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4. TITLE AND SUBTITLE A Tightly Coupled Non-Equilibrium Magneto-Hydrodynamic Model for Inductively Coupled RF Plasmas

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6. AUTHOR(S) A. Munafo, S.A. Alfuhaid, J.-L. Cambier, M. Panesi

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13. SUPPLEMENTARY NOTES Journal article published in the Journal of Applied Physics, Vol. #118, Issue #13, October 2015 PA Case Number: #15301; Clearance Date: 6/3/2015 © 2015 AIP Publishing LLC The U.S. Government is joint author of the work and has the right to use, modify, reproduce, release, perform, display, or disclose the work. 14. ABSTRACT The objective of the present work is the development a tightly coupled magneto-hydrodynamic model for Inductively Coupled Radio-Frequency (RF) Plasmas. Non Local Thermodynamic Equilibrium (NLTE) effects are described based on a hybrid State-to-State (StS) approach. A multi-temperature formulation is used to account for thermal non-equilibrium between translation of heavy-particles and vibration of molecules. Excited electronic states of atoms are instead treated as separate pseudo-species, allowing for non-Boltzmann distributions of their populations. Free-electrons are assumed Maxwellian at their own temperature. The governing equations for the electro-magnetic field and the gas properties (e.g. chemical composition and temperatures) are written as a coupled system of time-dependent conservation laws. Steady-state solutions are obtained by means of an implicit Finite Volume method. The results obtained in both LTE and NLTE conditions over a broad spectrum of operating conditions demonstrate the robustness of the proposed coupled numerical method. The analysis of chemical composition and temperature distributions along the torch radius shows that: (i) the use of the LTE assumption may lead to an inaccurate prediction of the thermo-chemical state of the gas, and (ii) non-equilibrium phenomena play a significant role close the walls, due to the combined effects of Ohmic heating and macroscopic gradients.

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A tightly coupled non-equilibrium magneto-hydrodynamic model for

Inductively Coupled RF PlasmasA. Munafo,1, a) S. A. Alfuhaid,1, b) J.-L. Cambier,2, c) and M. Panesi1, d)1)Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Talbot Lab., 104 S. Wright St.,

Urbana, 61801 IL2)Edwards Air Force Base Research Lab., 10 E. Saturn Blvd., CA 93524

(Dated: 7 May 2015)

The objective of the present work is the development a tightly coupled magneto-hydrodynamic model forInductively Coupled Radio-Frequency (RF) Plasmas. Non Local Thermodynamic Equilibrium (NLTE) effectsare described based on a hybrid State-to-State (StS) approach. A multi-temperature formulation is usedto account for thermal non-equilibrium between translation of heavy-particles and vibration of molecules.Excited electronic states of atoms are instead treated as separate pseudo-species, allowing for non-Boltzmanndistributions of their populations. Free-electrons are assumed Maxwellian at their own temperature. Thegoverning equations for the electro-magnetic field and the gas properties (e.g. chemical composition andtemperatures) are written as a coupled system of time-dependent conservation laws. Steady-state solutionsare obtained by means of an implicit Finite Volume method. The results obtained in both LTE and NLTEconditions over a broad spectrum of operating conditions demonstrate the robustness of the proposed couplednumerical method. The analysis of chemical composition and temperature distributions along the torch radiusshows that: (i) the use of the LTE assumption may lead to an inaccurate prediction of the thermo-chemicalstate of the gas, and (ii) non-equilibrium phenomena play a significant role close the walls, due to the combinedeffects of Ohmic heating and macroscopic gradients.

I. INTRODUCTION

Inductively Coupled Plasma (ICP) torches have widerange of possible applications which include depositionof metal coatings, synthesis of ultra-fine powders, gener-ation of high purity silicon and testing of thermal pro-tection materials for atmospheric entry vehicles.1,2 In itssimplest configuration, an ICP torch consists of a quartztube surrounded by an inductor coil made of a series ofparallel current-carrying rings (see Fig. 1).

FIG. 1. Example of ICP torch in operating conditions (mini-torch facility; credits von Karman Institute for Fluid Dynam-ics).

The radio-frequency (RF) currents running throughthe inductor induce toroidal currents in the gas which

a)Electronic mail: munafo@illinois.edub)Electronic mail: alfuhai2@illinois.educ)Electronic mail: jean luc.cambier@us.af.mild)Electronic mail: mpanesi@illinois.edu

is heated thanks to Ohmic dissipation.2,3 If the energysupplied is large enough, the gas flowing through thetorch can undergo ionization, leading to the formationof a plasma.

