J. Phys. Chem. 1991, 95, 145-154
145
SiH, 4- H, and AH,'(SIH,) by a Estimation of the Arrhenius Parameters for SIH, Nonlinear Regression Analysis of the Forward and Reverse Reaction Rate Data Harry K. Moffat,*.t Klavs F. Jensen,t and Robert W. Carr Department of Chemical Engineering and Material Science, University of Minnesota, Minneapolis, Minnesota 55455 (Received: January 17, 1990; In Final Form: July 10, 1990)
We have extended a calculational procedure for the estimation of Arrhenius parameters for unimolecular reactions by direct regression of RRKM-derived predictions of experimental data to include nonlinear regression of experimental data for the reverse reaction. Both the forward and reverse reaction experimental data may then be used to access the validity of kinetic parameters assumed in the RRKM model. Using recent direct measurements of the rate of SiH2 insertion into H1, a recent SiH, pyrolysis study in a static reactor, and earlier shock tube and static reactor SiH, pyrolysis studies, estimates for the high-pressure Arrhenius parameters for the SiH, decomposition reaction are obtained. The RRKM model is a good representation of the data over a 900 K temperature range, at pressures from 1 to 4000 Torr. The 1100 K Arrhenius expression is: log k( T ) (s-I) = (1 5.79 f 0.5) - (59.99 f 2.0) kcal mol-'/2.3RT. The use of a correction factor to the density of states due to the effects of anharmonicity is seen to lead to improvement in the agreement with the data. The large uncertainty in the preexponential was due to the existence of a range of permissible A factors which provided adequate fits. Since the use of the detailed balance principle necessitates estimation of the thermodynamic equilibrium constant, we have also determined estimates for AHro(SiH2)consistent with the above high-pressure Arrhenius parameters. AHfo(SiH,) is predicted to be 65.5 f 1 .O kcal mol-'. The current thermal kinetic experimental data precludes a more accurate determination of AHt(SiH2) due to the uncertainties in the temperature extrapolation of the high-pressure rate constant and in the bath gas collision efficiencies.
Introduction The thermal decomposition of silane has been of considerable interest both from the standpoint of its importance in silicon chemical vapor deposition processes and from its significance as the lowest member of the homologous series of silanes. There is general agreement that the first step in the homogeneous gas-phase reaction is the molecular elimination of H2, resulting in generation of silylene, SiH2. In their detailed study of 1966, Purnell and Walsh (PW)I pointed out that this is the lowest energy reaction path, so that, although their rate expression was consistent with both a free-radical chain reaction and a reaction in the pressure-dependent unimolecular falloff regime, they were able to favor the latter on energetic grounds. Subsequent experimentation by static bulb pyrolysis,2-5by competitive rate shock tube studies,2,6 and by laser-powered homogeneous pyrolysis7+' have all confirmed this finding. Despite the extensive experimental effort devoted to silane decomposition, reported rate constants have all been in the pressure-dependent unimolecular falloff regime. Arrhenius parameters have never been measured in close approach to either the high-pressure limit or the low-pressure bimolecular regime. Roenigk, Jensen, and C a d o have shown that nonlinear regression of RRKM theory on experimental data can be reliably used to predict high-pressure Arrhenius parameters. When rate constants for CH3NC isomerization that are wholly in the unimolecular falloff were fitted by nonlinear regression of RRKM theory, satisfactory extrapolation to the high-pressure limit was obtained, and the entire falloff curve was faithfully reproduced by RRKM model generated by the fitting procedure. This approach was employed by Roenigk, Jensen, and Carr (RJC)I1to fit falloff data on the thermal decomposition of silane (and disilane) and obtain the high-pressure Arrhenius parameters. In RJC's silane work," the PW static reactor experiments' and the shock tube experiments reported by Newman, Ring, and ONeal (NR0)6and Newman, Ring, ONeal, Lesksa, and Shipley (NORLS)2were fit. The RRKM model achieved good agreement with the values of the experimental rate coefficients from these investigations, while the predicted values of E" and A" varied over a range of about 3 kcal/mol and a factor of IO, respectively, * Author to whom correspondence should be addressed. 'Sandia National Labs, P.O. Box 5800, Albuquerque, N M 87185. 8 Present address: Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139.
depending upon the assumptions made about the temperature dependence of ( A E ( T ) ) , the average energy transferred per collision. A previous RRKM analysis by NORLS yielded a prediction of E" and A" encompassed by this range, but Arrhenius expressions where the activation energy from RJC closely matched that of NORLS gave preexponential factors that were smaller than NORLS, and when the two preexponentials matched, the NORLS activation energy was larger than that of RJC. When (AE(T ) ) was assumed to be independent of temperature, RJC's RRKM model underpredicted both experiments' activation energies and overpredicted PW's observed pressure dependence. Later, Erwin, Ring, and ONeal (ERO)4 did RRKM calculations, employing a 0.9 kcal/mol larger high-pressure activation energy and a 0.55 larger value for log A I " than RJC's temperature independent ( AE( 7')) calculations used. ERO compared their calculated rate constants to P W s static study at one pressure and three temperatures, their own static study at 200 Torr at two temperatures, and a reevaluation of NORLS's silane shock tube dissociation experiment at one temperature. Agreement with the absolute value of the rate constant was again good, with realistic values for the collision efficiencies being employed. However, agreement with the observed activation energies and pressure dependence was not as good. Their RRKM model again consistently underestimated the observed activation energy in all three experiments by 1-2 kcal/mol. Also, the power law pressure ( I ) Purnell, J. H.; Walsh, R. Proc. R . SOC.1966, 293, 543-560. (2) Neuman, C. G.; ONeal, H. E.; Ring, M. A,; Leska, F.; Shipley, N. In?. J . Chem. Kine?.1979, 11, 1167-1182. (3) Neudorfl, P.; Jordon, A.; Strausz, 0. P. J. Phys. Chem. 1980, 84, 338-339. (4) Erwin, J. W.; Ring, M. A.; ONeal, H. E. In?. J . Chem. Kine?.1985, 17, 1067-1083. (5) White, R. T.; Espino-Rios, R. L.; Rogers, D. S.; Ring, M. A.; O'Neal, H.E . In?. J . Chem. Kine?. 1985, 17, 1029-1065. ( 6 ) Newman, C. G.; Ring, M. A.; ONeal, H. E. J. Am. Chem. SOC.1978, 100, 5945-5946. (7) Longeway, P. A.; Lampe, F. W. J. Am. Chem. SOC.1981, 103, 6813-6818. Longeway, P. A.; Lampe, F. W. J. Phys. Chem. 1983, 67, 354-358. (8) Jasinski, J. M.; Estes, R. D. Chem. Phys. Lerf. 1985, 117, 495-499. (9) Bonsella, E.; Caneve, L. Appf. Phys. E 1988, 46, 347-355. (IO) Roenigk, K. F.; Jensen, K. F.; Carr, R. W. J. Phys. Chem. 1987, 91, 5726-573 1. (11) Roenigk, K. F.; Jensen, K. F.; Carr, R. W. J. Phys. Chem. 1987,91, 5732-5739. Roenigk, K. F.; Jensen, K. F.; Carr, R. W. J . Phys. Chem. 1988, 92, 4254.
0022-3654/9l/2095-0l45$02.50/0 0 1991 American Chemical Society
146 The Journal of Physical Chemistry, Vol. 95, No. I , 1991
Moffat et al.
