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Peak shifts in thermal desorption experiments due to vacuum time constants—a generalized solution
Peak shifts in thermal desorption experiments due to vacuum t i m e c o n s t a n t s a g e n e r a l i z e d solution* received2 February 1976 D Edwards Jr, BrookhavenLaboratory,Upton,New York 11973,U.S.A.
The shift in the maximum (Tin) of the pressure transient during a thermal desorption experiment from the maximum (To) of the sample desorption rate due to the influence of the vacuum time constant ~ is considered. It is shown that AT/AW, whereAT -- Tm - Tp and AW is the full width at half height of the sample desorption rate, is determined by the quantity f3r/AW where is the temperature sweep rate. A general solution of AT/AW as a function of [3flAW is given for 10 -3 < ~ / A W < 10 3 for both first and second order desorption transients. In addition an optimization criterion for thermal desorption experiments is described and the implications of this criterion considered. It is found in particular that the simultaneous requirements of negligibly low distortion with maximum measured signal amplitude are satisfied for fJr/dW =0.1 for an elementary first or second order desorption transient.
1. Introduction In thermal desorption experiments a gas is typically implanted into or adsorbed onto a sample which is contained in a vacuum system of volume V (liters) having a pump of pumping speed S (1. s - t ) . After the adsorption or implantation process is complete the sample temperature is increased with respect to time according to a defined schedule (only temperature vs time relations of the form T = To + St will be considered here) and the system gas pressure p(t) (torr) monitored during the subsequent desorption. In general due to the vacuum time constant z = V/S the pressure transient p(t) slightly lags in time behind the rate of sample desorption resulting in the maximum of p(t) occurring at a higher sample temperature T,, than that of the maximum Tp of the sample desorption rate. That is, the sample is at a higher temperature at the maximum of the pressure transient than it was at the maximum of the sample desorption rate. The influence of the vacuum time constant ~- on the position of the pressure transient has been recognized in a qualitative way by early investigators, t Somewhat later 2 the problem was more clearly defined and the delay in the maximum of p(t) ( T = T,,) from the maximum of - d n , / d t ( T = Tp), where ns(t) is the adsorbed number on the sample at time t, was numerically computed for a particular fl and y for several different values of the activation energy as a function of z for a first order desorption process. It was found in this study 2 that AT/Tp, where AT = Tm -- Tp, varied as ln(r/T o) over the range of parameters studied. More recently 3 series solutions have been developed for AT in terms of f17 for small [3r/AW where A W is the full width in K at the half height of --dns/dt. The implication of this work was that AT,--' fir for both first and second order desorption transients. Thus there appeared to be solutions to the problem of the shift in T,~ from To as a
* Work performed under the auspices of the U.S. Energy Research and Development Administration.
function of r either for particular values of the desorption parameters 2 or for a restricted range 3 (small ST/A IF) of these parameters. In this article a relation will be established between AT]To and /3r/A 14" for large ~r/A W for both first and second order desorption rate equations. By suitable manipulation of this relationship (and the one previously found for small ~r/AW (ref. 3)) it will be shown that the quantity AT/A W is a function only of fl~-/AW and a where ,z = E/kTp for any set of desorption parameters (E, y, S, ~') with the a dependence being quite weak. The use of computational techniques will be made to bridge the gap between the large and small /3~-/AW solutions yielding a general relation between AT[A W and fl~-[AW accurate over the known range 4 of the desorption parameters (y, E, /3, r). Finally a discussion will be given of the optimum choice of for a given (y, E, ~-) based on some o f the results of the time constant studies.
2. General F o r the system previously described, the differential equation relating the system pressure p(t) to the sample desorption rate is: d(pV) d-~ -
dns dt
(pV) z
(1)
where in the following only first or second order desorption rates are considered; i.e. dn~
d-~ = - yns e x p ( - E/kT)
dr/s
(first o r d e r )
(2)
(second o r d e r )
(3)
/7 2
= _ y:2__ e x p ( - E/kT) no
where ns is the number on the sample at time t, no is the initial adsorbed number, (y, E) are rate constants, T is the sample
Vacuum/volume26/number 3. PergamonPressLtd/Printedin GreatBritain
91
D Edwards Jr:
Peak shifts in thermal desorption experiments due to vacuum time constants--a generalized solution
temperature taken to be {T = To -F flt~ and k is Boltzmann's constant. For large flr/AW (_10), (i) may be integrated from t' = 0 to t' = t and the following equation obtained: 5 p(t) V = n o - n~(t) - noa(A W/,Sr)
(4)
At the maximum of p(t), dp/dtltm ~ 0; thus evaluating (I) at t,,, the following is obtained:
d',l p(t,.) V = - ~ mI-771
(5)
2.1 First order desorption
170
?E ~-./[u) e" ll-
=
=
[]r___~ 2.4464
r,,
(I Aw\
---
,40 + 35)
(I 1)
y is determined from (8), and using ( I I ), depends only on the parameters flr/A H/and a. After determining y, AT/A 14,"is found from (I0) and is seen to be a function only o f y and a. Thus A T / A W is entirely determined by the quantities f l T / A W and a for this large ~-solution. It should be noted that y is, in practice, found by iterating equation (8), the iteration requiring only ~ 8 steps for the solution of (8) to be accurate to ,~0. I °,/o.
