J. Phys. Chem. 1995, 99, 4531-4535
4531
Temporal Dependence of Collisional Energy Transfer by Quasiclassical Trajectory Calculations of the Toluene- Argon System V. Bernshtein,’ K. F. Lim,* and I. OrePrJ Department of Chemistry, Technion-Israel Institute of Technology, Haifa 32000, Israel, and School of Chemistry, University of Melbourne, Parkville, VIC 3052, Australia Received: August 8, 1994; In Final Form: November 23, 1994@
The average energy transferred per collision and its dependence on collision duration were evaluated by using quasiclassical trajectory calculations with valance force field intramolecular potential for toluene and pairwise Lennard-Jones intermolecular potential for argon-toluene interactions. The average energy transferred in up, down, and overall collisions were sorted according to the duration of the collisions. It was found that, on average, collision durations, for collisions lasting longer than zero, are 0.68 and 0.23 ps at 300 and 1500 K, respectively, and this is the time in which energy transfer takes place. Most collisions of duration longer than 0 are impulsive, and the number of complex-forming collisions is negligible. The average minimal distance at which the collisional event manifests itself in internal energy change in the molecule is -0.31 nm at both 300 and 1500 K. One in 800 collisions is a supercollision. The implications of these findings on energy transfer models are discussed.
Introduction Collisional energy transfer between a highly excited polyatomic molecule and an atom is of great importance for the basic understanding of chemical reactions in the gas phase.’-6 Energy up pumping and down pumping play major roles in unimolecular and bimolecular reactions as well as in physical processes where a photoexcited molecule decays by collisional energy transfer. The subject was reviewed recently in a number of articles.lP6 To explain the experimental results, energy transfer models based on an “artist’s view” of how a collisional process takes place were developed. Some of the older models were reviewed in ref 1; the more recent ones will be discussed below. Dashevskaya et a L 7 s 8 have developed the sequential direct encounter model in which energy is removed by a sequence of direct collisions, a chattering, of the atom with the excited molecules. On one hand, the heavier the atom, the larger the van der Waals forces, the longer the time the atom spends near the molecules, the larger the number of the chattering events. On the other hand, the heavier the atom, the slower it moves and the amount of energy transferred per direct collisions decreases. The balance between the processes, with the latter being more important, leads to the well-known experimental results where a plot of the average energy transferred per collision vs the boiling temperature of the gas shows a levelingoff for the heavy inert gases.g Tardy’O has investigated the collisional efficiencies of fluoroalkanes on the deactivator mass and modeled the leveling off effect discussed above, by simulating the collision as an encounter between a diatomic molecule containing a pseudoatom and a bath atom. Borgesson and Nordholm’ have developed a collisional energy transfer efficiency model for hot molecule and small and medium bath molecules. Lim and Gilbert12have developed a biased random walk model in which the energy of the atom-molecule undergoes diffusion during the collision. The value of (AE) is found by solving a Fokker-Plank diffusion equation. OreP has developed an empirical model where (AE) for hot polyatom-bath can be predicted from the boiling temperature of
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Technion-Israel Institute of Technology. University of Melbourne. Abstract published in Advance ACS Abstracts, February 15, 1995.
0022-365419512099-453 1$09.0010
the bath gas. All these models came into being because the statistical model in which energy is distributed statistically between the substrate and the bath predicts values of (A@ which are an order, and sometimes more, of magnitude larger than those found experimentally. In the past few years classical trajectory calculations have provided some insight into the energy transfer process.13-19 Average energy transferred per collision, collisional transition probabilities, and vibrational, rotational, and translational energy distributions in the energy transfer process were obtained from these studies. The effects of temperature and interatomic potentials were studied systematically. In the present study we wish to explore the temporal behavior of the collision process. We wish to obtain the average lifetime of the collision and to provide an answer to a nagging question: are there long-lived collision complexes which contribute significantly to the energy transfer process?
