5732
J . Phys. Chem. 1987, 91, 5732-5739
nearest the transition region of the falloff. This region lies in the 100-1000-Torr range. It is also interesting to note that although a better fit of the high-pressure data is obtained per data point than all other regions, confidence intervals from regression in that region are not necessarily more definite. Similar results were obtained by taking a total of 10 data points within each pressure range. Although uncertainties in the data and the limited size of the regressed samples could reduce the conclusiveness of this test, the results are entirely consistent with the previous analysis of the covariance matrix determinant.
Conclusions A general computational approach to analysis of unimolecular kinetic data has been demonstrated. The approach utilizes RRKM theory and an efficient regression technique in determining limiting high-pressure Arrhenius parameters from data in the falloff. Much of the arbitrariness involved in prediction of Arrhenius parameters is eliminated in this procedure. Considering its advantages and
the increasing availability of high-speed computers, the method should find preference in future RRKM analysis of unimolecular kinetics. Arrhenius parameters for methyl isocyanide isomerization have been reevaluated and agree well with previously published values. Resulting RRKM predictions are shown to agree well with experimental data. The assumption of exponential distribution of collisional energy transfer appears to allow satisfactory prediction of experimental data in this application. The temperature dependence of average energy transferred in collisions appears to be weak, if present at all, although inaccuracies in the experimental data preclude distinguishing its value. With the optimal experimental design technique presented here, it is demonstrated that more accurate parameter estimates should come from regression of data defining the curvature of the knee region of the falloff rather than data at the high- or low-pressure limits.
Acknowledgment. This work was partially supported by NSF DMR 83 07924.
Rice-Ramsperger-Kassel-Marcus Theoretical Predlctlon of High-pressure Arrhenius Parameters by Nonlinear Regression: Appllcation to Silane and Disllane Decomposition Karl F. Roenigk, Klavs F. Jensen, and Robert W. Carr* Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455 (Received: November 4, 1986; I n Final Form: February 25, 1987)
Arrhenius parameters are estimated for silane and disilane thermal decomposition reactions by direct regression of RRKM predictions on published static and shock-tube data. For silane decomposition, we find E , = 57.4-61.1 kcal/mol and log A , = 14.9-16.3, while for disilane we find E , = 51.1-52.5 kcal/mol and log A , = 15.2-16.2. The lower limiting values correspond to inclusion of negative temperature dependence in the collision efficiency, while the higher values correspond to inclusion of weak or negligible temperature dependence. The Arrhenius parameters for both silane and disilane decomposition differ substantially from previously published values. For silane, we predict preexponentials approximatelyan order of magnitude greater than the previous values for the same activation energy. For disilane, we find A , is roughly an order of magnitude higher than the literature values and E , is greater by more than 2 kcal/mol. Falloff curves for both silane and disilane decomposition are given. Implications of these results for the activation energy of SiH, insertion into H2 and SiH4 and for AHfo(SiH,) are discussed.
Introduction The chemistry of silanes, their decomposition, insertion, and isomerization reactions, is receiving increasing interest.'-4 This is in part due to practical motivations coming from the field of microelectronics m a n u f a ~ t u r e , ~where - ' ~ such reactions play im(1) White. R. T.: Esuino-Rios. R. L.: Ropers. D. S.:, Rinn. M. A,: O"ea1. H. E.'Int. J . 'Chem.'Kiiet. 1985; 17, 1029z1065. (2) Erwin, J. W.; Ring, M. A.; O'Neal, H. E. In?. J . Chem. Kine?. 1985, -17 1067-10x3 -----(3) Gordon, M. S.; Gano, D. R.; Binkley, J. S.; Frisch, M. J. J . Am. Chem. SOC.1986, 108, 2191-2195. (4) Gordon, M. S.; Truong, T. N.; Bonderson, E. K.J . Am. Chem. SOC. 1986, 108, 1421-1427. ( 5 ) Coltrin, M. E.; Kee, R. J.; Miller, J. A. J. Electrochem. SOC.1984, 131, 425-434. (6) Scott, B. A.; Olbricht, W. L.; Meyerson, B. A,: Reimer, J. A,; Wolford, D.J. J . Vac. Sci. Tech. A 1984, 2, 450-456. (7) Scott, B. A.; Plecenik, R. M.; Simonyi, E. E. Appl. Phys. Lett. 1981, 39,73-75. (8) Scott, B. A. Amorphous Hydrogenated Silicon; Pankove, J. I., Ed.; Academic: New York; 1983; Chapter 7, p 123. (9) Meyerson, B. S.; Scott, E.A.; Wolford, D. J. J. Appl. Phys. 1983, 54, 146 1-1465. (IO) Meyerson, B. S.; Gannin, E.; Smith, D. A.; Nguyen, T. N. J . Electrochem. SOC.1986, 133, 1232-1235. -I
.
