Thermochemical Non-Equilibrium in Thermal Plasmas

atoms

Article Thermochemical Non-Equilibrium in Thermal Plasmas

Arnaud Bultel 1,* , Vincent Morel 1 and Julien Annaloro 2

1 CORIA, UMR CNRS 6614, Normandie Université, 76801 St-Etienne du Rouvray, France; [email protected] 2 CNES, French Spatial Agency, 31400 Toulouse, France; [email protected] * Correspondence: [email protected]

Received: 30 November 2018; Accepted: 25 December 2018; Published: 1 January 2019

Abstract: In this paper, we analyze the departure from equilibrium in two specific types of thermal plasmas. The first type deals with the plasma produced during the atmospheric entry of a spatial vehicle in the upper layers of an atmosphere, specifically the one of Mars. The second type concerns the plasma produced during the laser-matter interaction above the breakdown threshold on a metallic sample. We successively describe the situation and give the way along which modeling tools are elaborated by avoiding any assumption on the thermochemical equilibrium. The key of the approach is to consider the excited states of the different species as independent species. Therefore, they obey to conservation equations involving collisional-radiative contributions related to the other excited states. These contributions are in part due to the influence of electrons and heavy particles having a different translation temperature. This ‘state-to-state’ approach then enables the verification of the excitation equilibrium by analyzing Boltzmann plots. This approach leads finally to a thorough analysis of the progressive coupling until the equilibrium asymptotically observed.

Keywords: non-equilibrium; collisions; radiation; planetary atmospheric entry; laser matter interaction

1. Introduction The question of the existence of the thermodynamic equilibrium is crucial in plasma physics [1,2]. Indeed, since the energy can be freely distributed over the different excited states, radiation can be easily emitted, which leads to discrepancies in terms of population with respect to the Boltzmann distribution. In addition, ionization or recombination deals with excited states whose population density cannot be easily estimated. Moreover, temporary species, whose density would be negligible in case of thermodynamic equilibrium, can be formed and can have a significant influence on the behavior of the plasma. Many experimental and theoretical works have been devoted to plasma physics and the main objective of this communication is not to overview the field. This would be necessarily incomplete. Conversely, it is more interesting to focus our attention on unusual situations to enrich the analysis. The CORIA laboratory in France, with the collaboration of the French spatial agency CNES (Centre National d’Etudes Spatiales), works on plasmas produced during the planetary atmospheric entry of spatial vehicles. The CORIA laboratory works also on laser-induced plasmas in the framework of the multi-elemental composition determination based on the laser-induced breakdown spectroscopy (LIBS) technique with the CEA (Commissariat à l’Energie Atomique et aux Energies Alternatives). In these two cases, the plasma can depart significantly from thermodynamic equilibrium, and analyzing this departure helps to enlarge our understanding of the global behavior of the plasmas. This is why we propose in the present communication to focus our attention on these situations.

Atoms 2019, 7, 5; doi:10.3390/atoms7010005 www.mdpi.com/journal/atoms Atoms 2019, 7, 5 2 of 16

As a result, the structure of the communication is separated in two main parts. The ﬁrst part is dedicated to planetary atmospheric entry plasmas and the second part is devoted to the laser-induced plasmas. In each part, the context is given as well as the main features of the related plasmas. Then, tools are presented to characterize the plasmas formed using relevant models.

2. Planetary Atmospheric Entry Plasmas

2.1. Context

Due to gravity, the speed of bodies coming from space can be high. Typically, this speed u1 reaches several km s−1. When the body approaches a planet having an atmosphere, the penetration of its upper layers generates the formation of a shock layer by compression around the forward part of the body [3,4]. The temperature in this layer increases significantly at levels reaching easily several 104 K. A gas to plasma transition takes place in the shock layer and leads to a strong heat transfer to the surface of the body. This heat transfer can be high enough to increase its surface temperature beyond the melting point and to cause the destruction of this surface. In the case of a manned-controlled mission, from the technological point of view, the covering of the spatial vehicle external surface by a thermal protection system (TPS) based on the use of ceramic materials is therefore mandatory. Literature reports many entries in the Earth [5] or in the Mars [6] atmospheres. They can be controlled in the case of the flight of spatial vehicles, or totally uncontrolled in the case of natural bodies. For Earth, the human missions are well controlled and are particularly illustrated by the supplying of the International Space Station (ISS) at an altitude of ∼ 410 km. One of the most impressive events of natural entry took place on 15th February 2013 when a meteorite crossed the sky of the city of Chelyabinsk in Russia before to crash on the ground. Many cameras filmed this event, which clearly put in evidence the strong level of temperature reached in the shock layer through the strong emitted radiance.

2.2. Inside the Shock Layer Figure1 illustrates the production of the shock layer (thickness of ∆ ∼ 5 cm in order of magnitude) due to the fast external gas motion relative to the surface of a spatial vehicle. In particular, a fluid particle is followed along its trajectory. Entering the shock layer by crossing the detached shock front, its volume is strongly decreased due to the increase in pressure. As a result, the temperature increases and provokes chemical non-equilibrium leading to the global dissociation and ionization of the flow. If the trajectory is close to the stagnation streamline, the fluid particle enters the boundary layer in which the plasma flow gives energy to the body surface (in x = ∆). In this part, the temperature decreases approaching the body, which leads to significant gradients. In this region, whose typical thickness is ∆ − δ ∼ 1 cm, the recombination occurs. Consequently, these recombined species will interact with the TPS. The thickness of these layers is pretty low. In addition, although the speed is decreased at the crossing of the shock front, the speed remains sufficiently high in the shock layer to prevent local thermodynamic equilibrium. Indeed, the three Damkhöler numbers—Da1, Da2, and Da3—defined as

τu τu τu Da1 = Da2 = Da3 = (1) τCR τMB τe−h are not much higher than unity. Since the speed is high, the value of the characteristic time scale for convection τu is weak. The time scale τCR for the collisional-radiative source term of the species is of the same order of magnitude, which leads to a Damkhöler number Da1 close to unity. The ﬂow is therefore out of chemical equilibrium (in case of chemical equilibrium, Da1 1) and not frozen (in case of frozen ﬂow, Da1 1). The time scale to reach a Maxwell–Boltzmann distribution for the translation (of electrons or heavy species) τMB is much shorter than τu, which leads to Da2 1. As a result, the translation equilibrium is reached. However, the coupling between electrons and heavy Atoms 2019, 7, 5 3 of 16

species is difficult: the time scale of coupling τe−h is then higher than τu and the third Damkhöler number Da3 is therefore lower than unity. Thus, the flow is out of thermal and chemical equilibrium, inAtoms other 2019 words, 7, x FOR the PEER flow REVIEW is in thermochemical non-equilibrium. 3 of 16

Figure 1. Global situation close to the TPS TPS of of a a spatial spatial vehicle vehicle showing showing the the structure structure of of the the shock shock layer, layer, the motion of fluid ﬂuid particles and the boundary layer. layer. ∆ is is thethe typicaltypical thicknessthickness ofof thethe shockshock layerlayer and ∆∆−훿− δ the the oneone ofof thethe boundaryboundary layer.layer.

