maximum and then it calculates in more detail its value in this area. With the set of values [r(s,,,),s,,,(p,)] obtained in such a way we determine the efficiency E by means of a cubic spline’ Sp3(s) with boundary conditions
~~Xs,&d ~0.5 andSPiCse~~(P,im)l= - 00. We have used the calculation procedure described above for evaluation and optimization of different vacuum systems with adsorbing surfaces. An example of its application to the case of cylindrical body systems is given in ref 1.
References
’ G Grigorov and K Tzatzov, Vacuum, 33, 139 (1983). ’ R W Hamming, Numerical Merhods /or Scienrisrs and Engineers, McGraw-Hill, New York (1973). ’ A A Kirilenko and V P Loginov, Zarubezhnaja Radioelekrronika. 2, 3 (1978).
511