Direct molecular simulation of oxygen dissociation across normal shocks

Abstract

We present direct molecular simulations (DMSs) of rovibrational excitation and dissociation of oxygen across normal shock waves. These are the first atomistic simulations of normal shock waves to rely exclusively on ab initio potential energy surfaces to describe the full collision dynamics (accounting for elastic, inelastic and reactive processes) in a dilute gas mixture of molecular and atomic oxygen. The simulated setup aims to reproduce two of the experimental conditions in the shock tube tests of Ibraguimova et al. (J Chem Phys 139:034317, 2013). We compare mixture composition and vibrational temperatures extracted from our simulations with those inferred from the shock tube tests and observe good agreement. In addition to this, we report macroscopic moments due to dissipative transport across the shock, i.e., species diffusion fluxes, viscous stresses and heat flux. Furthermore, we examine the distributions of vibrational and total internal energy of the $\hbox {O}_{2}$ molecules at several locations across the shock wave. We are able to follow the gradual transition from pre-shock to post-shock population distributions and, in both cases studied, find depletion of the high-energy tail of the internal energy distributions due to preferential dissociation from states close to the dissociation energy $D_0$. Finally, we extract molecular velocity distributions functions (VDF) of $\hbox {O}_{2}$ and O at selected locations across the shock to delimit the region where continuum breakdown occurs.

Appendices

Collision-speed dependent $b_\text {max}$ and collision cross section

In the DMS method we require the maximum impact parameter for two main purposes: First, to establish the range $0 \le b \le b_\mathrm{max}$ between which to randomly sample the impact parameter b in our trajectory calculations. Second, for defining the total collision cross section $\sigma ^\mathrm{I} = \pi \, b_\mathrm{max}^2$ to be used in calculating the collision frequency $\nu _{i} = \sum _{j = \{ \hbox {O}_{2}, \text {O}\}} n_j \langle \sigma ^\mathrm{I} \cdot g \rangle _{ij}$⁹ in each DMS cell.

The majority of prior DMS studies [25, 44, 45] have been carried out with a constant $b_\mathrm{max}$ (typically $6 \, \text {\AA }$). This practice can be traced back to QCT calculations of dissociating nitrogen [66] on similar ab initio potentials. It was adopted when DMS calculations were performed primarily to study vibrational excitation and dissociation in space-homogeneous, isothermal heat baths. During such studies the reservoir would be kept at a constant, relatively high gas temperature (typically $T = 5000 \, \mathrm {K}$ or higher), while the internal degrees of freedom, represented by the rotational and vibrational temperatures would rise over time from their low initial values. Simultaneously, a fraction of the most energetic collisions would cause molecules to dissociate. In such scenarios, the main effects that the DMS calculations were meant to capture were the rates of inelastic collisions between molecules, leading either to vibrational excitation, or dissociation of molecules. The logic behind choosing the range $0 \le b \le 6 \, \text {\AA }$ was that it would be wide enough for the Monte Carlo sampling routines to pick up most trajectories resulting in significant internal energy exchange and/or dissociation. Conversely, the reasoning was that, as the impact parameter was increased any further, most collisions would be of the glancing type, i.e., interactions in which the colliding molecules would not come into close enough contact to significantly alter their internal energy states. Since in these space-homogeneous reservoir studies the rate of elastic, or near-elastic collisions was of secondary concern, accurately capturing their numbers was not a priority, and the cap of $b_\mathrm{max} = 6 \, \text {\AA }$ was sufficient at the time.

By contrast, the shock wave calculations presented in this paper involve flow fields with steep gradients in chemical species concentration, velocity and temperature. Under such nonhomogeneous conditions, diffusive transport phenomena become important, and such processes are driven to a great extent by elastic collisions. Moreover, the range of gas temperatures covered by these calculations reaches much lower than in previous studies. In particular, the pre-shock gas enters the domain at room temperature, whereas behind the shock the temperature rises by several thousand kelvin. This affects the local collision rate in the gas: At room temperature, a large proportion of collisions occur at low relative speeds g. This allows the colliding molecules to “feel” each other’s influence over much longer times and effectively extends the range of influence of the inter-atomic potential compared to when the molecules fly past each other at high g. This, in turn, means that in order to reliably capture the effect of such low-speed collisions we must extend the range of impact parameters beyond $6 \, \text {\AA }$. Conversely, it means that at high collision energies we may be able to reduce $b_\mathrm{max}$ below $6 \, \text {\AA }$ and still capture a representative sample of all collisions. In order to accomplish this, we must effectively make $b_\mathrm{max}$ a function of g. It should be recalled that this issue was discussed early on in the development of the CT-DSMC/DMS method [30], and here we follow a similar approach.

Recall that the total cross section $\sigma _{ij}^\mathrm{I} ( g )$ is obtained by integrating the differential cross section $\sigma _{ij} = b / \sin \chi \left| \partial b / \partial \chi \right|$ over all post-collision scattering angles $\chi$ in the collision plane, or equivalently by integrating over all pre-collision impact parameters b:

$$\begin{aligned} \sigma _{ij}^\mathrm{I} ( g ) = 2 \pi \int _0^{\pi } \sigma _{ij} ( g, \chi ) \sin \chi \, \mathrm{d} \chi = 2 \pi \int _{0}^{\infty } b \, \mathrm{d} b, \quad i,j = \{ \hbox {O}_{2}, \text {O} \} \end{aligned}$$

(5)

In the context of classical trajectories, sampling the pre-collision impact parameter b is the more straightforward approach. However, the right-hand side of Eq. (5) implies that the integration interval actually extends from zero to $+\infty$ and means that, at least formally, the total cross section would become infinite. In practice, we cut off the integration at a suitable, but large enough $b_\mathrm{max}$, limiting the total cross section to a finite value $\sigma ^\mathrm{I} = \pi \, b_\mathrm{max}^2$. We thus disregard all trajectories at impact parameters greater than $b_\mathrm{max}$ under the assumption that they produce negligible deflection angles and thus do not noticeably affect the microscopic state of the gas. As just discussed, a reasonable value had so far been a constant $b_\mathrm{max} = 6 \, \text {\AA }$, regardless of the species pairing, or relative collision speed.