0022-3654 I87 I209 1-5732SO 1S O I O
portant roles in the chemical vapor deposition of pure or doped films of silicon. Such film growth processes are typically carried out at conditions where the dominant decomposition reactions are in their falloff regime. Accurate knowledge of the pressure dependence of these unimolecular reactions is therefore important to process characterization and d e ~ i g n . ~ ~ ' ~ Recent discussions on decomposition reactions of silanes center around the effects of surfaces in static experiments',"J* and the energetics of the reverse silylene insertion r e a ~ t i o n s . ~ , ~ J ~ - ' ~ Considerable insight into insertion reactions has been introduced through the recent works of Jasinski13and Inoue,14as well as from theoretical calculations of G ~ r d o n .However, ~~~ the details of surface effects are yet quite uncertain and possibly very complicated as pointed out by White et al.' Knowledge of the character of adsorption, surface species, and surface reactions awaits more careful experiments, although there are some proposed mechanisms" based on lower pressure (< 100 Torr) flow reactor ( 1 1 ) ONeal, H. E.; Ring, M. A. Chem. Phys. Lett. 1984,107,442-449.
(12) Purnell, J. H.; Walsh, R. Chem. Phys. Lett. 1984, 110, 330-334. (13) Jasinski, J. M. J. Phys. Chem. 1986, 90, 555-557. (14) Inoue. G.: Suzuki. M. Chem. Phvs. Lett. 1985. 122. 361-364. ( l S j Frey, H. M.;Walsh, R.; Watts, I. k.J. Chem. Soc., dhem. Commun. 1986. 1189-1191.
0 1987 American Chemical Societv
R R K M Prediction of Arrhenius Parameters studies.16 In any case, important to the discussions in general are the high-pressure unimolecular rate constants, which in many instances are employed in the estimation and discrimination of reaction energetics and pathway^.','^,'^,'^-^' Estimates of these Arrhenius parameters are based on extrapolation into the highpressure limit while obtaining agreement of RRKM predictions with data from static and shock-tube experiments. A careful analysis of the experimental data in the use of RRKM theory is therefore critical. A thorough RRKM analysis may compensate for the lack of kinetic data, which is the unfortunate situation for reactions of interest here. In this work, we employ a direct RRKM regression technique in the analysis of the available silane and disilane decomposition data. In a companion paper?2 we have successfully used nonlinear regression of RRKM theory on data for the well-known methyl isocyanide isomerization reaction.23 Included in the analysis is an approximate treatment of the collision efficiency and its general temperature dependence. For the case of methyl isocyanide, it was revealed that Arrhenius parameters found by regression of RRKM predictions were sensitive to data selected in limited experimental regions.22 To our knowledge, present Arrhenius parameters for silane and disilane decomposition are based primarily on shock-tube data, with demonstration of approximate, although perhaps adequate, agreement with lower pressure static experiments. This may give misleading results, since we have previously shown the sensitivity of RRKM predictions to both the choice and range of experimental data from which they are drawn. With this in mind, it was the objective of this work to determine Arrhenius parameters that best represent both the available static and shock-tube data for silane and disilane decomposition. The general procedure employed here for estimation of the Arrhenius parameters is discussed at length in a companion paper.22 Here, we present only those parts of the application that include collisional processes and data analysis.
Collisional Processes We assume here that the low-pressure rate constant may be written as a product of a strong collision rate constant and a collision efficiency. The collision efficiency summarizes all weak collision effects. The collision efficiency Pc is related to the average energy transferred in all up and down transitions ( A E ) . The assumption of an exponential collisional energy transfer distribution allows the analytic formulation
Expression 1 has been shown to lead to results similar to those in cases when the transfer distribution deviates significantly from exponential form, given that the average remains ( AE).24 The term FEand further considerations on the validity of eq 1 are dealt with in the companion paper.22 The temperature dependence of (A,??) is a current area of much interest and uncertainty.2e27 Most recent direct measurements
(16) Robertson, R.; Hills, D.; Gallagher, A. Chem. Phys. Lerr. 1984, 103, 397-404. (17) Robertson, R.; Gallagher, A. J. Chem. Phys. 1986,85, 3623-3630. (18) Bowrey, M.;Purnell, J. H. Proc. R. SOC.London, A 1971, 321, 341-359. (19) John, P.; Purnell, J. H. J. Chem. SOC.Faraday Trans. 1 1973, 69, 1455-1461. (20) Dzarnoski, J.; Rickborn, S. F.; ONeal, H. E.; Ring, M. A. Organometallics 1982, 1 , 1217-1220. (21) Olbrich, G.; Potzinger, P.; Reimann, B.; Walsh, R. Organometallics 1984, 3, 1267-1272. (22) Roenigk, K. F.; Jensen, K. F.; Carr, R. W. J. Phys. Chem., accom-
panying paper in this issue. (23) Schneider, F. W.; Rabinovitch, B. S.J. Am. Chem. SOC.1962, 84, 4215-4230. Fletcher, F. J.; Rabinovitch, B. S.;Watkins, K. W.; Locker, D. J. J. Phys. Chem. 1966, 70, 2823-2833. (24) Troe, J. J. Chem. Phys. 1977, 66, 4745-4757. (25) Forst, W. J. Phys. Chem. 1986, 90, 456-461.
