Calculation of Vibrational Relaxation Times Using a Kinetic Theory

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Calculation of Vibrational Relaxation Times Using a Kinetic Theory Approach Georgii P. Oblapenko J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b09897 • Publication Date (Web): 28 Nov 2018 Downloaded from http://pubs.acs.org on December 3, 2018

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The Journal of Physical Chemistry

Calculation of Vibrational Relaxation Times Using a Kinetic Theory Approach G. P. Oblapenko∗ Saint Petersburg State University, 7/9 Universitetskaya nab., 199034 St. Petersburg, Russia E-mail: [email protected]

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Abstract In the present work, a method for computation of vibrational relaxation times based on a kinetic theory definition is utilized to calculate vibrational relaxation times of molecules present in air (N2 , O2 and NO) in collisions with air species particles. An overview of available experiment VT relaxation time measurements, as well as quasiclassical trajectory (QCT) calculation results, and various empirical models, is given. Different inelastic cross-section models are used for the computation of the relaxation times, and their parameters are adjusted to fit the available experimental data and QCT results. It is shown that the proposed method of calculation can give a quantative and qualitative agreement with the available data in a wide range of temperatures; the obtained interaction parameters may be used not only for vibrational relaxation time calculation within a multi-temperature framework, but also for development of statespecific models for use in CFD and DSMC codes.

1 Introduction The multi-temperature approximation remain the most widely used framework for numerical simulation of non-equilibrium flows, due to its simplicity compared to a significantly more detailed state-specific approach (where each vibrational level is treated as separate chemical species), and the possibility to use the well-known Landau–Teller formula for the computation of changes in the vibrational energy of a molecular species due to vibrational–translation (VT) energy transitions. The Landau–Teller formula requires only the knowledge of vibrational relaxation times for the species in question; these times also depend on the collision partner. These relaxation times are commonly computed using the Millikan–White 1 formula, which has two notable drawbacks. The first is its linear dependence on T −1/3 (here T is the gas temperature), which leads to a significant underestimation of VT relaxation times at higher temperatures, and has led to the development of various corrective factors, the 2


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most widely-used being Park’s correction. 2 The other shortcoming of the Millikan–White correlation is its grounding in available experimental data — therefore, it may not necessary generalize to collision partners not considered in their original work. In air, where three molecular (N2 , O2 and NO) and two atomic (N and O) species are present, 15 different relaxation times need to be computed. If one also accounts for argon, one needs to compute 18 different relaxation times. The paper by Millikan–White only takes into account data on relaxation times for N2 -N2 , O2 -O2 and O2 -Ar collisions (other collisions are also considered in the paper, but the species are not present in a significant amount in air). Thus, its applicability to other species is of question. Both experimental measurements and quasiclassical trajectory (QCT) calculations have shown that for some collisions, such as O2 -O, O2 -N, N2 -O, NO-NO, the relaxation times are significantly smaller than those predicted by the Millikan–White expression, which is due to several “chemical” interactions between the collision partners. 3 The presence of these reactive interactions (the contribution of which to the vibrational relaxation rate can be of the same order of magnitude as the contribution of non-reactive interactions 4 ) prohibits the use of simpler models such as the SSH model 5 to correctly predict the vibrational energy transfer rate, as such semi-classical models only consider non-reactive collision processes. Another correction to the Millikan–White formula is a correction to account for the anharmonicity of the vibrational spectra. 6 A notable feature of this model is the dependence of the vibrational relaxation times on both the gas temperature and the vibrational temperature of the molecular species. However, since the derivations are based on SSH theory, the model suffers from the same drawbacks. Quasiclassical trajectory computations provide another way of computing VT relaxation times, but require extensive computational effort, and the results obtained via these methods need to be numerically approximated to be usable in CFD codes. The other drawback of QCT computations is their complexity with regards to molecule-molecule collisions; indeed, most works focus on simulating three-body systems (molecule-atom collisions), and only a

3



few recent papers apply QCT to simulation of inelastic molecule-molecule collisions. If one derives the Landau–Teller formula within the Chapman–Enskog framework, a kinetic theory definition of VT relaxation times may be introduced. 7,8 It has been shown that VT relaxation times computed using this method are correct in terms of their qualitative behaviour at higher temperatures, 8,9 and when used with a recently obtained modification of the Landau–Teller formula, can significantly improve the accuracy of numerical modeling of strongly non-equilibrium flows. 9–11 However, up to now, vibrational relaxation times computed using a kinetic theory definition have been considered only for binary mixtures of nitrogen and oxygen. The aims of the present work are to compute VT relaxation times for air species, using available experimental and QCT data to adjust the interaction parameters required for the calculation of vibrational relaxation times using the kinetic theory definition. The proposed method for computation of relaxation times can be used to increase the accuracy of multitemperature CFD simulations. By providing a possibility to adjust interaction parameters based on available VT relaxation time data, the method may also be applied to improve the quality of higher-fidelity simulations, such as state-specific CFD and DSMC computations, as well as coarse-grained reduced-order simulations. 12,13

2 Theory In previous work of the author, 7 within the framework of the generalized Chapman–Enskog method, a definition for VT relaxation times was obtained during the derivation of the Landau–Teller formula from a kinetic theory expression for vibrational relaxation rates via state-to-state rates averaged over non-equilibrium Boltzmann distributions over vibrational energies. The definition is similar to those previously obtained within the generalized Chapman– Enskog formalism for internal 14 and rotational relaxation times, 15 and takes on the following

