J. Phys. Chem. 1991,95, 5007-501 1 suggesting a complicated nature of the potential surface in the SI(possibly 2’ A,) state. The apparent similarity in the spectral features of the ion radicals and the TI signals may be explained by a simple MO scheme of HOMO and LUMO in the DPH (neutral) ground state. The assignment of the observed vibrational modes of the DPH ion radicals is in progress at the moment.
Conclusions The long-lived species observed in such polar solvents as acetone, ethanol, and M T H F was confirmed to be the cation radical by comparing the transient CARS spectrum with that of the 1,4dioxane solution of DPH containing DCNB as a strong electron acceptor. The formation of the charge-transfer exciplex in the 1,4-dioxane solution of DMA/DCNB was confirmed by comparing the CARS spectrum with that of the chemically produced DPH*-.
5007
The lifetime of the cation radicals produced in acetone, ethanol, and MTHF was derived as 1.5 ps, 4.0 ps, and 7 2 ns, respectively, indicating the role of polarity-dependent solvation in the stabilization of the species. The lifetime of the cation radical in the DPH/DCNB system and that of the anion radical in the DPH/DMA system were 60 and 110 ns, respectively, when the concentration of the electron donor or acceptor was 0.1 M in 1,Cdioxane. These lifetimes are longer by a t least 1 order of magnitude than the fluorescence lifetime of the solutions, indicating the formation of the geminated ion pairs or solvated free ions. Some evidence of both monophotonic and biphotonic processes of the DPH cation radical in the polar solvents such as acetone and methyltetrahydrofuran was deduced, suggesting the presence of competition between the ionization by electron ejection from a twephoton-excited state and the solvent-assisted photoionization from the S I state in acetone.
Collisional Energy Transfer at Hlgh Temperatures from the Biased Walk Model Robert G . Gilbert* School of Chemistry, Sydney University, NS W 2006, Australia
and I. Oref* Department of Chemistry, Technion-Israel Institute of Technology, Haifa 3200, Israel (Received: October 19, 1990)
A new expression for the probability, P(E,E’), of collisional energy transfer for large polyatomics at high temperatures is deduced by using the assumptions of the biased random walk (BRW) model [Lim, K. F.; Gilbert, R. G. J . Chem. Phys. 1986,84,6124] and of a Gaussian equilibrium population distribution. The latter approximation is shown to be very amrate at sufficiently high temperatures. Simple analytic expressions are obtained for P(E,E’), which show the correct limiting weak- and strong-collision forms as the duration of a collision approaches zero and infinity, respectively. Expressions arc as functions of initial energy E’. The expression found for the average energy-transfer quantities (A&), (A&), and (a2), for (AE,,,)explicitly shows the change of sign as E’is varied over a wide range of E’. The new BRW expression for P(E,E’) can be accurately approximated as Gaussian. The derived expressions will be of use in modeling high-temperature processes such as combustion.
Introduction An understanding of the rates of collisional energy transfer between a highly excited molecule and a bath gas is necessary for the interpretation of unimolecular and recombination rate coefficients and finds particular application in modeling studies, such as for combustion and atmospheric chemistry.’s2 The fundamental means of quantifying this energy-transfer process is as the rate coefficient R(E,E’) for collisional energy transfer from internal energy E’to E.’ Methods are now available for classical trajectory calculations of R(E,E’)and of its moments (only the first or second moment being required for computing quantities such as falloff curves). However, such trajectory calculations require computer codes and computational resources that are not readily available at present; nor do they provide physical insight per se. Tractable approximate models are therefore essential tools for practical applications. For the purposes of providing approximate models for collisional energy transfer, it is useful to reexpress R(E,E’) in terms of the probability per collision P(E,E’) P(E,E3 R(E,E?[Ml/ w (1) where [MI is the concentration of bath gas and o is an arbitrary (but phYsica1ly reference frequency’ A
* Author for correspondence. 0022-3654/91/2095-5007$02.50/0
number of models for P(EB’), applicable to various temperatures, bath gases, and excited molecules, have been given in the literature. However, no model presented hitherto has been properly applicable to large molecules and/or elevated temperatures, for reasons that are discussed in detail below. We present here the first model which is able to overcome these defects. We commence by briefly reviewing some well-known results for P(E,E’). Experimental variables determined by P(E,E’) or R(EB’) include falloff curves (the dependence of a rate coefficient on pressure) and the time evolution of the average energy. It is often found that these observables depend strongly on one moment of R(E,E’) or P(E,E’) and are less sensitive to other details of the functional form of these distributions. One such moment is the overall energy-transfer rate coefficient RE,,’,or equivalently, the mean energy transferred per collision, (PEaI1)= R F . ~ [ M ] / W RE,,’= L m ( E- E’)R(E,E’)d E
(2)
(3) ( I ) For example: Gilbert, R. G.; Smith, S. C. Theory o/Unimolemkrr and RecombiMrion Reoc?ionF;Blackwell Scientific: Oxford, U.K.,1990. Gilbert, R. G . In?. Rev. Phys. Chem., in press. (2) For example, Oref, I.; Tardy, D. C. Chem. Reo. 1990, 90, 1407.
