Rotational nonequilibrium in state-resolved models for shock-heated flows

Chemical Physics 398 (2012) 96–103

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Rotational nonequilibrium in state-resolved models for shock-heated flows M. Lino da Silva ⇑, J. Loureiro, V. Guerra Instituto Superior Técnico, Universidade Técnica de Lisboa, 1 Av. Rovisco Pais, 1049-001 Lisboa, Portugal

a r t i c l e

i n f o

Article history: Available online 8 September 2011 Keywords: Molecular Rotation Vibration Dissociation Thermodynamic nonequilibrium Rate coefficients Quasi-bound states Nitrogen

a b s t r a c t The effects of molecular rotation on the dynamics of high-temperature nonequilibrium flows is investigated in this work, with an application to nitrogen shocked flows. The overall manifold of rovibrational levels for the ground electronic state of N2 is firstly obtained through the solution of the radial Schrödinger equation over a high-fidelity potential curve. The obtained solutions predict the existence of 9510 states, including quasi-bound states which may spontaneously dissociate through tunneling effects. The lifetimes of such quasi-bound states are determined using a semiclassical method, and most of them are found to be long-lived. A semi-empirical rate for state-specific rotational translations has then been applied so as to examine the importance of rotational dissociation against vibrational dissociation, in equilibrium conditions. Rotational dissociation is found to be one to two orders of magnitude below vibrational dissociation, from low temperatures up to 100,000 K. The contribution of quasi-bound states to the overall dissociation rate is also found to be essentially negligible (increase of the rotational dissociation rate by a factor of 1.5–2 in the whole temperature range). Finally, as the full state-specific modeling of the interactions of this large number of rovibrational levels remains out of reach of the state of the art theories and computational resources, a simplified state-specific model for the simulation of shocked flows is proposed, treating vibration-translation and rotation-translation interactions in an uncoupled fashion. A Boltzmann equilibrium distribution is considered for each subset of rotational levels, for each of the molecular vibrational levels, at a characteristic rotational temperature T vrot . Rotational collisional numbers are calculated according to the method recently proposed by Park and confirm the rapid increase of the number of collisions needed for establishing rotational–translational equilibrium, as the gas temperature rises. The results of our simulations also confirm the claims by Park that for high temperatures (roughly above 12,000 K), rotational–translational equilibration actually occurs after vibrational-translational equilibration, contrary to the popular belief. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Low pressure, high-temperature plasmas such as atmospheric entry plasmas often experience a departure from equilibrium between the translational and rotational modes of the plasma chemical species, whereas this is seldom seen for most common plasmas, except for very low pressures, where the collision frequencies may become insufficient for quick equilibration. For these particular plasmas, the failure to establish quick equilibration between translational and rotational modes is generally a consequence of the significantly higher temperature ranges, which leads to decreasing timescales for exchange (rotational, vibrational, electronic) and chemical processes. Also, different processes timescales tend to be shorter and more closely spaced. For example, high-speed shock waves typical of hyperbolic atmospheric entries experience intense translational temperature jumps across the shock, leading to translational temperatures up to 50,000–

100,000 K, which favor the establishment of significant dissociation reactions in less than 1 ls [1]. It is then necessary to explicitly account for the interactions with the rotational modes of the molecular species of the flow. This mandates the calculation of the entire manifold of rovibrational levels and associated energies, followed by the development of a dataset for exchange reactions between these different levels. Several difficulties immediately arise, when compared to the more traditional works solely dealing with vibrational energy exchanges [2–5]. The first one is related to the increasingly large number of bound rovibrational levels, as the species atomic number becomes larger. For example, the H2 molecule ground electronic state only has a few hundreds of rovibrational levels [7], whereas the ground electronic state of N2 has nearly 10,000 rovibrational levels. Furthermore, some of these are quasi-bound levels,1 in the sense they lie above the dissociation limit of the molecular state, but are still bounded by a potential well (although

⇑ Corresponding author. Fax: +351 21 846 44 55. E-mail address: [email protected] (M. Lino da Silva). 0301-0104/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2011.08.014

1

Levels above the dissociation threshold, but limited by a potential barrier.