4


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VT form for the VT relaxation time τcd for collisions between molecules of species c and particles

of species d: 7 1 VT τcd

=

4kn ⟨( vibr )2 ⟩V T ∆Ec , mc cvibr,c cd

(1)

where k is the Boltzmann constant, n is the number density, mc is the mass of species c, cvibr,c is the specific vibrational heat of species c, ∆Ecvibr = (εci′ − εci ) /kT (here i and i′ denote the vibrational levels of the molecule of species c before and after the VT transition, correspondingly). The averaging operator is defined on the basis of the zero-order distribution function, which is given by the following expression: 15 (0) fcij

( ) ( m )3/2 n εcj mc c2c εci c c c = s exp − − − , 2πkT Zcint ij 2kT kT kTvc

(2)

where scij is the degeneracy of the molecular state with rotational level j and vibrational level i, Tvc is the vibrational temperature of molecular species c, and Zcint is the internal partition function defined as follows (a rigid rotator approximation is used throughout the present work): Zcint = Zcrot Zcvibr ,

(3)

where Zcrot and Zcvibr are the rotational and vibrational partition functions, correspondingly, defined as Zcrot

=

∑

(

scj

j

Zcvibr

=

∑ i

εcj exp − kT

) ,

(4)

( ) εci exp − c . kTv

(5)

5



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The averaging operator appearing in (1) is thus defined as: ( ⟨F ⟩VcdT

=

kT 2πmcd

)1/2 ∑ iki′ jlj ′ l′

scij sdkl Zcint Zdint

(

ˆ Fcij g03

−g02

exp

εcj εd εc εd − l − i c − kd − kT kT kTv kTv ′ ′

) ×

′

i j kl 2 × σcd,ijkl d Ωdg0 , (6)

where mcd = mc md / (mc + md ) is the collision reduced mass, g0 =

√

mcd /2kT g is the di′ ′

′

i j kl mensionless relative velocity, g is the relative velocity of the colliding particles, σcd,ijkl is

the cross-section of a VT transition in which the vibrational level i and rotational level j of a molecule of species c change to i′ and j ′ after colliding with a particles of species d with vibrational level k and rotational level l, the latter of which changes to l′ , and d2 Ω is the solid angle in which the relative velocity of particles after the collision appear. To compute the relaxation times, knowledge about the cross-sections is needed. We assume that the cross-sections do not depend on the vibrational level of the collision partner (for an assessment of the influence of variable molecular diameters on transport coefficients and vibrational relaxation rates, the reader is referred to 16,17 ), as well as the rotational states of the colliding particles; thus, we can write that ′ ′

′

i j kl σcd,ijkl = σcd,i→i′

(7)

and expression (6) is simplified (assuming that the function to be averaged does not depend on j: Fcij = Fci ): ( ⟨F ⟩VcdT

=

kT 2πmcd

)1/2 ∑ ii′

sci Zcvibr

(

ˆ Fci g03

exp

−g02

εc − ic kTv

) σcd,i→i′ d2 Ωdg0 .

(8)

It is important to note that this definition includes a dependence of the vibrational relaxation times not only on the gas temperature T , but also on the vibrational temperature Tvc of molecules of species c. However, since we intend to compare relaxation times with those given

6


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by the Millikan–White expression and various experimental data, we will assume Tvc = T . To compute the cross-sections, we assume that they can be presented in the following form σcd,i→i′ = Pcd,i→i′ σcd,el ,

(9)

where Pcd,i→i′ is the probability of the VT transition (during which the vibrational level of molecule c changes from i to i′ ) and σcd,el is an elastic total cross-section for the collision of particles of species c and d. The Variable Soft Sphere (VSS) model 18 was used to compute σcd,el , with interaction data taken from. 19 In the present work, the Forced Harmonic Oscillator (FHO) 20 is used to calculate the transition probabilities Pcd,i→i′ . The choice of model is due to several reasons: 1) its good agreement with quasi-classical calculations of rate coefficients in air and nitrogen 2) the presence of several parameters governing the model’s behaviour at low and high temperatures, which can be adjust to fit available experimental or QCT data. The FHO model contains three parameters — the depth E of the Morse potential well (which determines the behaviour of the model at low temperatures), the repulsive potential parameter α, and a steric factor SV T , which in the original work was proposed to be taken to be 1/2. Another approach to computation of the cross-sections is the use of QCT data — if one constructs an approximation of the computed rate coefficient, it is possible to use the inverse Laplace transform to obtain an expression for the cross-section. 21 The main drawback of such a method is the special care required when constructing the approximation, so as to avoid unphysical expressions for the cross-sections. 22 Recently, this approach has been applied to the computation of cross-sections of VT transitions and dissociation reactions for O2 -O and N2 -N collisions, 23 and these cross-sections have been implemented and validated in DSMC code. 24 One may avoid some of the difficulties associated with such an approach by first constructing a functional form of the cross-section with some unknown parameters, and then estimate these unknowns by comparing the resulting rates with available data. 25 Such an approach has been used for calculation of cross-sections of vibrational energy transitions 25 7



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and chemical reactions. 26 Thus, QCT data provides an alternative way for computation of relaxation times using definition (1); however, in the absence of such data, some theoreticallybased model, such as the FHO model, must be used. The question of choosing model parameters based on available experimental measurements and results of QCT simulations is considered in the next section. Finally, a recent development in the simulation and study of rarefied gas flows has been the so-called “direct simulation” method, 27 in which DSMC modeling of gas dynamics is coupled with a direct molecular simulation of the collisional process (instead of a probabilistic simulation). Thus, the method does not require pre-computed inelastic process cross-sections or reaction rates, and instead utilizes the potential energy surfaces of the collision pair directly. This method has also been used to study internal energy relaxation and dissociation processes in binary mixtures. However, the application of this method to complex air flows however may be hindered by its high computational cost.

