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Nonequilibrium internal energy distributionsduring dissociationNarendra Singha,1 and Thomas Schwartzentrubera

aDepartment of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455

Edited by R. D. Levine, The Fritz Haber Research Center, Jerusalem, Israel, and approved November 17, 2017 (received for review August 8, 2017)

In this work, we propose a model for nonequilibrium vibra-tional and rotational energy distributions in nitrogen using sur-prisal analysis. The model is constructed by using data fromdirect molecular simulations (DMSs) of rapidly heated nitrogengas using an ab initio potential energy surface (PES). The surprisal-based model is able to capture the overpopulation of high inter-nal energy levels during the excitation phase and also the deple-tion of high internal energy levels during the quasi-steady-state(QSS) dissociation phase. Due to strong coupling between internalenergy and dissociation chemistry, such non-Boltzmann effectscan influence the overall dissociation rate in the gas. Conditionsrepresentative of the flow behind strong shockwaves, relevantto hypersonic flight, are analyzed. The surprisal-based model cap-tures important molecular-level nonequilibrium physics, yet thesimple functional form leads to a continuum-level expression thatnow accounts for the underlying energy distributions and theircoupling to dissociation.

nonequilibrium distribution | surprisal analysis | high-temperaturethermochemistry | shock waves | hypersonic flows

H igh-temperature, chemically reacting gas systems are inher-ently nonequilibrium. As an example, dissociation reac-

tions are coupled to the vibrational energy of the dissociatingmolecule. Molecules in high vibrational energy states (high vlevels) are strongly favored for dissociation, leading to a popu-lation depletion compared with the corresponding equilibrium(Boltzmann) distribution. This depletion is balanced by nonreac-tive collisions within the gas that act to repopulate these high vlevels via translational–vibrational excitation. Rotational energystates (j levels) are similarly coupled to dissociation, although toa lesser extent than vibration. As a gas is undergoing dissociation,this combination of depletion due to dissociation and excitationdue to translational–internal energy transfer leads to a quasi-steady-state (QSS) where the internal energy distribution func-tions (both rotation and vibration) are time-invariant and non-Boltzmann. Furthermore, when a gas is rapidly heated, such asbehind a strong shockwave, high v levels can become overpop-ulated compared with the corresponding Boltzmann distributionas energetic collisions lead to multiquantum jumps in vibrationalenergy. Since high v and j levels are strongly favored for dissoci-ation, the precise nonequilibrium distributions of internal energycan significantly affect the gas dissociation rate.

Continuum models used to analyze reacting gas flows, suchas the Navier–Stokes equations extended for thermochemicalnonequilibrium (1, 2), model only average molecular energies.In some cases, it is appropriate to use a temperature T assumingequipartition of energy across available translational, rotational,and vibrational energy modes. In other cases, such as for hyper-sonic flows, thermal nonequilibrium is modeled by using a com-bined translational–rotational temperature (Ttr ) and a separatevibrational temperature (Tv ) (3–8). However, since these param-eters represent averages of the molecular energy distributions,they do not explicitly account for non-Boltzmann distributionsand their coupling to dissociation. Rather, when such multitem-perature models are parametrized by using experimental data,non-Boltzmann effects are captured empirically. As a result, the

parametrized model may not be accurate when extended to othernonequilibrium conditions beyond the limited experimental datait was fit to.

In this work, we develop a continuum-level model that nowexplicitly captures non-Boltzmann internal energy effects andcoupling to dissociation. This work is made possible by recentadvances in ab initio potential energy surfaces (PESs) forhigh-temperature air, by a direct molecular simulation (DMS)capability enabled by high-performance computing, and ulti-mately by a surprisal-analysis model formulation parametrizedby using the new ab initio data. We focus on modeling ther-mochemical nonequilibirum processes behind strong shockwaveswith application to hypersonic flows; however, the modelingapproach may be more widely applicable to other gas systems innonequilibrium.