Theory and Calculations Hamilton equations of motion were integrated by using a customized version of the Quantum Chemistry Program Exchange program Venus.20 Details of the calculations are given in ref 19. The initial translational and rotational energies were selected from the appropriate thermal distributions. The initial impact parameter was chosen randomly between 0 and its maximum value b,. The initial orientation of the colliding pair was randomly chosen by rotation through Euler’s angles. The internal energy was 30 000 cm-’ above the zero point energy of 27 974 cm-’. The intramolecular potential had four contribution~~~
obtained from Draeger’sZ1modified valence force field calculations of force constants given in ref 19. The functional form of the potentials of the harmonic stretchings, bendings, waggings, and torsions are given in ref 19. They reproduce the experimental normal mode frequencies, with the assumption of methyl free rotor, to a very good degree. The intermolecular potential was a Lennard-Jones pairwise potential 0 1995 American Chemical Society
Bemshtein et al.
4532 J. Phys. Chem., Vol. 99, No. 13, 1995 I
For the H-Ar interactions Aq = 9.419 65 x kJ mol-' nm12 and Bij = 1.3651 x kJ mol-' nm6 For the C-Ar interactions Au = 4.982 148 x kJ mol-' nm12 and Bg = 4.9369 x lop3kJ mol-' nm6. These correspond to atom-atom van der Waals radii of 0.335 and 0.355 nm for H-Ar and C-Ar, respectively, with atom-atom well depths of 42 and 104 cm-', respectively. The average energy transferred in a collision calculated from the trajectory calculations is given by
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where n is the moment of the energy transferred, is the Lennard-Jones collision cross section, s2(2~2)* is the collision integral, and N is the total number of trajectories. This was used to reproduce the experimental value of (AE). brei = 0.5546 nm at 300 K and 0.4450 nm at 1500 K where bref = /JL,(Q(22)*)
112.
A preliminary study was performed to assign proper values for b, and the minimal-distance cutoff where the minimal distance is defined as the distance between Ar and the closest atom in toluene. A systematic study of the effect of the maximum impact parameter, b,, on the value of (AE) was performed. Above a given value of b,, (AE) was found to be independent of the impact parameter. A value of b, = 1.1 nm was used in all trajectories. Impact parameters were therefore chosen randomly between 0 and 1.1 nm, Figure 1. The cutoff of the postcollision minimal distance was studied systematically between 0.8 and 1.6 nm. It was found that the value of (AE)d1 at distances above 0.8 nm was constant. The postcollision distance cutoff was therefore taken as 0.8 nm for all trajectories. This reduced the calculation time markedly compared with the upper limit of 1.6 nm, which was used as the initial minimal separation above which no energy is transferred. This can be seen in Figure 1 both from 300 and 1500 K. It should be clear that the impact parameter is the distance between the atom and the center of mass of the molecule while the minimal distance is the distance between the atom and the nearest atom to it in the molecule. The average energy transferred in all trajectories which lasted between time t and t At, N,, is given by
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(4) where AEt is the amount of energy transferred in a single collision which lasted in the time interval between t and t
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At.
We have developed a method by which we determine the initial and final moments of the collisional event, needed in eq 4, by the forward and backward sensing (FOBS) method. The beginning of the collisional interaction between the hot molecules and the bath atom was defined as the moment in which the intemal energy of the molecule changes by a quantity E . The termination of the interaction was defined by detecting, by tracing the trajectory backward, the instance after which no more changes in the intemal energy of magnitude E occurred. The two events bracketed the intermolecular interactions. The value of E was varied over a wide range until an optimum value was obtained. This was done to ensure that the collision duration, as chosen by FOBS, was independent of a particular value of
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Figure 2. Example of a rare long-lived trajectory which lasts 6.34 ps. The temperature is 300 K and AE = -123 cm-l. E . The trajectories were binned in the time domain according to the duration of the intermolecular interactions. Convergence of the values of (AE) was obtained after 4000 trajectories were calculated (at each temperature).
