- Email: [email protected]
Unified and generalized treatment of thermal desorption data
Unified and generalized desorption data received 14 November
treatment
of thermal
1973
M Smutek. J Heyrovsky Institute of Physical Chemistry and Electrochemistry, Prague, Czechoslovakia
Czechoslovak Academy of Sciences,
A unified treatment of formal mathematical relationships for thermal desorption from an energetically homogeneous surface is given for a wide class of heating regimes. The processing of experimental data is discussed.
Introduction
In experimental arrangements most widely used in thermal desorption, the primary output records reflect essentially the time dependence of desorption rates with negligible readsorption. The data are obtained in the form of peaks whose mathematical analysis allows us to estimate the principal apparent kinetic parameters of desorption: the order of the desorption kinetics n; the activation energy of desorption Q; and-to a much lesser degree of accuracy-the frequency factor k (the pre-exponential). The mathematical treatment is based generally on the Arrhenius equation, which may be written in its dimensionless form as N, = -(dtl/dt)
= kPe-x,
(1)
where 0 represents the relative degree of coverage and x = Q/RT
(2)
is the dimensionless activation energy of desorption. Essentially 0 should be given by the ratio of the actual coverage of the available surface to the total number of accessible adsorption sites. Less exactly, the equilibrium coverage at a given temperature To is used instead in the denominator. The equation (1) has been processed for constant Q and k values, implying a homogeneous surface and no lateral interactions, for the following heating regimes: linear’*‘:
I,‘: l/T = (l/T,)
exponential3
(l/a,)
= (l/T’)
= (dt/dT)
c biTZi, I
(3)
where zi are arbitrary exponents and generally a,>0 everywhere throughout the applied temperature range. Due corrections to the output records are assumed to have been performed, so that (de/dt) data are dealt with, subject only to experimental and evaluation scatter. In integral form, the considered class of heating regimes is given by t = to + b,ln(T/T,)
bi
c-
z,=]Z(
-
(‘pi
_
+ (l/T) Tori).
1
Thus, e.g., single-term relationships of type (4) (cf equation 3) mean the following heating regimes: 2 = 0:
linear in reciprocal
temperature
z = l/2:
linear
square
z=
exponential
1:
z = 3/2: linear z = 2:
T = T,, + a,t
hyperbolic
integer and also the variable 0 can be given a significance differing from the usual one, if a better fitting results therefrom. The considered heating regimes are of the form
in reciprocal
in square
(hyperbolic)
root of temperature
root of temperature
linear
The integration of (1) for the general heating regime (3), with use of (2), gives
- a,t
: T = T,exp(a,t)
as well as for some special forms of little practical importance.4 In the following, a larger class of heating schedules, including the summarized ones, will be considered and equation (1) will be solved in a unified way except for the singular cases. A deeper insight into the behaviour under different heating schedules is thus obtained, corrections accounting for deviations from the designed heating regime can be applied and generalized simple methods for estimating the parameters result.
where
Mathematical
E,(x) = 7 t -‘e-“‘dt I
treatment
The basic equation (1) is dealt with here as a formal correlation function with constant parameters k, n and Q, by which we try to fit best the experimental data. Thus n need not be an Vacuum/volume 24/number
4.
Pergamon Press LtdlPrinted
nfl 1
=_ n -
1 --1 ( P-l
(5)
1 e,“-l
>
= ln(&J@ for n = 1
(6)
defines exponential integrals for positive integral value of z; &(x) = e-“/x; and for negative z-values: E&x) = x(2-l) T(l -z;x), i.e. incomplete gamma functions result. For half-
in Great Britain
173
M Smufek: Unified and generalized treatment of thermal desorption data
integers, the integrals (6) can be reduced to erfc-functions, such as E,&x) = z/(n/x) erfc z/x, or Es&x) = 2e-“-22/(?rx) erfcz/x. From the condition (dz0/dt2) = 0 for the maximum desorption rate we obtain (7) z where the indices m denote the conditions at maximum. Equations (5) and (7) combine to give all the required relationships. Now, let us write 6,(x) = 1 - E,(x)/E,(x)
= 1 - xe’E,(x).
(8)
x(6x2 - 8zx + z2)
x(2x - z)
X
+
(x + 2)4 -
(x + zy
1
(9)
can be written approximately.’ Thus, for 212, the leading term z/(x+z) gives an error in E,(x) less than 3.5 per cent down to x = 5; the inclusion of the second term in (9) extends the range to x = 1.5, within the same margin of accuracy. The z-values need not be integers and even small negative values are admissible in (9). For z = 0, evidently 6,(x) = 0. Let us introduce further the abbreviations Y(i)(x) = (x,/x)“‘-“E~,(x)~E~,(x,)
(IO)
and the averages
1
i
1 ; w,le) f&*-l 1
=n_18”-‘-[
unless a heating regime close to the hyperbolic (z = 0) has been applied. Equations (12) and (7) combine to give the maximum condition in the form = (kT,&,)E,(x,)[l
+ (n- 1) Cl.
The relative coverages at a temperature T,, are given by
174
1 = i + (f~ -
i)5, - (~2-
T,
= 1 + j + (iii)
(13) with respect to
1) [(6y) - s>
# 1
ln(e/e,)
for
n z
- 1 + C) for n = 1.