rate, large r solution
Assuming that dndd/obeys the first order rate equation (2) it is known that: 2 In ns(t) --
In a similar way at} expression can be derived for fir~/T,. The result is:
+ ,'/(To)
(6)
2.2 First order desorption, small r solution. As has been previously indicated the solution for small f l r / A W is known and can be written a s : 3 fir 2 .~~ + a[(flrlA W) 2]}
AT = fir{ l
(12)
where t~ = - E f t < T , f(~) = 1 4- 21t~ -I- 61~2 -I 24/t~3 -I- . . . . where A(To) is a small constant and neglected. Evaluating (4) at t,. and using (2), (5) and (6) it can be shown that:
for a first order desorption transient. Equation (9) may be used to find a / T , and after dividing both sides of (12) by AW, the equation becomes:
In n,,t,,:t~ = - l n { l + ~ r e x p ( - E / k T , , ) }
A W -- A W
- a(AW/flr)
(7)
170
1.40 I -
1.2232
3.5
1 - - -
Equating (7) with (6) evaluated at t,, and using the well-known relation for a first order desorption transient: fiE kTp---~ = T exp(-E/kTp)
and again it is found that A T / A W is a function only of f l r / A W and a.
where, as previously mentioned, Tp is the temperature of the maximum of -dn.ddt, the following expression is derived: y
+ ~
exp(cg,l(I
--2 ln(l + y)
(8)
y) -- lnE, -- 2 ( I + o~
+ 6 [ (~l + Y ) ]
, - 2 4 [ (~1 ~ Y ) T + ' "
.]}
where AT I".
== E/krr
AW
-
/,.,
-
I + a A 14/
L
+
\fir
/ (14)
f(Pm) = 1 + 2/It,,, + 6~It,,,'- + 24/t6, 3
r,
ATp~Ifirst order
2.4464{1_= 1.40_¢+ 3.~}
(9)
Thus it can be shown that the normalized temperature shift A T / A W is given by: f2.4464 [
1.40 + -3.5Y} i_=~
; 92
i I +-Y-~f(t'').J +
g(~) = 1 + 2lot - 6/o~2 + 24/0c3
and as has been recently shown 6 for a first order desorption transient:
ar
Y -}.T.j'{In[
where
Now y can be written as: AT
2.3 S e c o n d order desorption--large "r solution. Using the second order rate equation for dn.,/dt together with a procedure similar to that described above for the first order desorption case, the following equation for y results for the large/h-/A W situation:
+ Tp .f(It,,)(I + y)2
T.,-Tp Y= r,
Y=
(13)
=,)j
(10)
=
I + y)
It can also be shown that for the second order desorption transient: fll~O[
T.
=
3fit
aW
3.5255(I - 2/~ + 6.9/~ 2)
and AW = y
( 1 - 2/~ + 6.9/~t 2)
D Edwards Jr: Peak shifts in thermal desorption experiments due to vacuum time constants--a generalized solution
Furthermore an equation (not given here) similar to (13) restllts from the small [3r/AW solution 3 for the second order desorption transient. Thus, as in the first order case, AT/AW is a function only of a, [3r/,xW for both the large and small ,Sr/A W solutions.
0
o /"
oo~
AT [
AWI=,Z0
3. Discussion Both as a check of our results in regions of flrlA W in which they should be valid and in order to indicate the radius of convergence of the various series solutions developed here, equation (I) has been mimetically integrated for a particular set of desorption parameters for various [3r/AW and the quantity AT/AW determined. The parameters used in the nunlerical integrations (for both first and second order) were fl : I K/s, 7", = 103 K, u = 30, n, :- I with 7 determined from the above quantities for the r.z-spective desorption process. Figure I shows a plot of A77AW as a function of/3r/A W for a first order desorption process with a = 30. The solid lines in Figure I are the small and large ~r/AW solutions derived from equations (13) and (10) respectively using a = 30. The darkened circles plotted in Figure I are the results of the numerically integrated solutions for a particular element
~Oil
K
--F~---'2"
/
/
0011 ....
00~00~101 -
-"
t
! / " ~ " ' - SMALL r SOLUTION
/ ....
AT
001 -
i
.........