Since we are primarily concerned with all those trajectories, which produce enough deflection to noticeably influence the transport fluxes, an alternative procedure is used instead to define a minim cut-off angle $\chi _\mathrm{min}$. We then retain all trajectories within the range $\chi _\mathrm{min} \le \chi \le \pi$ and disregard all, which lie below this threshold. In order to enforce this behavior, we must therefore replace the constant $b_\mathrm{max}$ with the function $b_{\mathrm{max}, \, ij} (g \, | \, \chi _\mathrm{min})$, i.e., the maximum impact parameter to produce the minimum deflection angle $\chi _\mathrm{min}$ as the upper limit for the impact parameter. As indicated, this function will implicitly depend on the colliding species pairing ij through the inter-molecular potential and on the relative collision speed g. From a physical viewpoint this is a more reasonable approach. However, from a practical viewpoint it complicates things, because for complex multi-body interaction potentials, such as the $\hbox {O}_{4}$ and $\hbox {O}_{3}$ ab initio PESs we use in this work, the relationship between deflection angle $\chi$ and impact parameter b is not easily known a priori.¹⁰ In fact, obtaining $b_{\mathrm{max}, \, ij} (g \, | \, \chi _\mathrm{min})$ is merely an intermediate step to computing the transport properties for $\hbox {O}_{2}$–$\hbox {O}_{2}$ and $\hbox {O}_{2}$–$\hbox {O}$ interactions. If done in the most rigorous manner, it would require extensive QCT calculations on the $\hbox {O}_{4}$ and $\hbox {O}_{3}$ potentials over the full range of collision parameters (including relative collision energies $\frac{1}{2} \mu _{ij} g^2$, but also pre-and post-collision internal energy states of $\hbox {O}_{2}$ molecules involved) until statistical convergence was established for the relevant transport cross sections $Q_{ij}^l (g) = 2 \pi \int _0^{b_{\mathrm{max},ij}(g)} (1 - \cos ^l \chi ) b \, \mathrm{d} b$, $(l = 1,2)$. Indeed, recent calculations of vibrationally resolved transport properties for the $\hbox {O}_{2}$–$\hbox {O}$ system based on the UMN PESs [43] by Subramaniam et al. [67] highlight the major effort involved. This is clearly too computationally costly and time-consuming for our current purposes and we resort to a much simpler approach instead.

We assume that we can describe elastic scattering between $\hbox {O}_{2}$–$\hbox {O}_{2}$, O–$\hbox {O}_{2}$ and O–O pairs with a spherically symmetric potential $\varphi _{ij} (r)$ in a sufficiently accurate manner, even after having replaced the multi-body ab initio potentials by a phenomenological one. A reasonable choice is to select a potential and model parameters that have been used to compute transport properties before. Here we have opted for the Lennard–Jones potential $\varphi _{ij}^\mathrm{LJ} (r) = 4 \varphi _{ij}^0 \left( (r_{ij}^0 / r)^{12} - (r_{ij}^0 / r)^{6} \right)$ with the species-dependent parameters $\varphi _{ij}^0$ and $r_{ij}^0$ taken from Capitelli et al. [68] (also see first two columns of Table 3).

The Lennard–Jones potential is convenient for our purposes, because it is able to reproduce both the repulsive forces at close range and the attractive forces at longer inter-molecular distances. We now compute the deflection angle as a function of relative collision speed and impact parameter through numerical integration of the expression:

$$\begin{aligned} \chi _{ij} ( g, b ) = \pi - 2 \, b \int _{r_\mathrm{min}}^{\infty } \frac{\hbox {d} r}{r^2 \sqrt{1 - \varphi _{ij}^\mathrm{LJ} (r) / \frac{1}{2} \mu _{ij} g^2 + (b / r)^2 }}, \quad i,j = \{ \hbox {O}_{2}, \text {O}\}, \end{aligned}$$

(6)

where $r_\mathrm{min}$ is the distance of closest approach between both particles and $\mu _{ij} = m_i \, m_j / (m_i + m_j)$ is the collision pair’s reduced mass. The deflection angle can be better visualized by plotting individual trajectories in the collision plane, as is done in Fig. 18. Here, a fixed velocity $g = 300 \, \mathrm {m/s}$ was chosen and two sets of trajectories at impact parameters $b_1$ and $b_2$ are shown with their corresponding deflection angles $\chi _1$ and $\chi _2$. Equation (6) yields the deflection angle directly and produces values outside the range [0,$\pi$] (or [0,$180^{\circ }$]) for trajectories, which may loop around the collision center at the origin. Such trajectories are produced by the action of the attractive part of the potential, and in extreme cases may result in several “orbits” around the origin. These values of $\chi$ must be “corrected” afterward to lie between 0 and 180$^{\circ }$. Thus, after numerically computing $\chi _{ij} ( g, b )$ within the ranges $30 \le g \le 30{,}000 \, \mathrm {m/s}$ and $0 \le b \le 210 \, \text {\AA }$ for all species pairings, we obtain curves of the type shown in Fig. 19. There we have plotted the corrected deflection angle $\chi _\mathrm{corr}$ as a function of b for three distinct values of g. At the highest value of g shown, the deflection angle starts from its maximum of $180^\circ$ at $b = 0$ and decreases from there in a monotonous fashion. This is the typical behavior for purely repulsive interactions. By contrast, at the lowest collision speed shown, several “kinks” appear in the corrected $\chi$-curve and the deflection angle rises again at larger values of b. These correspond to collisions, where the trajectories actually produce negative deflection angles, such as was the case for $\chi _2$ in Fig. 18. Such reversals in the deflection angle are due to the action of the weaker, attractive part of the potential and only appear at lower collision speeds. However, there they contribute to push the outermost intersection point between the $\chi$-curves and our arbitrarily chosen cut-off value $\chi _\mathrm{min} = 1^\circ$ (open circles in Fig. 19) to much larger values of b. It is precisely this outermost intersection point that marks the location of $b_\mathrm{max}$, subject to the cut-off angle at the relative collision speed.

We determine the values for $b_{\mathrm{max}, \, ij} (g \, | \, \chi _\mathrm{min})$ numerically for all three species pairings and plot them in the lower half of Fig. 20 as a function of g (solid lines in color). The dashed lines of corresponding color represent analytical fits to these curves of the type:

$$\begin{aligned} b_{\mathrm{max}, \, ij} (g, | \chi _\mathrm{min} = 1^\circ ) = \left\{ \begin{array}{l} A_{ij} \, g^{m_{ij}} \quad \text {if} \, g < g_{0, ij} \\ B_{ij} \, g^{n_{ij}} \quad \text {otherwise} \end{array} \right. , \qquad i,j = \{ \hbox {O}_{2}, \text {O} \}. \end{aligned}$$

(7)

The corresponding fit parameters for all species pairings have been collected in Table 3.

