PHYSICS OF FLUIDS 26, 046102 (2014)

Three-dimensional analytic probabilities of coupledvibrational-rotational-translational energy transferfor DSMC modeling of nonequilibrium flows

Igor V. AdamovichNonequilibrium Thermodynamics Laboratory, Department of Mechanical and AerospaceEngineering, The Ohio State University, Columbus, Ohio 43210, USA

(Received 23 February 2014; accepted 11 April 2014; published online 24 April 2014)

A three-dimensional, nonperturbative, semiclassical analytic model of vibrationalenergy transfer in collisions between a rotating diatomic molecule and an atom,and between two rotating diatomic molecules (Forced Harmonic Oscillator–FreeRotation model) has been extended to incorporate rotational relaxation and couplingbetween vibrational, translational, and rotational energy transfer. The model isbased on analysis of semiclassical trajectories of rotating molecules interacting bya repulsive exponential atom-to-atom potential. The model predictions are com-pared with the results of three-dimensional close-coupled semiclassical trajectorycalculations using the same potential energy surface. The comparison demonstratesgood agreement between analytic and numerical probabilities of rotational andvibrational energy transfer processes, over a wide range of total collision energies,rotational energies, and impact parameter. The model predicts probabilities ofsingle-quantum and multi-quantum vibrational-rotational transitions and is applica-ble up to very high collision energies and quantum numbers. Closed-form analyticexpressions for these transition probabilities lend themselves to straightforwardincorporation into DSMC nonequilibrium flow codes. C© 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4872336]

I. INTRODUCTION

Vibrational energy transfer processes in collisions of diatomic molecules play an important rolein gas discharges, molecular lasers, plasma chemical reactors, and high enthalpy gas dynamic flows.In these nonequilibrium environments, energy loading per molecule may be up to several eV, whiledisequipartition among translational, rotational, vibrational, and electronic energy modes of heavyspecies, and with the free electron energy, may be very strong. This results in development andmaintaining of strongly nonequilibrium molecular vibrational energy distributions, which inducea wide range of energy transfer processes among different energy modes and species, chemicalreactions, and ionization.1–4 The rates of these processes are strongly dependent on populationsof high vibrational levels of molecules, which at high temperatures are controlled primarily byvibration-rotation-translation (V-R-T) energy transfer,

AB(v, j1) + M → AB(w, j2) + M, (1)

where AB and M represent a diatomic molecule and a collision partner (another molecule or anatom), respectively, v and w are vibrational quantum numbers, and j1 and j2 are rotational quantumnumbers.

State-specific rates of vibrational relaxation processes, over a wide range of temperatures andvibrational quantum numbers, are needed for numerous nonequilibrium flow applications, such asradiation prediction in aerospace propulsion flows, in high altitude rocket plumes, and behind strongshock waves. For a number of diatomic species, such as H2, N2, and O2, vibrational relaxation

1070-6631/2014/26(4)/046102/19/$30.00 C©2014 AIP Publishing LLC26, 046102-1

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046102-2 Igor V. Adamovich Phys. Fluids 26, 046102 (2014)

rates have been predicted by semiclassical and quasiclassical trajectory calculations, using analyticatom-to-atom and ab initio potential energy surfaces.5–17 Recently, extensive databases incorporat-ing state-specific V-R-T energy transfer and dissociation rate coefficients for N2-N have also beendeveloped using quasiclassical trajectory calculations for ab initio potential energy surfaces.18–24

These databases have been used for modeling of nitrogen excitation and dissociation behind strongnormal shocks, in nonequilibrium nozzle flows, and in hypersonic flows over a cylinder.25–28 De-velopment of a comprehensive, complementary ab initio database for N2-N2 collisions remainsan open challenge. These databases also make possible thorough validation of analytic, physics-based, models of vibrational and rotational energy transfer, which provide insight into the energytransfer mechanism and lend themselves to being incorporated into existing nonequilibrium flowcodes.

State-specific V-T, V-V, and molecular dissociation rates in three-dimensional atom-moleculeand molecule-molecule collisions have been predicted analytically, using a nonperturbative ForcedHarmonic Oscillator–Free Rotation (FHO-FR) energy transfer model.29–31 V-T and V-V transitionprobabilities predicted by the analytic FHO-FR model are in very good agreement with numericaltrajectory calculations over a wide range of collision energies, for the same intermolecular inter-action potential. The FHO-FR model also reproduces the dependence of transition probabilities onindividual collision parameters such as rotational energy, impact parameter, and collision reducedmass. Finally, the model equally well predicts probabilities of single-quantum and multi-quantumtransitions, up to very high collision energies and quantum numbers. Although the FHO-FR modeldoes include the effect of rotational motion of molecules during the collision on vibrational energyprobabilities, the assumption of free rotation does not allow predicting rotational energy trans-fer. This limitation is critical for modeling of molecular dissociation from high vibrational levels,since rotational energy may well have a significant effect on overcoming the dissociation energythreshold.

