Understanding - sustained plasmas at intermediate pressure by 2D modeling and experiments

Citation for published version (APA): Georgieva, V., Berthelot, A., Silva, T., Kolev, S., Graef, W., Britun, N., Chen, G., van der Mullen, J., Godfroid, T., Mihailova, D., van Dijk, J., Snyders, R., Bogaerts, A., & Delplancke-Ogletree, M. P. (2017). Understanding microwave surface-wave sustained plasmas at intermediate pressure by 2D modeling and experiments. Plasma Processes and Polymers, 14(4-5), [1600185 ]. https://doi.org/10.1002/ppap.201600185

DOI: 10.1002/ppap.201600185

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Understanding Microwave Surface-Wave Sustained Plasmas at Intermediate Pressure by 2D Modeling and Experiments

Violeta Georgieva,* Antonin Berthelot, Tiago Silva, Stanimir Kolev, Wouter Graef, Nikolay Britun, Guoxing Chen, Joost van der Mullen, Thomas Godfroid, Diana Mihailova, Jan van Dijk, Rony Snyders, Annemie Bogaerts, Marie-Paule Delplancke-Ogletree

An Ar plasma sustained by a surfaguide wave launcher is investigated at intermediate pressure (200–2667 Pa). Two 2D self-consistent models (quasi-neutral and plasma bulk-sheath) are developed and benchmarked. The complete set of electromagnetic and fluid equations and the boundary conditions are presented. The transfor- mation of fluid equations from a local reference frame, that is, moving with plasma or when the gas flow is zero, to a laboratory reference frame, that is, accounting for the gas flow, is discussed. The pressure range is extended down to 80 Pa by experimental measurements. The electron temperature decreases with pressure. The electron density depends linearly on power, and changes its behavior with pressure depending on the product of pressure and radial plasma size.

Dr. V. Georgieva, G. Chen, Dr. J. van der Mullen, Dr. St. Kolev Prof. M.-P. Delplancke-Ogletree Faculty of , Sofia University, Blvd ‘‘James Bourchier’’ 5, 1164, UniversiteLibredeBruxelles, Ecole Polytechnique de Bruxelles, Sofia, Bulgaria Service4MAT,av.F.D.Roosevelt50-CP165/63,1050,Brussels,Belgium Dr. W. Graef, Dr. D. Mihailova, Dr. J. van Dijk E-mail: [email protected] Department of Applied Physics, Research Group EPG, Eindhoven Dr. V. Georgieva, A. Berthelot, Dr. St. Kolev, Prof.A.Bogaerts University of Technology, P.O. Box 513, 5600 MB, Eindhoven, the Department of Chemistry, Research Group PLASMANT, University Netherlands of Antwerp, Universiteitsplein 1, B-2610, Wilrijk-Antwerp, Belgium Dr. T. Godfroid, Prof. R. Snyders Dr. T. Silva, Dr. N. Britun, G. Chen, Prof. R. Snyders Materia Nova Research Center, av. Nicolas Copernic 1, 7000, Universite de Mons, ChIPS, av. Nicolas Copernic 1, 7000, Mons, Mons, Belgium Belgium

Plasma Process Polym 2016, DOI: 10.1002/ppap.201600185 ß 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim wileyonlinelibrary.com DOI: 10.1002/ppap.201600185 1 R Early View Publication; these are NOT the final page numbers, use DOI for citation !! V. Georgieva et al.

1. Introduction away from the wave launcher. The energy from the wave is transferred to the plasma by the charged particles, Microwave (MW) induced discharges are well known for especially by the electrons since they can follow the EM >> their high degree of non-equilibrium (Te Tgas), the high field and gain energy from it. The electrons transfer the level of applied energy absorption by plasma electrons, and energy to the heavy particles through collisions. relatively easy operation.[1–4] Based on the configuration Examples of SW plasma generators are the surfatron, and reactors and the way of coupling the MW energy into the the surfaguide launchers.[1–4] We have applied a surfaguide plasma, MW discharges can be divided into two general sustained discharge at intermediate pressure for plasma- types: localized, that is, plasma created in resonant cavity catalytic greenhouse gas conversion into valuable chem- reactors and plasma torches; and traveling-wave dis- icals.[21] Such an application requires careful investigation charges, that is, surface-wave (SW) plasma columns.[1–4] of the plasma column in order to achieve optimal They find application in a wide range of technology fields dissociation rates of molecular species as well as optimal such as surface treatment,[5,6] material processing and corresponding energy efficiency. Experimental measure- (nanoparticle) synthesis,[7–9] plasma diagnostics,[10] ments are not always possible at different positions of the plasma medicine (sterilization, skin treatment),[11–16] in plasma tube or on the catalytic surface, because of the solid atomic and molecular spectrometry,[17,18] and in dissocia- metallic parts surrounding the set-up in order to prevent tion of greenhouse gases.[19–21] microwave energy leakage (see below for the description of A SW sustained plasma, which is the topic of the present the experimental set-up in Section Experimental Set-Up research, is created by electromagnetic (EM) and the Computational Domain and Refs. [21,47]). Therefore, propagating along the boundary between the plasma computer modeling of the plasma region can bring a and a dielectric (usually a quartz tube).[1] The plasma SWs valuable insight into the process. Understanding the were discovered in the 1960’s.[22] The first application of EM plasma-wave interplay in the relatively simple Ar surface waves in the MW range for sustaining chemistry case is a first step towards understanding plasma was developed by Moisan et al. in 1974.[23] Since complex chemical plasmas in future. then SW sustained discharges attracted significant The plasma column in SW discharges is non-uniform attention due to their flexibility in terms of continuous both in the radial and axial direction. The strong coupling of or pulsed operation regimes, in a wide range of gas pressure the wave and the plasma requires considerable computa- (10 2–105 Pa), applied frequency (300 MHz–10 GHz), and tional resources even when a number of simplifications are geometry size and shape.[1–4] A considerable amount of applied. Initially, an approach of dividing the problem into research has been done in a number of groups leading to a two 1D problems was developed. The plasma is divided into vast number of publications (cf. Refs.[1–21] and [23], present- thin slabs in the axial direction, assuming that the local ing specific applications, Refs. [1–3] and [24–42], presenting axial gradients are negligible compared to the correspond- analytical or modeling research, and Refs.[1–3] and,[43–52] ing radial gradients.[24,25] The radial distribution of the presenting experimental or combined theoretical and plasma parameters and the wave propagation character- experimental research). Just to mention some of the groups istics are obtained from the plasma maintenance equations explicitly, we would like to refer to the pioneering work and the wave-field equation for given discharge operating on the theory of SW propagation in guiding structure conditions and power delivered per unit length at a given with different geometries and on the SW discharge axial position. Then the attenuation characteristic of the applications done in Montreal, Lisbon, Sofia, Bochum, and wave is found. The axial distribution of the power density Moscow, as well as in a number of collaborations is calculated from the wave energy balance equation. among these groups and thus forming the solid basis Merging the solution for each plasma slab allows deter- for further SW sustained plasma research and mining the spatial distributions.[24,25] Next, the kinetics of applications.[1,3,11,12,19,23–25,27–35,37,43–45] the SW discharges was investigated which resulted in The principle of operation of a SW plasma source is based similarity laws.[26,27] Further, the attention was turned on the fact that the plasma is the only dielectric medium again to the wave properties, which particularly determine e [28,29] which can have a negative , p, provided that the waveguided plasma properties. Analyzing the the plasma frequency is larger than the applied wave behavior of the space damping rate of the wave, an [1] frequency. When bounded to a dielectric with positive analytical solution for the axial distribution of the electron e e > e ; [28] permittivity d and provided that p d plasmas can density was found. The theory of the sustain surface wave propagation with evanescent fields on propagation and wave- plasma interaction are presented in both sides of the boundary.[1,53] Propagation of the surface detail in Refs. [1,3] and [29]. Other simplified approaches have wave ionizes the plasma and the wave is sustained by also been applied: (i) considering a 1D problem in the the produced plasma. Hence, a mutual plasma-wave axial direction, and assuming uniformity in the radial interaction is observed. The plasma column extends far direction[30–32] or applying a radial profile of Bessel-type of