Results Figures 2 and 3 illustrate the method of determining the duration of the intermolecular interaction, At. In all calculations, the value of the critical jump energy, E , was 70 cm-'. Figure 2 is an example of a long-lived collision complex which lasted 6.3 ps. The vertical arrows show the initial and final moments of interactions as determined during the trajectory. The energy transferred during that trajectory is 123 cm-'. Figure 3 is an example of a short-lived, impulsive collision. The duration of the collision is 0.14 ps and AE = 91 cm-'. As can be seen from the trajectory, there is no way one can define the initial and final moments of the collision by merely looking at the minimal separation of the bath atom from the closest atom of the hot molecule. The forward and backward sensing method developed here is a unique and consistent way of determining
J. Phys. Chem., Vol. 99, No. 13, 1995 4533
Toluene- Argon Collisional Energy Transfer 139
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Figure 3. Example of a typical short-lived trajectory which lasts 0.14 ps. The temperature is 300 K and AE = 119 cm-'. - 58200
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Figure 5. Histogram of the number of collisions vs collisions times at intervals of 0.1 ps at 300 K. The first bin is at t 0.0 ps, that is to say, those collisions in which the collision partners do not spend time next to each other. d indicates down, and up indicates up collisions.
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Figure 4. Trajectory (300 K) in which the minimal separation is almost constant for a very long time but energy exchange occurs during 0.2 PS.
the duration of a collision. The effectiveness of the FOBS method is demonstrated again in Figure 4, where the minimal distance hardly varies for a very long time. The actual change in the total internal energy of the molecule occurs suddenly and over a very short period of 0.2 ps (at 300 K). The widths at half-height of the initial and final internal energy peaks in Figures 2-4 determine the onset and termination of the collision and are chosen by the FOBS method by the numerical definition of E . Using the FOBS method, we have binned the trajectories according to their duration. Figure 5 shows the distribution of the trajectories at 300 K according to their collisional lifetimes. Panel a shows that most of the trajectories are very fast, that is to say, there is no energy exchange above E during the collision (they contribute negligibly to (A,?) as will be shown later). Panel b shows the number of trajectories that contribute to up collisions and to down collisions; obviously, the same trend as in a is obtained. Panel c shows the ratio of the number of trajectories that lead to down collisions, Nd, to the number of trajectories that lead to up collisions, Nu,,. The number of down collisions always prevail over that of up collisions. It should be born in mind that in the histogram the numbers of the long duration trajectories are very small compared with the shortlived collisions and are not amenable to statistical analysis. Now that the distribution of the number of trajectories as a function of collision duration is known, the cardinal question of what collisions contribute the most to energy transfer can be answered. This is depicted in Figure 6, which represents results obtained at 300 K. Panel a shows that the very short duration collisions, those that do not effect an energy change, E , do not contribute to (AE)all. From Panel b it can be seen that there is
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Figure 6. Histogram of the average energy transferred per collisions between t and t At, (AE,) vs time at 300 K. d indicates down, and up indicates up collisions. Note the different scales on the vertical axis.
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some up and down energy exchange at At = 0 but that it is negligible compared with energy exchanged during collisions of somewhat longer' duration. The main feature that emerges is that all energy transfer takes place in collisions that last between 0.1 and 0.8 ps. The values of (AE) from long duration collisions (At > 1.O ps) are statistically insignificant, and therefore their absolute values can be discarded, but the
Bernshtein et al.
4534 J. Phys. Chem., Vol. 99, No. 13, 1995 0.0 3000A1
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Figure 7. Histogram comparing the number of trajectories at intervals t and t At at 300 and 1500 K.
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Figure 8. Histogram of (AE,) vs time at 1500 K. Note the different scales on the vertical axis. 59000
conclusion that they do not contribute to the average energy transferred is of course valid. The average lifetime of the collision for collisions above At = 0 is defined by
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where N, are all collisions which last longer than At = 0 and amount to about 40% of all collisions. At 300 K (At) = 0.68 ps. Increasing the temperature of the ensemble from 300 to 1500 K has the following effects. The average energy transferred per collision increases from -127 cm-l at 300 K to 89 cm-l at 1500 K. This effect is consistent with previous results obtained in trajectory calculation^.^^ At 300 K the internal thermal energy of the toluene molecule is -700 cm-', well below the internal energy of the molecule so down collisions prevail. At 1500 K the internal thermal energy is 21 400 cm-', still below the value of the internal energy. Explanation of why up collisions prevail at high temperatures is deferred, pending further quantitative work on velocities within the collision sphere. Yet another factor which was found in this work is that the number of trajectories with At = 0 increases at high temperatures, Figure 7. The average collision lifetime at 1500 K is (At) = 0.23 ps. There is a reduction in the lifetime of the collision with an increase in energy, Figure 8. The incoming bath atom approaches faster at higher temperatures and is repelled faster at the repulsive part of the potential. The average minimal distance of the incoming bath atom from the nearest atom in the molecule at which energy exchange takes place is almost independent of the temperature being 0.31 nm both at 300 and 1500 K and independent of the duration of the trajectory. This is to be compared with the atom-atom V = 0 distance of 0.297 nm (H-Ar) and 0.317 nm (C-Ar) which gives an average value of 0.307 nm. These two numbers are identical for all practical purposes. It is possible therefore to define an energy transferring collision in a novel
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Figure 9. Example of a supercollision trajectory at 300 K. The duration of the collision is 0.13 ps and AE = 4747 cm-'.