Ab Initio ResultsRecently, for the purpose of studying air chemistry under hyper-sonic flow conditions, accurate PESs for N2–N2 and N–N2 col-lisions (9), O2–O2 (10) and O–O2 (11) collisions, N2–O2 col-lisions (12), and N2–O collisions (13) have been developed byusing methods from quantum mechanics. Given the spatial con-figuration of all atoms at any instant during a collision, themultidimensional PES is used to evaluate the force acting oneach atom. In this manner, a collision between molecules canbe time-integrated starting from an initial condition by usingthe classical equations of motion. The standard procedure isreferred to as quasi-classical-trajectory (QCT) analysis (14, 15),since, although the initial internal energy states may correspondto quantum states, the time integration is performed by usingclassical mechanics, and therefore postcollision internal energiesare not quantized.

Typically, QCT analysis is performed for large numbers ofcollisions with initial conditions corresponding to properly ran-domized atomic orientations (impact parameters) and corre-sponding to desired precollision translational energy and internalenergy states. Such QCT analysis is generally used to determine

Significance

Predicting the extent of air dissociation in thin shock layerscreated by hypersonic vehicles is challenging due to couplingbetween gas internal energy and chemical reactions. Thermo-chemical nonequilibrium flow models, capable of predictingthe heat flux and the flux of reactive atomic species to thevehicle surface, are critical for heat shield design. Whereasexisting continuum models do not account for non-Boltzmannrotational and vibrational energy distributions and couplingto dissociation, we develop a model based on recent ab initiocalculations that does.

Author contributions: N.S. and T.S. designed research, performed research, analyzeddata, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.1To whom correspondence should be addressed. Email: singh455@umn.edu.

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rovibrational state-specific transition rates (16–21). If a distri-bution of precollision states is imposed—for example, Maxwell–Boltzmann distributions—then thermally averaged excitation anddissociation rates can be computed as a function of temperaturefor instance as done by Bender et al. (22) (N2 + N2 and N2 + N),Meuwly and coworkers (23) [O(3P) + NO(2Π], and Varandas andcoworkers (24–26) [N (2D) +N2, N (2D) +NO(X 2Π)]. Apartfrom QCT, purely quantum dynamics approaches (27–29) canalso be used to obtain thermal rate constants, as done by Meuwlyand coworkers (29) to study the NO(2Π) + N(4S) interactionfor conditions relevant to hypersonic flows. However, to com-pute the evolution of a reacting gas, including non-Boltzmanninternal energy distributions, the set of state-specific transi-tion rates must be incorporated into master equation analysis.Unfortunately, full master equation analysis is computation-ally expensive, even for atom–diatom systems [≈ 107 rovibra-tional transitions for N–N2 collisions (30)] and is intractable fordiatom–diatom systems (≈ 1015 transitions for N2–N2 collisions).

Recently, this problem has been overcome by the capabil-ity to perform DMS of an evolving gas system, where the onlymodel input is the PES (or set of PESs), and precomputed statetransition rates are not required. Essentially, the DMS methodembeds trajectory calculations within a molecular simulation ofthe gas [specifically the direct simulation Monte Carlo (DSMC)particle method (31, 32)]. DMS maintains the accurate stochas-tic treatment for dilute gases exploited by DSMC; however, itreplaces stochastic DSMC collision models with trajectory calcu-lations performed on a PES. In this manner, a molecule’s post-collision state becomes its precollision state for a subsequenttrajectory calculation. The result is a direct simulation of rovibra-tional excitation and dissociation including all relevant physics.DMS is equal in accuracy to full master equation analysis usingrates from QCT calculations (33); however, DMS is tractable forthe full nitrogen system (both N–N2 and N2–N2 collisions). TheDMS method was originally proposed by Koura (34–36), andit is noted that similar approaches have been used by Bruehland Schatz (a sequential QCT approach) (37, 38) and Haseand coworkers (39) (a pure molecular dynamics approach). Themethod was implemented by using modern DSMC algorithms byNorman et al. (40) and extended to rotating, vibrating, and disso-ciating molecules by using ab initio PESs in a series of articles byValentini et al. (15, 33, 41, 42). The details of the DMS methodhave been summarized in a recent review article by Schwartzen-truber et al. (43).