dependence predicted by the RRKM model for ERO's static conditions was 0.67 at 702.4 K, while the experimentally observed power law dependence was 0.5. Frey, Walsh, and Wattsi2have also constructed an RRKM model for the SiH4 dissociation reaction and have used it to fit their pressure-dependent rate coefficients for the insertion of SiHz into Hz, obtained from a relative rate constant study. Observing that the resulting highpressure rate constant value, k-i"(300 K) = 1.1 X loi2cm3 mol-l s-l, was roughly in agreement with Jasinski and Chu's extrapolated value,I3 they used Jasinski's value of k-,"(300 K) to obtain a kinetics-based value of AHfo(SiHz). The wide range of satisfactory fits found by RJC and the substantial differences with the results of NORLS and ERO leave considerable uncertainty in A" and E". The RJC calculations, however, did not include new direct laser resonance absorption flash kinetic spectroscopy data on the reverse reaction, the insertion of SiH2 into H2, which has since appeared,13 or additional experimental data taken in static reactors3v4and a reevaluation of the shock tube silane dissociation e ~ p e r i m e n t . ~It seemed worthwhile to extend the nonlinear regression approach to include both forward and reverse reactions to see if parameter estimation for the silane reaction would be more accurate by the inclusion of these data. It also seemed important to develop the capability to handle reversible reactions as a matter of course. Pillingi4 and Keiffer et a1.I5 have recently shown the benefits of combining experimental data on association reactions over wide ranges of temperatures and pressures with RRKM theory and least-squares fitting methods in order to obtain consistent, accurate estimates for the limiting high- and low-pressure rate coefficients. The existing silane data encompass an unusually wide range of experimental conditions, with temperature ranging from room temperature to 1200 K, and pressures from 1 to 4000 Torr. It was of interest to see if a comprehensive theoretical model of the reaction could be achieved over the entire range and to check for consistency between experimental studies. In this paper, fixed transition-state theory was used to model Hz elimination from silane, using not only data of PW and NORLS, but also incorporating static decomposition data of ER04 and Neudorfl, Jodhan, and Strausz (NJS).3 The fitting procedure was extended to include the reverse reaction, where the data of Jasinski and Chu (JC)13 were regressed upon. Because there are no competing alternate reaction channels of significance, the insertion reaction will have the same pressure dependence as the unimolecular decomposition. There has been disagreement about the AH,O(SiH,), recently. Since this quantity is connected to the forward and reverse rate coefficients through the equilibrium constant and by detailed balance, it was included as a fitting parameter. Correlations in the fitting parameters (high-pressure Arrhenius parameters, AH?(SiH,), and collision efficiencies) lead to a range of high-pressure Arrhenius parameters that do a good job of fitting the experimental data. Special attention is paid to getting good model representation of the experimental activation energies, the pressure dependence of the silane unimolecular rate constants, and the low-pressure limiting termolecular rate coefficient of SiH2 + H2. The RRKM model gives a good representation of the experimental data spanning the whole range of pressure and temperature quoted above. The consequences for AH,0(SiH2) are explored. The results are presented in terms of parameters for Troe's Fcen,formalism,I6 which is useful for constructing accurate, computationally facile approximations for the reaction rate constant as a function of both temperature and pressure in large scale reaction4iffusion calculations that are needed in CVD and combustion mechanistic studies."
RRKM Method and Regression Procedure The reaction rate constant can be expressed by the following equation from RRKM theory, with nomenclature consistent with Robinson and Holbrook'8
(12) Frey, H. M.; Walsh, R.; Watts, 1. M. J. Chem. Soc., Chem. Commun. 1986, 1189-1191. ( 1 3 ) Jasinski. J.; Chu, J. 0. J. Chem. Phys. 1988, 88, 1678-1687. (14) Pilling, M. J . Inr. J . Chem. Kiner. 1989, 21, 267-291. ( I 5) Keiffer. M.; Pilling, M. J.; Smith, M. J. C. J. Phys. Chem. 1987, 91, 6028-6034. (16) Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1983,87, 161-169. Gilbert, R. G.: Luther, K.; Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1983,87, 169-177.
(17) Gardiner, W. C.; Troe, J. Rate Coefficients of Thermal Dissociation, Isomerization, and Recombination Reactions. In Combustion Chemisrry; Gardiner, W. C., Ed.; Springer-Verlag: New York, 1984. (18) Robinson, P. J.; Holbrook, K. A. Unimolecular Reactions; Wiley: New York, 1972. (19) Troe, J. J. Chem. Phys. 1977, 66, 4745-4757. (20) Heyman, M.; Hippler, H.; Troe, J . J . Chem. Phys. 1984, 80,
k , = Lt-[kT h
'1
Qi
exp[
$1
X
QiQz
where Lt is the reaction path degeneracy, Qlt is the partition function for adiabatic degrees of freedom in the transition state, Q, is the partition function for adiabatic degrees of freedom in the reactant, Q2is the partition function for all active degrees of freedom in the reactant, Eo is the critical energy, and E+ is the amount of energy above the critical energy. It was assumed in all cases that overall rotations were adiabatic, while vibrations were active. k,(Eo + E + ) , the rate constant for reaction of energized molecules, is determined from the statistical probability of the energized molecule being in an internal state near the transition state versus being in any other internal state.
where N*(Eo = E + ) is the density of states consisting of active degrees of freedom of the energized molecule and (W(E+)} is the sum of states of the active rotations and vibrations in the transition state whose total energy is less than E+. kz[M] is the LennardJones collision frequency. /3, is the low-pressure limiting collision efficiency. Troe's expression was used for the temperature dependenceIg (AE) -=-P C (3) 1 - PC'/Z FE(T)RT where FE
is a measure of the energy dependence of the density of states. Equation 1 implicitly assumed that weak collision effects at intermediate pressures could be taken into account by replacing the Lennard-Jones collision frequency by the product of the Lennard-Jones collision frequency with the low-pressure weak collision efficiency. ( A E ) is the average energy transferred in all up and down transitions. The temperature dependence of ( A E ) is a matter of current interest.20 For all of the work in this paper, ( A E ) was assumed to be independent of temperature. F is a factor that takes into account the change with energy of the energy levels in the adiabatic degrees of freedom between the transition state and the energized molecule.18 Waage and Rabinovitch'sz' expression was used for F
1853-1 860.
Arrhenius Parameters for SiH4 G SiH2
+ H2
The Journal of Physical Chemistry, Vol. 95, No. 1, 1991 147
TABLE I: Formulation of the Transition State: Freauencies for Silane and the Transition State alus ReDresentative Arrbenius Parameters
[SiH4],cm-l
general
log A’ ( I 100 K), s-I ,Ea-( 1100 K), kcal/mol
Ecri,,kcal/ mol Ecri,,-’kcal/mol M 0 ( O K), kcal/mol AH? (SiH2), kcal/ mol
2187 978 978 2183 2183 2183 910 910 910
Si-H str SiH, rock SiH, rock Si-H str Si-H str Si-H str SiH, rock SiH, rock SiH3 rock
reactn coord 993 1461fo 2108 2103 1461f 1328 9lOf 9lOf
Wtted.3
.
fixed.3b
fitted.1 15.13
15.79 59.99 55.88 0.54 55.34 65.49
15.90 60.68 56.43 0.47 55.96 66.11
58.03 54.91 0.18 54.73 64.88
reactn coord 993 586.4 2107 2102 586.4 1328 365.2 365.2
reactn coord
reactn coord
993 548.5 2107 2102 548.5 1328 341.7 341.7
993 877 2107 2102 877 I328 546 546
“f is a multiplicative factor that is allowed to vary in order to produce a specified preexponential factor. where the moments of inertia for the complex, I“, and the reactant, I , were taken along the axis containing the reaction coordinate. a( Eo)is Whitten-Rabinovitch’s correction factor calculated at the critical energy. E, is the zero point energy, and s is the number of vibrational degrees of freedom in the reactant. For the tight transition state initially assumed in this study, F is close to 1, being equal to only 0.96 at 300 K and 0.88 at 1200 K. In the highpressure limit the adiabatic degrees of freedom contribute a factor of Qlt/QI to the reaction rate constant, while in the low-pressure limit they contribute a factor of FQlt/QI. For highly excited molecules, especially small molecules where the number of quanta of energy per oscillator is large, the assumption that all oscillators are harmonic is not correct. As the energy per oscillator approaches the bond dissociation energy real oscillator energy levels become denser than their harmonic oscillator analogues. This fact only significantly bears on the calculation of the density of states of the reactant. The Fa,$,factor is a correction factor to the density of states of the reactant at an internal energy equal to the critical energy due to these anharmonic effects. From a simple expression based on Morse oscillators,22 Fanh = I .78 can be calculated. Because of the uncertainty in the potential energy curves, a more detailed treatment of the effects of anharmonicity in the oscillators is not warranted. The value of Fanh does not influence the high-pressure limiting rate constant, but the low-pressure limiting rate constant is proportional to Fan,,.Following ref 22, the low-pressure limiting reaction rate constant, klo(T),can be factored into well-defined contributions.
Here, F,( is the contribution due to the adiabatic external rotations and is equal to FQlt/QI. Note that for fixed high-pressure Arrhenius parameters (maintained by adjusting a set of transitionstate vibrational efficiencies), increasing the value of Fanhhas roughly the same effect as increasing the *loosenessn of the transition state, Le., the preexponential factor of the low-pressure limit. Therefore, the value of Fan,,was varied over a moderate range in the fitting study to test how a looser transition state might fit the experimental data. The method for constructing the transition state was based on RJC,”where BEBO correlations were used to assign frequencies in the transition state. Four frequencies were fit by one common multiplicative factor. The critical energy and common multiplicative factor were solved for given values for the high-pressure activation energy and preexponential factors, Le., two equations (21)Waage, E. V.; Rabinovitch, B. S. J . Chem. Phys. 1970, 52, 5581-5584. Waage, E.V.; Rabinovitch, B. S.J . Chem. Phys. 1970,53,3389. (22)Troe, J . J . Phys. Chem. 1979,83,114-126.
with two unknowns. The molecular structure and moments of inertia of the transition state were also taken from ref 11. Further details of the transition-state structure are given in Table I and in the two tables of the Appendix. A direct extended Beyer-Swinehart algorithm23was used to calculate sums and density of states for the activated complex, while Waage and Rabinovitch’s semiclassical method was used to calculate the density of states for the reactant.% The integration is carried out over E+, the energy above the zero point of the activated complex, not including the energy in the adiabatic rotations of the activated complex. Numerical integration was accurate to within 2%. The reverse reaction rate constant for SiHz H2 is related to the forward reaction rate constant by microscopic reversibility, where K l ( T ) is the equilibrium constant and AG,O(T) is the standard Gibbs free energy of reaction.