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M. Lino da Silva et al. / Chemical Physics 398 (2012) 96–103

As it is clear that the state-specific modeling of rotational levels for molecules such as nitrogen is still beyond reach of the presentday models and computational tools, it is important to examine how a sufficiently accurate model for rotational equilibration and dissociation can be achieved. This will be the purpose of this work. Firstly we will discuss the methods that are appropriate for the prediction of an adequate manifold of rovibrational levels for a specific electronic level. Indeed, the existence of quasi-bound levels raises some additional issues, related to the accountancy for the lifetimes of such levels, which can spontaneously dissociate through tunneling effects. Further, the most popular datasets of rovibrational levels have been obtained through semi-classical methods (such as the WKB approach [16,17]), although more detailed methods such as potential reconstruction methods coupled with an algorithm for the solution of the radial Schrödinger equation can provide more adequate solutions [8]. Secondly, this work will propose a computationally tractable model for the inclusion of rotational effects in state-resolved models for nitrogen. Here we will consider the restriction that the different rotational levels follow a Boltzmann distribution, so as to reduce the number of variables of the problem. Instead of accounting for the overall levels populations, we may simply account for a characteristic temperature, keeping the number of variables at an acceptable number. As we will see, this restriction is relatively reasonable for low quantum numbers J which have low energy spacings, but less so at higher J’s where such energy spacings quickly increase to be equivalent to vibrational energy spacings. Exchange processes are then considered between the rotational and the translational modes (T–R exchanges), according to the Landau– Teller equation. Such approach has been pioneered by Parker [18], who considered a simplified rigid rotator model to provide a rotational collision number Zr who defines the typical number of collisions necessary to reach R–T equilibrium. More recently, Park [19] updated this model in order to provide a more realistic value of Zr at higher temperatures, where increased rotational energy spacings (a consequence of molecular centrifugal distortion effects) lead to lower equilibration times, and hence higher Zr values. Additionally, Park has reviewed the different published stateto-state rates for J–J energy transfer. Here we propose to further update this model by accounting for the fact that the manifold of rotational levels differs for each vibrational quantum level v, an ubiquitous parameter in the different state-resolved dissociation models proposed in the literature. We will also examine the behavior of rotational dissociation processes, considering Park’s review of the corresponding state-resolved

(1) Accurate methods for the calculation of the manifold of rovibrational states and associated lifetimes of a diatomic molecule, including quasi-bound states. (2) An investigation of purely rotational dissociation, accounting for the effect of quasi-bound states. (3) An update of rotation–translation (R–T) relaxation models further extending the previous work from Park [19]. 2. Calculation of accurate manifold of rovibrational energy states The accurate energy calculation of the manifold of bound vibrational and rotational levels of a diatomic molecule cannot rely on the traditional approach of utilizing extrapolated polynomial expansions [8]. Instead we have reconstructed the potential VJ=0(r) of the ground electronic state of nitrogen, utilizing the RKR method as described in [8]. The potential for an arbitrary rotational number J can be obtained using the relationship 2

V J ðrÞ ¼ V J¼0 ðrÞ þ

h JðJ þ 1Þ ; 2lr 2

ð1Þ

where l is the reduced mass of the molecule. And the energies for the corresponding rovibrational levels can then be obtained by solving the radial Schrödinger equation [8]. The maximum rotational quantum number is then reached when the molecular potential no longer allows a local minima, becoming entirely repulsive. For the application of Eq. (1), we consider the accurate rotationless potential proposed by Le Roy [20]. The last potential curve allowing for discrete states occurs for J = 275. The potential curves for different rotational levels are reported in Fig. 1. 2.1. Quasi-bound states The obtained solutions allow for the existence of quasibound states, mostly for J > 0, although rotationless potentials may also have quasi-bound states as the result of interactions with other potential curves (avoided crossings). For higher rotational numbers, 5

2.5

x 10

2

−1

1.1. Objectives of this work

rates, with a particular emphasis of the contribution of quasibound states to the overall dissociation rate. The three topics of this work may therefore be summarized as follows:

Energy (cm )

such states can spontaneously dissociate through tunneling effects). The determination of such levels and their associated energies must also rely in more accurate potential reconstruction methods [8] instead of the traditional application of Dunham expansions. The second one lies with the calculation of the extensive set of state-resolved exchange rates between these overall levels. While a direct calculation and application of such a dataset is still within reasonable grasp for the levels of H2, as seen by the wealth of works on this topic [9–11], this is not generally possible for molecules such as N2. Here rates typically need to be calculated by increments of quantum number J (typically 5–10). The calculation methods also have to rely exclusively in Potential Energy Surface (PES) methods such as Quasi-Classical Trajectory (QCT) [12,13] or full quantum methods [14], although more simple models such as the Forced Harmonic Oscillator (FHO) can still extended for the case of rotational nonequilibrium, with the restriction that the selected intermolecular surfaces must be purely repulsive [15].