~T/,',w
I vS ,~,-/,',w
FORFIRSTORDER
-
J
.... NUMERICALSOLUTION
i
"
OI
I0-
iOz
IO3
flr/AW Figure 1. dT/,dW is plotted as a function of fl~-/dI,V, The solid lines refer to the series solutions for small and large ~T/d t4" whereas the darkened circles are the results of a numerical integration for a particular set of desorption parameters (see text).
(8 :-lK/s, etc.) in the set of desorption parameters. F o r flr/AW larger than 10 the % difference between the numerically integrated solution and that obtained fi-om equation (10) was less than 2 ~ , becoming quite small (~0.1 ~ ) for [3r/AW > 100. It is also evident fl'om Figure I that the small flT/AW series solution for A T / A / I / i s reasonably accurate for fl-r/AW < 0.2. The dashed line in Figure 1, determined from the numerically integrated solution, is seen to provide an interpolation in the intermediate region of flr/AW (0.2 -- 5.0) between the large and the small fl~/A W solutions. In this way a solution is formed for AT/AW over an extended range of ,Sr/A W for a = 30. The dependence of AT/A W on ct is shown in Figure 2 where
al< lao,oo,
O.OOC001
I I
011"
I I0
I I00
IO00
zxw Figure 2. In this figure is plotted the % deviation of dT/A W depending on the cheise of :z as a function of flrld 14/. although it is in fact quite unusual to find a very different from 30 ( ~ 5). It can be seen from Figure 2 that for flr[AW ~ 10 the possible deviations of ~ from 30 ( ± 10) could introduce a maximum error in the estimation of AT/AW from Figure ! of ---3 % where as for flr/A 14,"< 0.2 the corresponding uncertainty in AT/All," is less than 0.5%. Thus it is concluded that Figure 1 accurately represents the relation between AT/AW and flr/AW for a first order desorption process for the presently realizable (20 <_ a < 40) set of desorption parameters (E, ~,, fl, ~-). In Figure 3 the normalized temperature shifts for both first and second order desorption transients are compared. It should be emphasized that A W is the full width at half height of -dn.Jdt for the particular desorption transient considered. 6 It is seen in this figure that for small fl~-/AW(_< 0. l), AT/A W is independent of the peak shape as has been previously suggested? On the other hand for larger flT/AW (> 0.2) the normalized peak shift does depend on the desorption process. It is also apparent that it is quite difficult to force the first order peak shift AT to be larger than AW. This is essentially due to the fact that the decay of the high temperature side of the first order desorption peak is quite rapid. It should also be noted that for large fl~-/AW, (AT/AW) a In In ~r/AW for the first order transients--a very slowly varying function. It is also seen in Figure 3 that AT/A W for the second order desorption transient is more rapidly varying as a function o f flr/A 14/than the first order case for large fl~-/AIV. It is interesting
L f
SECONDORDER\ l0
/...~
'~J~ el
"FIRSTORDER
OOI
A W ==4o,,//A W ,=zo is plotted for a first order desorption rate, using equations (10) and (13), as a function of [J.r/AW. It should be noted that, as has been recently estimated, 6 it is possible for a to vary from 20 to 40 depending on the values of the desorption parameters
00[~( )01
001
Ot Br AW
1,0
I0
I00
I000
Figure 3. A comparison is made between ,47'//I W for first and second order desorption transients. 93
D E d w a r d s J r : Peak shifts in thermal desorption experiments due to vacuum time constants--a generalized solution
to note that for the second order case, equation (14), for large /3r/AW, AT/AW e In ,8~-/~1'1/, a result suggested by earlier work 2 for the first order case and implies that the previous study was not an asymptotic solution of AT/Tp for large ~- for the first order desorption transient but rather an intermediate solution between the small /h-/& I,V and large /3-r/AW limits. In Figure 4 is plotted both - I / n o cb~/dt (solid dots) and p(t) V/nor (open figures) for a particular second order desorption transient (/3 = IK/s, E]kTp -----30, Tp = 103 K, n,, = I, e = 3 >" 10X~/s) for various r (/3~-/AW) in order to show in a qualitative way the distortions (peak shift, peak amplitude reduction, etc.) introduced into the pressure measurements for various discrete magnitudes of /3r/APl,'. It is seen that for /3r/AW between 1-3 one has instrumentally introduced significant distortions into the measurement of - I / n o dn,/dt whereas for /3r/AW~ 0.1, p(t)V/nor is, apart from the slight shift in the peak position, quite closely approximating - l[no dn~/dt. It is, on the other hand, clear from Figure 3 that for Or/A W < 0.1 the normalized peak shift AT/AW is closely approximated by the small/3~-/A W series solution. Thus one is in a low distortion situation as qualitatively determined from Figure 4 if AT/A W is given by the small/3r/A W series solution.
low instrumental distortion, i.e. /3r/AI.I/:~ 0.3. The signal
p(t)" I/at the peak is given by (5): d nsl Vl,(
t,,,)
_ I_ dns
no di
6
- -
oJ-~:
t,
.