The physico-chemical modeling of the flow-field andelectromagnetic phenomena inside an ICP torch requires,in theory, the coupled solution of the Navier-Stokes andthe Maxwell equations. The numerical solution of thiscoupled system of partial differential equations representsa challenging task, due to the disparity between the flowand the electro-magnetic field time-scales.4 Since in themodeling of ICP facilities one is not normally interestedin resolving electro-magnetic field oscillations,5 displace-ment currents can be safely neglected without introduc-ing an appreciable error.2,6 This leads to a more tractableformulation, as it eliminates the speed of light from theeigenvalues of the governing equations.4

The first attempts to model the temperature andelectro-magnetic field distributions inside ICP torcheswere published in the 1960-1970’s. Examples are theworks of Freeman and Chase,7 Keefer et. al.,8 and theseries of papers by Eckert.9–12 In most of these references,the torch was approximated as an infinite solenoid andthe plasma generated was considered in Local Thermo-dynamic Equilibrium (LTE) conditions. This reduces theproblem to the coupled solution of the energy equationfor the gas (known as the Elenbaas-Heller equation7,13)and an induction equation for the electric field. Theinduction equation is formally identical to the one de-scribing induction heating of metals.14,15 The authorsuse the heat conduction potential (i.e. s =

∫

κ dT ), inplace of the temperature T , as thermodynamic variable.This choice allows one to hide the non-linearity of thegas (total) thermal conductivity κ and can partially alle-

2

viate numerical instabilities that may arise when solv-ing the discretized set of equations by means of aniterative procedure. As recognized by Pridmore andBrown,16 the use of the heat conduction potential be-comes less effective for molecular gases (e.g. air andnitrogen mixtures). The major developments achievedin the field of Computational Fluid Dynamics (CFD)during the 1970-1980’s, led to the possibility of solv-ing the ICP magneto-hydrodynamic equations in multi-dimensional configurations. Examples are given in thepapers by Boulos,17 Mostaghimi, Proulx Bouos and co-workers,18–23 Kolesnikov and co-workers,24–30 Chen andPfender,31 van den Abeele and Degrez32, and more re-cently Panesi et. al.33 and Morsli and Proulx.34 In theseworks, numerical solutions are obtained using an explicit

coupling approach, by solving independently the flow andthe electric field governing equations, and updating cou-pling terms after each iteration. As observed by van denAbeele and Degrez,32 the explicit coupling approach pre-vents the use of Newton’s method during the first it-erations and requires to resort to more conservative it-erative techniques (e.g Picard’s method32) at the begin-ning of the simulation. Convergence issues with Newton’smethod may also arise when solving the flow governingequations in time-dependent form. This is especially truewhen the current intensity in the inductor is updated tomatch the imposed value of the power dissipated in theplasma. The cause of the instability is most probablydue to the lagged update of the Joule heating term in theenergy equation, which is quadratic in the electric fieldamplitude.17,31,32

Most of the simulations available in the litera-ture assume that the plasma in the torch is in LTEconditions.17–19,31–33,35 This assumption is often justifiedby saying that, for the pressure values at which ICP fa-cilities are operated (e.g. ≈ 104 Pa and above), the colli-sional rate among the gas particles are sufficiently large tomaintain local equilibrium. A second, and more practicalreason, is the significant stiffness and CPU time reduc-tion compared to Non-LTE (NLTE) situations. More-over, in LTE conditions, the gas thermodynamic andtransport properties are only function of two independentstate variables36 (e.g. pressure and temperature). Hence,they can be easily tabulated to further reduce the com-putational time. Simulations performed by Mostaghimiet al.20,22 and by Zhang et al.37 (in Argon and air plas-mas, respectively) have shown, however, that the use ofthe LTE assumption may not always hold.

An accurate modeling of NLTE effects in ICP plas-mas can be achieved by means of State-to-State (StS)models.38–50 These treat each internal energy state as aseparate pseudo species, thus allowing for non-Boltzmanndistributions. Rate coefficients are usually obtainedthrough quantum chemistry calculations51–56 or throughphenomenological models providing a simplified descrip-tion of the kinetic process under investigation.57,58 State-to-State models provide a superior description comparedto conventional multi-temperature models, which are

based on Maxwell-Boltzmann distributions.59–62 How-ever, due to the large number of governing equations tobe solved, their application to multi-dimensional prob-lems can become computationally demanding.63–67

The purpose of the present paper is development of atightly coupled non-equilibrium model for ICP RF plas-mas. To alleviate the possible occurrence of numericalinstabilities, typical of an explicit coupling approach, thefollowing steps are taken: (i) the flow and the inducedelectric field governing equations are solved in a fully cou-pled fashion, and (ii) steady-state solutions are obtainedby means of a time-marching approach (as often done inCFD applications68). The governing equations are dis-cretized in space by using the Finite Volume method.Time-integration is then performed by means of a fullyimplicit method. As it is shown in the paper, the time-dependent formulation introduces a local relaxation inthe set of space-discretized equations, which enhancesconvergence significantly. The computational ICP frame-work developed in this work allows for the use of bothLTE and NLTE physico-chemical models. Non-LTE ef-fects are described based on a hybrid StS model. A multi-temperature (MT) formulation is used to account forthermal non-equilibrium between translation of heavy-particles and vibration of molecules. Excited electronicstates of atoms are instead treated as separate pseudo-

species. Free-electrons are assumed Maxwellian at theirown temperature.