These features have important consequences on the heat transfer to the surface. The presence of species not formed in case ofof chemicalchemical equilibriumequilibrium and the spectral radiance emitted by the plasma lead to additionaladditional contributions.contributions. TheThe local parietalparietal heatheat ﬂuxflux densitydensity isis thenthen givengiven byby

→ → ( ) ≈ k ∇ T · n + E ϕw ∑ i ⃗ i w ∑ γj βj ϕ(j) 휑 ≈Modes i 푘 ∇ 푇. 푛⃗ + Species j 훾 훽 휑 → → 3 (2) + R α τ ( r ) ε ( r ) cos θ d r dν ν t ν ν r2 (2) Frequency ν Plasma volume 푑푟 + 훼 휏(푟⃗) 휀(푟⃗) cos 휃 푑휈 −2 푟 and can easily exceed 1 MW m . In Equation (2), the first part of the right side refers to the influence ofand translation can easily andexceed internal 1 푀푊 modes 푚. In transfer, Equation the (2), second the first part part toparietal of the right catalysis side refers and the to the third influence part to k i T radiation.of translationi is and the thermalinternal conductivitymodes transfer, for the the mode secondwhose part to temperature parietal catalysis is i, γ jandis the the recombination third part to probability for species j, βj is the energy accommodation coefficient, αν is the spectral absorptivity, τν radiation. 푘 is the thermal conductivity→ for the mode 푖 whose temperature is 푇 , 훾 is the is the spectral transmittivity along r , r is the distance with the elementary volume d3r, θ is the angle recombination probability for species→ 푗 , 훽 is the energy accommodation coefficient, 훼 is the with respect to the normal vector n and ε is the spectral emission coefficient. spectral absorptivity, 휏 is the spectralw transmittivityν along 푟⃗, 푟 is the distance with the elementary The estimation of the wall heat flux density requires a detailed knowledge of the plasma upstream volume 푑 푟, 휃 is the angle with respect to the normal vector 푛⃗ and 휀 is the spectral emission thecoefficient. boundary layer. The CORIA laboratory develops models to go deeper in its understanding. In the upcomingThe estimation section, we of focus the our wall attention heat flux on density the work requires dedicated a detailed to the EXOMARS knowledge mission. of the plasma upstream the boundary layer. The CORIA laboratory develops models to go deeper in its 2.3. The EXOMARS Mission understanding. In the upcoming section, we focus our attention on the work dedicated to the EXOMARSRecently, mission. the European Space Agency (ESA) in cooperation with the Russian Space Agency (ROSCOSMOS) managed the EXOMARS mission whose first step was the landing on the ground of Mars2.3. The of EXOMARS a rover on boardMission of the Schiaparelli lander on 19th October 2016 [7]. The TPS of the lander was equipped with sensors called ICOTOM embedded in the COMARS+ housing whose role was to Recently, the European Space Agency (ESA) in cooperation with the Russian Space Agency provide information on the radiative signature in the infrared part of the spectrum [8]. The related (ROSCOSMOS) managed the EXOMARS mission whose first step was the landing on the ground of radiation is due to the production of hot CO and CO during the entry in the Martian atmosphere and Mars of a rover on board of the Schiaparelli2 lander on 19th October 2016 [7]. The TPS of the lander corresponds to the radiative contribution to ϕ in Equation (2) in the afterbody flow. This atmosphere was equipped with sensors called ICOTOM embeddedw in the COMARS+ housing whose role was to is indeed mainly composed of CO (95.97%), Ar (1.93%), and N (1.89%). Crossing the shock front, provide information on the radiative2 signature in the infrared part2 of the spectrum [8]. The related this mixture is then put at high temperature and pressure. The composition then changes since this radiation is due to the production of hot CO2 and CO during the entry in the Martian atmosphere composition does not correspond to chemical equilibrium. To estimate chemical relaxation time scales and corresponds to the radiative contribution to 휑 in Equation (2) in the afterbody flow. This atmosphere is indeed mainly composed of CO2 (95.97%), Ar (1.93%), and N2 (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 휏 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 Atoms 2019, 7, 5 4 of 16

τCR 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 km [9].

2.4. Modeling of the Shock Front Crossing Since the excitation equilibrium condition is not systematically fulﬁlled, the modeling is based on the solution of the excited states population density balance equation written as

h . i dy mXm Xm Xm = (3) dx ρ u assuming negligible the diffusion phenomena within the post-shock ﬂow at steady-state. In Equation (3), yX is the mass fraction of the species X on its excited state m whose mass is mX . m h . i m The collisional-radiative term Xm results from the influence of all the elementary inelastic/superelastic processes enabling a change in the population density [Xm]. The mass ﬂow density is written ρ. Equation (3) is coupled with the momentum balance equation

d(p + ρu2) = 0 (4) dx whose pressure p is calculated by the Dalton law p = ∑X,m[Xm] kB TX assuming a perfect gas-like behavior. Equations (3) and (4) are ﬁnally coupled with the energy balance of heavy particles

2 d e p ρ u −Q → − ε − Q − A + A + A = A e SE A RR (5) dx ρ ρ ρ 2 ρ u and of electrons 2 d e p ρ u Q → − ε + Q − e + e + e = A e RR A RR (6) dx ρ ρ ρ 2 ρ u

These equations must be considered separately because the energy per unit volume eA and ee for heavies and electrons depend on the heavy species temperature TA and on the electron temperature Te, respectively. The Dalton law p = pA + pe is obvious and the mass density results from the summation of the contributions of heavy species ρA and electrons ρe. Inelastic/superelastic elementary processes are considered through the term QA→e while the term QA−RR results from the influence of the radiative recombination whose emission coefficient is εRR. The spontaneous emission is related to the emission coefficient εSE. The complexity of the upstream flow composed of CO2,N2, 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 Ar2 dimers. The species taken into account in the resulting CoRaM-MARS collisional-radiative model are listed in Table1. 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. When the mixture crosses the shock front, a drastic reduction of the mean free path takes place due to the strong increase in mass density ρ. The typical thickness of a shock front is of several mean free paths. In a first approximation, we can consider this increase so rapid that this corresponds to a discontinuity. The Rankine–Hugoniot jump conditions at the shock front can then be used, where the chemistry is frozen. Electron temperature Te remains unchanged because the rare electrons are not perturbed by the shock front. Indeed, due to their very weak mass, the electrons are in a quite subsonic Atoms 2019, 7, 5 5 of 16

flow regime. Heavy species temperature TA is conversely strongly increased at a level incompatible with the upstream chemical conditions. Chemistry then starts, which modifies the composition as a result of the elementary processes, first due to heavy species impact, the electron density ne being too weak. When ne is high enough, the elementary processes are driven by the electron-induced collisions. These elementary processes are collisional (vibrational excitation, dissociation, electronic excitation, ionization, excitation transfer, neutral and charge exchanges, and backward processes driven by the detailed balance principle), radiative (spontaneous emission) or mixt (radiative recombination, photo-ionization, self-absorption). This underlying chemistry represents a set of around 106 elementary reactions [10].

Table 1. List of the species and their excited states involved in CoRaM-MARS, the CR model developed

at the CORIA laboratory for the CO2-N2-Ar mixtures.

