# Thermochemical Non-Equilibrium in Thermal Plasmas

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Planetary Atmospheric Entry Plasmas

#### 2.1. Context

#### 2.2. Inside the Shock Layer

#### 2.3. The EXOMARS Mission

_{2}and CO during the entry in the Martian atmosphere and corresponds to the radiative contribution to ${\phi}_{w}$ in Equation (2) in the afterbody flow. This atmosphere is indeed mainly composed of CO

_{2}(95.97%), Ar (1.93%), and N

_{2}(1.89%). Crossing the shock front, this mixture is then put at high temperature and pressure. The composition then changes since this composition does not correspond to chemical equilibrium. To estimate chemical relaxation time scales ${\tau}_{CR}$ of this mixture, we have developed models able to show how this relaxation occurs in a typical situation. Since the composition of the upstream atmosphere is well known, we focus our attention on the shock crossing when the heat transfer to the TPS is maximum, therefore over the first centimeters after the shock. This situation of “peak heating” corresponds to an altitude of $45\mathrm{km}$ [9].

#### 2.4. Modeling of the Shock Front Crossing

_{2}, N

_{2}, and Ar induces a very complex chemistry past the shock front. To be relevant, this chemistry must include enough species that can be formed in the post-shock conditions based on C, O, and N atoms. The pressure conditions are insufficient to produce Ar

_{2}

^{+}dimers. The species taken into account in the resulting CoRaM-MARS collisional-radiative model are listed in Table 1. This list involves 21 species and electrons, 1600 excited vibrational and electronic excited states. All the vibrational states of the molecular electronic ground states are taken into account to reproduce realistically the global dissociation processes.

#### 2.5. Some Results

_{2}. The collision frequency and the energy available in the collisions then start the chemistry.

_{2}($i,0,0$) vibrational states is displayed as a function of the position from the shock front. The vibrational distribution is far from being linear, that reveals a departure from vibrational excitation equilibrium.

_{2}is clearly observed. The dissociation degree is close to 0.01 at $1\text{}\mathrm{cm}$ after the shock front and to 0.35 at $\mathsf{\delta}\approx 5\text{}\mathrm{cm}$. Molecular and atomic ions start to be produced just after the dissociation of CO

_{2}. A maximum electron density of $1.8\times {10}^{19}{\text{}\mathrm{m}}^{-3}$ is obtained at a location close to $4\text{}\mathrm{cm}$ and corresponds to an ionization degree of $~4\times {10}^{-4}$. Even if this amount seems to be rather small, the electron density is nevertheless high enough to significantly influence the chemistry. Indeed, due to their weak mass, the efficiency of electrons in terms of inelastic/superelastic collisions is much stronger than the one of the heavy particles. However, this influence is reduced since the electron temperature ${T}_{e}$ is lower than the heavy particle temperature ${T}_{A}$.

_{2}resulting from our vibrational state-to-state approach. The total energy-dependent vibrational temperature has been also determined.

_{2}are the result of the thermal non-equilibrium ${T}_{e}\ne {T}_{A}$ and of the efficiency of electrons and heavies in terms of collisions. In addition, the electrons and heavy particles dynamics is deeply different since electrons are produced and heated behind the shock front while the heavies leave their energy along the flow where the global dissociation process takes place. This corresponds therefore to a strong non-equilibrium situation. This situation relaxes over typical length scales longer than the shock layer thickness as illustrated by Figure 5. The thermal non-equilibrium would be resorbed around $1\mathrm{m}$ from the shock front in case of infinite shock layer thickness. Since $\delta \approx 5\mathrm{cm}$, we conclude to a limit of the boundary layer departing from thermal equilibrium. This conclusion departs from the usual one considered for the case of Earth atmospheric entries [12].

## 3. Laser-Induced Plasmas

#### 3.1. Context

#### 3.2. Possible Non-Equilibrium Situation

#### 3.3. Tokamak and Tungsten

^{+}, and Rg

_{2}

^{+}can be found on their different excited states. The pressure in the shock layer can be high enough to promote the formation of the dimer molecule Rg

_{2}

^{+}. In the central plasma, electrons, W, W

^{+}, and W

^{2+}have been considered.