The Journal of Physical Chemistry, Vol. 91, No. 22, 1987 5733 of collisional energy transfer suggest a very weak negative dependence, if a n ~ ?and ~ *possibly ~ ~ a slight positive dependence for some mixtures with noble gases or triatomics.26 For example, direct measurements were made of collisional loss of energy from highly excited toluene in a variety of bath gases at temperatures from 300 to 1000 K.26 For the monatomic and diatomic bath gases, the gFneral observation was that temperature dependence was small, with ( A E ) a 7" and n = 0.0 f 0.2. For the larger bath gases, a temperature coefficient of n = -0.3 f 0.3 was found. The observed temperature coefficient for argon was approximately 0, while for methane a coefficient of n = -0.1 was reported. Similar weak temperature dependence was observed in experimental studies of collisional energy loss from cycloheptatriene, azulene, and CS2 from 300 to 1000 K.27 With the assumption of ( A E ) i= F ,we may rewrite eq 1 in a more convenient form: -=
1
Pco
Pc
-
6
"(
T & '
(2)
1 - 6 F "
In this equation, Pc0is a collision efficiency at some reference temperature To.This expression will be employed for estimation of the rate of the collisional deactivation as required in the RRKM treatment. Here, this rate is taken as the product of the LennardJones collision frequency and the collision efficiency, 0,. With this approach, we require only an estimate of the collision efficiency at the reference temperature, To,and then extrapolation to other temperatures is achieved through eq 2. This allows some exploration of the temperature dependence in (A,??) through the exponent n. Although error inherent in expression 1 may obscure otherwise weak temperature dependence in ( AE),this approach does allow some approximation of the effect of temperature dependence in collisional processes on resulting RRKM predictions. Regression Procedure The objective function of concern here is the weighted leastsquares error function between RRKM predictions ki and Ei and experimental data kei and Eei:
4=
E[ i
(yy+ (
(3)
The terms ukeiand uEeirefer to the standard errors in the experimental rate constants and activation energies, respectively. The local prediction of the activation energy comes from appropriate derivatives of the RRKM predictions of ki, with respect to temperature. The above objective function may include data over a wide range of temperatures and pressures, which would allow more accurate estimates of the high-pressure Arrhenius parameters. Included in the RRKM calculations are assumptions on the transition-state geometry and frequency assignment. Any difficulty in predicting data over a wide range of conditions may be some indication of need for improved treatment of the transition state. However, predictions are known to be quite insensitive to these variations, and the origin of the discrepancies may lie either in the overall interpretation of the kinetic data or in the inaccuracy of the molecular parameters within the calculations. The objective function in eq 3 may now be minimized with an appropriate algorithm, an RRKM model, and adequate experimental data. Here we perform regression on the magnitudes of the rate constants and their activation energies at the mean experimental conditions for which they are reported. The objective function is minimized for both silane and disilane decomposition cases with respect to the high-pressure Arrhenius parameters A , and E,, as well as the collision efficiencies at 300 K for the respective experimental collision pairs. These best fit parameters were then evaluated for varying values of the temperature exponent (26) Heyman, M.; Hippler, H.; Troe, J. J. Chem. Phys. 1984, 80, 1853-1 860. (27) Hippler, H. Eer. Bunsen-Ges. Phys. Chem. 1985, 89, 303-309.
5734 The Journal of Physical Chemistry, Vol. 91, No. 22, 1987
Roenigk et al.
1637A
1.637A
\
\
1516A
Figure 1. Transition state in silane decomposition. Figure 2. Transition state for 1,2 H shift disilane decomposition.
