Thermo-chemical dynamics and chemical quasi-equilibrium of plasmas in thermal non-equilibrium

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1 Thrmo-chmical dynamics and chmical quasi-quilibrium of plasmas in thrmal non-quilibrium Marc Massot, Bnjamin Graill and Thirry E. Magin Laboratoir EM2C, UPR 288 CNRS Ecol Cntral Paris, Franc Laboratoir d Mathématiqus d Orsay, UMR 8628 CNRS Univrsité Paris-Sud, Franc Aronautics and Arospac Dpartmnt, von Karman Institut for Fluid Dynamics, Blgium Abstract. W xamin both procsss of ionization by lctron and havy-particl impact in spatially uniform plasmas at rst in th absnc of xtrnal forcs. A singular prturbation analysis is usd to study th following physical scnario, in which thrmal rlaxation bcoms much slowr than chmical ractions. First, lctron-impact ionization is invstigatd. Th dynamics of th systm rapidly bcoms clos to a slow dynamics manifold that allows for dfining a uniqu chmical quasiquilibrium for two-tmpratur plasmas and proving that th scond law of thrmodynamics is satisfid. Thn, all ionization ractions ar takn into account simultanously, lading to a surprising conclusion: th innr layr for short tim scal (or tim boundary layr) dirctly lads to thrmal quilibrium. Global thrmo-chmical quilibrium is rachd within a short tim scal, involving only chmical ractions, vn if thrmal rlaxation through lastic collisions is assumd to b slow. Kywords: Kintic thory; Boltzmann quation; plasma ionization; thrmal nonquilibrium; singular prturbation mthod PACS: Jm; Dd INTRODUCTION Plasmas hav a broad fild of applications, such as air-brathing hyprsonic vhicls (plasma control for scramjt ngin), spaccraft atmosphric ntris (influnc of prcursor lctrons and prdiction of blackout phnomnon), highnthalpy wind tunnls (plasmatron, arc-jt, and shock tub facilitis), lightning phnomna, dischargs at atmosphric prssur, laboratory nuclar fusion and astrophysics. Graill t al.[3, 4] hav drivd basd on a multiscal Chapman- Enskog mthod a unifid fluid modl for multicomponnt plasmas by accounting for thrmal non-quilibrium btwn th translational nrgis of th lctrons and havy particls, such as atoms and ions, givn thir strong mass disparity. A 3-spcis plasma was considrd: lctrons, nutral particls, and ions ar dnotd by th indics, n, and i, rspctivly. Th full mixtur of spcis is dnotd by th st of indics S = {,n,i}, and th havy particls, by th st of indics H = {n,i}. Th ionization mchanism compriss th following ractions r ı : n + ı i + + ı, ı S. A rcurrnt topic in thortical works on plasmas is th drivation of a modifid Saha quation, dscribing systms in chmical quasi-quilibrium and thrmal non-quilibrium, with th consqunt dbat rgarding which of th forms of this quation is th corrct on to apply (s [1] and rfrncs citd thrin). In particular, Morro and Romo [7] and van d Sandn t al. [8] hav drivd an quation for th lctron-impact ionization raction basd on tchniqus issud from thrmodynamics of homognous systms at quilibrium. This approach is qustionabl for plasmas in both thrmo-chmical non-quilibrium, sing th strong coupling btwn chmical volution and thrmal xchang. In particular, it is not obvious to choos a suitabl st of constraints associatd with th optimization of th thrmodynamic functions. In this work, w propos to study both procsss of ionization by lctron impact, raction r, and by havy-particl impact, ractions r n and r i. W propos to xamin systms in chmical quasi-quilibrium and thrmal non-quilibrium by mans of a singular prturbation analysis, as opposd to a standard thrmodynamic approach, by xtnding th work of Massot [6] to thrmal non-quilibrium. This analysis is basd on a st of diffrntial quations drivd in [3] for th following physical scnario, in which th thrmal rlaxation bcoms much slowr than th chmical ractions. Th singular prturbation analysis, consistnt with th scal sparation associatd with this scnario, is usd to study th dynamics of th systm in two cass. First, lctron-impact ionization is invstigatd. Th dynamics of th systm rapidly bcoms clos to a slow dynamics manifold that allows for dfining a uniqu chmical quasi-quilibrium for two-tmpratur plasmas and proving that th scond principl is satisfid. Thn, all ionization ractions ar takn into account simultanously, lading to a surprising conclusion: whn ionization through

2 both lctron and havy-particl impact is considrd, th innr layr for a short tim scal (or tim boundary layr) dirctly lads to thrmal quilibrium. Thus, th global thrmo-chmical quilibrium is rachd within a short tim scal, involving only chmical ractions, vn if thrmal rlaxation through lastic collisions is not fficint and slow. To our knowldg, this approach shds som nw light on this mattr and has not bn usd prviously for such multicomponnt ractiv plasmas out of thrmal quilibrium. CONSERVATION EQUATIONS In this sction, w rviw th consrvation quations drivd in [3] for a spatially uniform plasma at rst in th absnc of xtrnal forcs. Thn, w introduc thrmodynamic functions and driv an ntropy quation for th singular prturbation analysis. Th drivation is basd on a multicomponnt Boltzmann quation with convntional lastic collision oprators and ractiv collision oprators writtn in trms of transition probabilitis [2]. A dimnsional analysis of th Boltzmann quation provids a small paramtr for th scal sparation, quantity ε = (m 0 /m 0 h )1/2, qual to th squar root of th ratio of th lctron mass to a rfrnc havy-particl mass. Th Knudsn numbr is assumd to scal as this paramtr, allowing for a continuum dscription of th systm. Th transition probabilitis ar linkd to diffrntial cross-sctions, allowing for a paramtrization of th ractiv collisions and a suitabl choic for th scaling lading to th Maxwllian raction rgim. In th multiscal Chapman-Enskog mthod, both th solution and th collision oprators ar xpandd in a sris of th ε paramtr, lading to two major rsults. First, nw xprssions ar drivd for th raction rat cofficints and zro-ordr chmical production rats for plasmas in thrmal nonquilibrium. Ths xprssions ar compatibl with th law of mass action. Th spcis formation nrgy is associatd with a tmpratur spcific to th ionization raction considrd. Consquntly, chmical ractions involving collision partnrs with populations distributd at distinct tmpraturs do not rsult only in changs for th mixtur chmical composition, but also rsult in hat xchang btwn th lctrons and havy particls. Scond, th st of drivd consrvation quations is compatibl with th first and scond laws of thrmodynamics. Enrgy and total dnsity ar consrvd and th ntropy production rat for ach typ of ionization raction is shown to b non-ngativ, involving a nw dfinition of th Gibbs fr nrgy for plasmas in thrmal non-quilibrium. At th lctron kintic tim scal (ordr ε 2 ), th lctron population thrmalizs to a quasi-quilibrium stat dscribd by mans of a Maxwll-Boltzmann distribution function at tmpratur T = 2 3 m T /k B ( ) 3/2 f 0 m = n xp( m ) c c, (1) 2πk B T 2k B T whr quantity m stands for th lctron mass; T, th lctron translational nrgy; n, th lctron numbr dnsity; k B, Boltzmann s constant; and c, th lctron vlocity. In contrast, havy particls do not xhibit any nsmbl proprty at this tim scal. At th havy-particl kintic tim scal (ordr ε 1 ), th havy-particl population thrmalizs to a quasi-quilibrium stat dscribd by mans of a Maxwll-Boltzmann distribution function at tmpratur T h = 3 2ρ h T h /(n hk B ) ( ) 3/2 fi 0 mi = n i xp( m ) i c i c i, i H, (2) 2πk B T h 2k B T h whr quantity n i = ρ i /m i stands for th numbr dnsity of spcis i, m i its mass, ρ i its mass dnsity, c i its vlocity, T h th havy-particl translational nrgy, ρ h th havy-particl mass dnsity, and n h = i H n i th havy-particl numbr dnsity. Th quasi-quilibrium stats givn in Eqs. (1) and (2) ar dscribd by mans of distinct tmpraturs for th lctrons and havy particls. At th macroscopic tim scal (ordr ε 0 ), th consrvation quations for th mass and global nrgy for th lctrons and havy particls ar drivd as d t ρ = m ω 0, d t ρ i = m i ω 0 i, i H, (4) d t (E ) = E 0 h + F r ω r 0 d t (E h ) = E 0 h F r ω r 0 + F r i ω r i0 + F r n ω r n0, (5) F r i ω r i0 F r n ω r n0. (6) with th global nrgis ar qual to th sum of th translational and formation nrgis E = ρ T + ρ U F, E h = ρ h T h + ρ j U F j. j H (3)

3 Th nrgy xchang cofficints by havy-particl impact ionization F r i = F r n = m U F, and by lctron-impact ionization, F r = m n U F n m i U F i allow to obtain th ionization nrgy from th rlation E = F r i F r. Th zroordr chmical production rats compris th contribution of th various chmical ractions ωi 0 = j S ω r j0 i, i, j S. Ths rats satisfy th proprty ω r i0 = ω r i0 i = ω r i0 n, i S, and can b xprssd in trms of th numbr dnsitis as ω r ı0 = K f r ı (T h,t ri )n n n ı K b r ı (T h,t,t ri )n i n n ı, ı S. Th tmpratur dpndnc for th dirct and rvrs rat cofficints is strongly connctd with th raction mchanism. Th ionization nrgy is providd by th catalyst at a raction tmpratur dfind as T r = T and T r ı = T h, ı H. Th translational nrgy transfrrd from havy particls to lctrons, is xprssd as E 0 h = 3 2 n k B (T T h ) 1 τ, whr τ is th avrag collision tim at which this nrgy transfr occurs. Using Eqs. (3) and (4), th total mass ρ = ρ + ρ h, th total charg, Q = q (n n i ), with th lctron charg q, and th total nrgy E = E + E h ar consrvd for th mixtur, i.., d t ρ = 0, (7) d t Q = 0. (8) d t E = 0. (9) Th systm volvs at constant total dnsity, total charg and total nrgy. It is important to mntion that no furthr assumption is mad on th intrnal variabls, dfind by Woods [9] as th mixtur composition and nrgy distribution among th spcis. In addition to th nrgy, othr rlvant thrmodynamic functions ar introducd. First, th ractiondpndnt Gibbs fr nrgy is dfind by th rlations ( ni ) ρ i g i = n i k B T i ln Q T i (T + T i ρ i U F i, i, j S, i) T rj with th translational partition function Q T i (T i) = (2πm ı k B T i /h 2 P) 3/2, ı, j S, and th tmpratur T i = T h, i H. Th spcis nthalpy is givn by ρ i h i = 5n i k B T i /2 + ρ i U F i, i S, and th spcis ntropy by s i = (h i g i )/T i, i S. Th mixtur ntropy rads S = j S ρ j s j. For ractiv plasmas, Gibbs rlation is found to b d t S = ϒ th + j S ϒ r j ch. Th ntropy production rat du to thrmal non-quilibrium is non-ngativ ϒ th = 3n (T T h ) 2 /(2T T h τ). Th ntropy production rats du to chmical ractions ϒ r i ch = gr i ω r i0 /T j H m j g r i j ω r i0 j /T h, i S, ar also non-ngativ, and th scond law of thrmodynamics is thus satisfid. CHEMICAL QUASI-EQUILIBRIUM FOR PLASMAS IN THERMAL NON-EQUILIBRIUM A compact vctorial notation is introducd for th systm of Eqs. (3)-(6). Th tmporal volution of th consrvativ variabl U = (ρ t,e,e h ) t, with th mass dnsity vctor ρ t = (ρ,ρ i,ρ n ), is dscribd by mans of th fiv-dimnsional dynamical systm Ω(U) = Ω ch(u) µ d t U = Ω(U), U(0) = U 0, (10) + Ω th (U), Ω ch (U) = ω r j0 (U)M r j ν, Ω th = E 0 (U)κ. (11) j S Th raction vctor in th composition spac rads ν t = (1,1, 1) R 3, and th raction vctor in th full composition and nrgy spac, ν t = (ν t,1, 1) R 5. Mass matrics ar dfind in ths two spacs as M = diag(m, m i, m n ) and M r j = diag(m, F r j, F r j ). Th sourc trm associatd to thrmal rlaxation Ω th involvs th vctor κ t = (0,0,0,1, 1). Th total mass dnsity rads ρ = ρ,u, whr symbol, dnots th uclidian scalar product, and U t = (1,1,1), th unit vctor in R 3. Th raction vctor spac is on-dimnsional, R = span{ ν }, and w dnot th