Species States 1 + 3 + 3 3 − CO2 X Σg (14 states (v1,v2,v3) with Ev < 0.8 eV, 106 states (i00,0j0,00k) with Ev > 0.8 eV), Σu , ∆u, Σu 1 + 3 + 3 3 3 − 1 − 1 1 3 3 3 + N2 X Σg (v = 0 → 67), A Σu ,B Πg,W ∆u, B’ Σu , a’ Σu , a Πg, w ∆u,G ∆g,C Πu,E Σg 3 − 1 1 + 1 − 3 3 + 3 − 1 + O2 X Σg (v = 0 → 46), a ∆g, b Σg , c Σu , A’ ∆u,A Σu ,B Σu , f Σu 1 + 3 3 − 1 3 + 3 1 3 1 + C2 X Σg (v = 0 → 36), a Πu, b Σg ,A Πu, c Σu , d Πg,C Πg, e Πg,D Σu NO X2Π (v = 0 → 53), a4Π,A2Σ+,B2Π, b4Σ−,C2Π,D2Σ+, B’2∆,E2Σ+,F2∆ CO X1Σ+ (v = 0 → 76), a3Π, a’3Σ+, d3∆, e3Σ−,A1Π,I1Σ−,D1∆−, b3Σ+,B1Σ+ CN X2Σ+ (v = 0 → 41), A2Π,B2Σ+,D2Π,E2Σ+,F2∆ + 2 + 2 2 + 4 + 2 2 + N2 X Σg ,A Πu,B Σu , a Σu ,D Πg,C Σu + 2 4 2 4 − O2 X Πg, a Πu,A Πu, b Σg + 4 − 2 4 2 + 2 4 − 2 + C2 X Σg , 1 Πu, Πu, 1 Σg , 2 Πu,B Σu , 1 Σu NO+ X1Σ+, a3Σ+, b3Π,W3∆, b’3Σ−, A’1Σ+,W1∆,A1Π CO+ X2Σ+,A2Π,B2Σ,C2∆ CN+ X1Σ+, a3Π, 1∆, c1Σ+ 4 ◦ 2 ◦ 2 ◦ 2 ◦ N S 3/2, D 5/2, D 3/2, P 1/2, . . . (252 states) 3 3 3 1 O P2, P1, P0, D2 . . . (127 states) 3 3 3 1 C P0, P1, P2, D2 . . . (265 states) 1 2 ◦ 2 ◦ 2 ◦ Ar S0, [3/2] 2, [3/2] 1, [1/2] 0, . . . (379 states) + 3 3 3 1 N P0, P1, P2, D2 . . . (9 states) + 4 ◦ 2 ◦ 2 ◦ 2 ◦ O S 3/2, D 5/2, D 3/2, P 3/2, . . . (8 states) + 2 ◦ 2 ◦ 4 4 C P 1/2, P 3/2, P1/2, P3/2, . . . (8 states) + 2 ◦ 2 ◦ 2 4 Ar P 3/2, P 1/2, S1/2, D7/2, . . . (7 states)

2.5. Some Results Due to the upstream conditions in terms of pressure, temperature and speed relative to the spatial vehicle when the peak heating occurs, the crossing of the shock front in x = 0 induces high values of temperature and pressure. Table2 gives the jump conditions. We can see that the temperature is clearly incompatible with an insigniﬁcant dissociation degree for CO2. The collision frequency and the energy available in the collisions then start the chemistry.

Table 2. Jump conditions in x = 0 due to the Rankine–Hugoniot assumption at 45 km altitude corresponding to the peak heating.

Variable Upstream Conditions Conditions in x = 0 Speed (m s−1) 5270 690 Mach number 26.4 0.34 Pressure (Pa) 7.6 6000 Temperature (K) 162 16,800

Figure2 displays the resulting distribution of the aerodynamic variables (pressure, mass density, and speed). Even if Equations (3)–(6) are available only in the region where the diffusion phenomena are negligible (typically before the boundary layer, for x < δ ≈ 5 cm), we have displayed all the Atoms 2019, 7, 5x FOR PEER REVIEW 66 of of 16 Atoms 2019, 7, x FOR PEER REVIEW 6 of 16 relaxation as we could observe in shock tubes, to see the final convergence of the flow. Since the relaxation as we could observe in shock tubes, to seesee thethe finalfinal convergenceconvergence ofof thethe flow.flow. Since the boundary layer starts at 훿 ≈ 5 푐푚, only the first centimeters of the solution displayed on Figure 1 are boundary layer starts at δ훿≈ ≈5 5 cm푐푚,, onlyonly thethe firstfirst centimeterscentimeters ofof thethe solutionsolution displayeddisplayed onon Figure1 1 are are relevant with respect to the real situation. relevant with respect toto thethe realreal situation.situation. Figure 2 clearly shows that the relaxation is still in progress at 훿 ≈ 5 푐푚. This is also clearly Figure2 2clearly clearly shows shows that that the the relaxation relaxation is still is instill progress in progress at δ ≈ at5 훿cm ≈. 5 This 푐푚. is This also is clearly also clearlyshown shown by Figure 3 where the Boltzmann plots of the CO2(푖,0,0) vibrational states is displayed as a byshown Figure by3 Figurewhere 3 the where Boltzmann the Boltzmann plots of the plots CO of (thei, 0, CO 0) vibrational2(푖,0,0) vibrational states is displayedstates is displayed as a function as a function of the position from the shock front. The2 vibrational distribution is far from being linear, offunction the position of the from position the shock from front.the shock The vibrationalfront. The distributionvibrational distribution is far from being is far linear, from thatbeing reveals linear, a that reveals a departure from vibrational excitation equilibrium. departurethat reveals from a departure vibrational from excitation vibrational equilibrium. excitation equilibrium.

Figure 2. Post-shock relaxation for the pressure, the mass density and the speed resulting from Figure 2. Post-shockPost-shock relaxation relaxation for for the the pressure, the mass density and the speed resulting from Equations (3)–(6). Since the diffusion phenomena are assumed negligible in these equations, the Equations (3)–(6). (3)–(6). Since Since the the diffusion diffusion phenomena phenomena are assumed are assumed negligible negligible in these in equations, these equations, the solution the solution corresponds to the real flow before 푥 =훿 ≈5 푐푚. correspondssolution corresponds to the real to ﬂow the real before flowx =beforeδ ≈ 5푥 cm. =훿 ≈5 푐푚.

Figure 3. Evolution with the position from the shock front (indicated on the right) of the Boltzmann Figure 3. Evolution with the position from the shock front (indicated on the right) of the Boltzmann plot of the CO 2((푖,0,0i, 0, 0)) vibrational vibrational states. states. The The vibrational vibrational excitation excitation energy energy relative relative to to the the ground ground state state plot of the CO2(푖,0,0) vibrational states. The vibrational excitation energy relative to the ground state is given in abscissa. The The dissociation dissociation energy energy of the CO 2 moleculemolecule is is reminded. reminded. is given in abscissa. The dissociation energy of the CO2 molecule is reminded. The way to the dissociation is mainly driven by the excited vibrational states close to the The way to the dissociation is mainly driven by the excited vibrational states close to the dissociation limit limit [11]. [11]. As As a aresult, result, these these distributions distributions prove prove that that the thedissociation dissociation equilibrium equilibrium is not is dissociation limit [11]. As a result, these distributions prove that the dissociation equilibrium is not reached. Indeed, the distribution of the species number densities displayed on Figure 4 illustrates the reached. Indeed, the distribution of the species number densities displayed on Figure 4 illustrates the Atoms 2019, 7, 5 7 of 16

Atoms 2019, 7, x FOR PEER REVIEW 7 of 16 not reached. Indeed, the distribution of the species number densities displayed on Figure4 illustrates thecurrent current global global dissociation dissociation process. process. The Theformation formation of CO of and CO andO resulting O resulting from from the dissociation the dissociation of CO of2 COis clearly2 is clearly observed. observed. The dissociation The dissociation degree degree is close is closeto 0.01 to at 0.01 1 cm at 1 aftercm after the shock the shock front front and andto 0.35 to 0.35at δ at≈ 5δ cm≈ 5. Molecularcm. Molecular and andatomic atomic ions ionsstart start to be to produced be produced just just after after the the dissociation dissociation of ofCO CO2. A2. Amaximum maximum electron electron density density of of1.8×1.8 × 101019 mm− 3 isis obtained obtained at at a alocation location close close to to 44 cmcm and corresponds to an an ionizationionization degreedegree ofof ∼~4×104 × 10−4.. Even Even if if this this amount amount seems seems to to be be rather small, the electronelectron density isis neverthelessnevertheless high high enough enough to to significantly significantly influence influence the chemistry.the chemistry. Indeed, Indeed, due todue their to weaktheir mass,weak themass, efficiency the efficiency of electrons of electrons in terms in of terms inelastic/superelastic of inelastic/superelastic collisions collisions is much is stronger much stronger than the onethan of the the one heavy of the particles. heavy particles. However, However, this influence this influence is reduced is reduced since thesince electron the electron temperature temperatureTe is lower푇 is lower than thethan heavy the heavy particle particle temperature temperatureTA. 푇.