^{+}excited state corresponds to an excitation energy of $9.23\mathrm{eV}$ in the NIST database while the ionization limit is $16.37\mathrm{eV}$ [16]. We have therefore assumed a hydrogen-like behavior up to the ionization limit. Moreover, to reduce the total number of excited states considered in the conservation equations, the classical lumping procedure has been performed. It consists in the grouping of states sufficiently close in terms of energy. The statistical weight of the grouped levels is taken as the summation of those of the individual levels [17].

#### 3.4. Collisional and Radiative Processes in the State-to-State Approach

^{+}, a lack of elementary data (Einstein coefficients) can be observed. This induces an underestimate of the radiative losses. The radiative recombination is also accounted for. In each layer, electrons and heavies are assumed Maxwellian, but at a different temperature. These particles collide with the different species on their excited states, which leads to their excitation, deexcitation, ionization, and recombination. Each elementary process is considered with its backward process using the detailed balance principle. The derived collisional-radiative model involves almost 550,000 elementary reactions, therefore an order of magnitude similar to atmospheric entry calculations. This number is lower than for atmospheric entry because a lower number of species is involved. In addition, except Rg

_{2}

^{+}for which the chemistry is simple since no vibrational state is considered, no molecule is concerned.

#### 3.5. Results

^{+}in the central plasma at $200$ and $300\mathrm{ns}$ at atmospheric pressure. Figure 12 displays those related to a $10\mathrm{Pa}$ argon gas.

^{2+}ions have a density of the order of ${10}^{17}{\mathrm{m}}^{-3}$, but temperature is too weak to influence the electron density ${n}_{e}$. Indeed, we have ${n}_{e}=\left[{W}^{+}\right]\cong 9\times {10}^{23}{\mathrm{m}}^{-3}$. At $300\mathrm{ns}$, the situation is almost the same. A weak decrease in ${n}_{e}$ can be observed. This means that the collisional frequency is high enough to maintain in time the plasma situation.

## 4. Conclusions

## Author Contributions

## Funding

^{3}, PICOLIBS project.

## Conflicts of Interest

## References

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**Figure 1.**Global situation close to the TPS of a spatial vehicle showing the structure of the shock layer, the motion of fluid particles and the boundary layer. $\Delta $ is the typical thickness of the shock layer and $\Delta -\delta $ the one of the boundary layer.

**Figure 2.**Post-shock relaxation for the pressure, the mass density and the speed resulting from Equations (3)–(6). Since the diffusion phenomena are assumed negligible in these equations, the solution corresponds to the real flow before $x=\delta \approx 5\mathrm{cm}$.

**Figure 3.**Evolution with the position from the shock front (indicated on the right) of the Boltzmann plot of the CO

_{2}($i,0,0$) vibrational states. The vibrational excitation energy relative to the ground state is given in abscissa. The dissociation energy of the CO

_{2}molecule is reminded.

**Figure 5.**Distribution of the electron temperature ${T}_{e}$, the heavy particle temperature ${T}_{A}$, and the post-processed energy-dependent vibrational temperature of CO

_{2}for the first (symmetric stretching) mode, second (bending) mode and third (asymmetric stretching) mode. The total energy-dependent vibrational temperature is also displayed. The limit of the boundary layer is located at $\delta \approx 5\mathrm{cm}$: in the red region, feature of the flow in case of infinite shock layer thickness.

**Figure 6.**Laser-induced plasma situation. The laser pulse is focused on the sample using a converging lens at an irradiance higher than the breakdown threshold. The ablated material expands according to a hypersonic regime and produces a shock wave propagating in the background gas. As a result, two layers are formed. The first one corresponds to the ablated material and the second one corresponds to the shock layer. These two layers are separated by a contact surface across which the diffusion phenomena can be considered as negligible in a first approximation.

**Figure 7.**Evolution of the pressure inside the central tungsten plasma and inside the shock layer (case of argon) for a classical laser ($10\mathrm{mJ}$, $30\mathrm{ps}$, $532\mathrm{nm}$)-induced plasma experiment at atmospheric pressure.

