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CONTRIBUTIONS TO T H E
125
to be 8.0 kcal/mol, similar to the 9.5 kcal/mol barrier for the present reaction. RRKM plus tunneling calculations were carried out for the reverse methoxy radical decomposition reaction. It was found that in order to reproduce these results using RRKM calculations without tunneling, the barrier had to be lowered by 3.5 kcal/mol. The calculated rate constant is well-represented (& 10%) over the temperature range 300-3000 K by the three-parameter fit k ( T ) = 2.96 X 10-22T3.29e-2200/T
VI. Summary and Conclusions We present high-level ab initio calculations of the reactant, saddle point, and product for the reaction H2 + BO H HBO. Contrary to previous estimates, this reaction is predicted to be exothermic, with a 0 K exothermicity of 6.4 kcal/mol and barrier height of 9.5 kcal/mol. The transition state is centrally located with the HH bond stretched by about 20% and the BH bond stretched by about 28%. The frequency of the doubly degenerate bending vibration at the transition state is found to be very low. As a consequence, this mode becomes active at relatively low energies. This is manifested as substantial upward curvature in the transition-state-theory Arrhenius plot at high temperatures. The Arrhenius plot is found also to be notably curved at low temperatures. This behavior is attributed to quantum mechanical tunneling. The computed rate is found to be well-described over a wide temperature range by a three-parameter fit. The rate calculations represent a prediction in the absence of experiment for this critical reaction in boron combustion.
-
- 1 0 ' " '
0
1
-IO 5 ~ ' ' ' " ' ' ~ ~ ~ ' ' " ' ' ' " 1 " ' ' 1 ' ' " IO I S 2 0 2 s 3 0 3 s 1000/T(K)
-
Figure 2. Breakdown of the various contributions to the computed activation energy for the reaction H2 + BO H + HBO (see text).
vibrational energy differences. The contributions due to the low-frequency degenerate bending modes and due to all the remaining modes of motion are shown as curves C and D, respectively. Curve C essentially contributes the classical equipartition value of 2kT even at low temperatures. Curve E is the tunneling contribution to the activation energy. At low temperatures, the activation energy is seen to be lowered by over 3 kcal/mol due to tunneling. It is interesting to compare this to the computed effect of tunneling in the addition of a hydrogen atom to formaldehyde.I8 The potential energy barrier in that case was found (18) Page, M.; Lin, M. C.; He, Y.; Choudhury, T. K., submitted for publication in J . Phys. Chem.
+
Acknowledgment. This research was supported by Naval Research Laboratory and by the Mechanics Division of the Office of Naval Research. Registry No. H2, 1333-74-0;BO, 12505-77-0;HBO, 2061 1-59-0. (19) Kawashima, Y.; Kawaguchi, K.; Hirota, E. Chem. Phys. Lerr. 1986, 131, 205.
(20) Kawashima, Y.;Endo, Y.; Kawaguchi, K.; Hirota, E. Chem. Phys. Leu. 1987, 135, 441. (21) Tanimoto, M.; Saito, S.; Hirota, E. J . Chem. Phys. 1986, 84, 1210.
Applications of Generating Functions to Polymerization Kinetics. 2. Laser Pulse Initiated Free-Radical Polymerizations K. W. McLaughlin,* D. 'D.Latham, C. E. Hoyle, and M. A. Trapp Department of Polymer Science, University of Southern Mississippi, Southern Station Box 10076, Hattiesburg. Mississippi 39406 (Received: September 29, 1988)
The molecular weight distributions generated by laser pulse initiated free-radical polymerization reactions are examined. By use of an elementary reaction mechanism, expressions for the time-dependent behavior of the product distribution are derived directly from the rate equations for propagation and termination. These expressions apply for the case of a symmetric photoinitiator with single-pulse initiation or for sufficiently low frequency multiple-pulse initiation. The derivation of the product distribution functions is accomplished by the use of generating functions. By dividing the overall product distribution into two components, the growing (living) chains and terminated (dead) chains, the predicted transition from a Poisson distribution at short polymerization times to a distribution much broader than the Schulz-Flory distribution can be easily explained. These results clearly suggest the origin of the multiple peaks recently reported in higher frequency pulsing experiments.
