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be definitively explained. However, the rate parameters of this reaction were obtained from the strictly linear Arrhenius plot observed at the higher temperatures of this study.
termination of the rate constant, at pressure near 1 torr, indicates that this rate constant is pressure independent, which supports an H atom transfer mechanism and not an addition one. For the C1 CH,CN, the temperature dependence of the rate constant has been determined on a rather extended temperature range, showing an unexpected temperature dependence which could not
+
Registry No. CH,CN, 75-05-8; OH radical, 3352-57-6;CI2,7782-
50-5.
Representation of Multistage Mechanisms In Detailed Computer Modeling of Polymerlzatlon Kinetics M. Frenklach* Department of Chemical Engineering, Louisiana State University, Baton Rouge, Louisiana 70803
and W. C. Gardiner, Jr. Department of Chemistry, University of Texas, Austin, Texas 78712 (Received: March 12, 1984)
A method for condensing the differential equations describing homogeneous polymer growth kinetics is described. It permits accounting for polymer mass growth in detailed kinetic modeling by adding a single set of kinetic equations for the elementary reactions augmenting polymer size. Flexibility for adjusting rates to change with concentration or time is retained. Growth
of nascent soot polymer is discussed as an example.
Introduction Modeling of complex homogeneous chemical processes by numerical integration of the kinetic equations for assumed reaction mechanisms has become a standard technique for gaining insight into the fluxes of chemical change and for deriving elementary reaction rate In its most fundamental form, such modeling requires one differential equation for each species concentration, each equation expressing a concentration derivative as a sum of mass-action terms for each of the elementary reactions producing or consuming that species. The number of species considered can be quite large if sophisticated numerical integration techniques are Nonetheless, in modeling polymerization kinetics15 it sooner or later becomes impractical to continue considering each stage of polymerization as a new species. We describe here a method of combining polymerization stages such that explicit account can be taken of all pathways for increasing the degree of polymerization. In place of the concentration growth of different size polymers, the total amount of polymer growth appears as the essential measure of reaction progress. (1) Symposium on Reaction Mechanisms, Models, and Computers, J . Phys. Chem. 81, 2309-2586 (1977). (2) W. C. Gardiner, Jr., J . Phys. Chem., 83, 37 (1979). (3) "Modeling of Chemical Reaction Systems", K. H. Ebert, P. Deuflhard, and W. Jager, Eds., Springer-Verlag,New York, 1981. (4) E. S . Oran and J. P. Boris, Prog. Energy Combust. Sei., 7, 1 (1981). (5) C. K. Westbrook and F. Dryer, Symp. (Int.) Combust., [Proc.],Z8th, 749 (1981). (6) F. Kaufman, Symp. (In?.) Combust., [Proc.],Z9th, 1 (1982). (7) K. H. Ebert, H. J. Ederer, and U. Stabel, Ber. Bumenges. Phys. Chem., 87, 1036 (1983). (8) M. Frenklach in "Combustion Chemistry", W. C. Gardiner, Jr., Ed., Springer-Verlag, New York, 1984, Chapter 7. (9) C. W. Gear, "Numerical Initial Value Problems in Ordinary Differential Equations", Prentice-Hall, Englewood Cliffs, NJ, 1971. (10) A. C. Hindmarsh, 10th World IMACS Congress on Systems Simulation and Scientific Computation, Montreal, Canada, 1982; also Lawrence Livermore Laboratory Report UCRL-87465, 1982. (1 1) T. R. Young and J. P. Boris in ref 1, p 2424. (12) P. Deuflhard, G.Bader, and U. Nowak in ref 3, p 38. (13) G. Bader, U. Nowak, and P. Deuflhard, International Conference on Stiff Computation, Park City, Utah, 1982. (14) D. T. Pratt, Spring Technical Meeting of the Combustion Institute (Western States Section), Pasadena, CA, 1983. (15) W. H. Ray in ref 3, p 337.
0022-365418412088-6263$01.50/0
Analysis We denote a polymeric species as P,, where the first index indicates degree of polymerization and the second refers to the specific chemical form of the s p i e s . The degree of polymerization i could be the number of monomer units incorporated, or some linear function thereof, as may be appropriate for a specific model. The choice of defining i is arbitrary and has no effect upon the results given below. For completeness of the mathematical formalism it will be assumed that each species may be formed from nonpolymeric species or decompose to these species at rates rij in irreversible elementary reactions. The value of each rij is assumed to be recomputed from the prevailing species concentrations at each stage of the numerical integration. Let the number of different species at each stage of polymerization be n. Then the rate law for the P, takes the form
-d[Pijl - dt
or combining the summations
j'#j
Since in computer modeling it is usually convenient to compute reaction rates separately, our further development is therefore based upon the multiple summations of eq 1. We use the notation &#l. hereafter to denote summation over j ' = 1 to j' = n omitting j' = J . In these equations the coefficients xjjt contain as factors the rate coefficientscharacterizing the process of converting species Pij to species Pv,. Similarly, the yjy contain as factors the rate coefficients for the processes that increase the degree of polymerization by one. In addition, the x's and y's may (and presumably do) contain functions of the nonpolymeric species concentrations (see Discussion). These are assumed to remain the same for all i, however, thus requiring that the rates of all j j'conversions are independent of the size of the polymer molecules. The last two summations in eq 1 contain j ' - j and j j' terms for the
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0 1984 American Chemical Society
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The Journal of Physical Chemistry, Vol. 88, No. 25, 1984
Frenklach and Gardiner
C H.
