Estimation of cross sections for energy transfer in bimolecular collisions

ChemicalPhysics North-Holland

177 (1993) 133-143

Estimation of cross sections for energy transfer in bimolecular collisions L.E.B. Bhjesson,

Jan Davidsson,

Nikola Markovic and Sture Nordholm

Department ofPhysical Chemistry, University of GiiteborgKTH, S-41296 Gdteborg, Sweden Received 7 April 1993

The proper definition of a cross section for energy transfer in bimolecular collisions is considered. Three models transforming smooth impact parameter dependence into hit or miss cross section form are discussed. The property dependence of the concept of a collision cross section is investigated. Translational energy cross sections (viscosity, diffusion, deflection energy) are found to be smaller than internal energy cross sections (thermal conductivity, thermal activation). An effective potential analysis of the concept of cross section shows that both radial and rotational collisions must be considered. Properly defined theoretical (effeo tive potential, MD simulation) and experimental cross sections for Brr-Brr show satisfactory agreement.

1. Introduction In order to describe unimolecular reactions by master equation as used in RRK and RRKM theory [ l-51 a statistical representation of the collisional energy transfer mechanism is required. A collision frequency w, or a cross section o, is obtained and the effect of the collision is described by an energy transfer kernel P( E, E’ ) which is a conditional probability density of final (after collision) energy E given an initial (before collision) energy E’. It is often the case that only the total energy transfer rate, i.e., obtained by summation over all impact parameters, is needed. However, this is not the case in chemical reaction rate theory where, for the same total energy transfer rate, it matters whether the collisions are weak or strong. The concept of collision “strength” (or energy transfer efficiency) is also important in the attempt to gain an intuitive understanding of the energy transfer mechanism. As it turns out collisions are not well defined. They fall in reality on a continuous scale of coupling strength from head on collisions (normally but not always the strongest) to glancing collisions where the interaction and energy transfer amounts to a small perturbation. Thus one must use some criterion of some degree of arbitrariness to distinguish collisions from noncollisions. The nonunique nature of this criterion has led to problems when compari0301-0104/93/$06.00

sons are made of results obtained from different sources. In particular, the comparison of cross sections obtained from experimental data, most often viscosity data, with data from molecular dynamics simulations is nontrivial. The normal simulation practice leads to a larger cross section and (in compensation therefore) “weaker” energy transfer per collision ( AE) . A detailed account of this problem has been given recently by Lendvay and Schatz [ 61 referring to a large body of simulation data. The problem of the proper definition of a collision cross section is neither new nor resolved in the literature in any affirmative way [ 6- 12 1. Our work to be described below is an attempt to remove some of the obstacles and at least partially resolve the issue. It is also meant to raise the awareness of the problem and the different mechanisms at work. We shall begin by reminding the reader of the now normal approach used by reaction kineticists for the last 30 years, at least, based on a Lennard-Jones pair potential and viscosity data. We then note that self-diffusion and thermal conductivity give rise to slightly different cross section values even for inert gases with larger differences for polyatomic molecules. Thus, we see that cross sections are expected to be property dependent. We then consider the typical impact parameter (b) dependence of the average energy transferred per collision, i.e., ( AE) b, as observed in classical trajec-

0 1993 Elsevier Science Publishers B.V. All rights reserved.

L.E.B. Bdrjessonet al. / ChemicalPhysics177 (1993) 133-143

134

tory calculations and present three models by which we can extract a cross section and a corresponding total average ( AE) . A simple geometrical argument is given to show how the dependence may arise for the translational energy contribution. We also introduce the concept of deflection energy in order to be able to compare internal energy and translational energy cross sections obtained by trajectory calculation. Molecular dynamics (MD) simulations have been carried out for Brz-Brz collisions to allow comparison of theoretical and experimental cross sections obtained by the different procedures described. In order to more systematically develop the normal interpretation of collisions in terms of a spherically symmetric pair potential, usually the Lennard-Jones ( 12-6 ) potential, we draw upon a recently presented [ 131 effective potential method. It is based on statistical mechanical arguments and produces a spherically symmetric potential which gives a good description of collision complex formation in a number of tested cases. However, in order to get a proper definition of collision cross section the criterion used to define a collision must allow for the underlying asymmetry and allow for both radial and rotational collisions. When this is done good agreement is obtained between experiment, simulation and effective potential theory.