Introduction The product distributions generated by polymerization reactions are frequently difficult to derive directly from the rate equations.' The derivation of such distributions, even for simple mechanistic (1) Tompa, H . In Comprehensive Chemical Kinetics; Bamford, C. H., Tipper, C. F. H., Eds.; Elsevier: New York, 1976; Vol. 14A, Chapter 7.
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schemes, can require sophisticated mathematical techniques.2-s The free-radical polymerization of olefins represents an important industrial process that frequently results in complex product (2) Peacock-Lopez, E.; Lindenberg, K. J . Phys. Chem. 1984, 88, 2270. (3) Peacock-Lopez, E.; Lindenberg, K. J . Phys. Chem. 1986, 90, 1725. (4) McLaughlin, K. W. Ph.D. Thesis, Texas A&M University, 1986. (5) Latham, D. D. Ph.D. Thesis, Texas A&M University, 1987.
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The Journal of Physical Chemistry, Vol. 93, No. 9, 1989
distributions.610 Recent investigationsinto the laser pulse initiated free-radical polymerizations of methyl methacrylate, lauryl methacrylate, and styrene have provided an entirely new route of probing the mechanisms for this class of reactions6-* The novel reaction conditions for laser pulse initiated free-radical polymerizations give rise to extremely complex product distributions at high pulse frequencies, while much simpler distributions are produced at very low pulse frequencie~.~?' These molecular weight distributions represent a permanent though indirect record of the reaction kinetics, which in turn are related to the reaction mechanism. In order to use the product distribution to test a proposed reaction mechanism, it is necessary to derive the molecular weight distribution function from the rate equations arising from the proposed mechanism. By imposing a pseudo-steady-state condition, Olaj and co-workers have constructed a distribution function for the high-frequency experiment.s-10 Olaj's function is applicable when the number of pulses is very large and is exact in the limit of infinite pulses. A general procedure for solving this type of problem in chemical kinetics using generating functions exists and has been recently outlined." In this paper, we report the derivation and analysis of the general, discrete distribution function obtained directly from the rate equations for an elementary free-radical polymerization mechanism.
Rate Equations One of the simplest free-radical polymerization mechanisms is shown in Scheme I. In this mechanism, a symmetric photoinitiator X,, upon absorbing a quantum of light, breaks into two free-radical species represented by X. The free radical can then react with the monomer M, generating a higher molecular weight free-radical species XM, where k, is the rate constant for propagation. This process can repeat itself until the free-radical species is a high polymer XMi. Eventually, two free-radical species will couple, forming an inert high polymer XMiX, where k, is the rate constant for termination. The distribution of inert XMiX species can be determined by gas chromatography for oligomerization and by gel permeation chromatography for polymerization. For SCHEME I
-
x* hv
initiation: propagation:
kP
X+M-XM
+ M kP XMi+I k, XMj-j + XMj XMiX XM,
termination:
2x
McLaughlin et al. equations. Only the initial concentration of free-radical species X, represented by Ao, is required. After the laser pulse, propagation and termination compete for the free radicals. Letting Ai represent the concentration of free-radical species composed of i monomers (Le., XMi) allows us to write the following rate equations
where Y is used to represent the monomer concentration and i I 1 for eq 2. Representing the concentration of XMiX species by B, allows us to write (3)
The in eq 3 arises from the definition of the rate constant for termination. In all subsequent work in this paper, the two rate constants will be assumed to be independent of chain length and the monomer concentration will be held constant. The value of Ai as a function of i gives rise to the distribution of living (Le., growing) polymer chains, while the value of B, as a function of i gives rise to the distribution of dead (Le., terminated) polymer chains.