The main objective of the present work is expanding the FHO-FR model to include rotationalrelaxation and coupling between vibrational, translational, and rotational energy transfer. The re-sultant set of closed-form, analytic V-R-T transition probabilities in atom-atom, and molecule-atomcollisions can be readily incorporated into state-specific DSMC flow codes, such as have beendeveloped recently.28, 32, 33

II. R-T AND V-R-T ENERGY TRANSFER MODELS

In the FHO-FR model, dynamics of collisions between a rotating symmetric diatomic moleculeand an atom, or collisions between two rotating diatomic molecules, was analyzed assuming that therotational motion was not affected by the collision. Thus, the only effect of rotation on the collisiontrajectory in the FHO-FR model is the periodic modulation of the repulsive, exponential, atom-to-atom interaction potential. As demonstrated in Ref. 29, this modulation makes the time-dependentinteraction more sudden, thus considerably increasing vibrational energy transfer probabilities. Theassumption of free rotation is similar to the basic premise of the semiclassical approach, whichdecouples the classical translational-rotational trajectory during a collision from the vibrationalmotion of the oscillator, which is treated quantum mechanically. Combining this approach with theexact solution of the Schrodinger equation predicting probabilities of vibration-translation (V-T)and vibration-vibration (V-V) energy transfer in a forced harmonic oscillator, valid for arbitraryvalues of initial and final vibrational quantum numbers,29, 30 and using frequency correction toaccount for anharmonicity, yields closed-form analytic expressions for V-T and V-V probabilitiesin three-dimensional atom-molecule and molecule-molecules collisions. These probabilities arein very good agreement with “exact” close-coupled semiclassical trajectory calculations for N2-N2, for the same intermolecular potential, over a wide range of collision energies (temperatures)and vibrational quantum numbers.29, 30 Also, it has been shown that FHO-FR model accuratelyreproduces the dependence of the numerical V-T probabilities on the rotational energies of themolecules. This demonstrates that modulation of the interaction potential by rotation is indeed thedominant effect of rotational motion on vibrational energy transfer (producing strong “R → V-T”coupling).

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046102-3 Igor V. Adamovich Phys. Fluids 26, 046102 (2014)

The assumption of free rotation used by the FHO-FR model, which does not have a significanteffect on the V-T and V-V rates, is no longer sufficient if rotational energy transfer during a collisionalso needs to be predicted. High-energy collisions of strongly vibrationally excited molecules,when a significant fraction of the total collision energy is transferred to the rotational energy mode(producing strong “V-T → R” coupling), are of particular interest since they may affect the rate ofmolecular dissociation. Basically, these collisions may result in strong contribution of centrifugalforces to dissociation.

Incorporating rotational energy transfer into the semiclassical FHO theory requires an an-alytic solution for the classical vibrational-rotational trajectory in three-dimensional collisionsof an atom and a molecule, or in collisions of two diatomic molecules. A previous semiclas-sical trajectory calculation study of rotational relaxation of N2 in collisions with Ar34 pointedout the existence of an approximate “stochastic” analytic solution for the rotational transitionprobabilities:

P( j0 → j) = 1√4π S

exp

[− ( j0 − j)2

4S

], (2)

where j0 and j are the initial and the final rotational quantum numbers. In Eq. (2), S is an inte-gral over the collision trajectory, which depends on the total collision energy, rotational energy ofthe molecule, impact parameter, and the direction of the angular momentum vector. In Ref. 34,S was evaluated numerically, based on an approximate analytic solution for the center-of-massdistance during the collision. The results suggested that the distribution of probabilities predictedby Eq. (2) may be fairly accurate in the limit of weak coupling, i.e., at large impact parame-ters. The possibility of evaluating the entire set of rotational transition probabilities in terms ofa single integral parameter of the trajectory is very attractive, although obtaining a general ana-lytic solution for the trajectory, for a realistic intermolecular potential, appears to be an intractableproblem.

Further analysis done in the present work shows that for the intermolecular potential dominatedby steep repulsion, such as U(R) ∼ A exp(−αR), where the spatial scale for the potential energychange, 1/α, is much less than the distance of closest approach, 1/α � R0, the trajectory integralS in Eq. (2) scales proportionally to the “effective kinetic energy” of collision, i.e., the energy ofrelative translational motion in the radial direction near the “maximum interaction point,” whereU(R0) reaches maximum,29, 30 as follows:

S( j0 → j, Et , b) =(√

ξ Et/B − j0)2

8

[1 −

(b

R0

)2]

. (3)

In Eq. (3), Et is the total collision energy, b is the impact parameter, ξ = 1 and ξ = 5/6 foratom-molecule and molecule-molecule, respectively. The lower value of ξ for molecule-moleculecollisions is due to energy storage in the rotation of the “projectile” molecule, such that a smallerfraction of total collision energy is available for the translational motion. As will be shown below,this simple correction is sufficient to account for the rotational motion of the “projectile” moleculeon rotational energy transfer in the “target” molecule, for a wide range of conditions. Energy andangular momentum conservation impose the following constraints on the final value of the rotationalquantum number after collision,

max

(0, Er0 − ξ Et

2

)< B j2 < Et − Ek0

2

Et = Ek0 + Er0 = Ek + Er , Er0 = B j20 , Er = B j2

, (4)

where Ek0, Ek, Er0, and Er are the initial and the final values of the kinetic (translational) and therotational energy of the “target” molecule.