TABLE I: Frequencies of Silane, Silene, and the Corresponding Transition State
[SiH41+, assignt VI
"2
u3 u4 US "6 Vl
Ua u9
SiH40 2187 978 978 2183 2183 2183 910 910 910
[SiH4]+ reaction coord 978 + (1008-978)nHHci( (978 X 2183)'/*fd ~4 + (2032 - 2183)n~H vg + (2022 - 2183)nHH (978 X 2183)'/*f UT + [(u~vHH)"~ - v7]nHHe uaf
vef
~ H H =
OSb 0
993 839 2108 2103 839 1328 523 523
a Silane frequencies from JANAF Thermochemical Tables, ref 40. *Resulting frequencies for transition state with H-H bond order = 0.5, consistent with A , predicted here as presented in Table 111 for n(static/shock) = 0/0 and with molecular parameters presented in Table IV. 'nHH is the bond order between the departing H2 group. dfis a factor, fitted to obtain correct between A , and entropy of activation. e ~ H His estimated by using the correlation uHH = uH2nHH0'52; uH2 = 4406 cm-I. /Vibrational frequencies of SiH2 are 2032, 1008, and 2022 cm-', as taken from ref 41.
n from eq 2. The predictions are summarized and discussed under Results. Application to SiH4 Decomposition Silane decomposition has repeatedly been shown to most probably involve H2elimination, and there have been no observed effects of radical reaction^.^*^^"' Recent experimental and theoretical studies have further confirmed that silane decomposition reaction paths involving H and SiH3 radicals are energetically u n f a ~ o r a b l e . ~ ~ ~ ~ ~ ~ - ~ ~ Calculations here are based on the reaction SiH,
-
SiH,
+ H,
(4)
In this work we assume a transition state which allows for partial formation of an H-H bond between the departing H2 fragment. The transition state may be conceived as in Figure 1. With the assumption of a bond order between the departing H2 group, bond-length correlations were used to fix the geometry of the ~ o m p l e x . ~In~the , ~ ~SiH2 group, the H-Si-H angle was assumed to be the same as that observed in silylene. Resulting Arrhenius parameters were shown to be relatively insensitive to choice of H-H bond order, nHH, near nHH= 0.5, and this value was assumed for all subsequent calculations. Free rotation is not permitted in the transition state. Frequency assignments are based in part on the use of bondorder correlations. Equations for the corresponding frequencies are given in Table I, as a function of a given nHH. Four remaining (28) Purnell, J. H.; Walsh, R. Proc. R . SOC.London, A 1966, 293, 543-561. (29) Newman, C. G.; O'Neal, H. E.; Ring, M. A,; Leska, F.; Shipley, N. Int. J . Chem. Kinet. 1979, 1 1 , 1167-1 182. (30) Bell, T. N.; Perkins, K. A,; Perkins, P. G.J . Phys. Chem. 1984, 88, 116-1 18. (31) Shin, S. K.; Beauchamp, J. L. J . Phys. Chem. 1986, 90, 1507-1509. (32) Ho, P.; Coltrin, M. E.; Binkley, J. S.; Melius, C. F. J . Phys. Chem. 1985,89, 4647-4654. (33) Robinson, P. J.; Holbrook, K. A. Unimolecular Reactions; Wiley: New York, 1972. (34) Johnston, H. S. Gas Phase Reaction Rate Theorv: Ronald: New York, 1966; Chapter 4.
frequencies, bends and twists, are arbitrarily fitted so as to predict the high-pressure Arrhenius preexponential. With these frequency assignments, the resulting fitted frequencies were of reasonable value considering similar bending and twisting modes of vibration in silane and disilane. Experimental data on silane decomposition, although limited, were drawn from two sources. Purnell and Walsh2*carried out static decomposition studies at conditions of 35-230 Torr and 650-700 K. They report the unimolecular rate constant kpW, which is dependent on pressure: 1 d[SiH41 kPW= = [SIH,] dt 1015.18i0.32 @/RT)1/2 exp[(-55.9 f l ) / R T ] (5a) log kpW = (13.93 f 0.32) - [(55.9 f 1)/2.303RT], at 133 Torr and 675 K (5b) The second set of data is from the shock-tube studies of Newman et al.,29at conditions of 4000 Torr and 1035-1 184 K. The rate constant reported for their conditions is log k,o = (13.33 f 0.56) - [(52.7 f 2.8)/2.303RT] (6) Error intervals in these constants correspond to 95% confidence (2a). Back-calculation of the reported confidence intervals leads to standard errors of 7% in the static rate constant and 13% in the shock rate constant.
Application to SizHs Decomposition The 1,2 H shift mechanism appears to be the most energetically favorable decomposition path for d i ~ i l a n e . This ~ ~ is confirmed in recent experimental studies as well.2' The reaction involves transfer of H and cleavage of the Si-Si bond to form the silylene and silane products: Si,H6 SiH2 + SiH4 (7)
-
The alternate reactions, H2 elimination to form silylsilylene or disilene, are not considered here. This is in part due to a lack of reliable experimental data. Data for these reactions come from disilane decomposition studies and indirect interpretation of the H2 elimination constant.20 The transition state of the 1,2 H shift reaction can be visualized as in Figure 2. In this case we again make use of the bond-order correlations in fixing bond lengths within the complex. The bond order of interest here, nHSi,is that between the shifting H and Si atoms and which goes to unity in the final silane product. This bond order is assumed to have a value of 0.5 in all subsequent calculations. The defining bond angles are assumed to change from those of disilane to those of silene and silane as nHSi goes to unity. Internal rotation is forbidden in the activated complex. However, there is hindered rotation in disilane. In evaluating the internal rotation partition function for the energized disilane parent, we used the classical approximation for a hindered rotor having an energy barrier V.36 This partition function is given in (8) and includes a correction factor for quantum effects at high
(35) Ho, P.; Coltrin, M. E.; Binkley, J. S.; Melius, C. F., submitted for publication in J . Phys. Chem. (36) Pitzer, K. S.; Gwinn, W. D. J . Chem. Phys. 1942, 10, 428-440.