Molecular Weight Distribution Function In order to obtain the molecular weight distributions for the living and dead chains, it is necessary to integrate the coupled differential equations (1)-(3). This is conveniently accomplished by introducing the following generating functions m
F,(t) = CZiAi
(4)
i=O
where Z is simply an indexing variable that is independent of time." Unless a specific time is required, the (t) after F, and G, will be omitted. The time derivative of F, is given by
---*
-
photoinitiated free-radical chain reactions, a single laser pulse generates an initial concentration of free-radical species more or less instantaneously. As time passes, the concentration of freeradical species decreases due to termination, eventually approaching zero. When a single laser pulse generates too little product to be accurately measured, many laser pulses will be required. If the laser pulse frequency is too high, extremely complex molecular weight distributions are For the purposes of this paper, low-frequency pulsing will be defined as that frequency for which the time interval between pulses corresponds to effective completion of the free-radical coupling reaction. Lower frequencies are also acceptable. A single laser pulse initiates polymerization by instantaneous formation of free-radical species from the photoinitiator. Consequently, the rate of initiation does not enter into the rate (6) Hoyle, C. E.; Trapp, M. A,; Chang, C. H.; McLaughlin, K. W.; Latham, D. D.Macromolecules 1989, 22, 3 5 . (7) Hoyle, C. E.; Trapp, M. A,; Chang, C . H. J . Polym. Sci., Polym. Chem. Ed., submitted for publication. (8) Olaj, 0.; Bitai, I.; Gleixner, G. Makromol. Chem. 1987, 188, 1689. (9) Olaj, 0.;Bitai, I.; Gleixner, G. Makromol. Chem. 1985, 186, 2569. (IO) Olaj, 0.;Bitai, I. Makromol. Chem., Rapid Commun. 1988,9, 275. ( 1 1 ) McLaughlin, K . W.; Latham, D. D.; Hoeve, C. A. J. M A T H / CHEMICOMP 88; Gaovac, A,, Ed.; Elsevier: New York, to be published.
Substituting eq 1 and 2 into eq 6 gives dF, m m _ - Ck,YZ(Zi-'A+l) - Ck,Y(Z'Aj) - Ck,(Z'Aj)CAj (7) dt j=] i=O i=l j=O m
m
Equation 7 can be simplified into the following form dF, dt
- = k,YZF,
- k,YF,
- k,F,FI
(8)
For 2 = 1 (9)
Integrating eq 9 from time = 0 to t gives F1
=
FI (0)
1
+ k,tF,(O)
Substituting eq 10 into eq 8 results in
_ dF, - k,YZF,
- kpYFz - k,F,
dt
1 + k,tF,(O)
It is now convenient to define two new variables U = k,Y I/ =
k,F1(0)
(12)
(13)
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 3645
Laser Pulse Initiated Free-Radical Polymerizations Expressing eq 11 in terms of U and V gives
(
dF,- - UZF, - UF, - L)F, dt
1
+ vt
(14)
By use of eq 20, eq 27 becomes
To obtain an expression for F, in terms of the kinetic parameters U and V requires that eq 14 can be integrated over time Expanding exp(2ZUt) leads to
Carrying out the integration yields
+ Vt)-l]
(16)
+ ZA, + Z2A, + ...
(17)
F, = F,(0)[$ufe-uf(l
Recalling that
F, = A.
then at t = 0, Ai = 0 for i > 0, since propagation has not yet begun,
so that = AO(0)
(18)
A term-by-term comparison of eq 22 and 29 leads to dBi dt
1
-= - W ( l 2
+ Vt)-2[
dBi = -WS(1 21
+ RS)-2
1 R LB'dBi = 2 W S I0 (1
The utility of the generating function now becomes clear. A term-by-term comparison of eq 17 and 20 leads to the expression Of Ai
+ .)-I[
-1
(Ut)'e+'
(21)
Equation 21 defines the distribution of the number of growing chains as a function of i in terms of the kinetic parameters summarized in Table I. To obtain an expression for the distribution of dead polymer chains, it is necessary to take the time derivative of eq 5
-dG,dt
-
CZ'-dBj j=o dt m
Substituting eq 3 into eq 22 gives
Equation 23 can be rearranged to
Expanding eq 24 yields
m
ZjAj(x,Zi-jAi-j)
+ ...I
f*J
Note that all the terms in parentheses in eq 25 equal
5 Z'A,
1-0
Therefore, eq 25 can be rewritten as
Substituting eq 4 into eq 26 gives
[ "r] - dR
(31)
Equation 31 must be integrated over time to obtain the distribution of dead chains
(ZUt)'
E-
eq 16 can be expressed as
Ai = W(1
(30)
Letting R = 2Ut and S = Vl2U results in
Letting Ao(0) = Wand recalling that $ut =
]
(2Ut)ie-2u1 i!