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046102-4 Igor V. Adamovich Phys. Fluids 26, 046102 (2014)

After averaging the square of the rotational energy transferred, predicted by Eqs. (2)–(4), overimpact parameter and rotational quantum numbers,

P( j0 → j, Et ) =1∫

0

P( j0 → j, Et , b) d

(b2

R20

)

⟨�E2

r (Et )⟩1/2 =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

√Etotal/B∫∫

0

(2 j0 + 1)(Er − Er0)2 P( j0 → j, Et ) d jd j0

√Etotal/B∫∫

0

(2 j0 + 1)P( j0 → j, Et ) d jd j0

⎞⎟⎟⎟⎟⎟⎟⎟⎠

1/2

= ξ

8Et

, (5)

the maximum value of the trajectory integral in Eq. (3), Smax, can be related to the root-mean-square(RMS) average rotational energy transferred in a collision:

Smax = ξ

8

Et

B=

⟨�E2

r (Et )⟩1/2

B= �E RM S

r

B. (6)

Thus, the rotational energy transfer probability distribution over the final rotational quantum numbersis determined by a single parameter, RMS average rotational energy transfer, i.e., by the rotationalrelaxation time.35 Note that the expressions for S and �Er

RMS do not contain explicitly the repul-sive exponential potential parameter, α. Basically, since 1/α � R0, rotational energy transfer canbe assumed to occur in hard sphere collisions. Finally, to account for detailed balance betweenthe forward and the reverse processes, P( j0 → j, Et, b)(2j0 + 1) = P( j → j0, Et, b)(2j + 1), theexpressions for S in the transition probabilities need to be “symmetrized” over the initial and thefinal rotational quantum states,

P( j0 → j, Et , b) =√

1

4π Sexp

[− ( j0 − j)2

4S

] √2 j + 1

2 j0 + 1, (2a)

S( j0 → j, Et , b) =(√

ξ Et/B − √j0 j

)2

8

[1 −

(b

R0

)2]

. (3a)

Note that symmetrization affects normalization of the rotational transition probabilities given by

Eq. (2a), and they need to be renormalized to ensure thatjmax∫jmin

P( j0 → j) d j = 1, where jmin and jmax

are determined by Eq. (4).For the exponential repulsive part of the intermolecular potential used in semiclassical trajectory

calculations to predict V-T and V-V energy transfer rates for N2-N210 and rotational relaxation cross

section for N2-N2,35 α = 4.0 A−1, the condition 1/α � R0 is satisfied, 1/αR0 ∼ 0.1. The closestapproach distance in Eq. (3a), evaluated for the potential used in Ref. 10 in previous work on FHOmodel,29, 30 is

R0(Etotal) ≈ 2.5 − 1

2αln

(Et (cm−1)

104

)0A . (7)

In the first approximation, a semiclassical V-R-T transition probability in atom-molecule collisionscan be expressed as a product of V-T and R-T probabilities, as follows:

PV RT (v, j1 → w, j2, Et , b) = PRT ( j1 → j2, Et , b) · PV T (v → w, Er , Et , b). (8)

In Eq. (8), symmetrized R-T probabilities are given by Eq. (2a), and V-T transition probabilitiesbetween vibrational quantum numbers v and w are predicted by the Forced Harmonic Oscillator–Free

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046102-5 Igor V. Adamovich Phys. Fluids 26, 046102 (2014)

Rotation (FHO-FR) model,29

PV T (v → w) ≈ (ns)s

(s!)2Qs exp

[− 2ns

s + 1Q − n2

s

(s + 1)2(s + 2)Q2

],

s = |v − w| , ns =[

max(v,w)!

min(v,w)!

]1/s

.

(9)

In Eq. (9), Q is the average number of vibrational quanta transferred per collision (a trajectory inte-gral), calculated using an approximate analytic trajectory for collisions of freely rotating molecules,29

Q(Et , Er , b) = θ ′

4θ

1

sinh2

[πω

αuγ (Et , Er , b)

] , (10)

γ (Et , Er , b) = max

⎡⎣0,

√(1 − Er

Et

) (1 − b2

R20

)− sin(2ϑ) cos(ϕ)

2

√Er

Et

⎤⎦ . (11)

In Eqs. (10) and (11), ω = |Ew−Ev |s�

is the average vibrational quantum for the transition v → w,

θ ′ = 4π2ω2mα2k , θ = �ω

k , m is the collision reduced mass, Er = B j1 j2 is the symmetrized rotationalenergy, u = √

2Et/m, and ϑ , ϕ are the rotation phase angles at the closest approach point.29 Althoughit is desirable to average the trajectory integral, Q, in Eqs. (10) and (11) over the randomly distributedrotation phase angles, 0 ≤ ϑ ≤ π and 0 ≤ ϕ ≤ π /2, it is challenging to do this analytically withsufficient accuracy, e.g., by expanding it in series near its maximum value, achieved at Er/Et = 0.2,b/R0 = 0, ϑ = 3π /4, and ϕ = 0.29 This difficulty occurs because V-T transition probabilitiesactually decrease at high values of Q, as can be seen from Eq. (9). Therefore, at high total collisionenergies (i.e., high u values in Eq. (10)), collisions with low values of γ , which correspond to highrotational energies (Er/Et ∼ 1), large impact parameters (b/R0 ∼ 1), and rotation in a plane nearlyperpendicular to the velocity vector (ϕ ∼ π/2) provide the dominant contribution to V-T relaxation.As demonstrated in Ref. 29, this effect causes complete breakdown of the one-dimensional FHOV-T energy transfer model at high energies, compared to the three-dimensional collision FHO-FRmodel.