2. Definition of collision cross section - the ergodic sphere

kinetic theory of gases The standard way of finding a collision cross section u is to represent the interactions by a LennardJones pair potential and use the parameters oU and eU with an Q(*,‘) collision integral [ 5,14,15 1, (1)

where fJHS =7&J

(2)

and

T*=kBT/eU. All these quantities

(3) may be readily obtained

(5)

where ( v) T is the thermal average velocity,

r=,/%Thm,

2.1. Traditional methods and methods from the

rJ=rJ&2(2’2)*( z-*) ,

bles [ 16- 19 ] or by analytical approximation [ 17,2 11. Following the suggestion of Kohlmaier and Rabinovitch [ 15,221, it has become conventional to use this method to calculate a collision cross section also in reaction rate theory. It was developed for monoatomic molecules interacting through a Lennard-Jones potential and designed to tit the cross section that corresponds to the viscosity. It has been shown [ 23,241 that this method may be extended to complicated polyatomic molecules such as butane and weakly polar molecules like nitrous oxide. However, in this case the Lennard-Jones parameters are just scaling parameters and should not be given the physical significance of the true potential. In general the temperature range over which this method applies is more limited for these complex molecules than for the monoatomic molecules. A better method of finding a cross section is to use the corresponding expressions for viscosity, thermal conductivity and the self-diffusion coefficient from the kinetic theory of gases [ 17-201, i.e.,

from ta-

(7)

and m is the particle mass, rl is the coefficient of viscosity, c, is the heat capacity per molecule, 1 is the coefficient of thermal conductivity, c is the concentration, D is the coefftcient of self-diffusion and T is the temperature. For a noble gas these two first cross sections almost coincide, but for a polyatomic molecule it has long been known that the cross section from the thermal conductivity is noticeably greater than that from the viscosity. For nitrogen it is x 30% greater, for carbon dioxide it is x 50% greater and for butane it is x 100% greater [20]. The cross section from the coefficient of self-diffusion is roughly 10% smaller than that from the viscosity. This is illustrated in table 1 for some molecules. There are some simple models which may be employed for trial cal-

135

L.E.B. BGrjessonet al. /Chemical Physics177 (1993) 133-143 Table 1 Cross sections (in AZ)obtained from the experimentally measured transport properties, viscosity, thermal conductivity and the diffusion coefficient, are shown for some gases at 273 K and 1 atm [ 201 (note that the cross sections in ref. [ 201 are given as the square of the collision diameter) Molecule

GSC.

G.U0ZI~

HZ

23.1 44.3 67.5 68.8 54.0 89.5 153.6

30.4 58.1 104.3 111.8 72.8 154.0 304.7

N2 co2 H20

CH. C2Hs

n-CsI,o

=IeEdLE 20.4 38.0 58.0 45.2 78.1 141.0

culations of the transport properties of polyatomic molecules such as the rough sphere [25] and the spherocylinder [ 26,271. These models are also only approximations but they do incorporate some of the features of polyatomic molecules such as internal energy, inelastic collisions and nonspherical interactions. They also show that the effects of internal energy changes upon the viscosity coefficient are small and the same result [ 28,291 is likely to be valid for a more realistic polyatomic molecule. The process of internal energy transfer of interest here should be more closely related to the thermal conductivity than to the viscosity, so that the conventionally employed cross sections will be too small for this purpose. With the advent of fast computers it has become possible to study collisional energy transfer by molecular dynamics simulations, i.e., by setting up potential models for interacting molecules and solving Newton’s equations of motion for a set of collision trajectories. For a trajectory calculation of the average energy transfer the maximum impact parameter, b is chosen sufficiently large so that trajectories bznd this value do not exchange significant amounts of energy [ 30 1. This value of b,, will be roughly 50% greater than the Lennard-Jones diameter aU{Ll(2,2)*}1/2(see, e.g., ref. [5], p. 236).

( AE) b, is smoothly fading away for large values of the impact parameter b. When does a collision change from a hit to a miss? Three models will be used to estimate such a cross section. Define the total energy transfer rate, rcE,as

OD K,=2n

I 0

dbb( AE)b.

(8)

Insisting that the total energy transfer rate, mains correct we then get

K~,

re(9)

KE=~E(~).

