AIAA-02-1373

________________* Researcher, Centro Studio Chimica dei Plasmi -CNR† Professor, Dipartimento di Chimica - Universita' di Bari‡ Professor, Dipartimento di Chimica - Universita' di BariCopyright © 2002 by M.Capitelli Published by the American Institute of Aeronautics and Astronautics, Inc. with permission

QUASICLASSICAL CALCULATIONS OF DISSOCIATION CROSS SECTIONS FOR THE PROCESS: O+O2(V)->3O: COMPARISON WITH PHENOMENOLOGICAL

THERMAL DATA

F.Esposito*, M.Capitelli† and C.Gorse†

Centro di Studio per la Chimica dei Plasmi del CNR

Dipartimento di Chimica - Universita' di Bari

Via Orabona 4, 70126 Bari, Italy.

AbstractComplete sets of dissociation cross sections for the process O + O2(v) -> 3O from each vibrational state v supported by the PES used for different rotational temperatures have been calculated by using a quasiclassical trajectory method (QCT) on the PES of Varandas and Pais. The results are globally compared with experimental thermal data. The relevant problems of computational load, quasibound states and multisurface factors are discussed.

IntroductionKinetic studies of atmospheric species are nowdays of large interest in fields as plasma chemistry, plasma physics, laser chemistry and dynamics of expanding flows. State to state vibrational kinetics requires the knowledge of rate coefficients involving each vibrational level of the ground electronic state of the molecule eventually averaged on a thermal distribution of rotational states [1]. In the application of DSMC (Direct Simulation) Monte Carlo method not simply rate coefficients but state to state cross sections are required, that is a much more detailed kind of information [2]. Being very large the amount of dynamical calculations required, it is mandatory to choose an appropriate computational method to obtain reliable results in a reasonable amount of time. To this end a quasiclassical trajectory method has been developed and used in our group to obtain cross sections for the following atom-diatom processes

M+M2(v,J)∅M+M2(v',j')

M+M2(v,J)∅3M

where v,j represent vibrational and rotational quantum numbers.In particular we have studied the system O-O2(v,j), limiting however our presentation to the dissociation process. Actually the rotational number does not

appear explicitly in our results, because we average the results on distributions at various rotational temperatures. We have used the potential energy surface (PES) of Varandas and Pais [3], the same PES used by Lagana' [4] in his calculations for vibrational energy transfer.

2.Method of calculationAs suggested by our preceding works on similar problems, concerning hydrogen and nitrogen collision induced dissociation [5,6], we used in this work quasiclassical calculations (QCT) in order to achieve results with reasonable amount of computational time, with a good global reliability for the specific problem. With over 6000 rovibrational states supported by the PES used, it has been necessary to adopt some techniques for drastically reducing computational time. The original code for QCT, totally written in our group in Bari [7], has been parallelized using the well known MPI [8] environment at CASPUR [9]. Lacking a dependence among the various trajectories, QCT code is practically perfect for parallelization: each processor receives a given batch of trajectories to calculate and can for example write directly its results in a file, with minimal information exchange with other processors, and consequently very high parallel performance. Using parallel computers is necessary in this case: on a single fast processor, this work would have required not less than four months of continuos calculations. But parallelization is not the only technical aspect of relevance to limit computational time. The choice of time step (TS) for trajectory integration is crucial: it should be neither too small (causing slow calculations), nor too large (causing wrong trajectories). Often one finds a compromise by choosing euristically a TS that gives a very high fraction of "good" trajectories (TJ) as a mean. The best way of judging a trajectory calculation is to back-integrate it or re-integrate it with smaller TS, reproducing respectively the initial condition or the final one. But both these methods are of course very time-expensive, and often one simply checks for total

8th AIAA/ASME Joint Thermophysics and Heat Transfer Conference24-26 June 2002, St. Louis, Missouri

AIAA 2002-3219

Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

American Institute ofAeronautics and Astronautics

2

energy conservation for each TJ and back-integration for one TJ in ten. Actually there are cases in which it can be much better to check each trajectory, starting with a time step relatively large, reducing it progressively if the test fails. For this work, we found extremely useful to start each TJ with a TS ∆to which is small enough to obtain more than 95% of "good" TJ. For each TJ, we perform a first integration with ∆to, a second integration from the same initial

conditions with ∆t1 = 3/4 ∆to, and check for final conditions with a small tolerance. If this test fails, we perform the third integration with ∆t2 = 3/4 ∆t1, and check with the second TJ, and so on up to (eventually!) the fifth integration. Contrarily to what one can guess, computational time for wrong trajectories is not proportional to the inverse of TS at all, as we can say after many experiments. On the contrary it grows much faster, because these trajectories wander for umpredictable times on a very structured PES as this one for O3.In this way, even if we have to spend large time for checking "difficult" TJs, really we get a globally faster calculation, because only rarely we spend much more time than about twice the minimum, corresponding to ∆to. Experimentally we can say that we spend from 20 to 90% of mean computational time more than for "good" trajectories (good at first test), but one should also realize that ∆to required in this method is quite larger than in the normal case of checking of one TJ in ten. Besides, the fraction of wrong TJs is extremely low (surely better than 0.1%), and in any case they are totally eliminated from cross section calculation (this is impossible for one in ten checking).We calculated cross sections from only one in five vibrational states to further reduce computational load, starting from v=0 to v=45, plus the last one v=46, which is of particular importance for dissociation. All the rotational states associated with each vibrational state has been considered (they have been calculated with WKB approximation from O2potential of Varandas and Pais[3]).