+
k-i(T,P) = ki(TJ’)/Kl(T)
(7)
This necessarily implies that the reverse reaction has the same falloff behavior as the forward reaction, barring the existence of additional product channels that may be open to the chemically activated silane. The relatively high Si-H bond strength, D(H,Si-H) = 90 kcal mol-’ 25 causes the Si-H dissociation channel to be unimportant. Using the estimate of log k,”( r ) = 15.57-93 kcal m o P / [ 2.3RT] for the high-pressure Arrhenius parameters showed that this channel was not significant for conditions corresponding to any of the experiments studied. Thermodynamic information for the vibrational frequencies of silylene came from ref 26 and for the geometry from ref 27. Thermodynamic information for silane and H2 came from the JANAF tables.28 The resulting thermodynamic functions are contained in Table V. The ACpo formula is valid between 300 and 1000 K. In the calculations, thermodynamic functions were computed directly from statistical mechanical formulas and not from the fit to the functions that are presented in Table V. The method used to form and minimize an objective function is an extension of that used by RJC.IO Partial minimization of (23)Stein, S.E.;Rabinovitch, B. S. J . Chem. Phys. 1973,58,2438-2445. (24)Whitten, G. 2.;Rabinovitch, B. S. J . Chem. Phys. 1964, 41, 1883-1883. (25)Walsh, R. Arc. Chem. Res. 1981,14, 246-252. (26)Fredin, L.;Hauge, R. H.;Kafafi, Z. H.; Margrave, J. L. J . Chem. Phys. 1985,87,3542-3545. (27)Dubois, I. Can. J . Phys. 1968,46,2485-2490. (28)JANAF Thermochemical Tables, 1978 Supplement J . Phys. Chem. ReJ Data 1978,7,193.
148 The Journal of Physical Chemistry, Vol. 95, No. 1 , 1991
the objective function is performed a t a fixed value of A” using the BFGS algorithm.29 Movement of A” is then done using a local quadratic interpolation of the objective function. The objective function consisted of the weighted least-squares error function between RRKM predictions, k, and Ei, and experimental data, ke,iand for the forward reaction, and RRKM predictions, kj, and experimental data, k:,i, for the reverse direction.
Moffat et al. SiH, 1 ,SiH2 SiH2
(9)
n, refers to the number of collision efficiencies included as fitting variables. The weighting factors, and o$, refer to standard errors in the experimental rate constants and activation energies. In some cases, however, these weights are modified to give a particular data set more influence in the objective function, if for example it is considered to be more reliable. In other cases, the standard errors even with a data set could only be guessed at. The sensitivity of the predictions to changes in the weighting factors and to whether the data set is expressed in terms of its individual data points or in terms of a reaction rate constant and activation energy was explored. The objective function was minimized with respect to changes in the high-pressure Arrhenius parameters, A,” and El“, the bath gas collision efficiencies at 300 K, and AH: (Si H2). Data Sets Used in the Regression Procedure for Silane Dissociation The mechanism for silane pyrolysis is now well understood to be mainly initiated via the homogeneous molecular H2 loss to form SiH2 and H2.The presence of an initiation reaction involving the surface initiated atomic loss of H is also postulated to contribute at lower temperatures and pressures than in the experimental data on the homogeneous reaction used here.30s3’ The range in temperature and pressure that experimental data on the silane decomposition reaction exists is large, representing contributions from room temperature recombination rate studies of SiH2 + H2, 700 K static reactor silane pyrolysis studies, and 1100 K shock tube silane pyrolysis studies. However, the reaction has never been experimentally measured in either its high-pressure limit or far into the bimolecular second-order region. Moreover, in both the static reactor pyrolysis and shock tube pyrolysis experiments, mechanistic complications create uncertainty in the derivation of rate constants for the initial silane decomposition reaction from an analysis of the pyrolysis products. Therefore, an RRKM analyses of the experimental rate constant parameters are essential as a consistency check. The range in temperature provided by the experimental data provides accuracy in the determination of transition-state parameters. The following are the experimental data included in the regression analysis from experiments on silane pyrolysis carried out in a static r e a c t ~ r . l - ~In? ~a very thorough and complete study, Purnell and Walsh (PW)’ conducted silane pyrolysis experiments in a static system. The experiments were carried out with pure silane between 35 and 230 Torr, and 651 and 703 K. Their results for the initial formation of principal gas products, Si,H, and H2, as a function of temperature and pressure, valid over the entire temperature and pressure regime encompassed by their study, were presented in terms of a single rate expression proportional to the initial silane concentration to the 3/2 power. PW assumed that the initial rate of formation of disilane could be related to the rate of the silane pyrolysis reaction by the following mechanism. (29) Beck, J. V.; Arnold, K. J. Parameter Estimation in Engineering and Science; Wiley: New York; 1977. (30) Scott, 8. A.; Estes, R. D,;Jasinski, J. M. J. Chem. Phys. 1988, 89, 2544-2549. (3 I ) Robertson, R.; Gallagher, A. J . Chem. Phys. 1986.85, 3623-3630. Robertson, R.: Gallagher, A . J . Chem. Phys. 1987.86, 3059-3060. O’Neal, H. E.: Ring, M. A . Chem. Phys. Leu. 1984, 107, 442-449.
-3
Si3HB
-
The initial reaction stoichiometry, which varied neither with pressure nor temperature, was found to be, 2.39SiH4 Si2H6 0. 13Si3Hs 1 .26H2, generally supporting the proposed mechanism. The initial rate of formation of disilane was equated with the rate of the initial silane H2 molecular dissociation reaction. The 3/2 power law dependence stems from the initial unimolecular reaction being in the “knee-region’’ of the falloff curve. Because PW’s rate expression formula spans such a large temperature and pressure range, the rate expression formula probably is not as accurate at predicting the experimental results at the periphery of the covered temperature and pressure range than at the center. Therefore, in the regression results, PW’s experimental data were s-I, represented by one reaction rate coefficient, k = 6.71 X and activation energy, Eelp= 55.23 f 0.5 kcal/mol at the mean conditions of their experiment, 675 K and 135 Torr, and one reaction rate coefficient, k(675 K,100 Torr) = 5.77 X s-I in order to include their reported pressure dependence. The relative standard errors in the reaction rates were 7.5%, following the discussion in RJC.IO Neuman et al. (NORLS)2 raised the point that the initial rate of disilane formation may not represent the rate constant for the initial silane decomposition reaction. For example, silylene may decompose by a unimolecular process to Si and H2. Then, silicon may insert into SiH, to create two silylene molecules in a chain branching reaction that eventually increases the disilane formation rate. Therefore, silylene trapping agents have been added to the bath gas in later static silane pyrolysis experiments in order to prevent secondary reactions ((-2) and others) from occurring. The most extensive study of these is Erwin, Ring, and O’Neal (ERO)4 who conducted static silane dissociation experiments around 200 Torr and 670 K with silane-acetylene mixtures in a propane bath gas. The concentration of acetylene was I O times greater than that of silane. Acetylene is an efficient scavenger for free radicals i~~ and, as it turns out, silylene. Chu, Beach, and J a s i n ~ k have recently obtained a rate constant for silylene reaction with C2H2 at 300 K and 5 Torr of 9.8 X IO-’’ cm3 molecule-’ s-’, nearly gas kinetic. Therefore, ERO’s assumption that the kinetics of silane loss in the presence of excess acetylene should be identical with the kinetics of the pure silane gas-phase initiation reaction appears valid. No disilane formation was observed under ERO’s conditions. The two experimental Arrhenius relations, which were used as the basis for including the experimental data in the objective function, were derived from Table I1 of their paper, which contained their raw rate constant data at 200 Torr and two measurements at 400 Torr: k(200 Torr,700 K) = 10’3.26exp[-54.37 exp[-55.14 kcal kcal mol-’/RT]; k(400 Torr,705 K) = 10’3,65 mol-’/RT]. In the fitting procedure, relative standard errors of 0.04 and 0.06 were assigned to the weighting factor for the reaction rates at 200 and 400 Torr, respectively, due to the quality of the data, while standard errors of 0.5 and 0.9 kcal mol-’ were assigned to the weighting factors for the activation energies. A direct comparison of E R O s rate constants with PWs is marred by the use of different bath gases in the two experiments. In general, the rate constants of ERO are slightly less than a factor of 2 lower than PW’s, but this could be accounted for by differences in the bath gas collision efficiencies. PW conducted their experiment in a silane bath gas, while ERO’s bath gas was propane. The two 400-Torr experimental points at 709.8 and 702.4 K along with their respective 200 Torr points indicate a power law pressure dependence of 0.47 and 0.49 for EROs data, closely agreeing with P W s findings. Agreement with other static reactor silane pyrolysis data, such as the static data in refs 2 and 3, are not presented in this paper. However, the rate constants obtained in these studies
t[(7) + ( y )+] [; (yj2]
i= I
+ SiH, 2 Si2H,
SiH2 + Si2H6
+
~[Eo~P,,,,Pc,,,~~,o(Si= H2)I
+ H2
+
.