1.5 J=275

1

0.5

J=0

0 0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Internuclear Distance (A) Fig. 1. N2(X) Potential curves for J = 0, 100, 150, 200, 250, and 275, from bottom to top.

M. Lino da Silva et al. / Chemical Physics 398 (2012) 96–103

potential curves with a potential well minima above the dissociation limit can also occur. Quasi-bound states are classically allowed. However, such states can spontaneously dissociate through quantum tunneling. The lifetime of a quasi-bound state can be calculated using a modified classical WKB method. We start by defining the phase integrals, with r0, r1, r2 the turning points of the potential curve at energy E [21,22]:

bðEÞ ¼

cðEÞ ¼

Z Z

r1

r0 r2

  1=2 2l E  V J ðrÞ =h dr;

ð2Þ

  1=2 2l E  V J ðrÞ =h dr;

ð3Þ

300

250

Rotational Number

98

200

Q−Bound

150

100

r1

Bound

50

which allow obtaining the classical one-way transit time for a particle in the potential well as tTR = h  db/dE, and a measure of the barrier opacity h(E) = exp(c). The lifetime of a quasi-bound state is then sqb = 2tTR/T, where T is the barrier transmission coefficient such that T(E) = (1 + h2)1 [21,22]. An example of a quasi-bound state and the corresponding turning points is presented in Fig. 2. The finite lifetime of a quasi-bound state immediately raises the question of whether such states can be considered to effectively exist and interact with their environment. An answer may be proposed considering the kinetic rate of the surrounding gas/plasma. Indeed, we may consider that such states are almost exclusively populated by molecular collisions (radiative transitions may occur, but only with the change of a rotational quanta, due to selection rules). Then, after the state is populated, it will spontaneously decay after a while if no further interactions occur in the meanwhile. However, if a second collision occurs before the state is decayed, it will necessarily participate in the interactions with its medium. Therefore, a quasi-bound state can be considered to interact with the surrounding gas if sqb > scoll or not if sqb < scoll, where scoll is the gas mean collisional time. When sqb ’ scoll, a detailed kinetic approach has to be considered, akin to the ones considered for radiative states (who can also spontaneously be destroyed through spontaneous and induced radiative emission). Applying the method of Stogryn and Baylis, lifetimes have been obtained for all the quasi-bound states predicted by the resolution of the radial Schrödinger equation on the recalculated potential curves for the ground electronic state of N2. Fig. 3 presents the

0 0

10

20

30

40

50

60

70

Vibrational Number Fig. 3. Manifold of bound and quasi-bound rovibrational levels for the ground electronic state of N2.

manifold of bound and quasi-bound levels of N2, and Fig. 4 presents the calculated lifetimes of the N2 quasi-bound states. A total of 7582 bound levels and 1928 quasi-bound levels are obtained. The calculation of the quasi-bound states lifetimes shows that most of them are very long-lived (s  1 s), with only a minority of very higher energy rovibrational states showcasing lifetimes below s = 1 ns. These results may be compared to the ones recently published by Jaffe et al. [23], showing trends very similar to ours, with a maximum vibrational level of 60 (61 in our work), a maximum rotational level below the dissociation limit of J = 210 (214 in our work), and a maximum rotational level of 279 (275 in our work), for an overall number of 9390 rovibrational states (9510 in our work). Finally we calculated the temperature-dependent rovibrational partition function for N2, with and without the inclusion of quasibound states. The calculated partition functions are reported in Fig. 5. We may verify that the influence of the quasi-bound states on the overall partition function is negligible at temperatures below

5

1.5

x 10

1.4

300 J=214 J=191

1.3 250

200

1.1 1 0.9

r0

r1

log10(τ)

−1

Energy (cm )

1.2

r2

0.8 0.7

J=164

J=133

150

100

J=98

50 J=58

0.6

J=21

0

0.5 0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

J=25

5.5

Internuclear Distance (A) Fig. 2. N2(X) Potential curve for J = 200 showing the wavefunction for the vibrational level v = 12 with the corresponding r0, r1, r2 turning points ( symbols from left to right).

v=50

v=60

−50 0

J=66

50

100

v=40

v=30

v=20

v=10

v=0

J=119

J=165

J=204

J=242

J=275

150

200

250

300

Rotational Number Fig. 4. Lifetimes (in logarithmic units) of the quasi-bound states of N2.