2.1 Note on the modified Landau–Teller formula The relaxation times given by formulas (1), (6) are intended for use with the modified Landau–Teller formula: 7

Rcvibr,V T =

∑ nd T c (T − T ) ρ c , c vibr,c v VT Tvc nτcd d

(10)

since their definition has been derived consistent with the derivation of expression (10). Here Rcvibr,V T is the vibrational energy relaxation rate due to VT transitions for species c. As mentioned previously, the use of the kinetic theory based method for computation of the relaxation times in conjunction with the modified Landau–Teller formula can lead to an improvement in the modeling accuracy for strongly non-equilibrium flows. 9–11 VT to It should be stressed that in formula (10), it is essential for the relaxation time τcd

be dependent on both the gas temperature T and the vibrational temperature Tvc — in case

8


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one uses a relaxation time that is independent of Tvc , the multiplier

T Tvc

should be omitted.

Otherwise, the modified Landau–Teller formula (10) becomes inconsistent with the model for the vibrational relaxation time. Conversely, if one wants to use relaxation times as defined by (1) in the Landau–Teller expression, one must compute them assuming that T = Tvc . In order to ease the implementation of the modified Landau–Teller formula in CFD solvers, it can be re-written using definition (1):

Rcvibr,V T = 4k

∑ ⟨( ) ⟩V T T c vibr 2 ) n , (T − T n ∆E c d v c Tvc cd d

This form allows one to pre-compute the averaging operator

⟨(

∆Ecvibr

(11)

)2 ⟩ V T

in the desired

cd

temperature range and store it in tabular form (since it is dependent only on two temperatures – T and Tvc ), which significantly improves the computational efficiency of the modified Landau–Teller formula.

3 Results & Discussion To determine the parameters in the FHO model, one needs data to which the computed relaxation times can be compared. As discussed previously, the Millikan–White correlation can not be used as a ground truth, especially if other data is available. In this section, an overview is given of existing experimental measurements and quasiclassical trajectory computations of vibrational relaxation times. The parameters in the FHO model (used to compute relaxation time using formula (1)) are adjusted to improve its agreement with the available data. Since the FHO model may not be physically valid for some interactions (see below), the obtained parameter values may have no direct physical meaning; however, the behaviour of the relaxation times remains physical in a wide temperature range. One should also note that while it would be reasonable to assume that for a given collision pair of species c and d, the FHO interaction parameters would be constant irregardless of which species is

9



undergoing a vibrational transition, in practice, a better agreement with the available data is obtained if one uses two sets of interaction parameters for each collision pair – one for the relaxation of molecular species c, and one for the relaxation of molecular species d; the difference between these two parameter sets is especially strong for collisions involving NO molecules. The code used to compute the relaxation times using definition (1) was written in Python and has been made available online at https://github.com/knstmrd/VT-relaxation, along with the obtained best-fit FHO model parameters.

3.1 Relaxation of O2 The relaxation times of oxygen in collisions with various air species have been extensively studied both experimentally and numerically. 10−1

O2+O2, FHO O2+O2, M-W

100

O2 +O2 , Ibraguimova

10−2

10−1

O2 +O2 , Owen

10−2

10−3

O2+Ar, FHO O2+Ar, M-W O2+Ar, Owen (exp.) O2+Ar, Boyd (QCT) O2+Ar, Boyd (QM)

10−3

lgpτOVT2

10−4

lgpτOVT2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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10−4

10−5

10−5

10−6

10−6

10−7

10−7 10−8

10−8 0.04

0.06

0.08

T −1/3

0.10

0.04

0.12

0.06

0.08

T −1/3

0.10

0.12

Figure 1: Relaxation times for collisions of oxygen molecules with oxygen molecules (left) and argon (right); vibrational relaxation times computed using definition (1) are shown by the “FHO” curves. Recent experimental data 28 on relaxation of O2 in collisions with O2 molecules shows a deviation from the linear dependence of the relaxation times on T −1/3 — at higher temperatures, the relaxation times behave non-monotonically, as seen in Figure 1. At lower temperatures, O2 -O2 vibrational relaxation times are well described by the Millikan–White expression, as seen also in experiments of Losev and Generalov, 29,30 as well as more recent 10


