8450
J. Phys. Chem. 1992, 96, 8450-8453
Collisional Energy Transfer in Highly Excited Molecules: Deuteration Effects David L. Clarke and Robert G. Gilbert* School of Chemistry, University of Sydney, Sydney, NSW 2006, Australia (Received: June 12, 1992)
Average energy-transfer values are reported from trajectory studies of Xe and He colliding with highly excited azulene-do and azulene-d8. The calculated isotope effect is small, in agreement with experiments on related systems. This suggests that the low-frequency modes are important in determining the amount of energy transfer. Furthermore, the fact that both calculated and observed changes with deuteration are small for helium suggests that the classical description of the energy transfer is adequate. The biased random walk model for collisional energy transferred is found to reproduce the observation of a small isotope effect. The calculated fraction and magnitude of supercollisions are in accord with extant data and are predicted to be relatively unaffected by deuteration.
Introduction
Trajectory calculations can be used to make qualitative and quantitative deductions on the nature of the distribution functions for the rate of collisional energy transfer that are necessary for the interpretation and prediction of unimolecular and recombination rate data.'-3 Comparison with experimental data for isotopic substitution is a useful tool for the theoretical dynamics, as it changes dynamical properties of the system without affecting the potential energy surface. Hence, extra information can be obtained with little extra effort. (It is even possible to use nonphysical isotopes to test mechanistic ideas.) The trajectory study of deuteration effects given in this paper supplement our earlier trajectory studies of collisional energy transfer of highly excited substrate interacting with a bath g a ~ . ~In-those ~ studies we used established methods to examine the details of collisions of monatomic bath gases with highly excited azulene. Theoretically, the field is still far from fully understood; trajectory methods, while showing promise, have been able to reproduce some, but by no means all, the results of "direct" experimenh2 Although it is possible to reproduce some trends and values measured experimentally, a worrying discrepancy occurs with the behavior of lighter bath gases He or Ne.' Without agreement over the range of bath gases an important advantage of trajectory techniques is partially lost-the ability to obtain answers for much more specific questions about the intimate processes that occur in collisions. A number of possible explanations have been suggested for the discrepancies, as follows. (1) There is the prospect of inaccuracies in the potential energy surface for the light bath gases. It seems likely that the Lennard-Jones repulsions currently used are too 'hard" (e.g., refs 8 and 9), and it has been shown both from trajectories7 and from model theoryIOthat softening the repulsive part of the potential significantly decreases the calculated energy-transfer values. (2) Gilbert and Zare" suggested that dynamical quantum effects could be behind the failure of the classical calculations. The effects arise from the same interference of partial waves that make the quantum scattering cross sections finite, and they suggested that the classical trajectory calculations would overestimate the energy transfer. Furthermore, they proposed an experimental test: the comparison of 3Heand 4Hewhere the large relative mass difference in an otherwise identical system should best display any quantum effects. (3) Another possibility is the difference between classical versus quantum initial distributions. Toselli and Barkerl2-l4carried out the test suggested by Gilbert and Zare and found that it was not important in systems of interest. However, they suggested, quite reasonably, another possible quantum effect: that which could arise from differences in classical and quantum probability statistics, because the quantum distribution of energy among a collection of coupled oscillators is different from the classical one