Plasma Process Polym 2016, DOI: 10.1002/ppap.201600185

2 ß 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim DOI: 10.1002/ppap.201600185 R Early View Publication; these are NOT the final page numbers, use DOI for citation !! Understanding Microwave Surface-Wave Sustained Plasmas ...

the electron density, which is obtained from an analytical ionization mechanism, the role of ambipolar diffusion solution of the electron density balance equation[33]; (ii) versus volume recombination.[3,41,55,56] and the coupling to calculation of the electron energy distribution function the electric field. According to the classification,[3,41] the (EEDF) from the Boltzmann equation[34]; (iii) applying a main electron loss mechanism is ambipolar diffusion for pR collisional radiative model (CRM) for plasma production in a (Pa mm) in the range of 10–1300. The next regimes are the uniform EM field[35]; (iv) coupling the CRM to the EM transition region when volume recombination (1300–2600) field equations in 1D (axial or radial) models.[36–38] More and molecular assisted recombination (2600–2 104) start elaborated, self-consistent, 2D models solving self- playing a role. The main electron production mechanism in consistently Maxwell’s equations describing the EM field the three regimes is step-wise ionization.[3,41] and the set of plasma fluid equations in ambipolar diffusion For the plasma source under study, pR is in the range approximation and neglecting sheaths, have been 700–2 104 Pa mm, which implies that stepwise ioniza- developed using academic or commercial plasma modeling tion, volume recombination, and molecular assisted frameworks. [39–42] These models were applied to atmo- recombination have to be taken into account. Therefore, spheric pressure cylindrical (surfaguide or surfatron)[40,41] the description of the Ar chemistry, that is, the different and to intermediate pressure coaxial microwave dis- species and the list of transitions between them, is of charges.[42] The self-consistent solution of the plasma- considerable importance for reliable calculation results. At wave equation set in 2D gives direct information on the the same time, the number of followed species should be plasma characteristics and electromagnetic field space kept limited in order to achieve a reasonable CPU time. We distributions and reduces considerably the assumptions consider a set of Ar chemistry consisting of Ar neutrals, þ þ that need to be made when the problem is considered in 1D. electrons, the atomic Ar and molecular Ar2 ions and two Moreover, adding the heavy particle energy transport excited states grouping the 4s and 4p levels each, and an equation to the set of equations is another advantage of the exhaustive list of reactions between the species. The set 2D models. Therefore, the 2D models are an important step was previously applied for the simulation of a gliding in the development of the SW sustained discharge arc discharge[57] and is presented in the Supporting modeling. Information. Despite its limitations, the ambipolar approximation has The developed simulation models have pressure been successfully applied in intermediate pressure limitations. When the pressure decreases, the mean free regimes.[16,36,42,54] A model that resolves the space-charge path of the electrons increases and at 100 Pa it is calculated sheath region has also been developed, however, it to be 4.5 mm (see Section Plasma Fluid Equation Set), which considers only the radial description, that is, 1D model.[38] is comparable with the radius (7 mm) of the plasma tube. Further elaboration of the 2D models is the consideration of Therefore, the lowest pressure limit at which the solution is a gas flow,[39–41] which introduces additional complexity in stable was found to be 200 Pa. The results from the models the system of the fluid equations as it will be explained are extended with measurements of the plasma character- below in the model description section. istics of the modeled set-up. Using a self-absorption method In this paper, we present and compare two 2D self- associated to optical emission spectroscopy (OES),[58] the consistent models of microwave surfaguide discharges metastable argon density and electron temperature, taking the gas flow into account and we extend the and the electron density are evaluated in the range simulation pressure range by experimental measurements. 80–170 Pa.[59] The intensity of the excited Ar atom emission The applied frequency is 2.45 GHz, the pressure is in the decreases whit increasing pressure similarly to the reported range of 80–2667 Pa, and the applied power varies from 50 measurements in literature.[49,52] Experimentally, we to 200 W in the simulation and experiment. The choice of found that the maximum pressure at which the optical the pressure range is based on the application of the signal is strong enough to define the Ar(4s) group density is [21] microwave discharges for CO2 dissociation. In addition, 170 Pa. The gas temperature is measured by adding N2 to we found that the investigation of Ar discharges sustained the Ar plasma.[60] The modeling and experimental results by a surfaguide wave launcher in the range 300–3000 Pa is are also compared with measurements, available in the scarce in the literature.[26] literature, in Ar surfaguide[26,37] and surfatron[49,50] One of the developed models assumes infinitely thin discharges at the position where the wave is launched sheaths and applies the ambipolar diffusion approxima- and for similar operating conditions. tion. The other model resolves the sheaths by solving The influence of the applied power and pressure on the Poisson’s equation. plasma characteristics is investigated both by simulations An important classification parameter to analyze the and by measurements. different plasma sources is the pressure-radius pR prod- The paper continues with a description of the uct.[3] Increasing the value of this product leads to changes experimental set-up and the corresponding simulation in the plasma behavior with respect to the nature of the domain (Section Experimental Set-Up and the

Plasma Process Polym 2016, DOI: 10.1002/ppap.201600185 ß 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.plasma-polymers.org 3 R Early View Publication; these are NOT the final page numbers, use DOI for citation !! V. Georgieva et al.