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experimental measurements by Owen. 31 Owen has also conducted experimental studies of vibrational relaxation of oxygen by argon atoms, and his results are in excellent agreement with previous measurements done by Camac, 32 and confirm the validity of the Millikan–White expression for oxygen-argon collisions at moderately high and low temperatures. The vibrational kinetics O2 -Ar system have also been studied using QCT and quantum mechanics, 33 and vibrational relaxation times for such collisions have been computed. At temperatures lower than 4000 K, the QCT method fails to provide data on relaxation times/transition rates due to the inability to accumulate sufficient statistics, but at higher temperatures the results are close to those predicted by the Millikan–White formula. At lower temperatures, relaxation times computed via the quantum mechanics method, are also in moderately good agreement with Camac’s experimental measurements and the Millikan–White fit. The model proposed in this paper provides an excellent agreement with the various experimental data for O2 -O2 collisions at temperatures lower than 8000 K, and slightly overestimates the relaxation times at higher temperatures; for O2 -Ar collisions, the discrepancy at higher temperatures is slightly higher, since the available data does not exhibit a non-monotonous behaviour at higher temperatures. 10−2

O2+O, FHO O2+O, M-W

O2+O, M-W, Park O2+O, Thivet O2+O, Kiefer & Lutz (exp.) O2+O, Breen (exp.) O2+O, Shatalov (exp.) O2+O, Kalogerakis (exp.) O2+O, Boyd (QCT) O2+O, Grover (DMS) O2+O, Esposito (QCT)

1063

1064

1065

10−3

O2+N, FHO O2+N, M-W

O2+N, M-W, Park O2+N, Thivet O2+N, Boyd

10−4

lgpτOVT2

1062

lgpτOVT2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10−5

1066 10−6 1067 10−7 1068 10−8 0.04

0.06

0.08

T 61/3

0.10

0.12

0.14

0.04

0.06

0.08

T −1/3

0.10

0.12

Figure 2: Relaxation times for collisions of oxygen molecules with oxygen atoms (left) and nitrogen atoms (right); vibrational relaxation times computed using definition (1) are shown by the “FHO” curves.

11



Vibrational relaxation times for O2 -O collisions are significantly lower than those for O2 -O2 collisions. This has been first noted by Kiefer and Lutz, 34 and later experiments, 28,35 as well as QCT simulations, 36 confirmed this finding. Recent experimental 28 and QCT 36 data also provides evidence of low O2 -O VT relaxation times, but predicts their monotonous growth with temperature, a trend not seen in the older experiments. It should be noted that the data of Breen 35 is almost temperature-independent, a trend also predicted by the QCT data in the same temperature range (1000-3400 K). Based on experimental data, Park has proposed 2,37 a new set of coefficients for use in the Millikan–White expression for the computation of τOV2T,O , while a new correlation (also grounded in experimental data) for relaxation times for O2 -O collisions has been proposed by Thivet. 38 Room temperature measurements of the rate coefficient of vibrational deactivation of molecular oxygen at the first vibrational level 39 also point towards a monotonous increase τOV2T,O with increasing temperature and a weak dependence of τOV2T,O on the gas temperature. The QCT calculations of Esposito et al. 40 are in a qualitative agreement with the other QCT data, and at lower temperatures are close to the experimental measurements; however, at high temperatures, the relaxation times are almost 1.5 orders of magnitude higher than those obtained by other authors. Finally, the direct simulation method has been used to compute the vibrational relaxation times for the O2 -O collision pair, 41 and the values obtained using such a method are up to an order of magnitude lower than those obtained by QCT, and are also somewhat lower than the experimental measurements of Breen and Kiefer & Lutz, however, the direct simulation results also point towards a very weak dependence of τOV2T,O on the gas temperature. The computations of relaxation times performed by Kulakhmetov et al. 25,42 are in good agreement with the results of Andrienko and Boyd, 36 since the same potential surfaces have been used; thus, their results are not shown on Figure 2. The model proposed in the present work provides a good agreement with the QCT data of Esposito et al. at low temperatures and reasonable agreement with the various experimental and numerical results in the mid-temperature range, while at higher temperatures, it behaves qualitatively similar to recent QCT data.

12


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It should be noted that the presented QCT results utilized the potential energy surface of Varandas and Pais, 43 which has been found to contain artifacts; 41 besides, the Varandas potential energy surface is only for the singlet state and the triplet and quintet states are ignored. Thus, the accuracy of the QCT data is somewhat questionable. QCT simulations have shown that O2 -N collisions are also effective 44 — the relaxation times for them are of a similar magnitude to those in O2 -O collisions. Park has also proposed new coefficients for the Millikan–White relation for this collision pair (to be more precise, Park has suggested new coefficients for the Millikan–White relaxation for N2 -O collisions and proposed using the assumption that τOV2T,N = τNV 2T,O ); while Thivet 38 has suggested using the assumption that τOV2T,N = τOV2T,O . It should be noted that recent QCT simulations and available experimental data show that Park’s assumption of τOV2T,N = τNV 2T,O is not valid, 45 the relaxation times of oxygen in collisions with atomic nitrogen being lower than relaxation times of nitrogen in collisions with atomic oxygen; while Thivet’s assumption leads to an underestimation of τOV2T,N . The kinetic theory-based model provides a reasonable agreement with the QCT data of Boyd at temperatures lower than 4500 K, but leads to a strong increase in the relaxation at higher temperatures. 10−1 10−2

O2+N2, FHO

O2+NO, FHO O2+N2, M-W O2+NO, M-W O2+N2, Boyd (QCT)