(32) Chu, J. 0.; Beach, D. B.; Jasinski, J. M. J. Phys. Chem. 1987, 91, 5340-5343.
Arrhenius Parameters for SiH,
~i
SiH2
+ H2
are in agreement with the two static pyrolysis studies employed. We included experimental data on the silane pyrolysis reaction from shock tube experiment^.^,^,^ In the original study NORLS2y6 shocked mixtures of SiH4, cyclopropane (as the internal standard), and toluene in an argon bath gas. They obtained the rate of the initial silane decomposition directly from the rate of SiH4 loss. The decomposition rates were observed to be independent of the initial silane concentration and of the presence of excess toluene, a free-radical scavenger. It was assumed that the SiH2 formed by the initial silane decomposition did not react further with SiH+ No disilane was observed in the products even though it would be expected thermodynamically. Product yields from the pyrolysis, AH2/ASiH4 = 1.85, and the lack of disilane as a product suggest that SiH, further decomposes either unimolecularly to yield Si and H2 or bimolecularly to yield H2 denuded disilenes without first reacting with SiH4. The details of the later stages of silane shock tube pyrolysis are unknown as are the mechanisms behind the formation of silicon dust observed in the experiments and certainly provide a potential source for systematic uncertainties in the experimental rate constants obtained by this technique. For example, SiH, might still react with SiH4 without forming Si2H6 via the chemically activated channel (4), SiH2
+ SiH,
-
Si2H6*2 H2 + HSiSiH,
which becomes competitive with the stabilization channel, (-2), a t high temperatures and low pressures and/or inefficient bath gas colliders such as argon. Silylsilylene, HSiSiH3, might then undergo an isomerization reaction to form disilene or unimolecularly decompose to Si2H2. Experimental Arrhenius parameters for the rate of SiH4 loss were derived in the paper. However, Erwin, Ring, and O’Neal’s later paper4 includes a recalculation of NORLS’s silane shock tube data,, due to an incorrect initial treatment of the temperature dependence of the bath gas collision efficiency. Their recalculated experimental data can be represented by the following Arrhenius expression, log k l (s-l) = 12.78-50.275 kcal mol-’/(ln IORT). We have used this value in our fitting study rather than NORLS’s value, because the original senior authors have reevaluated their data. A relative standard error of 0.1 was assigned to the rate constant and an absolute error of 0.5 kcal mol-I was assigned to the activation energy in the objective function formulation. Two shock tube rate s t ~ d i e sof~ silane ~ , ~ ~decomposition were not included in the objective function, because they were found to be inconsistent with all other experimental rate constant data for silane dissociation, including that for the reverse bimolecular reaction. Both of these shock tube studies were done at higher temperatures, 1400 and 1600 K, than Ring and ONeal’s shock tube studies. It is believed that the mechanism for silane pyrolysis at these elevated temperatures may be more complicated. Experimental data from Jasinski and Chu (JC),13 where the SiH2 insertion rate into H2, SiH,, and Si2H6was measured, were also included in the regression analysis. They obtained time-resolved measurements of the silylene removal rate by the technique of laser resonance absorption flash kinetic spectroscopy. The room temperature rate constant was measured at pressures ranging from I to 100 Torr in an helium bath gas. Different buffer gases were also compared at 5 Torr and were shown to have no significant influence on the rate constant. The pressure range studied fell into the “knee region” of the falloff curve. JC, however, presented an RRKM calculation based on RJC’s transition-state model that fit the pressure dependence of their data well and produced a high-pressure bimolecular reaction rate constant of k-,”(300 K) = 1.9 X 10l2cm3 mol-l s-l, a PI/*value of 45 Torr, and a lowpressure termolecular limiting reaction rate constant of k-,0(300 K,[M]) = 6.0 X 1Olo cm3 mol-’ s-I Torr-’. In the regression procedure, a subset of JC’s original points were used in the objective function. Relative standard errors of 13% and 7% were (33) Tanaka, H.;Koshi, M.; Matsui, H. Bull. Chem. SOC.Jpn. 1987, 60, 3519-3523. (34) Votintsev, V. N.;Zaslonko, 1. S.; Mikheev, V. S.; Smirnov, V. N. Kinet. Catal. USSR 1987, 27, 843-846.
The Journal of Physical Chemistry, Vol. 95, No. 1, 1991 149 n
dlo’
D
’
’
,
1
~
’
4
’
1
’
’
’
’
’
Id
’
’
.
j . - f i J
Id
Fressura (Torr)
Figure 1. Agreement of RRKM model with PW’s static reactor silane pyrolysis experiments (silane bath gas): (-) best fit with fitted collision efficiencies, fitted.3, Fanh= 3.0; (---) best fit with fixed collision efficiencies, fixed.3b. Fsnh = 3.0; fitted.1 model, fitted collision efficiencies, Fan,, = 1.0; Experimental data: (A) 700 K, (0)675 K; (0)= 650 K. (a-)
assigned to the weighting factors of the regression procedure below and above 10 Torr, respectively.
Results and Discussion Fitted 3(s:. Table IIa contains a summary of best fits versus changes in the Fa,,,,factor. In this study, the high-pressure A factor and activation energy, the SiH2 heat of formation, and all bath gas collision efficiencies were optimized. C#J is the resulting best value for the optimization parameter. In all cases the best fit to the silane bath gas collision efficiency was 1.O;i.e., silane is a strong collider with itself. Unfortunately, no two experimental studies in the set employed the same bath gases. PWs and ERO’s static pyrolysis studies employed a silane and propane bath gas, respectively. EROs shock tube study employed an argon bath gas. Jasinski and Chu’s SiH2 + H2 study employed helium, mostly. Jasinski and Chu tried other bath gases but found that the reaction rate constant did not depend upon the bath gas. However, as the authors pointed out, at 5 Torr and 300 K, the reaction rate constant is not fully into the low-pressure termolecular limiting regime, and the dependence of the reaction rate constant on the bath gas was at least partially masked. Also included in Table Ila is a representation of the fit of the numerical RRKM models to each data set. For the two static pyrolysis studies the results from the experimental data set are represented by the observed A factor, activation energy, and pressure dependence, as expressed by the local power law dependence, a t a mean temperature and pressure of each experiment. For the shock tube experiment, the experimental data is represented by the observed A factor and activation energy only, since the pressure dependence under shock tube conditions was not measured and is in any case roughly equal to 1.O. The experimental falloff curve measured by Jasinski and Chu is represented by the “experimentally determined” highpressure limiting rate constant, the value, and the low-pressure limiting termolecular rate constant. The term “experimentally determined” means that those numbers came from a local RRKM calculation that fit JC’s experimental points only. Representation of JC’s experimental results in this manner is more illustrative, because the values for high- and low-pressure limiting reaction rates are more amenable to interpretation than are the mixed falloff curve data (Figure 2). The derived high-pressure Arrhenius parameters and collision efficiencies of the Fan,,= 1 case, called fitted.1, closely agreed with the temperature independent ( A E ) results obtained by RJC.I’ RJC obtained log kI(n= 15.19-58.08 kcal mol-I/(2.3RT) at 1100 K with a critical energy of 54.82 kcal mol-’, while fitted.1 produced log k,(n= 15.13-58.03 kcal mol-Il(2.3RT) at 1100 K with a critical energy of 54.90 kcal mol-’. This agreement is in spite of the fact that NROLS’s shock tube experimental data
150 The Journal of Physical Chemistry, Vol. 95, No. 1 , 1991
Moffat et al.