Figure 4. 4.Distribution Distribution of of the the number number density density of the of different the different species species taken into taken account into in account the chemistry in the (seechemistry Table1 (see) behind Table the 1) shockbehind front. the shock Same front. as Figure Same2: as the Figure solution 2: the is relevant solution with is relevant respect with to the respect real situationto the real until situation 5 cm fromuntil the5 cm shock from front. the shock front.

Figure5 illustrates the distribution of temperatures. T and T have been plotted. We have also Figure 5 illustrates the distribution of temperatures. 푇A and 푇e have been plotted. We have also plotted the the distribution distribution of the of post-processed the post-processed values of values the energy-dependent of the energy-dependent vibrational temperature vibrational fortemperature each vibrational for each mode vibrational of CO mode2 resulting of CO from2 resulting our vibrational from our vibrational state-to-state state-to-state approach. approach. The total energy-dependentThe total energy-dependent vibrational vibrational temperature temperature has been alsohas been determined. also determined. This energy-dependent vibrational temperature TE (CO )_i for a mode i is derived from the This energy-dependent vibrational temperature 푇vib(퐶푂2)_푖 for a mode 푖 is derived from the vibrational energy per unit volume e (CO )_i by the equation vibrational energy per unit volume 푒vib(퐶푂2)_푖 by the equation Ev − E kBT (CO(2)_)i_ ∑∑[CO[퐶푂(i(,푖v, 푣)])·]∑. ∑g푔v E 퐸ve 푒 vib e (CO( ) i)= [CO[ (i (v)]E)] = v 2 v vib푒 퐶푂2 _ _푖 =∑ 퐶푂2 , 푖,푣 v퐸 = Ev (7) − v k TE (CO )_i B vib (2 )_ ∑∑v g푔v e 푒 푇 (퐶푂 ) In thethe casecase ofof thethe meanmean vibrationalvibrational temperaturetemperature Tvib(CO2), the summationssummations of EquationEquation (7) areare extended to the vibrational modes. On FigureFigure5 ,5, we we can can see see the the progressive progressive coupling coupling between between the threethe three modes modes with thewith distance the distance from thefrom shock the shock front. front. No one No is one perfectly is perfectly coupled coupled with the with translation the translation temperature temperature of electrons of electrons or heavy or particlesheavy particles before thebefore limit the of limit the boundary of the boundary layer. We layer. can also We observecan also that observe the temperature that the temperature departure betweendeparture the between first (symmetric the first (symmetric stretching) stretching) mode and themode second and (bending)the second mode (bending) is pretty mode low. is Thispretty is mainlylow. This due is to mainly the Fermi due resonance to the Fermi resulting resonance from the resulting energy diagram. from the The energy quasi diagram. resonance The between quasi theresonance related between states leads the to related easy excitation states leads transfer to easy between excitation them. transfer The third between (asymmetric them. stretching) The third (asymmetric stretching) mode is more difficult to excite and remains at temperature rather low. It is particularly interesting to see that, despite the thermal non-equilibrium between these states, they do not contribute in the same way to the mean vibrational temperature. Indeed, Figure 5 shows that this Atoms 2019, 7, 5 8 of 16 mode is more difficult to excite and remains at temperature rather low. It is particularly interesting to seeAtoms that, 2019 despite, 7, x FOR the PEER thermal REVIEW non-equilibrium between these states, they do not contribute in the 8 same of 16 way to the mean vibrational temperature. Indeed, Figure5 shows that this temperature follows the vibrationaltemperature temperature follows the ofvibrational the second temperature (bending) mode.of the Thissecond is mainly(bending) due mode. to the This degeneracy is mainly of due the statesto the ofdegeneracy the second of vibrational the states mode.of the Whilesecond the vibrational first and third mode. vibrational While the modes first and are notthird degenerated, vibrational themodes second are modenot degenerated, presents a degeneracy the second equalmode topresentsv2 + 1 resultinga degeneracy from equal the rotational to 푣 +1 motion resulting induced from bythe the rotational bending motion of the molecule.induced by One the finds bending further of the molecules molecule. per One unit finds volume further in the molecules related states per andunit thevolume global in vibrationalthe related temperaturestates and the follows global thevibrational one of this temperature mode. follows the one of this mode. TheThe differencesdifferences observedobserved onon FigureFigure5 5 between between the the vibrational vibrational modes modes of of CO CO22 areare thethe resultresult ofof thethe thermalthermal non-equilibriumnon-equilibrium T푇e6=≠푇TA and and ofof the efficiencyefficiency ofof electrons and heavies in terms of collisions. InIn addition, addition, the the electrons electrons and and heavy heavy particles particles dynamics dynamics is deeply is deeplydifferent different since electrons since are electrons produced are andproduced heated and behind heated the shockbehind front the while shock the front heavies while leave the theirheavies energy leave along their the energy flow where along the the global flow dissociationwhere the global process dissociation takes place. process This corresponds takes place. therefore This corresponds to a strong therefore non-equilibrium to a strong situation. non- Thisequilibrium situation situation. relaxes over This typical situation length relaxes scales over longer typical than length the shock scales layer longer thickness than asthe illustrated shock layer by Figurethickness5. The as thermalillustrated non-equilibrium by Figure 5. The would thermal be resorbed non-equilibrium around 1 mwouldfrom thebe resorbed shock front around in case 1 of푚 infinitefrom the shock shock layer front thickness. in case of Since infiniteδ ≈ shock5 cm, layer we conclude thickness. to aSince limit 훿 of ≈ the 5 푐푚 boundary, we conclude layer departing to a limit fromof the thermal boundary equilibrium. layer departing This conclusion from thermal departs equilibrium. from the usual This oneconclusion considered departs for the from case the of Earthusual atmosphericone considered entries for the [12 ].case of Earth atmospheric entries [12].

