Kinetics of C2 reactions during high-temperature pyrolysis of acetylene

Article abstract of DOI:10.1021/jp963373o

The kinetics of C₂ radical reactions during the first stage of acetylene high-temperature pyrolysis was studied by monitoring C, C₂ and C₃ radicals. Quantitative C₂ detection was performed by ring dye laser absorption spectroscopy, C atoms were measured by applying atomic resonance absorption spectroscopy, and C₃ radicals were monitored by their emission using a combination of a spectrograph and an intensified CCD camera system. The experiments were performed behind reflected shock waves and cover the temperature range of 2580-4650 K at pressures around 2 bar. In the first part of the study initial mixtures containing Ar with 5-50 ppm C₂H₂ were used. In this very low concentration range, rate coefficients for the following four reactions were determined: C₂H₂ + M ^k1? C₂H + H + M (R1); C₂H + M ^k2? C₂ + H + M (R2); C₂ + C₂ ^k3? C + C₃ (R3); C₂ + M ^k4? C + C + M (R4); where k₁ = 6.96 × 10³⁹ T^-6.06 exp(-67 130/T) cm³ mol^-1 s^-1, k₂ = 1.74 × 10³⁵ T^-5.16 exp(-57 367/T) cm³ mol^-1 S^-1, k₃ = 3.2 × 10¹⁴ cm³ mol^-1 s^-1, and k₄ = 1.5 × 10¹⁶ exp(-71 650/T) cm³ mol^-1 s^-1. Furthermore, this experiments indicate that a modification of the JANAF¹ thermodynamic data of either C, C₂ or C₃ seems to be necessary. In the second part, some experiments with relatively high initial acetylene concentrations up to 500 ppm C₂H₂ in Ar were carried out to check the validity of a more complex mechanism for the acetylene pyrolysis. Finally in a third part, a perturbation study was performed by adding 1000 ppm H₂ to the initial mixtures of Ar with 20 and 50 ppm of C₂H₂. For the most important perturbation reactions, C₂ + H₂ ^k5? C₂H + H (R5) and C₂H + H₂ ^k6? C₂H₂ + H (R6), rate coefficients of k₅ = 6.6 × 10¹³ exp(-4000/T) cm³ mol^-1 s^-1 and k₆ = 7.4 × 10¹⁴ exp(-3400/T) cm³ mol^-1 s^-1 were obtained.

Full text of DOI:10.1021/jp963373o

2138
J. Phys. Chem. A 1997, 101, 2138-2146
Kinetics of C2 Reactions during High-Temperature Pyrolysis of Acetylene
T. Kruse and P. Roth*
Institut f u¨ r Verbrennung und Gasdynamik, Gerhard-Mercator-UniVersit a¨ t Duisburg, 47048 Duisburg, Germany
X
ReceiVed: October 29, 1996; In Final Form: January 3, 1997
The kinetics of C
by monitoring C, C
spectroscopy, C atoms were measured by applying atomic resonance absorption spectroscopy, and C
2
radical reactions during the first stage of acetylene high-temperature pyrolysis was studied
and C radicals. Quantitative C detection was performed by ring dye laser absorption
radicals
2
3
2
3
were monitored by their emission using a combination of a spectrograph and an intensified CCD camera
system. The experiments were performed behind reflected shock waves and cover the temperature range of
2
5
580-4650 K at pressures around 2 bar. In the first part of the study initial mixtures containing Ar with
-50 ppm C
2
H
2
were used. In this very low concentration range, rate coefficients for the following four
k1
k2
reactions were determined: C
2
H
2
+ M S C
2
H + H + M (R1); C
2
H + M
\
-6.06
2
+ H + M (R2); C
2
+ C
2
k
k
4
3
3
9
3
-1
y
\
z C + C
3
(R3); C
) 1.74 × 10
2
+ M
\
1
) 6.96 × 10
T
-
1
35 -5.16
3
-1 -1
14
-1 -1
s , k
2
T
exp(-57 367/T) cm mol s , k
3
) 3.2 × 10 cm mol s , and k
4
) 1.5 ×
1
6
3
-1 -1
1
1
0 exp(-71 650/T) cm mol s . Furthermore, this experiments indicate that a modification of the JANAF
thermodynamic data of either C, C
relatively high initial acetylene concentrations up to 500 ppm C
validity of a more complex mechanism for the acetylene pyrolysis. Finally in a third part, a perturbation
study was performed by adding 1000 ppm H to the initial mixtures of Ar with 20 and 50 ppm of C . For
2
or C
3
seems to be necessary. In the second part, some experiments with
2
H
2
in Ar were carried out to check the
2
2 2
H
k
k
6
5
the most important perturbation reactions, C + H (R6), rate
2
+ H
2
3
y
\
z C
2
H + H (R5) and C
2
H + H
2
y
\
z C
2 2
H
13
-1 -1
14
3
-1
coefficients of k
5
) 6.6 × 10 exp(-4000/T) cm mol
s
and k
6
) 7.4 × 10 exp(-3400/T) cm mol
-1
s
were obtained.
Introduction
vacuum apparatus and can be evacuated down to pressures
-
8
below 5 × 10 mbar by a turbo-molecular pump. The shock
wave is formed after bursting of aluminum diaphragms, which
were 50 or 70 µm thick. Temperatures and pressures behind
the reflected shock waves were calculated, on the basis of one-
dimensional gas dynamic equations, from the speed of the
incident shock wave, which was measured by four piezoelectric
pressure gauges. The gases used were carefully mixed in a
stainless steel cylinder by the partial pressure method. They
were of the highest commercially available purity; i.e., Ar
Elementary reactions involving C2 radicals and other small
carbon species like C and C3 play an important role in the high-
temperature chemistry of hydrocarbons. Examples are soot
formation in hot flames, diamond synthesis from gas phase
pyrolysis, or the formation and decay of the recently discovered
C60 fullerene.2
-4
The pyrolysis of acetylene has been studied
in the last decades by numerous investigators, but up to now,
no quantitative measurement of C, C2, or C3 radicals has been
performed. Only in one kinetic study by Beck et al., C2 radicals
were measured during the acetylene pyrolysis, but in this
investigation C2 was monitored only by emission and thus only
qualitatively. With the development of tunable and narrow
bandwith CW ring dye lasers, the sensitive and quantitative
detection of many di- and some triatomic molecules by direct
absorption measurements became feasible.
