J. Phys. Chem. 1993,97, 9934-9941
9934
Importance of Energy Transfer and Lattice Properties in H-Atom Association with the (1 11) Surface of Diamond CMdc Accary, Philippe Barbarat, and William L. Haw' Department of Chemistry, Wayne State University, Detroit, Michigan 48202
Kenneth C. Hass Research Stafl, Physics Department, Ford Motor Company, Dearborn, Michigan 481 21 -2053 Received: April 13, 19936
This paper reports the results of a classical trajectory study of the dynamics of H-atom association with a radical site on the (1 11) surface of diamond. The association dynamics are affected by the potential between the H-atom and radical site, nonbonded interactions between the H-atom and the surface of the lattice, and the lattice vibrational frequencies. The sensitivity of the association probability to the lattice frequencies suggests that in a complete theory for association the dynamics of energy transfer from H-atom relative translation to the lattice modes must be considered. As a result, a capture theory like transition-state theory is expected to overestimate the association rate constant. The trajectories also show that once the H-atom associates there is a negligible initial transient in the ensuing dissociation of this H-atom from the lattice. The trajectory results are found to be sensitive to the treatment of zero-point energy. A quasiclassical trajectory calculation as performed here, which includes lattice zero-point energy, is expected to give a larger abstraction/addition rate constant ratio for a H-atom interacting with a diamond surface than does a molecular dynamics calculation, which does not include lattice zero-point energy.
I. Introduction An important new technology is the growth of diamond films by low-pressure chemical vapor deposition (CVD).14 The basic reaction mixture consists of H2, the predominant component, and a hydrocarbon feed gas such as CH4 or C2H2. Growth is initiated by heating the reaction mixture (e.g. by a hot filament maintained at -2500 K) to dissociate H2. Deposition occurs on a substrate heated to a lower temperature of 100&1300 K. The H-atoms produced by H2 dissociation abstract H-atoms from the diamond surface to create radical sites for further growth, i.e. Cd-H
+ 'H
H2 + 'C,
(1) In this reaction Cd represents a surface carbon atom. Of considerableimportance in modeling diamond film growth is the fraction of surface radical sites."' The fraction is related to the relative rates for reaction 1 and the recombinationreaction
Cd'
+
+ 'H
Cd-H
(2) Rate constants for reactions 1 and 2 have not been measured, and their values have been estimated by comparisons with gas-phase ~ t u d i e s . ~ ' ~ J In ~ J 5this manner, rate constants for H-atom association with secondary and tertiary carbon atoms on a diamond surface have been chosen by comparison with gas-phase rate constants for H-atom and alkyl radical association. These latter rate constants have not been measured but have been estimated from model collision and transition-state theory type calculations and by comparisons with other experimental rates.l"l8 Twodifferenttheoretical approaches have been used to calculate gas-phase A + B AB* association rate constants, i.e. capture and dynamical models.20 An assumption prevalent to the capture model is that inside a critical A + B separation all intramolecular degrees of freedom of the AB* moiety are highly coupled and therefore readily exchange energy so that a long-lived vibrationally/rotationally excited AB* metastable is formed. Transition-state theory is a capture theory, since it is assumed that --.*
-
Abstract published in Aduunce ACS Absfrucfs,September 1, 1993.