Fig. 1 shows an example DMS result where a zero-dimen-sional system containing 1 million N2 molecules is initial-ized corresponding to average rotational and vibrationalenergies of (〈εrot/kB 〉= 〈εv/kB 〉= 2000K ); however, the center-of-mass translational energies of the molecules is maintained atT = 20,000 K (by resampling/resetting at each timestep). Thisis representative of the conditions immediately behind a strongshockwave where the relative translational energy is very highand the internal energy of the gas requires a finite time to excite,ultimately leading to dissociation.

In Fig. 1, the average internal energies (〈εrot〉, 〈εv 〉) are seento increase and reach a QSS during which the gas continues todissociate. Fig. 2 shows the evolution of the vibrational energydistribution function for this simulation. As discussed previously,initially high v levels are overpopulated, and later during QSSthey are depleted, compared with the corresponding Boltzmanndistribution based on the average vibrational energy. Such deple-tion has been shown to reduce the computed dissociation ratein nitrogen by approximately three to five times compared withthat computed by QCT using a Boltzmann distribution (22, 33).This is due to strong coupling between vibrational energy anddissociation, shown by the circular symbols in Fig. 2. Specifically,by analyzing the molecules that dissociate during a DMS calcu-lation, it is evident that, per collision, the probability of disso-

ciating from a high v level is three orders of magnitude higherthan dissociating from a low v level. Therefore, even small vari-ations in the vibrational energy distribution can lead to notice-able differences in the overall dissociation rate. Nitrogen disso-ciation rates computed in QSS by the DMS method have beenshown to agree well with existing experimental data taken inshock-tube facilities (15, 42). Recent DMS results for oxygensystems (10) exhibit similar trends (44) and also agree well withexperimental data.

Below, we use recent DMS results to construct and vali-date a continuum-level model that captures the evolution ofnon-Boltzmann internal energy distributions and coupling todissociation.

Before proceeding, it is important to note that DMS operatesonly on the position and velocities of atoms (whether bondedwithin a molecule or not) and therefore makes no assump-tion about decoupling rotational and vibrational energy. Suchdecoupling is only performed as a postprocessing step to guidereduced-order modeling. In this work, we follow the vibrationalprioritized approach of R. L. Jaffe (45), explained in detail forour trajectory calculations in ref. 22. This vibrational prioritizedapproach is used, along with the PES, to determine (εv , εrot )and (v ,j ) from the positions and momenta of DMS atoms boundwithin molecules.

Model Framework: Surprisal AnalysisWe choose to model nonequilibrium internal energy distribu-tions using surprisal analysis (46, 47). The surprisal, I (i), is ameasure of deviance of an observed distribution f from a priordistribution f0. In our case, f is the nonequilibrium internalenergy distribution during excitation and QSS (seen in Fig. 2),whereas f0 is the equilibrium distribution corresponding to themaximum entropy state. Mathematically,

I (i) = − log

[f (i)

f0(i)

]. [1]

Essentially, I (i) is a measure of entropy deficiency, which is min-imized subject to constraints acting on the system. The surprisalcan be expressed in terms of constraints in the following manner:

I (i) =λ0 +

m∑r=1

λrAr (i), [2]

where Ar (i) are the set of m properties for the state i .Surprisal analysis (also known as information theoretic anal-

ysis) (46–48) has been used in many fields, including recently

Time [ s]

<i>

/kB [K

]

[N2]/

[N2] 0

0 0.001 0.002 0.003 0.004

5000

10000

15000

20000

0.2

0.4

0.6

0.8

1

T < rot>/kB

< v>/kB

[N2]/[N2]0

Fig. 1. Average energy evolution during isothermal excitation from DMSat T = 20,000 K. Dissociation is depicted by the fraction of N2 moleculesremaining. Simulation data were obtained by Valentini et al. (15).

