J. Phys. Chem. 1993,97, 1170611711
11706
Comparison of Canonical Variational Transition State Theory Rate Constants for H Atom Association with Alkyl Radicals and with the (1 11) Surface of Diamond Pbilippe Barbarat, Cedric Accary, and William L. Haw' Department of Chemistry, Wayne State University, Detroit. Michigan 48202 Received: June 21, I993@
+
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A model potential energy function developed previously for H CH3 CH4 association is extended, with transfer of parameters, to H atom association with other alkyl radicals and with the diamond (1 11) surface. Reaction path following calculations are performed to determine canonical variational transition state theory (CVTST) rate constants for these association reactions. The CVTST rate constants for H atom association with CzH5, i-C3H7, and t-C4H9 agree with experimental and/or estimated rate constants to within a factor of 2. This finding indicates it is not a severe approximation to assume transferability of potential energy parameters for different H atom and alkyl radical association reactions. Differences between the CVTST rate constants for these associations are discussed in terms of moment of inertia ratios between the transition state and reactants and frequencies for the transitional bending modes. The CVTST rate constant for H atom association with the diamond (1 1 1) surface is approximately 2 times smaller than that for H t G H 9 association, which results from a factor of 2 difference in reaction path degeneracies for these two associations and agrees with a kinetic model proposed previously [J.Phys. Chem. 1993,97,23].The H diamond (1 11) surface association rate constant is weakly sensitive to both the nonbonded potential between the associating H atom and H atoms attached to the surface and the lattice potential. The lattice partition function changes less than 10% in forming the association transition state. In contrast to these CVTST results, a recent trajectory study shows that the H atom diamond (1 11) surface association rate constant is sensitive to the lattice potential. This is because the transfer of the H atom relative translational energy to lattice vibration, which is necessary for association to occur, is sensitive to the lattice potential. Thus, CVTST may overestimate the H diamond (1 11) surface association rate constant, since it does not treat this energy-transfer process.
+
+
+
+
I. Introduction A critical issue in modeling diamond film growth by chemical vapor deposition (CVD)14 is the fraction of surface radical sites, which is assumed to be governed by the reactions
site in a prototype gas-phase reaction. Both models relate surface and gaseous rate constants as a basic assumption whose validity is still uncertain. Theoretical studies are expected to be particularly useful for studying the relationship between rate constants for H atom association with radical sites on diamond surfaces and gas-phase rate constants for H atom and alkyl radical association. Unfortunately, there have been very few studies of this type. For H CH3 CH4 association, classical trajectory and variational transition state theory calculations give nearly the same rate constant, which agrees with the experimental ~a1ue.l~ Similar studies have not been performed for H atom association with other alkyl radicals. Few theoretical calculations have been reported for H atom association with a radical site on a diamond surface.20 In the work presented here canonical variational transition state theory (CVTST) is used to calculate and, thus, compare rate constants for H atom association with the methyl, ethyl, isopropyl, and tert-butyl alkyl radicals and with a radical site on the (1 11) surface of diamond. CVTST has been widely used to calculate bimolecular rate constants for association reactions without potential energy barriers.21 The goal of this research is to acquire a deeper understanding between the relationship between H atom + alkyl radical and H atom + diamond surface association rate constants. H atom association with tert-butyl most closely resembles H atom association with the diamond (1 11) surface, and it is of particular interest to determine whether the rate constants for these two associations are similar.
+
where c d represents a surface carbon atom. In reaction 1 an H atom is abstracted from the diamond surface to create radical sites for further growth, and in reaction 2 an H atom associates with a radical site leading to an occupiedsurface site. The purpose of the present study is to provide rate constants for reaction 2 on the diamond (1 11) surface using canonical variational transition state theory (CVTST)e12 and tocompare the results with similar gas-phase rate constants. There have been no direct experimental measurements of the rate constant for reaction 2, and this rate constant has been chosen by comparison with experimental or estimated gas-phase rate constants for H atom and alkyl radical associations.1s18 Two approaches have been used to deduce the rate constant for reaction 2 from gas-phase rate constants for H atom and alkyl radical association. In the model proposed by Frenklach and Wang,13 the rate of the gas-phase reaction divided by the gas-phase collision rate is equated to the per site ratio of the surface reaction rate to collision rate. The surface collision cross section is assumed to be 2.6 Az/site. A different model has been proposed by Belton and Harris.14 Their assumption is that the reaction cross section per surface site is equal to the reaction cross section per equivalent Abstract published in Aduunce ACS Abstracts, October 15, 1993.
0022-3654/93/2097-11706$04.00/0
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II. Potential Energy Surfaces A. H Atom and Alkyl Radicals. The following four gas-phase recombination reactions are studied here: Q 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, No.45, 1993 11707
H Atom Association with Diamond (1 11) Surface
-.