each trajectory which crosses the transition state leads to association.21Recrossingsof the transitionstatearenot accounted for and transition-state theory gives an upper bound to the association rate constant.22 Dynamical theories for association attempt to treat the transfer of energy from A + B relative translation toAB* internal modes, which is required for association to occur. The criterion for association is not capture, but the actual formation of a long-lived AB* moiety. Thus, recrossings of the transition state can occur and the actual association rate constant may be less than that of transition-state theory. Presumably, if the dynamical calculations are quantum mechanical without extensive approximations,the correct value for the association rate constant would be obtained. To date, the dynamical calculations have involved classical trajectories.20 Extensive comparisons between different theoretical models and with experimenthave been reported for only one H-atom and alkyl radical association reaction, Le. H + CH3 CH4.23-2' Using a potential energy surface derived from ab initio calcul a t i o n ~ ?it~was found that transition-state theory and classical trajectories give the same H CH3 association rate constant.25 This rate constant agrees with that found by experiment.25 The agreement between transition-state theory, a capture model, and classical trajectories, a dynamical model, for the H CH3 association indicates the CH4 degrees of freedom are highly coupled inside the transition state, so that crossing the transition state is a sufficient criterion for association. In other words, intramolecular vibrational energy redistribution (IVR)28,29from the forming H 4 H 3 bond to CH3 vibrational/rotational degrees of freedom is sufficiently rapid that short-time recrossings of the transition state are unimportant. Using molecular dynamics30andan empirical analyticpotential energy function developed for hydrocarbons,31 Brcnner et a1.32.33 calculated the relative rates of reactions 1 and 2 versus temperature for the (1 1 1) surface of diamond. From these calculations they found that the abstraction/addition rate ratio increased with increase in temperature, with a value of -0.1 at 1200 K. In the work reported here the quasiclassical trajectory method is used to study the microscopic dynamics of H-atom association
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+
0022-3654/93/2091-9934$04.00/0 Q 1993 American Chemical Society
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H-Atom Association with the (1 11) Surface of Diamond
The Journal of Physical Chemistry, Vol. 97, No.39, 1993 9935
TABLE I: Models for the Diamond Lattice total total carbon carbon description atoms model name description atoms model name 1 layer 13 3-ring 4 rings (6 + 7)-ring 3 rings 2 layers 46 llayer 22 6-ring 13 rings 6 rings 3 layers 38 (3 + 1 + 3)-ring 2layers 22 (3 + I)-ring 7rings
with a radical site on the (1 11) surface of diamond. The methodology of these calculations differs from that of Brenner et a1.32-33in that a different empirical analytic potential energy function is used and zero-point energy motion of the lattice is included. The principal goal of this work is to determine whether a capture model is sufficient for calculating the rate constant for H-atom association with the surface. If not, specific couplings between the H-atom translational motion and lattice vibrations must be treated. Other goals of this work include determining how the association probability depends on the nonbonded interactions between the H-atom and the lattice, the size of the model required to represent the diamond lattice, the temperature of the lattice, and the treatment of the lattice zero-point energy motion in classical trajectory simulations. The transient dissociation probability of the associated H-atom is also studied.
W
II. Potential Energy Function The potential energy function used for this trajectory study is written as a sumof thelattice potential, Vbttia, a potential between a gas-phase H-atom and a surface radical site, V H , ~ and ~ ~ ,a potential describing the nonbonded interactions between this H-atom and the lattice, VH,nonbondd: (3) = Vlatticc + 'H,sitc + VH,nonbonded At the potential energy minimum, all carbon atoms in the lattice are assumed to conform to an exact tetrahedral bonding pattern. Hydrogen atoms are attached to the top layer of carbon atoms, if the resulting CH bond is perpendicular to the surface at the potential minimum. Hydrogen atoms are not added to any of the remaining exterior carbon atoms. The radical site is located at the middle carbon atom of the top layer. The equilibrium CC and CH bond lengths for the lattice are 1.545 and 1.086 A, respectively. Five different models are used to represent the diamond lattice. Each model is delineated by the number of layers of carbon atoms and the number of 6-membered carbon atom rings in each layer. The smallest model consists of a single layer with three rings and is called a 3-ring model. The largest model has 13 rings and two layers, with six rings in the first layer and seven rings in the second layer. It is called a (6 + 7)-ring model. The (3 1 3 ) h g model has three layers and a total of seven rings. The (6 + 7)-ring and (3 1 3)-ring models are depicted in Figure 1. The dimensions for the top layer of the lattice models are given in Figure 2. (This figure also illustrates two dimensionsof the 6-ring model.) The potential energy function for the lattice is represented by a sum of Morse functions for the CC and CH stretches and harmonic functions for the CCC and CCH valence angle bends. Harmonic force constants f for these four internal coordinates were taken from fits to the cyclohexane vibrational frequencies by Wiberg et al.34 The CH and CC bond energies D were equated to the H + tert-butyl and tert-butyl + fer?-butyl bond energies, respectively.3sJ6 The Morse exponential parameters for the CC and CH bonds were determined from the relationshipf- 2Dfi2. Potential parameters for the lattice are listed in Table 11. Vibrational frequencies for the different lattice models were calculated with the generalchemical dynamics computerprogram VENUS.37J* Distributionsof the vibrational frequenciesfor the smallest and largest models of the lattice are plotted in Figure 3.
+ +
+ +
Figure 1. (3 + 1 + 3)-ring and (6 + 7)-ring models for the diamond lattice. Shaded atoms represent carbon radicals.