way, namely, a collision that occures at a distance at which the internal energy changes by a quantity E. It should be pointed out that E has nothing to do with the energy transfer quantity AE, E is merely the tool by which collisions are defined.
Supercollisions Supercollisions are collisions which transfer an inordinate amount of energy in a single collisional These collisions were found e ~ p e r i m e n t a l l y ~and ~ - ~in ~ trajectory calculation^.^^^^^ Their contribution to chemical reactions and to the values of the average energy transferred per collision were r e p ~ r t e d . ~The ~ - ~present ~ study has found as well trajectories which indicate that supercollisions took place. Figure 9 shows a supercollision that takes place at 300 K. The AE of the collision is -2500 cm-l which is to be compared with the average energy transferred per collision of -127 cm-l, a 20fold increase compared to the average amount of energy transferred per collision. The duration of the collision is 0.13 ps. Figure 10 shows a trajectory which describes a supercollision at 1500 K in which 6300 cm-l is transferred in one event. Again it is a short trajectory which lasts about 0.14 ps. Some trajectories of supercollisions are of longer duration. Figure
Toluene- Argon Collisional Energy Transfer
J. Phys. Chem., Vol. 99, No. 13, I995 4535
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for polyatomic-polyatomic energy transfer. In summary, (a) most collisions are of very short durations At 0 and do not contribute to the energy exchange; (b) the average collision duration, of the nonzero duration collisions, is 0.68 ps at 300 K and 0.23 ps at 1500 K; (c) the minimal distance between the bath atom and one of the atoms of the molecule is -0.31 nm regardless of duration of the collisions and almost independent of temperature; and (d) supercollisions do occur. Some are impulsive with very short lifetimes and some with longer lifetimes. One in every 800 collisions was found to be a supercollision.
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Figure 10. Supercollision trajectory at 1500 K. At = 0.12 ps, and AE = 6491 cm-I.
References and Notes
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Acknowledgment. This work is supported by the Fund for Promotion of Research at the Technion (1.0.) and by the Center for Absorption in Science, Ministry of Immigrant Absorption (V.B.).
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Figure 11. Example of a supercollision trajectory at 300 K. The duration of the collision is 2 ps, and AE = -2842 cm-I.
11 shows a supercollision trajectory at 300 K with At = 2 ps. Out of five supercollision trajectories at 300 K, three were shortlived and two of longer duration.