H + CH, H + C2H,
CH,
(3)
C2H,
(4)
-
H +K3H,
C3H8
(5)
+ t-C,H,
i-C,H,o
(6)
Ij
-
Q
-
A model analytic function, similar to the one used to analyze H CHI CH4 association is used to represent the potential energy surfaces for these association reactions. The analytic function is written as
+
V = VR + V ( f )+ V(4)
3-ring mode1
(7)
where VR is the potential for the radical R, V(r) is the H-R radical potential, and V(4) is the H-R angular potential. Before describing the explicit form of the potential energy function, the geometry of the radicals must be considered. The tert-butyl radical is pyramidal with an angle of looo between a C-C bond and the C3, axis.23 For the calculations reported here this angle as well as all the tert-butyl valence angles was set to 1O9So, the tetrahedral value. This choice, as shown below, does not significantly affect the H + tert-butyl association rate constant, since the external rotational partition functions for the tert-butyl radical and the transition state, which depend on the molecular structure, nearly cancel in the CVTST calculations. The goal of this study is to determine how the H atom and alkyl radical association rate constant is affected by the size of the radical. Thus, models are also used for the methyl, ethyl, and isopropyl radicals in which all valence angles are set at the tetrahedral value. This approximation is most severe for the methyl radical, which is planar. Tocompare with thecalculations for a tetrahedral CH3, a calculation is also performed in which CH3 is kept planar. The C-H and C-C bond lengths for the radicals are 1.086 and 1.54 A, respectively. The same potential parameters are used for all the association reactions studied here, with a goal of determining how the size of the radical affects the association kinetics. By comparing the calculated rate constants with experimental values, this study also provides insight into the transferability of potential energy parameters for different H atom and alkyl radical association reactions. Theassociationrateconstant is expocted tobe relatively insensitive to the specific form of the radical potential V R ,and ~~ thus, VRis represented by a simplified MM3 forcefieldZ5in which only diagonal force constants are retained. The stretching force constants in mdyn/A units are as follows: HC, 4.74; CC, 4.49. The bending force constants in mdyn*A/rad2units are as folIows: HCH, 0.55; HCC, 0.59; CCC, 0.67. A H atom and alkyl radical (R) association rate constant is primarily influenced by the H-R radical potential V(r) and angular potential V(4). The former is given by the Morse function
-
W )= De(1 - exp[--be(r - re)lY
(8)
-
with De = 92 kcal/molZ6and re = 1.086 A. Avalue of Be = 2.412 A-2 is used to fit the H + CH3 CH4 association rate constant at 300 K,22and this value is also used here. The H-R angular potential is given by (9) where the 4, are the three valence angles for the associating H atom, e.g., for H + CH3 they are three H-C-H angles, while for H + CzH5 they are two H-C-H angles and one H-C-C angle. Theequilibriumangle+ois l09.So. Thequadratic forceconstants
u -l
1
Fiopn 1. Two models used for the diamond lattice. The shaded carbon atom is the radical site.
f+,are expressed as
f*l=&,
exp[-cr(r - re)*]
where re = 1.086 A and they4, are the above MM3 HCH and HCC bending force constants. When q 10is used in an analytic potential for the H CH3 CH4 system, a value of a 0.783 A-2 is found to fit the H CH3 association rate constant at 300 K.22 This value is used here. B. H Atom .adM.mond (111) Surface. The analyticpotential energy function used to describe an interaction between a H atom and a radial site on the (1 11) surface of diamond is similar to that described above for a H atom and alkyl radical interaction and is written as
+
V=
+
-
-
Yatticc + VH,site
+ Vnonbonded
(11)
The interaction term between the H atom and the radical site is vH,sitc
= v(r)
+ v(4)
(12)
where the radial and angular terms, V(r)and V(4).are the same as those above for an H atom and alkyl radical. In comparing eqs 11 and 12 with eq 7, one sees that ,it& replaces VRwith the additional Vnmbondd term for the H atom and surface potential. Two models of different sizes are used to represent Vht& in this work to investigate the effect of the lattice model on the rate constant. The first model contains two layers of carbon atoms and three 6-membered carbon atom rings and is referred to as the 3-ring model. The second model contains four layersof carbon atoms, six 6-membered carbon atom rings in the first and second layers, and w e n 6-membered carbon atom rings in the third and fourth layers. It is called the (6 7)-ring model. Both models are depicted in Figure 1. For each model Vhtd, is written as a sum of harmonic stretches and valence angle bends with MM3 force constants as used for the alkyl radicals. The term V , win q 11describesthe nonbondcd interactiom between the incoming H atom and the lattice. Three different models are used for V m w d . Two, identified as LNJl and LNJZ,
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Barbarat et al.