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M. Lino da Silva et al. / Chemical Physics 398 (2012) 96–103 −5

4000

10

Bound states Bound & Quasibound states

3500

−10

Dissociation Rate

Total Partition Function

10

3000 2500 2000

−15

10

−20

10

1500 −25

10

Experimental rate (top and bottom limits) Vibrational dissociation rate (FHO model) Rotational dissociation rate (B; B+QB)

1000 −30

10

500

3

4

10

5

10

10

Temperature (K) 0 10

20

30

40

50

60

70

80

90

−30

100

10

Temperature (kK) −8

10

Fig. 5. Comparison of the total partition function with and without the inclusion of quasi-bound states.

Dissociation Rate

3. Rotational dissociation

Dissociation Rate

30,000 K, with a very moderate increase up to 10% of the partition function at 100,000 K.

−35

10

−40

10

   1:346 1 þ ð1:5Ei =kB T tr Þ=1:26 295  K ij ¼ 1:757  1010 1 þ 1:5Ei =kB T tr T tr  1:67DEij :  exp  kB T tr

ð4Þ

The availability of such rate, coupled to the manifold of rotational levels determined in this work, allows estimating the importance of purely rotational dissociation, which occurs through processes such as:

N2 ðv ; J i Þ þ N2 ! N2 ðv ; J f > J diss Þ þ N2 ! N þ N þ N2 ;

ð5Þ

where the pair (v, Jf) belongs to the non-bound (continuum) states of Fig. 3. Furthermore, if we are to examine the influence of the quasi-bound states in the rotational dissociation dynamics, we may consider two different cases. In the first case, dissociation occurs for a transition from a bound state to either a quasi-bound state or the continuum (quasi-bound states are assimilated to dissociated states), and in the second case, dissociation occurs for a transition from a bound or quasi-bound state to the continuum. The thermal dissociation rate can be written as:

K rot d ðTÞ ¼

X X X v

J bound J diss i f

Neq Ji K JJ : f

ð6Þ

Akin to the approach considered by Park, we extrapolate the rate expression proposed by Rahn up to Ttr = 100,000 K, in order to obtain an overall dissociation rate for the temperature range

−50

10

1000

−12

10

−45

10

Rotational exchange rates for the N2 molecule have been recently reviewed by Park [19], accounting for a series of experimental and numerical works. Overall, Park found the semi-empirical rate proposed form Rahn and Palmer [24] to adequately fit experimental data, up to the temperature limit T = 1500 K where data is available. The proposed expression is capable of predicting multiquantum rotational rates, as it only depends on the translational temperature Ttr, the initial energy Ei, and the energy ratio DE between initial and final rotational levels. Eq. (4) reports the expression proposed by Rahn

−10

10

−14

1500

2000

Temperature (K)

2500

10

0

5

10

Temperature (K)

4

x 10

Fig. 6. Comparison between thermal vibrational dissociation (FHO model; – .) and rotational dissociation (–), with (upper full line; bound + quasi-bound B + QB), and without (lower full line; bound B) the accounting of quasi-bound states. The upper and lower bounds of published experimental dissociation rates are also reported (– –). The lower subfigures zoom into the low-temperature and high-temperature behavior of the rates.

Ttr = 100–100,000 K. Finally, we obtain two values for the rotational dissociation rate, with and without the inclusion of quasi-bound states. It is then useful to carry a comparison against purely vibrational dissociation rates, which have been proposed by the authors in the framework of previous numerical works, considering a stateto-state model based on the Forced Harmonic Oscillator approach [25,1]. For this comparison, we further include several experimental rates for dissociation in N2–N2 collisions, reported in Ref. [1], which can then provide upper and lower experimental bounds for the overall dissociation rate. These different dissociation rates are plotted in Fig. 6, with additional zooms on low and high-temperature dissociation trends. The examination of Fig. 6 leads to the drawing of several conclusions: The first one points to the validity of Rahn’s rate up to the high-temperature limit, as the rotational dissociation rates follow an Arrhenius trend, close but below the experimental limits. The second one is that purely rotational dissociation rates are systematically one to two orders of magnitude below the vibrational dissociation rates, which means that they can be safely neglected in equilibrium or close to equilibrium situations. The third conclusion points to a very limited influence of quasi-bound states on the dissociation dynamics, as the rotational rates with and without the accounting for the quasi-bound states are quasi-identical. Quasibound states only increase the dissociation rate for about a factor of 1.5–2, which is essentially negligible in view of the orders of magnitude variations for the overall dissociation rate. Again, these

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M. Lino da Silva et al. / Chemical Physics 398 (2012) 96–103

conclusions are essentially valid in thermodynamic equilibrium conditions.