at low energies. For experiments such as the extant data on azulene, where the internal energy is of the order of twice the zero-point energy and there are some higher-frequency modes separated from the rest, classical statistics distributes the energy equally over the oscillators, whereas quantum mechanically the energy is concentrated in the low-frequency modes. At higher energies, of course, the two distributions become identical, but the persistent question of classical mechanics remains: are we close enough to the classical limit to obtain useful results? The classical description of isolated d e n e is the same whatever the bath gas, and so Toselli and Barker's suggestion also requires that helium must interact in a sufficiently different manner such that the deficiencies of the classical energy distribution in azulene become important, while they were not with xenon. This difference was ascribed to the fact that, at a given translational energy (or temperature), helium has a significantly larger translational velocity, and therefore the collisions are more impulsive than the chattering collisions that have been s h ~ w n ' ~ to-dominate ~~ the energy transfer with heavier bath gases. Simple considerations based on linear atomldiatom collisions2' suggest that an impulsive collision will be relatively efficient at transferring energy because the substrate molecule will be at the spectator or impulsive limit. That is, the incoming bath gas moves so fast that it undergoes an elasticlike collision with a peripheral atom on the substrate molecule with the rest of the molecule acting as a spectator. The energy is transferred to/from the centerof-mass motion of the peripheral atom. After the bath gas has departed, the peripheral atom remembers it is part of a molecule and the added kinetic energy becomes vibrational energy. In the other limit the bath gas approaches so slowly it only sees the average position of the vibrations and hence undergoes an elasticlike collision with the whole of the substrate molecule, transferring energy to/from the molecular center of mass: this is the adiabatic limit. In principle, a system could progress from one limit to another not only by changing the translational energy of the bath gas but also by changing the vibrational frequencies. So Toselli and Barker12 proposed that their conjecture could be partially tested by comparing classical trajectory results of protonated and deuterated molecules with experiment; in part, this work performs such a tat. We have carried out such calculations with azulenedo and a~u1ene-d~ with both xenon and helium using both a full trajectory calculation and the approximate biased random walk (BRW) model.*O These azulene systems were chosen for comparison with our earlier trajectory and modeling work; while experimental information is not available for deuterated azulene, it is possible to compare against other experimental ~ystems'~-'~*~* such as the closely related toluene-do/toluene-d8. Trajectory Method The trajectory method was that of our earlier paper using the public domain program MARINER.23 The nth moment of the
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The Journal of Physical Chemistry, Vol. 96, No. 21, 1992 8451
Deuteration Effects on Collisional Energy Transfer TABLE I: TOWEnergy-Tronofer Values (em-') from Trajectories aod from the BRW Model for Azulew-de rad Azukaed8 Colliding with Helium and Xemf
azulene-do trajectories BRW He Xe
azulene-d8 trajectories BRW
(A,?)
-416 f 2gb -92 f 1 1
-30 -160
4 3 0 f 29 -107 f 1 1
-29 -155
TABLE Ik Experimental Energy-Transfer Values for Deuterated "I Undeuterated Substrate Molecules (A,?), cm-'
bath gas 'He 'He Ne Ar
Kr
He Xe
824 + 45 397 f 22
(AE2)"2
133 340
860 f 60 407 f 18
131 336
'All values were normalized using a hard-sphere diameter of 6.97 A for xenon and 4.49 A for helium. buncertaintiesrepresent the average deviation as determined by a bootstrapping proccd~re.~
rate of energy transfer between a thermal bath gas and an excited substrate is given by5*7924
RE'., = J-dP dq [E - ~ s u ~ ( ~ - ~ ) l " f ( P , s ) ~ l / ~ ~QI0) ~(ql (1)
where the integrals are over all of phase space whose positions and momenta are p and q, q, is the bath gas/substrate relative translational coordinate, qlo is a large value of this coordinate where trajectories are started, p l / p = u is the corresponding velocity,/(p,q) is the distribution function describing the appropriate initial conditions, HSubis the Hamiltonian for the substrate molecule, the initial and final energies of the substrate are E'and E, AEi is the energy change in the ith trajectory, b, is the maximum impact parameter considered, and N is the number of trajectories. These moments can for convenience be expressed in terms of the moments of the mean energy transferred per collision, ( AE"),related to the corresponding rates by a chosen collision frequency (or hard-sphere diameter) (e.g., ref 6). In the present calculations, the number of trajectories was 2000 and the statistical l"n * ty calculated by the bootstrap method described previously.5 The calculations were for azulene at an internal energy of 30644 cm-l with Xe at a single fmed t r a n ~ l a t ienergy ~~l of 1.2 kcal mol-' (-2RT at T = 300 K) and with He at a translational temperature of 300 K, conditions chosen to take advantage of previous trajectories results.