FigureFigure 5. DistributionDistribution of of the the electron electron temperature temperature 푇,T thee, the heavy heavy particle particle temperature temperature 푇, andTA, the and post- the post-processedprocessed energy-dependent energy-dependent vibrational vibrational temperature temperature of of CO CO2 for2 for the the first first (symmetric (symmetric stretching) stretching) mode,mode, secondsecond (bending)(bending) modemode andand thirdthird (asymmetric(asymmetric stretching)stretching) mode.mode. TheThe totaltotal energy-dependentenergy-dependent vibrationalvibrational temperaturetemperature is is also also displayed. displayed. The The limit limit of of the the boundary boundary layer layer is located is located at δ at≈ 훿5 cm ≈ 5: 푐푚 in the: in redthe region,red region, feature feature of the of flowthe flow in case in case of infinite of infinite shock shock layer layer thickness. thickness. 3. Laser-Induced Plasmas 3. Laser-Induced Plasmas 3.1. Context 3.1. Context The laser-induced plasmas correspond also to an interesting situation where thermodynamic non-equilibriumThe laser-induced plays an plasmas important correspond role. These also plasmas to an interesting are produced situation when a where (typically thermodynamic nanosecond) lasernon-equilibrium pulse reaches plays a sample an important at a spectral role. These irradiance plasmas of 10 are13– produced1014 W m− when2 higher a (typically than the nanosecond) breakdown 10 − 10 푊 푚 thresholdlaser pulse (see reaches Figure 6 a). sample The absorption at a spectral of the laserirradiance energy of leads to a strong increase higher in the than local the temperaturebreakdown threshold of the sample (see atFigure values 6). (of The the absorption order of several of the10 laser4 K) energy exceeding leads largely to a strong the conditions increase ofin 10 퐾 athe phase local change. temperature A plasma of the is indeedsample producedat values (of whose the pressureorder of several is initially very) exceeding high (of the largely order the of conditions of a phase change. A plasma is indeed produced whose pressure is initially very high (of the order of several 10 푃푎) with respect to the background gas one. The produced plasma then expands according to a hypersonic regime (the Mach number can reach values of the order of 25), produces a shock wave propagating in the background gas and cools mainly owing to the radiative losses. This leads to a physical object having lifetimes of the order of several 휇푠. Atoms 2019, 7, 5 9 of 16 several 1011 Pa) with respect to the background gas one. The produced plasma then expands according to a hypersonic regime (the Mach number can reach values of the order of 25), produces a shock wave propagating in the background gas and cools mainly owing to the radiative losses. This leads to a Atoms 2019, 7, x FOR PEER REVIEW 9 of 16 physical object having lifetimes of the order of several µs. TheThe atoms atoms and and ions ions composing composing the the plasma plasma are initially inside inside the the sample. sample. Once Once the the plasma plasma relaxed,relaxed, a cratera crater is is formed formed where where thethe laserlaser pulsepulse reached the the sample. sample. The The radiative radiative signature signature of ofthe the plasmaplasma can can therefore therefore give give valuable valuable informationinformation about the the composition composition of of the the sample. sample. This This explains explains whywhy this this laser–matter laser–matter interaction interaction isis atat thethe basisbasis of the laser-induced laser-induced breakdown breakdown spectroscopy spectroscopy (LIBS) (LIBS) techniquetechnique to to determine determine the the multi-elemental multi-elemental composition composition of of samples. samples. MeasuringMeasuring thethe spectralspectral radiance of relevantof relevant lines, it lines, is possible it is possible to derive to the derive relative the population relative population of the different of the species different if thermochemical species if equilibriumthermochemical conditions equilibrium are fulfilled conditions [13]. are fulfilled [13].

FigureFigure 6. 6.Laser-induced Laser-induced plasma plasma situation. situation. The laser pulse pulse is is focused focused on on the the sample sample using using a converging a converging lenslens at at an an irradiance irradiance higher higher than than the the breakdown breakdown threshold. threshold. The The ablated ablated materialmaterial expandsexpands accordingaccording to a hypersonicto a hypersonic regime regime and and produces produces a shock a shock wave wave propagating propagating in thein the background background gas. gas. As As a result,a result, two layerstwo are layers formed. are formed. The ﬁrst The one first corresponds one corresponds to the ablated to the material ablated and material the second and the one second corresponds one tocorresponds the shock layer. to the Theseshock twolayer. layers These are two separated layers are byseparated a contact by surfacea contact across surface which across the which diffusion the phenomenadiffusion phenomena can be considered can be considered as negligible as negligible in a ﬁrst approximation. in a first approximation.

3.2.3.2. Possible Possible Non-Equilibrium Non-Equilibrium Situation Situation TheThe pulse pulse duration duration playsplays anan importantimportant role role on on the the interaction. interaction. In Inthe the case case of ultrashort of ultrashort (푓푠 ( orfs 푝푠or)ps ) pulses,pulses, the the laser laser energy energy is directly is directly deposited deposited within within the material the material and leads and to leads thermal to non-equilibrium thermal non- becauseequilibrium electrons because and heavieselectrons have and notheavies enough have time not toenough be coupled. time to In be the coupled. case of Inns thelaser case pulses, of 푛푠 the endlaser of thepulses, pulse the is absorbedend of the by pulse the plasmais absorbed in expansion by the plasma with a in good expansion coupling with between a good electrons coupling and heavies.between In electrons addition, and thermal heavies. effects In addition, due to heatthermal diffusion effects within due to theheat material diffusion can within be observed the material for ns lasercan pulses be observed and can for produce 푛푠 laser micro-droplets, pulses and can contrary produce to micro-droplets, the case of the ultrashortcontrary to laser the case pulses of wherethe nanoparticlesultrashort laser can pulses be observed. where nanoparticles This explains can the be use observed. of fs laser This sources explains for the the use micromachining of 푓푠 laser sources devices. forIn the terms micromachining of ablation devices. precision, it is therefore better to use ultrashort laser pulses. According to the experimentalIn terms of setupablation used, precision, a nominal it is therefore ablation ratebetter as to low use as ultrashort some 10 lasernm pulse pulses.−1 canAccording be reached. to the experimental setup used, a nominal ablation rate as low as some 10 푛푚 푝푢푙푠푒 can be reached. This means that the matter forming the laser-induced plasma is in low amount. Typically, experiments This means that the matter forming the laser-induced plasma is in low amount. Typically, are performed by accumulating signals over a tenth of pulses. Then the net minimum ablation rate experiments are performed by accumulating signals over a tenth of pulses. Then the net minimum is of 100 nm. A depth profiling of the sample is therefore possible if the spectral radiance of the ablation rate is of 100 푛푚. A depth profiling of the sample is therefore possible if the spectral radiance relevant lines is high enough to provide significant results. We consider only picosecond laser pulses of the relevant lines is high enough to provide significant results. We consider only picosecond laser inpulses the upcoming in the upcoming sections. sections. As a result, As a the result, absorption the absorption of the laser of the pulse laser by pulse the expanding by the expanding plasma is totallyplasma avoided. is totally avoided. InIn some some specific specific applications, experiments experiments must must be performed be performed in low in pressure low pressure conditions. conditions. This Thiscorresponds corresponds to toin insitu situ measurements measurements if the if the sample sample cannot cannot be beremoved removed for for the the analysis. analysis. If the If the matter matter is radioactive or toxic, low-pressure experiments are often better to avoid any dissemination and to keep safe the environment. Then the plasma expands freely. The collision frequency collapses and the analysis of the situation in the light of the Damkhöler numbers performed in Section 2.2 then reveals that the equilibrium conditions are not satisfied. The LIBS determination of the multi- Atoms 2019, 7, 5 10 of 16 is radioactive or toxic, low-pressure experiments are often better to avoid any dissemination and to keep safe the environment. Then the plasma expands freely. The collision frequency collapses and the analysis of the situation in the light of the Damkhöler numbers performed in Section 2.2 then reveals that the equilibrium conditions are not satisfied. The LIBS determination of the multi-elemental composition of the sample cannot be directly and easily performed. Developing state-to-state approaches may be valuable in this context [14].