8000 7000

4. A compact approach for the modeling of rotational exchanges in state-resolved models Temperature (K)

6000

As it has been pointed out in the introduction, it is customary to consider that rotational levels follow a Boltzmann distribution at a characteristic rotational temperature Trot, which avoids resorting to a state-specific model for the calculation of these levels. The latter approach is likely to be favored in the future, but still remains far from tractable within the current state-of-the art computational material. As it has been discussed in the introduction, the essential parameter controlling the rate of T–R relaxation is the Landau– Teller equation

dErot Erot ðT tr Þ  Erot ðT rot Þ ; ¼ Z r tcoll dt

Zr ¼ 3=2

1 þ p2



Z1 r 1=2

 2 hrot þ p4 þ p hTrottr T tr

ð8Þ

ð9Þ

where q represents the normalized nonequilibrium populations of the initial and final states i and j, and z represents the number of collisions in the gas. We then define the equivalent rotational temperature as the characteristic temperature of a rotational Boltzmann distribution with the same total energy than the nonequilibrium distribution: 2

3000

1000 0 0

It can be verified that the solutions to the radial Schrödinger equation over Eq. (1) potential can be fitted to such an expression. 3 Through a coordinate transform of the time-dependent master equation.

100

200

300

400

500

600

Collision number z Fig. 7. Relaxation to T–R equilibrium at T = 8000 K. Z is the collision number at which T–Trot becomes 1/e of the initial temperature (here, Z = 80).

X

qðJÞEJ ¼

J

has been utilized throughout a large number of previous simulation works, considering Z 1 r ¼ 15:7 and 14.4 for N2 and O2 respectively. As the calculated values of Zr converge to Z 1 r at high temperatures, most of these works have pointed out to a very fast equilibration of rotational and translational modes of molecules. It has previously been pointed out by Park [19] that this approach is overly simplistic. The rotational collisional number is instead found to have a considerable temperature dependence, significantly increasing at higher translational temperatures. Indeed, the rotational energy spacings increase quadratically for high rotational quantum numbers J (Erot = Bv(J(J + 1))  Dv(J(J + 1))2).2 Also, at higher rotational temperatures there is an increasing number of degenerate rotational states (gJ = 2J + 1), which means that higher rotational quantum numbers will be the most populated, even for equilibrium Boltzmann distributions (the most populated level will be the one which yields the maximized individual partition function (2J + 1) exp(Erot/kBTrot), which corresponds to J = (kBTrot/2hcBv)1/2  1/2 if we assume Erot = Bv(J(J + 1))). Park acknowledged that this would lead to increased equilibration times, and he selected the ‘‘universal rate’’ by Rahn and Palmer [24] for the calculation of an updated effective collision number Z, up to a 128,000 K limit. This calculation considers setting a given translational temperature Ttr and calculating the relaxation of the rotational distribution function as a function of the number of molecule collisions in the gas,3 from an initial condition where only the J = 0 level is populated. The master equation applies Rahn’s rate dataset, and is written as [19]:

dqðiÞ X ¼ kði; jÞ½qðjÞ  qðiÞ; dz j

4000

2000

ð7Þ

where the effective rotational collision number Zr defines the average number of collisions needed to establish T–R equilibration. The expression of Eq. (8), proposed by Parker [18]

5000

X

qeq ðJÞEJ ¼

J

X ð2J þ 1Þ expðEJ =kB T rot Þ EJ : Q rot J

ð10Þ

A sample plot of the relaxation of the equivalent rotational temperature Trot towards the translational temperature Ttr, as a function of the number of gas collisions Z, is presented in Fig. 7, for Ttr = 8000 K. From such a plot it is then possible to determine several rotational collision numbers Z1, Z2, and Z3, which are representative of the early to latter dynamics of relaxation of Trot towards Ttr. A detailed discussion on the significance and applicability of these different collision numbers can be found in the paper by Park [19]. Following the discussion by Park, we consider Z2 to be the collisional number which better represents Zr, where Z2 corresponds to the collision number where (Ttr  Trot)/Ttr = 1/e (e-folding collision number). In this work, we further acknowledge that there is a subtle influence from vibration on the rotational energy level spacings. For example, in the case of diatomic molecules, the rotational constant Bv depends on the vibrational quantum number such that4:

Bv ¼ Be  ae



vþ

 2 1 1 þ be v þ þ  2 2

ð11Þ

As the rotational energy spacings decrease for higher vibrational levels, it is expectable that the rotational relaxation number Zr will be accordingly smaller. The master equation (Eq. (9)) has therefore been solved for the different vibrational levels subsets, yielding vibrationally-dependent Z vr values. As expected, we obtain the same increasing trend for higher translational temperatures, but with smaller values. Fig. 8 shows the temperature-dependent Z vrot , obtained for v = 0, 20 and 40. Table 1 further reports calculated values for Zr, compared with the values obtained by Park [19], and the old values from Parker [18]. We verify that the rotational collisional numbers obtained in this work closely follow the trends of those obtained by Park, being somehow larger. There is also a subtle influence of vibration on these values, which decrease as we reach higher vibrational levels. As expected, the values from Parker are very different from those obtained in these two works, with a very small variation from around 12 at 300 K to around 15.4 around 50,000 K. 4.1. Application to the simulation of nitrogen shock-heated flows 4

Again, rovibrational levels obtained in Section 2 can be fitted to such expressions.

M. Lino da Silva et al. / Chemical Physics 398 (2012) 96–103

tiquantum set of rates for N2–N2 collisions [25,1], coupled to the multiquantum QCT dataset proposed by Esposito et al. for N2–N collisions [3,12,13], with the appropriate manifold of 61 vibrational levels, as opposed to the traditional 45–47 levels obtained from polynomial expansions [8]. To examine the influence of these new vibrationally-specific relaxation numbers Z vr , the Landau–Teller equation (Eq. (7)) for rotational relaxation has been added to our previous vibrationally-specific state-to-state model. This updated model accounts for state-to-state vibrational relaxation through the usual master equation, relaxation for the rotational mode of each vibrational level through the Landau–Teller equation with the corresponding rotational collisional numbers Z vr , and energy conservation accounting for energy exchanges between vibrational, rotational and translational modes. The corresponding system of equations is written as:

10 3

Effective Coillision Number Z

v=0 v=20 v=40

10 2

10 1

10 0 2 10

101

10

3

10

4

10

5

Temperature (K)

X dN2 ðv i Þ ðK ji ÞM N2 ðv j ÞM  ðK ij ÞM N2 ðv i ÞM ¼ dt M¼N ;N 2

Fig. 8. Rotational collision number Z2 from T = 300 K to T = 50,000 K, for v = 20, and v = 40.

v = 0,

þ ðK rec;i ÞM N2 M  ðK diss;i ÞM N2 ðv i ÞM;

ð12Þ

v

dErot Evrot ðT tr Þ  Evrot ðT rot Þ ; ¼ dt Z vr t coll

Table 1 Comparison of rotation collision numbers Z2. T (K)

v=0

v = 20

v = 40

Park

Parker

300 1000 50,000

4.07 20.08 551

3.78 18.14 522

3.12 15.18 459

3.26 9.85 351

11.85 13.48 15.37

The recalculated rotational collisional numbers Z vr confirm the conclusions from Park, pointing towards a very slow rotational relaxation for high translational temperatures typical of shocked flows. Indeed, a comparison of the rotational relaxation time sr with the vibrational relaxation time sv, as proposed by Millikan and White [26], shows that rotational relaxation becomes slower than vibrational relaxation for Ttr > 12,000 K [19]. This trend inversion, for a translational temperature typically found in shocked flows, points towards the necessity of a full state-specific kinetic dataset, accounting for all the interactions between the over 9000 rovibrational levels of N2, including vibration–rotation (V– R) interactions. Some pioneering works of this type already exist for the case of molecular hydrogen [9]. However, the large number of rovibrational states of N2 (when compared to H2, with around 300 levels), and the difficulties of modeling diatom–diatom rovibrational transitions, still preclude the development of accurate and detailed databases of this kind, at least in the near-future. One is therefore restricted to more compact approaches such as grouping subsets of rotational states under the assumption of Boltzmann equilibrium distributions, reducing the modeled parameters in the master equation. Furthermore, the approach discussed here also considers a strict separation between the rotational and vibrational modes, without the accounting for vibration–rotation (V–R) interactions, for lack of available models. Under these assumptions, and in order to complement the previous work by Park, an hybrid state-to-state/Boltzmann nonequilibrium description has been developed and applied to the simulation of rotational and vibrational relaxation behind a shock-wave. A first complement from this work is the determination of individual rotational temperatures for each subset of rotational levels with identical vibrational quantum numbers (see Section 4). In addition to this, a series of previous works from our group have been devoted to the development of a vibrationally-specific model for the simulation of nitrogen shocked flows. Previous works have considered the Forced Harmonic Oscillator (FHO) approach for the development of a mul-

dE dEtr dErot dEv ib dEchem ¼ þ þ þ dt dt dt dt dt ! X v N2 ðv Þ dN2 X 3 dN2 dN2 ðv Þ ¼ kB T tr T rot Eðv Þ þ kB þ 2 N dt dt dt 2 v v  6 d2N kB T tr þ Edis ¼ 0: þ 2 dt