The paper is organized as follows. Section II describesthe physical model. The numerical method for solvingthe governing equations is given in Sec. III. Computa-tional results are presented in Sec. IV. Conclusions arediscussed in Sec. V.

II. PHYSICAL MODELING

This section describes the physical model developedfor the investigation of non-equilibrium effects in ICPRF plasmas. The non-equilibrium model for the ICPtorch is built based on the torch geometry displayed inFig. 2. To make the problem tractable, the followingassumptions are introduced:

(i) Constant pressure and no macroscopic streaming,

(ii) Charge neutrality and no displacement current,

(iii) Steady-state conditions for gas quantities (i.e.∂()/∂t = 0),

(iv) No gradients along the axial and circumferential di-rections (i.e. ∂()/∂z = 0, ∂()/∂φ = 0).

3

FIG. 2. Torch geometry and adopted reference frame.

A. Electro-magnetic field

The electromagnetic field inside the ICP torch is de-scribed by the Maxwell equations:

∇ ·E =ρc

ǫ0, ∇ ·B = 0, (1)

∇×E = −∂B

∂t, ∇×B = µ0J+ µ0ǫ0

∂E

∂t, (2)

where quantities E and B are the electric and magneticfields, respectively. Quantity ρc stands for the chargedensity. The current density J is assumed to obey Ohm’slaw J = σeE, with σe being the electrical conductiv-ity. Quantities ǫ0 and µ0 are the vacuum permittivityand magnetic permeability, respectively. The applica-tion of the simplifying assumptions just introduced to theMaxwell equations (1)-(2) leads to the induction equationfor the induced toroidal electric field:

∂

∂r

(

1

r

∂rEφ

∂r

)

= −µ0σe∂Eφ

∂t. (3)

Since the induced eddy currents which are responsiblefor the heating of the gas are induced by a primarycurrent whose intensity varies sinusoidally in time, itseems natural to seek for a monochromatic wave solution,Eφ = E exp(ıωt), where ω = 2πf (with f being the fre-quency of the primary current). To account for the pos-sible phase difference between the electric and magneticfields, the amplitude E is taken complex, E = Ere+ıEim.The substitution of E exp(ıωt) in Eq. (3) leads to:

0×∂rUem

∂t+

∂rFem

∂r= rSem. (4)

The electromagnetic (em) conservative variable, flux andsource term vectors are:

Uem =[

Ere Eim

]T, (5)

Fem =

[

∂Ere

∂r

∂Eim

∂r

]T

, (6)

Sem =

[

Ere

r2+ ωµ0σeEim

Eim

r2− ωµ0σeEre

]T

. (7)

Equation (4) must be supplemented with boundary con-ditions at the axis (r = 0) and at the torch wall (r = R,with R being the torch radius). On the axis, due to sym-metry, both components of the electric field must vanish:

Ere = 0, Eim = 0, at r = 0. (8)

The boundary condition at the torch wall is obtained asfollows. The amplitudes of the toroidal electric field andthe axial magnetic field are linked via:6

1

r

∂rE

∂r= −ıωB, (9)

where the amplitude B is taken complex. Immediatelyoutside the wall the magnetic field must be real and, sincethere is no plasma outside the tube, its value can onlydepend on the ICP operating conditions and characteris-tics. If the torch is long enough, the magnetic field at thetorch wall can be approximated with the expression foran infinite solenoid, B = µ0NIc, where quantities N andIc are the number of turns per unit-length and the am-plitude of the primary current. The evaluation of Eq. (9)at the torch wall and the use of the relation B = µ0NIcgives the wall boundary condition for the induced electricfield:

1

r

∂rEre

∂r= 0,

1

r

∂rEim

∂r= −ωµ0NIc, at r = R. (10)

B. Hydro-dynamics

The gas contained in the torch is made of electrons,atoms and molecules. Charged particles comprise elec-trons and positively singly charged ions. The set Sstores the chemical components, and the heavy-particlesare stored in the set Sh. The atomic and molecularcomponents are stored in the sets Sa and Sm, respec-tively. The previously introduced sets satisfy the rela-tions Sh = Sa ∪ Sm and S = e− ∪ Sh, where thesymbol e− indicates the free-electrons. The electroniclevels of the heavy components are stored in sets Iel

s

(with s ∈ Sh) and are treated as separate pseudo species

based on a StS approach.69 The notation si is used todenote the i-th electronic level of the heavy components ∈ Sh, with the related degeneracy and energy beinggelsi and Eel

si , respectively. A multi-temperature model isinstead used for vibration of molecules and translationof free-electrons (with the related temperatures being Tv

and Te, respectively).60 Rotational non-equilibrium ef-fects are disregarded.