4 A ρ n U U Orth B ρ i ρ FIGURE 1. Th raction simplx is th lin sgmnt AB in th composition spac (ρ,ρ i,ρ n ). Th total charg is assumd to b zro. Point A corrsponds to a fully nutral mixtur, and point B, to a fully ionizd mixtur. Th spac orthogonal to th raction simplx is spannd by th orthogonal basis (U,U Orth ). augmntd vctor spac R = span{ ν }. Th raction simplx, whr ρ livs, is th on-dimnsional affin subspac R = (ρ 0 + n 0 MR) (0, ) 3, whr quantity n 0 is a dimnsional numbr dnsity (s figur 1). In addition to ρ, w dfin ρ Orth = ρ,u Orth, whr th vctor U Orth = ( m i m n,m n + m,m i m ) t /m n is orthogonal to U as wll as orthogonal to Mν in R 3. Thn d t ρ Orth = 0 and quantity ρ Orth is invariant by th dynamical systm (10), as a rsult of th total mass and charg consrvation Eqs. (7) and (8). In this sction, w invstigat particular flow conditions for which th thrmal rlaxation trm Ω th is assumd to b much lowr that th chmical rlaxation trm Ω ch, i.., dnoting by µ a ratio btwn a chmical tim and a thrmal rlaxation tim which is supposd to b small with rspct to on, and a singular prturbation analysis of th dynamics of such a systm is carrid out in th limit µ 0. Ionization by sol lctron impact A simplifid cas, for which Ω = Ω = ω r 0 M ν, is first xamind. Th chmical mchanism compriss ionization only by lctron impact. Th dynamics of th systm, in th approximation of small µ paramtr, can b dcomposd into an innr tmporal layr involving only chmical ractions and an outr tmporal layr at chmical quasiquilibrium involving only thrmal rlaxation toward th uniqu global quilibrium dscribd in th prvious sction. W will first tackl th problm of th innr layr for U inn, th typical tim of which is dnotd by τ = t/µ. It satisfis th following st of quations: d τ U inn = Ω (U inn ) = ω r 0 (U inn )M r ν, U inn (0) = U 0. (12) For this tim scal τ, th innr layr is a rgular prturbation of th dynamics of th full systm at short tim scals, whr thrmal rlaxation dos not play any rol and fast ractions govrn th volution of th systm. Lt us mphasiz that, within th innr layr approximation, th translation nrgy of th havy particls is consrvd d τ (ρ h T h ) = 0, as wll as th total nrgy, so that w also hav d τ (E,r F ) = 0, whr th augmntd lctron nrgy is givn by th xprssion E,r F = ρ T + i S ρ i U F i. Th dynamics of th full original systm dos not possss additional invariants, but its dynamics can b approximatd, for short tim scals, to th on of th innr layr. Proposition 1. Lt us assum ρ 0 (0, ) 3, E 0 (0, ), E h0 (0, ), and undr som classical proprtis that can b found in [6], thr xists a smooth global in tim solution of th dynamical systm (12). Th spcis dnsitis ar positiv and thr xists two positiv tmpraturs, T 1 and T 2, bounding th tmpraturs: T 1 T inn (τ) T 2, T 1 Th inn(τ) T 2, for all µ. Th systm admits an ntropic structur, i.., it can b symmtrizd through th us of th ntropic variabl V inn and th systm satisfis a scond principl of thrmodynamics, i.. d τ σ(u inn ) is non-positiv; it xprsss th dcras of