11708 The Journal of Physical Chemistry, Vol. 97, No. 45, 1993
I k
Generalized normal coordinates may be defined for the motions orthogonal to the reaction path.33*34Harmonic frequencies for these normal coordinates are found by diagonalizing a projected mass-weighted Cartesian force constant matrix FP. As a result of the projection, FPhas seven zero eigenvaluescorresponding to infinitesimalrotations, translations, and motion along the reaction path. There are also 3N- 7 nonzero eigenvalues which give the harmonic frequencies for the vibrationsorthogonal to the reaction path. To calculate the free energy along the reaction path G*(s), the degrees of freedom are assumed to be separable. Thus, G*(s) becomes
\
30{
'
+
- 1 0 ~ " ' ~ ~ " ' " ' " " ' 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7
internuclear separation
(A>
-
Figure 2. H' + H interaction potential for the EXP6 (- - -),LNR (- -), and LNJl (-) potentials.
use the Lennard-Jones potential
+
+ +
For LNJl the H* H and H' C nonbonded interactions are represented by H' He and H' Nevan der Waals intera~tions,~' respectively, with C H - , H ~= 0.0119 kcal/mol, U H * , H ~= 3.21 A, C H - , N ~= 0.0437 kcal/mol, and U H - , N ~= 2.81 A. For LNJ2 the He H and H' C nonbonded interactions are represented by He He and He Ne van der Waals interactions, respectively, with ~ H ~ , H0.0215 ~ kcal/mol, U H ~ , H=:~ 2.65 A, C H ~ , N=~0.0437 kcal/mol, and U H ~ , N=~ 2.97 A. The third model called EXP6 uses the analytic potential energy function
+
+
+
+ +
V = a exp(-br) - c/r6 (14) and was derived by Williams and Star98 to represent nonbonded hydrogen interactions with hydrogen and carbon atoms in the experimentally determined crystal structures of 18 hydrocarbon molecules. For a H' H interaction the parameters are a = 2790.87 kcal/mol, b = 3.74 A-1, and c = 32.50 kcal*A6/mol. For a H' C interaction the parameters are a = 15651.29 kcal/mol, 6 = 3.67 A-l, and c = 136.95 kcal*A6/mol. A comparison of the LNJl, LNJ2, and EXP6 models for a H' H interaction is given in Figure 2.
+
+
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m. Canonical Variational Transition State Theory For a bimolecular reaction between a H atom and a reactant
R the canonical variational transition state theory (CVTST) rate c o n ~ t a n tis~ given ~ ~ . by ~~
where Q*,Qml,and QR are the partition functionsfor the transition state'svibrational/rotational degrees of freedom, H' + R relative translational motion, and the reactants' vibrational/rotational degrees of freedom, respectively, and Eo* is the difference in the zero-point energy levels for the transition state and the reactants H' R. In CVTST k in eq 15 is a minimum along the reaction path. This result is obtained by minimizing Q* (Le., maximizing the transition state's free energy) as a function of the reaction path. The reaction path is the steepest descent path in mass-weighted Cartesian coordinates connecting reactants with products.30-s2 The numerical procedure for finding the reaction path is to follow the negative gradient vector in mass-weighted Cartesian coordinates. Included among the propertiesfound from the evaluation of the reaction path, identified ass, are the reaction path classical potential U(s) and the molecular geometry versus the reaction path. From the latter, principal rotational moments of inertia are determined as a function of s for gas-phase reactions.
+
G*(s) = GZib(s)f v'(s) 6G;Js) (16) where 6 = 1 for gas-phase reactions and 6 = 0 for gas-surface reactions. Gib(s) is the free energy for the internal vibrational/ rotational degrees of freedom, C&(s) is the external rotational free energy, and v'(s) is the classical potential energy. Internal degrees of freedom in the variational transition state are identified as "conserved" or "transitional" modes. Conserved modes correspond to internal vibrational or rotational motions whose form is not significantly altered as reactants approach along the reaction coordinate. For H atom association with a radical site on the diamond (1 11) surface, the conserved modes are the lattice vibrations. In contrast, transitional modes correspond to motions whose form changes drastically in going fromreactants toproducts. In this work, methyl internal rotations for C2Hs i-CsH7, and t G H 9 are treated as conserved modes and assumed to be free rotors.3s Conserved vibrational modes and transitional modes are treated as quantum harmonic oscillators assuming separability between all vibrational modes. External rotation is only considered for the gas-phase associations. In previous studies of H atom and CH3 association, the transitional modes have also been treated as hindered rotors using the Hamiltonian which describes the long-range interaction between H and CHsas6 It was found that this method and the harmonic oscillator model used here give rate constants for H CHJ CH4 which agree to within 20%.21936