Thermodynamic properties

The gas pressure is computed as p = pe+ph, where thesymbol kB stands for Boltzmann’s constant. The par-tial pressures of free-electrons and heavy-particles are,respectively, pe = nekBTe and ph = nhkBT , where quan-tities ne and nh denote, respectively, the related the num-ber densities (with nh =

∑

s∈Shns and ns =

∑

i∈Iels

nsi).The gas total, rotational, vibrational and free-electronenergy densities are:

ρe =3

2p+ ρer + ρev +

∑

s∈Sh

∑

i∈Iels

nsi(Eelsi +∆Ef

s) (11)

4

ρer =∑

s∈Sm

nsErs(T ), ρev =

∑

s∈Sm

nsEvs (Tv), ρee =

3

2pe.

(12)Quantity ∆Ef

s stands for the formation energy (per par-ticle) of the heavy component s ∈ Sh. The average

particle rotational and vibrational energies (Ers and Ev

s ,s ∈ Sm ,respectively) are computed, respectively, accord-ing to the rigid-rotor and harmonic-oscillator models70.Thermodynamic data used in this work are taken fromGurvich tables71 (with the exception of the spectroscopicdata for the electronic levels taken from Ref. 43).

Chemical-kinetics

The NLTE kinetic mechanism for ICP RF plasmas de-veloped in this work accounts for:

(i) Excitation by electron impact,

(ii) Ionization by electron impact,

(iii) Dissociation by electron impact,

(iv) Dissociation by heavy-particle impact,

(v) Associative ionization.

The endothermic rate coefficients for electron inducedprocesses and associative ionization reactions are takenfrom the abba StS model.43–46 Those for dissociation byheavy-particle impact are taken from the work of Park.60

Reverse rate coefficients are obtained based on micro-reversibility.72,73

The mass production terms for free-electrons andheavy-particles are computed based on the zeroth-orderreaction rate theory.72,73 In what follows, the latter quan-tities are indicated with the notation ωe and ωsi , respec-tivelyThe energy transfer terms for the gas vibrational en-

ergy account for (i) vibrational-translational (vt) en-ergy exchange in molecule heavy-particle collisions, (ii)vibrational-electron (ve) energy exchange in moleculeelectron collisions, and (iii) the creation/destruction ofvibrational energy in chemical reactions (cv). The firsttwo energy transfer terms (indicated in what followswith Ωvt and Ωve, respectively) are evaluated based ona Landau-Teller model,74 while the chemistry-vibrationcoupling term (Ωcv) is computed by using the non-preferential dissociation model of Candler.75 The re-laxation times for vt energy transfer are computed bymeans of the modified formula of Millikan and Whiteproposed by Park.60 The energy transfer in molecule-electron inelastic collisions is considered only for N2. Thecorresponding relaxation time is taken from the workof Bourdon.76 The energy transfer terms for the free-electron gas account for energy exchange undergone byfree-electrons in (i) elastic collisions with heavy-particles(Ωel), (ii) inelastic electron induced excitation, ioniza-tion and dissociation processes (Ωin) and (iii) Joule heat-ing (Ωj). The expressions for the first two can be

found in Refs. 44–46. The (time-averaged) Joule heat-ing source term is obtained by averaging over a pe-riod the instantaneous Joule heating power and readsΩj = σe(E

2re + E2

im)/2.31,32

Transport properties and fluxes

Transport phenomena are treated based on the re-sults of the Chapman-Enskog method for the Boltzmannequation77 under the assumption that: (i) inelastic andreactive collisions have a no effect on the transport prop-erties and fluxes and (ii) the collision cross-sections forelastic scattering do not depend on the internal quantumstates.The translational component of thermal conductivity

is λt =∑

s∈ShαλsXs, where the mole fractions of the

heavy components are Xs = nskBT/p (s ∈ Sh). The co-efficients αλ

s are solution of the linear (symmetric) trans-port system for the translational thermal conductivity(see, for instance, Ref. 72 for the details). The con-tributions of the gas rotational and vibrational degreesof freedom to the thermal conductivity (λr and λv, re-spectively) are taken into account by means of the gener-alized Eucken’s correction.72 The thermal and electricalconductivity of the electron gas are:78,79

λe =75

8kB

√

2πkBTe

me

XeΛ22ee

Λ11eeΛ

22ee − (Λ12

ee )2, (13)

σe =3

2

e2

kB

√

2πkBmeTe

XeΛ11ee

Λ00eeΛ

11ee − (Λ10

ee )2, (14)

where the mole fraction of free-electrons is Xe =nekBTe/p and e = 1.602× 10−19 C is the electron charge.Quantities Λij

ee denote the Devoto collision integrals.78

The mass diffusion fluxes are found by solving theStefan-Maxwell equations under the constraints of globalmass conservation and ambipolar diffusion.78–81 The dif-fusion driving forces include only mole fraction gradi-ents. In view of the assumed independence of the elas-tic collision cross-section on the internal quantum states,the Stefan-Maxwell equations are solved for the diffusionfluxes of chemical components (Je and Js, s ∈ Sh, respec-tively). The mass diffusion fluxes for the internal levels(Jsi), are then found as shown in Ref. 6. The gas total,rotational, vibrational and free-electron heat flux are:

q =∑

s∈Sh

∑

i∈Iels

(

5

2kBT + Eel

si +∆Efs

)