In molecule-molecule collisions, parameter γ in Eq. (11) is calculated taking into account theeffect of rotation of both molecules on the collision trajectory, and thus on vibrational energy transferprobabilities,30

γ (Et , Er1, Er2, b) = max

[0,

√(1 − Er1

Et− Er2

Et

) (1 − b2

R20

)

− sin(2ϑ1) cos(ϕ1)

2

√Er1

Et− sin(2ϑ2) cos(ϕ2)

2

√Er2

Et

⎤⎦ . (11a)

In the present work, the projectile molecule was assumed to remain in vibrational level v′ = w′

= 0,30 and vibrational-rotational transitions in the projectile molecule were not evaluated explicitly.However, the present approach can be easily extended to predict probabilities of simultaneous VRTtransitions in both molecules, P( v,j1,v′, j1′ → w,j2,w′, j2′).

Thus, V-R-T transition probabilities are expressed in terms of two trajectory integrals, S andQ, given by Eq. (3a) and Eqs. (10)–(11a), respectively, each one of them being a function of totalcollision energy, rotational energy, and impact parameter. Note that V-T/R-T factorization used inEq. (8) does not fully account for V-R-T coupling since V-T probabilities in it are still calculated forfree-rotating molecules, and is therefore only a first-order approximation.

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046102-6 Igor V. Adamovich Phys. Fluids 26, 046102 (2014)

III. COMPARISON WITH TRAJECTORY CALCULATIONS

To assess the accuracy of the simple analytic solution for the rotational transition probabilitiesgiven by Eqs. (2a), (3a), and (4), the model predictions have been compared with the results ofsemiclassical trajectory calculations (a) between a nitrogen molecule and an atom of the same mass(spherical nitrogen) and (b) between two nitrogen molecules, using trajectory calculation codesADIAV and DIDIEX developed by Billing.36, 37 In both cases, the exponential repulsive interactionpotential used was the same as in previous work predicting FHO-FR vibrational energy transfer ratesin atom-molecule and molecule-molecule collisions.29, 30 The maximum impact parameter used inthe numerical calculations was R = 2.5 A. In the calculations, the total collision energy, Et, was keptconstant, while rotational and orbital angular momenta, as well as initial orientation of the collisionpartners, were selected randomly, for 104–105 trajectories.

Figure 1 compares transition probabilities in atom-molecule collisions for three different impactparameters, as well as averaged over the impact parameter, 0 ≤ b ≤ R0. It can be seen that theanalytic model reproduces the probability distribution over the final rotational quantum numbersquite well, although it underpredicts the probability for small changes in the rotational quantumnumber, j-j0. Figure 2 plots analytic and numerical rotational transition probabilities, averaged overthe impact parameter, for different values of the initial rotational energy. One can see that theprobability distribution over the final rotational quantum number is reproduced well by the analyticmodel, except for the case of a high rotational energy/low translational kinetic energy, Er/Et = 3/4,

FIG. 1. Comparison of analytic and numerical (104 trajectories) rotational energy transfer probabilities, for different impactparameter values. Atom-molecule, Et = 104 cm−1, Er/Et = 1/4, b = 0.5 Å, 1.0 Å, 2.0 Å, and averaged over 0.0–2.5 Å.

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046102-7 Igor V. Adamovich Phys. Fluids 26, 046102 (2014)

FIG. 2. Comparison of analytic and numerical (105 trajectories) rotational energy transfer probabilities averaged overthe impact parameter, for different initial rotational energies. Atom-molecule, Et = 104 cm−1, Er/Et = 0, 1/4, 1/2,and 3/4.

(Er/Et = 3). Basically, the trajectory calculations demonstrate that, as expected, rapidly rotatingmolecules with low translational energies behave as nearly isotropic particles (near “breathingspheres”), which makes rotational energy transfer very inefficient, such that the transition probabilitydrops rapidly with �j = j0 − j. For these trajectories, the approximate analytic expression for thetrajectory integral S, given by Eq. (3a), becomes less accurate (see Fig. 2(d)). Figure 3 compares thetransition probabilities averaged over the impact parameter and initial rotational quantum numbers(assumed to be distributed randomly between j0 = 0 and j0,max),

〈P(� j, Et )〉 =

j0,max∑j0=0

(2 j0 + 1)P( j0 → j, Et )