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v / d

f(v)

P(d|

v)

0.2 0.4 0.6 0.810-6

10-5

10-4

10-3

10-2

10-1

10-6

10-5

10-4

10-3

10-2

10-1

t = 0

t ~ tQSS

t ~ 2.9 x10-4 s

Fig. 2. Evolution of vibrational energy distribution during isothermal exci-tation from DMS at T = 20,000 K. Simulation data were obtained by Valen-tini et al. (15). Square symbols refer to the distributions at various timeinstants obtained from DMS. Dashed lines denote the Boltzmann distribu-tions evaluated at 〈εv〉/kB. Solid line with circle symbols denotes the proba-bility of dissociation (p(d|εv )) given vibrational energy of the molecule.

for cancer research (49). In the field of information theory, thedeviance is referred as Kullback–Leibler divergence (50). Thisinformation theoretic approach and its links to thermodynamicsare well documented in a review article by Levine (47). We usethis modeling approach to describe nonequilibrium distributionsof internal energy in a gas, which do not reach equilibrium dueto dynamical constraints acting on the system.

To evaluate the surprisal (Eq. 1), f (i) (where i→ v or j ) isdetermined directly from the DMS system molecules (atom posi-tions and momenta) using the vibrational prioritized approach(22, 45) as shown in Fig. 2. The equilibrium distribution f0(i)in Eq. 1 is obtained by the following joint (v , j ) probabilitydistribution:

f0(v , j ) =(2j + 1) exp [−εint(v , j )/kBT ]∑vmax

v=0

∑jmaxj=0(2j + 1) exp [−εint(v , j )/kBT ]

, [3]

where jmax is the maximum rotational quantum level, vmax is themaximum allowed vibrational quantum level for a given j , thedegeneracy of the rotational level is given by 2j + 1, and T isthe temperature. It is noted that vmax and jmax are coupled, sinceonly a certain number of vibrational levels are allowed for a givenrotational level. This information is computed specifically for theab initio PES used (9). The required marginal distributions arethen determined as f0(v)≡

∑j f (v , j ) and f0(j )≡

∑v f (v , j ).

Finally, for thermal nonequilibrium conditions present duringexcitation and during QSS, the value of T and the definition ofa unique equilibrium distribution is ambiguous due to rotation–vibration coupling (22). For our postprocessing analysis, we eval-uate f0(v) using the average vibrational energy (T ≡〈εv/kB 〉)and f0(j ) using the average rotational energy (T ≡〈εrot/kB 〉).Interestingly, this approach without the logarithm (referred to asthe relative nonequilibrium energy distribution) was used to fitsteady-state vibrational energy distributions of Br2 in an Ar heatbath (51).

Surprisals, I (v) and I (j ) from Eq. 1, corresponding to theDMS results for nitrogen excitation to 20,000 K (Figs. 1 and 2),are plotted in Figs. 3 and 4, respectively. Note that we choose toplot results in terms of energies, rather than in terms of quan-tum levels. The surprisal value is small for the low-energy states;however, it becomes significant in magnitude for higher-energy

states. The surprisal is seen to be negative when the averageinternal energy (〈εv 〉 in Fig. 3 and 〈εrot〉 in Fig. 4) is low, indi-cating overpopulation compared with the equilibrium distribu-tion. This corresponds to the early phases of excitation in Fig. 1.The surprisal becomes positive when the average internal ener-gies are high, corresponding to the QSS region in Fig. 2 whenthe gas is rapidly dissociating. What is most interesting is that thesurprisal is approximately linear over a wide range of internalenergy levels (0.15<εi/εd < 0.8). It is important to note fromFig. 2 that the population of molecules with εv/εd > 0.8 becomesvanishingly small, and therefore this portion of the distributionfunction has a limited effect on the overall system. The nonlin-ear nature at very high energies is a consequence of (i) when theaverage vibrational energy is low, the population of moleculesin high-energy states becomes vanishingly small; and (ii) suchhigh-energy vibrational states have strong coupling to rota-tional energy.

To be concise, we have only shown the case of excitation to20,000 K. However, DMS results have been analyzed for a rangeof temperatures between 10,000 and 30,000 K for nitrogen sys-tems involving only N2–N2 collisions and systems involving bothN2–N2 and N–N2 collisions (52). In all cases, the surprisal trendsare similar. The linear nature of the surprisal functions suggeststhat a simple model can be constructed to accurately representthe nonequilibrium internal energy distributions.