Jsims

− λt∂T

∂r+

qr + qv + qe, (15)

qr =∑

s∈Sm

Ers(T )

Jsms

− λr∂T

∂r, (16)

qv =∑

s∈Sm

Evs (Tv)

Jsms

− λv∂Tv

∂r, (17)

qe =

(

5

2kBTe

)

Jeme

− λe∂Te

∂r. (18)

5

Governing equations

The governing equations for the gas chemical compo-sition and temperature distribution in the ICP torch are:

∂rUg

∂t+

∂rFg

∂r= rSg. (19)

The gas (g) conservative variable, flux and source termvectors are:

Ug =[

ρe ρsi ρe ρev ρee]T

, (20)

Fg =[

Je Jsi q qv qe]T

, (21)

Sg =[

ωe ωsi Ωj Ωv Ωe

]T, (22)

i ∈ Iels , s ∈ Sh, with Ωv = Ωvt + Ωve + Ωcv and Ωe =

Ωel +Ωin +Ωj.The boundary conditions used for solving Eq. (19)

are a symmetry boundary condition at the axis and anisothermal non-catalytic boundary condition at the torchwall. Following the work of Mostaghimi et al.,20,22 an adi-abatic wall boundary condition is used for the vibrationaland free-electron temperatures.

III. NUMERICAL METHOD

The governing equations for the gas and the electro-magnetic fields are strongly coupled due to the presenceof the Joule heating term in Eq. (22) and the electricalconductivity in Eq. (7). This suggests to adopt a fullycoupled approach by casting Eqs. (4) and (19):

∂rΓU

∂t+

∂rF

∂r= rS, (23)

where the conservative variable, flux and source termvectors are now U = (Ug, Uem), F = (Fg, Fem) andS = (Sg, Sem), respectively. The matrix Γ in Eq. (23)reads:

Γ =

(

I(ns+nt)×(ns+nt) 0(ns+nt)×2

02×(ns+nt) 02×2

)

, (24)

where quantities I and 0 are, respectively, the identityand null matrices. Their number of rows and columnsare indicated by the first and second lower-scripts, re-spectively. The symbols ns and nt denote, respectively,the number of species and temperatures.

Spatial discretization

The application of the Finite Volume method to Eq.(23) leads to the following ODE governing the time-evolution of the conservative variables of cell j:68

Γ∂Uj

∂trj∆rj = −Resj . (25)

The right-hand-side residual reads:

Resj = rj+ 1

2

Fj+ 1

2

− rj− 1

2

Fj− 1

2

− Sj rj ∆rj , (26)

where the cell volume (length) and its centroid lo-cation are computed as ∆rj = rj+1/2 − rj−1/2 andrj = (rj+1/2 + rj−1/2)/2, respectively. The evaluation ofthe diffusive flux Fj+1/2 is performed by approximatingthe values and the gradients of a given quantity p (e.g.temperatures and electric field components) by meansof an arithmetic average and a second order central fi-nite difference, respectively. To facilitate the implemen-tation of the constant pressure constraint, the solutionupdate is performed on primitive variables (P) consistingof mass fractions, temperatures and electric field compo-nents, P = (ye, ysi , T, Tv, Te, Ere, Eim):

ΓTj∂Pj

∂trj∆rc = −Resj . (27)

The transformation matrix T can be obtained from thetime-derivative of the conservative variables (∂U/∂t) byexploiting the global continuity equation, ∂ρ/∂t = 0.

A. Temporal discretization

Equation (27) is integrated in time by means of thebackward Euler method:68

ΓTnj

δPnj

∆tjrj∆rj = −Res

n+1j , (28)

where δPnj = P

n+1j − P

nj . The local time-step ∆t is

computed based on the von Neumann number (ξ) as∆t = ξ/(2ρd), where quantity ρd stands for the spectralradius of the diffusive flux Jacobian matrix ∂F/∂U.68 Toadvance the solution from the time-level n to the time-level n + 1, Eq. (28) is linearized around the time-leveln. The outcome of the linearization is a block-tridiagonalalgebraic system to be solved at each time-step:68,82

Lnj δP

nj−1 +C

nj δP

nj +R

nj δP

nj+1 = −Res

nj , (29)

where the left, right and central block matrices are:

Lj =2 rj−1/2

∆rj +∆rj−1Aj− 1

2

, (30)

Rj =2 rj+1/2

∆rj +∆rj+1Aj+ 1

2

, (31)

Cj =

[

ΓTj

∆tj−

(

∂S

∂P

)

j

]

rj∆rj − (Lj +Rj), (32)

where quantity A stands for the (primitive) diffusive fluxJacobian matrix, A = ∂F/∂P. The block-tridiagonalsystem (29) is solved by means of Thomas’ algorithm68

and the solution updated at the next time-level, Pn+1j =

Pnj + δPn

j . This process is continued until steady-stateis reached.