103

101

10−3

N2+N2, FHO N2+O2, FHO N2+N2, M-W

N2+O2, M-W N2+N2, Jaffe (QCT) N2+N2, Grover (DMS) N2+O2, Boyd (QCT)

10−1

lgpτNVT2

10−4

lgpτOVT2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10−5

10−3

10−6

10−5

10−7 10−7

10−8 0.04

0.06

0.08

T −1/3

0.10

0.12

0.04

0.06

0.08

T −1/3

0.10

0.12

0.14

Figure 3: Relaxation times for collisions of oxygen molecules with nitrogen molecules and nitric oxide (left) and of nitrogen molecules with nitrogen and oxygen molecules (right); vibrational relaxation times computed using definition (1) are shown by the “FHO” curves. For molecule-molecule collisions of nitrogen and oxygen, recent QCT simulations 46 show 13



that O2 -N2 and N2 -O2 collisions are well-described by Millikan–White’s expression for temperatures lower than 9000 K, but at higher temperatures, the Millikan–White formula, even with Park’s correction, underestimates the relaxation times, especially for N2 -O2 collisions. Relaxation times given by equation (1) exhibit a non-monotonous behaviour at temperatures higher than 8000 K and are larger than the QCT results. At lower temperatures, the agreement with the QCT data is good, and the times are very close to those given by the Millikan–White expression, as seen on Fig. 3. No data is available for τOV2T,N O , and therefore, one can only estimate the relaxation times using the Millikan–White formula. Thus, for VT relaxation times of oxygen, the proposed model provides good agreement with the various available data in a wide temperature range, the only exception being τOV2T,N at temperatures higher than 8000 K. However, for simulation of various re-entry problems, at these temperatures, oxygen experiences a significantly stronger degree of dissociation than nitrogen, and the O2 +N collision might not have a noticeable effect on the overall relaxation rate, due to the low concentrations of the collision partners involved. For τOV2T,N2 and τOV2T,N O , the present model is constrained by a lack of data at low temperatures (and, in the latter case, the lack of any data at all), and further improvements can be made once QCT or experimental results for such collisions become available.

3.2 Relaxation of N2 For the N2 -N2 collision pair, QCT computations 47 show a deviation from the Millikan– White relaxation times at temperatures higher than 10000 K, a trend also present in the results obtained using the direct simulation method. 48 The model developed in the present work overestimates the relaxation times at temperatures higher than 10000 K, and exhibit a stronger non-monotonous behaviour, but at lower temperatures, a good agreement can be seen with Millikan–White’s expression and the direct simulation data. Experimental studies of N2 vibrational relaxation by oxygen atoms have shown a strong 14


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10.1

N2+O, FHO N2+O, M-W

103

N2+O, M-W, Park N2+O, Thivet N2+O, McNeal (exp.)

101

N2+O, Gordiets (exp. fit) N2+O, Esposito (QCT)

10/1

N2+N, FHO N2+N, M-W

N2+N, Thivet N2+N, Panesi (QCT) N2+N, Boyd (QCT) N2+N, Esposito (QCT) N2+N, Grover (DMS)

lgpτNVT2

101

lgpτNVT2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10.3

10/3

10.5

10/5

10.7

10/7

0.04

0.06

0.08

T .1/3

0.10

0.12

0.14

0.04

0.06

0.08

T

/3

0.10

0.12

0.14

−1

Figure 4: Relaxation times for collisions of nitrogen molecules with oxygen atoms (left) and nitrogen atoms (right); vibrational relaxation times computed using definition (1) are shown by the “FHO” curves. deviation from the Millikan–White expression, 49–51 indicative of a “chemical” effect, 50 moreover, the experimental data displays a very weak temperature dependence. 50,51 Again, a new correlation for relaxation times for N2 -O collisions has been proposed by Thivet, 38 while Park, as mentioned previously, has suggested using different coefficients in the Millikan– White expression. Gordiets has also proposed a fit to the experimental data in the form of a double Arrhenius fit. 52 While QCT computations of VT transition rates in N2 +O collisions have been performed, 53–55 they significantly underestimate the transition rates (and, as a result, overpredict the relaxation times) at lower temperatures, which is caused by the consideration of only adiabatic transitions — thus, a more precise treatment is required in order to obtain a good agreement between QCT and experimental data. Both Thivet’s and Gordiets’ fits provide a good agreement with the available experimental data, but strongly differ in their behaviour at higher temperatures. It is worth noting that the QCT computations performed by Esposito et al. 55 give a noticeably higher relaxation rate than the fit of Gordiets at temperatures higher than 12000 K (a region where the QCT calculations are reliable), thus leading to a lower vibrational relaxation time, which is in good agreement with the fit of Thivet. Relaxation times computed using definition (1) are in excellent agreement with the fit of Gordiets throughout the whole temperature range; on the whole, most of the