Figure 2. Dimensions (in A) of a 6-ring top layer. A 3-ring top layer consists of the three innermost rings.
TABLE Ik Parameters for the Potential Energy Function fcc (mdyn/A) (mdyn/A) fcm (mdyn A/rad2) fCCH (mydn A/rad2) fCH
Lattice Potential 4.456 D m (kcal/mol) 4.664 DCH(kcal/mol) 2.637 &C (A-I) 1.306 @CH (A-l)
70.0 92.0 2.14 1.93
H' + Radical Site Potential Dr (kcal/mol)
92.0 2.412
A
mdyn A/rad2) a (h~-~) f00
1.306 0.783
H'+ Lattice Nonbonded Potential ~H.,H (kcal/mol) W , H (A)
0.01 19 3.21
cp,c (kcal/mol) OH-,C (A)
0.0437 2.8 1
Though there are no nondiagonal potential energy terms in the lattice potential, when represented in internal coordinates, there is extensive coupling in the internal coordinate Hamiltonian for the lattice. The couplings are between momenta in the kinetic energy.39 These couplings are more important than the much smaller nondiagonal potential energy couplings and have been shown to be the coupling terms controlling IVR in hydrocarbon molecules.2S.29 Below, in section IV.A.4, it is shown that a model
9936 The Journal of Physical Chemistry, Vol. 97, No. 39, 199'3
Accary et al. H ATOM I
RADICAL SITE (S)
/
I
AIMING POINT (A)
I I I
' D I A M O N D (1 11) SURFACE
Figure 4. R and 0 coordinates used for choosing the aiming point.
the lattice is expressed as a sum of two-body Lennard-Jones terms, each of the form V(r) =
(;)"-
(;)"I
The parameters u and c between the associating H-atom and a hydrogen attached to the lattice are assumed to be the same as those for a H-atom and He-atom interaction. For the nonbonded interactions between the associating H-atom and the carbon atoms of the lattice, the u and c the parameters are equated to those of a H-atom and Ne-atom interaction. The values used for u and c are taken from the review of Scoles42 and are listed in Table 11.
III. Trajectory Calculations
v (cm-l) Distributionoflatticevibrational frequencies: (a) 3-ring model; (b) (6 + 7)-ring model.
with nondiagonal lattice couplings does not enhance energy transfer between the H-atom and the lattice. The potential energy function and parameters for H-atom association with the radical site were chosen by following previous work for the H CH3 CH4 system.22-25 The CH potential at the radical site is modeled by a Morse function with Dr = 92 kcal/mol (Le., the H + tert-butyl bond energy)35J6 and = 2.412 A-1. This is the value for /? which reproduces the experimental H CH3 association rate constant.m.41 As the H-atom adds to the radical site, three H-CC bends are created and the potential for each is given by
+
-
+
v, =f,(r)[O - 0012/2
(4)
where 60 is the tetrahedral angle 109.47O. The attenuation of the harmonic bend force constant is represented by
S(r) = 1.0; r < ro
(5)
S(r) = e x p [ - ~ ( r - r ~ ) ~r~2; ro
-
whererO=l.O86Aandu -0.783A-2. Thisvalueforareproduces the H + CHI CH4 association rate constant when the above value for is also used.m The parameters for the H + radical site potential are summarized in Table 11. The potential energy function for the nonbonded interactions between the associating H-atom and the hydrogen and carbon atoms of the top layer of
The trajectory calculations were performed with the general chemical dynamics computer program VENUS.37.38The analytic functions described in the previous section, which are used to build the system potential energy function, are standard options in VENUS. The procedures described below, for choosing the trajectory initial conditions, integrating Hamilton's equations of motion, and analyzing the trajectories, are also standard options in VENUS. A. Initial C ~ n d i t i In ~ ~this . simulation,quasiclassical initial conditions4- are chosen for the trajectories to mimic the experimental conditionsof low-pressurechemicalvapor deposition (CVD). The von Neumann rejection method" is used to sample the quantum mechanical thermal distribution
~ ( n ) e&v/W(l
- eAp/W)
(7) for each normal mode of the lattice. A random phase is then chosen for each mode to convert n for the mode to a normal mode coordinate and momentum. The normal mode eigenvector is then used to transform the normal coordinate and momentum to Cartesianmomenta and displacement coordinatts. A temperature of lo00 K is used for the lattice. As shown in Figure 4, two coordinates, R and 4, are used to define an aiming point (A) for the impinging H-atom. When the aiming point (A) is defined as the orthogonal projection of the H-atom on the lattice plane, R represents the distance from (A) to the radical site (S),and 4, the angle between R and a fixed radius on the lattice plane. Trajectories were performed with fixed values for R, Le., 0.0, 0.5, 1.0, 1.5, and 2.0 A. The angle I#J is randomly chosen from a uniform distribution between 0 and 27r. A relative translational energy of 2 kcal/mol is given to the H-atom so that it moves perpendicular to the surface toward the aiming point. This is the average translational energy for unidirectional motion at 2000 K. The H-atom is initially separated from the surface by 8 A, at which there is no interaction between the H-atom and the surface. A total of 50 trajectories are calculated for each value of R.