( j0,max + 1)2, (12)

for different values of the total collision energy. Again, it is apparent that the analytic model predictionfor the rotational energy transfer probability distribution over �j = j0 − j is in good agreement withthe results of the trajectory calculations, over a wide range of total collision energies, Et = 103–105

cm−1 (Et ≈ 0.12–12.0 eV). A closer look shows that at high collision energies, the analytic model un-derpredicts the average probability for the molecule to remain on the same rotational level, such that�j = j0 − j = 0, and overpredicts average probabilities of transitions with �j �= 0. Further analysis

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046102-8 Igor V. Adamovich Phys. Fluids 26, 046102 (2014)

FIG. 3. Comparison of analytic and numerical (105 trajectories) rotational energy transfer probabilities averaged overthe impact parameter and rotational energies, for different total energies. Atom-molecule, Et = 103, 104, 3 × 104, and105 cm−1.

shows that this occurs due to the contribution of molecules with high initial rotational energies (highj0 values), i.e., low translational energies, Ek = Et − Er0, for which the analytic model predictions areless accurate (see Fig. 2(d)). Finally, the “pointed arch” shaped probability distributions plotted inFig. 3 demonstrate that the use of linear or Gaussian scaling for rotational energy transfer prob-abilities, ln[P( j0 → j)] ∼ −A | j0 − j| or ln[P( j0 → j)] ∼ −A(j0 − j)2, would be simplistic andis not justified. Basically, for initially rapidly rotating molecules (high j0), the “spread” over thefinal rotational quantum numbers is much narrower compared to the molecules with lower initialrotational energies (e.g., compare Figs. 2(c) and 2(d)), which produces the pointed arch shape ofP( �j) distributions shown in Fig. 3.

Figures 4 and 5 compare the rotational transition probabilities in molecule-molecule collisions,for different initial rotation energies (Fig. 4), and different total collision energies (Fig. 5), showingtrends similar to the ones for atom-molecule collisions. These plots demonstrate that the effectof rotation of the “projectile” molecule on rotational energy transfer in the “target” molecule isaccounted for by reducing the total collision energy available for the translational motion (by aparameter ξ in Eq. (3a)). This also shows that, in the first approximation, the probabilities forsimultaneous rotational energy transfer in both collision partners can be factorized as follows, P( j0→ j, j0′ → j) ≈ P( j0 → j) · P( j0′ → j′).

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046102-9 Igor V. Adamovich Phys. Fluids 26, 046102 (2014)

FIG. 4. Comparison of analytic and numerical (104 trajectories) rotational energy transfer probabilities averaged overthe impact parameter, for different initial rotational energies. Molecule-molecule, Et = 104 cm−1, Er/Et = 0, 1/4, 1/2,and 3/4.

Figure 6 plots RMS average rotational energy transferred per collision vs. total collision en-ergy, both for atom-molecule and molecule-molecule. The results of the analytic model, plotted inFig. 6, have been corrected for the variation of the closest approach distance (i.e., total collisioncross section) with the total collision energy, given by Eq. (7), such that �Er

RMS predicted by theanalytic model (see Eq. (5)) slightly decreases with the collision energy,

�E RM Sr (Et )

Et= ξ

8

R0(Et )

2.50A

, (13)

in good agreement with the trajectory calculations. Summarizing, the results shown in Figs. 1–6demonstrate that the present analytic model of rotational energy transfer is in good agreement withatom-molecule and molecule-molecule trajectory calculations, reproducing accurately the depen-dence of state-specific rotational energy transfer probabilities on the change of rotational quantumnumber during collision ( j − j0), impact parameter, initial rotational energy (Er0 = Bj02), and totalcollision energy (Et).

RMS average rotational energy transferred per collision, �ErRMS, can be related to the rotational

relaxation collision number, Zrot, as follows:35

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046102-10 Igor V. Adamovich Phys. Fluids 26, 046102 (2014)

FIG. 5. Comparison of analytic and numerical (105 trajectories) rotational energy transfer probabilities averaged over theimpact parameter and rotational energies, for different total energies. Molecule-molecule, Et = 103, 104, 3 × 104, and105 cm−1.

1

Zrot= π

4

η

Pτrot≈ 5π

16

⟨⟨�E2

r (T )

T 2

⟩⟩⟨⟨

E2k0 sin2 χ

T 2

⟩⟩ ≈

∞∫0

⟨�E2

r (Et )⟩

T 2

(Et

T

)n

exp

(− Et

T

)d

(Et

T

)

∞∫0

⟨E2

k0 sin2 χ⟩

T 2

(Et

T

)n

exp

(− Et

T

)d

(Et

T

) . (14)

In Eq. (14), η and τ rot are dynamic viscosity and rotational relaxation time, respectively,

η ≈ 5kT

8

⟨⟨E2

k0 sin2 χ

T 2

⟩⟩ , (15)

τrot = 1

2N

⟨⟨�E2

r (T )

T 2

⟩⟩ = kT

2P

⟨⟨�E2

r (T )

T 2

⟩⟩ , (16)

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046102-11 Igor V. Adamovich Phys. Fluids 26, 046102 (2014)

FIG. 6. Comparison of analytic and numerical RMS average rotational energy transferred per collision. Left: atom-molecule,right: molecule-molecule.

