Models of Collisional Dissociation
R.G. Lord
Department of Engineering Science
Oxford University, England
Abstract. The paper presents an assessment of a model of collisional dissociation proposed by Lord for use in
DSMC calculations. Particular attention is paid to the phenomenon of vibrational favouring of dissociation.
Extensions to the original model are presented which attempt to include the influence of the rotational kinetic energy
of the molecule in promoting dissociation.
INTRODUCTION
A model of dissociation of diatomic molecules in intermolecular collisions has been proposed by Lord [1]. This
model is a development of the exact available energy or vibrationally linked chemistry model of Bird [2,3], which
models dissociation via Borgnakke-Larsen (B-L) type energy redistribution between translational and internal
degrees of freedom. However, Bird's model assumes the existence of discrete vibrational energy levels above the
dissociation limit, which is unrealistic; in fact, only levels below the dissociation energy are discrete, those above
forming a continuous spectrum. Application of the usual B-L procedures to the proper energy level scheme of an
anharmonic oscillator is not straightforward; because the potential energy remains finite as the atomic separation
becomes large, the area of phase space above the dissociation energy is infinitely larger than that below it and this
leads to the prediction of dissociation in any collision of sufficient energy. The problem is overcome in Lord's
model by introducing the constraint that only the kinetic energy of vibrational is redistributed in a collision, the
potential energy (and thus the atomic separation) remaining unchanged through the collision.
Another consideration is that, in both Bird’s model and Lord’s, it is assumed that a molecule dissociates when its
vibrational energy exceeds its dissociation energy. However, it seems at least arguable that it is the total internal
energy of the molecule, i.e. the sum of the rotational and vibrational energies rather than the vibrational energy
alone, which is the significant quantity in promoting dissociation, i.e. infinite separation of the orbiting particles.
Certainly, in the case of classical orbits under an inverse square law of force it is the total energy of the orbit, i.e. the
sum of radial and tangential kinetic energies plus potential energy, relative to the potential energy at infinite
separation, which determines whether the orbit is elliptic or hyperbolic. The situation for the interatomic potential of
an anharmonic oscillator is somewhat more complicated on account of the existence of quasi-bound orbital states
having energies greater than the dissociation energy [6]. It would appear necessary in this case to adopt a more
complicated criterion for dissociation, depending on both the energy and the angular momentum of the molecule.
Lord’s original model, when used in conjunction with the Morse model of intermolecular potential, has been
shown to predict equilibrium dissociation rate constants for nitrogen which are in excellent agreement with
published values [1]. However, as pointed out by Wadsworth and Wysong [4], the predicted rate constant is a
relatively poor measure of the validity of a model of dissociation, a much more sensitive criterion being whether the
model exhibits vibrational favouring of dissociation [4,5], i.e. whether, for a given total collision energy, the
probability of dissociation increases as the fraction of the energy in the vibrational mode increases. The model has
also been shown to lead naturally to vibrational favouring [1]; however further investigation of this phenomenon is
called for.
The present paper presents the results of more searching tests on Lord's original model and introduces some
possible extensions to include the effects of rotational kinetic energy. Particular attention is paid to the predicted
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dissociation probabilities from different vibrational levels at a given collision energy, the predicted variation of rate
constant with vibrational temperature at constant transrotational temperature and the predicted pre-collision
vibrational distributions of molecules which become dissociated.
DISSOCIATION PROBABILITIES
Figure 1 shows the predicted variation of dissociation probability of a nitrogen molecule in collision with another
nitrogen molecule at various total collision energies. The vibrational energy Ev and collision energy Ec are here
expressed as fractions of the dissociation energy Ed. The procedure used is the serial procedure in which the
vibrational energy of the second molecule does not contribute to the exchange. The collision energy thus consists of.
the translational kinetic energy of collision, the rotational kinetic energy of both molecules and the vibrational
energy (kinetic and potential) of the molecule under consideration; the vibrational potential energy is, however,
frozen and not available for exchange. The probability of dissociation is plotted against the vibrational energy of the
molecule before collision and can be seen to approach unity as this energy approaches the dissociation energy for all
three collision energies.
dissociation probability
1
0.1
0.01
0.001
0.0001
0.00001
0
0.2
0.4
0.6
0.8
1
vibrational energy Ev / Ed
Ec / Ed =1.1
Ec / Ed =1.2
Ec / Ed =1.3
FIGURE 1. Collisional dissociation probabilities of N2 molecules.
EFFECTS OF ROTATIONAL KINETIC ENERGY
In order to investigate the effects of the rotational kinetic energy of the molecule on dissociation, classical orbit
calculations were performed to determine apogee and perigee distances for various combinations of the energy and
angular momentum of a rotating Morse oscillator, infinite apogee distances being taken to indicate dissociation. A
typical set of results is shown in Figure 2, which shows a series of curves of vibrational energy versus atomic
separation for a single value of angular momentum H and several values of total internal energy Ei = Ev + Er. These
are compared with the Morse potential for nitrogen. Regions of the curves below the Morse potential would indicate
negative vibrational kinetic energy and are therefore impossible. The apogee and perigee distances thus lie at the
points of intersection of the vibrational energy curves with the Morse potential curve. As can be seen from the
graphs, the internal energy in this case must exceed the dissociation energy by some 18% before dissociation occurs.
On the basis of such calculations, it was found that the criterion for dissociation could in all cases be represented
quite accurately by means of a simple formula of the form Ei > Ed + kH 3, where k is a constant.