Model FormulationIdeally, a model for the nonequilibrium distributions shouldbe constructed by using average energy parameters only sothat the model can be incorporated into continuum-level anal-ysis. These parameters include the average internal energy permolecule, 〈εrot〉, and 〈εv 〉, and the average translational energyper molecule, (3/2)kBT = 〈εt〉, where T is the translational tem-perature of the gas.

QSS Distributions. For vibration, we propose the following modelform for the surprisal in the QSS regime:

− log

[f (v)

f0(v)

]= λ0,v +λ1,v

〈εt〉εd

v , [4]

where λ0,v and λ1,v are constants. The main feature of thissimple model is the linear dependence on v level and that the

v / d

I(v)

0.2 0.4 0.6 0.8

-4

-2

0

2

< v> = 6, 541 kB K< v> = 8, 530 kB K< v> = 12, 065 kB K< v> = 15, 993 kB K

Fig. 3. Surprisal vs. vibrational energy at T = 20,000 K.

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rot / d

I (ro

t)

0 0.2 0.4 0.6 0.8

-2

-1

0

1

2

3 < rot> = 6, 847 kB K< rot> = 13, 727 kB K< rot> = 15, 782 k B K< rot> = 18, 103 kB K

Fig. 4. Surprisal vs. rotational energy at T = 20,000 K.

constraint imposed by the dynamics of dissociation should bebased on the translational energy of the system (relative to thedissociation energy, εd ). DMS results clearly show that as the sys-tem translational energy increases, the degree of internal energydepletion increases. At temperatures close to 10,000 K, depletionis limited to the high-energy tails; however, at higher tempera-tures, depletion occurs over a wider range of vibrational energylevels (15, 42).

The modeled nonequilibrium distribution function for vibra-tion is then given by:

f (v) =C1,v f0(v) exp

[−λ1,v

〈εt〉εd

v

], [5]

where C1,v = exp[−λ0,v ] and can be obtained from the normal-ization condition,

∑vmaxv=0 f (v) = 1.

Since rotational energy is approximately linear with j (j + 1),as seen in Fig. 4, the surprisal is modeled as:

− log

[f (j )

f0(j )

]=λ0,j +λ1,j

〈εt〉εd

j (j + 1), [6]

where λ0,j and λ1,j are constants. The nonequilibrium distribu-tion function for rotation is then:

f (j ) =C j1,j f0(j ) exp

[−λ1,j

〈εt〉εd

j (j + 1)

], [7]

where C1,j = exp[−λ0,j ], which can be evaluated by using thenormalization condition,

∑jmaxj=0 f (j ) = 1.

Transient Distributions. We now extend the above surprisalapproach for QSS distributions to model the transient internalenergy distributions during excitation.

Interestingly, as seen in Figs. 3 and 4, we find the surprisalto be approximately linear even during the transient phase. Thismotivates the formulation of a second constraint acting on thesystem that causes a deviation from equilibrium. As pointed outby Levine (48) (refer to Eq. 2), the surprisal with multiple con-straints can be constructed as a linear combination of the con-straints. During the excitation phase, the system will continue toevolve as long as the internal energy modes are not in equilib-rium with the translational mode. Therefore, the average energygap between these modes (in an antisymmetric manner) is pro-

posed as the dynamical constraint. The resulting models for sur-prisal become:

− log

[f (v)

f0(v)

]= λ0,v +λ1,v

〈εt〉εd

v +λ2,v

(2

3

〈εt〉〈εv 〉

− 3

2

〈εv 〉〈εt〉

)ψv ,

[8]

and

− log

[f (j )

f0(j )

]= λ0,j +λ1,j

〈εt〉εd

j (j + 1)