ioio,
,H
c.*
Figure 1. A schematic diagram of the polymerization mechanism developed for soot formation in acetylene pyrolysis. In developing the reaction mechanism for polymerization up to ovalene it was found16 that generation of exterior aromatic rings (from PiI to Pi6) showed different kinetics than generation of interior ones (Pi6 to Pi, and Pi8 to in that the latter reactions become practically irreversible at the stage of coronene. Thus, for further extension it is convenient to consider a stage of polymerization as comprising addition of four acetylenes to give one exterior and two interior rings as shown. Scheme I
a
p1,3
p1,5
p1,4
# N
N
p1s6
N
-
p1,7
= pl,8 N
%,l
Pl,S
- -+
i-1 i and i i 1 processes, respectively, thus including only processes that increase the degree of polymerization. This essentially assumes irreversible covalent bond formation. It should be emphasized that the irreversibility of these steps is a necessary condition of the technique developed here. For i = 1 the third sum is zero, as [Po/] = 0 and all kinetic terms not involving polymer are contained in the r l J . In any real application, each j will be coupled only to a few j ’ in either sum, such that most of the x’s and y’s are zero (see example below). Forming the derivatives of the sums of the concentrations of each type of species j over all degrees of polymerization m
sj = i=1 E[Pij]
j = 1 , 2,
..., n
(3)
Equations 5 or 6 compose the rate laws for n sums of concentrations for polymeric species of n specific types. Combining these n differential equations with the differential equations for the nonpolymeric species (and for other variables, as appropriate) defines the kinetic model governing the time evolution of the system. Before considering total polymerization it will be helpful to write out eq 5 for a specific example. A recent studyI6 of soot formation during acetylene pyrolysis suggested that the main polymerization route proceeds as a repetitive sequence of primarily reversible steps with an irreversible step connecting the sequences (Figure 1). Following the formalism of eq 1, the process can be presented as shown in Scheme I where
x1,2= x3,4 = x7,8 = xio,s = X i i , 6 = k23[H] -k k-24[C,H]
k-21
gives
and after reversing the order of summation dSj _ - E r i j + j ’E# j xfjSjl- j ,E# j xJJ. SJ . j’*lCyjf,Sjf- j ’E# j yjjSj dt
-
or dSj
+
(5)
and k, and k-i are the rate coefficients for the forward and reverse
m
= C r i j + C (xjv + yfj)Sjt - (j$/xjf dt j=1 jt#j .~
. -
+ yjjT)JSj f o r j = 1, 2,
..., n (6)
(16) M. Frenklach, D. W. Clary, W. C. Gardiner, Jr., and S. E. Stein, Symp. (IN.) Combust., [Proc.],ZOth, to be published.
The Journal of Physical Chemistry, Vol. 88, No. 25, 1984
Representation of Homogeneous Polymerization
6265
directions, respectively, assigned to reaction class 1.16 The differential equations for the sum concentrations in this mechanism are m
D
m
m
m
Figure 2. A profile of polymer carbon growth at T = 1600 K and [C2H210= 4.0 X lo-' mol/cm3. Detailed information on the simulations is given in ref 16.
or
e..
.... These 12 differential equations are to be added to the set of differential equations that describes the rates of change in the concentrations of the species to be considered individually. Integration of the entire set of differential equations for appropriate initial conditions (including presumably Sj[t=O] = 0, J = 1, 2, ..., 12) generates the profiles of the species concentrations and the polymer concentration sums.