Conclusions The implications of the present findings are very interesting. The fact that most energy transferring events take place in a very short duration on a subpicosecond time scale indicates that at least for the atom-polyatom system no long-lived complexforming collisions take place. This finding excludes statistical theories which allow enough time for the internal energy to equilibrate with the collisional transition modes. The duration of the collisions in Figure 3 is 0.14 ps which is of the duration as that of a 230-cm-' mode. This time scale is shorter than intramolecular energy redistribution, IVR,32and serious redistribution of energy during collision cannot be expected. Some trajectories, however, do last a few vibrational periods, allowing for at least partial IVR. The question remains, how are large quantities of energy,'~~ 1000-3000 cm-', being transferred in normal collisions of polyatomic-polyatomic molecules? Energy must flow from the transition modes into the internal modes of the polyatomic bath collider. Here it is possible that a shortlived complex exists. Further trajectory calculations are required
(1) Oref, I.; Tardy, D. C. Chem. Rev. 1990, 90, 1407. (2) Gilbert, R. G. Adv. Chem. Kine?. Dynam. 2, in press. (3) Hippler, H.; Troe, J. In Bimolecular Reactions; Baggott, J. E., Ashfold, M. N. R., Eds.; The Chemical Society: London, 1989. (4) Tardy, D. C.; Rabinovitch, B. S. Chem. Rev. 1977, 77, 369. (5) Quack, M.; Troe, J. Gas Kinetics and Energy Transfer, Vol. 2, Specialist Periodical Report; The Chemical Society: London, 1977. (6) Barker, J. R., Ed. Vibrational Energy Transfer Involving Large and Small Molecules; JAI Press: in press. (7) Dashevskaya, E.; Nikitin, E. E.; Oref, I. J. Phys. Chem. 1993, 97, 9397. (8) Dashevskaya, E. I.; Nikitin, E. E.; Oref, I. J. Phys. Chem., in press. (9) Oref, I. J. Phys. Chem. 1992, 96, 6308. (10) Tardy, D. C. J. Chem. Phys. 1993, 99, 963. (1 1) Borgesson, L. E. B.; Nordholm, S. J. Phys. Chem. 1995, 99, 938. (12) Lim, K. F.; Gilbert, R. G. J. Chem. Phys. 1986, 84, 6129; 1990, 92, 1819. (13) Brown, N. J.; Miller, J. A. J. Chem. Phys. 1984, 80, 5568. (14) Breuhl, M.; Schatz, G. C . J . Chem. Phys. 1988, 89, 770; J. Phys. Chem. 1988, 92, 7223. (15) Lendvay, G.; Schatz, G. C. J. Chem. Phys. 1992, 96,4356; 1993, 98, 1034. (16) Lendvay, G.; Schatz, G. C. J. Phys. Chem. 1990, 94, 8864; 1992, 96, 3752. (17) Lim, K. F.; Gilbert, R. G. J. Phys. Chem. 1990, 94, 77. (18) Clarke, D. L.; Oref, I.; Gilbert, R. G.; Lim, K. F. J. Chem. Phys. 1992, 96, 5983. (19) Lim, K. F. J. Chem. Phys. 1994, 100, 7385. (20) Venus, Quantum Chemistry Program Exchange: Hase, W. L.; Duchovic, R. J.; Hu, X.; Lim, K. F.; Lu, D. H.; Pesherbe, G.; Swamy, K. N.; Vaneliude, S. R.; Wolf, R. J. Quantum Chem. Program Exchange Bull, to be submitted for publication. (21) Draeger, J. A. Spectrochem. Acta 1985, 41A, 607. (22) Pashutzky, A.; Oref, I. J. Phys. Chem. 1980, 92, 178. (23) Hassoon, S.; Oref, I.; Steel, C. J. Chem. Phys. 1988, 89, 1743. (24) Morgulis, I. M.; Sapers, S. S.; Steel, C.; Oref, I. J. Chem. Phys. 1989, 90, 923. (25) Mullin, A. S.; Park, J.; Chou, J. Z.; Flynn, G. W.; Weston, R. E. Chem. Phys. 1993, 53, 175. (26) Clarke. D. L.: Thomson. K. G.: Gilbert. R. G. Chem. Phvs. Let?. 1991,182, 357. (27) Lendvav. G.: Schatz. G. C. J. Phvs. Chem. 1990. 94. 8864. (28) Oref, L'Supercollisions. In Vibraiional Energy Transfer Involving Small and Large Molecules, Vol. 2; Advances in Chemical Kinetics and Dynamics; Barker, J. R., Ed.; JAI Press: in press. (29) Bemshtein, V.; Oref, I. J . Phys. Chem. 1993, 97, 12811. (30) Bemshtein, V.; Oref, I. J . Phys. Chem. 1994, 98, 3782. (31) Oref, I. Chem. Phys. 1994, 187, 163. (32) Oref, I.; Rabinovitch, B. S. Ace. Chem. Res. 1979, 12, 166. JP942085U