- 4
= J+&V~,)”
11 =Y(0) M 1 ,u = 1 + (n -
l)(j
1
(16a)
(l6b)
v(,,exp[l
1 (17a)
- (6y) + S) - 4cn- I) for n z
[
- j + (6y) - S) for n = 1.
(17b)
These shapes of desorption peaks are not affected by the initial values z, but in genera1 they depend both on (x, - x) and on x,. Yet for heating regimes close to the hyperbolic, z = 0, S-corrections are close to zero too, and y = y(,,) = exp(x, - x), so that universal shapes of desorption peaks in the coordinates (x, - x), proportional to (T,,,-’ --T-l), result: : + “,,-
‘“I”‘(“-~)
for
n _# 1;
- S for n = 1.
-.v)
for
12 = I.
(18)
The expressions (18) for the 1st and 2nd order of desorption kinetics have been obtained previously’s’. The symmetric shape for n = 2 has been given in the equivalent form
p= (12)
S = C biTmZi 6i(~,)/ C biTmZi, I I
forn
N, = MB,(a,x,~T,,)exp(
n=l
where in the averages S,,(x,,,) values are taken and F0 = y(xO). For the maximum desorption rate, the most important term in the averaged correction i = s -t- yy - (8~~) will be the first one,
n(e,/e)n-
(15)
[1 + (fl - l)~l”(“-‘) N,, = ~4damxmlT,,J nn/(nI I
p =yexp(l
nfl
l/e,“-’
= I.
(11)
with the components of l/a, at T,,, as the weighing factors. With these notations and using (7), equation (5) can be transcribed to the form
m
l)G for n # 1;
+ Cforn
This gives for the maximum desorption rate, with MB, representing the initial adsorbed amount corresponding to the desorption process under consideration,
p = f’
G = 1 biTn,Zi ui/ xbiTmZi, i I
(n-1)
= -1
are given by
x-l-z
’ - (x + zy
no
l = 1 + (n -
ln(0,/0,)
p = (NT/N,,,) = (~IRJ’exp(x,,,
2--
u-Q(Y-Y,)
n(e,/e,)“-
at T, results for
Finally, the relative rates of desorption
For sufficiently large x-values, 6,(x) zz
Specifically, the relative coverage remaining j=yO:
icosh (xyy-2*
Processing of data For estimates of parameters in the correlating equation (I), the behaviour of the peak maximum is analyzed most frequently. Here, the contribution of the G-term is negligible, reducing the corrective terms to s only. Analysis of the maximum condition (13). Most information,
in principle, can be drawn from the changes of the maximum condition (13) with varying experimental conditions. This expression indicates that the value T, usually does not depend too strongly on the actual heating regime, the dominating factor being the rate of temperature change a,,. = (dT/dt), at T,,,. For the first order desorption, n = 1, T,,, is totally independent of the actual course of heating and depends only on a,,,. Let us write (13) in the form u = -
log E,(x,)
(Ida)
= log(T,/a,)
(lab)
+ log[l + (n -
= log(x,e’“‘)
+ log k + (it I)S]
1)log 0,
(19)
M Smutek: Unified and generalized treatment of thermal desorption data
and try the power series approximation
Q zz (4.35737 u -
+ ...
x, = a, + a,u + a$
(20)
It can be shown readily that the linear correlation holds with sufficient accuracy over a wide range of x,-values. Let the best fit be defined as the one in which the deviations at the ends of the chosen x,-interval are the same and also equal the negatively taken largest misfit within the interval, this maximum being minimized. The calculations then lead to 2a,
x2 - XI
= x1 x2
Xl
1 + In
-
[ ul =
-
x1
x1(x2
+ -
Wx21xd Xl)
Wx2/xl)
-
x1
+
. ’
(x2 - x1)ln 10 x2
1
W21xl)
and the largest deviation is Amax =
z 1
x,ln
x1 - x,ln
x2 - (x2 - x1)
In
x2
-
Xl
-1
ln(x,lx,) x2
-
x1
+
In most cases of thermal desorption, the x,-values covered by the range 15 I x, I 35, which gives x, z
-2.03096
1
Wx2/xl)
will be
+ 2.20900 U; IAl _< 0.04265.
Even for the larger range of x, from 10 to 35, x, w -1.80693
+ 2.19271 u; [Al I 0.09143,
the linear correlation gives satisfactory results. Since x, = Q/RTm, we get from this linear for Q (Cal/mole):
correlation
Q x (4.38970 u - 4.0359)T,,,; IAQ] I 0.0853 T,,,; 15 5 x, I 35
(21a)
3.5907)T,,,; IAQl I 0.183 T,,,;