Table 3 Lennard–Jones parameters taken from Table 1 of Ref. [68] and resulting fit parameters for $b_{\mathrm{max}, ij} \left( g, | \chi _\mathrm{min} = 1^\circ \right)$ to be used with Eq. (7)

Fig. 18

Pair trajectories for $\hbox {O}_{2}$–$\hbox {O}_{2}$ Lennard–Jones potential in x, y-collision plane at $g = 300 \, \mathrm {m/s}$ for two different impact parameters $b_1$ and $b_2$ produce deflection angles $\chi _1$ and $\chi _2$

Fig. 19

Deflection angle $\chi _\mathrm{corr} (g,b)$ for $\hbox {O}_{2}$–$\hbox {O}_{2}$ Lennard–Jones potential. Values have been corrected to lie between 0$^{\circ }$ and 180$^{\circ }$. Dashed horizontal line marks chosen cut-off angle $\chi _\mathrm{min} = 1^{\circ }$. Circles mark outermost intersection of $\chi$ and $\chi _\mathrm{min}$ for every value of g (color figure online)

These fits follow one type of behavior at relative speeds below the chosen $g_{0, ij}$ and switch to the second, shallower slope at higher speeds. As can be seen in Fig. 20, toward the lower end of the collision speed range, the maximum impact parameter (and with it the total cross section) increases rapidly, and lies well above $6 \, \text {\AA }$ (horizontal dashed line). Conversely, toward higher collision speeds, the fitted $b_\mathrm{max}$-curves drop below the $6 \, \text {\AA }$-line. As a complement to the $b_\mathrm{max}$-curves, in the upper half of Fig. 20 we plot the function:

$$\begin{aligned} g \cdot f_{ij}^0 (g, T) = \frac{4}{\sqrt{\pi }} \left( \frac{\mu _{ij} g^2}{2 \mathrm {k_B} T} \right) ^{3/2} \exp \left( \frac{-\mu _{ij} g^2}{2 \mathrm {k_B} T} \right) , \quad i,j = \{ \hbox {O}_{2}, \text {O} \}, \end{aligned}$$

(8)

which governs the distribution of relative speeds in ij-collisions of a gas in thermal equilibrium. When evaluated for a particular species pairing, Eq. (8) reveals the range of collision speeds that are most probable at a given temperature T. As can be seen from Fig. 20, collisions at lower temperatures are weighted toward lower collision speeds. In particular, for a gas at room temperature, such as the free-stream gas in our shock calculations, this range lies roughly between 30 and $3000 \, \mathrm {m/s}$. The lower part of Fig. 20 shows that at this temperature $b_\mathrm{max}$ is significantly larger than $6 \, \text {\AA }$. Conversely, as the temperature is raised to $3000 \, \mathrm {K}$ or higher, the weight of the distributions shifts to the right, toward larger collision speeds. At these higher values of g, the $b_\mathrm{max}$-curves lie well below $6 \, \text {\AA }$, and the total cross section can be reduced accordingly without loss of accuracy. This noticeably reduces the number of required trajectory calculations in DMS cells belonging to the high-temperature post-shock region, and contributes a significant speedup of the calculation as a whole.

Fig. 20

Collision speed distribution $g \cdot f_{ij}^0 (g,T)$ as a function of species pairing and equilibrium temperature T (upper half). Species-pair-dependent maximum impact parameter as a function of collision speed g (lower half). Solid colored lines represent $b_\mathrm{max} (g, | \chi _\mathrm{min} = 1^{\circ })$ calculated numerically, corresponding dashed colored lines are analytical fits to the data. Black horizontal dashed line represents $b_\mathrm{max} = 6 \, \text {\AA }$ (color figure online)

With expressions for $\sigma _{ij}^\mathrm{I} (g) = \pi b_{\mathrm{max}, ij}^2 (g)$ known, and together with Eq. (8), it is possible to compute the corresponding thermally averaged collision cross sections $\langle \sigma \rangle _{ij} (T) = \langle \sigma _{ij}^\mathrm{I} \cdot g \rangle _{ij} / \langle g \rangle _{ij}$, where $\langle g \rangle _{ij} = \sqrt{8 \mathrm {k_B} T / \pi \mu _{ij}}$ is the thermally averaged collision speed for species pair ij. To this end, we first compute:

$$\begin{aligned} \langle \sigma ^\mathrm{I} \cdot g \rangle _{ij} (T) = \int _0^\infty \, \pi b_{\mathrm{max}, ij}^2 (g) \cdot g \, f_{ij}^0 (g, T) \, \mathrm{d} g, \quad i,j = \{ \hbox {O}_{2}, \text {O} \}, \end{aligned}$$

(9)

which for a constant $b_{\mathrm{max}, ij}$ yields simply $\langle \sigma \cdot g \rangle _{ij} (T) = \pi b_\mathrm{max}^2 \langle g \rangle _{ij}$. For the functional form of Eq. (7), one obtains the slightly more complex expression:

$$\begin{aligned} \langle \sigma ^\mathrm{I} \cdot g \rangle _{ij} (T) = \left[ \left( \frac{2 \mathrm {k_B} T}{\mu } \right) ^{m} \pi A^2 \gamma \left( m + 2, x_0 \right) + \left( \frac{2 \mathrm {k_B} T}{\mu } \right) ^{n} \pi B^2 \varGamma \left( n + 2, x_0 \right) \right] _{ij} \langle g \rangle _{ij}, \end{aligned}$$

(10)

where $\gamma \left( s, x \right) = \int _0^x t^{s-1} \exp ( -t ) \mathrm{d}t$ and $\varGamma \left( s, x \right) = \int _x^\infty t^{s-1} \exp ( -t ) \mathrm{d}t$ are the lower and upper incomplete gamma functions, respectively, and the intermediate integration limit is $x_{0, ij} = \mu _{ij} g_{0, ij}^{2} / 2 \mathrm {k_B} T$. Plugging in the values from Table 3 into Eq. (10) and dividing by $\langle g \rangle _{ij}$ produces the thermally averaged collision cross section for each species pairing for our Lennard–Jones fit. The resulting thermal cross sections for $\hbox {O}_{2}$–$\hbox {O}_{2}$ (solid red line), $\hbox {O}_{2}$–$\hbox {O}$ (solid green line) and O–O (solid blue line) using our speed-dependent $b_\mathrm{max}$ are shown in Fig. 21, along with the constant average cross section one obtains for $b_\mathrm{max} = 6 \, \text {\AA }$ (dashed black line). Strictly speaking, these curves are only valid in a gas at thermal equilibrium. Nevertheless, they allow one to quickly estimate the cross section across the flow field based on the local kinetic temperature. Moreover, Eq. (10) can be used to estimate the local collision rate and mean free path in the flow.