**Figure 9.**Same as Figure 7, but for the temperatures.

**Figure 10.**Same as Figure 8, but for the temperatures.

**Figure 11.**Boltzmann plots at $200$ and $300\mathrm{ns}$ for an argon gas at atmospheric pressure. The first ($7.86\mathrm{eV}$) and second ionization ($24.23\mathrm{eV}$) limits are indicated by a vertical blue line. Each state is represented by a square. We see the added states following a hydrogen-like assumption between $20$ and $24\mathrm{eV}$.

**Figure 12.**Same as Figure 11, but with an argon gas at $10\mathrm{Pa}$. The second ionization limit is not displayed since the corresponding number densities are very weak. The line interpolating the excited states just below the first ionization limit is plotted to easily estimate the departure from excitation equilibrium.

**Table 1.**List of the species and their excited states involved in CoRaM-MARS, the CR model developed at the CORIA laboratory for the CO

_{2}-N

_{2}-Ar mixtures.

Species | States |
---|---|

CO_{2} | X^{1}Σ_{g}^{+} (14 states (v_{1},v_{2},v_{3}) with E_{v} < 0.8 eV, 106 states (i00,0j0,00k) with E_{v} > 0.8 eV), ^{3}Σ_{u}^{+}, ^{3}Δ_{u}, ^{3}Σ_{u}^{−} |

N_{2} | X^{1}Σ_{g}^{+} (v = 0 → 67), A^{3}Σ_{u}^{+}, B^{3}Π_{g}, W^{3}Δ_{u}, B’^{3}Σ_{u}^{−}, a’^{1}Σ_{u}^{−}, a^{1}Π_{g}, w^{1}Δ_{u}, G^{3}Δ_{g}, C^{3}Π_{u}, E^{3}Σ_{g}^{+} |

O_{2} | X^{3}Σ_{g}^{−} (v = 0 → 46), a^{1}Δ_{g}, b^{1}Σ_{g}^{+}, c^{1}Σ_{u}^{−}, A’^{3}Δ_{u}, A^{3}Σ_{u}^{+}, B^{3}Σ_{u}^{−}, f^{1}Σ_{u}^{+} |

C_{2} | X^{1}Σ_{g}^{+} (v = 0 → 36), a^{3}Π_{u}, b^{3}Σ_{g}^{−}, A^{1}Π_{u}, c^{3}Σ_{u}^{+}, d^{3}Π_{g}, C^{1}Π_{g}, e^{3}Π_{g}, D^{1}Σ_{u}^{+} |

NO | X^{2}Π (v = 0 → 53), a^{4}Π, A^{2}Σ^{+}, B^{2}Π, b^{4}Σ^{−}, C^{2}Π, D^{2}Σ^{+}, B’^{2}Δ, E^{2}Σ^{+}, F^{2}Δ |

CO | X^{1}Σ^{+} (v = 0 → 76), a^{3}Π, a’^{3}Σ^{+}, d^{3}Δ, e^{3}Σ^{−}, A^{1}Π, I^{1}Σ^{−}, D^{1}Δ^{−}, b^{3}Σ^{+}, B^{1}Σ^{+} |

CN | X^{2}Σ^{+} (v = 0 → 41), A^{2}Π, B^{2}Σ^{+}, D^{2}Π, E^{2}Σ^{+}, F^{2}Δ |

N_{2}^{+} | X^{2}Σ_{g}^{+}, A^{2}Π_{u}, B^{2}Σ_{u}^{+}, a^{4}Σ_{u}^{+}, D^{2}Π_{g}, C^{2}Σ_{u}^{+} |

O_{2}^{+} | X^{2}Π_{g}, a^{4}Π_{u}, A^{2}Π_{u}, b^{4}Σ_{g}^{−} |

C_{2}^{+} | X^{4}Σ_{g}^{−}, 1^{2}Π_{u}, ^{4}Π_{u}, 1^{2}Σ_{g}^{+}, 2^{2}Π_{u}, B^{4}Σ_{u}^{−}, 1^{2}Σ_{u}^{+} |