10 I x, I 35.
(21’$
Inserting for u from (19) we have thus in principle the possibility to solve for all the three unknown parameters n, k and Q from at least three runs, provided that both a, and B0 span over a sufficient range. The solution reduces to the solution of equations linear in Q, (n-l) and log k and if more than three runs have been performed, reliability of the estimates can be evaluated by conventional methods of regression, or the assumption of the constancy of parameters can be checked. The first solution will be done without the last correction term in (19); in the iteration the first results will be used to estimate this correction, if necessary. The estimate of the k-value is least accurate, which is frequently worsened by the poorly defined number of adsorption sites and hence by the estimate of the relative coverage 8, at the beginning of the run. If the data indicate variability of parameters, there is no means to assign it a priori to specific parameters. In fact, this ambiguity is already incorporated in the basic equation (1). The mathematical treatment followed in this paper will also be no more correct. For a rapid orientation, the use of the simple diagram in Figure 1 can be recommended. It is based on the correlation (21a), with Q-values in kcal/mole an the x-axis and a logarithmic y-axis for the quantities u = u - log T, = log [(x,,,/T,)exp(x,)]. Straight lines T,,, are drawn as parameters, since for a given T, the relations Q = c,(T,,,) + c2(Tm)u(x,,,;Tm) hold. Moreover, at a given y-level (or v-level), the T,-lines are almost equidistant, making the interpolation very easy. Now, the T,-lines for the actually observed T,,,-values are drawn into the chart and using a transparent paper a drawing is made like the one in Figure 2, with actual x values for a,,,, and em,, in the same logarithmic scale as on the y-axis of the chart. In Figure 2, three runs are considered. Any number, however, can be drawn in, using one run as the reference. On the x-axis, the n-values are indicated, in an arbitrary scale. Then, the transparent paper is shifted (the n-axis parallel to the
Q kcal/male
Figure 1. The basic chart for evaluation
of desorption
parameters
from the position of the peak maximum Z’,,,. 175
M Smutek
Unified
and generalized
treatment
of thermal
desorption
data
given by the crossing of the said vertical with the Q-axis of the chart. Then horizontals to the Y-axis of the chart are drawn from the cross-points, the corresponding v-values are read and therefrom the log k-values are calculated according to
+2 ,,
logk
log iam/ _.__
0 I
10 12
, __.. 18
16
18
,~._~_ .~_~ 20
22
2L
26
2’8
T,, (condhons
= c + loga,
-
(n -
l)loge,
a,,,O,,)
(for all the runs and the established ranges of Q and n-values). Finally, the corresponding x,-values are calculated and used
;O--
to estimate the i-terms and to check if any correction is needed. It would result in shifts of the v-values, different for individual runs. Figure 2. The auxiliary sliding graph for evaluation of desorption parameters from the position of the peak maximum T,,,.
Analysis of the maximum peak height. The expressions (16) for the maximum desorption rate, derivable from the observed height of the desorption peak, provide us with a direct means for estimates of Q, if the order of desorption n has been already
Q-axis of the chart) along the T,,,,-lines, until-if ever-all the T,,,-lines on the chart cross the corresponding T,,,-lines on the sliding drawing approximately on the same vertical. This will be accomplished to about the same accuracy only over a limited range of the n-values on the n-axis and similarly over a limited range of Q-values on the underlying chart, which are
Table 1. Relative rates of desorption ‘A of max
for the hyperbolic heating regime.
First order Y
X,-X
Y
x,--x
Y
x,-x
20 25
0.01872 0.03822 0.05851 0.07968 0.10183
- 3.9782 -3.2644 -2.8386 -2.5297 -2.2845
0.01282 0.02633 0.04061 0.05573 0.07 I 80
-4.3565 -3.6369 -3.2038 -2.8873 -2.6339
0.00991 0.02044 0.03165 0.04364 0.05649
-4.6142 -3.8904 -3.4530 -3.1318 -2.8736
30 35 40 45 50
0.12507 0.14952 0.17536 0.20276 0.23196
-2.0789 - I .9003 - I .7409 .- I .5957 -1.4612
0.08893 0.10728 0.12702 0.14837 0.17157
-2.4198 -2.2323 -2.0634 -- 1.9081 - I .7627
0.07034 0.0853 1 0.10160 0.11941 0.13903
-2.6545 -2.4614 -2.2867 -2.1252 1.9730
55 60 65 70 75
0.26327 0.29708 0.33392 0.37449 0.41987
-1.3346 ~ I .2138 ~ 1.0969 -0.9822 -0.8687
0.19702 0.22515 0.25659 0.29222 0.33333
-- 1.6245 1.4910 ~~I .3603 - I .2302 - 1.0986
0.16083 0.18527 0.21303 0.24503 0.28269
~~I .8274 ~ I .6859 ~~1.5463 I .4064 ~ 1.2634
0.47167 0.53267 0.60934 0.71295 1.OOOOO
-0.7515 --0.6298 --0.4970 ~~0.3383 0.0000
0.38197 0.44164 0.51949 0.63451 I .OOOOo
-0.9624 -0.8172 -0.6549 - 0.4549 0.0000
0.32823 0.38560 0.46286 0.58203
-1.1140 -0.9529 p-o.7703 -0.5412 0.0000
5 10 15
80 85 90 95 100
176
determined. Again, G can be safely neglected and the bracketted term in (16) gives usually an unsubstantial correction, hardly ever exceeding 5 per cent (2nd order, low X, values and linear heating). Thus the actual heating regime is of little importance