15



available data for N2 -O VT relaxation times is close in the range from 273 to 15000 K. VT relaxation times for N2 -N collisions have been computed via QCT methods by Boyd 56 and Panesi, 57 and have been shown to be more than an order of magnitude lower than those predicted by the Millikan–White formula. While Thivet has suggested the assumption τNV 2T,N = τNV 2T,O , it can be seen from Fig. 4 that such an expression gives an incorrect temperature dependence of the relaxation time, underestimating it at temperatures lower than 3500 K and overestimating it at temperatures higher than 8000 K. Direct simulation of the N2 -N interaction has also been performed, 58 and the results are in good agreement with those obtained using QCT. The computations of Esposito 59 again give values of the relaxation time higher than those obtained by other authors (and are in fact quite close to the fit of Thivet), but the qualitative behaviour of these values is the same as obtained by Boyd, Panesi, and Grover. At lower temperatures, the results of computations done by Panesi and Esposito lie higher than the relaxation times given by the Millikan–White expression, however, since the Millikan–White formula may not be necessarily valid for the N2 +N system, it should not be considered a benchmark for the QCT-based results. The model proposed in the present work can be fitted with a reasonable degree of accuracy to the data given by Esposito; it also is in agreement with the results of other authors in the range from 2000 to 4500 K. 105 103

N2+NO, FHO N2+Ar, FHO N2+NO, M-W N2+Ar, M-W

101 10−1

lgpτNVT2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 33

10−3 10−5 10−7 0.04

0.06

0.08

T −1/3

0.10

0.12

0.14

Figure 5: Relaxation times for collisions of nitrogen molecules with nitric oxide and argon; vibrational relaxation times computed using definition (1) are shown by the “FHO” curves.

16


Page 17 of 33 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Unfortunately, no data is available for τNV 2T,N O and τNV 2T,Ar , and thus, one can only rely on the Millikan–White formula; however for nitrogen, apart from τNV 2T,O , the Millikan–White expression provides a reasonable approximation to the available data on relaxation times, and thus, one can assume that it provides a correct description of τNV 2T,N O and τNV 2T,Ar at moderately high and low temperatures. Thus, the parameters of the FHO model were chosen so that the relaxation times computed are close to those given by the Millikan–White formula at temperatures up to 4500 K, as shown on Fig. 5. Thus, for VT relaxation times of nitrogen, definition (1) in conjunction with the VSS and FHO models give good agreement with the various available data in a wide temperature range. The biggest scatter is exhibited by the QCT data for N2 +N relaxation times at temperatures higher than 8000 K. Therefore, further improvement of the model requires more thorough investigation of vibrational-translational energy transitions in N2 +N collisions at high temperatures; the relaxation time for this process may have a noticeable effect on modeling of strongly non-equilibrium nitrogen and air flows, in cases where nitrogen begins to dissociate before vibrational equilibrium has been reached. For τNV 2T,N O and τNV 2T,Ar , no data (experimental or results of numerical modeling) is available, and thus, the present model and the FHO interaction parameters may be refined further once data becomes available. However, the actual values for these relaxation times might not have a significant effect on post-shock flows, since argon and nitric oxide in such flows are present only in small quantities.

3.3 Relaxation of NO Relaxation times of NO in collisions with NO are known to be several orders of magnitude lower than those of nitrogen and oxygen. 60 Based on the experimental data of Wray, 60 Park proposed new coefficients for use in the Millikan–White expression for NO-NO collisions, and also suggested using the same values for relaxation times for all other collision partners (O2 , N2 , N, O). 37 However, some simulations show that using the original Millikan–White 17



expression for all collision pairs except NO-NO and keeping the values given by Park to T compute τNV O,N O gives better agreement with experimental measurements of temperature

and NO vibrational temperature profiles in nozzle flows. 61 Therefore, a further analysis of vibrational relaxation times of NO is required.

10.2

10.4

NO+NO, FHO NO+NO, M-W

101

NO, M-W, Park NO+NO, Moser (exp.)

NO+NO, Glanzer (exp., 1975) NO+NO, Glanzer (exp., 1977) NO+NO, Hancock (exp.) NO+NO, Horiguchi (exp.) NO+NO, Kamimoto (exp.) NO+NO, Wray (exp.)

10−1

NO+Ar, FHO NO+Ar, M-W NO, M-W, Park NO+Ar, Moser (exp.) NO+Ar, Glanzer (exp.) NO+Ar, Hancock (exp.) NO+Ar, Kamimoto (exp.) NO+Ar, Wray (exp.)

10−3

VT lgpτNO

100

VT lgpτNO

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 33

10−5 10.6

10−7 10.8 0.04

0.06

0.08

T .1/3

0.10

0.12

0.04

0.14

0.06

0.08

T −1/3

0.10

0.12

0.14

Figure 6: Relaxation times for collisions of nitric oxide with nitric oxide (left) and argon atoms (right); vibrational relaxation times computed using definition (1) are shown by the “FHO” curves. Experimental data by Wray 60 gives relaxation times for NO-NO collisions in the temperature range 1000–8000 K. Measurements by Moser and Hindelang, 62 performed in the temperature range of 2000–3500 K, also show that NO-NO relaxation times are more than one order of magnitude lower than those given by the Millikan–White formula, however, the values are higher than those obtained by Wray. It should be noted that the data given by Moser and Hindelang shows the increase in VT relaxation times with growing temperature, something that is not present in the data of Wray. Another set of data are the rate measurements performed by Glanzer 3,63 (in the temperature range 900–2700 K), which are in better agreement with the data of Wray, but exhibit an almost temperature-independent behaviour. The work of Glanzer 63 also provides an approximation for NO relaxation rates, based both on the experimental results obtained by the authors, as well as other available measurements. At low temperatures this data is very close to that of Wray. Relaxation times obtained by Kamimoto 64 are in close agreement with those of Glanzer. Room temperature 18