3.3. Tokamak and Tungsten The tungsten tiles of the divertor of a tokamak like WEST from the CEA Cadarache or ITER must be kept inside the machine as much as possible. A possible LIBS analysis of the fuel (hydrogen isotopes) contamination within these tungsten tiles must therefore be performed in low pressure conditions, at a maximum pressure of ∼ 10 Pa. In this context, the CORIA laboratory develops modeling tools similar to those developed for Section1. The structure of the plasma flow is close to the one developed around the surface of a spatial vehicle. The main difference results from the geometry of the flow. Fundamentally, the entry plasma is a 1D flow regarding the stagnation stream line. Conversely, the laser-induced plasma is a 3D flow but with a hemispherical symmetry and a radial dependency much higher than for the other coordinates. In a first approximation, this flow can be considered as made of two layers separated by a contact surface (see Figure6) separating the matter ablated from the sample and the shock layer. This shock layer corresponds to the background gas across which the shock front has propagated since the laser pulse. As a result, its external limit corresponds to the shock front. In a second approximation, we can assume these two parts as uniform [15]. For tokamak studies, we are working on tungsten. For comparison with laboratory studies, we focus our attention on the modeling of the behavior of tungsten laser-induced plasmas in rare gases. We have therefore elaborated collisional-radiative models based on state-to-state descriptions of tungsten and of the retained rare gas. The rare gas is denoted as Rg in the upcoming sections. In the + + shock layer, electrons and the species Rg, Rg , and Rg2 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 + + 2+ Rg2 . In the central plasma, electrons, W, W , and W have been considered. The knowledge of the electronic excited states structure of W is satisfactory. This is not the case for the ions. The last known W+ excited state corresponds to an excitation energy of 9.23 eV in the NIST database while the ionization limit is 16.37 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]. Table3 lists the species and the excited states finally accounted for tungsten and rare gas in the case the rare gas is argon. 230 different excited states are considered.

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 2 ◦ 2 ◦ 2 ◦ Ar S0, [3/2] 2, [3/2] 1, [1/2] 0, . . . (90 states) + 2 ◦ 2 ◦ 2 4 (1) shock layer Ar P 3/2, P 1/2, S1/2, D7/2, . . . (7 states) + 2 + Ar2 X Σu 5 5 5 5 W D0, D1, D2, D3, . . . (60 states) + 6 6 6 6 (2) central plasma W D1/2, D3/2, D5/2, D7/2, . . . (74 states) 2+ 5 3 5 3 3 W D0 . . . 4, P20 . . . 2, F1 . . . 5, H4 . . . 6, F22 . . . 4 (5 states) Atoms 2019, 7, 5 11 of 16

Atoms 2019, 7, x FOR PEER REVIEW 11 of 16 3.4. Collisional and Radiative Processes in the State-to-State Approach In the source terms of the related equations, the spontaneous emission is accounted for. As for Since the layers are assumed uniform, equations similar to Equations (3)–(6) written in the energy diagram of W+, a lack of elementary data (Einstein coefficients) can be observed. This hemispherical symmetry can be spatially integrated to obtain pure temporal equations. Coupled induces an underestimate of the radiative losses. The radiative recombination is also accounted for. with the shock propagation from the sample at a speed driven by the Rankine–Hugoniot assumptions, In each layer, electrons and heavies are assumed Maxwellian, but at a different temperature. These they lead to a system of non-linear ordinary differential equations whose solution can be derived. particles collide with the different species on their excited states, which leads to their excitation, In the source terms of the related equations, the spontaneous emission is accounted for. As for the deexcitation, ionization, and recombination. Each elementary process is considered with its energy diagram of W+, a lack of elementary data (Einstein coefficients) can be observed. This induces an backward process using the detailed balance principle. The derived collisional-radiative model underestimate of the radiative losses. The radiative recombination is also accounted for. In each layer, involves almost 550,000 elementary reactions, therefore an order of magnitude similar to atmospheric electrons and heavies are assumed Maxwellian, but at a different temperature. These particles collide entry calculations. This number is lower than for atmospheric entry because a lower number of with the different species on their excited states, which leads to their excitation, deexcitation, ionization, species is involved. In addition, except Rg2+ for which the chemistry is simple since no vibrational and recombination. Each elementary process is considered with its backward process using the state is considered, no molecule is concerned. 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 3.5. Results is lower than for atmospheric entry because a lower number of species is involved. In addition, + exceptWe Rg consider2 for which the classical the chemistry laser conditions is simple 10 since 푚퐽 no, 30 vibrational 푝푠 with a wavelength state is considered, of 532 푛푚 no molecule in argon atis concerned.atmospheric pressure. The ablated mass is then of the order of 10 푘푔 by pulse. The laser pulse duration is shorter than the typical time scale of expansion of the plasma and the deposited energy does3.5. Results not diffuse significantly within the sample. As a result, the pulse energy is totally given to the ablatedWe mass. consider Its initial the classical temperature laser conditions and pressure10 mJ are, 30 thenps withquite a high. wavelength They induce of 532 thenm subsequentin argon at evolutionatmospheric of the pressure. plasma. The ablated mass is then of the order of 10−10 kg by pulse. The laser pulse durationFigure is shorter7 illustrates than the the pressure typical time evolution scale of in expansion the central of plasma the plasma and in and the the shock deposited layer. Due energy to thedoes initial not diffuse pressure significantly of the central within tungsten the sample. plasma, As the a expansion result, the starts pulse around energy is1 푛푠 totally and giveninduces to the compressionablated mass. of Its the initial external temperature background and pressure gas whose are then pressure quite increases high. They in induce the shock the subsequent layer. The expansionevolution ofleads the plasma.to the decrease in the pressure of the central plasma until sufficient inversion with respectFigure to the7 illustrates shock layer. the Then pressure a recompression evolution in theof the central central plasma plasma and due in theto the shock shock layer. layer Due takes to placethe initial before pressure a coupling of the between central tungstenthe two layers plasma, from the the expansion pressure starts point around of view1 observedns and induces along the remainingcompression part of theof the external evolution. background gas whose pressure increases in the shock layer. The expansion leadsIn to the the framework decrease in of the the pressure present of assumptions the central plasma in terms until of flow sufficient continuity, inversion a minimum with respect pressure to the ofshock 10 푃푎 layer. can Then be considered a recompression for argon. of theFigure central 8 illustrates plasma duethe pressure to the shock evolution layer takesof the place layers before in this a case.coupling We see between that the the recompression two layers from does the not pressure take place. point The of view outside observed pressure along is too the low remaining to ensure part the of confinementthe evolution. of the plasma. Its lifetime is therefore considerably shortened.