Ek0 is the initial translational kinetic energy, χ is the scattering angle, n = 2 for atom-molecule andn = 3 for molecule-molecule collisions. For atom-molecule collisions,

⟨E2

k0 sin2 χ⟩ ≈ (

35

)2E2

t

⟨sin2 χ

⟩≈ 1

2

(35

)2E2

t . For molecule-molecule collisions,⟨E2

k0 sin2 χ⟩ ≈ 1

2

(37

)2E2

t . Combining this with the

expression for the average squared rotational energy transfer,⟨�E2

r

⟩ = (ξ

8

)2E2

t (see Eq. (5)), weobtain

Z A−Mrot ≈ 1

2

(24

5

)2

≈ 12 Z M−Mrot ≈ 1

ξ 2

1

2

(24

7

)2

≈ 8.5, (17)

both values nearly independent of temperature. These results are in good agreement with the presenttrajectory calculations, which predict Z M−M

rot ≈ 10 ≈ const for the repulsive exponential potentialused, in the range of T = 300-5000 K (see Fig. 7). Figure 7 plots Zrot for nitrogen, predicted using

FIG. 7. Comparison of rotational collision number in nitrogen predicted by the present trajectory calculations with previousresults.

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046102-12 Igor V. Adamovich Phys. Fluids 26, 046102 (2014)

FIG. 8. Effect of repulsive potential parameter α on rotational collision number, and on rotational and translational energytransfer in nitrogen. Et = 104 cm−3.

DIDIEX for a purely repulsive exponential potential with α = 4.0 A−1 29, 30 and for an attractive-repulsive potential.10 It can be seen that the effect of molecular attraction, which may considerablyreduce the collision number, is significant only at fairly low temperatures, below T ∼ 1000 K.Figure 7 also plots the rotational collision number predicted by molecular dynamics (MD)38 calcula-tions, using a more accurate Lennard-Jones potential energy surface. One can see that Zrot predictedby MD calculations is ∼30% lower than the values predicted by the semiclassical code using thepotential from Ref. 10, although both models exhibit the same trend for Zrot to level off at hightemperatures, when the effect of attractive forces becomes insignificant.

To estimate the sensitivity of the rotational collision number to the uncertainty in the repulsiveexponential potential parameter α, Fig. 8 plots Zrot, as well as RMS average values of

⟨�E2

r

⟩1/2and⟨

E2k0 sin2 χ

⟩1/2, calculated for a purely repulsive potential at Et = 104 cm−1, vs. α. The results of

these calculations show that varying the potential parameter primarily affects the viscosity, while theRMS rotational energy transfer per collision remains nearly constant over a wide range of repulsiveintermolecular potentials. As a result, the collision number decreases as α increases and the potentialapproaches the hard spheres potential, approaching the asymptotic value of Zrot ≈ 4. Thus, lowervalues of Zrot predicted in Ref. 37 are likely due to a stiffer repulsive potential (i.e., larger effectivevalue of α) used. Finally, Fig. 8 illustrates that rotational energy transfer at high temperatures isindeed almost insensitive to the intermolecular potential, as long as 1/α � R0, in good agreementwith the present analytic model,

⟨�E2

r

⟩1/2/Et = ξ

8 ≈ 0.1. The results of Figs. 7 and 8 demonstratethat, as expected, the use of a simple repulsive exponential potential to predict rotational relaxationat high temperatures (above T ∼ 1000 K) is fully justified.

The present rotational relaxation model reproduces the effect of rotational-translational nonequi-librium on rotational relaxation time, discussed in detail in Ref. 39. Figure 9 plots the average squaredrotational energy transferred in atom-molecule collisions,

⟨�E2

r

⟩, at the conditions when the ratio of

the initial rotational energy to the total collision energy, Er0/Et, is fixed, normalized on its value inequilibrium,

⟨�E2

r

⟩eq , i.e., when the initial rotational energy is randomly distributed between 0 and

Et, at Et = 103 cm−1. The results are shown both for the analytic model and for trajectory calcula-tions. From Fig. 9, it can be seen that

⟨�E2

r

⟩in collisions of non-rotating molecules (Er0/Et = 0) is

significantly higher than the equilibrium value, by about a factor of two. On the other hand,⟨�E2

r

⟩

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046102-13 Igor V. Adamovich Phys. Fluids 26, 046102 (2014)

FIG. 9. Average squared rotational energy transferred in atom-molecule collisions, normalized on its value in equilibrium,vs. ratio of the initial rotational energy to the total collision energy, Er0/Et. Et = 103 cm−1.

in collisions of rapidly rotating molecules (Er0/Et → 1) is much lower compared to the equilibriumvalue, as expected based on the results of Fig. 2. Since the rotational relaxation time is inverselyproportional to

⟨�E2

r

⟩(see Eq. (16)), τ rot in collisions of initially non-rotating molecules will be

shorter than near rotational equilibrium, by approximately a factor of two at this collision energy,and τ rot in collisions of rapidly rotating molecules will be significantly longer, consistent with thepredictions of the molecular dynamics model of Ref. 39.