!
..\:,,
"
t.,
5
,,P
2.4 T.
A IIt
it is [bund that the signal at the peak of the measured desorption transient is, (neglecting the small connection terms)
o
700
800
.."
...~--3 0
900 T(K} I000
",.
1100
" "~
:
1200
Figure 4. Both --1/n,, dn,/dt and l/nop(t)V/r are plotted for a particular second order desorption transient ( T , = 103K, ¢ = 30, /3 = i K / s ) for various/h-[A W. It is seen that the maximum of the function p(t) V/noz is occurring on the sample desorptio=n rate curve as required by equation (5) and that the flr/AW = 0.I i:urve has a very slight amount of instrumental distortion.
The region of intermediate and larger flr/A W (>_ 1) solutions seen in Figures 1 and 3 correspond to pressure measurement p(t)V/nor produced by a convoluted function of instrument response together with the elementary desorption transient. It should of course be clear that for large ~r]AW ( ~ 10) the quantity Vdp]dt will again yield a close approximation to --dnUdt (Ref. 7,3).
5. A n o p t i m i z a t i o n
criterion
Let us suppose that an elementary first order desorption transient is being measured in a system with time constant ~- and that it is desired to be in a region of/3~-/A W in which there is 94
f0.90 A-~-/ ,1o
first order rate equation
]0.89 fir no k A W
second order rate equation
=
(16)
0.1,
(17)
w h i c h implies that for any given (T,A W ) c o m b i n a t i o n , t3 can be
,
2
./.."
+cr[(flr/AW)2]}
Using (9),
/% W
I
•
dn~l .,. - f l ~ . ~ £ { I d t I%
,6T
I
"
and it can be shown for a lirst order desorption transient for small #flAW (-~ 0.3) that
--,--
x 3
r
But, as we have previously seen, to be in the low distortion region ~3flAW % 0.1. Thus to obtain the maximum signal at the desorption peak, /3-r/Al~ should be as large as possible (see equation (16)) but for a low distortion measurement ~3flAW < 0.1. Thus an optimized operating point is clearly obtained at:
V, ~'~,
___
-
d t It,,,
Vp(I,,,)
7
=
adjusted to satisfy the optimized condition. It should be noted that in most experiments one would simply decrease the signal: noise ratio by going to/3r/A W ~ 0.1 without any gain in information, i.e. at /3r/AW~ 0.1 one is already in a quite low distortion region (see Figure 4). Another feature of equation (16) is that for a system with arbitrary time constant r using the optimized sweep rate criterion (equation (17)) the optimized pressure "< volume signal at the peak maximum will be ~ 0.1 no independent of all system and desorption rate parameters, for both first and second order desorption transients. As an example of the above statements, if one has a sample in a 5 1 system with 10 t5 atoms s adsorbed in a first order desorption state then the pressure signal at the maximum of the pressure transient during the desorption will be ~ 6.2 × 10 -~ torr if the system is operated in an optimum manner.
6. S u m m a r y
The problem of the shift AT of the maximum of the pressure transient from the maximum of -dnUdt due to the vacuum time constant r has been considered. It has been shown that the normalized temperature shift AT/AW is a function only of /3r/AI,V and a for both first and second order desorption transients and the solution for AT/A W as a function of/3r/A W is given over an extended range of /3T/AIV. Use of numerical integration techniques have been made for the region of
D Edwards Jr: Peak shifts in thermal desorption experiments due to vacuum time constants--a generalized solution
JJr/AW not covered o p t i m i z a t i o n criterion c o n s t a n t studies has c e r n i n g the m e a s u r e d
in the analytic solutions. Finally an based on some of the results of the time been described a n d i m p l i c a t i o n s c o n pressure t r a n s i e n t s discussed.
Acknowledgement A l t h o u g h it s h o u l d be clear from the text of the paper, I would like to a c k n o w l e d g e the prior work of Dr P A R e d h c a d b o t h in p r o v i d i n g a clear definition of the present p h e n o m e n a a n d in closely d e t e r m i n i n g the final form of the generalized solution.
References E G Ehrlich, Ad~, Catalysis 14, 1963, 255. z p A Redhead, Vacmml, 12, 1962, 203. 3 D Edwards, Jr, J Vac Sci Technol, II, 1974, 1141. "~This essentially means that 20 - 7 <- 40. For a further discussion see Ref 6. 5 The symbol f~(J W/[Jr) means that terms of the order [J~/A14/are neglectcd in the equation. Thus for fJ~-/J I,V >_ 10, terms <0, I no are neglected in equation (4). " D Edwards, Jr, Surf Sol (in press). 7 See, for example, the work of E V Kornel~en in Rad Effects, 13, 1972, 227, and his other papers for a discussion of this measurement. The units used here are such that I torr i. = 3.21 ;< l019 atoms which is applicable to room temperature pressure measurements.