I .3554
Second order
Third order
1.9613
0.3041 0.4265 0.5207 0.6012 0.6736
I .5760 1.9250 2.2642 2.6180 3.0000
0.4549 0.6549 0.8172 0.9624 I .0986
2.0973 2.2350 2.3764 2.5235 2.6784
0.7407 0.8042 0.8656 0.9256 0.9852
3.4221 3.8973 4.4415 5.0757 5.8284
1.2302
45 40 35 30 25
2.8436 3.0223 3.2188 3.4392 3.6926
I .045 I 1.1060 1.1697 I .2352
20 15 10 5
3.9943 4.3725 4.8897 5.7439
1.3849 1.4753
95 90 85 80 75 70 65 60 55 50
1.5318 1.6832 I .8244
1.3063
I .5871 1.7481
I .m I .7783 2.3202 2.8976 3.5524 4.3197
0.5757 0.8417 I .0639 1.2671 1.4632
I .7627
5.2401 6.3673 7.7765 9.5774 I 1.9363
I .6563 I .8512 2.051 I 2.2594 2.4796
6.7405 7.8730 9.3213 11.2444 13.9282
1.9081 2.0634 2.2323 2.4198 2.6339
15.1165 19.5551 26.0220 35.9791 52.4857
2.7158 2.9732 3.2589 3.5829 3.9605
17.9443 24.6261 37.9737 77.9872
2.8873 3.2038 3.6369 4.3565
82.8660 148.495 335.998 1348.50
4.4172 5.0006 5.8171 7.2067
1.3603 I .49lO
1.6245
M Smutek: Unified and generalized treatment of thermal desorption data
and mainly the temperature gradient a, at T,,, matters (note that T, increases with a,, as stated above). Rewriting (16) we obtain
nn/(n-
1)
f@) = [, + (n _ ,)$1/V)’
. exp(1 - 5).
The Q-estimate obtained in this way may either serve to corroborate the results of the evaluation of the peak position, or to reduce the number of runs to two at different initial coverages. The knowledge of n and Q makes then possible to estimate log k, either from (21) or from the chart Figure 1. Analysis of the peak shape. The analysis of the reduced desorp-
tion rates p as a function of l/T allows in principle to obtain the kinetic parameters Q and n from one single run. Apart from the obvious requirement that the record of the peak be sufficiently accurate and the due corrections have been applied, it is necessary that at least a significant part of the curve, including its maximum, can be assumed as free of interferences from the neighbouring peaks. For a general heating regime, the analysis of the shape is somewhat involved, as the shape depends not only on the order of desorption n, but to a lesser degree on the value of x,,, as well. Most simple is the procedure for the hyperbolic heating regime, z = 0 (or some regime close to it), when the universal shape for each value of n results, independent of x,. In this case, the analysis of the two half-widths at different p-levels enables us to estimate the desorption order from the assymetry, i.e. from the appropriate ratios. Once the parameter n estimated, the comparison of the differences (l/T,,,) - (l/T,) for selected p-values with the corresponding (x,-x,)-values gives the estimates of Q/R. A multiple set of estimates of n and Q is thus available, allowing to establish the confidence intervals, or possibly the inadequacy of the primary postulate of constant parameters in (1). Sometimes, however, it would not be possible to decide between the inconstancy of, say, Q, and the effect
of a hidden near-by desorption peak, or at least the decision will be more or less arbitrary. When such an analysis of the peak shape is possible, one run will be sufficient to characterize the desorption process. The peak height will corroborate the Q-estimate and the T,,,-position will give the log k-estimate, provided B0 is known, or the reaction of desorption is first order. In Table 1 the universal p-values are given as functions of y = exp(x, - x) and of (x, - x) for n = 1, 2, and 3, respectively. The inspection of the tabulated data indicates that the ascending branches, especially for lower p-values, exhibit a rather similar course, apart from being shifted with respect to x,. Thus they are not particularly suited for estimates of the order of desorption. With increase in the z-index, characterizing the heating regime, the peak shapes become in the x-scale more slender, the descendent branch being affected more strongly than the ascendent one. The deformation of peaks increases with increase in z and n and with decrease in x,. Computations have shown that for a first order desorption, the assymetry of the peak does not change appreciabiy even under most unfavorable conditions (linear heating, z = 2 and x, = 10). For the second order desorption the deformation is somewhat more pronounced, the descent is steeper than the ascent. Nevertheless, decision between the possible desorption orders by the assymetry criterion is not impaired when other than the hyperbolic heating regime is used. The narrower shape of the peaks, however, simulates-when data of Table 1 are applied-a smaller activation energy of desorption. This is not serious for 1st order desorption, where the lowest estimate (z = 2, x, = 10) of Q still makes 91 per cent of the true value. For 2nd order desorption, the disagreement can be more severe impairing the simple use of Table 1 for estimates of Q. References
1P A Redhead, Vacuum, 12, 1962, 203. 2 G Carter, Vacuum, 12, 1962, 245. 3 G Carter, W A Grant, G Farrell and J S Colligon, Vacuum, 18, 1968, 263. 4 G Carter and D G Armour, Vacuum, 19, 1969,459. 5 Natl. Bureau of Standards, Handbook of Mathematical Function (1968).