Figure 7. EvolutionEvolution of of the the pressure pressure inside inside the the central tungsten plasma and and inside the shock layer (case of argon) for for a a classical classical laser laser (10 (10 푚퐽mJ, ,3030 푝푠ps,, 532 nm푛푚)-induced)-induced plasma experiment experiment at at atmospheric atmospheric pressure. pressure. Atoms 2019, 7, 5 12 of 16

In the framework of the present assumptions in terms of ﬂow continuity, a minimum pressure of 10 Pa can be considered for argon. Figure8 illustrates the pressure evolution of the layers in this case.Atoms We 2019 see, 7, that x FOR the PEER recompression REVIEW does not take place. The outside pressure is too low to ensure 12 of 16 the conﬁnementAtoms 2019, 7 of, x FOR the plasma.PEER REVIEW Itslifetime is therefore considerably shortened. 12 of 16

Figure 8. Same as Figure 7, but with an argon pressure of 10 푃푎. FigureFigure 8. 8.Same Same as asFigure Figure7 7,, but but with with anan argonargon pressure of of 10 Pa.푃푎. These trends can be also observed on the temperature evolutions of the different layers. Figure These trends can be also observed on the temperature evolutions of the different layers. Figure9 9 illustratesThese trendsthe results can be for also an observedargon gas on at the atmospheric temperature pressure evolutions and ofFigure the different 10 those layers. obtained Figure at illustrates the results for an argon gas at atmospheric pressure and Figure10 those obtained at 10 Pa. 109 illustrates푃푎. It is interesting the results to seefor onan Figureargon 9gas that at the atmospheric thermal coupling pressure resulting and Figure from 10the those elastic obtained collisions at It isis10 interestingefficient 푃푎. It is in interesting the to seecentral on to Figureplasmasee on 9Figure duethat to the9 thethat thermal high the thermallevel coupling of couplingpressure. resulting resulting This is from not from thethe the case elastic elastic for collisions the collisions shock is efﬁcient in the central plasma due to the high level of pressure. This is not the case for the shock layer layeris efficient where in 푇 the≠푇 central along plasma almost due the to complete the high evolution.level of pressure. In the caseThis ofis anot 10 the 푃푎 case pressure for the for shock the wherebackgroundlayerTe where6= TA gas, along푇 ≠푇 the almost thermal along the almost coupling complete the is complete evolution.satisfactory evolution. In in the the case central In ofthe a plasma 10casePa ofpressure until a 10 a 푃푎 characteristic for pressure the background for time the gas,ofbackground the the thermal order gas, of coupling 100 the 푛푠 thermal. is Before satisfactory coupling this time, inis thesatisfactory the central evolution plasmain the is centralquite until the aplasma characteristic same until as the a characteristic time one of obtained the order time at of 100atmosphericofns the. Before order thispressure of time,100 푛푠 until the. Before evolution푡 < 30 this 푛푠 . time,is The quite plasma the the evolution same evolves as is theindependently quite one the obtained same from asat atmospheric thethe onepresence obtained pressureof the at untilbackgroundatmospherict < 30 ns .gas. pressure The Then, plasma until the evolves pressure푡 < 30 푛푠 independently has. The sufficiently plasma evolves from decreased, the independently presence and the of collision the from background the frequency presence gas. isof tooThen,the theweakbackground pressure to ensure has gas. sufﬁcientlythe Then, coupling the decreased, betweenpressure 푇has and and sufficiently the 푇 collision. Electron decreased, frequency density isand is very too the weak weak collision and to ensure thefrequency energy the coupling ofis thetoo weak to ensure the coupling between 푇 and 푇 . Electron density is very weak and the energy of the betweenplasmaT e isand mainlyTA. Electron stored density in the iskinetic very weak energy and due the to energy expansion. of the plasma Internal is mainlyenergy stored collapses: in the plasma is mainly stored in the kinetic energy due to expansion. Internal energy collapses: kinetictemperature energy due푇rapidly to expansion. decreases. Internal energy collapses: temperature TA rapidly decreases. temperature 푇rapidly decreases.

FigureFigure 9. 9.Same Same asas Figure 77,, butbut forfor the temperatures. temperatures. Figure 9. Same as Figure 7, but for the temperatures. Atoms 2019, 7, 5 13 of 16 Atoms 2019, 7, x FOR PEER REVIEW 13 of 16

Figure 10. Same as Figure8 8,, but but forfor thethe temperatures. temperatures.

Our state-to-state approach enables the analysis of the departure from excitation equilibrium. + Figure 11 displaysdisplays the the Boltzmann Boltzmann plots plots of of the the excited excited states states of ofW Wand and W+ Win thein central the central plasma plasma at 200 at 200and and300 300 푛푠 ns at atmospheric at atmospheric pressure. pressure. Figure Figure 12 12displays displays those those related related to toa 10 a 10 푃푎 Pa argon argon gas. gas. On FigureFigure 11 11,, we we clearly clearly see see that that equilibrium equilibrium is reached.is reached. The The distribution distribution is perfectly is perfectly linear. linear. We canWe 2+ 17 −3 alsocan seealso that see temperature that temperature is high is since high W sinceions W have2+ ions a density have a of density the order of of the10 orderm ,of but 10 temperature 푚, but + − is too weak to influence the electron density n . Indeed, we have n = [W ] =∼ 9 × 1023 m 3. At 300 ns, temperature is too weak to influence the e electron density 푛e . Indeed, we have 푛 = [푊 ] ≅ the situation is almost the same. A weak decrease in n can be observed. This means that the collisional 9×10 푚 . At 300 푛푠, the situation is almost the esame. A weak decrease in 푛 can be observed. frequencyThis means is that high the enough collisional to maintain frequency in time is high the enough plasma situation.to maintain in time the plasma situation. When the the argon argon background background gas gas pressure pressure is decreased is decreased at 10 Pa at , the10 푃푎 situation, the situation is deeply is modified. deeply Wemodified. can see We that can the see main that slope the main of the slope distribution of the distribution of neutral of or neutral ionic excited or ionic states excited is more states negative. is more Thenegative. excitation The excitation temperature temperature is therefore is therefore lower. lower. Electron Electron density density is decreased is decreased with with respect respect to the to 21 −3 atmosphericthe atmospheric pressure pressure situation. situation. Indeed, Indeed, the the electron electrondensity density reaches 55×10× 10 m푚 at 200200 ns푛푠,, andand 21 −3 1.41.4× ×10 m푚 at at 300300ns 푛푠.. Moreover, Moreover, we we can can see see thatthat thethe distributiondistribution departsdeparts from a linear behavior. behavior. The excited excited states states just just below below the ionizationthe ionization limit limiton a 2 oneV ainterval 2 푒푉 interval are satisfactorily are satisfactorily coupled according coupled toaccording a linear distribution, to a linear distribution, whereas the lowerwhereas excited the lower states haveexcited a fluctuating states have behavior a fluctuating increasing behavior with time.increasing We are, with in thetime. present We are, case, in in the a strong present recombination case, in a strong situation recombination where the recombination situation where induces the arecombination satisfactory coupling induces a of satisfactory the involved coupling excited of statesthe involved at number excited density states valuesat number higher density than values those expectedhigher than due those to the expected lower excited due to states. the lower This veryexcited common states. behaviorThis very has common been already behavior observed has been in otheralready situations. observed In in an other ionization situations. situation In an similar ionization to post-shock situation flowssimilar observed to post-shock in atmospheric flows observed entries, thein atmospheric ionization induces entries, the the fast ionization depopulating induces of the the excited fast depopulating states of atoms of closethe excited to the ionizationstates of atoms limit, whichclose to leads the ionization to a depletion limit, of which these statesleads into termsa depletion of density. of these Then, states this in is terms the exact of density. symmetric Then, case. this is theOver exact the symmetric whole energy case. diagram, the distribution cannot be linear. We have also to analyze the influenceOver the ofwhole radiation. energy At diagram,200 ns, the orderdistribution of magnitude cannot be of linear. the number We have density also to of analyze the excited the [ ] [ ] states close to 5 eV is low. We have Wk =∼ 1016–1017 m−3 whereas Wk =∼ 1019–1020 m−3 for argon influence of radiation. At 200 푛푠, the ordergk of magnitude of the numbergk density of the excited states [ ] [ ] atclose atmospheric to 5 푒푉 is pressure.low. We have In these low≅ 10 density − 10 conditions, 푚 whereas theinfluence ≅ 10 of − radiation 10 푚 is formuch argon more at significant. Radiation causes departures from a linear distribution whose linearity cannot be recovered atmospheric pressure. In these low density conditions, the influence of radiation is much more by the collisional coupling. At 300 ns, the situation is worse since the collisional frequency is collapsing. significant. Radiation causes departures from a linear distribution whose linearity cannot be These behaviors have important consequences regarding the LIBS diagnostic. In case the LIBS recovered by the collisional coupling. At 300 푛푠, the situation is worse since the collisional frequency experiments are performed at atmospheric pressure, the equilibrium is rapidly obtained. As a result, is collapsing. the estimate of the excited states population density and of the electron density is sufficient to derive These behaviors have important consequences regarding the LIBS diagnostic. In case the LIBS the ground state number density. Using the same number of Boltzmann plots as the number of different experiments are performed at atmospheric pressure, the equilibrium is rapidly obtained. As a result, species, the composition of the plasma, therefore of the sample, can be identified. the estimate of the excited states population density and of the electron density is sufficient to derive Atoms 2019, 7, x FOR PEER REVIEW 14 of 16 Atoms 2019, 7, x FOR PEER REVIEW 14 of 16 Atomsthe ground2019, 7, 5 state number density. Using the same number of Boltzmann plots as the number14 of of 16 the ground state number density. Using the same number of Boltzmann plots as the number of different species, the composition of the plasma, therefore of the sample, can be identified. different species, the composition of the plasma, therefore of the sample, can be identified. In the case when the experiments are performed at low pressure, the analysis is considerably InIn thethe case case when when the the experiments experiments are are performed performed at low at low pressure, pressure, the analysis the analysis is considerably is considerably more more difficult. Indeed, it is not directly possible to derive the ground states number density from the difficult.more difficult. Indeed, Indeed, it is not it directly is not directly possible possible to derive to thederive ground the ground states number states number density fromdensity the from analysis the analysis of the radiation produced during the deexcitation of the excited states. The distribution of ofanalysis the radiation of the radiation produced produced during the during deexcitation the deexcitation of the excited of the states. excited The distributionstates. The distribution of the excited of the excited states departs from excitation equilibrium. Then, the development of collisional-radiative statesthe excited departs states from departs excitation from equilibrium.excitation equilibrium. Then, the Then, development the development of collisional-radiative of collisional-radiative models models based on state-to-state approaches is therefore mandatory, except if known samples are basedmodels on based state-to-state on state-to-state approaches approaches is therefore is therefore mandatory, mandatory, except if known except ifsamples known are samples available are available whose composition similar to the one to be obtained have been previously determined. In whoseavailable composition whose composition similar to similar the one to to the be obtainedone to be haveobtained been have previously been previously determined. determined. In that case, In that case, the composition will be directly derived from comparisons with these calibrated samples. thethat composition case, the composition will be directly will be derived directly from derived comparisons from comparisons with these with calibrated these samples.calibrated samples.