95
The shift in the maximum (Tin) of the pressure transient during a thermal desorption experiment from the maximum (To) of the sample desorption rate due to the influence of the vacuum time constant ~ is considered. It is shown that AT/AW, whereAT -- Tm - Tp and AW is the full width at half height of the sample desorption rate, is determined by the quantity f3r/AW where is the temperature sweep rate. A general solution of AT/AW as a function of [3flAW is given for 10 -3 < ~ / A W < 10 3 for both first and second order desorption transients. In addition an optimization criterion for thermal desorption experiments is described and the implications of this criterion considered. It is found in particular that the simultaneous requirements of negligibly low distortion with maximum measured signal amplitude are satisfied for fJr/dW =0.1 for an elementary first or second order desorption transient.
1. Introduction In thermal desorption experiments a gas is typically implanted into or adsorbed onto a sample which is contained in a vacuum system of volume V (liters) having a pump of pumping speed S (1. s - t ) . After the adsorption or implantation process is complete the sample temperature is increased with respect to time according to a defined schedule (only temperature vs time relations of the form T = To + St will be considered here) and the system gas pressure p(t) (torr) monitored during the subsequent desorption. In general due to the vacuum time constant z = V/S the pressure transient p(t) slightly lags in time behind the rate of sample desorption resulting in the maximum of p(t) occurring at a higher sample temperature T,, than that of the maximum Tp of the sample desorption rate. That is, the sample is at a higher temperature at the maximum of the pressure transient than it was at the maximum of the sample desorption rate. The influence of the vacuum time constant ~- on the position of the pressure transient has been recognized in a qualitative way by early investigators, t Somewhat later 2 the problem was more clearly defined and the delay in the maximum of p(t) ( T = T,,) from the maximum of - d n , / d t ( T = Tp), where ns(t) is the adsorbed number on the sample at time t, was numerically computed for a particular fl and y for several different values of the activation energy as a function of z for a first order desorption process. It was found in this study 2 that AT/Tp, where AT = Tm -- Tp, varied as ln(r/T o) over the range of parameters studied. More recently 3 series solutions have been developed for AT in terms of f17 for small [3r/AW where A W is the full width in K at the half height of --dns/dt. The implication of this work was that AT,--' fir for both first and second order desorption transients. Thus there appeared to be solutions to the problem of the shift in T,~ from To as a
* Work performed under the auspices of the U.S. Energy Research and Development Administration.
function of r either for particular values of the desorption parameters 2 or for a restricted range 3 (small ST/A IF) of these parameters. In this article a relation will be established between AT]To and /3r/A 14" for large ~r/A W for both first and second order desorption rate equations. By suitable manipulation of this relationship (and the one previously found for small ~r/AW (ref. 3)) it will be shown that the quantity AT/A W is a function only of fl~-/AW and a where ,z = E/kTp for any set of desorption parameters (E, y, S, ~') with the a dependence being quite weak. The use of computational techniques will be made to bridge the gap between the large and small /3~-/AW solutions yielding a general relation between AT[A W and fl~-[AW accurate over the known range 4 of the desorption parameters (y, E, /3, r). Finally a discussion will be given of the optimum choice of for a given (y, E, ~-) based on some o f the results of the time constant studies.
2. General F o r the system previously described, the differential equation relating the system pressure p(t) to the sample desorption rate is: d(pV) d-~ -
dns dt
(pV) z
(1)
where in the following only first or second order desorption rates are considered; i.e. dn~
d-~ = - yns e x p ( - E/kT)
dr/s
(first o r d e r )
(2)
(second o r d e r )
(3)
/7 2
= _ y:2__ e x p ( - E/kT) no
where ns is the number on the sample at time t, no is the initial adsorbed number, (y, E) are rate constants, T is the sample
Vacuum/volume26/number 3. PergamonPressLtd/Printedin GreatBritain
91
D Edwards Jr:
Peak shifts in thermal desorption experiments due to vacuum time constants--a generalized solution
temperature taken to be {T = To -F flt~ and k is Boltzmann's constant. For large flr/AW (_10), (i) may be integrated from t' = 0 to t' = t and the following equation obtained: 5 p(t) V = n o - n~(t) - noa(A W/,Sr)
(4)
At the maximum of p(t), dp/dtltm ~ 0; thus evaluating (I) at t,,, the following is obtained:
d',l p(t,.) V = - ~ mI-771
(5)
2.1 First order desorption
170
?E ~-./[u) e" ll-
=
=
[]r___~ 2.4464
r,,
(I Aw\
---
,40 + 35)
(I 1)
y is determined from (8), and using ( I I ), depends only on the parameters flr/A H/and a. After determining y, AT/A 14,"is found from (I0) and is seen to be a function only o f y and a. Thus A T / A W is entirely determined by the quantities f l T / A W and a for this large ~-solution. It should be noted that y is, in practice, found by iterating equation (8), the iteration requiring only ~ 8 steps for the solution of (8) to be accurate to ,~0. I °,/o.