Fig. 21

Thermally averaged collision cross sections as a function of equilibrium temperature. Solid colored lines represent curves obtained with $b_{\mathrm{max}, ij} (g, | \chi _\mathrm{min} = 1^{\circ })$ from Eq. (10) and the analytical fit parameters for each species pair from Table 3. Black dashed line shows cross section for $b_\mathrm{max} = 6 \, \text {\AA }$. Dotted colored lines: equivalent VHS cross sections based on parameters from Wysong et al. [14] (color figure online)

To wrap up this discussion, we should recall that the total cross section fulfills a slightly different function in the DMS method than in standard DSMC. In both methods, $\sigma ^\mathrm{I}$ is used to determine the number of pairs to test for, and subsequently determine their probability of collision (e.g., by means of Bird’s NTC [18], or some equivalent scheme). In DSMC, any accepted pairs will then typically collide according to the Variable Hard Sphere [69] (VHS), or Variable Soft Sphere [70] (VSS) models. In any case, both particles involved in a DSMC collision will definitely have their velocities changed, transferring a finite amount of momentum and energy in the process. In DMS, although the pair selection procedure is the same, the final velocities (and internal energies) of both particles are always determined by on-the-fly classical trajectory calculations on the inter-atomic PES, as opposed to a prescribed scattering law. The final velocities of both collision partners are thus governed by the PES, together with the impact parameter b randomly sampled from within the interval $[0,b_\mathrm{max}]$. For small b, trajectory calculations will typically yield larger scattering angles and consequently greater transfer of momentum and energy. For larger b, trajectories will more likely be of the glancing type, only slightly changing both particle’s velocities. Although such trajectories are individually less effective in changing the overall gas state, they become more abundant as the impact parameter increases. Thus, their overall effect must be accounted for in the DMS collision selection scheme. In the limit of very large b though, a trajectory calculation will produce no change in both particle’s states, effectively becoming a fly-by. Such fly-by trajectories, though still costly in terms of computational time, do not contribute at all to the transfer of momentum and energy in the gas and one wishes to avoid them by imposing a reasonable upper limit for b. Thus, in DMS the total cross section $\sigma ^\mathrm{I} = \pi b_{\mathrm{max}}^2$ needs to be “conservatively large” to account for the momentum and energy transferred in the less effective glancing-type collisions, but not “too large” so as to avoid producing too many fly-by trajectories. Our arbitrary choice was to set the cut-off at a $b_{\mathrm{max}}$, which would include all trajectories producing deflections with $\chi \ge 1^{\circ }$ on the Lennard–Jones potential. As a consequence the total cross sections in our DMS calculations are noticeably larger than equivalent ones used in DSMC, because in the latter method every single collision is more efficient in transferring momentum and energy. The difference is clearly seen in Fig. 21, where the thermally averaged cross sections applicable to our DMS calculations (solid lines) are plotted together with equivalent ones based on the VHS parameters of Wysong et al. [14] (dotted lines).

Fig. 22

Effect of varying maximum impact parameter on shock structure: Temperatures (continuous lines, left ordinate axis) T, $T_\mathrm{rot}$, $T_\mathrm{vib}$ (K) and collision rate per unit volume (dotted lines, right ordinate axis) $N_\mathrm{coll}$ ($\mathrm{s}^{-1} \cdot \mathrm{m}^{-3}$) obtained with $b_\mathrm{max} = 6 \, \text {\AA }$ (gray) vs. species-pair-dependent $b_\mathrm{max} \left( g, | \chi _\mathrm{min} = 1^{\circ } \right)$ (orange) (color figure online)

It should also be reiterated that, although the functional dependence of $b_{\mathrm{max}, ij} (g \, | \, \chi _\mathrm{min} = 1^{\circ })$ (and the derived total cross section $\sigma _{ij}^\mathrm{I} = \pi b_{\mathrm{max}, ij}^2$) in this section was calculated for a Lennard–Jones potential and the parameters of Ref. [68], the actual trajectory calculations in our DMS code are still done exclusively on the UMN ab initio PESs [41,42,43] for oxygen. As mentioned before, this is a valid approach as long as $\chi _\mathrm{min}$ is chosen to be conservatively small (simultaneously making $b_\mathrm{max}$ conservatively large). Thus, relying on the spherically symmetric Lennard–Jones potential should be understood as being merely a convenient, quicker way to obtain reasonable estimates for the associated $b_\mathrm{max}$. If the aforementioned conditions are met, all thermochemistry and transport phenomena observed in the gas are still determined by the properties of the ab initio PESs and not those of the Lennard–Jones potential.

The overall effect of using our collision-speed-dependent $b_\mathrm{max}$, as opposed to the constant one of $6 \, \text {\AA }$, on the simulations is illustrated by the two sets of DMS results in Fig. 22. Here we plot profiles of temperature (solid lines, left ordinate axis) and estimated average collision rate per unit volume $N_\mathrm{coll}$ (dotted lines, right ordinate axis) across the high-enthalpy shock. The solution for $b_\mathrm{max} = 6 \, \text {\AA }$ (in gray) exhibits a much thinner shock front than the one with the collision-speed-dependent $b_\mathrm{max}$ (in orange). As would be expected, the greatest differences in the temperature profiles are observed in the upstream portion of the shock, where the gas temperature is lowest and the variable $b_\mathrm{max}$ becomes noticeably larger than $6 \, \text {\AA }$. This difference in the choice of $b_\mathrm{max}$ and the associated collision cross section $\pi b_\mathrm{max}^2$ also has an effect on the estimated collision rate in the gas, as shown by the dotted lines. Whereas in the colder, upstream part of the flow field the collision rate for the $6 \, \text {\AA }$-case (in gray) is about half that of the variable $b_\mathrm{max}$ one (in orange), the situation is reversed downstream of the shock front, in the high-temperature region. This is consistent with Fig. 20, where it can be seen that the variable $b_\mathrm{max}$ for all three species pairs fall below the $6 \, \text {\AA }$-mark at the higher collision speeds more common at temperatures of 3000 K and above, thus ultimately yielding smaller collision cross sections and associated collision rates.