H-Atom Association with the (1 11) Surface of Diamond
0.9
c
& m
8E z
E2
The Journal of Physical Chemistry,Vol. 97, No. 39, 1993 9937
-
0.80.70.6-
0.50.4-
g
03-
9
030.1
-
V."
3-RING
(3+1)-RING (J+I+J)-RING
6-RING
(6+7).RING
0.0
0.5
Figure 5. Association probability at R = 0 for different lattice models. The calculations are for perpendicular attack, a H-atom translational energy at 2000 K, and a lattice temperature of lo00 K. Zero-point vibrational energy is added to the lattice modes. Uncertainties are for one standard deviation.
B. Integrating Hamilton's Equations of Motion. Hamilton's equations of motion
are numerically integrated over 0.5 ps using a fourth-order RungeKutta integrator followed by a sixth-order Adams-Moulton integrator. The calculations were carried out on an IBM RISC6000 high-performance workstation at Wayne State University. A integration step size of 0.5 X s is used for all the calculations. C. Analysis. In this study, the association of an H-atom with the lattice is assumed to occur if the relative motion between the H-atom and the radical site exhibits three or more inner turning points. This choice seemed appropriate since the probability of association did not change when the criterion for an event was changed from two or more inner turning points to three or more inner turning points, Le., every trajectory with two inner turning points also had three inner turning pints. An association probability for orthogonal H-atom attack is determined from each set of 50 trajectories. Uncertainties reported are for one standard deviation.
IV. Results and Discussion A. Association Probability. 1. R = 0. Setting R = 0 corresponds to a head-on collision of the gaseous H-atom with the lattice surface site. At this aiming p i n t , 50 trajectories were run for each of the five models. The data depicted in Figure 5 show that the association probability is not particularly sensitive to the size of the model used to represent the lattice. All the values for the association probability are statistically the same, except the value for the 6-ring model, which is slightly lower. It is worthwhile noting that the 3-ring model with only one layer has the same association probability as do the 2- and 3-layer models. Animations of trajectories indicated the reduced association probability for the 6-ring model resulted from its proclivity for bending and, thus, significantly distorting the geometry near the radical site. Evidencefor the importanceof this global bending motion comes from the increase in the sticking probability when one layer is added to the 6-ring model. Inspection of Figure 5 reveals no apparent differencebetween the results with the 2-layer and 3-layer models. Thus, a model with more than two layers does not seem necessary for this study. For head-on collisions with the surface site, the association probability is about 0.65. Thus, 35% of the trajectories which acquire a CH separation less than the C H equilibriumbond length
1.5
1.0
DISTANCE FROM RADICAL SITE
MODELS
2.0
(A)
+
Figure 6. Association probability versus R for the (3 1)-ring (M) and (6 + 7 ) h g (A) models. Same initial conditions and uncertainties as in Figure 5.