(25)
+ RS)-2[
-1
Rie-R dR
(32)
This integration can be accomplished numerically.12 The numerical integration of eq 32 defines the distribution of the number of dead chains as a function of i in terms of the kinetic parameters summarized in Table I.
Results and Discussion For the reaction conditions under consideration, the molecular weight distribution of the growing polymer chains at any given polymerization time obeys the Poisson distribution. The Poisson distribution function f(i) = i!
(33)
can be easily identified in eq 2l.I3 The remaining terms in eq 20 describe a function that is decreasing monotonically with time. This function corresponds to a reduction in the number of living chains due to termination. After a growing chain undergoes termination, it then belongs to the distribution of dead chains, given by eq 32. This distribution represents the chains formed by termination summed over the time interval of 0 to t. Figure 1 illustrates the qualitative features of the living and dead distributions. Weight fraction distributions are shown since this is what is measured by gas chromatography or gel permeation chromatography. At any given time, termination occurs by coupling two living chains that belong to the same Poisson distribution. This leads to an approximate doubling of the chain length. As a result, the dead distribution extends beyond the living distribution to a point roughly double in molecular weight. This is clearly illustrated in Figure 1. As time passes, the number of chains in the living distribution decreases, while the number of chains in the dead distribution increases. This decay in the living distribution relative to the dead distribution is shown in Figure 2. The living peak shifts to higher molecular weights linearly with time. As the living peak shifts to higher molecular weights, it uncovers the peak in the distribution of the dead polymer. At long enough polymerization times, the peak due to the living polymer will become insignificant. At this limit, only the broad, dead distribution will remain. Figure 3 illustrates how the limiting (12) Equation 32 was numerically evaluated by using a Romberg integration. ( 1 3) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953; Chapter 8.
McLaughlin et al.
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TABLE I: Definitions of the Kinetic Parameters Used in the Derivation of the Molecular Weight Distribution Function
parameter W
U Y
R S
definition number of propagating free-radical species generated by a laser pulse, A. k,Y, rate constant for propagation times monomer concentration (which is being held constant) k,F,(O) or k,W, also a constant 2Ut or 2k,Yt, a constant times time V / 2 U or k , W / 2 k p Y ,a constant
10-
8 -
hgl0(hlonomer Units per Chain)
Figure 3. Limiting theoretically predicted weight fraction distribution (-) for R = m and S = 0.02 overlayed for comparison with a SchulzFlory distribution (---). 0 200
0
400
6W
800
\IOSOIIER CSITS P E R C U I \
Figure 1. Typical theoretically predicted weight fraction distributions for the living (---) and dead (-) distributions: R = 350 and S = 0.02.
I
*R=liO
R=ia
n
\IOIO\IER L I I T S PER C W I Y
Figure 2. Theoretically predicted weight fraction distributions for S = 0.02 and selected values of R,illustrating the time dependence of eq 21 and 3 2 .
dead distribution is clearly broader than the Schulz-Flory distribution. If chain transfer to a monomer replaces the free-radical coupling reaction in Scheme I, a Schulz-Flory distribution would be observed for the dead polymer." The composite molecular weight distribution obtained from eq 21 and 32 describes the transition from a Poisson distribution to the very broad distribution of dead chains which results after a single laser pulse. Very broad, simple distributions have been reported for very low frequency laser pulse initiated free-radical polymerizations which are qualitatively consistent with the theoretically predicted distribution for dead chain^.^,^ Though an analytical solution for the position of the dead peak is unavailable due to the need for numerically integrating the time derivative of B,, by inspection it appears that the dead peak is equal to 1
+ s-1.