177
treatment
of thermal
1973
M Smutek. J Heyrovsky Institute of Physical Chemistry and Electrochemistry, Prague, Czechoslovakia
Czechoslovak Academy of Sciences,
A unified treatment of formal mathematical relationships for thermal desorption from an energetically homogeneous surface is given for a wide class of heating regimes. The processing of experimental data is discussed.
Introduction
In experimental arrangements most widely used in thermal desorption, the primary output records reflect essentially the time dependence of desorption rates with negligible readsorption. The data are obtained in the form of peaks whose mathematical analysis allows us to estimate the principal apparent kinetic parameters of desorption: the order of the desorption kinetics n; the activation energy of desorption Q; and-to a much lesser degree of accuracy-the frequency factor k (the pre-exponential). The mathematical treatment is based generally on the Arrhenius equation, which may be written in its dimensionless form as N, = -(dtl/dt)
= kPe-x,
(1)
where 0 represents the relative degree of coverage and x = Q/RT
(2)
is the dimensionless activation energy of desorption. Essentially 0 should be given by the ratio of the actual coverage of the available surface to the total number of accessible adsorption sites. Less exactly, the equilibrium coverage at a given temperature To is used instead in the denominator. The equation (1) has been processed for constant Q and k values, implying a homogeneous surface and no lateral interactions, for the following heating regimes: linear’*‘:
I,‘: l/T = (l/T,)
exponential3
(l/a,)
= (l/T’)
= (dt/dT)
c biTZi, I
(3)
where zi are arbitrary exponents and generally a,>0 everywhere throughout the applied temperature range. Due corrections to the output records are assumed to have been performed, so that (de/dt) data are dealt with, subject only to experimental and evaluation scatter. In integral form, the considered class of heating regimes is given by t = to + b,ln(T/T,)
bi
c-
z,=]Z(
-
(‘pi
_
+ (l/T) Tori).
1
Thus, e.g., single-term relationships of type (4) (cf equation 3) mean the following heating regimes: 2 = 0:
linear in reciprocal
temperature
z = l/2:
linear
square
z=
exponential
1:
z = 3/2: linear z = 2:
T = T,, + a,t
hyperbolic
integer and also the variable 0 can be given a significance differing from the usual one, if a better fitting results therefrom. The considered heating regimes are of the form
in reciprocal
in square
(hyperbolic)
root of temperature
root of temperature
linear
The integration of (1) for the general heating regime (3), with use of (2), gives
- a,t
: T = T,exp(a,t)
as well as for some special forms of little practical importance.4 In the following, a larger class of heating schedules, including the summarized ones, will be considered and equation (1) will be solved in a unified way except for the singular cases. A deeper insight into the behaviour under different heating schedules is thus obtained, corrections accounting for deviations from the designed heating regime can be applied and generalized simple methods for estimating the parameters result.
where
Mathematical
E,(x) = 7 t -‘e-“‘dt I
treatment
The basic equation (1) is dealt with here as a formal correlation function with constant parameters k, n and Q, by which we try to fit best the experimental data. Thus n need not be an Vacuum/volume 24/number
4.
Pergamon Press LtdlPrinted
nfl 1
=_ n -
1 --1 ( P-l
(5)
1 e,“-l
>
= ln(&J@ for n = 1
(6)
defines exponential integrals for positive integral value of z; &(x) = e-“/x; and for negative z-values: E&x) = x(2-l) T(l -z;x), i.e. incomplete gamma functions result. For half-
in Great Britain
173
M Smufek: Unified and generalized treatment of thermal desorption data
integers, the integrals (6) can be reduced to erfc-functions, such as E,&x) = z/(n/x) erfc z/x, or Es&x) = 2e-“-22/(?rx) erfcz/x. From the condition (dz0/dt2) = 0 for the maximum desorption rate we obtain (7) z where the indices m denote the conditions at maximum. Equations (5) and (7) combine to give all the required relationships. Now, let us write 6,(x) = 1 - E,(x)/E,(x)
= 1 - xe’E,(x).
(8)
x(6x2 - 8zx + z2)
x(2x - z)
X
+
(x + 2)4 -
(x + zy
1
(9)
can be written approximately.’ Thus, for 212, the leading term z/(x+z) gives an error in E,(x) less than 3.5 per cent down to x = 5; the inclusion of the second term in (9) extends the range to x = 1.5, within the same margin of accuracy. The z-values need not be integers and even small negative values are admissible in (9). For z = 0, evidently 6,(x) = 0. Let us introduce further the abbreviations Y(i)(x) = (x,/x)“‘-“E~,(x)~E~,(x,)
(IO)
and the averages
1
i
1 ; w,le) f&*-l 1
=n_18”-‘-[
unless a heating regime close to the hyperbolic (z = 0) has been applied. Equations (12) and (7) combine to give the maximum condition in the form = (kT,&,)E,(x,)[l
+ (n- 1) Cl.
The relative coverages at a temperature T,, are given by
174
1 = i + (f~ -
i)5, - (~2-
T,
= 1 + j + (iii)
(13) with respect to
1) [(6y) - s>
# 1
ln(e/e,)
for
n z
- 1 + C) for n = 1.