Total Polymer The three sources of change in each species concentration sum are direct production from monomers, interconversion of species with the same degree of polymerization, and increase in the degree of polymerization (eq 5). Each of these reactions is associated with a defined increase in the amount of polymer measured somehow, for example by molecular weight or number of monomer residues converted to polymer. Let the increases in the measures from the three. sources be Am;, Am;., and A ~ j t The . first of these is in principle increasing without limit with increasing irealistically, however, the corresponding rii are going to be nonzero only for i = 1 or 2. The rate of increase of total polymer in species j is formed by including the increases in the measures in eq 5 m
dMj/dt = CAmbrij + i= 1
C Am;qjtPj, - jzjAm$xjjSj j'# j
c AmFgy,Sy -
j'zj
+
Am$yjySj (7) j'#j
from which the total amount of polymer grows as n -
n
dM/dt = C(dMj/dt) = j= 1
2
J-I j + j
(Arn&xyj
e cAm;jrij +
j=1
+ Arnj,yyj)Sy - j=1
(Am$xjjl
+ Am$yjy)Sj
j'#j
In order to apply eq 8 to the example introduced above we need only specify the Am values. Using molecular weight as the measure of amount of polymer would give AmX= -1 for H-atom abstractions, AmX= 26 for C2Hzadditions, and Amx = Amy = 25 for CzH additions. If number of carbon atoms is used instead
as the measure of amount of polymer one has Amx = 0 for H-atom abstractions and AmX = Amy = 2 for Cz additions. The value of Ami,l is equal to the molecular weight of or the number of carbon atoms in Pl,l. In our example of soot formation,16 it is appropriate because of the size of the reaction mechanism in relation to the required computing time to make the transition from detailed to summary kinetics at coronene (CZ4Hl2),thus Am;,, = 300 amu or 24 carbon atoms. With the carbon atoms measure and the above values of Am, eq 8 takes for our example the form
where M is the total number of carbon atoms accumulated in species beginning with coronene. Equation 9 with the initial condition M[t=O] = 0 is to be added to the previous set of differential equations of the species concentrations and the concentration sums Sj. An example of a polymer growth curve for the soot formation mechanism using this polymerization model is shown in Figure 2.
Discussion The equations derived above are based on assumptions that are general enough to apply to a variety of polymerization reactions. One could also adapt the equations to test for side effects such as loss of degree of polymerization or termination reactions. The two limitations that restrict the generality of this formalism are its linearity in polymeric species concentrations and the i independence of x and y factors. For particular cases, these two limitations can be explored in modeling experiments by using the results obtained with the summation formulas as a first-order model and constructing nonlinear global models or i-dependent models to see how large the effects may be. It may be presumed that large effects of nonlinearity would appear for high i values, where doubling reactions of smaller polymers could increase polymer size more rapidly than the sequential linear processes. For very high i values it is also to be assumed that the polymer growth model itself will change, which again can readily be tested by global modeling. The seriousness of the restriction that the x and y factors must be i independent will depend on the specific polymerization process described. For small degrees of polymerization, decreases would be expected on the grounds of diminished encounter frequency for larger molecules with adducts. As the degree of polymerization increases, so would the expected x and y values, as the threedimensional diffusive (or collisional) process applicable to smaller
J. Phys. Chem. 1984, 88, 6266-6271
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molecules becomes a faster two-dimensional diffusive process of adduct on the surface of growing polymer molecules. Specific experiments to evaluate the size of these changes, and their effect upon the modeling, would be required for any application where it is felt that they may be important.
Conclusions A simple formalism for including polymerization reactions in detailed reaction modeling has been discovered. It permits ac-
counting for the growth of amount of polymer in the system without increasing the computational burden significantly. Its limitations of size-independent reactivity and linear growth can be tested by global modeling experiments. Acknowledgment. Our modeling research was supported by NASA-Lewis Research Center, Grant NAG 3-477 and Contract NAS 3-23542, and the Robert A. Welch Foundation. Registry No. Acetylene, 74-86-2.
Mechanism of Tritium-Atom-Promoted Isotope Exchange in the Benzene Ring: Application to Tritium Labeling of Biologically Important Aryl Compounds M. F. Powell,+H. Morimoto, W. R. Erwin, B. E. Gordon, and R. M. Lemmon* National Tritium Labeling Facility, Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720 (Received: April 6, 1984)
Reaction of thermal tritium atoms, generated by microwave activation of T2gas, with benzene and biphenyl was studied at - 4 0 and -196 "C. The saturation reactions (Le., benzene -,cyclohexane-t6)predominated over isotope exchange (Le., benzene benzene-t) at -196 "C. However, significant exchange labeling occurred at --50 "C, with a concomitant reduction in the yields of saturated products. This reversal in labeled product yields at the different temperatures is due, in part, to the faster rate of H expulsion from the intermediate cyclohexadienyl radical at -50 "C and to the increased mobility of the warmer matrix that retards multiple T. reactions with the same aryl molecule by covering up singly tritiated intermediates. The less volatile aryl compound, biphenyl, was labeled in a diffusionallyactive matrix of either benzene or cyclohexane, whereas it could not be labeled otherwise.