3 Cross sectionsIn order to achieve reliable results for translational temperatures from 300K to 10000K, we chose a translational energy interval from 0.001 to 3 eV, with continuos distribution. The results of the dynamical calculations are rotationally averaged vibrational-depending cross sections with a given rotational temperature (this last can be chosen at will at the trajectory analysis stage):

σdiss(v) = Qr−1 gjj∑ (2 j +1) ⋅

exp(−Ev, j / kTrot ) σdiss(v, j)(1)

with gj being nuclear degeneracy (as well known, 16O2 can have only odd rotational states), and Qrbeing rotational partition function. The sum is over the rotational quantum number j. The energy axis is discretized with 300 bins, with very large statistical error on dynamical results, because of small number of TJ in any specific bin. This procedure is useful if coupled with an efficient method of data smoothing. In this work a weighted least square (LS) procedure has been used, with weights decreasing with the distance from the point of interest. These weights w(ε) as a function of the translational energy ε are obtained in this way:

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3

v=20v=25v=30v=35v=40

v=17v=22v=27v=32v=37

Å2

translational energy (eV)

Fig.1a. Rotationally averaged initial vibration depending dissociation cross sections for a rotational temperature of 500K.

w(ε) = exp[ -|ε-εο| /∆]. These weights are applied to the points used in the LS procedure in the neighbourhood of any given point of energy εο, in

order to give more importance to points nearest to εοon the energy axis. We apply three times this LS

American Institute ofAeronautics and Astronautics

3

procedure consecutively, with an increasing value of ∆ (0.03, 0.05 and 0.1 eV) empirically determined.As explained in the preceding section, we calculate cross sections from actually only one in five vibrational states. Cross sections for intermediate values of initial vibrational quantum number are obtained by linearly interpolating the logarithm of the cross sections between to two nearest vibrational values. It is important in this case to preserve the thresholds of dissociation cross sections: this should be very difficult on a linear scale, because the threshold should be invariably equal to that one of the higher v-value cross section. Exploiting the fact that we are using rotationally averaged cross section with a minimal temperature value of 300K, actually no cross section is exactly zero, because of the presence of high lying rotational states (eventually with low weight), therefore we can succesfully interpolate the logarithm of the cross section on the whole energy axis.

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3

v=20v=25v=30v=35v=40

v=17v=22v=27v=32v=37

Å2

translational energy (eV)

Fig.1b As in 1a for a rotational temperature of 5000K.

In fig.1a there are some examples of dissociation cross sections from v=17 to v=40, at rotational temperatures Trot=500K, while in fig.1b Trot=5000K. In both cases the effect of rising the rotational temperature is clear, but the thresholds of cross sections from low lying states are displaced towards low energy values in a dramatic way. It should be considered that the number of rotational states associated with a vibrational state v decreases when v is increased, therefore dissociation cross sections from low lying vibrational states have generally very large dependence on rotational temperature. In both fig. 1a and 1b some examples of interpolated cross sections are shown (from vibrational states not multiple of 5). In these linear plots interpolated cross sections appear to have reasonably correct interpolated thresholds, as a consequence of the procedure adopted.

4 Rate CoefficientsIn fig.2 we show some rate coefficients obtained from calculated cross sections. A seventh order polynomium is used in logarithmic scale to describe accurately their values at different ro-translational temperatures, starting from rate coefficients obtained directly from calculated cross sections. Rate coefficients are obtained in this way:

kdiss(v, Trotrans ) = (k3Trotrans3 π µ / 8)−1/ 2 ⋅

dE∫ E exp(−E / kTrotrans ) σv,Trotrans(E)

(2)

We have chosen to equal rotational and translational temperatures because this case is of practical importance, due to generally not dissimilar relaxation times for these two degrees of freedom. It is also possible to average the results (2) on a Boltzmann distribution for the vibrational levels at the temperature Tv = T:

kdiss(T ) =

Qv−1

v∑ exp(−Ev , j = 0 / kT) kdiss

(v,T )

(3)

obtaining the thermal dissociation rate coefficient.It is important to point out an aspect of particular importance when considering the whole ladder of vibrotational states of a biatomic molecule, as in this case for O2. There are many states lying over the dissociation limit ("quasibound states", QB) classically stable because limited by the rotational barrier, but quantally unstable because of tunneling through the barrier. These states have lifetimes in a range from femtoseconds to seconds. The problem is if a QB state should be considered bound (or not). As a consequence, one should obtain (or not) also the dissociation cross section from that state, should include (or not) it into the partition function for rate calculation, and no dissociation results (and