- 4
= J+&V~,)”
11 =Y(0) M 1 ,u = 1 + (n -
l)(j
1
(16a)
(l6b)
v(,,exp[l
1 (17a)
- (6y) + S) - 4cn- I) for n z
[
- j + (6y) - S) for n = 1.
(17b)
These shapes of desorption peaks are not affected by the initial values z, but in genera1 they depend both on (x, - x) and on x,. Yet for heating regimes close to the hyperbolic, z = 0, S-corrections are close to zero too, and y = y(,,) = exp(x, - x), so that universal shapes of desorption peaks in the coordinates (x, - x), proportional to (T,,,-’ --T-l), result: : + “,,-
‘“I”‘(“-~)
for
n _# 1;
- S for n = 1.
-.v)
for
12 = I.
(18)
The expressions (18) for the 1st and 2nd order of desorption kinetics have been obtained previously’s’. The symmetric shape for n = 2 has been given in the equivalent form
p= (12)
S = C biTmZi 6i(~,)/ C biTmZi, I I
forn
N, = MB,(a,x,~T,,)exp(
n=l
where in the averages S,,(x,,,) values are taken and F0 = y(xO). For the maximum desorption rate, the most important term in the averaged correction i = s -t- yy - (8~~) will be the first one,
n(e,/e)n-
(15)
[1 + (fl - l)~l”(“-‘) N,, = ~4damxmlT,,J nn/(nI I
p =yexp(l
nfl
l/e,“-’
= I.
(11)
with the components of l/a, at T,,, as the weighing factors. With these notations and using (7), equation (5) can be transcribed to the form
m
l)G for n # 1;
+ Cforn
This gives for the maximum desorption rate, with MB, representing the initial adsorbed amount corresponding to the desorption process under consideration,
p = f’
G = 1 biTn,Zi ui/ xbiTmZi, i I
(n-1)
= -1
are given by
x-l-z
’ - (x + zy
no
l = 1 + (n -
ln(0,/0,)
p = (NT/N,,,) = (~IRJ’exp(x,,,
2--
u-Q(Y-Y,)
n(e,/e,)“-
at T, results for
Finally, the relative rates of desorption
For sufficiently large x-values, 6,(x) zz
Specifically, the relative coverage remaining j=yO:
icosh (xyy-2*
Processing of data For estimates of parameters in the correlating equation (I), the behaviour of the peak maximum is analyzed most frequently. Here, the contribution of the G-term is negligible, reducing the corrective terms to s only. Analysis of the maximum condition (13). Most information,
in principle, can be drawn from the changes of the maximum condition (13) with varying experimental conditions. This expression indicates that the value T, usually does not depend too strongly on the actual heating regime, the dominating factor being the rate of temperature change a,,. = (dT/dt), at T,,,. For the first order desorption, n = 1, T,,, is totally independent of the actual course of heating and depends only on a,,,. Let us write (13) in the form u = -
log E,(x,)
(Ida)
= log(T,/a,)
(lab)
+ log[l + (n -
= log(x,e’“‘)
+ log k + (it I)S]
1)log 0,
(19)
M Smutek: Unified and generalized treatment of thermal desorption data
and try the power series approximation
Q zz (4.35737 u -
+ ...
x, = a, + a,u + a$
(20)
It can be shown readily that the linear correlation holds with sufficient accuracy over a wide range of x,-values. Let the best fit be defined as the one in which the deviations at the ends of the chosen x,-interval are the same and also equal the negatively taken largest misfit within the interval, this maximum being minimized. The calculations then lead to 2a,
x2 - XI
= x1 x2
Xl
1 + In
-
[ ul =
-
x1
x1(x2
+ -
Wx21xd Xl)
Wx2/xl)
-
x1
+
. ’
(x2 - x1)ln 10 x2
1
W21xl)
and the largest deviation is Amax =
z 1
x,ln
x1 - x,ln
x2 - (x2 - x1)
In
x2
-
Xl
-1
ln(x,lx,) x2
-
x1
+
In most cases of thermal desorption, the x,-values covered by the range 15 I x, I 35, which gives x, z
-2.03096
1
Wx2/xl)
will be
+ 2.20900 U; IAl _< 0.04265.
Even for the larger range of x, from 10 to 35, x, w -1.80693
+ 2.19271 u; [Al I 0.09143,
the linear correlation gives satisfactory results. Since x, = Q/RTm, we get from this linear for Q (Cal/mole):
correlation
Q x (4.38970 u - 4.0359)T,,,; IAQ] I 0.0853 T,,,; 15 5 x, I 35
(21a)
3.5907)T,,,; IAQl I 0.183 T,,,;