The Journal of Physical Chemistry, Vol. 91, No. 22, 1987 5735
RRKM Prediction of Arrhenius Parameters TABLE II: Frequencies of Disilane and the Transition State
assignt VI
2163
UZ
920 432 128 2154 844 2179 2179 940 940 379 379 2155 2155 94 1 94 1 628 628
"3 "4
v5 u6 Y7
us
[Si2H61
Si2H60
[SiZH61t9 nHSi = 0.5b
ul(l - x)047;x < 0.494' 2 1 8 7 ~ O . x~ ~>: 0.494
1585
~2
949 0 128 2087 927 2105 2181 925 925 94 94 2169 2169 925 960 155 155
+ (978 - V~)X''
reaction coord u4
+ (2022 - U ~ ) X + (1008 - Vs)X "7 (2032 - ~ 7 ) x U S + (2183 - ~ 7 ) x ~g + (910 - U S ) X uIO + (910 - ~ g ) x "5
u6
experimental error of 7.1% in their rate constant values. Assuming this error, our reanalysis of their data yields the constant log kDRO' = (12.85 f 0.33) - [(42.9 f 0.9)/2.303RT]
(12)
These 95% (20) confidence intervals in eq 12 appear more reasonable than those reported. Unfortunately, no discussion is presented in their shock-tube study that explains the large activation energy confidence interval. For consistency, the rate constant given in (1 2) from our reanalysis will be used in its range of application for subsequent determination of high-pressure Arrhenius parameters.
Results High-pressure Arrhenius parameters found from regression of VI 1 VIJf experimental data for silane and disilane decomposition are listed "I2 Y12f in Table 111. Also listed with these are the previously reported "I3 ~ 1 + 3 (2183 - ~ 1 3 ) ~ Y14 "14 + (2183 - ~ 1 4 ) ~ values. Several case studies are considered which varied the l'l5 "I5 + (910 - "I5)X temperature dependence in ( AE)through the exponent n as deY16 Y16 + (978 - "16)x scribed in eq 2. Also considered were the cases neglecting all "17 V I S temperature dependence in the collision efficiency, Le., p, = VI8 "Id constant. Pertinent values of moments of inertia and LennardJones parameters for the various chemical species are listed in From ref 37. Frequencies of transition state with H-Si bond order nHSl = 0.5: consistent with predicted A , presented in Table I11 for Table IV. Collision efficiencies reported in Table I11 were obis a n(static/shock) = 0/0 and molecular parameters in Table IV. tained by fitting of the data, and their values do have the expected normalized bond order used here for frequency interpolation: x = (nHSI relative ordering. Table V contains the experimental data - n ~ s ~ o ) / ( 0 . 9-9 nHs,o), nHSiO = 1.36 X corresponds to Si2H6 evaluated from reported static condition rate constants for silane configuration. Linear relations imply linear change from frequencies and disilane. RRKM predictions are also listed in parentheses in disilane to those of silene or silane. ePower dependences from below the experimental values. bond-order correlation u = Y (n = 1)n"; a = 0.47 and 0.43 for Si-H In our analysis of the silane decomposition data, we find disand Si-Si vibrations, respectively; see ref 34. ffis the factor which is agreement with the previous Arrhenius parameter estimates for fitted to obtain correct prediction of A,. Frequencies uI1, u12,uI7, and uls correspond to bending modes. all cases of assumed temperature dependence in ( AE).We would have expected our 0, = constant I1 case to have given estimates V relative to R T . Here, cris the frequency associated with the most similar to the previous values. However, although we do internal rotation and is given by the Einstein relation. predict a similar activation energy for this case, our preexponential is larger by an order of magnitude. Where we do predict a similar preexponential, our activation energy is less than the previous (9) hui,/kT = 4.463( L, 10401,~ values by at least 1 kcal/mol. The best agreement in the activation energy is seen to lie somewhere between the n(static/shock) = 0.5/0.5 and 0, = constant I 1 cases. It is uncertain how to In expressions 8 and 9, Io is the modified Bessel function of zero attribute the discrepancies since no discussion was offered in the order, I , is the reduced moment of inertia of the two groups about earlier work on the assumptions involved in the RRKM calcuthe axis of internal rotation, and u, is the symmetry number of lations. However, there are two likely explanations for the difthe internal rotation. The quantity Qfis the familiar partition ferences. First, the differences may come from the earlier workers function for free internal rotation, and as in the case of all other giving greater weight