6

Notice that the discretized time derivative in Eq. (32)plays the role of a relaxation term. Setting this termto zero (i.e. infinite time-step) is equivalent to solvethe steady-state form of the governing equations (23)by means of Newton’s method. Preliminary calculationsperformed in LTE conditions indicated that this strategycan easily lead to numerical instability problems in theinitial transient of the simulation.

IV. APPLICATIONS

The gas contained in torch consists of molecular nitro-gen and the related dissociation and ionization products,S = e−, N, N2, N

+2 , N

+. Simulations have been per-formed by means of the abba StS model43–46 and theMT model developed by Park.60

The torch radius, the number of coils per unit-lengthand the wall temperature are set to 0.08m, 50, and 350K,respectively. The current intensity of the primary circuitis found from the solution by imposing that the dissipatedpower (per unit-length) in the plasma:

P = 2π

∫ R

0

Ωj r dr, (33)

is equal to a fixed value P0. To match the conditionP = P0 at steady-state, the current intensity is multi-plied by the scaling factor γ =

√

P0/P after updatingthe solution at the new time-step. This approach wasoriginally introduced by Boulos17 in the 1970’s, and ithas been used since then by other investigators.18,31–33,37

In the present work, no scaling is applied to the electricfield (as opposed to explicit coupling methods32), due tothe use of a fully coupled formulation.In order to assess the influence of the ICP operating

conditions on non-equilibrium effects, different values ofpressure, frequency of the primary circuit and dissipatedpower have been adopted (see Table I).

TABLE I. Adopted values for the pressure, frequency of theprimary circuit and dissipated power (per unit-length) in theplasma.

Quantity Units ValuesPressure (p) Pa 3000, 5000 and 10 000Frequency (f) MHz 0.1, 0.5, 1 and 2.5Dissipated power (P0) MW/m 0.2, 0.3, 0.35 and 0.4

A. Performance of the coupled numerical formulation

The coupled numerical method developed in Sect. IIIhas been applied over a wide spectrum of operating con-ditions (given in Table I) in both LTE and NLTE condi-tions. In all cases treated in this work, no need arose forthe use of techniques to cope with numerical instabilities

such as under-relaxation factors, conservative iterativestrategies (e.g. Picard’s method) or choice of a smart

initial guess for the solution.To demonstrate in practice the robustness and the

effectiveness of the proposed computational method, aMT NLTE simulation has been chosen as working ex-ample. The operating conditions are: p = 10 000Pa,f = 0.5MHz and P0 = 0.35MW/m. For the sake of sim-plicity, an isothermal wall boundary conditions has beenused for all temperatures. The solution has been initial-ized with a uniform equilibrium distribution at 7500K,with both the real and imaginary electric field compo-nents set to 0.1V/m. The initial value of the currentintensity was 250A.

0 20 40 60 80 100Iter. number

120

130

140

150

160

I c(A

)

(a)

0 20 40 60 80 100Iter. number

-10

-8

-6

-4

-2

0

err

L2

(T)

(b)

FIG. 3. Time-history of (a) current intensity and (b) nor-malized L2 norm of the relative error on the translational-rotational temperature for the MT NLTE model (p =10 000Pa, f = 0.5MHz, P0 = 0.35MW/m; isothermal bound-ary condition for all temperatures).

In the early stages of the numerical simulation, stronggradients in temperature and chemical composition formin correspondence of the wall, due to the isothermal

7

boundary condition imposed. Despite the challenges im-posed by the problem, the initial value of the von Neumannumber was set to to 1× 105, and increased interactively(every 25 iterations) up to 5× 108. Such large valueswere possible due to the adoption of a fully implicit time-integration method. Figure 3 shows the time-history ofthe current intensity and the L2 norm of the relative erroron the translational-rotational temperature:

errnL2(T ) =

√

√

√

√

1

nc

nc∑

j=1

(δTnj )

2, (34)

where quantity nc denotes the number of cells, andδTn

j = Tn+1j − Tn

j . A converged solution is achieved

in less than 100 iterations (see Fig. 3(b)), with a reduc-tion of more than nine orders of magnitude in the relativeerror for the temperature. In practice, the solution is al-ready converged after 80 iterations, as can be observedfrom the current intensity time-history (see Fig. 3(a)).Figure 4 shows the temperature distribution along

the torch radius. Due to the efficient energy exchangein N2-e

− interactions,60,76,83 the vibrational and free-electron temperature profiles are essentially indistin-guishable. This feature has been observed in all NLTEsimulations performed. This is the reason for the use ofthe notation Tve in Fig. 4 (and also in what follows).

0 0.02 0.04 0.06 0.08r (m)

0

2000

4000

6000

8000

10000

T, T

ve (

K)

FIG. 4. MT NLTE temperature distribution (p = 10 000Pa,f = 0.5MHz, P0 = 0.35MW/m; unbroken line T , dashed-lineTve; isothermal boundary condition for all temperatures).