TABLE II: Fits of RRKM Model to the Experimental Data as a Function of the Fd Factor (a) Case Where the Weak-Collision Collision Efficiencies Are Allowed To Be Fit by the Regression Technique exDt fitted.1 fitted.l.78 litted.3 fitted.4a fitted.4b fitted.4c log high-pressure A factor, 15.13 15.23 15.79 15.69'" 15.94 16.24' log s-I ( T = 1100 K) high-pressure activation energy, 58.03 58.59 59.99 59.97 60.45 60.96 kcal/mol. T = 1100 K critical energy, kcal/mol 55.29 55.88 55.99 54.90 56.16 56.33 4.0 4.0 3.0 4.0 I .o 1.78 Fanh 6.668 5.325 6.90 52.42 8.061 13.45 4 0.562 0.251 1 0.301 0.256 0.248 0.4303 Bc,ars0n(300K) 0.435 0.399 0.554 0.4161 0.4832 0.423 Bc.ptopanc(300 K) 1 .O' 1.O' I .O' 1 .o 1.0' 1 .O' Pc,si1ane(300K) 1 .o 1.o 0.948 0.642 0.7 154 1.o fic.h~lium(~~0 K, 65.74 AHf0298(StH2),kcal/mol 64.88 65.09 65.77 65.49 65.63 Jasinski, Chu (300 K, helium bath gas) k-l"(300), cm3/(mol s) 0.192E+ 13 0.241E+13 0.185E+13 0.178E+ I3 0.177E+13 0.189E+I 3 0.190E+l3 fli2(300 K), Torr 45 73 35 34 33 38.6 37 k-,'(300), cm-'/(mol s Torr) 6.07E+IO 4.23E+10 5.46E+10 7.26E+30 6.97E+lO 7.24E+IO 7.98E+lO Purnell, Walsh, 675 K, 135 torr 0.6838-4 0.606E-4 0.658-4 0.683E-4 0.699E-4 0.7038-4 0.703E-4 k, (s-'1 53.94 54.65 55.77 activation energy 55.25 55.44 55.92 55.85 0.61 0.52 0.54 0.67 pressure dependence 0.5 0.48 0.53 Erwin, Ring, O'Neal, Static, 700 K, 200 Torr k, s-' 1.93E-4 1.90E-4 1.92E-4 3.89E-4 1.90E-4 I ME-^ 1.90E-4 52.91 53.50 54.15 54.42 activation energy 54.38 54.51 54.59 pressure dependence 0.59 0.62 0.56 0.59 0.55 0.5 0.63 Erwin, Ring, O'Neal, Shock, 1100 K, 3200 Torr k. s-' 616 618.9 590.1 604.2 602 603.4 584.7 activation energy 50.76 50.62 50.58 50.28 50.21 50.89 50.60
log high-pressure A factor, log s-I ( T = 1100 K ) high-pressure activation energy, kcal/mol ( T = 1100 K ) critical energy, kcal/mol
(b) Case Where the Weak-Collision Efficiencies Are Fixed at Physically Reasonable Values eXDt fixed.1 fixed.2 fixed.3a lixed.36 fixed.3~ 14.62 14.82 14.78 15.90' 16.72' 60.68
61.97
57.53
55.00
55.54 2 60.75 0.481 1' 0.7179' 0.9230' 0.6360' 65.45
55.72 3 63.07 0.481 I + 0.7 179' 0.9230' 0.6360' 65.53
56.435 3 68.23 0.481 1 0.7179' 0.98' 0.636'
56.91 3 116 0.481 1' 0.7179' 0.9 230 0.6360' 66.46
5.69 4 80.35 0.481 I' 0.7179' 0.9230' 0.6360' 65.44
66.11
'
8.0
20.70 0.163 0.290 1 .O'
0.258 65.71 0.165E+I 3 29 6.678+10 0.7 1 I E-4 56.28 0.31 1.93E-3 54.76 0.42 520.8 51.54
56.75
58.20
'
55.96
fixed.8 14.16
58.1 I
110.2 0.481 1' 0.7179' 0.9230' 0.6360' 65.13
4
BCerpan(300 K), ( A E ) = 1 kcal/mol Pc,propanc(300 K), ( A E ) = 3 kcal/mol /3c,silanc(300 K), ( A E ) = 15 kcal/mol &hclium(300 K), ( A E ) = 2 kcal/mol AHf,029&3iH2),kcal/mol Jasinski, Chu, (300 K, helium bath gas) k-,"(3OO), cm3/(mol s) Pli2(300 K). Torr k_,'(300), cm3/(mol s Torr) Purnell, Walsh, 675 K, 135 Torr k, s-' activation energy pressure dependence Erwin, Ring, ONeal, static 700 K, 300 Torr k, s-l activation energy pressure dependence Erwin, Ring, O'Neal, shock, I100 K, 3200 Torr k, s'' activation energy
59.17
fixed.4 14.48
57.15 1
Fanh
fitted.8 15.17
55.70
a
164.5 0.481 17 0.7179' 0.9230' 0.6360' 65.30
0.192E+13b 0.279E+ I3 0.1978+13 0.1 63E+13 0.218E+13 0.369E+13 O.l43E+l3 o.ioaE+13 30 ao 45 40 9 55 120 18 6.07E+10 3.49E+IO 5.06E+10 6.658+10 5.42E+lO 4.91 E+10 7.878+10 1.28E+I I 0.683~-46 55.25 0.5
0.47E-4 53.85 0.58
0.473 E-4 54.82 0.468
0.484~-4 55.26 0.39
0.48E-4 55.94 0.56
0.42OE-4 55.95 0.69
0.4948-4 55.37 0.31
0.51E-4 55.64
1.938-4 54.38 0.5
0.207E-3 53.47 0.59
0.21 3E-3 54.50 0.49
0.2198-3 54.99 0.33
0.208E-3 55.38 0.52
0.200E-3 55.47 0.65
0.222E-3 55.16 0.25
0.228~-3 55.51 0.14
616 50.28
429.8 50.3 I
529.3 51.63
575.6 52.42
760.8 51.89
868.6 51.45
554 53.01
540.2 54.04
"Daggers indicate variables that were fixed and thus not allowed to be optimized. *E+13 3 XlO" and E-4
was replaced with ERO's shock tube experimental data, and experimental data from ERO's static pyrolysis experiment and JC's silylene insertion study were added to the objective function. However, other RRKM modeling attempts have employed higher values of the high-pressure preexponential factor and activation energy to achieve good agreement with the shock tube data and static silane pyrolysis experiments, combined. For instance, ERO used an RRKM model with high-pressure Arrhenius parameters of log k l ( T ) = 15.49-57.90 kcal mol-'/(2.3RT) at 675 K, which produces log k l ( T ) = 15.74-58.88 kcal mol-'/(2.3RT) at 1100 K. Fitted.1 is consistent with log k , ( T ) = 14.84-56.94 kcal mol-'/(2.3RT) at 656 K. The fact that the predicted high-pressure Arrhenius parameters did not change very much, when new experimental data were added, demonstrates that ERO's static decomposition study and JC's silylene insertion study are in general agreement with the previous static decomposition study of PW and the shock tube study of NORLS. It is also because three
0.18
X104.
additional fitting parameters, the propane and helium bath gas efficiencies and AHf"(SiH2), have been added. Any particular disagreement in the overall rate constant between EROs and PWs predicted absolute rate constants could be taken up by the value of the propane bath gas efficiency. The value of &pmpane(300K), 0.554, by fitted.1 is somewhat lower than expected, since it predicts ( AE) = 1.4 kcal mol-I. The absolute rate constants of NORLS's shock tube data and ERO's shock tube data agree fairly well, too. NORLS quoted experimental Arrhenius parameters of log k , ( T ) = 13.33-52.70 kcal mol-I/(2.3RT) at 1100 K and 4000 Torr produce k l ( 1 1 00 K,4000 Torr) = 722 s-I, while ERO's Arrhenius parameters of log k , ( T ) = 12.78-50.275 kcalm01-~/(2.3RT) at 1100 K and 3200 Torr produce kl(llOO K,3200 Torr) = 616 s-l. Though ERO's and NORLS's shock tube activation energies differ by 2.5 kcal mol-', fitted.1 closely agrees with the activation energy in ERO's shock tube Arrhenius parameters, while RJC significantly underpredicted NORLS's activation energy. RJC also
Arrhenius Parameters for SiHl
G
SiH2 + H2
The Journal of Physical Chemistry, Vol. 95, No. I, 1991 151
€ 8Y 1 & 8Y
I
20.0
40.0
80.0
,
80.0
,
100.0
Pressure (Torr) Figure 2. Agreement of RRKM model with Jasinski and Chu's SiH2 + H2rate constant measurements in a helium bath gas: (-) best fit with K) = 0.948; fitted collision efficiencies, fitted.3, Fan,,= 3.0, Bc,hclium(300 ( - - - ) best fit with fixed collision efficiencies, fixed.3b, Fanh = 3.0, ~,heli,m(300 K) = 0.636; (-) fitted.1, fitted collision efficiencies, Fanh = 1 .O, &hclium(300 K) = 0.554;(a) experimental data. overpredicted the pressure dependence and underpredicted the activation energy under the conditions of PW's static pyrolysis experiment. This is characteristic of overestimating the degree of falloff from the high-pressure limit. The results of the fitted.1 fitting run show the same behavior. PW's predicted activation energy is more than 1 kcal mol-' greater than the experimental value. It should be noted that the predicted activation energy of the numerical model could be made to agree with the measured values in the static pyrolysis studies by increasing the value of the critical energy. However, this leads to a trade-off with the observed pressure dependence in the same pyrolysis studies. Additionally, as Figure 2 shows, i.e., the pressure at which k I T,P) ( = '/2k-im(T), is higher than JC's experimental data would indicate. Also, the value of the low-pressure limiting termolecular reaction rate constant is underpredicted by 50%, even though the fitting procedure has maximized the &helium value to a physically unrealistic 1 .O. The value of the low-pressure limiting termolecular reaction rate constant is proportional to &,hc]ium. Therefore, fitted. 1 is underpredicting the preexponential factor of the strong collision low-pressure limiting termolecular reaction rate constant, which leads to the fitting procedure assigning a &helium value that is too high in order to try to compensate. These two effects combine to make fitted.1 underpredict the sharpness of the "knee" in the JC's experimentally derived bimolecular reaction rate constant versus pressure curve, as shown in Figure 2. Increasing the value of Fanh has the effect of decreasing the degree of falloff, all else staying constant, by decreasing the estimated value of the microscopic rate constant, ka(E+),at each energy value, E+, above the critical energy. Fanh should be significantly different from unity for small molecules, like SiH4, where the number of quanta per oscillator is large. Additionally, a greater contribution from overall rotations that one would expect from a looser transition state has the same effect as Fanhdoes on the preexponential factor of the low-pressure limit. The lowpressure limiting reaction rate constant is directly proportional to the value of Fan,,. Therefore, Fanhwas varied from 1 to 8 in the fitting procedure. The columns of Table IIa are arranged in increasing value of Fanh. It is found that a value for the anharmonicity correction factor of 3.0 minimized 4. Reasonable values of 4 were obtained with Fanh= I .78, the value of the anharmonic correction factor predicted by a simple Morse oscillator model.22 I f the anharmonicity correction is not taken into account, Table IIa demonstrates that the main source of error is that the activation energy and the pressure dependence of the static pyrolysis results and the P'12, k-,"(300 K), k-10(300 K) values from Jasinski's experiment cannot be duplicated. As the value of Fanhis increased, the best fit curves are better able to fit both the activation energy and pressure dependence of the static pyrolysis experiments. Since