Diatomic potential and thermodynamic properties of $\hbox {O}_{2}$

In Sect. 3.1 it was discussed how the post-shock equilibrium conditions were estimated by applying the Rankine–Hugoniot jump relations for a reacting gas mixture. This requires knowledge of the thermodynamic properties (species heat capacities, enthalpies and chemical equilibrium constants) for the mixture in question. As mentioned in Sect. 3.1, the precise values of these properties determine the predicted equilibrium state behind the shock and, depending on the thermodynamic source data, can produce different answers (e.g., compare rows labeled “CEA” and “UMN PES” in Table 1). Here we summarize how the thermodynamic properties for the ab initio UMN PESs [41,42,43] were derived from the associated diatomic potential $V_\mathrm{O-O} (r)$. Keep in mind that these potential surfaces were constructed only for trajectory calculations between oxygen molecules and atoms in their ground electronic states. Thus, the properties calculated in this section also only apply to mixtures comprised of $\hbox {O}_{2}(\hbox {X} ^3\varSigma ^{-}_g)$ and $\hbox {O}(^3\hbox {P})$, with all constituting atoms assumed to be of the most common ${}^{16}\hbox {O}$ isotope. With $V_\mathrm{O-O} (r)$ known (pairwise term $V_\mathrm{PA}(r)$ from Sec. B of Paukku et al. [41]), we first compute the effective diatomic potential:

$$\begin{aligned} V_\mathrm{eff} (r, J) = V_\mathrm{O-O} (r) + \frac{J \left( J + 1 \right) \hbar ^2}{2 \, \mu _\mathrm{O-O} \, r}, \qquad J \in \mathcal {J}_{\hbox {O}_{2}}, \end{aligned}$$

(11)

where r is the distance between both O-nuclei, $\mathcal {J}_{\hbox {O}_{2}} = \{ 0,1, \ldots , J_\mathrm{max} \}$ is the set of discrete rotational quantum numbers spanning all energy levels supported by the potential. The factor $\hbar = \mathrm {h_P} / 2 \pi$ is the reduced Planck constant and $\mu _\mathrm{O-O} = m_{\text {O}}/2$ is the reduced mass of the two nuclei constituting the oxygen molecule. This effective diatomic potential yields 6115 discrete rovibrational energy levels for oxygen with discrete energies $V_{(J,v)}$ in the Wentzel–Kramers–Brillouin approximation. Out of the full set, 4581 bound rovibrational levels possess energies below $D_0$, whereas the remaining 1534 quasi-bound levels lie above the dissociation limit. Each level is associated with a unique pair of rotational and vibrational quantum numbers (J, v) ranging from $J_\mathrm{min} = 0$ to $J_\mathrm{max} = 240$ and $v_\mathrm{min} = 0$ to $v_\mathrm{max} = 44$. When measured from the bottom of the rotationless potential curve $V_\mathrm{O-O} (r)$, the lowest-lying rovibrational level is found at an energy $V_{(J=0,v=0)} = 0.0982 \, \mathrm {eV}$, whereas the energy for the two O-atoms at infinite separation lies at $V_\infty = 5.2113 \, \mathrm {eV}$ above the well minimum. We therefore obtain a dissociation energy equal to $D_0 = V_\infty - V_{(J=0,v=0)} = 5.113 \, \mathrm {eV}$ from the lowest-lying rovibrational level.

Figure 23 shows several effective potential curves as a function of the inter-nuclear distance for a selection of rotational quantum numbers J. The rovibrational energies $V_{(J,v)}$ of individual levels are shown as horizontal lines bounded by the effective potential curve on both sides. These locations mark the inner and outer turning points ($r_\mathrm{min}$,$r_\mathrm{max}$) for the molecule’s vibrational motion at that particular energy. Several of the rovibrational levels on the $J = 0$-curve (red lines) have been labeled with their vibrational quantum numbers v. Notice that the range spans the full $v = 0 - 44$ for the rotationless potential, but becomes narrower as J is increased. The spacing between the first and second vibrational levels on the $J=0$-curve yields a “characteristic temperature for vibration” (equivalent to the one in the harmonic oscillator model) of $\theta _\mathrm{vib} = (E_{(v=1,J=0)} - E_{(v=0,J=0)})/\mathrm {k_B} = 2251.4 \, \mathrm {K}$. Conversely, a “characteristic temperature for rotation” (equivalent to what is used in the rigid rotor approximation) is found to be $\theta _\mathrm{rot} = (E_{(v=0,J=1)} - E_{(v=0,J=0)})/(2 \mathrm {k_B}) = 2.0676 \, \mathrm {K}$. These values are in close agreement with the ones from literature [71]. But it should be kept in mind that the energies obtained with these simpler models only agree with the PES for the lowest-lying rovibrational levels and become less accurate for higher (v, J).

Fig. 23

Effective diatomic potential curves as a function of inter-nuclear distance r and rotational quantum number J

Figure 24 shows how the rovibrational energies are distributed across rotational and vibrational quantum numbers. Notice that a maximum vibrational quantum number $v_\mathrm{max}(J)$ exists for each rotational level, or conversely, there exists a particular $J_\mathrm{max}(v)$ for each vibrational level. These upper limits for (J, v) pairings are a consequence of the constraints imposed by the effective diatomic potential, beyond which no stable molecule can be formed. As would be expected, low (J, v) correlate with the lowest-lying energy levels and their energies tend to rise when both J and v increase. Notice the red line in Fig. 24 separating bound levels to its left from quasi-bound levels to its right. Notice also that levels with the highest overall energy are found in a combination of very low v, but very high J quantum numbers (dark orange region). These tend to be molecules exhibiting the most pronounced potential energy barriers (e.g., see local maxima in orange ($J=200$) and black ($J=240$) curves of Fig. 23).

Fig. 24

Distribution of rovibrational energy levels $E_{(J,v)}$ of $\hbox {O}_{2}$ over rotational and vibrational quantum numbers J and v. Color-coded contour levels show approximate distribution of energy as a function of (v, J) coordinates. Red dash-dotted line at $D_0 = 5.113 \,\mathrm{eV}$ separates bound levels to the left from quasi-bound levels to the right (color figure online)

With the rovibrational energy levels $V_{(J,v)}$ and dissociation energy $D_0$ known, we are able to compute the thermodynamic properties of molecular oxygen in the ground electronic state. For convenience, we define a new energy scale as $E_{(J,v)} = V_{(J,v)} - V_{(J=0,v=0)}$, which shifts the origin to the lowest-lying rovibrational level, i.e., $E_{(J=0,v=0)} = 0 \, \mathrm {eV}$. For molecular oxygen in the ground state (i.e., $\hbox {O}_{2}(^3 \varSigma _g^-)$) we have $Q_{\hbox {O}_{2}}^\mathrm{el} = 3$. The electronic partition function for atomic oxygen in the ground state (i.e., triplet oxygen) is $Q_{\text {O}}^\mathrm{el} = 5 + 3 \exp \left( -227.7 \, \mathrm {K} / T \right) + \exp \left( -326.6 \, \mathrm {K} / T \right)$ when one accounts for spin-orbit coupling. However, the PESs used in this work explicitly neglect spin-orbit coupling, which implies that the fine structure states $\text {O} (^3P_2)$, $\text {O} (^3P_1)$ and $\text {O} (^3P_0)$ are degenerate. Thus, $Q_{\text {O}}^\mathrm{el}$ simplifies to $5 + 3 + 1 = 9$. Any further contributions of electronic excited states of atomic, or molecular oxygen to the thermodynamic properties derived from the UMN PESs are assumed to be zero.