5
(99.9999%), H2 (99.9999%), and C2H2 (99.6%). In a few
experiments the acetylene was passed through an activated
carbon filter to remove the traces of 0.4% acetone, but this
experiments showed no difference from the ones with unpurified
acetylene. This is in agreement with earlier studies (see, e.g.,
refs 6 or 7 that also showed no significant influence of acetone
impurities on the kinetics of acetylene pyrolysis at low relative
concentrations.
In the present study, C2 radicals were detected by ring dye
laser absorption spectroscopy (RDLAS) using selected rotational
3
The laser spectrometer (see Figure 1), used to measure the
C2 concentration, consists of a CW ring dye laser from Spectra
transitions in the (1-0) and (0-0) bands of the C2 (d Πg r
3
a Πu) Swan system. To reveal the reaction mechanism, some
+
Physics, Model 380D, pumped by an Ar laser. To realize
additional experiments were made by recording C and C3
concentrations by atomic resonance absorption spectroscopy
single-mode operation, the frequency was controlled and
stabilized by a reference interferometer located in a feedback
loop. For a further improvement of the detection limit, an
external intensity stabilization was employed. The dye laser
operates under these conditions with a line width below 500
kHz and an intensity fluctuation below 1%. The laser beam
was coupled into the measurement plane of the stock tube via
an optical fiber and was split into probe and reference beam
prior to the shock tube. Two Si photodetectors were used in
the differential technique to monitor the C2 absorption. The
time absorption of about 3 µs is limited by the beam diameter
and the shock speed.
(ARAS) and emission spectroscopy, respectively. The combi-
nation of this spectroscopic methods with the shock tube
technique allowed direct observation of several C2 reactions.
Experimental Section
The experiments were performed behind reflected shock
waves in a stainless steel shock tube of 80 mm inner diameter;
see Figure 1. The tube is constructed as heatable ultrahigh-
X
Abstract published in AdVance ACS Abstracts, February 15, 1997.
S1089-5639(96)03373-7 CCC: $14.00 © 1997 American Chemical Society

Kinetics of C2 Reactions
J. Phys. Chem. A, Vol. 101, No. 11, 1997 2139
Figure 1. Experimental setup.
The C2 concentrations were measured at two different
wavelengths in the d Πg r a Πu Swan system:
C atoms were detected at λ ) 156.1 nm by atomic resonance
absorption spectroscopy (ARAS). A microwave discharge lamp,
which was operated at 6 mbar in a mixture of He containing
1% CO, was used as the source for CI spectral radiation. A
McPherson 0.5 m vacuum monochromator with a 100 µm slit
combined with a photoelectron multiplier was used to select
the respective C atom line. Since the spectral shapes of the
lines emitted by the discharge lamp are not known precisely,
the relation between the measured absorption and corresponding
C atom concentration was obtained by calibration experiments
based on thermal decomposition of CH4 at temperatures T g
3800 K. For example, an absorption of 50% corresponds to a
3
3
(
1) at λ ) 467.440 nm corresponding to the center of the
three overlapping rotational lines R1(37), R2(36), and R3(35) in
the (1-0) band,
(2) at λ ) 516.646 nm corresponding to the band head of
the (0-0) band, consisting of nine overlapping rotational lines.
The line positions were taken from ref 8, and the pressure
broadening was checked by scanning each line in frequency
-
1
steps of 0.02 cm in a series of shock-tube experiments and
was found to be in agreement with ref 9. Additional measure-
ments of C2 performed at four other lines in the (1-0) and (0-
1
2
-3
0
) bands were in good agreement with the earlier described
experiments.
C atom concentration of about 8 × 10 cm .
The time-resolved emission during C H thermal decomposi-
2
2
The oscillator strengths of the C2 Swan system, measured in
the previous decades in numerous experiments, show some
discrepencies. Most recent results by refs 10-12 obtained from
lifetime measurements recommend a value around f00 ) 0.032,
which is significantly higher than the older value of f00 ) 0.025
deduced from C2 emission behind shock waves (see, e..g., refs
tion in the wavelength range from 330 to 600 nm was recorded
with a combination of a grid spectrograph and an intensified
CCD camera. The spectral resolution was about 2 nm and the
time resolution was 20 µs during a measurement time of 1 ms.
A detailed description of this system is given in ref 16.
1
3 or 14. To overcome this difficulty, we have performed
Results
oscillator-strength measurements of the C2 Swan system in some
shock tube experiments; see also ref 15. The very low detection
limit of less than 0.1 ppm C2, which could be achieved with
the RDLAS at λ ) 516.646 nm in the (0-0) band head, allows
us a direct calibration of the C2 absorption cross section and
therewith determination of the oscillator strength. At very low
C2H2 concentrations of 2 ppm or less and temperatures above
The experiments on acetylene pyrolysis were performed with
mixtures of argon containing 5-500 ppm C2H2 at temperatures
580 K e T e 4650 K and pressures between 1.5 and 2.7 bar.
In an additional series of experiments 1000 ppm H2 was added
to initial mixture containing 20 or 50 ppm C2H2. The present
results can be grouped in three parts:
2
(a) experiments in mixtures containing 5-50 ppm C2H2,
3500 K it can be assumed, even without a detailed knowledge
where C2 concentrations (RDLAS), C concentrations (ARAS),
and C3 concentrations (CCD) were measured,
of the kinetics, that the C2 formation due to two successive H
atom abstractions is much faster than any subsequent C2
consumption reactions. Thus, the maximum C2 concentration
must be equal to the initial C2H2 concentration. In a series of
(
b) experiments in mixtures containing 100-500 ppm C2H2,
where only C2 was measured, and
1
5 shock tube experiments with initial C2H2 concentrations
between 0.5 and 2 ppm an oscillator strength of f00 ) 0.037 (
.002 was determined, which is about 15% higher than the value
(c) perturbation experiments in mixtures of argon with 20 or
50 ppm C2H2, where the formation of C2 was perturbed by the
addition of 1000 ppm H2 to the initial mixture.