Computational Domain), a presentation of the two 2D models (Section Description of the Models) and of the plasma diagnostics methods used in the experimental measurements (Section Plasma Diagnostics). The results are presented and discussed in Section Results and Discussion. Finally, conclusions are given in Section Conclusion.

2. Experimental Set-Up and the Computational Domain

The experimental set-up for the microwave plasma generation is shown in Figure 1. Symmetric surface waves are launched by a surfaguide operating at 2.45 GHz in the continuous regime. The discharge is

generated in a quartz tube with a 7 mm inner radius R0, surrounded by a polycarbonate tube with a 16 mm inner radius. The quartz and polycarbonate tubes have each 3 mm thickness. The inner tube is cooled by an oil flow at Figure 2. 2D axisymmetric computational domain. The metal 10 8C. The metallic grid, which surrounds the plasma boundaries are drawn in blue and confine the computational EM region. The plasma region is located in the inner quartz tube tubes and forms a Faraday cage, has a radius of 50 mm. and is coloured purple. There are two metal rings which confine the electro- magnetic field in a region of 31 cm along the discharge tube (see also Figure 2 below, where the corresponding The description of surface waves propagating at the computational domain is shown). The center of the plasma-dielectric boundary requires, generally speaking, a quartz tube is positioned in the gap. The gas 3D approach. It is found that the product of the applied mixture injected from the top of the system is regulated frequency and the plasma tube radius defines the propaga- by electronic mass flow controllers. Further details of the tion mode of the surface wave. [1,44] At the conditions under experimental surfaguide system used in the present study,thisproductis1.715 GHz.cmandhencethesurfaguide [47] research can be found in ref. The operating launches symmetric surface waves propagating in trans- conditions under study are the following: pressure verse magnetic (TM) mode, that is, azimuthally symmetric range 80–2667 Pa, applied power range 50–200 W, and (m ¼ 0) waves. Therefore, taking into account also the a flow rate of 125 sccm. cylindrical symmetry of the plasma tube, we can describe the system by a 2D axisymmetric model. The computational domain corresponding to the experi- mental set-up is presented in Figure 2. In the developed models, the fluid description of the plasma is coupled to a self- consistent solution of Maxwell’s equations. Maxwell’s equa- tions are solved in the region called EM region. In Figure 2 this is given by the region surrounded by the blue lines denoting the metal boundaries (either a metal grid or metal rings at the plasma tube). The plasma fluid equations are solved in the plasma region, that is, the inner quartz tube (see Figure 2). The difference in the two models comes from the different descriptions of the plasma. In the first model the sheath is assumed infinitely thin and is neglected, thus assuming quasi- neutral plasma in the complete volume. This model will be referred to as quasi-neutral (QN) model further in the text. It is developed in the modeling framework PLASIMO.[39,41,42,61–63] The second model considers the formation of the sheath next to the tube wall, and hence it is referred to as plasma bulk-sheath Figure 1. Schematic representation of the surface-wave (PS) model below. The PS model is developed within the 1 [64] microwave set-up. commercial software COMSOL Multiphysics ref.

Plasma Process Polym 2016, DOI: 10.1002/ppap.201600185

4 ß 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim DOI: 10.1002/ppap.201600185 R Early View Publication; these are NOT the final page numbers, use DOI for citation !! Understanding Microwave Surface-Wave Sustained Plasmas ...

3. Description of the Models equations the curl equations (Maxwell-Ampere’s and Faraday’s laws) predict the wave propagation. Therefore, We shall start with the features common to both models. these two equations are solved in the EM region in order The following assumptions apply: to calculate the electric, ~E, and magnetic, H~,fieldvectors. For symmetric surface waves propagating in TM mode, [65] 1. The EEDF is assumed to be Maxwellian ref. the non-zero electric and magnetic field components in a w When the EEDF is close to the Maxwell-Boltzmann cylindrical coordinate system (r, ,z)areEr, Ez,andHw, ~ distribution, the average electron energy hie is propor- respectively, with the wave-vector k directed along the [1] tional to the electron temperature, Te : hie ¼ 3kBTe=2, z-axis. 1 where kB is the Boltzmann constant in J K and Te is The electromagnetic field equations are solved in the expressed in K. Very often the electron energy is frequency domain, assuming that all harmonic compo- expressed in eV (1 eV ¼ 1.6022 10 19 J). Hence, the nents above the fundamental frequency can be neglected, relation between the average electron energy and that is, assuming a sinusoidal signal. Therefore, the phasor electron temperature becomes[66] notation FðÞ¼r; w; z; t Re½ FðÞ r; w; z expðivtÞ is used and the time dependency is removed from Maxwell’s equa- 3 [41,68] hie ½¼eV kT ½eV; ð1Þ tions, giving 2 e r~ ¼ ~; ð Þ E ivm0H 2 where k is 1 when the electron temperature is expressed in eV and is often omitted. ~ ~ ~ rH ¼ J þ ivere0E: ð3Þ 2. All heavy species are assumed to have a Maxwellian e e distribution. The dielectric permittivity r is a scalar, 0 and m0 are the 3. The plasma is characterized by two temperatures: the permittivity and permeability of free space, v is the wave 2 electrons have a temperature Te and all heavy particles angular frequency, and i is the imaginary unit, i ¼1. The 6¼ : have one temperature Th. In general Te Th relative magnetic permeability of the waveguide system is 4. We consider a weakly ionized collisional plasma, which taken as 1.[29] The ion mobility is low compared to the means that the collision frequency of electrons and ions electron mobility, so we can neglect the ion current and with neutral atoms greatly exceeds the collision hence the total current density is equivalent to the electron frequency of these particles with one another. Hence current density. The electron current density~J is calculated the electron-electron (e-e) and ion-ion collisions are not from Ohm’s lawintroducing a complex conductivity s^:[41,68] considered in the models. However, the e-e collisions are taken into account within the solution of the BE in ~J ¼ s^~E; ð4Þ Bolsig þ [67] and thus in the calculation of the e-Ar reaction rate coefficients. 5. The flow is laminar, that is, relatively low values of the e2 n ðÞr; z s^ðÞ¼r; z e : ð5Þ Reynolds number. me ½nmðÞþr; z iv 6. In the QN model, we consider a stationary case, that is,

zero time-derivatives of all plasma characteristics. In the where ne(r,z) is the electron number density, me and e are