0 1991 American Chemical Society
5008 The Journal of Physical Chemistry, Vol. 95, No. 13, 1991
Gilbert and Oref
of a reacting molecule, in a unimolecular reaction) are near the average energy of the ensemble; this is often the case, for example, in high-temperature systems. A m P ( E , E ’ )d E = 1 (4) Other models for P(E,E? have included ergodic approaches, has been invoked. Another suitable moment is the mean-square wherein some degree of ergodic energy transfer between excited or equivalently the mean-square energy rate coefficient molecule and bath gas is assumed. Such approaches have been transferred per collision ( M z=) Rs,2[M]/u,which are defined used by Schlag and co-workers,’ Lin and Rabinovitch? Bhattaby expressions similar to eqs 2 and 3 with ( E - Eq2 replacing ( E charjee and F ~ r s tOref,Io ,~ Troe,” and Nordholm and co-work- E ’ ) . Given R(E,E’), experimental observables such as a falloff ers.I2J3 Some of these models yield functional forms for P(E,E’) curve or the time evolution of .the average energy of a highly and readily evaluated expressions for (A&), etc. The “IECT” excited molecule can be obtained by solving the appropriate master model of Nordholm and Schranz” in particular is able to give equation. quite acceptable accord with experiment where the bath gas is One can discern several separate theoretical problems with a polyatomic containing, say, more than four atoms. However, regard to energy transfer: none of these models is able to give acceptable agreement with 1. One goal is to find functional forms for R(E,E’) or P(E,E’) data for the important case where the bath gas is monatomic or which accurately reflect the underlying physical processes. This diatomic. is desirable because (a) experimental observables such as falloff A recently developed model is the “biased random walk” curves, etc., are not completely insensitive to the form of P(E,E’), (BRW) treatment.IeI6 This model is based on precepts (which and the dependence can be experimentally significant (e.g., a will be discussed in detail later) emanating from trajectory change as large as 50% is possible on going from an exponential data.1k18 Moreover, expressions have been derived16which enable to a Gaussian P(E,E? with the same (AEall)); (b) methods for this model to be used for simple estimates of quantities such as obtaining experimental information on this functional form are (Mall) which give quite good accord with trajectory data for available? and (c) improved models for the functional form may monatomic bath gases. However, the model as presented thus help to elucidate the underlying mechanisms of energy transfer. far also assumes an exponential f ( E ) for some aspects of its 2. Another goal is to estimate the parameters in these functional derivation (as discussed in detail below). This cannot be accurate forms (or alternatively, one of the above moments such as (Mal)) for systems a t high temperatures. Such systems are of great from first principles and/or from experimental data. Given such technical importance, for example, in modeling combustion. In information (either for the collision partners in question or for the present paper, we present an extension to this model which similar ones), one can predict rate data. overcomes this difficulty. 3. A third goal is to find quantities such as (Mall) and ( M 2 ) Model Development for a given functional form. If the form of P(E,E’) is specified (e.g., as exponential), then knowing one moment (e.g., ( M 2 ) ) The BRW model has been discussed extensively elsewhere,lvlc16 completely specifies all other moments. This is useful, for example, and only a cursory treatment will be given here. Its starting point (a) where one has a theory which yields a value of (AI?) whereas is the observation, an outcome of trajectory studies,Icl8 that the an approximate solution of the master equation is available in internal energy of the highly excited molecule varies in an apterms of (AEall);(b) where one has an experimental value of parently random fashion during the collision with the bath gas. ( AEaII)(which is an experimentally observable quantity) and The collision duration, for monatomic or other small bath gases, wishes to know the parameters for P(E,E’) to use in