rate, large r solution
Assuming that dndd/obeys the first order rate equation (2) it is known that: 2 In ns(t) --
In a similar way at} expression can be derived for fir~/T,. The result is:
+ ,'/(To)
(6)
2.2 First order desorption, small r solution. As has been previously indicated the solution for small f l r / A W is known and can be written a s : 3 fir 2 .~~ + a[(flrlA W) 2]}
AT = fir{ l
(12)
where t~ = - E f t < T , f(~) = 1 4- 21t~ -I- 61~2 -I 24/t~3 -I- . . . . where A(To) is a small constant and neglected. Evaluating (4) at t,. and using (2), (5) and (6) it can be shown that:
for a first order desorption transient. Equation (9) may be used to find a / T , and after dividing both sides of (12) by AW, the equation becomes:
In n,,t,,:t~ = - l n { l + ~ r e x p ( - E / k T , , ) }
A W -- A W
- a(AW/flr)
(7)
170
1.40 I -
1.2232
3.5
1 - - -
Equating (7) with (6) evaluated at t,, and using the well-known relation for a first order desorption transient: fiE kTp---~ = T exp(-E/kTp)
and again it is found that A T / A W is a function only of f l r / A W and a.
where, as previously mentioned, Tp is the temperature of the maximum of -dn.ddt, the following expression is derived: y
+ ~
exp(cg,l(I
--2 ln(l + y)
(8)
y) -- lnE, -- 2 ( I + o~
+ 6 [ (~l + Y ) ]
, - 2 4 [ (~1 ~ Y ) T + ' "
.]}
where AT I".
== E/krr
AW
-
/,.,
-
I + a A 14/
L
+
\fir
/ (14)
f(Pm) = 1 + 2/It,,, + 6~It,,,'- + 24/t6, 3
r,
ATp~Ifirst order
2.4464{1_= 1.40_¢+ 3.~}
(9)
Thus it can be shown that the normalized temperature shift A T / A W is given by: f2.4464 [
1.40 + -3.5Y} i_=~
; 92
i I +-Y-~f(t'').J +
g(~) = 1 + 2lot - 6/o~2 + 24/0c3
and as has been recently shown 6 for a first order desorption transient:
ar
Y -}.T.j'{In[
where
Now y can be written as: AT
2.3 S e c o n d order desorption--large "r solution. Using the second order rate equation for dn.,/dt together with a procedure similar to that described above for the first order desorption case, the following equation for y results for the large/h-/A W situation:
+ Tp .f(It,,)(I + y)2
T.,-Tp Y= r,
Y=
(13)
=,)j
(10)
=
I + y)
It can also be shown that for the second order desorption transient: fll~O[
T.
=
3fit
aW
3.5255(I - 2/~ + 6.9/~ 2)
and AW = y
( 1 - 2/~ + 6.9/~t 2)
D Edwards Jr: Peak shifts in thermal desorption experiments due to vacuum time constants--a generalized solution
Furthermore an equation (not given here) similar to (13) restllts from the small [3r/AW solution 3 for the second order desorption transient. Thus, as in the first order case, AT/AW is a function only of a, [3r/,xW for both the large and small ,Sr/A W solutions.
0
o /"
oo~
AT [
AWI=,Z0
3. Discussion Both as a check of our results in regions of flrlA W in which they should be valid and in order to indicate the radius of convergence of the various series solutions developed here, equation (I) has been mimetically integrated for a particular set of desorption parameters for various [3r/AW and the quantity AT/AW determined. The parameters used in the nunlerical integrations (for both first and second order) were fl : I K/s, 7", = 103 K, u = 30, n, :- I with 7 determined from the above quantities for the r.z-spective desorption process. Figure I shows a plot of A77AW as a function of/3r/A W for a first order desorption process with a = 30. The solid lines in Figure I are the small and large ~r/AW solutions derived from equations (13) and (10) respectively using a = 30. The darkened circles plotted in Figure I are the results of the numerically integrated solutions for a particular element
~Oil
K
--F~---'2"
/
/
0011 ....
00~00~101 -
-"
t
! / " ~ " ' - SMALL r SOLUTION
/ ....
AT
001 -
i
.........
~T/,',w
I vS ,~,-/,',w
FORFIRSTORDER
-
J
.... NUMERICALSOLUTION
i
"
OI
I0-
iOz
IO3
flr/AW Figure 1. dT/,dW is plotted as a function of fl~-/dI,V, The solid lines refer to the series solutions for small and large ~T/d t4" whereas the darkened circles are the results of a numerical integration for a particular set of desorption parameters (see text).