PS model the time-derivatives are not neglected. the electron mass and charge, respectively, and vm(r,z)is The time steps are calculated by COMSOL based on the total elastic momentum transfer frequency between the convergence of the solver for each time step (i.e., if electrons and heavy particles. The equation of the complex convergence, the time step is reduced, otherwise the conductivity is valid in a cold plasma approximation, which time step is increased). Since the PS model runs until a means that the electron thermal velocity is small compared steady state solution is found, the time dependence has to the wave phase velocity.[69] Surface waves were obtained little influence and hence, we may assume that in both only as slow waves, and hence,pffiffiffiffiffiffiffiffiffiffi the phase velocity is smaller ¼ = e models the solution is time independent. than the light speed c 1 0m0 in vacuum due to the interaction with the surrounding dielectric with a relative, e [3] or dielectric, permittivity r. The dielectric permittivityp offfiffiffiffi quartz is close to 4, hence the wave phase velocity c= er is 1.5 108 ms 1. The electron thermal velocity under 3.1. Electromagnetic Equations investigation is in the order of 105–106 ms 1. Both models solve the same set of equations describing Combining equation (2), (3), and (4), the wave equation the surface electromagnetic waves. From Maxwell’s for the electric or magnetic field in the general three-

Plasma Process Polym 2016, DOI: 10.1002/ppap.201600185 ß 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.plasma-polymers.org 5 R Early View Publication; these are NOT the final page numbers, use DOI for citation !! V. Georgieva et al.

3 dimensional form can be readily derived. For the surface power. The absorbed power density QOhm (W m )is waves in TM mode, we solve the following system of computed as equations for the three non-zero components Er, Ez, and  [41,68] 1 ~ ~ 1 ~2 Hw Q ¼ Re J E ¼ Reðs^Þ E : ð9Þ Ohm 2 2

@Ez @Er ¼ ivm Hw; ð6Þ [63] [64] @r @z 0 The manuals of PLASIMO and COMSOL supply further information on the equations and the numeri- cal procedure for solving them. @H i w ¼ ve be E ; ð7Þ @z 0 r r Here we present the deposited power density (Figure 3) and the electric field (Figure 4) calculated by both models when we assume as input uniform @ 1 rHw b plasma properties and momentum transfer collision i ¼ve0erEz; ð8Þ r @r frequency in the entire plasma volume (so called 1-step run). b where be ¼ e is is defined as a complex permittivity, e r r e0 v r being 1 for plasma. The other materials are described Figure 3 shows a good agreement in the power only by the real component, e , which is 1 for air, 3.96 r density absorbed in the plasma region calculated by for quartz, 2.75 for oil, and 2.8 for the polycarbonate the two models. The maximum value in front of the material. wave-guidegapiscalculatedtobe6.73 107 and The following boundary conditions (BCs) apply for the 6.27 107 Wm 3 in the QN and PS models, respec- electric and magnetic fields: tively. As it is expected, the power is mainly deposited along the plasma- dielectric boundary. A ¼ (i) At the axis of symmetry: Er and Hw 0. good agreement is found also in the calculated EM (ii) At material interfaces: field components. Figure 4 shows Ez (a) and Er (b) in the plasma region calculated by both models. The a. Between different dielectric materials and at the plasma-dielectric boundary: continuity of the @ Ez ¼ tangential components Ez and Hw, that is, @r 0 @ Hw ¼ : and @r 0 In addition, there is no free surface charge, which means that for the displacement ~ ¼ e ~ electric field D1 r E the following BC applies: ~ ~ ~ D1 D2 n ¼ 0: Hence, the normal component of

the electric field, which is Er in our case, satisfies er1 Er1 ¼ er2 Er2 at both sides of the boundary e between materials with relative permittivity r1 e and r2, respectively. b. Between a dielectric and a metal: a perfect electrical conductor BC is applied, which means that the tangential component is 0. We assume that the metal is a perfect conductor and the electric field is reflected at the metal boundaries. Therefore the corresponding tangential compo-

nent of the electric field, which is Ez along a

boundary parallel to the z-axis and Er along a boundary parallel to the r-axis, is set to 0. Figure 3. Power density deposition in the plasma region calculated in the QN model and in the PS model by one-step run with the following input data: uniform electron density of (iii) At the waveguide gap: an excitation BC, which 1020 m 3; uniform electron temperature of 13,000 K (1.12 eV), specifies an initialization value of E . The scale of the z uniform gas temperature of 1000 K, applied power of 100 W, incoming electric field is arbitrary and it is iteratively pressure of 1000 Pa, and a gas flow rate of 125 sccm. The coloured re-scaled to match the input and the absorbed scale is the same for the two plots.

Plasma Process Polym 2016, DOI: 10.1002/ppap.201600185

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Figure 4. The axial Ez (a) and radial Er (b) components (real values) of the electric field calculated in the QN and PS models. The input data and operating conditions are the same as in Figure 3.