In the first model the collision cross section is given the value ai’) = xbk,,, from the maximum impact parameter b,,. The average energy change per collision, (AE) (I), IS chosen to give the correct total energy transfer rate with this collision cross section. This is the most commonly used model in molecular dynamics simulations where b,, is determined so that the total energy transfer rate is converged to within a few percent. The problem with the first method is that it will overemphasize the weak collisions for large impact parameters. In the second model we take the opposite point of view. We let the head-on collision define the average energy change ( AE) c2)= ( AE) o, the average energy transfer when the impact parameter is 0. The collision cross section, ai’), is chosen to give the correct total energy transfer rate with this average energy change. It is clear, however, that in representing the typical collision strength (AE-value) at fixed total energy transfer rate the first and second models represent extremes where in the first they are too weak and in the second too strong. Thus the third model should give a balanced average for ( AE) . This is obtained as (U) m =27c

I

m dbbw(b)(AE),

2x IJ

0

dbbw(b),

(10)

0

2.2. Cross section models accountingfor soft edges

where the weight factor for ( AE) (3) is chosen to be

How to define a collision cross section, cr,, and an average energy change per collision, (M) , when the average energy transfer is a function of the impact parameter? Consider a case when this function,

w(b) = (M),

.

(11)

This weight implies that the importance of the collisions at impact parameter b is directly proportional to the importance of these collisions to the energy

L.E.B. B&jesson et al. /Chemical Physics177(1993) 133-143

136

transfer rate [9]. It follows that the third, and preferred, cross section is obtained as @=KJ(AE)(3).

Energy transfer in red. u&s ( n = 7 ) .

(12)

This definition of ( AE) (3) and ah3) is consistently based on the energy transfer rate through the two conditions that (i) the total energy transfer rate is conserved by the model, (ii) in calculating the average ( AE) in the model the weight factor is the contribution of those collisions at fixed b to the energy transfer rate. Note that the first two models satisfy the first condition but not the second. Figs. 1 and 2 illustrate the application of these three models to an ensemble of collisions showing typical b-dependence of the energy transfer. Define a function:

0.6.

u.2.

u, U

,

a2

,

u.4

,

u.6

,

u.6

,

1.0

,

1.2

.‘,

1.4

1.6

,

1.8

2.0

.

2.2

*

2.4

:6

Square of impact parameter in red. unik

Fig. 2. As in fig. 1 but the average energy transfer is plotted against the square of the impact parameter to give a better understanding of how the three models represent the collision cross sections (see text).

(m)b

=(hE),{exp[~(b14~r-1)l+1)-L,

(13)

where ( AE) Oand bhalfare scaling parameters and (Y is a parameter which determines the shape of the curve. This function gives a good fit to the impact parameter dependence of the average energy transfer from molecular dynamics simulations. In fig. 1 the average energy transfer is plotted against the impact parameter in the “reduced” units (A&,, and bhalfto give an idea of how the three models represent the

Energy transfer in red. units ( a = 7 1.

I

0

0

I

0.2

0.4

a6

as

1.0

13

1.4

1.6

1.8

I

2.0

Impact parameter in red. units.

Fig. 1. A typical b-dependence of the energy transfer with the three different average energy changes per collision and their corresponding collision cross sections are shown (see text).

average energy transfer per collision and the corresponding collision cross sections. In fig. 2 the average energy transfer is plotted against the square of the impact parameter to give a better understanding of how the two models represent the collision cross sections. Model 3 clearly gives the best and most balanced rectangular fit to the energy transfer function
2.3. The ergodic sphere The ergodic sphere is a very simple model which indicates one reason for the difference in size of the cross sections, i.e., the conservation of tangential translational energy. Consider classical scattering in a center of mass reference frame. A molecule in thermal equilibrium with m internal degrees of freedom is colliding with a target molecule with n internal degrees of freedom. Initially the target molecule has an internal energy of Ei. The first molecule is moving with a momentumpi which corresponds to a translational energy of 3k,T/2 and is then scattered with a momentum pf. Assume now an ergodic collision producing a microcanonical energy redistribution where the two internal energies and the translational energy which corresponds to the normal component Ofpi take part. This energy is allocated among the internal degrees of freedom and the “normal fraction” of the

L.E.B. Btirjessonet al. / ChemicalPhysicsI77 (1993) 133-143

three translational degrees of freedom. Define (e) as the allocated energy per degree of freedom, i.e.,

(e>=

E,+(m+3cos2a)kBT/2 n+m+3cos2a

’