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IV. Trrnsition State Properties and Association Rate const8nts The results of CVTST calculations at 1O00, 1500, and 2000 K are given in,Table I for association with the alkyl radicals and in Table I1 for association with the diamond (1 11) surface. Compared in these tables are the H-C bond length and frequencies for the transitional modes at the transition state and the CVTST and experimental rate constants. The reactant three relative translational degrees of freedom are transformed into reaction path motion and two transitional bending modes during the association. For H atom association with a symmetric top species, the transitional bending modes are degenerate. Evolution of the transitional bending mode frequency along the reaction path is shown in Figure 3 for H atom association with the (6 + 7)-Mg model. The frequencyincreasesdramatically as the H atom approaches the carbon radical. The transition statepropertiesgiveninTablesIandIIshowthat theH-Cdistanoe shortens and the transitional mode frequency increases as the temperature increases. These are general findings for association rea~tions.2~~~~ A. H Atom d Alkyl R8dicd Rate Constants. The CVTST rate constants reported here were determined using the same H-C radical potential V(r), eq 8, and H-C angular potential V($),eq 9, for each of the H atom and alkyl radical associations. A comparison of the transition state properties in Table I shows that thedifferent association reactionshavea similar H-Cdistance at the transition state r* and, thus, a similar transition state potential energy EO*.However, the reactions have different moment of inertia ratios It/Z and different transitional mode bending frequencies. The transition state/reactant external rotational partition function ratio, which equals the square root
The Journal of Physical Chemistry, Vol. 97,No.45, 1993 11709
H Atom Association with Diamond (1 1 1) Surface
TABLE I: Transition State Properties and Rate Constants for H Atom and Alkyl Radical Association H + C2Hs H + CC3H7 H + CH3 2000 1000 1500 2000 1000 1000 1500 1500 2000 T (K) 4 (0 PIP ut
2.53 2.79 2.64 4.6 4.4 5.1 469 332 406 469 332 406 -3.0 4.3 -5.6 10.6 11.0 11.4 28.0/21.1,3 10.4,' 20.01
(cm-l)C
Eo$ (kcal/mol)d k(CVTST)' &(experimental or estimated)
2.81 1.8 228 302 -2.9 5.2
2.66 1.7 284 371 4.1 5.4 3.4'
2.55 1.6 332 427 -5.4 5.8
2.82 1.2 221 228 -2.8 4.6
2.66 1.2 280 282 -4.1 4.8 2.0,' 2.4m
2.55 1.1 330 338 -5.4 5.0
H + t-C4H9 1000 2.84 1.2 211 211 -2.7 5.0
1500 2.68 1.1 267 267 -3.9 5.4 2.4"
2000 2.57 1.1 299 299 -5.2 5.6
a H-C bond length a t the transition state. Ratio of the product of the moments of inertia for the transition state and radical. Transitional bending mode frequencies a t the transition state. Difference in the classical potential energy between the transition state and reactants. e Canonical variational transition state theory rate constants in lOI3 cm3mol-l s-l. Though the radicals are pyramidal in this model study, rapid inversion about the radical center is assumed for thesc elevated temperatures. This gives rise to a reaction path degeneracy of two. If inversion was not allowed, the reaction path degeneracy would be one. /Experimental value from ref 37 for the 300-600 K range. The uncertainty in the rate constant is +1.6 and 4 . 6 a t 300 K and +4.2 and -1.7 a t 600 K. 8 Experimental value from ref 38 for the 300-2200 K range. N o uncertainties are cited. Experimental value from ref 41 for 300-1000 K. Uncertainty factor of 2. Experimental value from ref 39 for 600 K. The rate constant a t 300 K is 14.1 X 10') cm3 mol-' 5-1. N o uncertainties are cited for either rate constant. Experimental value from ref 40 for 1200 K. N o uncertainties are cited. Experimental value from ref 42 for 298 K. Uncertainty factor of 1.4. 'Estimated value from ref 43. Uncertainty factor of 3.2. Estimated value from ref 44. uncertainty factor of 2. Estimated value from ref 45. Uncertainty factor of 2.
*
TABLE Ik Transition State Prowrties and Rate Constants for H Atom Association with the Diamond (111) Surface ~~
~
~~
3-ring model LNJI
T (K)
4 (0 ut(~m-')~
Eot k(CVTST)d
1000 2.68 314 -3.9 1.0
LNJZ
1500 2.48 426 -6.1 0.8
2000 2.35 508 -8.3 0.7
~
(6 + 7)-ring model
~
1000 2.82 221 -3.1 2.2
1500 2.63 272 -4.6 2.2
2000 2.50 325 6.2 2.1
1000 2.81 266 -3.0 1.6
LNJI 1500 2.63 337 -4.4 1.6
2000 2.51 414 -5.7 1.6
1000 2.87 201 -2.8 2.5
LNJZ 1500 2.70 261 -4.1 2.6
~
_
_
_
EXP6 2000 2.58 308 -5.3 2.7
1000 2.86 212 -2.9 2.4
1500 2.68 274 -4.3 2.5
2000 2.56 317 -5.5 2.5
0 H-C bond length a t the transition state. Degenerate transitional bending mode frequency a t the transition state. Difference in classical potential energy between the transition state and reactants in kcal/mol. Canonical variational transition state theory rate constant in 1013cm3 mol-I S-I.
1504 2.5
2.1
2.6
I
2.8
H---C distance
2.9
(A)
Figure 3. Degenerate transitional bending harmonic frequency versus H-C distance along the reaction path for H atom association with the
+
-
-
(6 7)-ring lattice model with the nonbonded potentials EXP6 (- -), LNJZ (- -), and LNJl (-).