5 th ntropy σ purly du to th chmical raction. Thr xists a uniqu chmical quasi-quilibrium point U q = (ρ q,e q,e q h ), whr ρ q is in th raction simplx, such that th sourc trm vanishs Ω ch (U q ) = 0 or quivalntly ω r 0 (U q ) = 0, or quivalntly V q (U q ) (M r R ). Th quasi-quilibrium composition and nrgis ar smooth function of (ρ 0, ρ0 Orth, (ρ h T h ) 0, (E,r F ) 0 ) which ar invariant by th dynamical systm (12). Th linarization of th sourc trm at U q has non-positiv ignvalus and xactly on ngativ ral ignvalu. Th mathmatical ntropy production from chmical ractions admits zro as a strict maximum at U q ovr th raction simplx. Finally, th uniqu chmical quasi-quilibrium is asympotically stabl and attracts th long tim bhavior of th dynamical systm (12). Following [6], th fast chmical dynamics which lads to chmical quasi-quilibrium provids us with th ability of partitioning th systm (10) into fast and slow variabls. Th fast variabl U Fast is simply dfind as a projction; lt us dnot U Fast = (Π Fast ) t U, whr Π Fast = M r ν is th projction matrix, up to a mtric, onto th raction vctor spac. In fact, in our particular simpl cas, it is asy to dscrib th basis of (M r R ) sinc it corrsponds to th four invariant variabls of our dynamical systm: a 1 = (U t,0,0) t for th consrvation of mass, a 2 = (0,0,0,1,1) t for th consrvation of total nrgy, a 3 = (U Orth,t,0,0) t for th consrvation of ρ Orth and a 4 = (0,U F i,uf n,1,0) t for th consrvation of augmntd lctron nrgy. Ths vctors form a basis which was dnotd Π = [a 1,a 2,a 3,a 4 ] in [6]. Following [6] th orthogonality rlations satisfid at chmical quasi-quilibrium by th ntropic variabl dfins, onc a basis of (M r R ) is chosn, th slows variabl which is dnotd by U = (Π ) t U. W will thn naturally hav R 5 = M r R span{ a i, i [1,4] }.. From thr, th outr layr can asily b dfind: which can also b rwrittn : d t U,out = (Π ) t Ω th (U q (U,out )), (13) d t ρ = 0, (14) d t E = 0, (15) d t ρ Orth = 0, (16) ( d t E,r F = Eh 0 U q( ρ 0,ρ0 Orth,(ρ h T h ) 0,E,r F ) ), (17) with th hlp of th slow variabl E,r F,q lft invariant by th fast chmical raction. This last quation dscribs rathr straightforwardly th fact that th chmical quasi-quilibrium will volv owing to hat xchang and convrg toward th uniqu global quilibrium point. Proposition 2. Th outr layr follows a scond principl of thrmodynamics, that is, d τ σ out 0, whr σ out = σ(u out ). In addition, th global quilibrium point dfind in th prvious sction is asymptotically stabl and th dynamics of th outr layr convrgs toward this point. Thus, w can compltly charactriz through a singular prturbation analysis th dynamical bhavior of th systm in th limit of small µ. W do not provid th dtails of th proof omittd hr for two rasons. First th principl of such an analysis was alrady providd in [6]; scond, it is not th scop of th prsnt contribution to focus on mathmatical background, but rathr to focus on th physics of th considrd phnomna. It can thn b provd that th dynamics of (10) can b approximatd in th following way: U = U,out + O(µ), U Fast = U Fast q (U,out ) + O(xp( δ t/µ)) + O(µ), whr both th innr layr whr U inn = U 0 and U Fast inn convrg toward U Fast q and th outr layr U,out with U Fast q (U,out ) satisfy a scond principl of thrmodynamics. Such an xpansion provids a vry prcis sns to th notion of chmical quasi-quilibrium in th framwork of thrmal non-quilibrium bcaus it dscribs th outr layr, that is th slow dynamics, of (10) through thrmal rlaxation, whras th raction oprats in tmporal boundary layrs associatd to th tim ratio µ. Lt us mphasiz that th sam study can b conductd in th framwork of th ionization by th sol havy particls, lading to th sam typ of rsults. As a conclusion, for this cas of a singl ionization raction through lctron impact, w hav bn abl to idntify and charactriz th two-tmpratur chmical quasi-quilibrium. Th purpos of th following subsction is to conduct th sam kind of analysis in th framwork of th whol st of thr ionization ractions.