Prior to reaching the limiting distribution, the sharp, living, Poisson distribution is still present in significant quantity. Olaj has suggested that the unusual molecular weight distributions generated by high-frequency laser pulsing are due to the introduction of many free radicals by a second laser pulse prior to the disappearance of the living peak. The molecular weight distribution defined by eq 21 and 32 is consistent with Olaj's results and conclusions.* Hoyle and co-workers have recently reported similar results with methyl methacrylate and lauryl methacrylate which strongly suggest the same conclusion for high-frequency laser pulse initiated free-radical polymerization reaction^.^,^ The sudden introduction of many A,, species by a second laser pulse prior to the disappearance of the living peak, produced by the first pulse, results in the coupling of many first-pulse chains with second-pulse chains. The coupling of chains generated by different pulses will change the molecular weight distribution of the dead polymer by introducing a new peak which will start at the molecular weight achieved by the first-pulse chains at the time of the second pulse. The multimodal distributions that have been reported for high-frequency laser pulse initiation are closely related and qualitatively similar to the theoretically predicted semiliving molecular weight distributions (see Figure 2) produced by a single p ~ l s e . The ~ , ~ distribution function derived in this paper represents the starting point for the derivation of the molecular weight distribution function appropriate for the high-frequency laser pulse initiated free-radical polymerizations. Conclusions Using a simple mechanistic scheme, we have derived the molecular weight distribution resulting from a laser pulse initiated free-radical polymerization reaction. The constraints on the derivation make the distribution applicable to single- or lowfrequency laser pulsing. A composite product distribution was obtained which possessed a living Poisson distribution that decayed with time. The dead distribution was found to be broader than the Schulz-Flory distribution. Due to the mechanism for termination, part of the dead distribution actually exceeds the molecular weight of the living polymer chains. The transition from the Poisson distribution to the broader dead distribution is fully defined as a function of polymerization time, the rate constants for propagation and termination, the monomer concentration, and the free-radical concentration. The procedure used to obtain this distribution can be repeated for many other types of mechanisms so that the product distribution functions for a wide range of
J. Phys. Chem. 1989, 93, 3647-3649 reaction mechanisms can be derived, thereby allowing the observed product distributions to be used for elucidation of a reaction mechanism. Furthermore, the distribution function derived here provides a starting point for the quantitative analysis of the complex behavior recently reported for high-frequency laser pulse initiated free-radical polymerizations. Due to the complexity of
3647
the high-frequency experiment, it will be treated as a separate case in a subsequent paper, as will the influence on the distribution of subtle changes in the reaction mechanism. Acknowledgment. This work was supported by the National Science Foundation Grant DMR 85- 14424 (Polymer Program).
Infrared Multiphoton Decomposition of Hexafluorobenzene Investigated by Diode Laser Kinetic Spectroscopy: Detection of CF and CF, Ko-ichi Sugawara,* Akio Watanabe, Yoshinori Koga, Harutoshi Takeo, Kenzo Fukuda, Jiro Hiraishi, and Chi Matsumura National Chemical Laboratory for Industry, Tsukuba, Ibaraki, 305 Japan (Received: October 12, 1988)
Infrared multiphoton decomposition of hexafluorobenzene (C6F6) was investigated by diode laser kinetic spectroscopy. Two transient species, CF and CF2, were detected, and the time evolution of their signals was observed. The signal of CF rose within 10 p s after a C 0 2 laser pulse and decreased with lifetime of about 200 p s , while that of CF2 rose slowly as CF decayed out. Final products observed by an FTIR spectrometer were mostly C2F4and C6FsCF3.These experimental results suggested that CF was produced at the early stage of the reaction and that CF2 was produced by the reaction of CF with a certain fluorine-containing species and then decayed out through the dimerization and/or the reaction with C6F6.