0
obtained from the lifetime measurements. For the evaluation
of the later described kinetic experiments we used our own f00
value, although the kinetic results are not significantly different
when using the recent-literature value. With the knowledge of
the oscillator strength together with the pressure-broadening
factor, the absolute C2 concentration could directly be deter-
mined from the measured absorption by applying the Lambert-
Beer law.
Typical C2 concentration profiles belonging to the first group
of experiments are shown in Figure 2 for three different
temperatures and an initial C2H2 concentration of 50 ppm diluted
in Ar. The directly measured ring dye laser absorption profiles
were converted via the Lambert-Beer law into concentrations
based on f00 ) 0.037. The noisy drawn lines belong to the
experiments, and the dotted lines represent computer simula-
tions, which will be discussed later. After a fast rise, all profiles

2140 J. Phys. Chem. A, Vol. 101, No. 11, 1997
Kruse and Roth
Figure 4. Measured and calculated C, C
pyrolysis of 10 ppm acetylene diluted in Ar.
2
, and C
3
concentrations during
2
Figure 2. Measured and calculated C concentration profiles at three
different temperatures obtained during pyrolysis of 50 ppm C
in Ar.
2 2
H diluted
Figure 5. C
different initial C
2
concentration profiles obtained from experiments with
concentrations at temperatures around 3500 K
2 2
H
Figure 3. Emission spectrum of C
at a reaction time of 200 µs. Initial mixture: 10 ppm C H diluted in
2 2
Ar.
2
and C
3
during acetylene pyrolysis
and pressures of about 2 bar, in comparison with computer simulation.
concentration by the RDLAS, the C3 concentration can be
calibrated at the time τ by
drop rapidly to about 85% of the peak concentration. In the
two experiments with temperatures below 4000 K, and C2
concentration reaches a nearly constant level after the initial
peak, whereas the signal of the T ) 4450 K experiment shows
a further decrease within the observation time.
(
C3
τ)
3
C3
(
τ)
I
I
fel,C3
ν
exp(-hcν /kT)
[C3]
C
3
)
(
C2
τ)
(τ)
3
[C2]
∑i fii,C ii,C2
ν
exp(-hcνii,C2/kT)
2
To clarify the origin of the initial peak in the C2 absorption,
additional experiments with ARAS and the CCD camera were
performed. Figure 3 shows an emission spectrum obtained
during the thermal decomposition of 10 ppm acetylene in argon
at T ) 3450 K. It was recorded at a reaction time of 200 µs.
Two main features are of importance: the strong emission of
whereby the sum must be taken over the progression of the C2
bands, and νC and νii,C are the respective mean band wave-
3
2
numbers of C3 and C2. The electronic oscillator strength fel,C3
was taken from ref 18 and fii,C was calculated from the measured
f00 oscillator strength of C2 using the Franck-Condon factors
qii,C2 given in ref 19. After calibration of the C emission against
2
3
3
the C2(d Πg r a Πu) Swan bands at λ > 430 nm and the weak
1
1 +
emission of the C3(A Πu r X Σ ) bands in the spectral range
g
3
3
60 nm < λ < 420 nm.17 These emission spectra can be
C3 concentration at one fixed time τ, the proportionality factor
evaluated in order to get quantitative C3 concentration profiles.
between emission and concentration of C3 is known for the
whole time scale of this specific shock tube experiment. A
typical result of a C3 concentration profile obtained from an
emission signal is shown in Figure 4 together with a C2 profile
and a C atom concentration profile (see open circles) obtained
by ARAS. The experimental conditions are 10 ppm C2H2, T
For this purpose, the emission signals ꢀ(λ) of the ∆V ) 0
1
1
+
g
progressions of C2 and the A Πu r X Σ transition of C3 were
integrated with respect to the wavelength at a fixed time τ for
(
C2
τ)
each experiment. The integrated emission intensities I and
(
C3
τ)
I
at a fixed time τ were defined as
)
3450 K, and p ) 2.2 bar. The decrease in the C2
5
22 nm
420 nm
concentration at early reaction time is accompanied by an
increase in C3 and C atom concentration. At later reaction times,
the concentration of all three species reach a nearly constant
level. The computer simulation shown in Figure 4 as dotted
lines will be discussed later.
(
C2
τ)
(τ)
(τ)
3
(τ)
I
)
∫
ꢀ (λ) dλ
I
)
∫
ꢀ (λ) dλ
C
480 nm
350 nm
(τ)
with ꢀ (λ) being the wavelength dependent emission at t ) τ.
With the knowledge of the simultaneously measured C2

Kinetics of C2 Reactions
J. Phys. Chem. A, Vol. 101, No. 11, 1997 2141
The results of Figure 4 also clearly indicate a strong formation
of both C and C3 during the time of about 100 µs, when the
initial C2 peak decreases to a nearly constant concentration level.
It seems quite obvious to interpret this behavior by the
bimolecular reaction:
3
C + C h C + C
(R3)
2
2
3
Furthermore, the observed additional decrease in the C2
concentration at temperatures T > 4000 K during the whole
observation time of 700 µs (see Figure 2) suggests a dissociation
of C2:
4
C + M h C + C + M
(R4)
2
Figure 6. Measured and calculated C
different temperatures obtained from the pyrolysis of 20 ppm C
diluted in Ar perturbated by the addition of 1000 ppm H
2
concentration profiles at four
For the computer simulation of the measured C2 time histories,
this simple mechanism was extended by additional reactions
listed in Table 2, which have, however, only a very minor
influence for initial C2H2 concentrations of 50 ppm or less.
A problem is that the experimentally observed steady state
2 2
H
2
.
The second group of pyrolysis experiments was performed
in mixtures containing 100-500 ppm C2H2 diluted in Ar. The
reason for chosing this higher initial concentrations was to verify
existing kinetic models for C2H2 pyrolysis by comparing
calculated C2 profiles with measured ones. Three typical
measured C2 profiles from a total of about 10 experiments are
shown in Figure 6; see noisy lines. At time t ≈ 0 C2 increases
very rapidly, but different from the low C2H2 concentration
experiments, the initial C2 peak structure diminishes with
increasing initial C2H2 concentration nearly completely. Fur-
thermore, a second slower C2 increase appears, which reaches
a flat maximum or a constant level after some 100 µs. The
result of all experiments together with the experimental condi-
tions are summarized in Table 1.