TABLE IIk Association Probability versus Lattice Temperature for Different Values of R association probability T(K) 0.OA 0.5 A 1.0 A i.sA 500 750 lo00 1250 1500 2000
0.72 i 0.06 0.68 i 0.07 0.68 & 0.07 0.66* 0.07 0.50 0.07 0.46 iO.07
*
0.86 i 0.05 0.72* 0.06 0.62f 0.07 0.72 0.06 0.62* 0.07 0.64& 0.07
*
* *
0.86 0.05 0.84 0.05 0.88 i 0.05 0.82 i 0.05 0.66 i 0.07 0.72 i 0.06
~~
0.00 0.04+ 0.03 0.04 0.03 0.04k0.03 0.06 0.03 0.04 i 0.03
*
2.0 A 0.00 0.00 0.00 0.00 0.00 0.00
simply bounce off the lattice and do not associate. It is of interest to note that transition-state theory (TST)21identifies these latter trajectoriesas reactive events,sincethey crossthe transition state.27 Because TST gives an upper bound to the association rate constant,22 these trajectory results indicate that TST may overestimate the actual rate constant for H-atom associationwith the (1 11) surface of diamond. 2. R > 0. Calculation of the association probability was extended to R = OS, 1.0, 1.5, and 2.0 A for the (3 + 1)-ring and (6 + 7)-ring 2-layer models. Figure 6 shows that the association probability versus R is the same for the two models except at R = 1.5 A, where it is 20% larger for the (3 + 1)-ring. Figure 2 shows that, for R > 1.45 A, the aiming point for the sampling can be out of the surface covered by the (3 + 1)-ring model. Thus, a slightly lower association probability might have been expected at R = 1.5 A for the (3 + 1)-ring model. It is significant that the association probabilities for the (3 + 1)-ring and (6 7)-ring models are so similar. The large decrease in the association probability when R increases from 1.0 to 1.5 A is caused by the surface H-atoms. From computer animations it was observed that the gaseous hydrogen atom closely approachesthe lattice hydrogensand strong intermolecular repulsive forces repel the gaseous hydrogen from the surface. Reactive trajectories are observed when at least two lattice hydrogens push the gaseous H-atom toward the radical site. Such events occur with a low probability. No association occurred for R 1 2.0 A for either the (3 + 1)-ring or (6 + 7)-ring model. 3. Effect of Lattice Temperature. The effect of the lattice temperature on the association probability was studied by also performing calculations versus R for the (6 + 7)-ring model at T = 500, 750, 1250, 1500, and 2000 K in addition to 10oO K. The results are listed in Table 111. For small R, the trend is for the association probability to decrease as the lattice temperature is increased. The association probability is less sensitive to the lattice temperature for larger R. It is of interest to note that Brenner et a1.32.33 found the same trend. However, their calculations differ from ours in that their lattice and translational
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Accary et al.
9938 The Journal of Physical Chemistry, Vol. 97, No. 39, 1993
that a valence-force potential parameterized to fit the diamond phonon spectrum49gives association probabilities versus R nearly the same as those reported here (Figure 7) for the model with the frequencies reduced by a factor of 2. (This valence-force potential has both diagonal and nondiagonal internal coordinate potential energy terms.) The transfer of energy from CH stretch modes in highly excited polyatomic molecule^^^^ has been described in terms of sequential and concurrent nonlinear resonance~.S*~5~ The H-atom lattice relative translational motion can be viewed as a low-frequency vibration, and a nonlinear resonance model may be able to explain why the transfer of energy from translation to lattice vibration depends on the lattice vibrational frequencies. Ewinga has used a momentum gap model to interpret energy transfer between high-frequency intramolecular modes and low-frequency intermolecular modes of van der Waals molecules. Such a model may also explain the dependence of energy transfer from the lowfrequency H-atom lattice relative translational motion to the high-frequency lattice vibrations. In future work it would be of interest to apply both the nonlinear resonance model and the momentum gap model to the energy-transfer dynamicsof H-atom association with diamond surfaces. 5. Variationof the Nonbonded Potential. The above results were obtained with the nonbonded potential described in part 11, referred to as the H / H e H / N e potential. For this potential, interactions between the gaseous H-atom and surface hydrogens and carbons are represented by Lennard-Jones H-atom/He and H-atom/Ne potentials, respectively. To investigate the possible influenceof the nonbonded potential on the associationprobability, another Lennard-Jones potential was used, for which the H-atom interactions with the surface hydrogens and carbons are given by He/Heand Ne/Ne terms. This nonbonded potential is identified as He/HeNe/Ne. Its parameters were also taken from the review by S c o l e as ~ ~follows: ~ Q.,H = 0.0215 kcal/mol, UH*,H = 2.65 A, Q . , ~ = 0.0437 kcal/mol, UH.C = 2.65 A. A comparison of these two Lennard-Jones potentials in Figure 8 shows that the H / H e H / N e potential is the more repulsive. Thus, one might expect a larger association probability for the He/He-He/Ne potential, especially at large distances from the radical site, where the role of a nonbonded potential is more significant. Figure 9, in which theassociation probability is plotted as a function of R for both potentials, presents results that are in agreement with this consideration. These calculations are for the (6 + 7)-ring lattice model. At R = 1.5 A, 80% of the trajectories lead to association with the He/HeHe/Ne potential while only 10% association is observed with the H / H e H / N e potential. The latter potential gives trajectories where the gaseous H-atom is easily repelled from the surface because of the strong repulsions generated by the hydrogens on the lattice, whereas with the He/He-He/Ne potential, which is less repulsive, the H-atom is able to come close enough to the surface to actually "sense" the radical site and then associate. Since the association probability is sensitive to the nonbonded interactions, representation of these interactions will have to be further refined before an accurate association probability versus R can be calculated. In this regard, we are currently performing high-levelab initio calculationsto determine accurate nonbonded intermolecularpotentials between a diamond surface and various gaseous species including the H-atom. B. Residence Lifetime of the Associated H-Atom. A total of 1000trajectories were calculated for the results presented above fora latticetemperatureof lOOOK (alllatticemodelsareincluded in this set of trajectories). Lifetimes were computed for these trajectories with two or more inner turning points for which dissociation occurred before the end of the simulation (Le., 0.5 ps). Out of the 1000 trajectories, 576 association events were observed, of which only 3 1 dissociated. Results are depicted in Figure 10,where each histogram gives the number of dissociations
+
...