a numerical is short (of the order of a picosecond or less) so that the energy solution of the master equation. Because the falloff curve can transfer is unlikely to be ergodic. The apparant randomness depend on the functional form for P(E,E’) as well as on one of suggests that the energy-transfer process might be treated as a its moments (say, (AI&)), for such purposes one needs as type of diffusion in “energy space”. Such diffusion must be subject physically realistic form for P(E,E? as possible. to microscopic reversibility. That is, in the description of the I n the present paper, we address the first and third of these motion of the molecule as a Brownian particle in energy space, problems, providing expressions that are valid at high temperathe probabilities of up versus down collisions are not the same but tures, where no previous description can be properly applied. instead are related by the constraint of microscopic reversibility: There have been a number of models suggested for P(E,E’); P(E,E? f ( E ? = P(E’,E) f ( E ) (5) the subject has recently been reviewed,IJ and so only a brief outline will be given here. The commonest of these is the exponentialThis biases the random motion, and forces particular relations down model. Very often, this model for P(E,E’) is coupled with between the up ( E > E’) and down ( E < E’) transition probaan exponential model for the equilibrium population,f(E). This bilities: hence the name “biased random walk”. latter approximation has some mathematical advantages: it leads Given that the energy-transfer process during the collision is to simple approximate solutions to the master equation, and random, the probability distribution should be governed by an analytical or semianalytical relations between moments such as equation describing diffusion in an external field (the detailed ( AEaII)and ( M 2 (as ) deduced by Troe and others,“ Barker and balance constraint). This is the Smoluchowski equation (someGolden? and Tardy and Rabinovitch6). However, the exponential times also referred to as the Fokker-Planck equation, although model for P(E,E’)has no physical basis. Moreover, the assumption the latter term is also used for a more complete description) of an exponentialfiE) does not apply to large molecules at elevated temperatures. One case where this breakdown is important is when - = a D ( E a)sz + z ( E ) B ] as the treatment is unable to take into account the “inversion” effect. at aE Recall that (AE,,,) is an experimental observable. The inversion effect is the change in sign of (A&), which is a function of the initial energy E’, as E’is varied. Inversion occurs as E’approaches (7) Serauskas, R. V.; Schlag, E. W. J . Chem. Phys. 1966,45, 3706 and references therein. the average energy. Taking such inversion into account is im(8) Lin, Y. N.; Rabinovitch, B. S. J . Phys. Chem. 1970, 74, 3151. portant if the energies under consideration (eg., the average energy
where, in eq 3, the normalization condition
-[
(3) For example, (a) King, K. D.; Nguyen, T. T.; Gilbert, R. G. Chem. Phys. 1981,61,221. (b) Hassoon, S.;Oref,I.; Steel, C . J. Chem. Phys. 1988, 89, 1743. Morgulis, J.; Sapers, S.;Steel, C.; Oref, I. J . Chem. Phys. 1989, 90,923. Luther, K.; Reihs, K. Ber. Bumen-Ges. Phys. Chem. 1988, 92,442. (4) Troe, J . J . Chem. Phys. 1977,66,4745. Gilbert, R. G.; Luther, K.; Troe, J . Ber. Bunsen-Ges. Phys. Chem. 1983, 87, 169. (5) Barker, J . R.; Golden, R. E. J . Phys. Chem. 1984.88, 1012. (6) Tardy, D. C.; Rabinovitch, B. S.(a) J . Phys. Chem. 1985, 89, 2442 (b) J . Phys. Chem. 1986, 90,1187.
(9) Bhatacharjec, R. C.; Forst, W. Chem. Phys. 1978, 30, 217. (10) Oref, I. J . Chem. Phys. 1982, 77, 1253. ( 1 1) Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1973, 77, 665. (12) Nordholm, S.;Freasier, B. C.; Jolly, D.L. (a) Chem. Phys. 1977,25, 433; (b) Chem. Phys. 1978, 32, 161, 169. (13) Schranz, H. W.; Nordholm, S.Inr. J . Chem. Kinet. 1981, 13, 1051. (14) Gilbert. R. G. J . Chem. Phys. 1984,80, 5501. ( 1 5) Lim, K. F.; Gilbert, R. G. J . Chem. Phys. 1986,84,6 124. (16) Lim, K. F.; Gilbert, R. G. J . Chem. Phys. 1990, 92, 1819. (17) Date, N.; H a w W. L.; Gilbert, R. G. J. Phys. Chem. 1984,88,5 135. (18) Lim, K . F.; Gilbert, R. G. J . Phys. Chem. 1990. 94, 77.