400.0 500.0
800.0
700.0 800.0
9W.0
1wO.O
1100.0 1200.0
Temperature (K) Figure 3. High-pressure silane dissociation rate constant versus temperature, plotted as k,"( r)/kl-DArd(r ) versus T, where kImaxd(r) = 1015,55 exp[-59.01 kcal mol-l/RT] is the high-pressure rate constant expression for fitted.3 at 744.2 K. (-) fitted.3 RRKM model with Ea"(llOOK) = 59.99 kcal mol-'; (---) fixed.3b RRKM model with &-(I 100 K ) = 60.68 kcal mol-'; (-) fitted.1 RRKM model with E,"(1 100 K ) = 58.03 kcal mol-'. the low-pressure limiting termolecular reaction rate constant, kWI0( T , [ M ] )is proportional to Fanh, the estimate for kI0( T,[M]) increases as Fanh is increased until it overpredicts the k..,O(T,[M]) value based on the extrapolation from JC's data. If Fanh is increased to too high a value the pressure dependence of the static pyrolysis experiments are underpredicted. The value, representative of a characteristic pressure at which falloff occurs, is too low to fit JC's falloff curve. Also, the activation energies of both the static and shock tube pyrolysis experiments are overpredicted, due to an underprediction of how far the rate constants are into the falloff regime. Fanhvalues of between 1.78 and 4.0 provide the best fits. Table IIa also contains an example of the parameter correlation generically observed in trying to fit the high-pressure SiH4 Arrhenius parameters. Three different fitting runs with Fanh= 4.0 are shown. Two of them do not represent true minimums in the objective function curve. However, even though the high-pressure preexponential factor changed by over a half decade, the value of the objective function hardly changed. The high-pressure activation energy compensated for the change in the high-pressure preexponential factor. This parameter correlation holds for the lower Fan,,values as well, and helps to explain the difference in the high-pressure preexponential factors that were used by RJC and ERO to fit basically the same experiments on the silane dissociation reaction. A range of acceptable high-pressure preexponential factors exists for which an adequate RRKM model for the static and shock tube experimental data can be constructed, as long as the high-pressure activation energy is modified in a compensating manner. Figure 3 displays k,"( T)/kimrmd( 7"), where klmrmd( r ) is the high-pressure Arrhenius expression for the fitted.3 fitting run at 744.2 K, which we have called the best fit. Also, plotted in Figure 3 is the fitted.1 run, which had an 1100 K high-pressure preexponential factor of log A , = 15.13 s-I, and the fixed.3b run which had an 1100 K high-pressure preexponential factor of log A , = 15.90 s-I. Even though the high-pressure activation energies of the two fitting runs, fitted.1 and fixed.3b, differ by 2.65 kcal/mol, the high-pressure rate constants have nearly the same values over the 650-1100 K range, Le., the temperature range of the silane dissociation experimental data. The differences in the rate constant can be compensated by small changes in the estimates for the collision efficiencies. The two RRKM models do create substantial differences in the their predicted k,"(300 K) values, which in turn modifies the AHfo(SiH2) values needed to fit JC's k-,"( 7") value. However, uncertainties in AHfo(SiH,) prevent differentiation in the 1 kcal/mol accuracy range of the high-pressure activation energy of the forward SiH4 dissociation reaction based on JC's extrapolated value for k-,"(300 K).
152 The Journal of Physical Chemistry, Vol. 95, No. 1, 1991 Note that, while the preexponential factor cannot be nailed down accurately by the fitting procedure, AHfo(SiH2)varies over a more limited range. Even though the high-pressure forward activation energy varies by 1 kcal/mol and the high-pressure A factor by half a decade in the Fanh = 4 fitting runs presented in Table Ila, AHfo(SiH2)varies by only 0.14 kcal/mol. There is more variation in AHfo(SiH2)across changes in Fan,,. This is due to the change in the amount of falloff predicted at the experimental points, which in turn alters the prediction of the equilibrium constant. Table Ila demonstrates that, over the expected range of Fan,,.1 .O-4.0, RRKM models, which differ by a decade in their high-pressure A factor, are able to produce reasonable fits to the experimental data. However, over this same range of Fanh, AHfo(SiH2)varied only from 64.88 to 65.77 kcal/mol. Neudorfl and S t r a u s have ~ ~ ~done kinetic studies to determine the high-pressure Arrhenius parameters for decomposition of methylated silanes. The Arrhenius parameters at 675 K for dimethylsilane decomposition to H2 and dimethylsilylene and for methylsilane decomposition to H2 and methylsilylene were log k , (s-l) = 14.30-68.00 kcal mol-'/(2.3RT) and log k, ( S I ) = 14.95-63.20 kcal mol-I/(2.3R7'), respectively. The same type of transition state probably occurs for methyl-substituted silane dissociation as for silane dissociation, so this study may be used to estimate the A factor for silane dissociation. As has already been pointed out by NORLS,2 a continuation in this trend of methyl substitution would put the preexponential factor for silane decomposition around log A = 15.5 at 675 K. The best fit Fanh = 3 RRKM model in Table IIa, fitted.3, has a high-pressure preexponential factor of log A , = 15.48 at 675 K. Fixed &'s. One troubling aspect of the results in Table IIa is the unrealistic collision efficiencies that result from the regression procedure, especially the helium collision efficiency that is used in JC's 300 K SiH2 + H2 reaction rate study. For the fitted.3 RRKM model, &,,(300 K) = 0.301 results in an average energy transferred per collision, ( A E ) ,of 0.43 kcal/mol and in &,,on( I IO0 K) = 0.1 1. 300 K) = 0.435 results in (AE) = 0.814 kcal/mol and in @c,ppane(700 K) = 0.25. While the argon value agrees with normal expectations based on values obtained in other systems,36 the propane ( A E ) vaule appears to be too low. The ( A E ) value predicted for the helium collision efficiency from JC's experiment is clearly too high. However, it only takes a factor of 2 change in the estimate for the low-pressure limiting termolecular insertion reaction rate constant to produce reasonable values of ( L I E ) . Considering the uncertainties in the critical energy, AHfo(SiH2),and the possibility of a more substantial contribution to the low-pressure limiting reaction rate constant from the overall rotational degrees of freedom that would accompany a looser transition-state model, this factor of 2 is readily possible. Therefore, we undertook a series of runs where all collision efficiencies were fixed at physically realistic values. ( AE) was fixed at I , 2, 3, and 15 kcal/mol for argon-, helium-, propane-, and silane-silane, respectively. Table IIb contains the results of these runs, as Fanhis varied from 1 to 8. For most values of Fanh, the preexponential factor which minimized the objective function was around log A," = 14.4 at 1100 K, resulting in an unphysically tight transition state. Low A factors are preferred, when the collision efficiencies were fixed, because the helium collision efficiency could not be increased to compensate for the increased falloff occurring under JC's SiH2 + H2 conditions that is the result of a looser transition state. As the A factor is increased, &(E+) increases, leading to greater falloff. However, as is shown for the Fan), = 3 case, the objective function when the preexponential factor was fixed at high values, Le., log A," = 15.90 at 1100 K, which produces log A," = 15.56 at 675 K, was only slightly higher than the minimum value for the objective function (68.23 versus 63.07). This is another example of the highly correlated nature of the data. Even preexponential factors up to log A," = 16.7 at 1100 K did a credible job of fitting the data for the Fanh = 3.0 case. Changes in the preexponential factor were (35) Neudorfl, P. S.; Strausz, 0. P. J . Phys. Chem. 1978,82, 241-242. (36)Tardy, D. C.;Rabinovitch, B. S. Chem. Reo. 1977, 77, 369-408.