+
+
of S/Z, decreases in going from H CH3 to H t-C4H9. For H t-C4H9 association S/Z is approximately unity and has only a small contribution to the reaction rate. The transitional mode bending frequencies decrease in going from H CH3 to H + t-CdH9, so that H t G H 9 has the largest partition function for the transitional bending modes. The transitional mode bending frequencies are largest for H + CH3 CH4 association, since this reaction only forms H-C-H bends which have higher frequencies than the H-C-C bends which are formed for the other three associations. The above transition state properties translate into similar CVTST rate constants for H atom association with C2H5, i-CsH7, and t-C4H9 and a H CH3 CH4 CVTST association rate constant which is 2 times larger. Table I shows that the agreement between the CVTST and experimental rate constants is best for H C2H5, while the CVTST rate constant for H CH3 is 2 times smaller than the average of the experimental values and for H LC3H.l and H K 4 H 9the CVTST rate constants are a factor of 2 larger than the experimental estimate. However, given the uncertainties in
+
+
+
-
+
+
+
+
+
-
the experimental/estimated rate constant~,3'~~ there is overall good agreement with the CVTST and experimentaland estimated rate constants, which shows that it is not a severe approximation to assume identical radial and angular potentials for different H atom and alkyl radical associations. Nevertheless, to obtain exact agreement between theory and experiment for these association reactjons, one expects that it will be necessary to modify the assumption that each reaction has the same radial and angular potential. The CVTST rate constants were calculated using model alkyl radicals with tetrahedral valence bond angles. Since this approximation is most severefor the methyl radical, it is of interest to determine the effect of using a planar methyl radical (the actual structure) in the CVTST calculations. Using a potential energy function for the H atom and planar CHI system which is identical to that given above for the pyramidal CH3 system, except an out-of-plane bend force constant fa = 9.55 X lC3 mdyn.A/rad2 is included for the planar CH3, gives a CVTST rate constant of 8.9 X 1013,8.5 X lo", and 8.4 X 10') cm3mol-' s-I at T = 10o0, 1500,and 2000 K, respectively. These rate constants are slightly smaller than those in Table I for the pyramidal CH3 model, because the frequency of the transitional bending modes at the transition state is slightly larger for the planar CH3 model. Though the structure used for CHI affects the CVTST rate constant for H + CH3 association, these calculations show that it is not a dominating issue. By using the same radial potential for each of the association reactions, it is assumed that each association has the same De of 92 kcal/mol. This assumption is most severe for H CH3 association, which has an actual De of 110.6 kcal/moLN IncreasingDefrom 92.0to 1 10.6 kcal/mol for H + CH3association makes the potential more attractive, which increases the CVTST rate constant by approximately 30%. For the above planar CH3 model k is 11.9 X 11.3 X 1013,and 11.0 X lOI3 cm3 mol-' s-' for T = 1O00, 1500,and 2000 K,respecti~ely.'~These rate constants are similar to those in Table I for the nonplanar CH3 model with De = 92 kcal/mol. B. H Atom and Diamond (1 11) Surface Rate Constants. The results given in Table I1 show that the rate constant for H atom
+
_
_
-
11710 The Journal of Physical Chemistry, Vol. 97, No. 45, 1993
t 5
200
-.
I
LJ"
I
'
2.5
2.6
2.1
H---C distance
2.8
2.9
(A)
Figure 4. Harmonic vibrational frequencies in the 150-350-~m-~ range for the H + (6 + 7)-ring lattice model versus H-C distance along the association reaction path. Solid lines are singly degenerate frequencies,
and dashed lines are doubly degenerate frequencies. Frequencies for vibrations of the same symmetry do not cross (refs 33 and 34). The degenerate frequency that is increasing is for the transitional bending modes.