With this in mind, the temperature-dependent enthalpies per unit mole of $\hbox {O}_{2}$ and O can then be computed as:

$$\begin{aligned} {\hat{H}}_{\hbox {O}_{2}}(T)&= \frac{5}{2} {\bar{R}} \left( T - T_\mathrm{ref} \right) + \mathcal {N}_A \, \left( \frac{A_{\hbox {O}_{2}}^\mathrm{int}(T)}{Q_{\hbox {O}_{2}}^\mathrm{int}(T)} - \frac{A_{\hbox {O}_{2}}^\mathrm{int}(T_\mathrm{ref})}{Q_{\hbox {O}_{2}}^\mathrm{int}(T_\mathrm{ref})} \right) + \Delta _f {\hat{H}}_{\hbox {O}_{2}, T_\mathrm{ref}}^0 \end{aligned}$$

(12)

$$\begin{aligned} \text {and} \quad {\hat{H}}_{\text {O}}(T)&= \frac{5}{2} {\bar{R}} \left( T - T_\mathrm{ref} \right) + \Delta _f {\hat{H}}_{\text {O}, T_\mathrm{ref}}^0, \end{aligned}$$

(13)

where ${\bar{R}} = \mathcal {N}_A \, \mathrm {k_B} = 8.314 \, \mathrm {J \, mol^{-1} \, K^{-1}}$, or $1.987 \times 10^{-3} \, \mathrm {kcal \, mol^{-1} \, K^{-1}}$ is the universal gas constant, $\mathcal {N}_A = 6.022 \times 10^{23} \, \mathrm {mol^{-1}}$ is Avogadro’s number and $T_\mathrm{ref} = 298.15 \, \mathrm {K}$ is a conveniently chosen reference temperature. The temperature-dependent function:

$$\begin{aligned} Q_{\hbox {O}_{2}}^\mathrm{int}( T ) = \sum _{v \in \mathcal {V}_{\hbox {O}_{2}}} \sum _{J \in \mathcal {J}(v)} \left\{ g_{\hbox {O}_{2}}^\mathrm{NS}(J) \, g_J \exp \left( \frac{-E_{(v,J)}}{\mathrm {k_B} T} \right) \right\} \end{aligned}$$

(14)

is the partition function for the coupled rotational and vibrational degrees of freedom of $\hbox {O}_{2}$, along with its derived functions $A_{\hbox {O}_{2}}^\mathrm{int}( T )$ $= \sum _{v \in \mathcal {V}_{\hbox {O}_{2}}} \sum _{J \in \mathcal {J}(v)}$ $\{ g_{\hbox {O}_{2}}^\mathrm{NS}(J) \, g_J \, E_{(v,J)}$ $\exp ( -E_{(v,J)} /\, \mathrm {k_B} T ) \}$ and $B_{\hbox {O}_{2}}^\mathrm{int} ( T )$ $= \sum _{v \in \mathcal {V}_{\hbox {O}_{2}}} \sum _{J \in \mathcal {J}(v)}$ $\{ g_{\hbox {O}_{2}}^\mathrm{NS}(J) \, g_J \, E_{(v,J)}^2$ $\exp ( -E_{(v,J)} /\, \mathrm {k_B} T ) \}$. Rotational levels are have degeneracy $g_J = 2J + 1$ and the degeneracy due to nuclear spin of $\hbox {O}_{2}$ molecules is coupled to their rotational state. It is $g_{\hbox {O}_{2}}^\mathrm{NS}(J) = 0$ for even J and $g_{\hbox {O}_{2}}^\mathrm{NS}(J) = 1$ for odd J. Thus, oxygen molecules populating even J levels are quantum-mechanically forbidden and only levels with odd J are actually populated. Therefore, the sums over J in computing the functions $Q_{\hbox {O}_{2}}^\mathrm{int}$, $A_{\hbox {O}_{2}}^\mathrm{int}$ and $B_{\hbox {O}_{2}}^\mathrm{int}$ should be carried out only over the odd values of J. The nuclear spin degeneracy of atomic oxygen is $g_{\text {O}}^\mathrm{NS} = 1$. The heats of formation at $T_\mathrm{ref} = 298.15 \, \mathrm {K}$ have been set to $\Delta _f {\hat{H}}_{\hbox {O}_{2}, T_\mathrm{ref}}^0 = 0 \, \mathrm {kcal \, mol^{-1}}$ and $\Delta _f {\hat{H}}_{\text {O}, T_\mathrm{ref}}^0 = \frac{1}{2} D_0 \, \mathcal {N}_A = 58.96 \, \mathrm {kcal \, mol^{-1}}$, respectively. These values are in close agreement with the thermodynamic data of the NASA Glenn database, on which CEA [49] relies. The specific heat capacities at constant pressure ${\hat{C}}_{p, \hbox {O}_{2}} = \partial {\hat{H}}_{\hbox {O}_{2}} / \partial T$ and ${\hat{C}}_{p, \text {O}} = \partial {\hat{H}}_{\text {O}} / \partial T$ then are:

$$\begin{aligned} {\hat{C}}_{p, \hbox {O}_{2}}(T) = \frac{5}{2} {\bar{R}} + \frac{\mathcal {N}_A^2}{{\bar{R}} T^2} \left[ \frac{B_{\hbox {O}_{2}}^\mathrm{int}( T )}{Q_{\hbox {O}_{2}}^\mathrm{int}( T )} - \left( \frac{A_{\hbox {O}_{2}}^\mathrm{int}( T )}{Q_{\hbox {O}_{2}}^\mathrm{int}( T )} \right) ^2 \right] \quad \text {and} \quad {\hat{C}}_{p, \text {O}} = \frac{5}{2} {\bar{R}} \end{aligned}$$

(15)

respectively.