Results and Discussion Table I shows the energy-transfer values for total energy for both azulene and azuleneds with xenon and helium. The values for d e n e & are slightly different from those reported previously5 owing to the greater number of trajectories used here, but it is useful to note that the smaller number of trajectories gives for most purposes a satisfactory estimate of the energy transfer. The most obvious remark to be made is that there is relatively little difference between protonated and deuterated azulene, even for helium where effects might have been most striking. Although the ( AE) values for He are in serious disagreement with experiment, it is felt in the present context that what is important is that the relative change upon deuteration with He bath gas is as small as with Xe; the discrepancy with experiment we ascribe to the potential function. As will be discussed later, this suggests that the problem of the classical probability distributions cannot intrude greatly in this system and, most importantly, cannot be the principal cause of the anomalously large energy transfer for light bath gases. While the helium results are in error from an *external" source (most likely the interatomic potential energy surface), we feel that the relative change can provide meaningful conclusions about deuteration. Nevertheless, in the following caution is made in making inferences from the trajectory results for helium.
Xe
toluene-do"
toluene-d8"
-60 f lb -62+ 1 -77 f 2 -112 f 3 -110 f 4 -124 f 6
-53 -60 -87 -126 -126 -138
f
2
f 1 f 3 f 3 f
4
+5
(A,?), cm-'
bath gas Ne Xe
isopropyl bromide-&' -130 f 20" -155 f 15
isopropyl bromide-d,' -110 -170
f f
20 30
Reference 14. Uncertainties represent f2a statistical errors. Reference 22. Uncertainties represent 80%confidence limits. A further comparison is made, in Table 11, with experimental values of (AE)for a number of systems. One must always be careful in comparisons across substrate molecules, bath gases, and energies, but it is certainly clear that across substrates and bath gases there is still only a small change upon deuteration. The comparison between experimental data for azulene and toluene should be the most important, for not only are the molecules similar but also the internal energies in experimental studies are comparable: in azulene at 30.5 X lo3 cm-' and toluene at 24.5 X lo3cm-I, the internal energy is about the same fraction of the total energy (internal plus zero-point energy). The dependence of ( M )upon the internal energy at such energies is significant but not especially strong. It is encouraging that the relative decrease in (AE)upon deuteration seen among the heavier bath gases seen experimentally is almost the same as seen in our trajectory results for xenon. This is additional confirmation that we are obtaining at least a reasonable description for heavy bath gases. However, we feel that the most important point to be gained from these comparisons is that the change is small for both theory and experiment. Consider now the changes in internal energy. The most obvious effect of deuteration on the dynamical behavior of azulene is the change in the vibrational frequencies. The CH stretching frequencies and to a lesser extent the CH bending modes change the most, decreasing by a factor of 2-II2while the low-frequency modes are relatively unchanged. Now,we find that the internal (principally vibrational) energy transfer is relatively unaffected by deuteration, and so it is reasonable to conclude, as have othe r ~ , ~that ~ ,energy ~ ~ ,transfer ~ ~ principally involves the lowerfrequency modes. This does not mean that the higher-frequency modes are unimportant: we have seen in our previous examination of individual azulene-do/xenon trajectories4 that the hydrogens are intimately involved. It seems reasonable for the energy to be removed from the most deformable modes-the stiffer modes essentially acting as intermediaries, passing the impulse along. In this sense a CD group may be little different from a CH. Another case supportive of this view is the much larger (between a factor of 8 and 20) average energy transfer in hexafluorobenzene/Ar collisions compared to benzene/Ar.26-28 Here the higher-frequency modes involved in the CH stretching and bending are completely removed, creating more low-frequency modes and correspondingly increasing the amount of energy transferred. Such a description of the collision process provides an explanation of why the errors inherent in using a classical energy distribution appear to be relatively unimportant in collisional energy transfer. Under classical mechanics there is an large excess of energy in high-frequency modes separated from the bulk of the modes, such as the CH stretches. However, these appear to play a small part in determining the size of the energy transfer. It is the effect of the concomitant energy deficit in the lower-frequency modes that should be more important. However, because of their lower frequencies, these modes will have a relatively large amount of energy in them and hence should be adequately described classically.
Clarke and Gilbert
8452 The Journal of Physical Chemistry, Vol. 96, No. 21, 1992
W Hdazulene-dg
W Xe/azuleneQ E! Xe/azulene-dg
AE (cm-1)
Figure 1. Histograms of number of trajectories with a given energy transfer AE for Xe at a translational energy of 1.2 kcal mol-'. The internal energy was 30644 cm-l, and the total number of trajectories was 2000. Values of AE used here are the from the raw trajectory data, rather than being scaled to a hard-sphere diameter. Inserts show magnification of supercollision region.