(14)

where 3 cos2a is the “normal fraction” of the three translational degrees of freedom. The molecules have a hard diameter d and were approaching each other with an impact parameter b. The value of cos201 is given by cos2~= 1- (b/d)2. The average internal Ei - n ( e) , as a function then becomes

(15) energy loss, ( AE,nt)b= of the impact parameter b

How does the collision cross section for internal energy transfer relate to that for translational energy transfer playing a dominant role in other phenomena such as diffusion and viscosity? In order to address this question we make a single extension of the ergodic sphere analysis above. Define the deflection momentum as &=pf-pi which corresponds to a transfer” “deflection energy . The “deflection energy transfer”, (a&“) b, as a function of the impact parameter b then becomes (m&l

)b

=;k,Tcos2a

(l+J2olk,T)‘.

(17)

This “deflection energy transfer” is not a real physical energy transfer but is rather a model of momentum transfer. It is introduced to compare its collision cross section from viscosity. The collision cross section of the internal energy transfer can then be compared with that from the thermal conductivity. The collision cross sections for the internal energy trans-

137

fer and the so-called deflection energy transfer can be evaluated with model 3 above. If reasonable values are used for the parameters, the cross section for the internal energy transfer will always be greater than that for the deflection energy transfer. For two colliding diatomic molecules this model appears to give a value about 25% greater for the internal energy transfer and if both collider and target are larger molecules this model will give a value typically 35% greater.

3. Comparison

with simulation

and experiment

The transfer of energy in the collision between two bromine (Br,) molecules has been simulated by a highly energized target molecule interacting with a thermalized projectile molecule. We use the Morse potential

~(r)=~,{expI-P(r-r,)l-1}2,

(18)

with D .=192kJ/mol,/?=1.94A-‘andr,=2.28Ato model the intramolecular bond potential in Br2. The intermolecular potential is obtained by superimposing the atomic pair potentials represented by LennardJones potential functions, i.e. O(r)=4e[

(a/r)12-

(o/r)6]

,

(19)

with e/kg= 170 K and a~3.6 A. The simulation is carried out entirely within classical mechanics. No attempt is made to prepare the internal degrees of freedom of the diatomics to correspond to particular quantum vibrational-rotational eigenstates. Such “quasiclassical” siniulation is often used but suffers from the lack of an appropriate and consistent “quasiclassical” final state treatment. The molecular dynamics algorithm used in this study is based on a three atomic algorithm described earlier [ 3 1 ] and extended to treat four particles as well. It uses the integration routine RADAU [ 321, a 15th order Gauss-Radau method, which provides high accuracy for well-behaved analytical potentials and which has performed well in previous trajectory calculations [ 33 1. For all selections of initial conditions the energy conservation IAEl /E,, was typically lo-” and time reversal at the end of a trajectory allows the initial condition to be recovered. The initial conditions of the target molecule were obtained by an efficient microcanonical sampling rou-

L.E.B. B&jesson etal. /ChemicalPhysics 177(1993) 133-143

138

tine [ 341 combined with Monte Carlo stepping in the configuration space [ 35 1. Initially the target molecule is at a well-defined internal energy of 180 kJ/ mol just short of the dissociation energy of 192 kJ/ mol. It is in internal microcanonical equilibrium which means that the energy is distributed roughly evenly between the rotation and vibration of the molecule. The initial conditions of the projectile molecule as well as the initial translational energy are selected according to the canonical ensemble at a specified temperature which will be referred to as the medium temperature. These temperatures are chosen to be 50, 100, 160, 300, 1000, 1500 and 3000 K, which span the most relevant range. The initial distance between the center of mass of the two molecules is 20.52 A, i.e. they are sufficiently separated to be treated as noninteracting. The impact parameter is sampled in the range Ob,,, where the choice of b,,, is discussed below. In order to increase our accuracy, the b values were importance sampled from a probability density P(b) P(b)=exp[

-2.3(b/b,,,)‘]

.