B. Assessment of non-equilibrium effects

Before discussing in detail NLTE effects, LTE simula-tions have been performed and compared with the MTNLTE results to assess the extent of the departure fromequilibrium. The pressure is set to 10000Pa as, for thisrelatively high value, LTE conditions are often assumed.6

The frequency and the dissipated power are 0.5MHz and0.35MW/m, respectively.

0 0.02 0.04 0.06 0.08

r (m)

0

2500

5000

7500

10000

T (

K)

(a)

0 0.02 0.04 0.06 0.08

r (m)

0

10

20

30

40

50

60

Ωj[M

W/m

3]

(b)

FIG. 5. Comparison between the LTE and the MT NLTEtemperature (a) and Joule heating (b) distributions (p =10 000Pa, f = 0.5MHz, P0 = 0.35MW/m; unbroken lineLTE, dashed line NLTE).

Figure 5 compares the LTE and NLTE translational-rotational temperature and Joule heating distributions.Close to the wall, the curvature of the LTE tempera-ture distribution changes sign. This is a consequenceof the non-monotone behavior of the equilibrium totalthermal conductivity of the working gas (nitrogen). Thesame trend is also found for the MT solution, though lesspronounced. The MT model predicts that the gas is inthermal equilibrium close to the axis (see Fig. 4). Inboth the LTE and NLTE simulations, the temperatureis maximum on the axis, due to the absence of radiativelosses in the plasma.9,10 Overall, the LTE solution pre-dicts higher temperature values, with the difference beingmaximum on the axis. This is a general trend observed inall the simulations performed in this work (and also in themulti-dimensional results obtained by other investigatorsin Refs. 6 and 37). In NLTE conditions, the temperatureis lower because the plasma is heated over a wider regioncompared to LTE conditions. This is confirmed by the

8

Joule heating distribution shown in Fig. 5(c). Analogousconclusions can be drawn when comparing with the StSNLTE model adopted in this work.The results in Fig. 5 demonstrate that, even at rela-

tively high pressures, the LTE assumption can lead to asevere overestimation of the gas temperature and, in gen-eral, to an inaccurate prediction of the thermo-chemicalstate of the gas. This is further confirmed by the com-parison for the mole fractions given in Fig. 6. It isworth recalling that in the present work, the effects ofmacroscopic flows (which enhance non-equilibrium) areneglected.

0 0.02 0.04 0.06 0.08r (m)

10-3

10-2

10-1

100

Xs

N2

e-, N

+

N

FIG. 6. Comparison between the LTE and the MT NLTEmole fraction distributions (p = 10 000Pa, f = 0.5MHz, P0 =0.35MW/m; unbroken line LTE, dashed line NLTE). Due tothe low concentration of N+

2 (not shown), the mole fractionsof e− and N+ are essentially indistinguishable in both LTEand NLTE conditions.

C. Influence of frequency and power on non-equilibrium

In order to assess the influence of operating conditionson non-equilibrium phenomena, it was decided to per-form a parametric study on the frequency of the primarycircuit and the power dissipated in the plasma. Figures 7-8 show the results of this investigation for the MT NLTEmodel at p = 5000Pa. Increasing the frequency (Fig. 7)has the effect of narrowing the extent of the skin-depth,which is the zone over which most of the power is dissi-pated by Ohmic heating. This can be seen from the re-lation for the skin-depth14,15, δ = (σeπµ0f)

−1/2. The re-duction of the skin-depth induces sharper gradients closeto the wall, thereby enhancing non-equilibrium. Thesefindings are in accordance with the observations reportedby Mostaghimi et al.22 for NLTE Argon plasmas. The in-crease of the dissipated power (Fig. 8) has an oppositeeffect. As can be observed from the results, higher powerlevels favor the establishment of thermal equilibrium con-ditions.

0 0.02 0.04 0.06 0.08

r (m)

0

2500

5000

7500

10000

12500

T, T

ve (

K)

FIG. 7. MT NLTE temperature distributions for differentvalues of the frequency of the primary circuit: unbroken linef = 0.1MHz, dashed line f = 0.5MHz, dotted-dashed linef = 1MHz, dotted line f = 2.5MHz (p = 5000Pa, P0 =0.35MW/m; unbroken line T , line with symbols Tve).

0 0.02 0.04 0.06 0.08

r (m)

0

2500

5000

7500

10000

T, T

ve (

K)

FIG. 8. MT NLTE temperature distributions for differ-ent values of the dissipated power (per unit-length) in theplasma: unbroken line P0 = 0.2MW/m, dashed line P0 =0.3MW/m, dotted-dashed line P0 = 0.35MW/m, dotted lineP0 = 0.4MW/m (p = 5000Pa, f = 0.5MHz; unbroken lineT , line with symbols Tve).

D. Comparison between the StS and the MT predictions

This section compares the predictions obtained by theStS and MT NLTE models. Figure 9 shows the compar-ison in terms of temperatures and N mole fraction. Forboth the StS and MT solutions, decreasing the pressurehas the effect of enhancing thermal non-equilibrium. Inthe case of the MT model, the translational-rotationaltemperature is maximum on the axis and decreasesmonotonically when approaching the wall.