Page 19 of 33

T measurements performed by Horiguchi, 65 Hancock 66 give very low values for τNV O,N O ; in fact,

these values are somewhat lower than the higher-temperature data mentioned previously. The various data for these relaxation times is shown on Fig. 6. Thus, it can be concluded, that for NO-NO collisions, the relaxation times are extremely low and do not change significantly with increasing temperature. At temperatures higher than 8000 K, no data is currently available to assess their behaviour. Relaxation times calculated using definition (1) agree well with the available data in a wide temperature range. Moser and Hindelang, Glanzer, Wray, as well Kamimoto, also measured vibrational energy transfer rate constants for NO-Ar collisions, and their data shows that relaxation times for this collision pair are up to three orders of magnitude higher than for NO-NO collisions. T has been measured by Hancock, and is also significantly larger At room temperature, τNV O,Ar T than τNV O,N O . Again, the data of Wray shows a much stronger temperature dependence of

the vibrational relaxation times than the other experimental measurements. The proposed model is in moderately good agreement with the data, which exhibits a strong degree of scatter, as seen on Fig. 6. Insufficient data is available at higher temperatures to accurately adjust the FHO interaction parameters. 101

10−1

100

NO+O2, FHO NO+O2, M-W

NO, M-W, Park NO+O2, Bohn (exp.) NO+O2, Boyd (QCT)

10.2

10.4

lgpτVT

10−3

VT lgpτNO

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10−5

10.6

10−7

10.8

0.04

0.06

0.08

T −1/3

0.10

0.12

NO+O, FHO

NO+O, M-W NO+O, M-W, Park NO+O, Glanzer (exp.)

NO+O, Hwang (exp.) NO+O, Caridade (QCT), 2008 NO+O, Caridade (QCT), 2018 NO+O, Sharma (QCT) NO+O, Schinke (QCT) NO+O, Boyd (QCT)

0.04

0.14

0.06

0.08

T .1/3

0.10

0.12

0.14

Figure 7: Relaxation times for collisions of nitric oxide with molecular oxygen (left) and atomic oxygen (right); vibrational relaxation times computed using definition (1) are shown by the “FHO” curves.

19



For NO-O2 collisions, measurements of VT rate coefficients at room temperature 67 show that the relaxation times are very close to those for NO-NO collisions, and Parks collision partner-independent fit of NO relaxation times agrees well with the experimental results (though, obviously, no conclusion can be made concerning their temperature dependence). The QCT simulations, 68 however, give results very close to the original Millikan–White expression, overestimating the relaxation times for NO-O2 collisions (assuming the validity of Park’s fit) by more than 2 orders of magnitude, as shown on Fig. 7. However, in the QCT simulations the potential surfaces for NO-O2 and NO-N2 interactions were constructed empirically, and thus, may not be an adequate descriptor of such interactions. Thus, the room temperature measurements and Park’s assumption were used to adjust the parameters of the FHO model. Measurements of VT rate coefficients for NO-O collisions have been performed by Glanzer, 3 Hwang. 69 QCT data is also available, 4,70–72 which is in good agreement with both roomtemperature and shock-tube experimental measurements. The recent QCT modeling of Boyd 68 also confirms the trend of the relaxation times for such collisions with regards to temperature, but the computed times are lower than those predicted by other QCT computations, as well as those measured in experiments. The most recent results of Caridade et al. 72 are in good agreement with the experimental data of Hwang. For NO-O collisions, the VT relaxation times computed using the approach proposed in the present work are in a good agreement with the experimental and QCT data in a wide temperature range. The various measured and computed relaxation times for NO-O collisions are presented on Fig. 7. The only data available for NO-N2 and NO-N collisions (apart from that given by the Millikan–White formula and its collision partner-independent form for NO proposed by Park) is the QCT data of Boyd; 68 however, as mentioned previously, the NO-N2 data might not necessarily be correct due to an empirical treatment of the interaction potential. It can be seen on Fig. 8 that the QCT data is much closer to the values given by the original

20


Page 20 of 33

Page 21 of 33

100 10−1 10−2 10−3

NO+N, FHO

NO+N2, FHO NO+N, M-W NO+N2, M-W NO, M-W, Park NO+N, Boyd (QCT) NO+N2, Boyd (QCT)

lgpτVT

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10−4 10−5 10−6 10−7 10−8 0.04

0.06

0.08

T −1/3

0.10

0.12

0.14

Figure 8: Relaxation times for collisions of nitric oxide with molecular and atomic nitrogen; vibrational relaxation times computed using definition (1) are shown by the “FHO” curves. T Millikan–White formula (though τNV O,N displays a stronger temperature dependence) rather 2

than those given by Park. 37 Since the QCT data for the relaxation times for NO-N2 collisions is somewhat suspect, the FHO interaction parameters were fitted to provide to the original Millikan–White formula instead; while for NO-N collisions, the QCT data was used, and relaxation times computed using the FHO model and definition (1) provide a good agreement with it. It can be concluded that despite NO vibrational relaxation being strongly influenced by chemical effects, it is possible to adjust the FHO parameters to be in good agreement with the various data. Analyzing the QCT and experimental data, one can also conclude that NO relaxation times exhibit a strong degree of variability with respect to both their quantitative and qualitative behaviour; thus, the use of simplifying assumptions such as those of Park (namely, using a single, collision partner-independent value for NO relaxation times) can lead to a decrease in the modeling accuracy. While QCT data exists for NO+O2 and NO+N2 collisions, the potentials used are based on empirical assumptions, and therefore, further investigation of these collisions is needed in order to improve the accuracy of the present model.