The average of a b-dependent evaluated as

(20) quantity

f( b) is then

(21) where the sum goes over all trajectories. The energy transfer rate coefficient kE is defined as kE=o(AE)/(Ei-2kBT),

( AEint ) b with (AEM) b. ( AEi,t) b is the average internal energy loss per collision as a function of the impact parameter b given by (24)

(mint)b=Ei-(mf)b,

where E, is the initial internal energy of the target molecule and ( Ef) b is the average final internal energy of the target molecule as a function of the impact parameter b. Define the deflection momentum as Ap=pf -p,, with pf and p, the final and initial momenta of the projectile molecule, respectively, which energy transfer” corresponds to a “deflection (A,!&,) b which also is a function of the impact parameter b given by (25)

(&,fl)b=(I&12)/2PL

where ,Uis the projectile-target reduced mass. Fig. 3 shows both the average internal energy transfer and the average deflection energy transfer per collision as a function of the impact parameter at 160 and 1500 K. All results from the molecular dynamics simulations are fitted to the function in eq. ( 13) above. The scaling parameters ( AE) ,, and bhdf depend strongly on the temperature but the parameter c~which determines the shape of the curve is almost a constant. There is reason to believe that this parameter mostly depends on the pair of colliding molecules. For collisions of Br, against Br2 the parameter (Y is about 7

(22)

Energytransfer

in Id/mot.

50,

where AE is the internal energy change of the target molecule, i.e. Ei - Ef, with Ei and Ef the initial and final energies of the target bromine molecule respectively and o is the hard sphere collision frequency given by w= (8nk, T/p)“2b&,,

,

1500 K

40

30

mtern. ISCU K

(23)

where p is the projectile-target reduced mass. The mass of the bromine atom is taken to be 79.916 au. The choice of b,,, is based on the change of kE with b and that kE levels off at higher b when the energy transfer becomes negligible. The b values are divided into subintervals of width 0.8 A and if the change in kE between two subintervals is smaller than l%, b,, is set to be the midpoint between the two subintervals. The present study is focused on comparing

1 dcllect.

0

t

2

3

4

5

6

Impact parameter b (A

)

7

9

Fig. 3. The average energy transfer per collision as a function of the impact parameter is shown for Br2-Br, collisions investigated by molecular dynamics at 160 and 1500 K (see text ).

L.E.B. BSrjessonetal. /ChemicalPhysics 177(1993) 133-143

for the internal energy transfer and about 5 for the deflection energy transfer. The value of these parameters may depend on the energy content of the relaxing molecule. However, Lendvay and Schatz [ 61 have investigated some systems and shown that the shape of the curves for their systems is independent of energy within statistical uncertainty at 300 K. That means that the collision cross sections determined at a high energy are also appropriate at lower energy. In table 2 the cross sections from the experimentally measured transport properties, viscosity, thermal conductivity and the diffusion coefficient, are shown for gaseous Brz at some temperatures. The cross sections are calculated by eqs. (4)-(7) with data from ref. [ 21. The values in table 2 have an uncertainty of less than 5%. In table 3 the cross sections from molecular dynamics simulations of the internal energy transfer in collisions of Br, against Br2 at some temperatures are shown. The cross sections are evaluated by models 1, 2 and 3. In table 4 the cross sections are shown as in table 3 but for the average deflection energy transfer per collision. The values in Table 2 Cross sections (in A’) obtained from the experimentally measured transport properties, viscosity, thermal conductivity and the diffusion coeffkient, are shown for gaseous Br2 [ 171

139

Table 4 Cross sections (in AZ) obtained here from molecular dynamics simulations of ( AEden) in collisions of Br2 with Br, are shown. The cross sections are evaluated by models 1,2 and 3 (see text )

50 100 160 300 1000 1500 3000

170 110 90 70 40 40 30

390 340 290 240 160 130 130

CROSS SECTIONS IN

270 200 160 120 70 60 60

AZ FORBrs.

250

l

04 T(K)

c&SC

QL~0”d.

50 100 160 300 1000 1500 3000

189 162 122 69 61 52

226 196 158 91 86 86

Gclr~ff.

112 62 56 48

Table 3 Cross sections (in AZ) obtained here from molecular dynamics simulations of (AE,,,,) in collisions of Br2 with Br2 are shown. The cross sections are evaluated by models 1, 2 and 3 (see text) T(K)

fJC’

oli?

o/$

50 100 160 300 1000 1500 3000

390 340 290 240 160 130 130

220 140 120 90 60 60 60

300 220 180 140 90 80 80

loo

Temperature

(K )

1000

Fig. 4. Cross sections as functions of the temperature for Br, are shown, i.e., the conventionally employed one from the viscosity ( 0 ) and the one from the thermal conductivity ( n ) (see table 2). Also shown are cross sections from molecular dynamics simulations of (A&,,) (+) and (AI&) ( + ) evaluated by model 3 (see tables 3 and 4). The cross section from the effective potential (A ) using definition 3 is also shown (see table 5).

tables 3 and 4 have an uncertainty which is estimated to be loo/o. About half of this uncertainty originates from the simulations and the other half originates from the fitting procedure. To make it easier to compare these different cross sections from tables 2,3 and 4 some of them have been plotted against the temperature in fig. 4. By inspecting these tables and this figure two observations may be made. ( 1) The values of the cross sections for the deflection energy transfer evaluated by model 3 are very close to those for the viscosity.