On the other hand the free-electron temperature increases, reaches a maximum and then decreases until it reaches the value determined from the adiabatic bound-

9

0 0.02 0.04 0.06 0.08

r (m)

0

2000

4000

6000

8000

10000

T, T

ve

(K)

(a)

0 0.02 0.04 0.06 0.08

r (m)

0

2000

4000

6000

8000

10000

T, T

ve

(K)

(c)

0 0.02 0.04 0.06 0.08

r (m)

0

2000

4000

6000

8000

10000

T, T

ve

(K)

(e)

0 0.02 0.04 0.06 0.08

r (m)

0

0.2

0.4

0.6

0.8

1

XN

(b)

0 0.02 0.04 0.06 0.08

r (m)

0.2

0.4

0.6

0.8

1

XN

(d)

0 0.02 0.04 0.06 0.08

r (m)

0.4

0.6

0.8

1

XN

(f)

FIG. 9. Comparison between the translational-rotational temperature (left) and the N mole fraction (right) distributionspredicted by the StS and the MT NLTE models at different pressures: (a)-(b) p = 10 000Pa, (c)-(d) p = 5000Pa, (e)-(f)p = 3000Pa (f = 0.5MHz, P0 = 0.35MW/m; in (a), (c) and (e) unbroken line T StS, dashed line Tve StS, dotted-dashed lineT MT, dotted line Tve MT; in (b), (d) and (f) unbroken line StS, dashed line MT).

ary condition. This behavior is due to the balance be-tween the Joule heating (which heats up the electron gas)and the energy loss in elastic and inelastic collisions, and

chemical reactions. The peak location of the free-electrontemperature moves towards the wall when decreasing thepressure. This is a consequence of the Joule heating dis-

10

tribution (not shown in Fig. 9) which becomes sharperand clustered to the wall at lower pressures. It is worthnoticing that, in the StS simulation, (i) the translational-rotational temperature no longer exhibits a monotonebehavior and (ii) the axis values of both temperaturesare systematically lower than those predicted by the MTmodel. The differences observed between the StS andMT temperature distribution have an effect, as it shouldbe expected, on the chemical composition (see Figs. 9(b),9(d) and 9(f)).Figure 10 shows the normalized population of the elec-

tronic levels of N on the torch axis (circles), in the mid-point of the torch (squares) and at the wall (triangles)at different pressures. The population exhibit significantdistortions from a Boltzmann shape only close to the wall(where recombination occurs). Deviations from a Boltz-mann distributions are more significant when increasingthe pressure due to higher recombination.

V. CONCLUSIONS

A tightly coupled magneto-hydrodynamic solver forthe study of the weakly ionized plasmas found in RFdischarges has been developed. A hierarchy of thermo-physical models have been added to the solver to modelthe non-equilibrium effects in atomic and molecular plas-mas. These include LTE, multi-temperature and themore sophisticated State-to-State models. The govern-ing equations for the flow and electromagnetic fieldshave been written as a system of coupled time-dependentconservation-laws. Steady-state solutions have been ob-tained by means of an implicit Finite Volume Method.Results obtained by using a multi-temperature and

State-to-State models have shown that the LTE assump-tion does not hold and that its use can lead to a wrongprediction of the thermo-chemical state of the gas. Theanalysis of the temperature distribution in the torch indi-cated that non-equilibrium plays an important role closeto the walls, due to the combined effects of Ohmic heat-ing, and chemical composition and temperature gradi-ents. The accurate study of the population of excitedelectronic states has shown that, in view of the absenceof a macroscopic gas flow, non-Boltzmann distributionsare limited to a narrow region close to the torch wall.

ACKNOWLEDGMENTS

Research of A. Munafo was supported by the Univer-sity of Illinois at Urbana-Champaign Starting Grant. Re-search of M. Panesi and S. A. Alfuhaid was supported bythe AFOSR Summer Faculty Fellowship Program. Re-search of J.-L. Cambier was sponsored by AFOSR grant14RQ13COR (PM: M. Birkan).

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0 2.5 5 7.5 10 12.5 15Ei(eV)

1012

1013

1014

1015

1016

1017

1018

1019

1020

1021

1022

1023

ni/gi(m

-3)

(a)

0 2.5 5 7.5 10 12.5 15Ei(eV)

1012

1013

1014

1015

1016

1017

1018

1019

1020

1021

1022

1023

ni/gi(m

-3)

(b)

0 2.5 5 7.5 10 12.5 15Ei(eV)

1012

1013

1014

1015

1016

1017

1018

1019

1020

1021

1022

1023

ni/gi(m

-3)

(c)

FIG. 10. Normalized population of the electronic levels of Nat different pressures: (a) p = 10 000Pa, (b) p = 5000Pa,(c) p = 3000Pa (f = 0.5MHz, P0 = 0.35MW/m; line withcircles r = 0m [axis], line with squares r = 0.04m, line withtriangles r = 0.08m [wall]).

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