It thus appears that we might think of azulene as a mixed case with both low- and high-frequency modes. The high-frequency modes are in the adiabatic range of energy transfer, and the lower frequencies modes are closer to the spectator limit as described above. This gives the agreeable property that energy transfer is more efficient out of the low-frequency modes, and therefore the higher-frequencymodes take on the role of a (stiff) intermediary. What are the other consequences of this view? Firstly, this suggests that the energy transfer should become more efficient with decreasing bath gas mass, since for a given temperature the translational velocity increases, leading to more impulsive collisions. This is however not the case for a ~ u l e n e . ~ The answer to this is that the above considerationsapply to single impulsive events, and heavier bath gases are much more involved in chattering collisions than are the lighter ones. It would then seem reasonable to extend our picture so that the chattering collisions follow the same principles as the impulsive collisions: they are simply a sequence of impulsive collisions (as is indeed the essence of the BRW model). That is, the efficient energy transfer is still out of the low-frequency modes with the CH stretches playing a small part. The isotopic comparisons we make should then be relatively independent of the fact that they involved in chattering collisions. When azulene is deuterated, the low-frequency modes are essentially unchanged but the CH stretching modes, where the classical overabundance of energy is concentrated, drop by about 700 cm-1.29The question becomes; is this drop in frequency large enough so that all these modes become essentially low frequency? This would imply that energy transfer would be slightly more efficient out of the CD modes, and the classical excess of energy in these modes becomes significant, leading to error. The trajectories show little change in energy transfer for either bath gas so the CD modes appear to behave adiabatically as well. Comparisons of the histograms of the individual trajectories for total energy transfer for both xenon and helium are given in Figures 1 and 2. These plots give a crude energy-transfer probability function and show that the function does not appear to change appreciably upon deuteration, given that the average values are shifted slightly.
Supercollisions Supercollisions are relatively rare occurrences (typically 1-2%), seen both in e ~ p e r i m e n tand ~ ~ ~in~trajectory ' cal~ulations,4*~~ in which extremely large amounts of energy are transferred. Detailed examination of trajectories reveals4Jothat their occurrence requires two dynamical events to take place at the same time: (1) a substrate atom is squashed between the bulk of the substrate and the bath gas (an event which is of short duration but occurs fairly commonly in a typical collision and results in a large energy flow between bath gas and substrate; usually however this flow is reversed as the squashed atom recoils, resulting in no overall significant net energy flow), while at the same time (2) the centers of mass of bath gas and substrate move in such a way that the
El Hehzulene-dg
AE (cm-1)
Figure 2. As for Figure 1, but for He at a translation temperature of 300 K.
squashed atom is taken away from close interaction with the bath gas so that the usually temporary larger energy flow arising from this squashing remains permanent. The present trajectory data show a supercollision regime (magnified in the inserts in Figures 1 and 2), as seen pre~iously."~~ From the present trajectory results, it appears that the fraction of supercollision events, which can be gauged by the expanded section of each plot, is also not greatly affected by deuteration. This is particularly interesting since there appears to be an important role for the hydrogen in supercollisions, such as being abnormally compressed by the bath gas against bulk of the substrate. While this might suggest that deuteration would have a significant effect on the number of supercollisions, it could also be taken to imply that the outer (squashed) atoms are merely intermediaries in supercollisions. Our prediction of little change in the number of supercollisions upon deuteration should be amenable to experimental test. Comparison witb BRW Model The BRW "model B" is an approximate description for the