+λ2,j

(2

3

〈εt〉〈εr 〉− 3

2

〈εr 〉〈εt〉

)ψj (j + 1), [9]

where λ2,v , λ2,j , and ψ (odd integer) are additional constants.Therefore, the final model for nonequilibrium internal energy

distribution functions is:

f (i) = C1,i f0(i)× exp

[−λ1,i

〈εt〉εd

Γi

]× exp

[−λ2,i

(2

3

〈εt〉〈εi〉− 3

2

〈εi〉〈εt〉

)ψΓi

], [10]

where i corresponds to the internal mode (i → v or j ), Γv = v ,Γj = j (j + 1), and f0(i) is calculated by using Eq. 3 as previ-ously described. Both overpopulation and depletion terms followsurprisal functions that are linear in v, or j(j + 1). In addition,the translation energy 〈εt〉/〈εd〉 controls the depletion resultingfrom dissociation, whereas the energy gap 〈εt〉−〈εi〉 controls theoverpopulation during rapid excitation. Given this physics-basedfunctional form, the free parameters are then determined bycomparison with baseline ab initio results obtained from DMS.

Results and DiscussionBy using published DMS nonequilibrium distribution results (15,33, 42), corresponding to nitrogen systems over a range of tem-peratures, the best-fit model parameter values were determinedto be: λ1,v = 0.080, λ2,v =−7.3 × 10−3, λ1,j = 4.33 × 10−4,λ2,j = 1.00× 10−4, and ψ= 3.

Using our simplified model with these parameter values, wenow compare with the results from DMS. Using only the aver-age internal energy values, 〈εrot〉, 〈εv 〉, and 〈εt〉, at a specific timefrom DMS, we compare the corresponding nonequilibrium dis-tributions from our simplified model with the actual distributionscomputed by DMS.

For the 20,000 K vibrational excitation distributions (Figs. 2and 3), the model predictions are compared with DMS resultsin Fig. 5. The simple model captures the overpopulation of highv levels during excitation, when a large gap exists between 〈εv 〉and 〈εt〉. As the gas vibrationally excites and reaches a QSS,the degree of thermal nonequilibrium decreases, and the modelpredicts that the depletion of high v levels becomes the dom-inant term. The corresponding rotational energy distributionspredicted by the model are compared with the DMS results inFig. 6. The overall trends and level of agreement are similar tothat found for vibrational energy.

Capturing such non-Boltzmann physics is important for mod-eling the flow behind strong shockwaves. Under certain hyper-sonic flight conditions, >50% of the gas dissociation in front ofa spacecraft can occur in the transient excitation phase imme-diately behind the shockwave, with the remainder occurring inthe QSS phase (16). Predicting the extent of dissociation in theshock layer surrounding a hypersonic vehicle is crucial for pre-dicting thermal protection system (i.e., heat shield) performance.To first order, dissociation converts thermal energy into chemi-cal energy through bond-breaking, which significantly lowers theshock-layer temperature and convective heating to the surface.At the same time, the production of reactive atomic species (suchas N, but most importantly O) has a first-order effect on the

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v / d

f(v)

0.2 0.4 0.6 0.810-6

10-5

10-4

10-3

10-2

10-1< v>= 12, 065 kB K< v>= 15, 993 kB K< v>= 6, 541 kB K

Fig. 5. Nonequilibrium vibrational energy transient distributions from DMS(15) (symbols), proposed model (solid lines), and Boltzmann distribution(dashed line) for T = 20,000K.

degradation of the heat shield material. Due to the strong cou-pling between vibrational energy and dissociation, even smallchanges in the vibrational energy distribution function (whenviewed on a logarithmic scale) can result in noticeable changesin the overall extent of dissociation (15, 42).

Continuum Modeling of DissociationThe integrated continuum dissociation rate coefficient (kd ) isdetermined by integrating p(d |εt , i), the probability of dissoci-ation given a particular state i and relative collision energy εt ,over the distribution of relative energies and states. Without theintegration constant, this expression has the following form (53):

kd(T , 〈εrot〉, 〈εv 〉) ∝vmax∑v=0

jmax∑j=0

∫ ∞0

p(d |εt , εrot , εv )f0(εt)f (εrot)f (εv )dεt . [11]

When formulating continuum models, such integration is usu-ally performed assuming equilibrium distributions, not only forthe translational energy f0(εt), but also for the internal energydistributions. However, the new model now enables integrationover more accurate nonequilibrium internal energy distributions(Eq. 10), which capture key physics such as overpopulated anddepleted high-energy tails.