140

L.E.B. Biirjessonet al. /Chemical Physics177(1993) 133-143

(2) The values of the cross sections for the internal energy transfer evaluated by model 3 are greater than those for the deflection energy transfer but smaller than those for the thermal conductivity.

4. The effective potential method The traditional method of determining an energy transfer cross section starts by mapping a molecular interaction which is generally asymmetric onto a spherical symmetric Lennard-Jones potential. One might ask whether this is at all appropriate and how one might more systematically obtain a spherically symmetric representation of a molecular interaction. In order to at least partially answer this question we shall consider the use of “effective” interaction potentials. The effective potential method, developed for calculating the average interaction between an ion or atom colliding with a diatomic molecule and from it the collision complex formation frequency [ 13 1, has been extended to handle the case of two diatomic molecules. The asymmetric interaction potential between the two nonvibrating molecules is mapped onto a spherically symmetric effective potential by equating the canonical fluxes obtained using the two potentials, i.e.,

n = 7 corresponding to rotation of the two rigid linear molecules and relative translation. The triple integrals over the angular variables are evaluated using a Gauss-Legendre method, while the integrals over kinetic energy are solved analytically and the whole equation is solved numerically for U,, using the Newton-Raphson method. Once the effective potential is known and it is decided on how to define a collision (see below) the collision cross section can be calculated utilizing the “Langevin” method, i.e., by computing the maximum impact parameter allowing the system to pass over the centrifugal barrier and reach a given separation. The procedure is repeated for a number of collision energies in order to obtain the cross section as a function of energy. Finally, a collision rate coefficient is calculated, 4

k(T)= (2rcp)‘12(kB~)W

s

co

x dE, ~(EdE, exp(-4IbT)

,

(27)

0

where p is the reduced mass of the colliding system. A thermal cross section may also be obtained by dividing the rate coefficient by the average collision velocity,

4 T) a(T)=

co

X

dee(n--L)/2 e-8& I max(O,-V)

[ - B&T(R)I 00

= 4~ exp X

de E(“-‘)/2e-Bc I max(O,-U&)

(26)

Note that bound states, i.e., states with E
(8kBT/np)“2’

Three definitions of a collision based on the center of mass separation between the fragments, R,,, have been used here. The first definition requires the colliding partners to reach the point where the effective potential is repulsive and equal to k& The second definition requires the fragments to reach the point where UeR=O, i.e., the point where the inner repulsive part passes through zero and becomes attractive. The third definition just requires the molecules to reach a given center of mass separation which here is fixed to 4.5 8, for all temperatures. The corresponding thermal cross sections for Br, at some temperatures are shown in table 5. The results show great variations reflecting the hidden spherical asymmetry of the underlying interaction. Definitions 1 and 2 above are quite sound for inert gases but miss the fact that collisions for asym-

L.E.B. B&iesson et al. /Chemical Physics177(1993) 133-143 Table 5 Cross sections (in AZ) obtained from the effective potential method for Br, are shown (see text). Definition 1 uses R,=R( UeR=ksT), definition 2 uses R,=R( U.,=O), definition 3 usesR,=4.5 8, T(K)

%kf 1

Gf.2

Gef.3

50 100 160 300 1000 1500 3000

15.6 13.9 14.6 15.4 16.8 16.9 16.4

39.6 40.8 42. I 48.1 64.8 68.4 13.2

213 198 152 106 62.5 58.0 55.5

metric molecules may be both radial and rotational. Definitions 1 and 2 do not account for the latter and therefore misrepresent the cross section by increasing underestimation as the temperature decreases. This problem can be cured by use of definition 3 which, in a crude way, accounts for rotational collisions by assuming that they will occur if the molecules approach to within 4.5 A, a distance where the asymmetry is estimated to arise for the Brz-Brz interaction. We note that this simple device yields quite good agreement with both trajectory simulations and experiment.