Figure 10 compares V-R-T transition probabilities in atom-molecule collisions, averaged overthe impact parameter and initial rotational quantum numbers (assumed to be distributed randomlybetween j0 = 0 and j0,max, as has been done for purely rotational relaxation in Fig. 3),

〈PV RT (� j, v → w, Et )〉 =

j1,max∑j1=0

(2 j1 + 1)PV RT (v, j1 → w, j2, Et )

( j1,max + 1)2, (18)

for different values of the total collision energy. The plots shown in Fig. 10 are for vibrationaltransitions for �v = w-v = 0–3, i.e., from v = 0 to w = 0–3. Once again, the analytic modelprediction for the V-R-T probability distribution over �j = j0 − j is in good agreement with theresults of the trajectory calculations, over a wide range of total collision energies, Et = 103–105 cm−1

(Et ≈ 0.12–12.0 eV). In Figure 10, significant separation between the distributions of V-R-T proba-bilities over the rotational quantum number change at Et < 105 cm−1 is due to V-T probabilities inthis range of total collision energies being very low, P( v = 0 → w = 0) P( v = 0 → w = 1) P( v= 0 → w = 2) P( v = 0 → w = 3). Note that the “pointed arch” distribution observed for purelyrotational relaxation, P( v = 0, j1 → w = 0, j2), is not duplicated for V-R-T relaxation processes,P( v = 0, j1 → w = 1–3, j2), in the collision energy range Et ∼ 103 − 3 × 104 cm−1 (see Fig. 10).A more detailed analysis shows that this effect is due to contribution of molecules with very highinitial rotational energies ( j0 values), i.e., near “breathing spheres”, for which V-T probabilities aresignificantly lower than for moderately rotating molecules, due to absence of strong modulation ofthe interaction potential by rotational motion.29, 30 At a high collision energy, Et = 105 cm−1, thiseffect is no longer very pronounced, and distributions over �j = j0 − j become more similar, for bothpure R-T and V-R-T relaxation processes. The difference in shape between analytic and trajectoryV-R-T probability distributions over �j for �v = 1–3 at Et = 3 × 104 cm−1, evident from Fig. 10,is caused by factorization of V-R-T transition probabilities into R-T and V-T probabilities in Eq. (8),

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046102-14 Igor V. Adamovich Phys. Fluids 26, 046102 (2014)

FIG. 10. Comparison of analytic and numerical (105 trajectories) V-R-T energy transfer probabilities (v = 0 → w = 0–3),averaged over the impact parameter and initial rotational quantum numbers, for different total energies. Atom-molecule,Et = 103, 104, 3 × 104, and 105 cm−1.

PV RT (v, j1 → w, j2) = PRT ( j1 → j2) · PV T (v → w), since both purely R-T analytic probabilities,obtained in the present work, and “free rotation” V-T analytic probabilities, predicted in Ref. 29, arein very good agreement with trajectory calculations.

Figure 11 plots V-R-T transition probabilities in molecule-molecule collisions, averaged overthe impact parameter and initial rotational quantum numbers of the target molecule, as given byEq. (18), as well as over initial and final rotational quantum numbers of the projectile molecule,as given by Eq. (11a). Figure 11 shows the results obtained only for three values of total collisionenergy, Et = 104, 3 × 104, and 105 cm−1, since extensive trajectory calculations for molecule-molecule collisions necessary to accumulate sufficient statistics for individual vibrational-rotationaltransitions (105 trajectories in Fig. 11) are computationally expensive, especially at low collisionenergies. From Fig. 11, it can be seen that the difference between analytic and trajectory V-R-Tprobability distributions over �j for �v = 1–3 at Et = 3 × 104 cm−1, also apparent from Fig. 10,becomes more pronounced. Also, at Et = 105 cm−1 it is clear that the present model overestimatesrotational energy transfer probabilities for �v = 0 and underestimates them for �v = 3, althoughV-T probabilities for transitions �v = 0–3 (summed over the rotational quantum numbers) at thisenergy are predicted quite accurately, within ∼30%. More detailed analysis shows that this effectis due to contribution of rotational energy transfer in the “projectile” molecule to the vibrationalexcitation of the “target” molecule at high collision energies, which so far has not been accountedfor in the present model.

Figure 12 compares V-T transition probabilities in atom-molecule collisions (i.e., V-R-T prob-abilities for �v = 0–3 plotted in Fig. 10, averaged over the impact parameter as well as over initial

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046102-15 Igor V. Adamovich Phys. Fluids 26, 046102 (2014)

FIG. 11. Comparison of analytic and numerical (105 trajectories) V-R-T energy transfer probabilities (v = 0→ w = 0–3,v′ = w′ = 0), averaged over the impact parameter and initial quantum numbers, for different total energies. Molecule-molecule,Et = 104, 3 × 104, and 105 cm−1.

and final rotational quantum numbers),

〈PV T (v → w, Et )〉 =j2,max∑j2=0

j1,max∑j1=0

(2 j1 + 1)PV RT (v, j1 → w, j2, Et )