C concentration at later reaction times (see Figures 2 and 4) is
2
significantly higher than the calculated value. An example is
given in Figure 7. The dotted line represents the calculated C2
concentration, which is for times t > 100 µs significantly below
the measured steady state value. This is due to the JANAF
1
thermodynamic data for C, C , and C , which are equal to those
2
3
given in ref 20. For the example given in Figure 7 these
literature data yield an equilibrium constant of Kc,3 ) 10. The
best fit to the experiments however can be obtained with an
equilibrium constant of Kc,3 ) 1.9, which increases the
importance of the backward reaction R3; see Figure 7. This
can be achieved by either increasing the Gibbs free energies of
C or C by 11 kcal or by decreasing the value of C by 5.5 kcal
3
2
The last group of experiments includes perturbation of C2H2
pyrolysis by H2. Four examples of measured C2 profiles in
mixtures of Ar with 20 ppm C2H2 and 1000 ppm H2 are shown
in Figure 6. The peaks at t ) 0 and t < 0 are schlieren signals
caused by the incident and reflected shock wave when passing
through the laser beam. The structure of the C2 signals is very
different from those of the H2-free mixtures. A direct com-
parison of measured C2 profiles obtained at identical temper-
atures of T ) 3380 K from a mixture of 20 ppm C2H2 diluted
in argon with Figure 6) and without (Figure 7) addition of H2
clearly illustrates this effect. As a result of the H2 addition,
the initial rapid increase in the C2 formation (see Figure 7) is
very moderate in Figure 6 and the initial peak at t ≈ 40 µs
disappears completely. Also the maximum amount of C2
formed is different in Figures 6 and 7 by a factor of 1.8. In
general the H2 addition attenuates the C2 formation during
pyrolysis of C2H2. The results of all experiments together with
the experimental conditions are also summarized in Table 1.
or by a combination of both. A change in the Gibbs free energy
can be obtained either by a change in the enthalpies or the
entropies of the species. Since the necessary variation in ∆G°
is constant over the studied temperature range between 2580
and 4650 K, it seems near at hand to attribute it to changes in
the enthalpies rather than in the entropies of the species.
2
1
Recently Urdahl et al. determined experimentally a value for
the heat of formation of C2, which is by about 4 kcal/mol lower
than the JANAF value. A lower value was also recommended
2
2
in a theoretical study. Therefore and for reasons discussed
1
later, we think that a reduction of the JANAF value for the
-
1
enthalpy of C2 by about (5.5 ( 0.5) kcal mol corresponding
0
to a change in the heat of formation from ∆ H
) 200.2
f
298.15
-
1
0
-1
kcal mol to ∆ H
) 194.7 kcal mol is most likely to
f
298.15
fullfill the requirements of the equilibrium constant of reaction
R3.
With the above mechanism and the modification in the
thermodynamic data of C , it is possible to fit the measured C2
2
profiles. It is clear that the rapid initial C2 increase is mainly
sensitive to reactions R1 and R2. A value for the rate coefficient
Discussion
k1 is given by ref 23 for the temperature range 1850 K e T e
It seems useful to discuss the experimental observations
during pyrolysis of C2H2 within the meaning of the three groups
of experiments introduced earlier.
The very rapid formation of C2 during high-temperature
pyrolysis of acetylene at low relative C2H2 concentrations (group
I experiments), as shown in Figures 2 and 4, can kinetically be
understood by a successive abstraction of H atoms:
500 K on the basis of experiments of Frank and Just.6 In a
3
first attempt we used this rate coefficient and modified only k2
to fit the initial C2 increase. As indicated in Figure 8 the initial
C2 increase could not be fitted satisfactorly, especially for
temperatures below 3200 K. Moreover, the resulting rate
coefficient k2 necessary to fit the further parts of the C2 profiles
showed an extraordinary small activation energy around 67 kcal,
which is only 57% of the reaction enthalpy for R2. We therefore
started to fit the measured C2 profiles by varying both rate
coefficients k1 and k2 of the above mechanism. This results in
nearly perfect fits of the experimental traces, even for the colder
experiments; see Figure 8. The resulting rate coefficients k1
1
C H + M h C H + H + M
(R1)
(R2)
2
2
2
2
C H + M h C + H + M
2
2

2
142 J. Phys. Chem. A, Vol. 101, No. 11, 1997
Kruse and Roth
2
Concentration Profiles Obtained during
TABLE 1: Experimental Conditions and Characteristics of Measured C
High-Temperature Pyrolysis of Acetylene Highly Diluted in Ar
T (K)
p (bar)
[C
2
H
2
]
0
(ppm)
tpeak (µs)
[C
2
]
peak (cm-3)
[C
2
]
100µs (cm-3
)
[C
2
]
300µs (cm-3
)
[C
2
]
600µs (cm-3
)
1
1
1
1
1
1
2
2
3
3
3
3
12
12
12
3082
3225
3431
3501
3552
3610
2.24
2.15
2.05
2.27
2.16
2.09
5.