electric field does not penetrate deeply as is expected The macroscopic value of a particle physical property in the case of surface wave propagation.[1,3] xðÞ~r;~v; t can be found by multiplying xðÞ~r;~v; t with the number of particles integrating over the velocity space and 3.2. Plasma Description in Local and Laboratory dividing by the number of particles inside the volume 3 ; ðÞ~; 3 : Reference Frames element d r which is na r t d r Z Plasmas contain a very large number of particles interact- hiðÞ~;~; ¼ 1 ðÞ~;~; ðÞ~;~; 3 : ð Þ x r v t a ~ x r v t f a r v t d v 11 ing with each other. Therefore, one approach to describe naðÞr; t v such a system is the statistical approach. The plasma is considered as a continuum medium and the equation of For example, for xðÞ~r;~v; t 1 we obtain the number motion is solved for a system of particles considered as a density, that is, Equation (10); for xðÞ~r;~v; t ~v; we obtain whole. From classical mechanics we know that the the macroscopic average or flow (directed) velocity ~u ð~r; tÞ instantaneous state of a particle can be specified by its a ~ of particles type a. Therefore, it is necessary to know the position rðx1; x2; x3Þ and velocity ~nðvx ; vx ; vx Þ: If we 1 2 3 distribution function f ðÞ~r;~v; t in order to calculate the consider a finite volume element d3rd3v, and the number a macroscopic variables. The distribution function can in of particles type a at a given time t is d6N ð~r;~v; tÞ in this a principle be found by solving the Boltzmann equation (BE). ð~;~; Þ¼ [69] volume element, the distribution function f a r v t Its general form is 6 ~~ d Naðr;v;tÞ 3 3 is defined as the density of particles a in the 6d  d rd v @f ~F df phase space at time t. [69,70] The macroscopic variables such a þ ~v rf þ r f ¼ a ; ð12Þ @t a m v a dt as number density, flow velocity, kinetic pressure, thermal a coll energy flux, and so on, can be considered as average values  df a ~ where d istherateofchangeoffa due to collisions, F of the particle physical quantities involving the collective t coll behavior of large number of particles. The number density is the microscopic force acting on each particle with mass ~ 6 ~ ~ naðr; tÞ is defined as the number of particles, d Naðr; v; tÞ; ma. The left side of Equation (12) is the total derivative of 3 ðÞ~;~; inside the volume element d r of the configuration space, f a r v t with respect to time. If there are no collisions, irrespective of velocity [69] the total derivative is 0. Hence the BE is a statement of Z Z conservation of density of particles a in the phase space. ð~; Þ¼ 1 6 ð~;~; Þ¼ ð~;~; Þ 3 : ð Þ A plasma is a mixture of various species a (a denoting na r t 3 d Na r v t f a r v t d v 10 d r v v electrons, ions, or neutrals in ground or excited states) and

Plasma Process Polym 2016, DOI: 10.1002/ppap.201600185 ß 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.plasma-polymers.org 7 R Early View Publication; these are NOT the final page numbers, use DOI for citation !! V. Georgieva et al.

apart from the specific properties given above we may also types of diffusion (due to external fields, density define quantities for the plasma bulk.P For example, the total gradients or temperature gradients). On the other hand, ¼ numberP density is defined by: n ana; the averageP mass an important plasma property is the absolute tempera- ¼ 1 ¼ ¼ mP n anama; the average mass density rm anama ture of species, which is a measure for their thermal ; a rma and the average mass or bulk flow velocity of energy and is expressed by their velocity component due the plasma, ~u,by to random, chaotic, motion. Therefore, we present the X general form of the transport equations in both velocity 1 ~u ¼ r ~u ð13Þ representations in the Appendix, Section Transport a ma a rm equations in local and laboratory reference frames. The particle velocity can be presented with respect to the ~ particle flow velocity, ua, or with respect to the plasma (gas) flow velocity, ~u 3.3. Plasma Fluid Equation Set

~ ~ ~ v ¼ ua þ ca; ð14Þ 3.3.1. Specific Density Balance Equation

QN model: ~v ¼ ~u þ~c ; ð15Þ a0  ð Þ ~ ~ d na ~ hi~ ¼ r:ðnauÞDarna ¼ ; ð18Þ where ca is the random or thermal velocity, and ca a dt ~ coll 0 ; ca0 is the alternative random particle velocity relative to ~ [69] ~ u: The average value of ca0 is the mean velocity of each ~ ~ which is Equation (A3) from the Appendix if neglecting @=@t particle type a, called the diffusion velocity wa ¼ hica0 ;, a and applying Fick’s law for the neutral diffusion flux and and has the meaning of the average particle velocity in a ambipolar diffusion approximation for the ions. It is solved reference frame moving with the plasma. Thus the relation ~ for (N 2) types of particles. The electron density is equal to between the average particle velocity, ua; and the total P the sum of the (positive) ion densities: n ¼ n : Effective plasma flow velocity, ~u; is given by e i i diffusion coefficients are used for the neutral species in

hi~ ¼ ~ ¼ ~ þ ~ ð Þ excited state and the ambipolar diffusion coefficient DA v a ua u wa 16 P [Equation (A39) from the Appendix] for the ions. Further ~ ¼ ~ From Equation (13) we have rmu a rma ua information on the diffusion coefficients used in the QN X [41,68] model can be found in . ¼ r ð~u þ w~ Þ a ma a PS model: and hence  X @ ð Þ na þ ~ r þ r~ ~G ¼ d na ; ð Þ r w~ ¼ 0 ð17Þ u na a;dif 19 a ma a @ t dt coll For a collision-dominated plasma, it is more convenient to consider the particle velocities in a local frame of which is Equation (A3) from the Appendix if neglecting ~ ~ reference moving with the average velocity of the the term nar:u: Equation (19) is solved for (N1) types [69,70] plasma. In many books on plasma physics, the of particles. The electron and ion diffusion fluxes governing equations are derived in a reference frame ~G ¼ ~ ¼ ~ rðÞ; ¼ ; ; a;dif naw a naZamaE Dana a e i , are calcu- moving with the plasma or considering that there is no lated based on the drift-diffusion approximation, applying plasma (gas) flow.[66,69,70] However, often the plasma Equation (A36) and (A37). The electron mobility is (gas) flow velocity cannot be neglected in comparison calculated from BOLSIGþ.[67] The mobility for the ions is with the diffusion velocity, as will be discussed in Section taken from[72] Pressure range extension by experimental measure- ments. The transformation of the governing equations : : 5 1 01 10 Th 4 2 1 1 m þ ¼ 1:52:10 m V s ; ð20Þ from one velocity representation to the other (i.e., the Ar p 273:16 same as transforming the reference frames from a local reference frame, moving with the plasma, to a laboratory ~ reference frame, in which the plasma moves with u)is m þ ¼ 1:2 m þ : ð20aÞ Ar2 Ar not always trivial. In addition, on the one hand, the transport plasma properties are governed by the The electron and ion diffusion coefficients are calculated transport of the species and their energy by different from Einstein’s relation.