6 Ionization by lctron and havy-particl impact In this sction, w only hav to tackl th problm of th innr layr. Starting from th sam initial conditions as th full systm, it satisfis th following st of quations: d τ U inn = Ω(U inn ), U inn (0) = U 0, whr th chmical sourc trm is dfind by (11). Onc again, it rprsnts th dynamics at short tim scals whr thrmal rlaxation dos not play any rol, but whr th thr fast ractions govrn th volution of th systm. It is important to mntion that for this configuration, w do not hav th consrvation of translation nrgy of havy spcis d τ (ρ h T h ) 0, but w still hav th consrvation of total nrgy. Proposition 3. Lt us assum ρ 0 (0, ) 3, E 0 (0, ), E h0 (0, ), and undr som classical proprtis that can b found in [6], thr xists a smooth global in tim solution of th dynamical systm (12). Th spcis dnsitis ar positiv and thr xists two positiv tmpraturs, T 1 and T 2, bounding th tmpraturs: T 1 T inn (τ) T 2, T 1 Th inn(τ) T 2. Th systm admits an ntropic structur, i.., it can b symmtrizd through th us of th ntropic variabl V inn and th systm satisfis a scond principl of thrmodynamics, i.. d τ σ(u inn ) is non-positiv; it xprsss th dcras of th ntropy σ purly du to th chmical raction. Thr xists a uniqu chmical quilibrium point U q = (ρ q,e q,e q h ), whr ρ q is in th raction simplx, such that th sourc trm vanishs Ω ch (U q ) = 0 or quivalntly ω r i0 (U q ) = 0, for all i S, or quivalntly V q (U q ) (M r ir ), for all i S. Howvr, this chmical quilibrium satisfis th xtra proprty : T q = T q h, i.., ionization by lctron and havyparticl impact with diffrnt tmpraturs lads to fast tmpratur rlaxtion and th global chmical and thrmal quilibrium is rachd within th innr layr. Thus th quilibrium composition and nrgis ar smooth functions of (ρ 0,ρ Orth 0,E 0 ) which ar invariant by th dynamical systm (12). Th mathmatical ntropy production from chmical ractions admits zro as a strict maximum at U q ovr th raction simplx. Finally, th uniqu chmical quasi-quilibrium is asympotically stabl and attracts th long tim bhavior of th dynamical systm (12). CONCLUSIONS Basd on kintic thory, w hav proposd a unifid dscription of th thrmodynamic stat of plasmas in thrmal and chmical non-quilibrium, thus xtnding th work of Woods [9], in which th non-quilibrium ffcts ar tratd sparatly in trms of intrnal variabls. Th full thrmodynamic quilibrium stat of th systm, undr wll-dfind and natural constraints, can b studid by following th approach usd in [2] and [6]. Our rsults ar complmntary to th consrvation quations and transport flux xprssions drivd by [4] for non-homognous plasmas in th prsnc of xtrnal forcs, bcaus w provid adquat chmical sourc trms to b addd to th zro-ordr driftdiffusion/eulr st of quations or to th first-ordr drift-diffusion/navir-stoks st of quations, in particular, with a dscription of th Kolsnikov ffct for multi-componnt plasmas [5]. REFERENCES 1. Giordano, D. and Capitlli, M. Physical Rviw E (2001). 2. Giovangigli, V. Multicomponnt flow modling. Birkhäusr, Boston, Graill, B, Magin, T. E., and Massot, M., Procdings of th Summr Program 2008, Cntr for Turbulnc Rsarch, Stanford Univrsity, (2008). 4. Graill, B, Magin, T. E., and Massot, M. Mathmatical Modls Mthods Applid Scincs 19(4) 527 (2009). 5. Kolsnikov, A. F. Tchnical Rport 1556, Institut of Mchanics. Moscow Stat Univrsity, Moscow, Massot, M. Discr. Cont. Dyn. Systms-Sris B (2002). 7. Morro, A. and Romo, M. J. Plasma Phys (1988). 8. van d Sandn, M. C. M., Schram, P. P. J. M., Ptrs, A. G., van dr Mulln, J. A. M. and Krosn, G. M. W. Phys. Rv. A (1989). 9. Woods, L. C. Th thrmodynamics of fluid systms. Oxford, U.K., Oxford Univrsity Prss, 1986.