155
105
61
6.2 × 10
8.2 × 10
1.0 × 10
1.1 × 10
1.2 × 10
1.3 × 10
5.4 × 10
8.2 × 10
8.5 × 10
7.8 × 10
8.0 × 10
9.5 × 10
4.8 × 10
4.7 × 10
4.5 × 10
5.2 × 10
5.7 × 10
6.0 × 10
3.0 × 10
4.0 × 10
4.2 × 10
5.0 × 10
5.5 × 10
5.9 × 10
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
5
5
5
5
5
49
43
40
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
3
2
2
2
3
3
3
3
3
3
3
3
3
2
3
3
2
11
11
11
11
11
12
12
12
12
12
13
12
12
12
13
13
13
13
13
13
13
13
13
12
13
13
12
11
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
13
12
12
13
13
12
13
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
13
12
12
13
13
12
12
12
11
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
580
580
580
626
685
775
824
850
866
018
053
062
113
150
281
293
345
365
450
523
652
773
830
061
155
380
575
2.45
2.45
2.45
2.51
2.27
2.49
2.41
2.12
2.35
2.12
2.31
2.40
2.40
2.06
1.95
1.99
2.40
2.17
2.20
2.02
2.09
2.02
1.98
0.76
1.92
1.80
0.75
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
540
520
520
490
450
305
235
225
250
150
100
130
113
105
80
2.3 × 10
2.5 × 10
2.5 × 10
2.8 × 10
2.9 × 10
5.2 × 10
5.8 × 10
5.4 × 10
4.9 × 10
6.8 × 10
1.1 × 10
8.5 × 10
9.6 × 10
9.0 × 10
1.1 × 10
1.3 × 10
1.6 × 10
1.5 × 10
1.9 × 10
1.8 × 10
2.1 × 10
2.2 × 10
2.2 × 10
9.2 × 10
2.4 × 10
2.5 × 10
9.3 × 10
1.5 × 10
2.0 × 10
2.0 × 10
2.0 × 10
4.5 × 10
2.2 × 10
3.0 × 10
2.8 × 10
2.5 × 10
6.2 × 10
1.1 × 10
7.9 × 10
9.5 × 10
9.0 × 10
1.1 × 10
1.1 × 10
1.3 × 10
1.2 × 10
1.4 × 10
1.2 × 10
1.3 × 10
1.3 × 10
1.2 × 10
6.8 × 10
1.3 × 10
1.1 × 10
5.1 × 10
1.3 × 10
1.4 × 10
1.4 × 10
1.8 × 10
2.3 × 10
5.1 × 10
5.4 × 10
5.0 × 10
4.8 × 10
5.5 × 10
6.2 × 10
6.1 × 10
6.4 × 10
6.0 × 10
6.5 × 10
6.7 × 10
8.4 × 10
8.3 × 10
1.4 × 10
8.2 × 10
1.2 × 10
1.2 × 10
1.3 × 10
5.0 × 10
1.2 × 10
5.0 × 10
2.7 × 10
2.2 × 10
2.4 × 10
2.4 × 10
2.5 × 10
2.5 × 10
4.1 × 10
3.9 × 10
3.8 × 10
3.7 × 10
4.0 × 10
5.2 × 10
4.9 × 10
5.6 × 10
5.1 × 10
6.1 × 10
6.3 × 10
8.4 × 10
8.4 × 10
1.4 × 10
9.0 × 10
1.2 × 10
1.2 × 10
1.2 × 10
4.8 × 10
8.5 × 10
1.2 × 10
8.0 × 10
73
55
57
40
40
31
26
23
30
14
10
18
1
1
1
3
3
3
13
13
13
13
13
13
13
13
13
3
3
3
382
558
734
2.18
2.15
1.98
20
20
20
45
30
23
2.1 × 10
2.8 × 10
3.2 × 10
1.6 × 10
1.8 × 10
1.9 × 10
1.3 × 10
1.7 × 10
2.1 × 10
1.3 × 10
1.7 × 10
2.2 × 10
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
12
12
12
13
13
13
13
13
13
13
13
13
13
13
13
13
12
12
12
12
2
2
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
945
999
175
483
650
745
808
931
976
990
085
189
280
403
437
450
518
609
634
2.14
2.23
2.23
2.11
1.92
1.97
2.01
2.09
2.04
1.98
1.98
1.89
1.85
1.83
1.83
1.82
1.71
1.60
1.58
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
121
118
63
32
22
18
16
12
12
11
16
11
10
9
1.7 × 10
1.7 × 10
2.4 × 10
3.8 × 10
4.6 × 10
5.2 × 10
5.8 × 10
6.8 × 10
6.8 × 10
6.9 × 10
7.3 × 10
7.5 × 10
7.7 × 10
7.3 × 10
7.4 × 10
7.2 × 10
7.8 × 10
7.4 × 10
7.2 × 10
1.6 × 10
1.5 × 10
2.1 × 10
2.9 × 10
3.7 × 10
4.0 × 10
4.4 × 10
4.5 × 10
4.5 × 10
4.7 × 10
5.8 × 10
5.8 × 10
5.5 × 10
5.0 × 10
4.8 × 10
4.9 × 10
4.0 × 10
3.5 × 10
3.0 × 10
1.1 × 10
1.2 × 10
1.6 × 10
3.2 × 10
4.2 × 10
4.5 × 10
4.7 × 10
4.7 × 10
4.6 × 10
4.9 × 10
4.9 × 10
4.0 × 10
3.2 × 10
1.9 × 10
1.6 × 10
1.7 × 10
8.0 × 10
3.0 × 10
2.0 × 10
1.0 × 10
1.2 × 10
1.6 × 10
3.5 × 10
4.2 × 10
4.3 × 10
4.8 × 10
4.8 × 10
4.2 × 10
4.0 × 10
3.0 × 10
1.6 × 10
1.0 × 10
3.1 × 10
2.5 × 10
2.7 × 10
1.0 × 10
9
9
9
11
7
5 × 10
11
6
2 × 10