Figure 11. Boltzmann plots at 200 and 300 푛푠 for an argon gas at atmospheric pressure. The first FigureFigure 11.11. Boltzmann plots at 200 and and 300300ns 푛푠for for an an argon argon gas gas at at atmospheric atmospheric pressure. pressure. TheThe ﬁrstfirst (7.86 푒푉) and second ionization (24.23 푒푉) limits are indicated by a vertical blue line. Each state is ((7.867.86eV 푒푉)) andand secondsecond ionizationionization ((24.2324.23eV 푒푉)) limitslimits areare indicatedindicated byby aa verticalvertical blueblue line.line. EachEach statestate isis represented by a square. We see the added states following a hydrogen-like assumption between 20 representedrepresented byby a square. We We see see the the added added states states following following a hydrogen-like a hydrogen-like assumption assumption between between 20 and 24 푒푉. 20and and 24 24 푒푉 eV..

Figure 12.12.Same Same as as Figure Figure 11 ,11, but but with with an argon an argon gas at gas10 Pa at. The10 푃푎 second. The ionizationsecond ionization limit is not limit displayed is not Figure 12. Same as Figure 11, but with an argon gas at 10 푃푎. The second ionization limit is not sincedisplayed the corresponding since the corresponding number densities number are densities very weak. are very The weak. line interpolating The line interpolating the excited the states excited just displayed since the corresponding number densities are very weak. The line interpolating the excited belowstates just the firstbelow ionization the first limitionization is plotted limit to is easily plotted estimate to easily the estimate departure the from departure excitation from equilibrium. excitation states just below the first ionization limit is plotted to easily estimate the departure from excitation equilibrium. 4. Conclusionsequilibrium. 4. Conclusions 4. ConclusionsOur objective was to give the main information regarding the underlying physics involved by two examplesOur objective of plasma was to flows give departing the main frominformation thermochemical regarding equilibrium. the underlying The casesphysics of the involved planetary by Our objective was to give the main information regarding the underlying physics involved by atmospherictwo examples entry of plasmas plasma and flows of departingthe laser-induced from thermochemical plasmas have been equilibrium. discussed. The cases of the two examples of plasma flows departing from thermochemical equilibrium. The cases of the planetaryModeling atmospheric strategies entry have plasmas been detailed and of forthe twolaser-induced particular plasmas situations. have For been the atmosphericdiscussed. entry planetary atmospheric entry plasmas and of the laser-induced plasmas have been discussed. plasmas,Modeling we have strategies focused have our been attention detailed on for the two Martian particular missions situations. of the EXOMARSFor the atmospheric type. For entry the Modeling strategies have been detailed for two particular situations. For the atmospheric entry laser-inducedplasmas, we have plasmas, focused we our have attention detailed on the the properties Martian missions of the plasma of the EXOMARS flow produced type. during For the a laser- LIBS plasmas, we have focused our attention on the Martian missions of the EXOMARS type. For the laser- experimentinduced plasmas, on a divertor we have tungsten detailed tile. the Since properties the excitation of the plasma equilibrium flow producedcondition is during not fulfilled, a LIBS induced plasmas, we have detailed the properties of the plasma flow produced during a LIBS the only relevant way is to develop a state-to-state description of the species, i.e., to consider each Atoms 2019, 7, 5 15 of 16

excited state of the species as an independent variable and to solve the conservation equations in this framework. This requires the elaboration of a collisional-radiative model taking into account the different elementary processes at the level of each state. This strategy needs enough numerical means since the number of excited species and of elementary processes is often prohibitive. The results show that the excitation equilibrium can be observed if the collisional frequency is high enough to overcome the perturbative role of radiation. As a result, even in case of thermal non-equilibrium, the observed excitation equilibrium takes place at the translational temperature of the main collision partner.

Author Contributions: The authors have jointly contributed to the work described in this paper. Funding: This work has been carried out within the framework of the ICOTOM project funded by the French Space Agency CNES (Centre National d’Etudes Spatiales). This work has been also carried out within the framework of the French Federation for Magnetic Fusion Studies (FR-FCM) and of the Eurofusion consortium, and has received funding from the Euratom research and training programme 2014–2018 and 2019–2020 under grant agreement no. 633053. This work has been also carried out within the framework of the TRANSAT project and has received funding from the Euratom Research and Training Programme 2014–2018 under grant agreement no. 754586. The views and opinions expressed herein do not necessarily reflect those of the European Commission. This work has been also funded by the French ANR (Agence Nationale de la Recherche) through the programme “Investissement d’Avenir” (ANR-10-LABX-09-01), LabEx EMC3, PICOLIBS project. Conflicts of Interest: The authors declare no conflict of interest.

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