Moffat et al.
t
1 400.0 600.0 800.0 700.0 600.0 000.0 1000.0 1100.0 1200.0
Temperature (K) Figure 4. High-pressurereverse silane dissociation rate constant versus temperature: (-) fitted.3 RRKM model with AHID(SiH2)= 65.49 kcal/mol; ( - - - ) fixed.3b RRKM model with AHro(SiH2) = 66.1 1 kcal/mol; fitted.] RRKM model with AHfP(SiH2)= 64.88 kcal/mol. (e-)
partly compensated by changes in the activation energy and silylene heat of formation. One of the main sources of additional error that occurred for the fixed 8, case that did not occur for the fitted P, cases was in trying to accommodate the absolute rate constants of both PW's and ERO's static pyrolysis study. PW's absolute rate constants were underpredicted by 50% in the fixed.3b fitting run case. The agreement of fixed.3b with PW's data is also plotted in Figure 1. The disagreement is entirely due to the larger value of &pro ,,(300 K) = 0.718, which produces &propne(700K) = O S $ employed in fixed.3b than is employed in the fitted.3 case, where Pc, ,,(300 K) = 0.435 (Pc,propne(700 K) = 0.25). Note that N O R B a l s o did some careful static silane dissociation measurements. They concluded, though they used an extrapolation of PW's rate constant to a pressure outside of PW's limits, that spurious secondary reactions may have increased PW's rate constants by a factor of 2. Tests with propylene used as a silylene scavenger were employed in this endeavor. If PW's rate constants were influenced by secondary reactions, then they would agree more closely with the absolute rate constants of fixed.3b, and thus, P W s and E R O s static pyrolysis results could be fit with a larger propanesilane collision efficiency that is more characteristic of the number of atoms in the propane molecule. It seems that a factor of 2 is the limit by which these systematic uncertainties in the static experiments can be narrowed down to, given the uncertainties in the static silane pyrolysis kinetic mechanism. The silylene heat of formation predicted by the fixed collision efficiency cases was, in general, slightly higher than the fitted collision efficiency cases. For example, fixed.3b had AHfo(SiH2) = 66.1 1 kcal/mol, but fitted.3 had AHfo(SiH2)= 65.49 kcal/mol, even though both models used the same Fan,, value and almost the same high-pressure preexponential factor. Because the fixed and Pcargon? collision efficiency cases used larger values of the degree of falloff under static and shock tube conditions was less than the fitted collision efficiency case. This allowed the fixed collision cases to use a larger activation energy for the same A factor. In turn, this caused the extrapolated k 1 " ( 3 0 0 K) values in the fixed collision case to be less and thus AHfo(SiH2)to be greater. Figure 3 demonstrates these points by plotting klm(7') for the fitted.3 and fixed.3b cases. Figure 4 plots k-,"(7') versus T . The AHt(SiH2) value used in the construction of k-,"(7') was that predicted by the optimization scheme. Evaluation of AHf0(SI'H2).Variations in the structure of the transition state can alter the value of the activation energy's temperature dependence. In particular, a fixed, tight transition-state model, such as the one used here, is known to lead to higher curvature in an Arrhenius plot than experimental results would indi~ate,~'while a loose transition-state model whose (37) Wardlaw. D.M.; Marcus, R.A. J . Phys. Chem. 1986,90,5383-5393.
Arrhenius Parameters for SiH4 2 SiH2
+ H,
structure changes with temperture produces a straighter Arrhenius plot. This effect is important in determining the uncertainty in AH?(SiH2). The static silane dissociation experiments are conducted at 675 K, while Jasinski's recombination experiments were conducted at 300 K. In order to find the equilibrium constant at 300 K, on which AHfo(SiH2)is based, it is necessary to extrapolate the forward rate constant from 675 to 300 K. For example, the fitted.3 RRKM model yielded a high-pressure rate constant of k,'(r) = 0.4319 X IO9T2.O8' exp[-55.88 kcal mol-'/RT]. At 675 K this implies that k,"(T) = 0.280 X 10l6 exp[-58.68 kcal mol-'/RT]. The assumption that the activation energy does not change with temperature and is equal to the value at 675 K would roduce a rate constant at 300 K, kImnxd(300K) = 0.496 X 10-2ps-1, that is a factor of 2.5 lower than the rate constant at 300 K, kl"(300 K) = 0.124 X s-l produced by the fitted.3 RRKM model. This would demand an upward revision of 0.546 kcal mol-' in the calculated AG10(300K) and, therefore, an upward revision of AH?(SiH2) to 66.04 kcal mol-' from 65.46 kcal mol-'. If the calculation of AHfo(SiH2)were based on the equilibrium constant at 675 K and the assumption that the reverse rate constant does not change with temperature from 300 to 675 K, then another estimate for AH?(SiH,) may be obtained. The assumption that k-,"(300 K) = k-'"(675 K) is quite possible considering that the predicted critical energy in the reverse direction is 0.47 kcal mol-' and variational transition-state models for reactions on a type 11 potential energy surface (Le., with no activation energy in the reverse direction) are known to produce flatter reverse rate constant versus temperatures curves than fixed transition-state models d0.'4-38 For comparison purposes, Figure 4 plots k-,"( 7") versus T produced by our RRKM calculations for three different cases, with widely varying assumptions concerning the high-pressure preexponential factor. The critical energy for the reverse reaction was calculated from the computed AHlO(OK) and the critical energy for the forward dissociation reaction via the formula: AH,O(O K) = Elmt- ElQ". Even though E-IQit is small, K,"( T ) increases markedly with temperature due to the curvature in the Arrhenius parameters for the dissociation reaction created by the RRKM model. The assumption of a temperature-independent kIm( r ) leads to a revision of K1"(675 K) from 3.75 X 10l2to 1.78 X 10l2 cm3 mol-' s-I. This would imply a downward revision of 1.00 kcal mol-' to AG10(300K) and a downward revision of AH?(SiH,) from 65.46 to 64.46 kcal mol-'. Uncertainties in the temperature extrapolation of the forward and reverse rate constants therefore create an uncertainty in AHy(SiH2) of roughly 1 kcal mol-'. Figure 4 also demonstrates that the predicted activation energy in the reverse direction at 300 K is nearly 0 kcal/mol. Actually, fitted.3 predicts 0.30 kcal/mol at 300 K. This result agrees with recent experimental measurements of the activation energy for the SiH2 D2 reaction done by Baggot et al.,39 who measured k5"(300 K) = ( I . 13 f 0. IO) X 10I2cm3 mol-' s-' and an activation energy of 0.0 f 0.3 kcal/mol. In contrast to the undeuterated reactions, measurement of the SiHz loss rate constant for reaction 5 is expected to produce the high-pressure reaction rate constant, due to the presence of alternate channels for dissociation for SiH2D2*.
+
5
SiH2D2z= SiH,
+ D,
The predicted silylene heat of formation agrees well with recent direct experimental measurements and a b initio molecular structure calculations. Berkowitz et a1.@obtained AH?298(SiH2) = 65.2 f 0.7 kcal/mol as the lower of two possible results from ionization and appearance potential measurements. Curtis and Pople4' obtained 63.5 kcal/mol, while Gordon et al.42obtained (38) Wagner, A. F.; Wardlaw, D.M. J. Phys. Chem. 1988,92,2462-2471. (39) Baggott, J. E.; Frey, H. M.; King, K. D.; Lightfoot, P. D.; Walsh, R.; Watts, I. M. J . Phys. Chem. 1988, 92, 4025-4027. (40) Berkowitz, J.; Green, J. P.; Cho, H.; Ruscic, B. J . Chem. Phys. 1987, 86, 1235-1248. (41) Curtis, L. A.; Pople, J. A. Chem. Phys. Lett. 1988, 144, 38-42. (42) Gordon, M. S.;Gano, D. R.; Binkley, J. S.;Frisch, M. J. J. Am. Chem. SOC.1986, 108. 2191-2195.
The Journal of Physical Chemistry, Vol. 95, No. I, 1991 153 TABLE 111: Fd Parameters for the Fitted Collision Efficiency Run, fitted.3, Where F ., = 3.P
[
k"(T) = 0.825 X lO9F.Ooexp -55.98;;l/mol
1,
s-'
k&gm(T) = 0.329 X 1028T2.456 exp K( r ) = 4967 T-0.4445exp F,,,(r)
[
= 1.206 exp 8&]
+ exp[
F]
- 0.206 exp[
&]
Strong-collision third-body coefficients (use to adjust ko,(r) for bath gas; T dependence is weak): 300 K: argon = 1.0; helium = 1.30; propane = 1.71; silane = 1.41 Weak bath gas collision efficiencies (use Troe's formula for the temperature dependence): 300 K: argon = 0.301; helium = 0.948; propane = 0.435; silane = 0.99 1.07 (300 K), 1.12 (500 K), 1.17 (700 K), FE(T): 1.23 (900 K), 1.29 (1100 K) for example: koa,,,( T ) =
0.113
X
103'T3.s93exp
Range of applicability: 300
[
-58.79F;l/mol
< T < 1 100 K.