1
1
1
1
1
1
1
1
1
1
3
3
3
3
3
3
4
4
4
4
13
13
13
13
13
13
14
14
14
14
13
13
13
13
14
14
14
14
14
14
13
13
13
13
14
14
3
3
3
3
3
3
3
3
3
3
489
572
510
509
430
360
250
490
605
785
2.10
2.17
2.14
2.12
2.21
2.01
2.06
2.13
1.90
1.83
75
26
22
22
22
22
24
40
18
13
11
4.6 × 10
6.2 × 10
6.3 × 10
6.3 × 10
7.9 × 10
9.1 × 10
1.2 × 10
1.5 × 10
1.6 × 10
1.7 × 10
3.9 × 10
5.0 × 10
6.2 × 10
6.1 × 10
8.8 × 10
9.5 × 10
1.6 × 10
2.5 × 10
2.4 × 10
3.2 × 10
4.9 × 10
6.3 × 10
7.9 × 10
7.3 × 10
1.1 × 10
1.3 × 10
2.0 × 10
2.8 × 10
2.5 × 10
3.7 × 10
5.1 × 10
6.3 × 10
8.5 × 10
8.7 × 10
1.2 × 10
1.5 × 10
100
125
150
200
250
500
500
500
500
14
2.2q × 10
14
14
14
2.7 × 10
2.6 × 10
3.4 × 10
T (K)
p (bar)
[C
H ]
2 2 0
(ppm)
[H
2
]
0
(ppm)
t
peak (µs)
[C
2
]
peak (cm-3)
[C
2
]
100µs (cm-3
)
[C
2
]
300µs (cm-3
)
[C
2
]
600µs (cm-3
)
1
1
11
11
2
3
3
3
3
890
380
550
830
990
2.13
2.22
1.85
1.91
1.82
20
20
20
20
20
1000
1000
1000
1000
1000
1.0 × 10
3.2 × 10
6.3 × 10
1.4 × 10
1.8 × 10
2.0 × 10
8.1 × 10
1.3 × 10
1.5 × 10
7.2 × 10
7.0 × 10
1.2 × 10
1.1 × 10
5.1 × 10
1.5 × 10
12
12
13
13
12
13
13
12
12
1
1
1
3
3
3
12
13
13
395
195
110
1.3 × 10
1.6 × 10
1.8 × 10
1
1
1
1
2
2
2
2
12
13
13
13
13
13
13
13
13
13
13
12
3
3
3
3
3
3
128
378
440
440
700
810
2.23
2.19
1.97
1.97
1.95
1.83
50
50
50
50
50
50
1000
1000
1000
1000
1000
1000
1.9 × 10
6.3 × 10
9.5 × 10
9.5 × 10
1.8 × 10
2.0 × 10
6.9 × 10
1.7 × 10
2.0 × 10
2.0 × 10
2.5 × 10
1.6 × 10
1.3 × 10
2.7 × 10
2.7 × 10
2.7 × 10
1.2 × 10
4.0 × 10
1
1
3
3
13
13
260
150
2.6 × 10
2.3 × 10

Kinetics of C2 Reactions
J. Phys. Chem. A, Vol. 101, No. 11, 1997 2143
TABLE 2: Simplified Reaction Mechanism for the
High-Temperature Pyrolysis of Acetylenea
3
A/cm
A
E /
-
1
reaction
+ M h C
H + M h C
+ C h C + C
+ M h C + C + M
+ H h C H + H
H + H h C + H
f 4C(s)
mol s-1
n
(cal/mol) ref
3
9
1
2
3
4
5
6
7
8
9
C
C
C
C
C
C
C
2
2
2
2
2
2
4
H
2
2
H + H + M 6.96 × 10 -6.06 133 400. s.t.
35
2
+ H + M
1.74 × 10 -5.16 114 000. s.t.
1
1
1
1
3
1
1
1
1
1
1
1
4
6
3
4
2
3
3.20 × 10
1.50 × 10
6.60 × 10
7.40 × 10
8.00 × 10
5.00 × 10
7.00 × 10
4.00 × 10
2.00 × 10
1.20 × 10
1.00 × 10
1.20 × 10
1.20 × 10
1.20 × 10
1.20 × 10
1.20 × 10
1.20 × 10
1.10 × 10
1.00 × 10
1.10 × 10
1.00 × 10
1.00 × 10
5.00 × 10
1.00 × 10
1.20 × 10
1.00 × 10
1.20 × 10
1.50 × 10
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
2.4
0.
0.
0.
0.
0.
0.
0.
0.
2.4
0.
0.
0. s.t.
142 400. s.t.
8 000. s.t.
6 700. s.t.
0. s.t.
0. est.
0. est.
12 000. 29
0. 29
2
2
2
2 2
H
4
4
4
4
4
4
4
C + CH h C
2
+ H
3
C + C
2
H h C
h C
+ H
1
1
1
1
1
1
1
1
1
1
2
2
0 C + C
2
H
2
3
H + H
+ H
1 C
2 C
3 C
4 C
5 C
6 C
7 C
8 C
9 C
2
2
2
2
2
2
2
4
4
+ CH h C
3
+ C
+ C
+ C
+ C
+ C
+ C
+ C
+ C
2
2
4
4
4
6
2
6
H h C
h C
H h C
H h C
H
H h C
H h C
4
+ H
0. 28
0. 28
0. 28
0. 28
0. 28
0. 28
0. 28
0. 28
0. 29
0. 29
0. 29
0. 28
0. 28
0. est.
0. 28
0. 28
0. 28
H
2
4
H + H
2
H + C
+ H
4
Figure 7. Simulation of C
2
concentration profiles with original and
14
14
14
6
modified thermodynamic data for an experiment with 20 ppm C
2 2
H
2
h C
6
H + H
diluted in Ar.
2
H + C
+ H
6
1
1
4
4
6
H h C
4
H + C
6
and k2 obtained from a total of 38 experiments are summarized
in Figures 9 and 10. The points for k1 show a weak non-
Arrhenius behavior, which can in the lower temperature regime
14
14
0 CH + H h C + H
2
1 CH + CH h C + H + H
2
14
22 CH + H h CH + H
2
2
-1
13
be represented by an activation energy of about 104 kcal mol .