0.0
0.5 1.o 1.5 DISTANCE FROM RADICAL SITE (A)
2.0
Figura7. Sensitivityof theassociationprobabilitytothelatticevibrational frequencies: A, standard vibrational frequencies (see text); 0 ,standard frequencies increased by a factor of 2; standard frequencies decreased by a factor of 2. Same initial conditions and uncertainties as in Figure 5. Calculations are for the (6 + 7 ) h g lattice model.
temperatures were the same. Here, we have maintained a translational energy of 2000 K and varied the lattice temperature. 4. Variation of Lattice Vibrational Frequencies. For the gaseous H-atom to associate with the surface radical site, its initial translational energy must be transferred to the lattice vibrational modes, Le., included with these modes are the CH stretch and CCH bending motions of the lattice hydrogens. To determine whether this energy-transfer process is sensitive to the frequencies for the lattice vibrations, two additional calculations were performed by either decreasing or increasing all the lattice force constants by a factor of 4 for the (6 7)-ring model. The effect of this scaling is to decrease or increase all the lattice vibrational frequencies by a factor of 2. The results in Figure 7 show that the association probability is sensitive to the lattice frequencies. By increasing the lattice frequencies by a factor of 2, there are 22% more reactive trajectories at R = 0, and when decreasing the lattice frequencies by a factor of 2, the association probability decreases by 22%. These results suggest that the coupling between the translational mode of the gaseous H-atom and the vibrational modes of the lattice are particularly sensitive to thelatticefrequencies,andalsosuggestthat thelatticevibrations have to be correctly treated in order to obtain an accurate association probability. Since zero-point energy is added to the (6 + 7)-ring lattice model in the trajectory initial conditions, increasing the lattice vibrational frequencies also increases the initial classical energy of the lattice. At 1000 K the average total classical energy (i.e., one including zero-point energy) of the lattice is 343, 416, and 638 kcal/mol for lattice frequencies decreased by a factor of 2, normal lattice frequencies, and lattice frequencies increased by a factor of 2. In a classical canonical ensemble, a higher average energy is associated with a higher temperature. As shown above, increasing the lattice classical energy (Le. classical temperature), by increasing the zero-point energy, leads to a larger association probability. Comparing this result with that of the previous section, in which the temperature for the quantum mechanical thermal vibrational energy distribution [eq. 71 is varied, shows that the association probability cannot be simply related to a classical temperature but depends on the vibrational frequencies for the lattice as well as the temperature for a quantum mechanical thermal distribution. It is of interest to identify which of the above models for the lattice vibrational frequencies is most representative of bulk diamond. To answer this question, a series of trajectory calculations have been initiated using different force Weld models which have been proposed for diamond and diamond-like materials.4 Preliminary results from these calculations show
+
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The Journal of Physical Chemistry, Vol. 97, No. 39, 1993 9939
H-Atom Association with the (1 11) Surface of Diamond
'* 1
I"
I
98-
7-
8
6-
10-
F
8 w 8
54-
32-
8-
6-
8
1-
1
0'
I
1.7
1.9
2.1
2.3
2.5
2.7
2.9 3.1
3.3
INTERNUCLEAR SEPARATION
3.5
I
3.7 0
(A)
0.00 0.05
0.10
0.15
0.20
0.25
0.30 0.35 0.40 0.45 0.50
TIME (ps)
Figure 10. Number of dissociations from the lattice within 0.05-ps time intervals for the 576 trajectories which associate. Sameinitial conditions as in Figure 5.