-
Introduction The reaction of discharge-activated tritium gas with organic compounds often results in tritium isotope exchange labeling,l particularly when the target compound contains the phenyl group.* Limited insight as to the labeling mechanism@) has been provided from earlier ESR and reaction product studies using substituted benzenes. Cyclohexadienyl-type radicals were observed for reaction of activated H2 gas with polystyrene and substituted polystyrene^,^,^ L-phenylalanine, tyrosine,^ and benzene Both the reaction of atomic hydrogen with several substituted benzenes and the reaction induced by y-irradiation also resulted in cyclohexadienyl radical formatiom8 In contrast, the reaction study of atomic tritium with substituted benzenes has been carried out by labeled product a n a l y ~ i s . ~ Although the amount of tritium incorporation was often high, most of the label was found in the fully hydrogenated (tritiated) cyclohexyl ring or in the polymeric compounds produced by undesirable side reactions. Little tritium was found in the parent compound. This observation can be easily rationalized when one considers the experimental conditions used. Most T-labeling experiments were carried out by reacting a stream of T. atoms with the upper few monolayers of solid target matrix at -196 O C , such that [T.] >> [target molecules], and therefore, it is not surprising that saturation (a reaction involving several tritium atoms) predominated over substitution (reaction of a single Tper target molecule). For example, reaction of tritium atoms with benzoic acid afforded primarily cyclohexanecarboxylic acid with only small amounts (1-2%) of tritiated benzoic acid.9 Similarly, reaction of T. with benzene resulted in the formation of significant amounts of c y c l ~ h e x a n e - t ~ . ' ~ In contradistinction to the "saturation labeling" observed in solid matrices, the gas-phase reaction of hydrogen atoms with benzene afforded additional c6 products such as cyclohexadiene and cyclohexene." This suggests that the saturation-to-exchange ratio t Institute of Pharmaceutical Sciences, Syntex Research, 3401 Hillview Ave., Palo Alto, CA 94304.
of the products may be controllable by use of the appropriate experimental Conditions. Little emphasis, however, has been placed on delineating the factors affecting the isotope exchange process with microwave activated tritium gas. Our recent studies reported herein fill this void with evidence for (i) the stepwise addition of T. to the phenyl group and subsequent expulsion of Ha and (ii) the superiority of labeling in a "warm", diffusionally active matrix rather than an immobile one at -196 "C.
Experimental Section Spectral grade benzene and cyclohexane were obtained from Mallinckrodt and were used without further purification. Benzene-& 1,3-~yclohexadiene,and 1,4-~yclohexadienewere purchased from Aldrich Chemical Co. Chromatography standards toluene, ethylbenzene, n-propylbenzene, n-hexylbenzene, and 0-, m-, and p-xylene were obtained from Polyscience Corp. Chromatographic quality cyclohexene was from Matheson Coleman and Bell. Tritium gas was generated by heating uranium tritide as described earlier.9 The labeling apparatus used previously9 was modified (Figure 1) to permit isotope labeling at temperatures where benzene has an appreciable vapor pressure (> -60 O C ) . (1) For example, see: Lee, R.; Ehrenkaufer, R. L.; Wolf, A. P. J . Labeiied Compd. Radiopharm. 1963, 13, 359-62. Hembree, W. C.; Wolf, A. P.; Ehrenkaufer, R. L.; Lieberman, S . Methods Enzymol. 1975, 37, 313-21. (2) Ehrenkaufer, R. L.; Wolf, A. P.; Hembree, W. C.; Lieberman, S.J . Labeiied Compd. Radiopharm. 1977, 13, 367. (3) Wall, L. A.; Ingalls, R. B. J . Chem. Phys. 1964, 41, 1112. (4) Tiedeman, G. T.; Ingalls, R. B. J . Phys. Chem. 1967, 71, 3092. (5) Liming, F. G.; Gordy, W. Proc. Natl. Acad. Sci. U.S.A. 1968, 60, 794-801. (6) Chachaty, C.; Schmidt, M. C . J. Chim. Phys. Phys.-Chim. Biol. 1965, 62, 527-35. (7) Morgan, C. U.; White, K. J. J . Am. Chem. SOC.1970, 92, 3309-12. (8) Campbell, D.; Symons, M. C. R.; Verma, G. S. P. J . Chem. SOC.A 1969, 2480-3. 49) Gordon, B. E.; Peng, C. T.; Erwin, W. R.; Lemmon, R. M. Inr. J. Appl. Radiat. Zsot. 1982, 33, 715-20. (10) Gordon, B. E.; Otvos, J. W.; Erwin, W. R.; Lemmon, R. M. Znt. J . Appl. Radiat. Isof. 1982, 33, 721-24. (11) Knutti, R.; Biihler, R. E. Chem. Phys. 1975, 7, 229-43.
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