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The Ar(4s) and Ar(4p) diffusion coefficients are adopted for charged particles, De DA: Therefore, it is difficult from[73] to estimate if the second term in the heat flux is justified to be neglected. Further investigations are 1 : : 20 2 foreseen. ¼ ¼ 1 16 10 Th 2 1: ð Þ DArð4pÞ DArð4sÞ m s 21 nAr 300 The electron thermal conductivity le is calculated according to Devoto (third order approximation).[74] inel Both models: The neutral gas (Ar atoms in ground state) The energy gain or loss through inelastic collisions Qeh [41,68] density is found from the ideal gas law. The diffusion of is obtained from neutral species inexcited state is described by Fick’s law (see X Qinel ¼ R De ; ð24Þ Equation [A41] in the Appendix). The contribution of eh l l l diffusion due to the temperature gradients is neglected. The net production of species a due to different collisional- where Rl is the rate constant for the inelastic collision l,and radiative processes and reactions is calculated based on the De l is the average energy exchange per collision. The energy Ar chemistry set presented in the Supporting Information. el [41,68] lost through elastic collisions Qeh is obtained from X el ¼ 3 2me ðÞ ; ð Þ Qeh nenes kB Te Th 25 3.3.2. Specific Energy Balance Equations s6¼e 2 ms

a. Electron energy balance equation where ves is the elastic momentum transfer between electrons and species s. QN model PS model: 3 r~:ðÞþn k T ~u n k T r~:~u 2 e B e e B e @ ðÞþr:~G þ ~G :~ þrðÞÞ l rðk T ne e e;dif E e B e @t ¼ Q Qinel Qel : ð22Þ ¼ inel el ðÞ~:r : ð Þ Ohm eh eh Qeh Qeh u ne 26

The electron energy balance is derived from Equation It is obtained from Equation (A13a) from the Appendix. The (A13a) from the Appendix taking into account 0 time- electron heat flux vector, Equation (A32), can be transformed to derivatives. The partial pressure is given by the equation  5 ~ of state for an ideal gas, Equation (A16). The Ohmic ~q þ p w~ þ n e w~ ¼ ene m E e e e e e e 3 e heating, QOhm, is calculated by Equation (9) because the 5 erðD neÞ; ð27Þ electron current density is equivalent in both velocity 3 e representations when the plasma is neutral (see Equation [A28] from the Appendix). The right-hand side where ne ¼ neee is the electron thermal energy density of Equation (A13a) is the loss of electron energy in elastic expressed in [eV m 3]. Equation (A36), (A42), and (A43) elas inel; and inelastic collisions, Qeh and Qeh respectively. 3 3 and the relation p ¼ nekBTe ¼ eneðee½eVÞ are used in The terms ðp w~ þ n e w~ Þ in the electron heat flux in 2 e 2 e e e e e the derivation of Equation (27). The electron energy flux Equation (A32), which are due to the transformationÀÁ to ~ Ge is introduced the laboratory reference are equal to 5 k T D rn : 2 B e A e  The expression can be easily found by applying Equation ~ ~ Ge ¼ne meE rðÞDene ; ð28Þ (A11), (A16), and (A38) from the Appendix. This term is ¼ 5 ; neglected in the QN model. Using that le 2 Dene the 5 5 electron heat flux in case of gas flow and ambipolar where me ¼ m and De ¼ D : Hence the term ÀÁ3 e 3 e diffusion approximation is equal to r~: ~ þ ~ þ e ~ qe pew e ne ew e in Equation (A13a) is equal to  0 5 5 ~ ¼ rð Þ r 5 qe Dene kBTe kBTeDA ne r~ ~ þ ~ ¼ r~G ð Þ 2 2 qe pewe e e 29 5 rTe rne 2 ¼ nekBTeðDe þ DA Þ: ð23Þ 2 Te ne The Ohmic heating, which is the same as in Equa- r r Typically in plasma Te ne :[70] As discussed in the tion (22), is expressed by the electron diffusion flux Te ne ~G ¼ ~ Appendix, Section Ambipolar diffusion approximation e;dif newe

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¼ ~ ~ ¼ ~G ~: ð Þ Qohm neewe E e e;dif E 30 The contribution in the heat flux vector due to the transformation to a laboratory reference frame is Finally, applying Equation (27) –(30), the electron energy neglected (the last term in the left-hand side of balance equation has the form presented in Equa- Equation (32)). tion (26). Solving it, we find the electron energy density To make Equation (32) applicable for atoms and molecules, we use the volumetric heat capacity at constant ne, and hence the average electron energy ee: The P 1 3 1 volume: C ; ¼ 6¼ Esns in [J m K ], where the electron temperature, Te, is calculated from the average v h Th s e electron energy by Equation (2). The difference with the internal energy per particle Es is given by the number of electron energy balance equation solved in the QN degrees of freedom, f, and the energy associated to 1 degree 1 : ¼ 1 model, that is, Equation (22), is that in the PS model the of freedom: 2 kBTh Es f 2 kBTh f depends on the number term in the heat flux vector accounting for the of atoms in a particle. For a monoatomic particle, f ¼ 3(3 transformation to a laboratory reference frame in case degrees of translational motion) and for a diatomic particle of a plasma flow is taken into account. The other f ¼ 5 (3 degrees of translational freedom þ 2 degrees of difference between the two equations is the first term rotational freedom). The volumetric! heat capacity at considering the time dependence. However, as X ¼ ¼ discussed above, the solution in the PS model is constant pressure ph ns kBTh is defined as: Cp;hTh 6¼ steady-state, and neglecting time dependence in the ! s e X QN model is completely justified (see Section Validation Cv;hTh þ ns kBTh: Applying all definitions above, the of the assumptions in the QN model below). Therefore, s¼i;n we consider that accounting for the time variation in the heavy particle energy balance can be written in the form PS model does not influence the solution for the given in Equation (31). conditions under study. The thermal conductivity coefficient lh is obtained based [75] b. Heavy particle energy balance equation on ref.