association with the diamond (1 11) surface is weakly sensitive to temperature but sensitive to the nonbonded potential and particularly sensitive to the lattice model when using the most repulsive nonbonded potential LNJl. When using the 3-ring lattice model with LNJ1, there are significant changes in the low-frequency lattice vibrations between the separated reactants and transition state. These frequency changes result from buckling of the lattice, induced by the repulsive nonbonded interactions. This buckling and these frequency changes become less important for the calculations with the 3-ring model when the less repulsive nonbonded potential LNJZ is used. However, to obtain unambiguous CVTST rate constants for H atom association with the (1 11) diamond surface, a lattice model with multiple layers is needed to prevent the lattice surface from buckling along the reaction path. Results obtained with the (6 + 7)-ring model show negligiblechanges in the latticevibrational frequencies as the H atom approaches the transition state. In Figure 4 frequencies in the 150-350-~m-~range for the H atom and (6 + 7)-ring model/LNJ2 system are plotted as a function of the H-C distance. Values for this distance at the transition states are given in Table 11. Figure 4 shows that the only appreciable frequency change is that for the transitional bending mode. The value for this frequency and the potential energy difference between the reactants and the transition state is what influences the association rate constant. The ratio of the lattice vibrational partition function at the transition state and at the reactants, Le., q$attia/qhttiaris between 0.91 and 0.99 for the calculations with the (6 7)-ring model in Table 11. Though the CVTST rate constants calculated with the (6 7)-ring model are rather insensitiveto the form of the nonbonded potential, there are still some interesting relationships between the rate constants for the three nonbonded potentials. At the Variational transition states, the distance between the incoming H atom and the six neareat hydrogens on the surface is 3.0 & 0.1 A, depending on the nature of the nonbonded potential and the temperature. The plots in Figure 5 of the H'/H nonbonded potential, for H'-H distances in the range 2.2-3.6 A, show that the LNJ2 potential is the least repulsive and the LNJl potential the most repulsive at the transition state. As shown in Figure 3, the more repulsive the potential used for V a o n ~ dthe d higher the frequency for the transitional bending modes at the transition state. Considering the relationship between the transitionalmodes frequency at the transition state and rate constant, one expects therateconstant todecreaseas therepulsivenessof the nonbonded interactions increases. This trend is observed for the CVTST rate constants in Table 11. Also presented in Figure 5 are the resultsof a6 initlocalculations of hydrogen interactionsin methanedimem" If thesecalculations
+
+
Barbarat et al. I h
P
I;
9
c h P 0
E
&I
.-
I
0
c)
8
i
-0.1 2.2 ~
2.4
2.6
2.8
3.0
3.2
3.4
3.6
internuclear separation (A) Figure 5. Plots of the H'/H nonbonded potentials in the vicinity of the variational transition states: EXP6 (- - -), LNJZ (- -), LNJl (-),and ab initio results from ref 48 (A).
-
are representative of H atom nonbonded interactions with bonded hydrogens on a diamond (1 11) the LNJl nonbonded potential used here is most accurate in the vicinity of the transition state and should give the most accurate harmonic frequencies for the transitional bending modes. Thus, if a harmonic analysis is sufficientfor calculating the partition function for the transitional bendingmodes, LNJl shouldgivethemostaccuraterateconstant. However, LNJl appears to become too repulsive at short He-H separations, and if anharmonicity is found to be of some significance in calculating the CVTST rate constant for H atom association with the diamond (1 11) surface, LNJl may give a rate constant that is too small. It is also of interest to consider the possible effect of anharmonicity on the relationship between the CVTST rate constants calculated with the LNJZ and EXP6 nonbonded potentials. Since LNJZ is less repulsivein thevicinity of the transition state, it has the larger harmonic CVTST rate constant as found here. However, for short H'-H separations, LNJZ becomes more repulsive than EXP6 (see Figures 2 and 5 ) , and thus, LNJ2 may have the smaller anharmonic CVTST rate constant. The slight dependence of the H + diamond (1 11) surface rate constant on temperature is the same as found for the H atom and alkyl radical association rate constants in Table I. If the reaction path degeneracy of two for H + t-C4H9and one for H diamond (1 11) are taken into account, the former reaction is seen to be a good model for the latter. It is of interest to note that the median H atom + diamond (1 11) surface rate constant of 2.2 X 1013cm3mol-' s-I computed at 1000 K with the (6 + 7)-ring model yields a reactive cross section of 0.8 A*; i.e., k = u ( u ) , where ( u ) = ( 8 k ~ T / ? r m ~ ) ~ / ~ .
+