( j1,max + 1)2, (19)

vs. total collision energy. As expected, the agreement between the analytic model and the trajec-tory calculations is quite good, over a wide range of total collision energies, Et = 103–106 cm−1.Again, the difference between the analytic and numerical probabilities is due to V-R-T probabilityfactorization in Eq. (8), i.e., not accounting for full coupling between rotational and vibrationalmodes. As can be seen from Fig. 13, which compares atom-molecule V-T transition probabili-ties for v = 40 → w = 35–40 (�v = v-w = 0–5), similar level of agreement is obtained forV-T transitions of initially highly vibrationally excited molecules, for v > 0. Figure 14 plots thesame results for molecule-molecule collisions, for which the agreement with the trajectory cal-culations is not as good, especially for multiquantum transitions with �v = 3, where additionalanalysis is needed to account for coupling between vibrational and rotational modes. The results of

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046102-16 Igor V. Adamovich Phys. Fluids 26, 046102 (2014)

FIG. 12. Comparison of analytic and numerical (104 trajectories) V-T energy transfer probabilities in atom-moleculecollisions (v = 0 → w = 0–3), averaged over the impact parameter as well as initial and final rotational quantum numbers,vs. total collision energy.

Figs. 12–14 demonstrate that the present analytic model generates a complete, self-consistent set ofV-R-T transition probabilities in atom-molecule and molecule-molecule collisions, PV RT (v, j1 →w, j2), predicting vibrational and rotational energy transfer vs. total collision energy, as well as ini-tial and final vibrational and rotational quantum numbers. Although the present results are obtainedfor purely repulsive interaction potential, they are fully applicable for modeling of nonequilibriumflows behind strong shock waves, when the effect of attractive forces on energy transfer rates isnegligible.

FIG. 13. Comparison of analytic and numerical (104 trajectories) V-T energy transfer probabilities in atom-moleculecollisions (v = 40 → w = 35–40), averaged over the impact parameter as well as initial and final rotational quantum numbers,vs. total collision energy.

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046102-17 Igor V. Adamovich Phys. Fluids 26, 046102 (2014)

FIG. 14. Comparison of analytic and numerical (104 trajectories) V-T energy transfer probabilities in molecule-moleculecollisions (v = 0 → w = 0–3, v′ = w′ = 0), averaged over the impact parameter as well as initial and final rotational quantumnumbers, vs. total collision energy.

IV. SUMMARY

A three-dimensional semiclassical analytic model of vibrational energy transfer in collisionsbetween a rotating diatomic molecule and an atom, and between two rotating diatomic molecules(Forced Harmonic Oscillator–Free Rotation model), has been extended to incorporate rotational re-laxation and coupling between vibrational, translational, and rotational energy transfer. The model isbased on analysis of classical trajectories of rotating molecules interacting by a repulsive exponentialatom-to-atom potential. The model predictions are compared with the results of three-dimensionalclose-coupled semiclassical trajectory calculations using the same potential energy surface. The com-parison demonstrates good agreement between analytic and numerical probabilities of rotational andvibrational energy transfer processes, over a wide range of total collision energies, rotational ener-gies, and impact parameter. Specifically, rotational relaxation time (RMS average rotational energytransferred) predicted by the present model is within 10%–20% of trajectory calculation resultsfor the same interaction potential, and individual R-T relaxation probabilities are typically within50%–100% of the trajectory calculation results, except for collisions of very rapidly rotatingmolecules. For V-R-T relaxation probabilities, the difference between the present model predic-tions and the trajectory calculations is within a factor of 2 for atom-molecule collisions and upto about a factor of 3 for multi-quantum transitions in molecule-molecule collisions. The mainsource of the difference is the approximate nature of coupling between vibrational energy modeand rotational energy modes of one or two collision partners, assumed in the present approach. Themodel predicts probabilities of single-quantum and multi-quantum vibrational-rotational transitionsand is applicable up to very high collision energies and quantum numbers. Closed-form analyticexpressions for these transition probabilities lend themselves to straightforward incorporation intoDSMC nonequilibrium flow codes.

Future work will focus on taking into account coupling between vibrational and rotational energymodes of both colliding molecules and predicting state-specific rates of molecular dissociation fromhighly excited vibrational-rotational energy states. Also, detailed comparison with state-specificV-R-T databases for N2-N18–21 and N2-N2, obtained using ab initio potential energy surfaces, isdesirable. Although vibrational and rotational energy transfer at high collision temperatures is likelyto be dominated by the repulsive part of the interaction potential, such as used in the present work,

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046102-18 Igor V. Adamovich Phys. Fluids 26, 046102 (2014)

the role of atom exchange reactions at these conditions may well be significant, and needs to bequantified.

ACKNOWLEDGMENTS

This work has been supported by (U.S.) Air Force Office of Scientific Research (USAFOSR)Grant No. FA9550-11-1-0075 “Exploration of critical nonequilibrium processes in high speedviscous dominated flows: Nonequilibrium hypersonic shock layer flows” and FA9550-12-1-0439“Nonequilibrium molecular energy coupling and conversion mechanisms for efficient control ofhigh-speed flow field”. The author would like to thank Dr. Marco Panesi and Dr. Tom Schwartzen-truber for helpful technical discussions.

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