-14 . 1.7 1.9
'
'
2.1
9
2.3
.
8
2.5
'
8
2.7
.
9
2.9
'
' 3.1
'
INTERNUCLEAR SEPARATION
' . 3.3 3.5 8
'
I 3.7
(A)
Figure 8. Nonbonded intermolecular potentials (see text): (a) (-) for H/He and (- -) for He/He; (b) (-) for H/Ne and (- -) for He/Ne.
-
-
a diamond lattice heated to 1000 K. If the A factor is assumed to be that for CH4 dissociation,6* i.e., 6 X 1015 s-1, and E, is equated to the lattice CH bond energy of 92.0 kcal/mol, k can be estimated as 5 X 10-5 s-l, which gives an average residence time of 0.2 X 105 s for a H-atom which is dissociating from a lattice with a 1000 K canonical Boltzmann distribution of energy. Thus, the short-time unimolecular transient observed in the trajectory simulation arises from nonequilibration of the energy release as the H-atom associates. Apparently, there is some initial local heating in the vicinity of the radical site. C. Treatment of Zero-Point Energy Motion. The effect of zero-point energy (ZPE) on the association probability was investigated using the (6 + 7)-ring model and the H/He-H/Ne nonbonded potential. A total of 50 trajectories were run for each R in the range &2 A. Three series of calculations are reported here and are referred to as ZP1, ZP2, and ZP3. In ZP1, no zero-point energy is added in any latticevibrational mode, before adding additional quanta for the 1000 K Boltzmann distribution (eq. 7), which gives an average lattice energy of 123 kcal/mol. In ZP2, zero-point energy is initially added to all modes except the CH stretches, before adding the lo00 K Boltzmann distribution. This amounts to a total average energy of 377 kcal/mol, of which 254 kcal/mol is zero-point energy. The ZP3 calculation is the quasiclassical calculation reported in the previous sections, where zero-point energy is added to each lattice mode (for a total of 293 kcal/mol), before adding additional quanta in accord with the Boltzmann distribution in eq 7. The total average lattice energy for this calculation is 4 16kcal/mol. As displayed in Figure 1 1 , the association probability increases by as much as 30% in changing the calculation from ZP3 to ZP1. There is no significant difference between the ZP3 and ZP2 calculations. These results show that the association probability is independent of zero-point energy added to the CH stretch modes but dependent on the addition of zero-point energy to the remaining lattice vibrational modes. The insensitivity of the association probability to zeropoint energy in the CH stretch modes may result from short-time adiabaticity for these modes. The increase in the association probability, as the classical energy for the remaining lattice vibrations is decreased, is in accord with the temperature dependence of the association probability (Table 111). V. Conclusion The work presented here has provided insights into the effect of the potential energy surface on the probability of H-atom association with a radical site on the (1 11) surface of diamond, the importance of unimolecular transients when the H-atom associates, and the manner in which different treatments of the lattice zero-point energy affect the association probability. Each of these topics is summarized below.
A w r y et al.
9940 The Journal of Physical Chemistry, Vol. 97, No. 39, I993
*1n.n O-.e 0.0
0.5 1.0 1.5 DISTANCE FROM RADICAL SITE (A)
2.0
Figure 11. Effect on the association probability of adding zero-point energy to the lattice vibrational mod- in the trajectory initial conditions: 0 , normal calculation with zero-point energy added to each mode; A, zero-point energy added to all lattice modes except the CH stretches; 0 , no zero-point energy is added. Same initial conditions and uncertainties as in Figure 5, except the treatment of zero-point energy is varied. Calculations arc for the (6 7)-ring lattice model.