collisional-energy-transfer process that has shown to provide reasonable qualitative and quantitative agreement with the trajectory data for the azulene/Xe system without any semiempirical parameters. Implementation is trivial (and is available in the public domain UNIMOL program suite33),requiring only a numerical quadrature. The basic assumption of the mode110J6917920 is that the energy-transfer event is controlled by a gradual drift in the internal energy occuring as this internal energy shows multiple fluctuations during the collisions arising from the chattering nature of the bath gas/substrate dynamics: a Brownian-like diffusion in internal energy. This pseudorandom drift is constrained by microscopic reversibility. The model has been shown to give quite acceptable accord with both experimental and trajectory data in many circumstances.1o The effect of deuteration on energy transfer in the azulene/Xe system provides a further test of the adequacy of BRW. The significance of the comparison with trajectory data should be stressed as the BRW model uses essentially the same potential surface as the trajectories but a p proximates the classical dynamics. For lighter bath gases, however, the agreement of BRW with trajectories is poorer, probably resulting principally from the fact that the basis for the model is that the collisions are of a chattering nature; nevertheless, the behavior of the BRW for azulene-d8/He may still provide some insight. The comparison of trajectory data with BRW results is given in Table 1. The small size of the isotope effect in either ( AE) and ( M2) 1/2 is successfully reproduced by the BRW model even though the actual values of the ( AE) are slightly larger in the case of xenon and significantlysmaller in the case of helium. The fact that predicted changes on deuteration are smaller and of the wrong sign should not be judged too harshly as the origin of the these changes is a small and subtle one which is beyond the simplified dynamical assumptions made in the BRW treatment. In the context of the influence of high- and low-frequency modes on the energy transfer, it is noted that in the BRW model highfrequency modes, while giving rise to the rapid energy fluctuations
J. Phys. Chem. 1992,96, 8453-8461 that give rise to the model, are not predicted to be especially important in the overall energy transfer (in the same way that individual solvent molecule/particle collisions in macroscopic Brownian motion do not contribute much to the overall diffusive motion): indeed, in the most primitive version of the model, the average energy transfer is Seen to be completely independent of the highest-frequency modes of the substrate.1°
Conclusions Trajectory calculations and the BRW model agree with experiments on related systems: that there should be only a small isotope effect in the energy transfer between highly excited azulene and monatomic bath gases. This suggests that the low-frequency modes have an important role to play in determining the amount of energy transfer. In particular, the lack of any strong effect with helium suggests that classical calculations can provide an adequate representation, and the poor accord between experiment and trajectory results for light bath gas= is due to the poor description of the interaction potential. The results also predict that isotopic substitution should not have a strong effect on the fraction or the magnitude of supercollisions. Acknowledgment. The support of the Australian Research Grants Scheme is gratefully acknowledged. We also appreciative of interesting discussions with Professor John Barker and for his providing us with preprints of his work, as well as helpful interactions with Dr. Kieran Lim. Registry No. Xe, 7440-63-3; He, 7440-59-7; D2,7782-39-0; azulene, 275-51-4.
References and Notes (1) Tardy, D. C.; Rabinovitch, B. S . Chem. Reo. 1977, 77, 369. (2) Oref, I.; Tardy, D. C. Chem. Rev. 1990, 90, 1407. (3) Gilbert, R. G.; Smith, S.C. Theory of Wnimolecular and Recombination Reactions; Blackwell Scientific: Oxford, 1990. (4) Clarke, D. L.; Thompson, K. C.; Gilbert, R. G. Chem. Phys. Lerr. 1991, 182, 357.