V. Mscussion This research and that presented previously50have provided insights into the potential energy surface, rate constant, and dynamics for H atom association with the (111) surface of diamond. Each of these topics is considered below. As given by eq 11, the potential energy surface for H atom association with a radical site on the (1 11) surface of diamond is written as a sum of the lattice potential, Vhttia, a nonbonded potential, V n a b n d d , and an interaction potential between the H atom and the radical site, VHCite. This latter potential VHrite is the sum V(r) + V(~#I), which are radial and angular terms, respectively. The results presented here show that the harmonic CVTST rate constant for H atom association with the radical site is nearly insensitive to vatti= if a large enough model is used for the lattice. The insensitivity of the harmonic CVTST rate constant to Vhttiw results from negligible changes in the lattice harmonic vibrational frequencies as the H atom approaches the radical site and forms the variational transition state,
H Atom Association with Diamond (1 11) Surface If a proper lattice model is used, the harmonic CVTST rate constant is found to be only weakly sensitive to Vnonhndd. For the three nonbonded potentials used here there is seen to be a 50% variation in the CVTST rate constants. Increasing the repulsivenessof the nonbonded potential increases the frequency for the transitional bending modes and increases the potential energy along the reaction path. Both of these effects decrease theCVTSTrateconstant for H atomassociationwith thediamond surface. The functional form used here for the radial and angular potentials, V(r)and V(4),was chosen from a previous fit to the H CHs CH4 association rate constant.22 It is shown that these forms for V(r)and V(4)give CVTST rate constants for H C2H5, H + i-C3H7, and H + t-C4H9associations which agree with experimental and/or estimated rate constants, given their uncertainties. This result suggests that V(r) and V(4) may be transferable between different H atom and alkyl radical association reactions. This proposal is currently under further investigation by determining an ab initio V(r) V(4)potential for H t-C4H9association. The V(r)and V(4) potentials used for the H atom and alkyl radical associations are also used for the H atom and diamond (1 11) surface potential. Therateconstant for H atom association with a radical site on the surface is found to be only a factor of 2 smaller than that for H t-C4H9association,i.e., thedifference in the reaction path degeneracies. This result is consistent with the model proposed by Harris and c o - w o r k e r ~ . ~ ~ J ~ For the H atom to associate with the diamond lattice the translational energy of the H atom must transfer to the lattice vibrational modes (Le., T V energy transfer). However, since CVTST is a capture theory for association,21CVTST does not consider this T V process. Instead, it is simply assumed that each H atom which passes the variational transition state associates. In a previous trajectory study of H atom association with the diamond (1 11) surface,50it was found that not every H atom associates,which attains the H-C equilibrium bond length; i.e., some trajectories which pass the variational transition state do not undergo sufficient T V energy transfer for association to occur. This result suggests that CVTST overestimates the association rate constant. However, the probability of energy transfer to the lattice was found to be particularly sensitive to the lattice vibrational frequencies, and an accurate association rate constant cannot be obtained from a trajectory calculation until the correct Vi,, is known. Work is in progresss1to develop an accurate model for Vhth. A comparisonof CVTST and trajectory association rate constants determined from this potential will give a transition state recrossing correction factor to the CVTST rate constant. Finally, for H + CHs CH4 association CVTST and classical trajectories give the same rate constant,19 which indicates inefficient energy transfer and recrossing of the variational transition state are unimportant for this reaction. One reason for possible efficient energy transfer for H + CH3, but not for H diamond (1 1l), is that the former has a T R energytransfer pathway in addition to that for T V. The importance of T R energy transfer for ion-molecule associations is welld o c ~ m e n t e d . ~In~future . ~ ~ work it seems worthwhile to study the relative importance T V and T R energy transfer and compare CVTST and trajectory rate constants for H atom and alkyl radical association reactions.
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Acknowledgment. This research was supported by the Ford Motor Co. Scientific Research Laboratories, the Institute for Manufacturing Research (IMR) at Wayne State University, and the donors of the Petroleum Research Fund, administered by the American Chemical Society. The authors thank Michael Frenklach and Stephen Harris for stimulating discussions and acknowledge the important contributions of Ken Haas of the Physics Department of Ford Motor Co. Scientific Research Laboratories.