10 I x, I 35.
(21’$
Inserting for u from (19) we have thus in principle the possibility to solve for all the three unknown parameters n, k and Q from at least three runs, provided that both a, and B0 span over a sufficient range. The solution reduces to the solution of equations linear in Q, (n-l) and log k and if more than three runs have been performed, reliability of the estimates can be evaluated by conventional methods of regression, or the assumption of the constancy of parameters can be checked. The first solution will be done without the last correction term in (19); in the iteration the first results will be used to estimate this correction, if necessary. The estimate of the k-value is least accurate, which is frequently worsened by the poorly defined number of adsorption sites and hence by the estimate of the relative coverage 8, at the beginning of the run. If the data indicate variability of parameters, there is no means to assign it a priori to specific parameters. In fact, this ambiguity is already incorporated in the basic equation (1). The mathematical treatment followed in this paper will also be no more correct. For a rapid orientation, the use of the simple diagram in Figure 1 can be recommended. It is based on the correlation (21a), with Q-values in kcal/mole an the x-axis and a logarithmic y-axis for the quantities u = u - log T, = log [(x,,,/T,)exp(x,)]. Straight lines T,,, are drawn as parameters, since for a given T, the relations Q = c,(T,,,) + c2(Tm)u(x,,,;Tm) hold. Moreover, at a given y-level (or v-level), the T,-lines are almost equidistant, making the interpolation very easy. Now, the T,-lines for the actually observed T,,,-values are drawn into the chart and using a transparent paper a drawing is made like the one in Figure 2, with actual x values for a,,,, and em,, in the same logarithmic scale as on the y-axis of the chart. In Figure 2, three runs are considered. Any number, however, can be drawn in, using one run as the reference. On the x-axis, the n-values are indicated, in an arbitrary scale. Then, the transparent paper is shifted (the n-axis parallel to the
Q kcal/male
Figure 1. The basic chart for evaluation
of desorption
parameters
from the position of the peak maximum Z’,,,. 175
M Smutek
Unified
and generalized
treatment
of thermal
desorption
data
given by the crossing of the said vertical with the Q-axis of the chart. Then horizontals to the Y-axis of the chart are drawn from the cross-points, the corresponding v-values are read and therefrom the log k-values are calculated according to
+2 ,,
logk
log iam/ _.__
0 I
10 12
, __.. 18
16
18
,~._~_ .~_~ 20
22
2L
26
2’8
T,, (condhons
= c + loga,
-
(n -
l)loge,
a,,,O,,)
(for all the runs and the established ranges of Q and n-values). Finally, the corresponding x,-values are calculated and used
;O--
to estimate the i-terms and to check if any correction is needed. It would result in shifts of the v-values, different for individual runs. Figure 2. The auxiliary sliding graph for evaluation of desorption parameters from the position of the peak maximum T,,,.
Analysis of the maximum peak height. The expressions (16) for the maximum desorption rate, derivable from the observed height of the desorption peak, provide us with a direct means for estimates of Q, if the order of desorption n has been already
Q-axis of the chart) along the T,,,,-lines, until-if ever-all the T,,,-lines on the chart cross the corresponding T,,,-lines on the sliding drawing approximately on the same vertical. This will be accomplished to about the same accuracy only over a limited range of the n-values on the n-axis and similarly over a limited range of Q-values on the underlying chart, which are
Table 1. Relative rates of desorption ‘A of max
for the hyperbolic heating regime.
First order Y
X,-X
Y
x,--x
Y
x,-x
20 25
0.01872 0.03822 0.05851 0.07968 0.10183
- 3.9782 -3.2644 -2.8386 -2.5297 -2.2845
0.01282 0.02633 0.04061 0.05573 0.07 I 80
-4.3565 -3.6369 -3.2038 -2.8873 -2.6339
0.00991 0.02044 0.03165 0.04364 0.05649
-4.6142 -3.8904 -3.4530 -3.1318 -2.8736
30 35 40 45 50
0.12507 0.14952 0.17536 0.20276 0.23196
-2.0789 - I .9003 - I .7409 .- I .5957 -1.4612
0.08893 0.10728 0.12702 0.14837 0.17157
-2.4198 -2.2323 -2.0634 -- 1.9081 - I .7627
0.07034 0.0853 1 0.10160 0.11941 0.13903
-2.6545 -2.4614 -2.2867 -2.1252 1.9730
55 60 65 70 75
0.26327 0.29708 0.33392 0.37449 0.41987
-1.3346 ~ I .2138 ~ 1.0969 -0.9822 -0.8687
0.19702 0.22515 0.25659 0.29222 0.33333
-- 1.6245 1.4910 ~~I .3603 - I .2302 - 1.0986
0.16083 0.18527 0.21303 0.24503 0.28269
~~I .8274 ~ I .6859 ~~1.5463 I .4064 ~ 1.2634
0.47167 0.53267 0.60934 0.71295 1.OOOOO
-0.7515 --0.6298 --0.4970 ~~0.3383 0.0000
0.38197 0.44164 0.51949 0.63451 I .OOOOo
-0.9624 -0.8172 -0.6549 - 0.4549 0.0000
0.32823 0.38560 0.46286 0.58203
-1.1140 -0.9529 p-o.7703 -0.5412 0.0000
5 10 15
80 85 90 95 100
176
determined. Again, G can be safely neglected and the bracketted term in (16) gives usually an unsubstantial correction, hardly ever exceeding 5 per cent (2nd order, low X, values and linear heating). Thus the actual heating regime is of little importance