For the whole temperature range 2580 K e T e 3830 K a good
fit to the experiments is possible by the modified Arrhenius
expression:
23 C
2
2
2
2
2
2
2
2
2
2
2
3
3
4
H + C
H + C
H + C
H + C
H + C
H + C
H + C
2
2
2
4
4
6
6
H h C
H h C
H
H
H
H h C
H h C
2
H
2
+ C
2
1
1
1
4
4
3
2
2
2
2
2
4 C
5 C
6 C
7 C
8 C
4
H + H
2
2
2
h C
h C
h C
4
4
6
H
2
+ H
H + C H
H
2 2
14
2
+ H
13
6
+ C
+ C
2
H
H
2
39
-6.06
exp(-67 130/T) cm mol s-1
3
-1
14
k ) 6.96 × 10 T
29 C
2
6
2
0. 28
1
1
1
1
3
4
4
3
3
3
3
3
0 C
1 C
2 C
3 C
4 C
35 C
6 C
H
2
H
2
H
2
H
2
+ C
2
2
4
6
H
H
H h C
H h C
2
h C
h C
4
4
H
H
2
+ H
2
42 700. 28
56 000. 28
0. 28
+ C
+ C
+ C
2
3
+ H 1.50 × 10
Our results for k1 at 3000 K is by a factor of 2 higher than the
value of Frank and Just.6
The data points for the rate coefficient k2, summarized in
Figure 10, can be represented by the Arrhenius expression:
6
H
H
2
2
2
+ H
+ H
1.20 × 10
1.20 × 10
1.00 × 10
6.31 × 10
2.00 × 10
4.07 × 10
1.20 × 10
1.20 × 10
1.20 × 10
1.00 × 10
H 1.00 × 10
1.00 × 10
1.00 × 10
2.00 × 10
4.07 × 10
14
14
8
0. 28
0. 29
0. 29
0. 28
200. 28
0. 28
0. 28
0. 28
0. 28
0. 28
0. 28
0. 28
0. 28
200. 28
0. 28
0. 28
H + H h C
+ H h C
H + H h C
3
+ H
1
1
5
3
3
H
2
3
H + H
+ H
2
3
4
2
k ) 1.1 × 10 exp(-41 868/T) cm mol s-1
15
3
-1
37 C H + H h C H + H
4
4
4
4
2
4
2
2
14
3
3
4
8 C
9 C
0 C
H + C
H + C
H + C
4
4
6
H h C
h C
H h C
4
+ C
8
4 2
H
1
4
H
2
H
2
+ H
The relatively low activation energy of 83 kcal favors the
suggested reduction in the heat of formation for C2. A modified
Arrhenius expression similar to that of k1 based on the heat of
14
4
+ C
6
H
2
1
1
1
1
1
5
3
3
3
4
3
41 C H + C H h C H + C
4
4
4
6
6
6
6
6
8
4
2
6
42 C
43 C
4 C
5 C
6 C
H + C
H + C
H + H h C
H + H h C
H
2
h C
4
H
2
+ C
6
-
1
H h C
4
+ C
+ C
+ H
+ H
8
H
2
2
reaction for reaction R2 of 114 kcal mol would result in a
temperature exponent of -5.16.
4
4
4
4
2
H
6
2
The rate coefficients of the bimolecular reaction R3 and the
thermal decomposition reaction R4 were also determined by
computer fittings from the measured C2 concentration profiles.
Individual results obtained are summarized in Figures 11 and
H + H
6
2
h C
6
H
2
1.00 × 1013
47 C H + C H h C H + C
6
6
2
6
1
3
7
48 C
H
6 2
h C
6
H + H
6.00 × 10
3
4
5
5
5
9 H + C
4
+ M h C
4
H + M
1.74 × 10 -5.5
0. 28
8
0 C
1 C
2 C
4
3
2
H
2
h C
+ M h C
+ C + M h C
4
H + H
+ C + M
+ M
1.30 × 10
4.00 × 10
2.2 122 600. 28
1
2. They can be represented by the Arrhenius expressions:
1
2
6
7
2
0.
150 000. 29
0. 28
2
4
1.00 × 10 -3.0
k ) (3.2 ( 0.4) × 10 cm mol s-1
1
4
3
-1
14
3
53 CH + M h C + H + M
1.90 × 10
0.
0.
66 968. 35
55 820. 29
79 030. 29
76 610. 29
39 420. 29
60 000. 29
0. 28
14
5
4 CH
2
+ M h C + H + M
2
1.15 × 10
k ) (1.5 ( 0.5) × 10 exp(-71 660/T) cm mol s-1
16
3
-1
55 CH + M h CH + H + M 2.88 × 1014 0.
2
4
15
5
5
5
6 CH
3
+ M h CH
2
+ H + M 1.58 × 10
0.
0.
0.
14
7 C
8 C
3
H + M h C
3
3
+ H + M
1.58 × 10
The rate coefficient k3 was found to be independent of
temperature. The activation energy of reaction R4 of 142 ( 5
15
H
2
+ M h C
3
H + H + M 1.00 × 10
23
59 C + C H + M h C H + M 2.09 × 10 -3.0
2
4
6
-
1
14
kcal mol is in good agreement with published values for the
C2 bond dissociation energy; see, e.g., refs 21 and 24-27, and
the values resulting from the thermodynamic data of C and C2.
All computer simulations for obtaining the given rate coefficients
k1-k4 were performed both with the reaction mechanism R1-
R4 and with the more extended mechanism listed in Table 2.
The results were nearly the same.
The interpretation of the group II experiments with higher
initial C2H2 concentrations of 100-500 ppm (see examples in
Figure 6) needs an extension of the thermal decomposition
mechanism. We started with a system of 145 elementary
reactions, mainly taken from Kiefer et al., refs 28 and 29. By
careful sensitivity analysis, the 60 reactions listed in Table 2
60 H
2
+ M h H + H
2.23 × 10
/RT).
0.
96 080. 34
a
n
Rate coefficient k ) AT exp(-E
A
were found to have influence on the measured C2 profiles.
Reactions R1-R4 are identical to those discussed earlier, and
the rate coefficients of R5 and R6 will be discussed later. It
was found that the second, slower C2 increase observed in Figure
5 is mainly sensitive to the reactions R12, R52, R19, R36, and
R14 involving the species C2, C2H, C4, C4H, C6, and C6H. Since
many of the sensitive reactions have equilibrium constants close
to 1, the thermodynamic properties, i.e., the Gibbs free energies,
of this species become very important for these experimental
conditions. The thermodynamic properties at 3500 K of some

2144 J. Phys. Chem. A, Vol. 101, No. 11, 1997
Kruse and Roth
Figure 11. Rate coefficient of the reaction C
2
+ C
2
3
h C + C .
Figure 8. Measured and calculated C
at relatively low temperature. The dotted line shows the best simulation
using the literature value for k varying only k . The dashed lines shows
the best simulation fitting obtained by varying both k and k
2
concentration profile obtained
1
2
1
2
.
Figure 12. Rate coefficient of the reaction C + Ar h C + C + M.
2
TABLE 3: Thermodynamic Data at Room Temperature
and 3500 K of Some Important Species Used for the Kinetic
Modeling
Figure 9. Rate coefficient of the reaction C
M. The dotted line corresponds to the Arrhenius expression and the
solid line to a modified Arrhenius expression (see text).
2
H
2
+ M h C H + H +
2
S°
S°
H°
H°
3500
2
98.15
3500
298.15
species (cal/mol/K) (cal/mol/K) (kcal/mol) (kcal/mol)
ref
20
20, 1
s.t.