In fact, if analytical expressions are used for f0(v) and f0(j ), inplace of Eq. 3, then we have shown that an analytical result canbe obtained for the continuum rate coefficient (54), having thefollowing form:

kd(T , 〈εrot〉, 〈εv 〉) =AT η exp

[− εdkBT

]∗Grot ∗Gv . [12]

This is the standard Arrhenius rate coefficient expression forthermal equilibrium conditions (i.e., based on T ); however, twoadditional “controlling functions” appear. The exact form of Grot

and Gv depend on the probability model, p(d |εt , εrot , εv ); how-ever, the important point is that they are functions of averageenergies only; Grot =Grot(T , 〈εrot〉), and Gv =Gv (T , 〈εv 〉). Thefunctions Grot and Gv act to adjust the Arrhenius rate coefficientaccording to coupling with the local rotational and vibrationalenergy of the gas.

As an example, consider a continuum simulation solving foraverage energies, T , 〈εrot〉, 〈εv 〉, in the flow immediately behinda strong shockwave. The spatial evolution of internal energyand dissociation chemistry behind the shock would be analo-gous to the temporal evolution seen in Fig. 1. Consider the gasconditions at t ≈ 2.9× 10−4 in Fig. 1, where T > 〈εrot〉>> 〈εv 〉.In this case, Gv would act to lower the Arrhenius rate expres-sion since, although T is high, the gas is not vibrationallyexcited (〈εv 〉 is low), and dissociation should be delayed. Suchan incubation period for dissociation behind strong shocks isan important effect that has been experimentally measured byHornung and coworkers (55). However, although 〈εv 〉 is low,the high v levels are substantially overpopulated as seen inFig. 2, and these high v levels will contribute to dissociation.Therefore, while some models may overpredict the dissocia-tion rate by accounting only for T , other models that accountfor only the average vibrational energy, 〈εv 〉, may predict vir-tually no dissociation. In contrast, the new model will predicta more accurate dissociation rate since it contains a represen-tation of the underlying internal energy distribution functions.Since the model is formulated in terms of average internal ener-gies only, it can be incorporated into state-of-the-art compu-tational fluid dynamics simulations (2) of nonequilibrium re-acting flows.

ConclusionsTo summarize, we propose a model for rotational and vibrationalenergy distributions, for gases in nonequilibirum, that enablescoupling between internal energy and dissociation chemistry.Development of the model leverages recent advances in the con-struction of ab initio PESs for high-energy collisions in air (9,10, 12, 13), as well as the recent capability to perform DMS ofnonequilibrium gas systems by using only a PES (15, 40, 42). Wefind that a simple model, based on surprisal analysis, is able topredict the nonequilibrium internal energy distributions duringrapid rovibrataional excitation and dissociation of nitrogen gas.The surprisal-based model is able to accurately capture both theoverpopulation of high-energy levels during the excitation phase,as well as the depletion of high-energy levels in the QSS disso-ciating phase. With an accurate representation of the underly-ing internal energy distributions, this enables state-specific reac-tion probabilities to be integrated over the new nonequilibirum

rot / d

f (ro

t)

0 0.2 0.4 0.6 0.8

10-810-710-610-510-410-310-2

< rot> = 18, 103 kB K< rot> = 6, 847 kB K

Fig. 6. Nonequilibrium rotational energy transient distributions from DMS(15) (symbols), proposed model (solid lines), and Boltzmann distribution(dashed line) for T = 20,000K.

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distributions (instead of integrating over equilibrium Boltzmanndistributions). The result is an analytical continuum-level ratecoefficient expression that contains information about the under-lying nonequilibrium distributions of internal energy and theircoupling to dissociation. The model was parametrized for nitro-gen dissociation, but could potentially be parameterized for

other gases and extended for modeling other systems wherenonequilibrium coupling between internal energy and chemicalreactions is important.

ACKNOWLEDGMENTS. This work was supported by Air Force Office of Sci-entific Research Grant FA9550-16-1-0161 and was also partially supportedby Air Force Research Laboratory Grant FA9453-17-1-0101.

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