NO^{+} | X^{1}Σ^{+}, a^{3}Σ^{+}, b^{3}Π, W^{3}Δ, b’^{3}Σ^{−}, A’^{1}Σ^{+}, W^{1}Δ, A^{1}Π |

CO^{+} | X^{2}Σ^{+}, A^{2}Π, B^{2}Σ, C^{2}Δ |

CN^{+} | X^{1}Σ^{+}, a^{3}Π, ^{1}Δ, c^{1}Σ^{+} |

N | ^{4}S°_{3/2}, ^{2}D°_{5/2}, ^{2}D°_{3/2}, ^{2}P°_{1/2}, … (252 states) |

O | ^{3}P_{2}, ^{3}P_{1}, ^{3}P_{0}, ^{1}D_{2} … (127 states) |

C | ^{3}P_{0}, ^{3}P_{1}, ^{3}P_{2}, ^{1}D_{2} … (265 states) |

Ar | ^{1}S_{0}, ^{2}[3/2]°_{2}, ^{2}[3/2]°_{1}, ^{2}[1/2]°_{0}, … (379 states) |

N^{+} | ^{3}P_{0}, ^{3}P_{1}, ^{3}P_{2}, ^{1}D_{2} … (9 states) |

O^{+} | ^{4}S°_{3/2}, ^{2}D°_{5/2}, ^{2}D°_{3/2}, ^{2}P°_{3/2}, … (8 states) |

C^{+} | ^{2}P°_{1/2}, ^{2}P°_{3/2}, ^{4}P_{1/2}, ^{4}P_{3/2}, … (8 states) |

Ar^{+} | ^{2}P°_{3/2}, ^{2}P°_{1/2}, ^{2}S_{1/2}, ^{4}D_{7/2}, … (7 states) |

**Table 2.**Jump conditions in $x=0$ due to the Rankine–Hugoniot assumption at $45\mathrm{km}$ altitude corresponding to the peak heating.

Variable | Upstream Conditions | Conditions in $\mathit{x}=0$ |
---|---|---|

Speed (${\mathrm{m}\text{}\mathrm{s}}^{-1}$) | 5270 | 690 |

Mach number | 26.4 | 0.34 |

Pressure ($\mathrm{Pa}$) | 7.6 | 6000 |

Temperature ($\mathrm{K}$) | 162 | 16,800 |

**Table 3.**List of the species and their excited states involved in the CR model CoRaM-Ar and CoRaM-W developed at the CORIA laboratory for the laser-induced plasmas on W in a rare gas (here for argon).

Plasma Layer | Species | States |
---|---|---|

(1) shock layer | Ar | ^{1}S_{0}, ^{2}[3/2]°_{2}, ^{2}[3/2]°_{1}, ^{2}[1/2]°_{0}, … (90 states) |

Ar^{+} | ^{2}P°_{3/2}, ^{2}P°_{1/2}, ^{2}S_{1/2}, ^{4}D_{7/2}, … (7 states) | |

Ar_{2}^{+} | X^{2}Σ_{u}^{+} | |

(2) central plasma | W | ^{5}D_{0}, ^{5}D_{1}, ^{5}D_{2}, ^{5}D_{3}, … (60 states) |

W^{+} | ^{6}D_{1/2}, ^{6}D_{3/2}, ^{6}D_{5/2}, ^{6}D_{7/2}, … (74 states) | |

W^{2+} | ^{5}D_{0…4}, ^{3}P2_{0…2}, ^{5}F_{1…5}, ^{3}H_{4…6}, ^{3}F2_{2…4} (5 states) |

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**MDPI and ACS Style**

Bultel, A.; Morel, V.; Annaloro, J.
Thermochemical Non-Equilibrium in Thermal Plasmas. *Atoms* **2019**, *7*, 5.
https://doi.org/10.3390/atoms7010005

**AMA Style**

Bultel A, Morel V, Annaloro J.
Thermochemical Non-Equilibrium in Thermal Plasmas. *Atoms*. 2019; 7(1):5.
https://doi.org/10.3390/atoms7010005

**Chicago/Turabian Style**

Bultel, Arnaud, Vincent Morel, and Julien Annaloro.
2019. "Thermochemical Non-Equilibrium in Thermal Plasmas" *Atoms* 7, no. 1: 5.
https://doi.org/10.3390/atoms7010005