to their shock-tube results, while obtaining rotation partition functions, symmetry numbers are omitted. only rough agreement with an extrapolated value of the constant Recent experimental and theoretical studies of disilane put the value of the energy barrier, V, a t approximately 1 k c a l / m 0 1 . ~ ~ ~ ~ * determined by Purnell and Walsh.28 In our analysis, we required agreement of RRKM predictions with both the shock-tube and Frequency assignment is again based in part on use of bondstatic experiments, subject to appropriate experimental error order correlations, with analogies to resulting frequencies in the weighting. Further, we only fit RRKM predictions with the final products. Equations for the frequencies of the transition state Purnell and Walsh constant at conditions for which it was reported; are given in Table 11, as a function of nHSi. RRKM predictions were fitted to the reported rate constant and Two sets of data are available for disilane decomposition. These its activation energy at the mean experimental conditions. The are the static pyrolysis work of Bowry and Purnell,18 along with previous workers demonstrated rough agreement of their RRKM the shock-tube study carried out by Dzarnoski et aL20 Bowry and predictions with an extrapolated value of the Purnell and Walsh Purnell studied disilane decomposition a t conditions of 23-100 constant. Apart from these differences in treatment of the data, Torr and 556-613 K. They report the rate constant k g pat their discrepancies might also arise from the differing assumptions on conditions: collisional processes. log kBp = (14.52 f 0.36) - [(49.2 f 1.1)/2.303RT] (10) Variation of the temperature dependence in @, is seen from Table I11 to lead to significantly different results. It should be Confidence intervals for this constant correspond to roughly a 10% noted that for all cases of n(static/shock), as well as the case of standard error. p, = constant I1, activation energies predicted for the experiDzarnoski et aL20 performed shock-tube studies at conditions mental conditions were significantly less than those observed. The of 2300-2700 Torr and 850-1000 K. At these conditions, the observed values were as previously listed in eq 5b and 6 , E,,,,,, workers reported the rate constant = 55.9 and Eshock = 52.7, whereas the predicted values all fell log kDRo = 12.83 - [(42.796 f 3.13)/2.303RT] (11) = 53.9 and Eshock= 50.5. Only when @, was allowed around values greater than unity were the experimental activation energies Although no confidence interval is reported on the preexponential, correctly predicted. This may indicate some error in the assumed the 2a interval for the activation energy seems large for comLennard-Jones parameters, but it is unlikely that their uncertainties parative shock-tube studies. In their work, they report an average should result in such a large discrepancy. The results indicate that the collision cross section would be at least effectively twice the Lennard-Jones value. A resolution of this problem was sought (37) Durig, J. R.; Church, J. S . J . Chem. Phys. 1980, 73, 4784-4797. by variation of the transition-state structure, although this is rather (38) Sax, A. F. J . Comput. Chem. 1985, 6, 469-477. U¶
VI0
$)(
5736 The Journal of Physical Cheqistry, Vol. 91, No. 22, 1987
Roenigk et al.
TABLE III: Predicted and Previously Reported Arrbenim Parameters for Wane and Didme Decomposition
limiting hiah-messure Arrhenius Darameters
previous estb
15.55 f 0.3
59.60 f 0.5
55.9
15.19 15.27 15.35 15.36 15.13 15.02 14.88 16.21 16.28
58.08 58.24 58.36 58.38 57.95 57.73 57.41 59.71 61.07
54.82 54.85 54.85 54.86 54.80 54.77 54.73 55.09 56.38
21.13 21.10 20.13 20.07 21.14 22.00 21.93 18.15 0.07
14.36 f 0.02
48.81 f 0.3
15.35 15.36 15.58 15.59 15.35 15.17 15.19 16.20
5 1.06 5 1.06 51.53 51.52 5 1.07 50.70 50.73 52.54
49.55 49.54 49.93 49.91 49.55 49.25 49.27 50.76
0.36 0.86 0.11 0.13 0.17 0.21 0.75 0.94
our
n(static/shock), &(static/shock) O.O/O.O, 0.99/0.61 0.5/0.0, 0.99/0.60 0.0/0.5, 0.99/0.43 0.5/0.5, 0.99/0.43 -0.5/0.0, 0.99/0.62 O.OJ-0.5, 0.99/0.78 -0.5/-0.5, 0.99/0.80 0,= const 5 1, 1.00/0.19 0,= const, 4.08/0.36
previous estd our est?f
n(static/shock), P,(static/shock) O.O/O.O, 0.36/0.035 0.5/0.0, 0.28/0.035 0.0/0.5, 0.29/0.018 0.5/0.5, 0.22/0.018 -0.5/0.0, 0.44/0.035 O.O/-0.5, 0.43/0.067 -OS/-0.5, 0.51/0.067 0,= const, 0.075/0.0047
X lo-* x 10-4 x 10-2 x 10-3 X lo-' x 10-2 X