The Journal of Physical Chemistry, Vol. 95, No. 13, I991 5009
Collisional Energy Transfer from the Biased Walk Model where B(E,r;E’) is the probability of the excited molecule having internal energy E during the collision at time t, the collision having started with initial energy E’, D is the “diffusion coefficient” in energy space (a formal definition has been given by Lim and Gilbert15J6)and z is a quantity which takes into account microscopic reversibility: the “bias”. One might suppose that both z and D could have an arbitrary dependence on E. However, so as to have a viable approximate model, possible E dependences of D and z are either ignored (as in the original treatmenti4)or approximated (as in the present treatment). Moreover, it is assumed in the model that the collision starts instantly at time t = 0 and finishes abruptly at the collision duration t = 1,. Of course, in actuality, a ‘collision” can only be defined for hard spheres, but the assumption of a sharp start and finish has been shown16 to be an acceptable approximation. P(E,E? is then identified as B(E,t=t,;E’). In the original treatment,I4 it was assumed that z(E) and D(E) are both independent of E. Solution of eq 6, subject to eq 5, then leads to the simple result
P(E,E? =
[
-b[t - c’ exp(-bt,)12 b 2 r 6 [ 1 - exp(-bt,)] ]Ii2exp[ 2b[ 1 - exp(-bt,)]
]
(11)
where 6 = Dbz and t’ = a + bE‘. The quantities u and b will be determined from the requirement of microscopic reversibility, eq 5. Substituting eqs 9 and 1 1 into eq 5 yields: a = -bE
(12)
exp(-bt,) = (D/28b) - 1
(13)
Combining eqs 11-1 3 and substituting for t yields the required expression for the collisional energy-transfer probability -[E - E - (E’4$(1
&I2
- q2)
1
(14) where q = exp(-bt,). The P(E,E’) of eq 14 obeys both the normalization condition, eq 4, and microscopic reversibility, eq
5. As will be seen, eq 14 is approximately Gaussian in AE = E
- E’. Some insight can be gained by first noting (as will be seen
where the energy-independent quantity z is be given by z(E) z = -a Inf(E)/dE (8) In actuality, of course, the right-hand side of eq 8 varies with E. Assuming z is constant implies thatf(E) is exponential. This implies inter alia that A E ) is monotonically decreasing. This is an acceptable approximation for all except large reactant molecules and/or high temperatures, but obviously is inapplicable to energies which are near the maximum in A E ) which is important, for example, at higher temperatures. Consider now situations wheref(E) cannot be treated as exponential in the range of interest. This defect can be partially remedied by employing eq 7 only for downward transitions (E < E’), and evaluating P(E,E’) for upward transitions through microscopic reversibility, Le., from eq 5 with the exactAE), rather than an exponential approximation. Since the P(E,E’J so obtained obey microscopic reversibility exactly, they yield the correct sign change in ( AEaII)(indeed, a public program for the solution of the master equation19makes use of this trick). This device always yields solutions of the master equation (giving the unimolecular rate coeMicient as the largest eigenvalue of the appropriate matrix) which take exact amount of microscopic reversibility (with all the important mathematical ramifications therefrom) for a given functional form of P(E,E’) for E < E’. However, this treatment does not fully take into account the dynamical requirement of microscopic reversibility in deducing the form of P(E,E’) for E < E’. Hence this cannot yield an accurate model for P(E,E’) for situations where f ( E ) cannot be approximated as exponential. We now show that this defect can be rectified by assuming a different functional form for A E ) and simultaneously changing the form of z(E). The first of these steps is to replace the assumption of an exponentialAE), which cannot show a maximum, with a Gaussian form A E ) = (27r1I2c)-’exp[-((E
- E)/2cI2]
(9)
which can show the required maximum. Here c and E are constants to be determined from the actual population distribution. The second step in our extension of the BRW model is that, instead of being independent of E, z is taken to be linear in E: z(E) = u + bE (10) It will be seen that, given theAE) of eq 9, the assumption of eq 10 enables microscopic reversibility to be satisfied completely. Making the transformation c = u bE, and using eq 10, eq 6 can now be solved by Fourier transformation. The result is
+
~
( ! 9 ) Smith,
S.C.; Jordan, M. E.; Gilbert, R. C. UNIMOL program suite.
Available from the authors (Department of Theoretical Chemistry, Sydney University, N.S.W.2006, Australia).
later) that bt,