The Journal of Physical Chemistry, Vol. 97, No. 45, 1993 11711
References and Notes (1) Derjaguin, B. V.; Fedoseev, D. V. Sci. Am. 1975, 233, 102. (2) Derjaguin, B. V.; Fedoseev, D. V. Russ. Chem. Rev. 1984,53,435. (3) DeVries. R. C. Annu. Rev. Mater. Sci. 1987.. 17. 161. (4) Messier, R.; Badzian, A. R.; Badzian, T.; Spear, K. E.;Bachmann, P.; Roy, R. Thin Solid Films 1987, 153, 1. (5) Angus, J. C.; Hayman, C. C. Science 1988. 241, 913. (6) Spear, K. E. J. Am. Ceram. Soc. 1989, 72, 171. (7) Yarbrough, W. A.; Messier, R. Science 1990, 247, 688. (8) Angus, J. C.; Wang, Y.; Sunkara, M. Annu. Rev. Mater. Sci. 1991, 21,221. (9) Truhlar, D. G.; Garrett, B. C. Acc. Chem. Res. 1980, 13, 440. (10) Laidler, K. J.; King, M. C. J . Phys. Chem. 1983, 87, 2657. (1 1) Truhlar, D. G.; Hase, W. L.; Hynes, J. T. J . Phys. Chem. 1983,87, 2664. (12) Steinfeld, J. I.; Francisco, J. S.;Hase, W. L.Chemical Kinetics a d Dynamics; Prentice-Hall: Englewood Cliffs, NJ, 1989; p 308. (13) Frenklach, M.; Wang, H. Phys. Rev. B 1991, 43, 1520. (14) Belton, D. N.; Harris, S . J. J . Chem. Phys. 1992, 96, 2371. (15) Frenklach, M. J . Chem. Phys. 1992, 97, 5794. (16) Frenklach, M. Phys. Rev. B 1992, 45, 9455. (17) Harris, S.J.; Belton, D. N. Thin Solid Films 1992, 212, 193. (18) Hams, S.J.; Goodwin, D. G. J. Phys. Chem. 1993, 97, 23. (19) Hu, X.;Hase, W. L. J. Chem. Phys. 1991, 95, 8073. (20) (a) Brenner, D. W.; Robertson, D. H.; Carty, R. J.; Srivastava, D.; Garnson, B. J. MRS Symp. Proc. 1992, 278, 255. (b) Brenner, D. W.; Harrison, J. A. Am. Ceram. Soc. Bull. 1992, 71, 1821. (21) Hase, W. L.; Wardlaw, D. M. In Bimolecular Collisions; Baggott, J. E., Ashfold, M. N., Us.; Burlington House: London, 1989; p 171. (22) Hu, X.;Hase, W. L. J . Phys. Chem. 1989, 93, 6029. (23) Ycahimine, M.; Pacansky, J. J. Chem. Phys. 1981, 74, 5168. (24) Thevariational transitionstate for H + CH, associationhas properties which closely resemble those of the reactants H + CH,. Thus, the vibrational partition function for the CH, moiety is nearly the same and almost cancels for the reactants and the transition state; i.e. see refs 19 and 22 and Hase, W. L.; Mondro, S.L.; Duchovic, R. J.; Hirst, D. M. J . Am. Chem. Soc. 1987, 109,2916. (25) Allinger, N. L.; Yuh, Y. H. J. Am. Chem. Soc. 1989, 111, 8551. (26) The H-C(CH,), bond dissociation energy is 92 kcal/mol. CRC Handbook of Chemistry andPhysics, 65th ed.;Weast, R. C., Ed.; CRC Res: Boca Raton, FL, 1984. (27) Scoles, G. Annu. Rev. Phys. Chem. 1980,31,90. (28) Williams, D. E.; Starr, T. L. Comput. Chem. 1977, I , 173. (29) Hase, W. L. J. Chem. Phys. 1976,64,2442 and references therein. (30) (a) Fukui, K.; Kato, S.;Fujimoto, H. J . Am. Chem. Soc. 1975, 97, 1. (b) Kato, S.;Kato, H.; Fukui, K. ibid 1977, 99, 684. (31) Schaefer, H. F. Chem. Br. 1975,11,227. (32) Ishida, K.; Morokuma, K.; Komornicki, A. J. Chem. Phys. 1977,66, 2153. (33) Miller, W. H.; Handy,N. C.;Adams, J. E. J. Chem. Phys. 1980,72, 99. (34) (a) Kato, S.;Morokuma, K. J . Chem. Phys. 1980, 73, 3900. (b) Morokuma, K.; Kato, S. In Potential Energy Surfaces and Dynamics Calculatiom; Truhlar, D. G., Ed.; Plenum: New York, 1981; p 243. (35) Themethodused to treat themethylinternalrotationsisnotimportant for these calculations, since the form of these internal rotations changes very little in moving along the reaction path from the reactant asymptotic limit to the transition state. Thus,a methyl internal rotational partition function cancels for the reactants and transition state in calculating the association rate constant. (36) Aubanel, E. E.; Wardlaw, D. M. J. Phys. Chem. 1989, 93, 3117. (37) Brouard, M.; Macpherson, M. T.; Pilling, M. J. J. Phys. Chem. 1989, 93, 4047. (38) C o b , C. J.; Troe, J. Z . Phys. Chem. (Munich) 1990,167, 129. (39) Stewart, P. H.; Smith, G. P.; Golden, D. M. Int. J. Chem. K i m . 1989, 21, 923. (40) Warnatz, J. In Combustion Chemistry;Springler-Verlag: New York, 1984; p 243. (41) J. Phys. Chem. Ref. Data 1992, 21, 513. (42) J. Phys. Chem. Ref. Data 1992, 21, 531. (43) Warnatz, J. In CombustionChemistry;Springler-Verlag: New York, 1984; p 307. (44) J. Phys. Chem. Ref. Data 1988, 17. 932. (45) J. Phys. Chem. Ref Data 1991, 20, 31. (46) Duchovic, R. J.; Hase, W. L.; Schlegel, H. B. J. Phys. Chem. 1984, 88, 1339. (47) For the planar CHI model used here with De = 110.6 kcal/mol the association rate constant is 1.5 X 10" cm) mol-l s-* at 300K. This association rate constant would be the same as the value of 1.4 X loL4cm)mol-' s-l calculated by Hu and in their model study, if a H- -C-H bending force constant of 0.5938 mdyn.A/rad2 had been used instead of the MM3 value used here of 0.55 mdyn.A/rad2. (48) Williams, D. E.;Craycroft, D. J. J . Phys. Chem. 1987, 91, 6365. (49) Ab initio calculations are currently under way to determine accurate nonbonded interactions for H atom interaction with a diamond lattice. (50) Accary, C.; Barbarat, P.; Hase, W. L. J. Phys. Chem., in press. (51) Accary, C.; Barbarat, P.; Hase, W. L. To be published. (52) Hase, W. L.; Darling, C. L.;Zhu,L. J. Chem. Phys. 1992,96,8295. (53) Hase, W. L.; Cho, Y. J. J. Chem. Phys. 1993, 98, 8626. I
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