In Fig. 25a we compare the species enthalpies derived from the PESs with the output of NASA Glenn’s CEA (labeled CEA). The PES curves are plotted using continuous lines (red for $\hbox {O}_{2}$, blue for O), the NASA Glenn data are shown as dash-dotted lines. Notice that for $\hbox {O}_{2}$ the PES and CEA curves closely agree up to about 3000 K, but then begin to diverge. The CEA curves (dash-dotted red lines), which account for the additional contribution of electronic excited states of $\hbox {O}_{2}$, lie consistently above the corresponding PES curves. For comparison, we also plot the properties of $\hbox {O}_{2}$ as estimated from a combination of rigid rotor and simple harmonic oscillator models with $\theta _v = 2251 \, \mathrm {K}$ in gray. This model further underestimates the energy content of $\hbox {O}_{2}$ relative to the UMN PES and CEA predictions. For atomic oxygen (blue lines), there exist slight discrepancies between the UMN PES and CEA predictions at very low temperature and greater ones at high temperatures. Recall that, implicit in Eqs. (12)–(15) is our assumption that for atomic oxygen only translational energy contributes to the enthalpy/heat capacity, since the current PESs cannot account for electronic excited states of O-atoms and because spin-orbit coupling is being ignored. Therefore, the CEA-curves (dash-dotted blue lines) lie above the PES-curves (solid blue lines) over much of the temperature range, reflecting the contributions from a variety of higher-lying electronic states of O.

The equilibrium constant is required in the calculation of the $\hbox {O}_{2}$,O-mixture composition at the post-shock equilibrium conditions. When expressed in terms of the equilibrium partial pressure ratio of $\hbox {O}_{2}$ and O, it is given by [71]:

$$\begin{aligned} K_p^{\hbox {O}_{2} \rightleftharpoons 2\text {O}} \left( T \right) = \left[ \frac{p_{\text {O}}^2}{p_{\hbox {O}_{2}}} \right] ^\mathrm{eq} = \mathrm {k_B} T \frac{\left[ Q_{\text {O}}^\mathrm{tra} \left( T \right) \, g_{\text {O}}^\mathrm{NS} \, Q_{\text {O}}^\mathrm{el} \right] ^2}{Q_{\hbox {O}_{2}}^\mathrm{tra} \left( T \right) Q_{\hbox {O}_{2}}^\mathrm{int} \left( T \right) Q_{\hbox {O}_{2}}^\mathrm{el}} \exp \left( - \frac{D_0 \, \mathcal {N}_A}{{\bar{R}} T} \right) , \end{aligned}$$

(16)

where the translational mode partition functions per unit volume are $Q_{\hbox {O}_{2}}^\mathrm{tra} = ( 2 \pi \, m_{\hbox {O}_{2}} \mathrm {k_B} T / h_\mathrm{P}^2 )^{3/2}$ and $Q_{\text {O}}^\mathrm{tra} = ( 2 \pi \, m_{\text {O}} \mathrm {k_B} T / h_\mathrm{P}^2 )^{3/2}$, respectively, and $h_\mathrm{P}$ is Planck’s constant. In Fig. 25b we compare the equilibrium constant derived from the PESs (continuous red line) with the NASA Glenn (black dash-dotted line) curve. Both agree closely over most the temperature range shown. Slightly larger discrepancies in $K_p$ are observed when energy levels based on the rigid rotor + harmonic oscillator approximations are used for $Q_{\hbox {O}_{2}}^\mathrm{int}$ in Eq. (16) instead (gray line).

Fig. 25

a Species enthalpies ${(\mathrm{kcal\,mol}^{-1})}$ for $\hbox {O}_{2}$ and O for UMN PESs based on Eqs. (12) and (13), respectively (solid red and blue lines), vs. NASA Glenn CEA database (dash-dotted lines) vs. rigid rotor + harmonic oscillator approximation for $\hbox {O}_{2}$ (gray line). (b) Equilibrium constant $K_p^{\hbox {O}_{2} \rightleftharpoons 2\text {O}}$ (atm) for UMN PESs using Eq. 16 (red) vs. NASA Glenn CEA database (black line) vs. rigid rotor + harmonic oscillator approximation for $\hbox {O}_{2}$ (gray line) (color figure online)

Macroscopic flow field variables extracted from DMS samples

Here we summarize the expressions used to compute all macroscopic flow parameters discussed in Sect. 3. These steady-state profiles are based on the particle samples in each DMS cell, accumulated over all sampling steps. The species mass densities plotted in Figs. 3 and 4 are calculated as:

$$\begin{aligned} \rho _i = W_p \, m_i \, [N_i]_\mathrm{cell} / V_\mathrm{cell}, \qquad i = \{ \hbox {O}_{2}, \text {O} \} \end{aligned}$$

(17)

where $W_p = N^\mathrm{real} / N^\mathrm{sim}$ is the “particle weight factor” relating the number of real gas molecules to DMS simulated particles, $m_i$ is the molecular mass of species i (e.g., $m_{\hbox {O}_{2}} = 5.314 \times 10^{-26} \, \mathrm {kg}$, $m_{\text {O}} = 2.657 \times 10^{-26} \, \mathrm {kg}$), $[N_i]_\mathrm{cell}$ is the number of particles of species $i = \{ \hbox {O}_{2}, \text {O} \}$ in the given DMS cell and $V_\mathrm{cell}$ is the cell volume. The mixture mass density becomes $\rho = \sum _{i = \{ \hbox {O}_{2}, \text {O} \}} \{ \rho _i \}$, while the corresponding number densities follow as: $n_i = \rho _i / m_i$ and $n = \sum _{i = \{ \hbox {O}_{2}, \text {O} \}} \{ n_i \}$, respectively. The Cartesian components of flow velocity (Fig. 17) are calculated in every DMS cell as:

$$\begin{aligned} u_\nu = \frac{1}{\rho } \sum _{i = \{ \hbox {O}_{2}, \text {O} \}} \Bigl \{ \rho _i \langle c_\nu \rangle ^i \Bigr \}, \qquad \nu = \{ x, y, z \}, \end{aligned}$$

(18)

where $\langle c_\nu \rangle ^i = \sum _\mathrm{cell} \left\{ c_\nu \right\} / [N_i]_\mathrm{cell}$ represents the cell-average of all particle velocity components along Cartesian direction $\nu$ belonging to species i. Average velocity components for each mixture component in DMS are simply $u_\nu ^i = \langle c_\nu \rangle ^i$ for $\nu = \{ x, y, z \}$. The components of the species diffusion velocities are obtained as $U_\nu ^i = u_\nu ^i - u_\nu$. The diffusion fluxes of both mixture components in Fig. 9 are then obtained as $j_\nu ^i = \rho _i \, U_{\nu }^i$. The kinetic pressure tensor is the mass-weighted sum over all mixture components’ second-order velocity moments. In DMS its Cartesian components are calculated as:

$$\begin{aligned} \mathcal {P}_{\nu \eta } = \sum \limits _{i = \{ \hbox {O}_{2}, \text {O} \}} \biggl \{ \rho _i \Bigl ( \langle c_\nu \, c_\eta \rangle ^i + u_\nu \, u_\eta - \langle c_\nu \rangle ^i \, u_\eta - u_\nu \, \langle c_\eta \rangle ^i \Bigr ) \biggr \}, \qquad \nu ,\eta = \{ x, y, z \}, \end{aligned}$$