American Institute ofAeronautics and Astronautics

4

dissociation results) if the system ends in that state. The problem is if in a real collision the system "sees" a bound state or not: this depends on the kinetic behaviour, that is on the specific conditions in which the measure is taken. If the collision frequency, directly linked to pressure and collision cross section, is sufficiently high, a certain fraction of QB states are essentially stable (equivalently, their lifetimes are much higher that mean time between two consecutive collisions), another fraction is surely dissociated for the opposite reason, and the remaining are "undecided". Only a kinetic study, based on specific experimental conditions, can give results near to the real behaviour of the system. We plan to produce this kind of result in the future. As a simple comparison, in this work we obtained both rate coefficients with QB (all QB stable) and without QB (all QB dissociated at any time).

10-25

10-23

10-21

10-19

10-17

10-15

10-13

10-11

10-9

0 10 20 30 40

T=300KT=500KT=1000KT=2000KT=5000KT=10000K

cm3/s

quantum vibrational number

Fig.2. Dissociation rate coefficients as function of initial vibrational states for the ro-translational temperatures indicated in the legend.

Another problem of large importance is the fact that, considering high lying rovibrational states of the ground electronic state of O2 only, one necessarily neglects rovibrational states of other electronic states having similar energies. This approximation can be not small. In order to take into account in a very simplified way this problem, we have assigned a multiplicity to each vibrational level v summing the multiplicities of all the stable electronic states (see [10]) with minimum energy lower than that one of the v state, divided by the multiplicity of the ground state (a triplet). This procedure is an extension of the method suggested by Nikitin in [11], where the correction is applied to thermal rate coefficient (not depending on vibration). It is based on the assumption of the existence of an equilibrium condition among rovibrational states belonging to different electronic states but with similar energies, equilibrium that one supposes not significantly distorted by the dissociation process.

10-34

10-32

10-30

10-28

10-26

10-24

10-22

10-20

10-18

10-16

10-14

10-12

0.2 0.4 0.6 0.8 1

GardinerParkthis workthis work-QB

cm3 /s

1000/T

Fig.3. Arrhenius plot of the comparison of experimental dissociation rate coefficient as function of 1000/T with theoretical results of this work with

American Institute ofAeronautics and Astronautics

5

("this work QB") and without ("this work") quasibound states.

4. Results and conclusionsThe results concerning rate coefficients are shown in fig.3 in an Arrhenius plot, where the calculated dissociation thermal rate coefficient (3), obtained with and without QB states, multiplied by the variable vibration-depending factor explained in the preceding section, is compared with the fitted experimental data of (a) Park[12] and (b) Gardiner [13]. Globally the agreement is satisfactory particularly for the second set of data, being better for higher temperature values. Theoretical data are lower than experimental ones for the lowest temperature values and slightly larger in the opposite case. As it is clear from the figure, considering stable all the QB states gives quite lower values of rate coefficient than in the other case for all temperature values. Actually there is significant discrepancy in the available experimental data, therefore this comparison of fig.3 does not appear as decisive, but it seems that the model with variable factor without quasibound states could give the most accurate description. On the other side, only a kinetic study calibrated on specific experimental conditions can make possible a reliable comparison with experimental data.

AcknowledgementsThis work has been supported by MIUR (contract 2001031223_009) and by ASI (contract ASI/CNR 1/R/073/01 "Modellizzazione di processi elementari in flussi ipersonici")

References[1] M.Capitelli, B.F.Gordiets, N.Osipov and C.M.Ferreira “Plasma kinetics in atmospheric gases” Springer Verlag 2000 [2] D.Bruno, M.Capitelli and S.Longo, Chem.Phys.Lett. 289, 141 (1998)[3]A.J.C.Varandas and A.A.C.C.Pais, Mol.Phys.(1988) 65, 843. The numerical code has been kindly obtained from authors[4] A.Lagana', E.Garcia, T.Martinez, 'Quasiclassical and Quantum Rate Coefficients for the O+O2Reaction', Chemistry Department, University of Perugia, Italy[5] F.Esposito, M.Capitelli, Chem. Phys. Lett. 303 1999 636–640.[6] F.Esposito, M.Capitelli, Chem.Phys Lett. 302, 49 (1999)[7] F.Esposito, PhD Thesis, University of Bari, 1999[8] MPI is an industry standard Applications Programming Interface (API) for sending and receiving messages between nodes of a parallel computer.[9] Consorzio interuniversitario per le applicazioni di Supercalcolo per Universita' e Ricerca, Roma.[10] R.P.Saxon, B.Liu, J.Chem.Phys. 67, 5432 (1977)[11] E.E.Nikitin, Theory of Elementary Atomic and Molecular Processes in Gases, Clarendon Press -Oxford 1974[12] Park, C., AIAA 24th Thermophysics Conference, AIAA-89-1740 [13] Gardiner, W.C. Combustion Chemistry, Springer-Verlag, Berlin, 1984