X
lo-'
"Evaluated as described in eq 3.b bRef 29. CEvaluatedat 1110 K. dRef 20. eEvaluated at 925 K. /Reference temperature for pc0 = 300 K. TABLE I V Molecular Properties and RRKM Parameters for Silane, Disilane, and Corresponding Transition States'
SiH, [SiH4]+ Si2H, [Si2H61+ moment products I,I&, amu.A2) 180.6 201.2 0.1098 X lo6 0.1186 X lo6 I,, amuA2 3.184 Lennard-Jones param
A elk 0,
4.084 207.6
4.828 301.3
L f = 6 and 18 for silane and disilane decomposition, respectively. Barrier to internal rotation in disilane, V = 1.0 kcal/mol, v4 = 128 cm-'. Internal rotation symmetry number, ui = 3, for disilane. For argon, uLJ = 3.542 A; c / k = 93.3 K. Ratio of complex to silane moments on reaction coordinate = 1.408. Ratio of complex to disilane moments on reaction coordinate = 1.141.
TABLE V Experimental Static Data and RRKM Predictions
SiH, p, Torr
T = 650
data" (predictions);b 40 7.62 X 10" 7% error (5.83) 135 230
1.40 X (1.34) 1.83 X (1.79)
k , s-' 675 3.71 X loT5 (2.70) 6.83 X low5 (6.31) 8.91 X (8.49)
700 K 1.62 X lo4 (1.12) 2.97 X (2.64) 3.88 X lo4 (3.58)
k. s-' 585 1.50 X lo-, (1.50)
613 K 1.05 X (1.03)
Si,H,
dataC(predictions);b 62 10% error
T = 556 1.66 X (1.64)
"Ref 28. bConsistent with n(static/shock) = 0/0 results in Table 'Ref 18.
111.
arbitrary and the calculations are known to be quite insensitive to such changes. A recent theoretical prediction3 of the transition-state structure permits calculation of the product of the principal moments, (IAZBZc)+ = 241.2 ( a m ~ A * ) This ~ . is a 20% change from the value listed in Table IV for our original complex.
We have carried out regression of the data based on this proposed transition state for the case of n(static/shock) = O/O. For this bent asymmetric complex, Lt = 12. Except for the fitted frequencies, results with this altered complex are essentially identical with those from our original model. For example, compared to the results in Table 111, the preexponential is lowered by IO%, the activation energy is decreased by 0.25 kcal/mol, and the collision efficiencies are unchanged. With these parameters, the predicted activation energies at experimental conditions were also unchanged from our original results. An alternative explanation of the discrepancy may lie in the interpretation of the experimental data, in particular the static experiments. We are not prepared to speculate on this point in this work. It would be desirable to have additional experiments in order to confirm those earlier and with which to improve these predictions. Agreement of RRKM predictions with the static work of Purnell and Walsh is demonstrated in Table V. All predictions discussed here correspond to the case of n(static/shock) = 0/0 unless otherwise stated. The predictions are shown to compare measurably well over the entire range of conditions, although the predicted effect of pressure is noticeably stronger. Predictions of the shock-tube data are shown in Figure 3. Here again agreement with the data is quite good. From the general curves presented in Figure 4, it is evident that silane decomposition is well into its falloff at common conditions of interest. Differences in prediction of the rate constant for the results of the n(static/shock) = 0/0 and (I, = constant I1 cases in a silane bath are shown in Figure 5. Here it is seen that agreement between the two treatments of (I, is within 25% at a pressure of less than 1 atm and moderate temperatures. Prediction of activation energy over the same conditions in a silane bath is displayed in Figure 6. Also shown in this figure are the predictions from the (I, = constant I1 case at 500, 800, and 1100 K. Again, the two cases agree well at pressures of less than 1 atm. Results for disilane are also listed in Table 111. Here again there is considerable disagreement with the previous estimates. Since the experiments were carried out much nearer their highpressure limit than in the silane studies, it should be expected that transition-state structures and perhaps details of the RRKM treatment will play a larger role in determining k,. Near the high-pressure limit, differing assumptions in collisional processes become less important. Since these calculations were done, we received a preprint of a manuscript reporting new experimental
The Journal of Physical Chemistry, Vol. 91, No. 22, 1987 5737
RRKM Prediction of Arrhenius Parameters
100
75
.