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(5) Clarke, D. L.; Oref, 0.;Gilbert, R. G.; Lim, K. F. J. Chem. Phys. 1992, 96, 5983. (6) Lim, K. F.; Gilbert, R. G. J . Phys. Chem. 1990, 94,72. (7) Lim, K. F.; Gilbert, R. G. J . Phys. Chem. 1990,94,77. (8) Buck, U.; Kohl,K.H.; Kohlhase, A.; Faubel, M.;Staemmler, V. Mol. Phys. 1985, 55, 1255. (9) Buck, U.; Kohlhase, A,; Secrest, D.; Phillips, T.; Scoles, G.; Grein, F. Mol. Phys. 1985,55, 1233. (10) Gilbert, R. G. Inr. Reo. Phys. Chem. 1991, 10, 319. (11) Gilbert, R. G.; Zare, R. N. Chem. Phys. Leu. 1990,167,407. 112) Toselli. B. M.: Barker. J. R. Chem. Phvs. Lett. 1990. 174. 304. (13) Toselli,’B. M.;Brenner,.J. D.;Yerram, M:L.; Chin, W. E.; King, K. D. J . Chem. Phys. 1991.95, 176. (14) Toselli, B. M.;Barker, J. R. J . Chem. Phys. 1991, 95, 8108. (15) Date, N.; Ha=, W. L.; Gilbert, R. G. J. Phys. Chcm. 1984,845135, (16) Gilbert. R. G. J . Chem. Phvs. 1984.80. 5501. (17) Lim, K. F.; Gilbert, R. G. k . Chem. Phys. 1986,84, 6129. (18) Lim, K. F.; Gilbert, R. G.; Brown, T. C.; King, K. D. Inr. J . Chem. Kinet. 1987, 19, 373. (19) Lim. K. F. Ph.D. Thesis, University of Sydney, 1988. (20) Lim, K. F.; Gilbert, R. G. J. Chem. Phys. 1990, 92, 1819. (21) Levine, R. D.; Bernstein, R. B. Molecular Reaction Dynamics and Chemical Reacriuity; Oxford University: New York, 1987. (22) Brown, T. C.; King, K. D.; Gilbert, R. G. Inr. J . Chem. Kiner. 1987, 19, 851. (23) Hue, W. L.; Lim, K. F. Program package MARINER (a general
Monte Carlo classical trajectory computer program). Available from second author, Department of Chemistry, University of New England, NSW 2351, Australia. (24) Porter, R. N.; Raff, L. M. In Dynamics of Molecular Collisions; Miller, W.H., Ed.; Plenum Press: New York, 1976; Vol. B; p 1. (25) Yardley, J. T. Inrrduction to Molecular Energy Transfer, Academic Press: London, 1980. (26) Nakashima, N.; Yoshihara, K. J . Chem. Phys. 1983, 79, 2727. (27) Ichimura, T.; Mori, Y.; Nakashima, N.; Yoshihara, K. J. Chem. Phys. 1985,83, 177. (28) Ichimura, T.;Takahashi, M.;Mori, Y. Chem. Phys. 1987,114, 111. (29) Chao, R. S.;Khanna, R. K. Spectrochim. Acra 1977. H A , 39. (30) Hassoon, S.;Oref, I.; Steel, C. J . Chem. Phys. 1988, 89, 1743. (31) Lthnannsr6ben. H. L.; Luther, K. Chem. Phys. Lerr. 1988, 144,473. (32) vndvay, L.; Schatz, G. C. J. Phys. Chem. 1990.94, 8864. (33) Gilbert, R. G.; Smith, S.C.; Jordan, M. J. T. UNIMOL program suite (calculation of falloff curves for unimolecular and recombination reactions); 1991; available directly from the authors: School of Chemistry, Sydney University, NSW 2006, Australia.
Thermodynamics of Micellization at Charged Interphases within the Framework of the Phaseseparation Model P. Nikitas,* S. Sotiropoulos, and N. Papadopoulos Loboratory of Physical Chemistry, Department of Chemistry, University of Thessaloniki, 54006 Thessaloniki, Greece (Received: September 16, 1991)
An application of classical thermodynamics is made in the case of surface phase transitions occurring throughout a charged
interphase. This kind of surface phase transitions characterizes the aggregation of micelle-forming surfactants within the interphase formed between electrolyte solutions and an ideally polarized electrode, at least at concentrations higher than the critical micelle concentration. On the basis of a multilayer model for the electrical double layer, the thermodynamic conditions for its stability and the reversibility of a pbseparation proc~sswhich extends along the double layer are established. It is further shown that a rigorous but sufficient thermodynamic criterion for a surface micellization to take place across a charged interphase is the existence of two deformed peaks in the differential capacity versus applied potential curves on both sides of the adsorption maximum. The peaks may have comers and/or abrupt vertical segments and they may be split into two or more parts. Moreover, due to hysterisis phenomena their shape may depend on the potential scan rate and scan direction. Some expected deviations from the above criterion are also discussed.
I. Introduction The thermodynamic description of the properties of micellar systems in bulk aqueous solutions has attracted much attention for many years. The most crucial points have been the prediction of a surface tension minimum and the calculation of the ther0022-3654/92/2096-8453$03.00/0
modynamic functions of micellization. The interpretation of the interfacial properties of micellar solutions is usually based on either the phase separation’+ or the mass model. The possibility of the formation of a new phase consisting of micelles, above the critical micelle concentration Q 1992 American Chemical Society