ð13Þ

ð14Þ

We have also considered the more simplified expression for rotational relaxation (Eq. (8)) instead of Eq. (13), using the rotational collisional numbers from Parker in order to compare the obtained results with those issued from the new model discussed in this work. Finally, a nitrogen shockwave with an initial Ttr = 50,000 K and Tvib,rot = 195 K, in the pressure conditions of an Earth entry (75 km altitude), has been simulated in the state to-state approach, considering rotational nonequilibrium for each of the 61 individual vibrational levels. The obtained results are presented in Fig. 9. From the analysis of Fig. 9, one may confirm the conclusions already hinted at by this work and the previous work of Park. Considering the Parker collisional number, one finds a very fast equilibration of the rotational and translational modes around 1 ls, whereas vibration-translation equilibration (for the lower vibrational levels) only occurs at 10 ls. If we consider instead the rotational collision numbers recalculated from this work, we find that vibration-translation equilibration occurs a little bit sooner than 10 ls (due to a lower decrease of Ttr), but rotation-translation equilibrium now occurs around 100 ls. Furthermore, a noticeable lag for the equilibration of the rotational temperature of the higher vibrational levels is observed, when compared to the equilibration of the v = 0 rotational temperature. This is consistent with the lower values for Z vr reported in Table 1. Keeping in mind that this is a model where the rotational and vibrational modes are only coupled to the translational mode (through the state-specific V–T rates and the Landau–Teller equation for R–T energy exchanges), without any vibration–rotation (V–R) exchanges, one may confirm the previous Park findings that the conventional approach of a very fast equilibration of the rotational and translational modes is not valid. Instead the picture is much more complex, and these simulated results essentially point towards the need for an explicit state-specific model of the interactions between the overall rovibrational levels of N2. More importantly, approaches for the modeling of V–R interactions are lacking and should be developed

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M. Lino da Silva et al. / Chemical Physics 398 (2012) 96–103 4

x 10 5 4.5

Temperature (K)

4 3.5 3 2.5 2 Parker

Trot

0→1

Tvib

1.5

0

0→1

1

Trot

Tvib

60

Trot

0.5 0 −10 10

10

−8

10

−6

10

−4

10

−2

Time (s) Fig. 9. Relaxation behind a shock wave at 50,000 K in the typical conditions of an Earth re-entry. (– –): Single Trot with Parker rotational relaxation number, (–): 61 T vrot with the rotational relaxation numbers issued from this work. T 0!1 v ib represents the equivalent vibrational temperature for the populations of the v = 0 and v = 1 Parker levels. T rot represents the rotational temperature using Parker model; T 0;60 rot represent the temperatures of the sub-rotational levels of respectively the v = 0 and v = 60 levels.

in the future. In the meanwhile, approaches such as the one outlined in this work may help bridging the gap towards this kind of more complete descriptions. Indeed, the examination of the evolution of the translational temperature Ttr shows that the inclusion of a more simplified R–T model (Parker model) or a more complex R–T model (this work) leads to significant differences in the energy exchanges. As dissociation processes are strongly dependent on Ttr, it is important to update state-specific models for shocked flows in order to (1) include the modeling of rotation-translation nonequilibrium; (2) consider an approach such as the one outlined in this work (or at least considering the updated Zr values proposed by Park), so as to obtain a better reproduction of experiments. 5. Discussion This work has tackled with several questions related to the accounting of rotational levels in state-specific models. Modeling has been carried for the conditions of shocked flows, where high translational temperatures may be reached, also leading to the population of higher rotational levels. Nitrogen flows have been considered here, as they are typical of many applications such as plasma chemistry or the modeling of spacecraft Earth entries. The selection of the N2 molecule as the subject of study is also advantageous compared to H2. Indeed the former molecule is much heavier and accordingly accommodates a larger number of internal levels (by about a factor of 30), whereas the H2 molecule has very large spacings of its internal levels, which makes it a special case among the modeling of molecular kinetics, compared to the N2 molecule, which can be more representative for the tackling of such problems. The first task was the definition of an adequate set of rovibrational levels. The determination of the overall manifold of internal levels of a molecule is a topic which is not restricted to this work and which has been previously tackled, regarding the development of vibrationally-specific kinetic models [8]. Here we have straightforwardly reapplied the approach of recalculating the potential curve for the ground state of N2, with the addition of the centrifugal potential from rotational motion. The solutions