5. Discussion and conclusions It is probable that the traditionally used cross sections from the viscosity are too small to give a true value of the cross sections for the internal energy transfer. First remember that the values of the cross sections for the deflection energy transfer evaluated by model 3 are very close to those for the viscosity. That is quite reasonable because both describe a capacity to transfer momentum. The next observation established that the values of the cross sections for the internal energy transfer evaluated by model 3 are greater than those for the deflection energy transfer but smaller than those for the thermal conductivity. This is a more complicated result to explain. But if the cross sections for the viscosity also describe the capacity to transfer translational energy and when the cross sections for the thermal conductivity describe the capacity to transfer energy, both internal and translational, then the result seems to be reasonable.

141

Why does the average energy transfer as a function of the impact parameter, ( AE) b, fall off much faster for deflection energy than for the internal energy (see fig. 3)? Consider a naive model of a molecule. It should be important to hit near the center of mass to transfer large amounts of momentum or translational energy but a peripheral hit can transfer considerable amounts of rotational energy and even some vibrational energy. This observation could be the start for a more realistic theory. In this work we have studied the bromine molecule but its different property related cross sections such as the traditionally used cross sections from the viscosity and the cross sections for the internal energy transfer evaluated by model 3 are too close to reliably distinguish considering their error bars. Brown and Miller [ 81 have investigated the system H20-He at 800 K by classical trajectory calculations. They found that the value of the cross sections for the internal energy transfer was approximately 35 A2 and this is 24% greater than that for the viscosity which is 28.3 A*. They draw the conclusion that this difference does not lead to unacceptable error in most cases. However their system is still very small and for a larger system the difference is likely to be greater. Lendvay and Schatz [ 6 ] have investigated some larger target molecules like CS2 and SF6 whose different cross sections are more clearly separated. We have used their results to evaluate the cross sections for the internal energy transfer with model 3 above and for comparison calculated the cross sections from the viscosity. This is illustrated in fig. 5. For SF6 as target molecule and He, Ar and Xe as colliders the difference between the two cross sections is quite clear and amounts to about 1001. For CS2 as target molecule and H2, CO and CH4 as colliders the difference between the two cross sections is more complicated. For the pair CS,-CO, the cross section for the internal energy transfer is even smaller than that from the viscosity. In this case our fitting procedure in ( 13 ) does not work well with the results from Lendvay and Schatz for that pair showing a rather abrupt drop of ( AE) ,, at intermediate b values. For all pairs of target and medium molecules shown in fig. 5 the cross section according to our model 1 (or the similar definitions used by Lendvay and Schatz) is much larger (by a factor between 1 and 5) than either the traditional viscosity based or our model 3 estimate. Since

L.E.B. BSrjessonet al. /Chemical PhysicsI77 (1993) 133-143

142

cal Development for a grant allowing this work to be undertaken.

References [ 1] P.J. Robinson and K.A. Holbrok, Unimolecular reactions (Wiley, New York, 1972).

121W. Forst, Theory of unimolecular reactions (Academic

J

SF6-He

SF6-Ar

SF6-Xe

CS2-H2

C&-CO

CS,-C%

Fig. 5. Cross sections for some mixtures at 300 K are shown (see text). Dark bars are the conventionally employed cross sections from the viscosity. Light bars are cross sections from molecular dynamics simulations (see ref. [6] ) of (A&t) evaluated by model 3.

the value for (A,?) is (for a given energy transfer rate) inversely proportional to the cross section we see that the discussion of collision efficiency is entirely dependent on the proper definition of the cross section. In closing we note that by using the traditional method with the Lennard-Jones parameters o,_, and eU and an Q(**‘) collision integral to calculate a collision cross section chances are one will underestimate the cross section and overestimate the average internal energy transfer (AE). It is important to use a sound definition of cross section accounting for the smooth and property dependent falloff of energy transfer with impact parameter b. We believe our model 3 is preferable. It gives consistent and plausible results for the Br,-Br2 collisions considered here. Further detailed comparison for more molecules is needed. Even though a cross section leads the mind to a spherically symmetric object we should not forget the asymmetry of real molecules. In future work we hope to return to consider the use of diffusion and thermal conductivity data rather than viscosity to estimate collisional energy transfer cross sections.