21



Page 22 of 33

3.4 Influence of vibrational temperature As mentioned previously, the relaxation times as defined by (1) are functions not only of the gas temperature T , but also of the vibrational temperature Tvc of the molecular species for which the relaxation time is being computed. This dependency makes definition (1) inconsistent with the Landau–Teller formula, as discussed previousl. It is of interest to see how varying the ratio Tvc /T affects the relaxation times, and in the present section, results of computations performed using various fixed values of Tvc /T are presented. 10−5

10−1

10−6

10−2

10−7

10−3

10−8

10−4

10−9

10−5

10−10

10−11 0.04

O2+O, FHO, Tv/T = 1 O2+O, FHO, Tv/T = 1.5 O2+O, FHO, Tv/T = 2 O2+O, FHO, Tv/T = 0.5 O2+O, FHO, Tv/T = 0.25 0.05

0.06

N2+N2, FHO, Tv/T = 1 N2+N2, FHO, Tv/T = 1.5 N2+N2, FHO, Tv/T = 2 N2+N2, FHO, Tv/T = 0.5 N2+N2, FHO, Tv/T = 0.25

10−6

0.07

0.08

0.09

0.10

10−7 0.04

0.05

0.06

0.07

0.08

0.09

0.10

Figure 9: Relaxation times for O2 +O (left) and N2 +N2 (right) computed using definition (1) for different values of the ratio Tv /T . O2 +O and N2 +N2 relaxation times were chosen as they exhibit very different qualitative behaviour, as seen on figures 2 (left) and 3 (right). We see that at high temperatures, increasing the ratio Tv /T leads to a decrease in the computed vibrational relaxation times; this is due to an increase in the population of the upper vibrational levels, where the vibrational relaxation rates are higher. However, for low temperatures and ratios of Tv /T less than 1, the relaxation times decrease rapidly with decreasing temperature and can even exhibit a non-monotonous behaviour (see figure 9 (right)). This may be explained by the fact that at different temperatures, the FHO model predicts vibrational de-excitation rates that vary in their dependence on the vibrational level number. As can be seen, varying the ratio of Tv /T does not lead to any unphysical behaviour of the relaxation times.

22


Page 23 of 33 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3.5 Interaction parameters The interactions parameters for the FHO model used in the present work and obtained on the basis of comparisons with existing data on VT relaxation times are given in the tables below. These parameters are the α repulsive potential parameter, the Morse potential well depth E, and steric factor SV T . The different rows correspond to different molecular species for which the vibrational relaxation is considered, while different columns correspond to different collision partners. Table 1: Best-fit values of β for air mixtures, Å−1 N2 O2 NO

N2 3.9 4.1 4.4

O2 3.9 4.3 6.75

O Ar NO N 4 4.6 7.3 4 4.1 7.3 2.6 3.6 4.4 5 7.9 5

Table 2: Best-fit values of E for air mixtures, K N2 O2 NO

N2 O2 NO N O Ar 6 2 1 500 4 1 150 40 150 10 17000 200 20 1500 4500 200 16000 1100

Table 3: Best-fit values of SV T for air mixtures N2 O2 NO

N2 0.9 1/3 0.9

O2 NO N O 0.95 0.75 0.99 0.175 0.99 1/3 0.25 0.2 0.2 0.03 0.32 0.06

Ar 0.8 0.99 0.1

4 Conclusions A kinetic theory-based definition of VT relaxation times is used to calculate the vibrational relaxation times for air molecules in collisions with air species (including argon atoms) in a wide temperature range. The parameters of the interaction model used to compute the prob-

23



ability of vibrational-translational energy transitions are adjusted to provide better agreement with available experimental and numerical data. The set of parameters obtained in the present work can be used to compute vibrational relaxation times for air mixtures in a wide temperature range and improve the fidelity of CFD modeling of strongly non-equilibrium air flows, especially when used in conjunction with the modified Landau–Teller formula. The parameters obtained may also be used to state-specific VT transition rates and cross-sections. While the model for the vibrational transition probability used in the present work may not be suitable for some of the interactions considered, since it cannot capture complex quantum chemistry effects, it can be used in the absence of more accurate state-specific models for the VT transition cross-sections and rates.

5 Acknowledgments This work has been supported by the Russian Foundation of Basic Research, project no 18-01-00493. The author would also express his gratitude to Prof. Kustova E.V. of the Saint-Petersburg State University for helpful discussions and advice.

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O2+O, FHO O2+O, M-W

1062

O2+O, M-W, Park O2+O, Thivet O2+O, Kiefer & Lutz (exp.) O2+O, Breen (exp.) O2+O, Shatalov (exp.) O2+O, Kalogerakis (exp.) O2+O, Boyd (QCT) O2+O, Grover (DMS) O2+O, Esposito (QCT)

1063

1064

1065

lgpτOVT2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1066

1067

1068

0.04

0.06

0.08

T 61/3

33

0.10

0.12


0.14

Calculation of Vibrational Relaxation Times Using a Kinetic Theory

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