(8 :-lK/s, etc.) in the set of desorption parameters. F o r flr/AW larger than 10 the % difference between the numerically integrated solution and that obtained fi-om equation (10) was less than 2 ~ , becoming quite small (~0.1 ~ ) for [3r/AW > 100. It is also evident fl'om Figure I that the small flT/AW series solution for A T / A / I / i s reasonably accurate for fl-r/AW < 0.2. The dashed line in Figure 1, determined from the numerically integrated solution, is seen to provide an interpolation in the intermediate region of flr/AW (0.2 -- 5.0) between the large and the small fl~/A W solutions. In this way a solution is formed for AT/AW over an extended range of ,Sr/A W for a = 30. The dependence of AT/A W on ct is shown in Figure 2 where
al< lao,oo,
O.OOC001
I I
011"
I I0
I I00
IO00
zxw Figure 2. In this figure is plotted the % deviation of dT/A W depending on the cheise of :z as a function of flrld 14/. although it is in fact quite unusual to find a very different from 30 ( ~ 5). It can be seen from Figure 2 that for flr[AW ~ 10 the possible deviations of ~ from 30 ( ± 10) could introduce a maximum error in the estimation of AT/AW from Figure ! of ---3 % where as for flr/A 14,"< 0.2 the corresponding uncertainty in AT/All," is less than 0.5%. Thus it is concluded that Figure 1 accurately represents the relation between AT/AW and flr/AW for a first order desorption process for the presently realizable (20 <_ a < 40) set of desorption parameters (E, ~,, fl, ~-). In Figure 3 the normalized temperature shifts for both first and second order desorption transients are compared. It should be emphasized that A W is the full width at half height of -dn.Jdt for the particular desorption transient considered. 6 It is seen in this figure that for small fl~-/AW(_< 0. l), AT/A W is independent of the peak shape as has been previously suggested? On the other hand for larger flT/AW (> 0.2) the normalized peak shift does depend on the desorption process. It is also apparent that it is quite difficult to force the first order peak shift AT to be larger than AW. This is essentially due to the fact that the decay of the high temperature side of the first order desorption peak is quite rapid. It should also be noted that for large fl~-/AW, (AT/AW) a In In ~r/AW for the first order transients--a very slowly varying function. It is also seen in Figure 3 that AT/A W for the second order desorption transient is more rapidly varying as a function o f flr/A 14/than the first order case for large fl~-/AIV. It is interesting
L f
SECONDORDER\ l0
/...~
'~J~ el
"FIRSTORDER
OOI
A W ==4o,,//A W ,=zo is plotted for a first order desorption rate, using equations (10) and (13), as a function of [J.r/AW. It should be noted that, as has been recently estimated, 6 it is possible for a to vary from 20 to 40 depending on the values of the desorption parameters
00[~( )01
001
Ot Br AW
1,0
I0
I00
I000
Figure 3. A comparison is made between ,47'//I W for first and second order desorption transients. 93
D E d w a r d s J r : Peak shifts in thermal desorption experiments due to vacuum time constants--a generalized solution
to note that for the second order case, equation (14), for large /3r/AW, AT/AW e In ,8~-/~1'1/, a result suggested by earlier work 2 for the first order case and implies that the previous study was not an asymptotic solution of AT/Tp for large ~- for the first order desorption transient but rather an intermediate solution between the small /h-/& I,V and large /3-r/AW limits. In Figure 4 is plotted both - I / n o cb~/dt (solid dots) and p(t) V/nor (open figures) for a particular second order desorption transient (/3 = IK/s, E]kTp -----30, Tp = 103 K, n,, = I, e = 3 >" 10X~/s) for various r (/3~-/AW) in order to show in a qualitative way the distortions (peak shift, peak amplitude reduction, etc.) introduced into the pressure measurements for various discrete magnitudes of /3r/APl,'. It is seen that for /3r/AW between 1-3 one has instrumentally introduced significant distortions into the measurement of - I / n o dn,/dt whereas for /3r/AW~ 0.1, p(t)V/nor is, apart from the slight shift in the peak position, quite closely approximating - l[no dn~/dt. It is, on the other hand, clear from Figure 3 that for Or/A W < 0.1 the normalized peak shift AT/AW is closely approximated by the small/3~-/A W series solution. Thus one is in a low distortion situation as qualitatively determined from Figure 4 if AT/A W is given by the small/3r/A W series solution.
low instrumental distortion, i.e. /3r/AI.I/:~ 0.3. The signal
p(t)" I/at the peak is given by (5): d nsl Vl,(
t,,,)
_ I_ dns
no di
6
- -
oJ-~:
t,
.
!