Acknowledgement

We thank the Swedish National Board for Techni-

Press, New York, 1973 ). 13 1H.O. Pritchard, Quantum theory of unimolecular reactions (Cambridge Univ. Press, Cambridge, 1984). I4 1M. Quack and J. Troe, in: Gas kinetics and energy transfer, Specialist Periodical Report, Vol. 2 (The Chemical Society, London, 1977) p. 175. [ 51 R.G. Gilbert and S.C. Smith, Theory of unimolecular and recombination reactions (Blackwell, London, 1990). [6] G. Lendvay andG. Schatz, J. Phys. Chem. 96 ( 1992) 3752. [ 71 G. Lendvay and G. Schatz, J. Chem. Phys. 98 ( 1993) 1034. [S] N.J. Brownand J.A. Miller, J. Chem. Phys. 80 (1984) 5568. [ 91 D.L. Jolly, B.C. Freasier and S. Nordholm, Chem. Phys. 2 1 (1977)211. [lo] W.L. Hase, N. Data, L.B. Bhuiyan and D.G. Buckowski, J. Phys. Chem. 89 (1985) 2502. [ 111 Xiche Hu and W.L. Hase, J. Phys. Chem. 92 ( 1988) 4040. [ 121 H. Hippler, H.W. Schranz and J. Troe, J. Phys. Chem. 90 (1986) 6158. [ 131 N. Markovic and S. Nordholm, Chem. Phys. 135 ( 1989) 109. [ 141 J.I. Steinfeld, J.S. Francisco and W.L. Hase, Chemical kinetics and dynamics (Prentice Hall, Englewood Cliffs, 1989) p. 260. [ 151 G.H. Kohlmaier and B.S. Rabinovitch, J. Chem. Phys. 38 (1963) 1692. [ 161 J.O. Hirschfelder, CF. Curtis and R.B. Bird, Molecular theory of gases and liquids (Wiley, New York, 1964). 17 ] R.C. Reid, J.M. Prausnitz and T.K. Sherwood, The properties of gases and liquids, 3rd Ed. (McGraw-Hill, New York, 1977). 18 ] S. Chapman and T.G. Cowling, The mathematical theory of non-uniform gases, 3rd Ed. (Cambridge Univ. Press, Cambridge, 1970). [ 191 G.C. Maitland, M. Rigby, E.B. Smith and W.A. Wakeham, Intermolecular forces. Their origin and determination (Clarendon Press, Oxford, 198 1). [ 20 ] W. Kauzmann, Kinetic theory of gases (Benjamin, New York, 1966) p. 209. [21] J. Troe, J. Chem. Phys. 66 (1977) 4745,4758. [22] D.C. Tardy and B.S. Rabinovitch, Chem. Rev. 77 (1977) 369. [23] Y. Abe, J. Kestin, H.E. Khalifa and W.A. Wakeham, Ber. Bunsenges. Physik. Chem. 83 ( 1979) 27 1. [24] J. Kestin, S.T. Ro and W.A. Wakeham, J. Chem. Phys. 56 (1972) 5837. [25] F.B. Pidduck, Proc. Roy. Sot. A 101 (1922) 101.

L.E.B. Bkjesson et al. /Chemical PhysicsI77 (1993) 133-143 [26] C. Muckenfuss and CF. Curtis, 3. Chem. Phys. 29 (1958) 1257. [27] S.I. Sandler and J.S. Dahler, J. Chem. Phys. 44 (1966) 1229. [28] E.A. Mason and L. Monchick, J. Chem. Phys. 36 ( 1962) 1622. [ 291 L. Monchick, K.S. Yun and E.A. Mason, J. Chem. Phys. 39 (1963) 654. [ 301 J. Davidsson, S. Nordholm and L. Andersson, Chem. Phys. Letters 191 (1992) 489. [ 3 1] N. Markovic and S. Nordholm, Chem. Phys. 134 ( 1989) 69.

143

[32] E. Everhart, in: Dynamics of comets, their origin and evolution, eds. A. Carusi and G.B. Valsechi (Reidel, Dordrecht, 1985) p. 185. [33] J. Davidsson and G. Nyman, J. Chem. Phys. 92 ( 1990) 2407. [ 341 E.S. Severin, B.C. Freasier, N.D. Hamer, D.L. Jolly and S. Nordholm, Chem. Phys. Letters 57 (1978) 117. [ 351 H.W. Schranz, S. Nordholm and G. Nyman, J. Chem. Phys. 92 (1991) 1487.