TABLE I V Other Physical Parameters Used in RRKM Model SiH, Ar He 0, .A 4.084 3.54 2.55 5.12 elk, K 207.6 93.3 10.22 231 I,, (reactn coord) 0.946 X 0.4734 X g cm2 IY
4
0.946 X
0.946
X
1.272 X 1.602 X
g cm2 g cm2
Lt = 6
65.3 kcal/mol from theoretical structure calculations. Gordon et al. predict a transition state that lies 1.7 kcal/mol above SiH2 + Hz, a long-range minimum of 3.1 kcal/mol from the separated SiH2 H2 atoms, and an overall endothermicity of 55.2 kcal/mol. Our best fit calculations, fitted.3, predicts a transition state 0.47 kcal/mol above the separated atoms and an overall endothermicity of 55.34 kcal/mol. It is encouraging that kinetic studies are in such good agreement with theoretical structure calculations. Closer agreement in the predicted energy barrier from the RRKM calculations cannot be expected considering the simplicity of the tight transition-state model used. Curtis and Pople have reviewed the theoretical studies and direct measurements of AHfo(SiH2) from ionization and appearance potentials@*43 and proton affinity studies.44 With the exception of the Shin and Beauchamp proton affinity they can be made to be consistent with a AH?(SiH2) value in the range of 63.9-65.7 kcal/mol. Frey, Walsh, and WattsI2 have also recently examined the thermal kinetic data on the silane dissociation SiH2 H2 reaction as well as the disilane dissociation/SiH4 + SiH2 reaction and obtained Mf0(SiH2)= 65.3 & 1.5 kcal/mol. We have obtained essentially the same AHfo(SiH2)in this study, 65.46 f 1.0 kcal/mol, and have demonstrated the uncertainties involved in extrapolating the high-pressure rate constant obtained in the SiH4 pyrolysis studies to the room temperature, where Jasinski and Chu and I n o ~ e ' s ~ ~
+
+
(43) Boo, B. H.; Armentrout, P. B. J . Am. Chem. SOC.1987, 109, 3549-3559. (44) Shin, S.K.; Beauchamp, J. L. J . Phys. Chem. 1986.90, 1507-1509. (45) Inoue, G.; Suzuki, M. Chem. Phys. Leu. 1985, 122, 361-364.
154 The Journal of Physical Chemistry, Vol. 95, No. I. 1991
Moffat et al.
TABLE V Thermodynamic Variables Used in the Calculation of the Equilibrium Constant AHr'(298 S(298 K )
K) '
cpocn QI0 (72
03 04
QS
"C','(T)/R =
0,
SiH4 8.2 48.19
SiH2
H2
49.55
1.5845 1.2838-2 -1.8 I6E-6 -4.5 13E-9 2.08 I E-1 2
4.1132 -2.5038-3 1.28 1 E-5 -1.2528-8 4.01583-32
0.0
Si2H6
HSiSiH,
31.21
19.1 65.56
74.06 67.36
3.4604 3.101 E-4 -8.3 568-1 8.630E-IO -2.2678-13
1.337 I 3.6928-2 -3.8878-5 2.5568-8 -7.5898-1 2
2.9402 2.1448-2 -2.2408-5 1.5398-8 -4.7838-12
+ a 2 T + a 3 p + a4T3+ asp.
recombination rate studies were carried out. The value of AH?(SiH,) obtained from kinetic studies of SiH4 pyrolysis and SiH, + H2 is in excellent agreement with both theory and nonkinctic dcterminations. F,,,, Parameters. The results of the "best fit" calculations, fittcd.3, arc presented in Table Ill in terms of Troe's Fccnt factorization f o r ~ n a t . ' ~This . ~ ~ allows k l (T,P) and kI(T,P) to be calculated with a great deal of accuracy over extended T and P ranges. Carc must be taken in extending the extrapolation to temperatures far beyond those used in its construction. The method is based on constructing a correction factor, F( T,x), to KI,LH( T , f ) ,the reaction rate constant obtained by assuming a Lindeman-Hinshelwood mechanism. kI.LH(T?P) - - x (10) k,"(T) 1 +x where x=
kl0(T)[M1
kl"(T) Then, thc correction factor is applied to k,,,,(T,P) to yield k I ( T.P). k , ( T , P ) = k,,,,(T,P) F ( T , x ) (1 1) Though loosely based on the results from Kassel integrals, F(x,T) is essentially empirically determined. Following TroeI6 log F(x,T) = Fbroad(X) log F c e n t ( T ) (12) Fbroad(X) =
log x - 0.12 0.85 - 0.67 log Fcent )2]-1 log x - 0.12 0.65 - 1.87 log Fccnt )2]4
log x
> 0.12 (13)
log x
e 0.12
In the current implementation, k,"( T),K( T), and kIo,( T), where k l 0 , , ( T ) is the low-pressure limiting rate constant obtained by assuming strong collisions, were fit to the predictions of the RRKM model and thermodynamics, using an extended Arrhenius dependence. A nonlinear least-squares algorithm was employed to regress on 15 data points equally spaced in temperature to yield the parameters in the extended Arrhenius rate constant expressions. Then, the parameters in the FCcn,( T ) expression (a, b, c) were obtained in the same manner from a nonlinear least-squares regression on RRKM predictions of k,(T,x=l) at equally spaced temperature intervals. k,O( T ) is obtained from kIo,( T ) via klo(T ) = p,( T)kIo,( T). The temperature dependence of p,( T ) is given by eq 3. Usually in Troe's factorization approach, kIo(T ) is fit to an extended Arrhenius expression. However, since calculations for scvcral bath gases are used here, it would have been necessary to include kIo(T ) expressions for each bath gas, since the temperature dependence of p,( T ) can vary tremendously between strong and weak colliders. Instead, a single correction factor is supplicd in Table IV to correct the strong collision rate constant calculated assuming an argon bath gas for the actual bath gas. (46) Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1974, 78, 478-488.
This approach neglects any difference in the temperature dependence of the Lennard-Jones collision rate constant between argon and the actual bath gas. This difference is small. For example, this strong collision coefficient for silane at 300 K is 1.41 and at 1000 K is 1.30. This approach has the advantage of separating out the weak collisional effects from other contributions to kIo(7').
Summary The nonlinear regression technique used in this paper provides a powerful tool in analyzing experimental kinetic rate constants on a reaction or system of reactions to determine key rate constant parameters. In this case, we have shown that, by assimilating experimental rate constant measurements on both the forward and reverse SiH4 dissociation reaction, employing all known thermochemical information on the reaction, and regressing on two key parameters of the transition-state structure and the weak collision energy-transfer parameters, a unified picture of the SiH4 dissociation reaction can be constructed. It is shown that the experimental data on the forward dissociation reaction, both static and shock tube, and on the reverse association reaction are consistent with one another. The predictive power of this technique has been shown by using it to regress on the silylene heat of formation. The pitfalls of this method lie in the parameter correlation problems encountered, we believe, in most unimolecular thermal kinetic rate constant experiments and in the sensitivity of the weak collision efficiencies to deficiencies of the transition-state model employed here. Unless the rate constants are very near the high-pressure limit, the high-pressure activation energy cannot be sufficiently determined, because changes in the high-pressure preexponential factor and weak collision efficiency work in a compensating manner. This situation can lead to aphysical transition states turning out to create the lowest value of the objective function, even though reasonable transition states only yield slightly greater values of the objective function. No systematic study of weak collision efficiencies have been done on the silane dissociation reaction. This has compounded the problem encountered in the regression procedure. Because the temperature range covered by the experiments is so large and a simple fixed transition-state model has been used, uncertainties in the temperature extrapolation of the high-pressure rate constants are large. This uncertainty feeds directly into the prediction of the weak collision efficiency of a bath gas used in a particular experiment and is compounded when that bath gas is not used in other experiments. Also, small changes in the predicted high-pressure rate constant create drastic changes in the predicted ( M )values. This is the reason for the abnormally high prediction for the helium bath gas efficiency in JC's experiment. It would be interesting to see if a variational transition-state formulation of the highpressure Arrhenius parameters would result in a more accurate determination of the high-pressure Arrhenius parameters, AHfo(SiH2),and weak collision efficiencies. Acknowledgment. We are grateful to Karl Rcenigk for his work in formulating the optimization procedure. This work was supported in part by NSF DMR 87 04355 and by the Minnesota Supercomputer Institute. Appendix The physical parameters used in the RRKM calculations are given in Tables IV and V.