for the Schrödinger equation then yield the manifold of rovibrational levels for the molecule ground electronic state. As it has been discussed, bound states above the dissociation limit are predicted to exist. Such quasi-bound states can spontaneously dissociate through quantum tunneling, which leads to the question to whether such states can be said to exist or not. Indeed this is a question which has been subject to much debate among the scientific community, with contradictory arguments either pointing towards the need to account for the existence of such levels or not. In this work, we have considered a semiclassical model for calculating the lifetimes of the overall quasi-bound levels which are predicted for the N2 molecule. The obtained results points towards most of these states being long lived (s > 1 s). The existence of such states which can spontaneously transition towards another state (dissociated state) does not imply any conceptual difficulty per se. In this sense they are akin to radiative states, which can also be short-lived (e.g. s = 1 ls  1 ns). Quasi-bound states may then be modeled just like radiative states through the adequate expressions in the master equation. A more convenient (if somehow oversimplified) approach is setting the collisional time of the flow scoll as a threshold to the existence or not of those quasi-bound states, as discussed in Section 2.1. A small but non-negligible influence of quasi-bound states has also been shown to exist, regarding the thermodynamic properties (partition functions) of the N2 molecule. For the calculation of such properties, it seems reasonable to account for the overall quasibound states, including the very short-lived ones. Indeed thermodynamic functions by principle are time-independent and, again invoking the case of radiative states, one should not discard any discrete state on the argument that it is short-lived. Another relevant conclusion of this work is the limited interplay of quasi-bound states over the dissociation dynamics of N2. Indeed, such levels have been found to contribute very few to the overall rotational dissociation, as they are rather sparsely populated, even at high temperatures. This is not to say the quasi-bound states can safely be neglected, as there may be gas conditions where such states play an important role in chemical kinetics. One may immediately think of recombining flows, where more or less detailed descriptions of the near-dissociative levels lead to very different results [6]. Generally speaking, it is also observed that rotational dissociation remains one to two orders of magnitude below vibrational dissociation, for thermal equilibrium conditions. Finally, the problem of modeling the relaxation of the rotational mode has been treated in an uncoupled fashion, restricted to R–T interactions. A state-specific model for V–T and dissociation reactions is then applied in a separate way. This is made necessary due to the lack of information on V–R interactions, but also due to the excessive computational burden of explicitly modeling the overall rovibrational modes of N2. It is indeed necessary to reduce the number of modeled variables through the assumption of a Boltzmann distribution at a characteristic rotational temperature Trot. Here we have mostly extended a previous work by Park [19], acknowledging that rotational energy level spacings are subtly influenced by vibrational levels. As the rate for rotational energy exchanges [24], reviewed and validated by Park, depends on these energy spacings, so will the rotational collision number Z vr calculated in Section 4. We concluded this work with the development of a hybrid state-to-state/Boltzmann relaxation model (respectively for the vibrational and rotational modes), including this approach for rotational relaxation. This model has been applied to the simulation of a sample 50,000 K N2 shockwave typical of an atmospheric entry. The results confirmed the previous claims by Park that the assumption of a very fast equilibration of the translational and rotational modes of the N2 molecule, as obtained using the old Parker results, is essentially wrong. Instead rotation–translation equilibrium lags behind vibration–translation equilibrium. Fur-

M. Lino da Silva et al. / Chemical Physics 398 (2012) 96–103

thermore, simulated results – using the old Parker approach and the approach described in Section 4 of this work – showed that a significant impact on the simulated translational temperatures (and hence dissociation/recombination dynamics) is observed. It is therefore clear that a full state-specific description of the overall molecular rovibrational levels should be actively pursued. In the meanwhile, approaches such as he ones outlined and applied in this work may already help improve the detail and validity of state-specific models for the simulation of shocked flows.

6. Conclusions This work has studied high-temperature rotational relaxation and dissociation processes in a diatomic gas with an application to nitrogen shocked flows. As with previous recent works on this topic, rotational relaxation has been found to lag behind vibrational relaxation, contrary to the normally accepted view, which is only valid roughly below T = 12,000 K. Purely rotational dissociation processes have also been investigated and found to be at least two orders of magnitude below vibrational dissociation. Finally, the additional set of rotational quasi-bound states of the N2 molecule has been determined, along with their lifetimes. Most of these states are short-lived and contribute marginally to rotational dissociation processes (which are already very low). In the situation of a dissociating flow, the simple accounting for those states in the partition functions seems sufficient, without any further need to include these in chemical kinetics schemes.

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