Figure 3 shows the effect of three different dissociation criteria on the predicted dissociation probability
distribution for nitrogen, namely
(i)
(ii)
(iii)
Ev > Ed
Ei = Ev + Er > Ed
Ei = Ev + Er > Ed + kH3
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The conditions are the same as for Figure 1, except that the results apply to a single value of the collision energy Ec,
namely Ec = 1.1Ed. Not surprisingly, the main effect of including the rotational energy in the energy required for
dissociation is to increase the dissociation probabilities from all vibrational energy levels. Including the effects of
superenergetic bound states reduces these probabilities to a level between those obtained using criteria (i) and (ii).
1
Ev / Ed
0.8
0.6
0.4
0.2
0
0.5
1
1.5
2
2.5
r/r0
Morse
Ei / Ed = 1
Ei / Ed = 1.18
Ei / Ed = 1.27
Ei / Ed = 1.09
FIGURE 2. Effect of angular momentum on dissociation of N2 molecules.
dissociation probability
1
0.1
0.01
0.001
0.0001
0.00001
0
0.2
0.4
0.6
0.8
1
vibrational energy Ev / Ed
Ev > Ed
Ev + Er > Ed
Ev + Er > Ed + k H^3
FIGURE 3. Effects of rotational kinetic energy on dissociation probabilities for nitrogen, Ec / Ed = 1.1.
In these calculations, the value of the angular momentum H was calculated from the post-collision rotational
kinetic energy of the molecule and the atomic separation during impact (obtained as part of the procedure of
separation of the pre-collision vibrational energy into its kinetic and potential components and assumed unchanged
through the collision). The pre-collision rotational energy was selected from the total collision energy, less the
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vibrational energy of the molecule under consideration, according to the equilibrium (Borgnakke-Larsen)
distribution for the appropriate number of degrees of freedom. One consequence of using the two dissociation
criteria involving rotational energy is that it is possible for molecules to be pre-dissociated (i.e. for the dissociation
criterion to be satisfied before the collision actually occurs). Some runs were therefore made in which such predissociated molecules were discounted, but no significant differences in the results were observed despite the fact
that in some cases the number of pre-dissociated molecules exceeded that of the molecules dissociated in the
collision.
DISSOCIATION RATES
The effect of the various dissociation criteria on the forward dissociation rates in O2 – monatom collisions is
demonstrated in Figure 4, which shows the variation of predicted forward dissociation rate constant with vibrational
temperature at a constant transrotational temperature of 10000K for the three different critieria. It is clear that the
original model again gives the greatest vibrational favouring.
rate constant (m3/s)
1E-16
1E-17
1E-18
1E-19
0
2000
4000
6000
8000
10000
vibrational temperature (K)
Ev > Ed Ev + Er > Ed Ev + Er > Ed + k H^3
FIGURE 4. Vibrationally favoured dissociation in O2 – monatom collisions.
PRE-COLLISION VIBRATIONAL DISTRIBUTIONS
Figure 5 shows pre-collision vibrational distributions f (v | D) predicted by the model of oxygen molecules which
become dissociated in O2-monatom collisions at a temperature of 12000K. Here v is the pre-collision vibrational
quantum quantum level of the molecule and D indicates that the molecule becomes dissociated in the collision. The
results are compared with the quasi-classical trajectory (QCT) results of Mizobata [7] for the O2-A system, as
reported by Wysong and Wadsworth [8]. The agreement is not particularly good and, disappointingly, seems best
for the original model, the results for the extended models having the wrong trend with vibrational level.
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distribution function f(v|D)
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
10
20
30
40
50
60
vibrational level
Ev > Ed
Ev + Er > Ed
Ev + Er > Ed + k H^3
QCT
FIGURE 5. Pre-collision vibrational distributions of dissociated O2 molecules .
CONCLUSIONS
It has to be admitted that attempts to include the effects of rotational kinetic energy in the modelling of
collisional dissociation have proved somewhat less than successful, in that the original model appears to exhibit
greater vibrational favouring and to give results which are in closer agreement with other calculations. This is
particularly puzzling in view of the apparent reasonableness of the assumptions used in the most refined model.
Further investigations might provide some clue as to the reasons for this but, in the meantime, there would appear to
be little point in adopting the more complicated model.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
Lord, R.G., “Modelling Dissociation of Diatomic Gases using the Morse Potential” in Rarefied Gas Dynamics 20, edited by
Ching Shen, Peking University Press, Beijing, 1997, pp. 180-185.
Bird, G.A., “Molecular Gas Dynamics and the Direct Simulation of Gas Flows” , Oxford University Press, Oxford, 1994, pp.
241-251.
Carlson, A.B. and Bird, G.A., “Implementation of a Vibrationally Linked Chemical Reaction Model” in Rarefied Gas
Dynamics 19, edited by J.K. Harvey and R.G. Lord, Oxford University Press, Oxford, 1995, pp. 434-440.
Wadsworth, D.C. and Wysong, I.J., “Examination of DSMC Chemistry Models: Role of Vibrational Favoring” in Rarefied
Gas Dynamics 20, edited by Ching Shen, Peking University Press, Beijing, 1997, pp. 174-179.
Wadsworth, D.C. and Wysong, I.J., Phys. Fluids, 9, 12, 3873-3884 (1997).
Blais, N.C. and Truhlar, D.G., J. Chem Phys., 70, 2962 (1979).
Mizobata, K., "An Analysis of Quasiclassical Molecular Collisions and Rate Processes for Coupled Vibration-Dissociation
and Recombination" AIAA 97-0131, 35th Aerospace Sciences Meeting (1997).
Wysong, I.J. and Wadsworth, D.C., “Effect of DSMC Dissociation Models on a Stagnationi Streamline Flowfield” AIAA
97-2513, 32nd Thermophysics Conference (1997).
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