(19)

where $\langle c_\nu \, c_\eta \rangle ^i = \sum _\mathrm{cell} \left\{ c_\nu \, c_\eta \right\} / [N_i]_\mathrm{cell}$ represent the cell-averaged products of the Cartesian velocity components of all particles belonging to species $i = \{ \hbox {O}_{2}, \text {O} \}$. The mixture static pressure is the average of the three normal stress components: $p = \frac{1}{3} \left( \mathcal {P}_{xx} + \mathcal {P}_{yy} + \mathcal {P}_{zz} \right)$. The components of the viscous stress tensor are then obtained as $\tau _{\nu \eta } = p - \mathcal {P}_{\nu \eta }$, for $\nu ,\eta = \{ x, y, z \}$ and the mixture kinetic temperature follows from the equation of state for an ideal gas: $T = p \, / \, \left( n \, \mathrm {k_B} \right)$. The profiles for $T_\mathrm{rot}$ and $T_\mathrm{vib}$ reported in Figs. 5 and 6 are obtained as implicit solutions to a nonlinear system, such as the one defined by Eqs. (27)–(30) in Ref. [57]. Analogous to what was discussed in our recent DMS studies for nitrogen [36], this requires extracting the complete discrete rovibrational population distribution over all $\hbox {O}_{2}$-molecules in every DMS cell. Finally, the Cartesian components of the translational energy heat flux are:

$$\begin{aligned} q_\nu ^\mathrm{tra}= & {} \sum _{i = \{ \hbox {O}_{2}, \text {O} \}} \biggl \{ \frac{\rho _i}{2} \Bigl [ \langle \left| {\varvec{c}} \right| ^2 c_\nu \rangle ^i - \langle \left| {\varvec{c}} \right| ^2 \rangle ^i u_\nu + \left| {\varvec{u}} \right| ^2 \left( \langle c_\nu \rangle ^i - u_\nu \right) \ldots \nonumber \\&+ \sum _{\eta = \{ x, y, z \}} \left\{ 2 u_\nu \, u_\eta \langle c_\eta \rangle ^i - 2 \langle c_\nu \, c_\eta \rangle ^i u_\eta \right\} \Bigr ] \biggr \} \, \quad \nu = \{ x, y, z \} \end{aligned}$$

(20)

where $\langle \left| {\varvec{c}} \right| ^2 c_\nu \rangle ^i = \sum _\mathrm{cell} \{ (c_x^2 + c_y^2 + c_z^2) \, c_\nu \} / [N_i]_\mathrm{cell}$ and $\langle \left| {\varvec{c}} \right| ^2 \rangle ^i = \sum _\mathrm{cell} \{ (c_1^2 + c_y^2 + c_z^2) \} / [N_i]_\mathrm{cell}$, are again understood to be cell-averages taken over all particles belonging to species $i = \{ \hbox {O}_{2}, \text {O} \}$. The corresponding components of the internal energy heat flux are:

$$\begin{aligned} q_\nu ^\mathrm{int} = \sum _{i = \{ \hbox {O}_{2}, \text {O} \}} \biggl \{ n_i \left( \langle c_\nu \, E^\mathrm{int} \rangle ^i - u_\nu \, \langle E^\mathrm{int} \rangle ^i \right) \biggr \}, \qquad \nu = \{ x, y, z \}, \end{aligned}$$

(21)

where $\langle c_\nu \, E^\mathrm{int} \rangle ^i = \sum _\mathrm{cell} \left\{ c_\nu \, E^\mathrm{int} \right\} / [N_i]_\mathrm{cell}$ represents the average flux in the laboratory frame of internal energy of species $i = \{ \hbox {O}_{2}, \text {O} \}$ along Cartesian direction $\nu$, while $\langle E^\mathrm{int} \rangle ^i = \sum _\mathrm{cell} \left\{ E^\mathrm{int} \right\} / [N_i]_\mathrm{cell}$ is the average internal energy of species $i = \{ \hbox {O}_{2}, \text {O} \}$ itself. The overall “internal” energy collects contributions of each particle’s rotational and vibrational energies, plus an offset to account for the heats of formation:

$$\begin{aligned} E_i^\mathrm{int} = E_i^\mathrm{rot} + E_i^\mathrm{vib} + E_i^0, \qquad i = \{ \hbox {O}_{2}, \text {O} \} \end{aligned}$$

(22)

The first two contributions in Eq. (22) are only relevant for molecular species. For the DMS particles representing $\hbox {O}_{2}$, we calculate the rotational and vibrational energies directly as described in Sec. 2.5 of Ref [56]. In analogy to other recent DMS studies [35, 36], we have chosen to set the origin of the vibrational energy scale at $V_{v=0,J=0}$, i.e., the oxygen molecule’s potential energy in its lowest rovibrational energy level, and this is consistent with our discussion in “Appendix B.” The third term in Eq. (22) accounts for the formation energies. We set $E_{\hbox {O}_{2}}^0 = 0$ and $E_{\text {O}}^0 = D_0 / 2$, respectively, to make the definitions of $\langle E_{\hbox {O}_{2}}^\mathrm{int} \rangle$ and $\langle E_{\text {O}}^\mathrm{int} \rangle$ consistent with the ones for enthalpy introduced in “Appendix B.”

Thus, all macroscopic flow properties of interest can be entirely reconstructed from instantaneous, or time-accumulated samples of the quantities $[N_i]_\mathrm{cell}$, $\langle c_\nu \rangle ^i$, $\langle c_\nu \, c_\eta \rangle ^i$, $\langle \left| {\varvec{c}} \right| ^2 c_\nu \rangle ^i$, $\langle \left| {\varvec{c}} \right| ^2 \rangle ^i$, $\langle E^\mathrm{int} \rangle ^i$ and $\langle c_\nu \, E^\mathrm{int} \rangle ^i$. Using Eqs. (17)–(21) is especially convenient in DMS, because we only require samples gathered in the laboratory frame of reference, rather than in a frame moving with the local flow velocity. This avoids the need to calculate peculiar velocities for each particle and makes it possible to defer the calculation of the flow velocity and other moments depending on it to a separate post-processing stage.