I .3554
Second order
Third order
1.9613
0.3041 0.4265 0.5207 0.6012 0.6736
I .5760 1.9250 2.2642 2.6180 3.0000
0.4549 0.6549 0.8172 0.9624 I .0986
2.0973 2.2350 2.3764 2.5235 2.6784
0.7407 0.8042 0.8656 0.9256 0.9852
3.4221 3.8973 4.4415 5.0757 5.8284
1.2302
45 40 35 30 25
2.8436 3.0223 3.2188 3.4392 3.6926
I .045 I 1.1060 1.1697 I .2352
20 15 10 5
3.9943 4.3725 4.8897 5.7439
1.3849 1.4753
95 90 85 80 75 70 65 60 55 50
1.5318 1.6832 I .8244
1.3063
I .5871 1.7481
I .m I .7783 2.3202 2.8976 3.5524 4.3197
0.5757 0.8417 I .0639 1.2671 1.4632
I .7627
5.2401 6.3673 7.7765 9.5774 I 1.9363
I .6563 I .8512 2.051 I 2.2594 2.4796
6.7405 7.8730 9.3213 11.2444 13.9282
1.9081 2.0634 2.2323 2.4198 2.6339
15.1165 19.5551 26.0220 35.9791 52.4857
2.7158 2.9732 3.2589 3.5829 3.9605
17.9443 24.6261 37.9737 77.9872
2.8873 3.2038 3.6369 4.3565
82.8660 148.495 335.998 1348.50
4.4172 5.0006 5.8171 7.2067
1.3603 I .49lO
1.6245
M Smutek: Unified and generalized treatment of thermal desorption data
and mainly the temperature gradient a, at T,,, matters (note that T, increases with a,, as stated above). Rewriting (16) we obtain
nn/(n-
1)
f@) = [, + (n _ ,)$1/V)’
. exp(1 - 5).
The Q-estimate obtained in this way may either serve to corroborate the results of the evaluation of the peak position, or to reduce the number of runs to two at different initial coverages. The knowledge of n and Q makes then possible to estimate log k, either from (21) or from the chart Figure 1. Analysis of the peak shape. The analysis of the reduced desorp-
tion rates p as a function of l/T allows in principle to obtain the kinetic parameters Q and n from one single run. Apart from the obvious requirement that the record of the peak be sufficiently accurate and the due corrections have been applied, it is necessary that at least a significant part of the curve, including its maximum, can be assumed as free of interferences from the neighbouring peaks. For a general heating regime, the analysis of the shape is somewhat involved, as the shape depends not only on the order of desorption n, but to a lesser degree on the value of x,,, as well. Most simple is the procedure for the hyperbolic heating regime, z = 0 (or some regime close to it), when the universal shape for each value of n results, independent of x,. In this case, the analysis of the two half-widths at different p-levels enables us to estimate the desorption order from the assymetry, i.e. from the appropriate ratios. Once the parameter n estimated, the comparison of the differences (l/T,,,) - (l/T,) for selected p-values with the corresponding (x,-x,)-values gives the estimates of Q/R. A multiple set of estimates of n and Q is thus available, allowing to establish the confidence intervals, or possibly the inadequacy of the primary postulate of constant parameters in (1). Sometimes, however, it would not be possible to decide between the inconstancy of, say, Q, and the effect
of a hidden near-by desorption peak, or at least the decision will be more or less arbitrary. When such an analysis of the peak shape is possible, one run will be sufficient to characterize the desorption process. The peak height will corroborate the Q-estimate and the T,,,-position will give the log k-estimate, provided B0 is known, or the reaction of desorption is first order. In Table 1 the universal p-values are given as functions of y = exp(x, - x) and of (x, - x) for n = 1, 2, and 3, respectively. The inspection of the tabulated data indicates that the ascending branches, especially for lower p-values, exhibit a rather similar course, apart from being shifted with respect to x,. Thus they are not particularly suited for estimates of the order of desorption. With increase in the z-index, characterizing the heating regime, the peak shapes become in the x-scale more slender, the descendent branch being affected more strongly than the ascendent one. The deformation of peaks increases with increase in z and n and with decrease in x,. Computations have shown that for a first order desorption, the assymetry of the peak does not change appreciabiy even under most unfavorable conditions (linear heating, z = 2 and x, = 10). For the second order desorption the deformation is somewhat more pronounced, the descent is steeper than the ascent. Nevertheless, decision between the possible desorption orders by the assymetry criterion is not impaired when other than the hyperbolic heating regime is used. The narrower shape of the peaks, however, simulates-when data of Table 1 are applied-a smaller activation energy of desorption. This is not serious for 1st order desorption, where the lowest estimate (z = 2, x, = 10) of Q still makes 91 per cent of the true value. For 2nd order desorption, the disagreement can be more severe impairing the simple use of Table 1 for estimates of Q. References
1P A Redhead, Vacuum, 12, 1962, 203. 2 G Carter, Vacuum, 12, 1962, 245. 3 G Carter, W A Grant, G Farrell and J S Colligon, Vacuum, 18, 1968, 263. 4 G Carter and D G Armour, Vacuum, 19, 1969,459. 5 Natl. Bureau of Standards, Handbook of Mathematical Function (1968).
177