Introduction Infrared-laser-induced reactions with a pulsed CO, laser have been extensively studied for many molecules that absorb the infrared light in the 9-11-pm region. The reactions of the molecules that do not appreciably absorb the light have been made possible by adding an infrared sensitizer which strongly absorbs the C 0 2 laser light without participating in the chemical reactions. As sensitizers, sulfur hexafluoride (SF,) and silicon tetrafluoride (SiF4) have been widely used. Infrared multiphoton absorption of hexafluorobenzene (c6F6), which was thought to be another candidate for the sensitizer, has been investigated by several workers.’” Their results showed, however, that C6F6 was decomposed by the C 0 2 laser quite easily and that this molecule could be used as the sensitizer only when the fluence of the CO, laser was low. Recently, Koga et aL6 found that a considerable amount of CsF6 was decomposed by the CO2 laser with fluence and consequently, they recommended that as low as 0.7 J when this molecule was the fluence should be below 0.3 J used as the sensitizer. Duignan et al.4 investigated the decomposition reaction of C6F6 with high laser fluence, in the range and reported the observation of C2 and C3 as 100-900 J transient species by emission spectroscopy. The final gaseous products identified in their study were CzF4, C6FSCF3,C3F6,C4Fs, and C2F6. Their results strongly suggested that, besides C? or C3, CF, or other fluorocarbon species should exist as the transient species. It is of interest to know how is decomposed by the C 0 2 laser. To clarify this problem in further detail, we have studied the laser-induced reactions of C6F6and similar molecules (1) Speiser, S.; Grunwald, E . Chem. Phys. Left. 1980, 73, 438. (2) Starov, V.; Selamoglu, N.; Steel, C. J. Am. Chem. SOC.1981, 103,
7276. (3) Duignan, M. T.; Garcia, D.; Grunwald, E. J . Am. Chem. SOC.1981, 103, 7281. (4) Duignan, M. T.; Grunwald, E.; Speiser, S. J . Phys. Chem. 1983,87, 4387. (5) Mele, A.; Salvetti, F.; Molinari, E.; Terranova, M. L . J . Phofochem. 1986, 32, 265. ( 6 ) Koga, Y . ;Serino, R. M.; Chen, R.; Keehn, P . M. J. Phys. Chem. 1987, 91, 298.
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TABLE I: Observed and Calculated Wavenumbers of CF, N’ 16
K.‘
6 6 1 4 6 13 6 12 6 1 1 6 1 0 6 9 6 8 6 I 6 15
transition K,’ N” K.” 10 9
16 15
8 1 4 7 13 6 12 5 1 1 4 1 0 3 9 2 8 1 1
5 5 5 5 5 5 5 5 5 5
K,” 11 10 9 8 7 6 5 4 3 2
wavenumber/cm-’ obsd calcd” 1253.221 1253.258 1253.292 1253.323 1253.350 1253.375 1253.396 1253.415 1253.431 1253.447
1253.219 1253.256 1253.290 1253.321 1253.349 1253.375 1253.397 1253.417 1253.435 1253.450
“Calculated by use of the molecular constants reported in ref 13. C6FsX (X = H, CI, Br, I) at various laser fluences. From the detailed analysis of these reactions, the lowest decomposition path of C6F6was found to be C-F bond breaking, i.e., C6F5-t F. The results of this experiment will be published elsewhere.’ We have also applied diode laser kinetic spectroscopy to investigate the decomposition processes of C6F6. The usefulness of this technique was shown in our previous works on the detection of the transient species CF, in the infrared-laser-induced reaction of CHCIF; and the determination of the rate constants for reactions of CF2.9 We have succeeded in observing the transient species and time evolution of their signals in the infrared multiphoton decomposition (IRMPD) process of C6F6 by this technique. The detailed results and discussion on the reaction mechanism will be given in this paper.
Experimental Section The description of equipment and techniques has been presented previous1y.*-l0 In this experiment gaseous C6F6 was flowed (7) Watanabe, A.; Koga, Y.; Sugawara, K.; Takeo, H.; Fukuda, K.; Matsumara, C.; Keehn, P. M., to be published. (8) Sugawara, K.; Nakanaga, T.; Takeo, H.; Matsumara, C. Chem. Phys. Lett. 1986, 130, 560. (9) Sugawara, K . ; Nakanaga, T.; Takeo, H.; Matsumara, C. Chem. Phys. Lett. 1987, 134, 347.
0 1989 American Chemical Societv