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There are three general features of the potential energy surface which affect the association probability: (1) the interaction between the associating H-atom and the radical site, (2) nonbonded interactions between this H-atom and the lattice’s surface, and (3) the latticevibrational frequencies. From studies of H-atom and alkyl radical association,Q it is expected that the two propertiesof the H-atom + radical site potential which affect the association probability are the attractiveness of this potential and the attenuation of the He-CC bending forces which are formed during the association. With respect to the second surfacefeature, the calculations reported here show that, by making the H-atom lattice surface nonbonded potential less repulsive, it is not as necessary to direct the H-atom toward the radical site to attain association. Thus, decreasingthe repulsivenessof the nonbonded potential increases the overall association probability. A particularly interesting finding from this study is the sensitivity of the association probability to the values for the lattice vibrational frequencies. This result indicates that a capture theory for association like transition-state theory may overestimate the rate constant for H-atom association with a radical site. A model for calculating this rate constant must explicitly include the probability of energy transfer from H-atom translation to lattice vibration. A resonance energy transfer model used to interpret intramolecularvibrationalenergy redistribution in highly excited molecules58159 and/or a momentum gap model used to interpret energy transfer in van der Waals molecules@Jmay be applicableto the energy-transferprocess involved when a H-atom associates with a diamond lattice. With respect to our study, it is interesting to note that in D2 physisorption on cold Cu surfaces the surface structure profoundly influences the sticking probability.62 This has been interpreted in terms of structure-specific differences in the Dz-lattice vibration (Le. phonon) couplings. When the H-atom associates with the diamond lattice, the newly formed CH bond is highly energized. If this energy does not rapidly transfer toother vibrations, there may be a significant short-time transient in the dissociation probability of this newly attached H-atom. This study indicates that such a transient is unimportant. Only 3% of the associated H-atoms constitute an initial transient dissociation. There is a continuing uncertainty in the proper treatment of zero-point energy in classical trajectory simulations.6~5 If the classical motion is chaotic, the zero-point energy is free to move around the molecular system, which is a physically incorrect result, In a quasiclassical trajectory simulation zero-point energy is added to each mode in the initial conditions. However, based on
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classical/quantum correspondence arguments, it has been argued that zero-point energy should not be added.M.66 Nevertheless, recent theoretical studies of intramolecular energy transfer from highly excited CH bond~6~@ indicate the best agreementbetween classical and quantum dynamics is obtained when zero-point energy is included in the classical calculations. In the calculations reported here, it is found that adding zeropoint energy to the lattice in the trajectory initial conditions decreases the association probability. This is not a generalfinding for association reactions. In trajectory studies of Li+ + H2O and C1- + CH3Cl as~ociation,6~*6~ adding zero-point energy to a molecule increases the association rate constant. For H + CH3 association, zero-point energy has a negligible effect on the association rate.’O It is of much interest to determine the physical origin of these different effects of zero-point energy on classical trajectory association rate constants. In a molecular dynamics trajectory study, zero-point energy is not added to the system and the system’s temperature is determinedfrom the classical kinetic energy. For a quasiclassical trajectory study, as performed here, zero-point energy is first added, before sampling the quantum mechanical thermal vibrational energy distribution, eq.7. Even if thesame potential energy surface is used for a H-atom interacting with a diamond lattice, these two different trajectory methods are expected to give different values for the ratio of the H-atom abstraction and association rate, i.e., reactions 1 and 2. As shown in this work, including zero-point energy decreases the association rate. In contrast, the abstraction rate may increase if zero-point energy is added in the initial conditions. Abstraction rates are often enhanced by excitation of the rupturing bond,71 and this may be thecase for H-atom abstractionfromthediamondlattice. Adding energy to the CH bond of the lattice is not expected to decrease the abstractionrate. Thus, from these effects of zero-point energy on H-atom association and abstraction, one expects that, using the same potential energy surface, a quasiclassical trajectory simulationshould give a larger abstraction/association ratio than does a molecular dynamics simulation. Finally, it is worthwhile noting that, in trajectory simulations of direct gas-phase reactions such as H + H2 H2 H, the quasiclassical approach for adding vibrationalenergy in the initial conditions is referr red'^.'^ over the molecular dynamicsapproach of treating the reactant vibrationalenergy distribution classically. However, additional work clearly needs to be done to establish the proper approach for sampling initial conditions in trajectory simulations of gas-surface reactive collisions.
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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, administeredby the American Chemical Society. The authors wish to thank Professor Michael Frenklach for stimulating discussions. References and Nota (1) Derjaguin, E. V.; Fedoseev, D. V. Sci. Am. 1975, 233, 102. (2) Derjaguin, E.V.; Fedoseev, D. V. Russ. Chem. Reu. 1984,53,435. (3) DeVries, R. C. Annu. Rev. Muter. Sei. 1987, 17, 161.
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