..\:,,
"
t.,
5
,,P
2.4 T.
A IIt
it is [bund that the signal at the peak of the measured desorption transient is, (neglecting the small connection terms)
o
700
800
.."
...~--3 0
900 T(K} I000
",.
1100
" "~
:
1200
Figure 4. Both --1/n,, dn,/dt and l/nop(t)V/r are plotted for a particular second order desorption transient ( T , = 103K, ¢ = 30, /3 = i K / s ) for various/h-[A W. It is seen that the maximum of the function p(t) V/noz is occurring on the sample desorptio=n rate curve as required by equation (5) and that the flr/AW = 0.I i:urve has a very slight amount of instrumental distortion.
The region of intermediate and larger flr/A W (>_ 1) solutions seen in Figures 1 and 3 correspond to pressure measurement p(t)V/nor produced by a convoluted function of instrument response together with the elementary desorption transient. It should of course be clear that for large ~r]AW ( ~ 10) the quantity Vdp]dt will again yield a close approximation to --dnUdt (Ref. 7,3).
5. A n o p t i m i z a t i o n
criterion
Let us suppose that an elementary first order desorption transient is being measured in a system with time constant ~- and that it is desired to be in a region of/3~-/A W in which there is 94
f0.90 A-~-/ ,1o
first order rate equation
]0.89 fir no k A W
second order rate equation
=
(16)
0.1,
(17)
w h i c h implies that for any given (T,A W ) c o m b i n a t i o n , t3 can be
,
2
./.."
+cr[(flr/AW)2]}
Using (9),
/% W
I
•
dn~l .,. - f l ~ . ~ £ { I d t I%
,6T
I
"
and it can be shown for a lirst order desorption transient for small #flAW (-~ 0.3) that
--,--
x 3
r
But, as we have previously seen, to be in the low distortion region ~3flAW % 0.1. Thus to obtain the maximum signal at the desorption peak, /3-r/Al~ should be as large as possible (see equation (16)) but for a low distortion measurement ~3flAW < 0.1. Thus an optimized operating point is clearly obtained at:
V, ~'~,
___
-
d t It,,,
Vp(I,,,)
7
=
adjusted to satisfy the optimized condition. It should be noted that in most experiments one would simply decrease the signal: noise ratio by going to/3r/A W ~ 0.1 without any gain in information, i.e. at /3r/AW~ 0.1 one is already in a quite low distortion region (see Figure 4). Another feature of equation (16) is that for a system with arbitrary time constant r using the optimized sweep rate criterion (equation (17)) the optimized pressure "< volume signal at the peak maximum will be ~ 0.1 no independent of all system and desorption rate parameters, for both first and second order desorption transients. As an example of the above statements, if one has a sample in a 5 1 system with 10 t5 atoms s adsorbed in a first order desorption state then the pressure signal at the maximum of the pressure transient during the desorption will be ~ 6.2 × 10 -~ torr if the system is operated in an optimum manner.
6. S u m m a r y
The problem of the shift AT of the maximum of the pressure transient from the maximum of -dnUdt due to the vacuum time constant r has been considered. It has been shown that the normalized temperature shift AT/AW is a function only of /3r/AI,V and a for both first and second order desorption transients and the solution for AT/A W as a function of/3r/A W is given over an extended range of /3T/AIV. Use of numerical integration techniques have been made for the region of
D Edwards Jr: Peak shifts in thermal desorption experiments due to vacuum time constants--a generalized solution
JJr/AW not covered o p t i m i z a t i o n criterion c o n s t a n t studies has c e r n i n g the m e a s u r e d
in the analytic solutions. Finally an based on some of the results of the time been described a n d i m p l i c a t i o n s c o n pressure t r a n s i e n t s discussed.
Acknowledgement A l t h o u g h it s h o u l d be clear from the text of the paper, I would like to a c k n o w l e d g e the prior work of Dr P A R e d h c a d b o t h in p r o v i d i n g a clear definition of the present p h e n o m e n a a n d in closely d e t e r m i n i n g the final form of the generalized solution.
References E G Ehrlich, Ad~, Catalysis 14, 1963, 255. z p A Redhead, Vacmml, 12, 1962, 203. 3 D Edwards, Jr, J Vac Sci Technol, II, 1974, 1141. "~This essentially means that 20 - 7 <- 40. For a further discussion see Ref 6. 5 The symbol f~(J W/[Jr) means that terms of the order [J~/A14/are neglectcd in the equation. Thus for fJ~-/J I,V >_ 10, terms <0, I no are neglected in equation (4). " D Edwards, Jr, Surf Sol (in press). 7 See, for example, the work of E V Kornel~en in Rad Effects, 13, 1972, 227, and his other papers for a discussion of this measurement. The units used here are such that I torr i. = 3.21 ;< l019 atoms which is applicable to room temperature pressure measurements.
95