20
20
20, 1
30
H
C
27.4
37.8
47.6
49.6
48.0
56.7
54.2
60.4
63.5
57.9
74.1
39.6
50.1
70.1
52.1
171.3
194.7
135.0
54.2
68.0
187.5
224.4
178.4
113.0
233.5
219.6
311.7
268.6
360.7
326.6
C
C
C
C
C
C
C
C
C
2
2
2
3
3
4
4
6
6
H
H
79.7
88.2
83.2
91.8
111.8
117.8
113.2
154.1
2
196.0
163.5
247.1
192.0
282.2
213.2
H
H
H
s.t.
30
30
20
This can be achieved, e.g., by increasing the entropy of C4 by
-
1
-1
5
.7 cal mol
K
or 6% compared to the value given by
3
0
Burcat. The possibility of a too low literature value of the
C4 entropy was already discussed by Kiefer et al. on the basis
of experiments performed by Heath et al.
2
8
3
1
An example
illustrating the different effects is shown in Figure 13. The
experimental conditions are T ) 3492 K, p ) 2.14 bar, and
500 ppm C2H2 diluted in Ar. The noisy line comes from the
experiment, the dotted line represents a calculation with the
Burcat value for the entropy of C4, and the dashed line was
calculated with the modified C4 data. A nearly perfect agree-
ment, see the dashed dotted line, with the experimental profile
even at later reaction time was achieved by assuming the
formation of solid carbon. Following the suggestion of Kiefer
Figure 10. Rate coefficient of the reaction C
M. Dotted line, Arrhenius expression; solid line, modified Arrhenius
expression.
2
H + M h C + H +
2
important species used for our kinetic modeling are shown in
Table 3. Again a satisfactory fit of the measured C2 profiles
could only be obtained with the earlier suggested change in the
thermodynamics of C2. An additional modification of the
thermodynamic properties of C4 was also suitable. The best
fit was achieved by reducing the Gibbs free energy of C4 by
3
2
et al., reaction R7 was introduced which converts in an
irreversible way C4 into solid carbon. Similar good agreement
-
1
30
about 20 kcal mol compared to the value given by Burcat.

Kinetics of C2 Reactions
J. Phys. Chem. A, Vol. 101, No. 11, 1997 2145
Figure 13. Comparison of different models used to describe the
Figure 15. Rate coefficient of the reaction C
2
2 2
+ H h C H + H.
2 2 2
measured C concentration for an experiment 500 ppm C H diluted
in Ar, T ) 3490 K and p ) 21.4 bar.
2 2 2 2
Figure 16. Rate coefficient of the reaction C H + H h C H + H.
the Figures 15 and 16. The points can be approximated by the
Arrhenius expressions:
2
Figure 14. (upper) Measured and calculated C profiles during the
k ) 6.6 × 10 exp(-4000/T) cm mol s-1
1
3
3
-1
pyrolysis of 50 ppm C
2
H
2
diluted in Ar enriched with 1000 ppm H
perturbation
concentration
2
.
5
(lower) Normalized sensitivity coefficients of the H
2
k ) 7.4 × 10 exp(-3400/T) cm mol s-1
1
4
3
-1
reactions R5 and R6 describing the influence on the C
for this experiment.
2
6
The rate coefficient k5 has not been measured before in the high-
was also obtained for the other experiments; see Figure 6, dashed
lines. The best fit rate coefficient used during the computer
temperature regime, but the value is in accordance with an
28
estimation by Kiefer et al. Our value for k6 is at 3000 K by
3
-1
simulation was k7 ) (8 ( 3) × 10 s .
factor of 20 higher than the value recomended by Baulch et
The last group of results includes five experiments of 20 ppm
C2H2 diluted in Ar and five experiments of 50 ppm C2H2 diluted
in Ar enriched with 1000 ppm H2. We again started our
computer simulation on the basis of the complete mechanism
consisting of 145 reactions. It is obvious that now reactions
R5 and R6
33
28
al. An estimated value by Kiefer et al. and a value obtained
from the backward reaction of R6 measured by Frank and Just6
are relatively close to our value.
The nearly perfect agreement of the simulations with the
experimental profiles (see Figure 5) for the whole measurement
time of 1000 µs could only be obtained by including the
complete mechanism of Table 2. The slow disappearance of
the C2 at later reaction times in the higher temperature
experiments is mainly due to reactions involving H, C, C2, and
CH as, for example reactions, R8, R9, and R53. The rate
coefficients of this three reactions and the rate coefficients of
reactions R12 and R25 were adjusted in a range of (30%
compared to the values given in Table 2 to obtain best agreement
between experiment and simulation. For the H2 dissociation,
which is also very sensitive to the C2 profiles at later reaction
time, the rate coefficient recommended by Baulch et al.34 was
used. Again, the thermodynamic properties of the species listed
in Table 3 proved to be very important, and we found that the
best fit was again possible with the above suggested changes
in the thermodynamics of C2 and C4.
k5
C + H y
\
z C H + H
(R5)
(R6)
2
2
2
k6
C H + H y\ z C H + H
2 2
2
2
have a high sensitivity to the C2 concentration. As can be seen
in Figure 14, both reactions are responsible for the drastic
slowdown of the C2 rise, whereby reaction R6 is most sensitive
in the first 50 µs and reaction R5 in the following 200 µs. Thus,
both rate coefficients can be evaluated seperately on the basis
of computer fittings of the experimental C2 concentrations by
varying the rate coefficients k5 and k6. Individual results
obtained from fittings at early reaction time are summarized in

2146 J. Phys. Chem. A, Vol. 101, No. 11, 1997
Kruse and Roth
Conclusion
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low concentration range of 50 ppm C2H2 or less, a simplified
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4
(
(
(
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data of either C, C2, or C3 was revealed and tentatively
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(15) Kruse, T.; Roth, P. To be published.
f
298.15
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(
16) Von Gersum, S.; Kruse, T.; Roth, P. Ber. Bunsen-Ges. Phys. Chem.
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45 reactions was checked and reduced by sensitivity analysis
(
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(
1
(
(
Acknowledgment. The authors thank Mrs. C. Kmiecik, Mrs.
N. Schl o¨ sser, and Mr. L. Jerig for their help in conducting the
experiments. We also thank Prof. Th. Just, DLR Stuttgart, for
helpful discussion. The financial support of the Deutsche
Forschungsgemeinschaft is gratefully acknowledged.
(
5
(
(
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2
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