4
500K
*
1100
Iu Y
8
50 '
F
25
t 1 .o
0.9
0.8
-25 10-1
"I
too
lo1
*
,
.... . . ....,. , . "I
I
lo2
lo3
LLy
lo4
Pressure, Torr
1OOOK Figure 3. RRKM prediction of shock-tube data29for silane decomposition in an argon bath, with Arrhenius parameters from Table 111 for n(static/shock) = O/O.
Figure 5. Comparison of rate constant predictions for silane decomposition in a silane bath with results from Table 111; n(static/shock) = 0/0 (k'), and @, = constant I 1 ( k ) .
56
Ea
54
(kcalhnol) 52
50
F 1100 K
-101 10.'
- '.."." loo
Pressure, Torr Figure 6. Prediction of activation energy for silane decomposition in a silane bath at 100-deg intervals with results from Table 111. Curves with opened symbols are calculated with results for P, = constant 5 1. Remaining curves are for n(static/shock) = O/O. ' *'''"J
lo1
'
'.uJLJ
OI*
lo3
Pressure, Torr Figure 4. Silane decomposition falloff (a) and rate constant (b) predictions with results from Table I11 for n(static/shock) = O/O;silane bath.
data on disilane thermal decomposition from 538 to 587 K at 10-500 Torr.43 In that work, RRKM falloff calculations based on log A , = 15.7 and 8, = 1 were found to fit the data with E , = 52.2 kcal/mol. The calculated Arrhenius parameters reported in Table 111 are in much better agreement with these new values than with the Arrhenius parameters reported in ref 18 and 20. Furthermore, a higher value of E , for disilane is consistent with
AHfo(SiH2)= 68 kcal/mol, obtained from a variety of sources and discussed below. Since the values in Table 111 were obtained from the earlier rate constants of ref 18 and 20, we conclude that nonlinear regression is a superior method for obtaining accurate Arrhenius parameters from data in the falloff. It is interesting to note that, in contrast to the silane analysis, temperature dependence in 8, is much less critical, at least in regard to the resulting Arrhenius parameters and the quality of the data representation. All temperature-dependence cases listed in Table I11 were found to yield equally excellent predictions of the disilane experimental data. This suggests that a larger data base is needed for more detailed discrimination of preferred treatment of /3,. Figure 7 illustrates the prediction of the
Roenigk et al.
5738 The Journal of Physical Chemistry, Vol. 91, No. 22, 1987
50
t
500 K
Ea (kcal/mol)
I
1.o
0.9
1.1
1.3
1.2
Pressure, Torr
1ooorr Figure 7. RRKM prediction of disilane decomposition shock-tube data,20 with Arrhenius parameters from Table 111 for n(static/shock) = O/O; argon bath.
1oo 10-1
-!i
10-2
P
S 10-3 1od 1o - ~ 10-1
loo
lo1
lo2
lo4
lo3
1
-15
'
'"'d d -
IO-1
loo
*
10l
IO*
/
io3
to4
Pressure, Torr Figure 8. Disilane decomposition falloff (a) and rate constant (b) predictions with results from Table 111 for n(static/shock) = O/O, disilane
bath. shock-tube experiments for the case of n(static/shock) = O/O. Prediction of the static data with the same temperature dependence in 3/, is demonstrated in Table V. In both sets of data, agreement is seen to be good. Further predictions of the rate constants are shown in Figure 8, and the corresponding activation energy is presented in Figure 9.
Figure 9. Prediction of activation energy for disilane decomposition in a disilane bath at 100-deg intervals with results from Table 111. Curves are for n(static/shock) = O/O.
The falloff curves, presented for silane and disilane in Figures 4 and 8, respectively, may be useful in the design of additional kinetics experiments. Data that should most improve accuracy in the current estimates of the high-pressure parameters will come from conditions defining the curvature of the transition region. This is a subject dealt with in more detail in the accompanying work.22 It would be illustrative to consider the interpretation of other kinetic data with respect to results found here under varying assumptions of temperature dependence in p,. Our predictions appear to bracket the activation energy for silane decomposition between 58 and 61 kcal/mol. This is a difference of 1.5 kcal/mol on either side of the previously established estimate and is sufficient to alter previous conclusions about the insertion-activation energy, E,,. For example, in the work of Jasinski,13 the absolute rate constant for the insertion of SiH, into D2 was measured at 300 K and 1 Torr, and E , I1 kcal/mol was predicted. This constraint was estimated with the use of the previously reported high-pressure preexponential for the forward reaction29 and recent entropy data on ~ i l y l e n e . ~The ~ constraint was offered on the basis of the assumption that the rate constant for SiH2 insertion into D2 (kD) was an adequate approximation to the limiting high-pressure rate constant for SiH2 insertion into H2(kH/kDnot