PS model:

@ Th ~ QN model: r c þ r c ~u:rT þrðÞ l rT m p @t m p h h h ÀÁ ¼ Qelas þ Qinel; ð33Þ r~: ~ ~:r þrðÞ r eh eh Cp;hThu u ph lh kBTh ¼ elas þ inel ; ð Þ Qeh Qhh 31 where cp is the specific heat capacity at constant pressure, ¼ þ ; Theequationisderivedbasedonthefollowingsteps. cpTh cvTh rmRTh and cv is the specific heat capacity at As mentioned in the list of approximations, all heavy constant volume in [J kg 1 K 1]. particles are considered to have one temperature Th and Similar to the QN model, one energy balance equation to have a Maxwellian energy distribution. Therefore, is solved for all heavy particles and the contribution in wesolveoneenergybalanceequationfortheheavy the heat flux vector due to the transformation to a particles by summing Equation (A13a) from the laboratory reference frame is neglected. The Equation Appendix over all species except for the electrons and (A13a) from the Appendix is summed over all species neglecting the time derivatives, the viscosity, and the ion and the electron pressure is neglected. Therefore, the current density. pressure of the heavy particles is assumed equivalent to X X the Ar gas pressure and the thermal conductivity and r~ 3 ~ þ r~ specific heat capacity at constant pressure of Ar gas are 6¼ ns kBThu 6¼ ns kBTh 2 s e s e used for the values of l and c , respectively (COMSOL[64] ~ þrðÞ r h p u lh kBTh ¼ ¼ X database). We apply the ideal gas law p nkBTh 5 r k T D rn r RT ; and express the internal energy by e ¼ c T:The 2 s6¼e B h s s m h v ¼ elas þ inel ; ð Þ ðr~ ~Þ ~ Qeh Qhh 32 terms containing u and the time dependence of u are neglected. where the subscript ‘‘h’’ refers to ‘‘heavy particles’’. The change of the energy is due to gain through elastic collisions 3.3.3. Continuity Equation for the Plasma elas; with electrons, Qeh and gain or loss through inelastic Both models solve Equation (A4) from the Appendix, inel ; collisions with heavy particles, Qhh in which the internal neglecting the time-derivative in the QN model, in order energy is converted into heat.[39] to calculate the plasma (gas flow) velocity.

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3.3.4. Momentum Balance for the Plasma longer than the sheath width, which is in the order of the Debye length, all ions entering the sheath, reach the wall Both models solve Equation (A9) from the Appendix, and we assume that they recombine at the wall. The neglecting the time-derivative in the QN model, in order mean free path li ¼ 1=nArsArþ Ar for the ions is calculated to define the total pressure. The viscosity tensor for the Ar 3 [68] to be 4 10 mm, which is indeed greater than the gas is calculated based on Equation (A31). calculated Debye length of 7.5 10 4 mm at 1000 Pa. The corresponding data at 100 Pa are 4 10 2 mm and 3.3.5. Poisson’s Equation 3.3 10 3 mm.Hence,theionfluxatthewallisdefined It is solved in the PS model to obtain the plasma potential w from

r @ Dw ¼ : ð34Þ G ðÞ¼ ni j ¼ : ð Þ e i R0 DA nivBohm 36 0 @r R0

The electrons which enter the sheath can reach the 3.3.6. Boundary Conditions wall only if their initial energy exceeds eDf: This condition defines the electron flux at the wall by The BCs for solving the system of the transport equations in integrating over the velocity component normal to the both models are presented in Table 1. In addition, the wall and for electron energy exceeding the plasma specific BCs for each model are presented below. potential.[70] Applying that the electron and ion densi- ties are equal at the plasma bulk-sheath boundary layer, QN model: The BC for the flux of the excited Ar atoms to the the electron flux in the absence of current towards the wall is found from[66] wall is[70] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffi 1 8k T ðÞR  G ðÞ¼g ðÞ B h 0 ; ð Þ D s R0 ns R0 p 35 kBTe e w 4 mAr GeðÞ¼R0 niðÞR0 exp : ð37Þ 2pme kBTe where s denotes Ar(4s) or Ar(4p), g is the sticking coefficient The potential difference Dw in the sheath is found from and is equal to 1, that is, all excited species are de-excited equating the ion and electron fluxes at the wall, that is, once reaching the quartz tube wall. Equation (36) and (37),[70] The BCs for the electron temperature and ion flux at rffiffiffiffiffiffiffiffiffiffiffiffi the wall are derived from the Bohm sheath theory.[66,70] k T m Dw ¼ B e ln i : ð38Þ A sheath is formed next to the wall, and in that region e 2pme the quasi-neutrality is not fulfilled. Since due to ¼ ambipolar removal the walls acquire a negative charge, For the typical conditions under study (Te 1.0–1.4 eV see the electric field in the sheath is directed towards the the results in the next section) the potential difference is wall. The ions are accelerated in the pre-sheath to a calculated to be 57V. velocity,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi which is equal to the Bohm velocity The BC for Te at the plasma bulk-sheath boundary layer is [66] [70] vBohm ¼ kBTe=mi: Ifthemeanfreepathismuch found from

Table 1. BCs for solving the system of transport equations in the QN and PS models.a)

Inlet Outlet Axis of symmetry Tube wall

bÞ @c @c @c Th ¼ 283K ¼ 0 ¼ 0 ¼ 0 @z @z @r ~ ~ ~ ~ 1 ~: ¼ ~: ¼ ~: ¼ ~: ¼ ; n G e 0 n G e 0 n G e 0 n G e 2 ve thne ~ ~ ~ ~ ~: e ¼ ~: e ¼ ~: e ¼ ~: e ¼ 5 ; e; n G 0 n G 0 n G 0 n G 2 ve thn   cÞ @uz @uz ¼ 2 ¼ 0 ¼ 0 uz 0 ðÞ¼2F r @z @r uz r 2 1 pR0 R0

@2p ¼ @p @p ¼ p ppump ¼ 0 ¼ 0 @z2 0 @r @r a) ~ ~ b)c ; ; : c)F All expressions for G eand G e denote BCs in the PS model only. ns6¼e; Ar Te Th is the gas flow rate and R0 is the discharge tube radius; ur ¼ 0:

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 compare the electron, that is, plasma, frequency with @ @ Te 1 mi ne the average electron-heavy particle collision frequency. le j ¼kBTeðÞR0 2 þ ln DA j ; @ R0 @ R0 r 2 me r The electron plasma angular frequency is given by qffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð39Þ 2 [66] v ½rad s 1¼ e ne ¼ 56:4 n ½m 3: The electron p mee0 e 19 20 3 provided that there is no gas flow perpendicular to the wall. density is in the order of 10 –10 m , and hence the plasma frequency ¼ vp=2p is calculated to be in the Applying ne ¼ ni at the plasma bulk-sheath boundary layer 10 1 and Equation (36), the BC for the electron temperature range 2.8–9 10 s , which is almost 2 orders of becomes magnitude higher than the average total electron—Ar ;  collision frequency neAr calculated to be in the range 8 1 @T 1 m 2–20 10 s . The latter is calculated from neAr ¼ l ðÞR e j ¼ k T ðÞR n ðÞR v 2 þ ln i : e 0 R0 B e 0 i 0 Bohm @r 2 me nArseArve;th taking the following values: the Ar density is 22 23 3 ð40Þ 2.65 10 and 2.65 10 m when the pressure is 100 and 1000 Pa, respectively. The electronqffiffiffiffiffiffiffiffiffi thermal velocity ve;th The sheath in the QN model is assumed to be infinitely pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 8kBTe 5 is calculated by ve;th½ms ¼ ¼ 6:7x10 Te½eV ; thin and hence the BC for the electron temperature derived pme at the plasma bulk-sheath boundary layer can be applied as setting Te to 2 and 1 eV at a pressure of 100 and 1000 Pa, the corresponding BC at the wall for the electron energy respectively (see below for the simulation results and balance equation in the QN model, i.e., Equation (22) solved experimental measurements of Te). The average total s 20 2 with respect to Te. electron—Ar cross-section eAr is in the order of 10 m . The expression for the Bohm velocity and the BC for the Therefore, in the pressure range under study this electron temperature derived above are valid when there is assumption is valid. one type of positive ions. In the Ar plasma under study we It is important to calculate also the electron mean free þ [66] ; =n : may neglect the Ar2 ions, since their density is 3 orders of path, which is expressed by ve th eAr Applying the þ magnitude lower than the density of the main Ar ions (see values above, the electron mean free path is calculated below in Section Results and Discussion). If there is more to be 0.335 and 4.5 mm at pressures of 1000 and 100 Pa, than one type of positive ion, it is assumed that each type of respectively. ion satisfies the Bohm criterion[54] and an effective ion mass Next, the plasma is considered quasi-neutral in the can be introduced based on the different type of ions and complete volume and therefore, sheaths are infinitely thin. corresponding densities. The typical values of the electron temperature and density 20 3 The BC for the heavy particle flux at the tube wall in the are 1 eV and 10 m , respectively, at theqffiffiffiffiffiffiffiffiffiffi conditions under [64] e PS model is calculated from ¼ 0kBTe 4 study. Hence, the Debye length l 2 is 7.5 10 D nee sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mm. The sheath width is in the order of l ;[71] and hence ðÞ D G ðÞ¼ g 1 ðÞ8kBTh R0 it can be indeed neglected in comparison with the s R0 ðÞ = ns R0 1 g 2 4 pms plasma tube radius of 7 mm. þ ðÞ; ð Þ ZsnsmsErHEr 41

where g is the sticking coefficient, equal to 1 for each type s of heavy particles. The second term is different from 0 only 4. Plasma Diagnostics ðÞ for the ions: Zs is the charge ofthe corresponding ion, HEr is the Heaviside function (0 if Er is negative, 1 otherwise). In this work the OES measurements have been performed in 1 the direction perpendicular to the discharge axis, collecting The expression ðÞ1g=2 is the Motz-Wise correction and we the light from a particular volume of interest using an have found that it has a negligible effect for the conditions Andor iStarDH740-18F-03 ICCD camera (see Figure 1). under study. Figure 5a shows the emission spectra of Ar acquired in the range of 250–850 nm. As expected for typical Ar plasmas, the most intense emission lines 3.4. Validation of the Assumptions in the QN Model correspond to the transition from the 4p ! 4s (or in In the QN model the time-derivative of the variables is Paschen notation: 2p ! 1s) levels producing the emission neglected. This simplification is valid when the peaks in the 660–1150 nm range. We can also observe the 5p characteristic time of variation in plasma parameters ! 4s (or in Paschen notation: 3p ! 1s) transitions emitted greatly exceeds the time between collisions.[70] Since only in the wavelength range 400 < l (nm) < 470. Below the the electrons can respond quickly to any perturbation of Paschen notation is used since it is more convenient in the quasi-neutrality or to a time varying electric field, we notation of the detected emission lines.

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Figure 5. Emission spectrum from a pure Ar microwave surfaguide discharge (a). Inlet: Zoom of the 3p ! 1s (Paschen notation) emissions. ! ! An example of combining optical transitions 2py 1sx (4p 4s) for the calculation of the Ar(1sx) densities (b).

The absolute population of the different atomic levels in system (FPS) of N2 (via 5% N2 admixture to the discharge) [60] the 1s group (formed by two metastable 1s5 and 1s3 and two was utilized. resonant levels 1s2 and 1s4) was determined using so-called The plasma emission in the OES measurements is passive spectroscopy, based on the study of the self- collected by an optical fiber covering a cylindrical plasma absorption of photons generated in 2p ! 1s transitions. volume with a height of 1 cm. The fiber is positioned at 4 cm The self-absorption method has been extensively stud- above the waveguide gap. In order to compare the plasma ied[76,77] and widely used in various spectroscopic measure- characteristics calculated by the models with the experi- ments.[78–80] This method is suitable for absolute density mentally measured values, the simulated plasma charac- measurements if the population of the lower state of a teristics are averaged radially in a cylinder with height 1 cm particular optical transition is high enough to provide a at the same position in the simulation domain, that is, at traceable optical thickness. In this case, the reabsorption of 26 cm below the inlet (see Figure 2). emission lines is taken into account via the concept of an [79] escape factor, which turns the emission transitions 5. Results and Discussion dependent on the metastable and resonant absolute population. This allows building a system of equations 5.1. Comparison of the Plasma Characteristics, by combining line ratios between the intensities of two Calculated by the QN and PS Models emission lines produced by de-excitation to two 1s levels from two higher 2p excited levels (see Figure 5b for a typical The output of the models gives information on the spatial case). If one repeats this method for another pair of levels distribution of a number of characteristics, that is, the and pair of lines, all 1s populations can be determined.[59] species density and temperature, electric and magnetic Further details on evaluating the excited state densities, field intensity, deposited power density, electron-heavy and the electron density and temperature based on the particle collision frequency, particle diffusion and thermal emission spectra are given in ref.[59] conductivity coefficients, etc. The following operating

The relative error of Te has been estimated based on the conditions are used as input: applied power of 100 W, so-called error propagation formulas[81] as a function of the pressure 1000 Pa, and gas flow rate of 125 sccm. Figure 6 relative errors of the line intensity ratio (dR) and normalized presents the calculated electron density ne (a), electron ðÞð Þ : þ metastable state density dN 1s5 Based on the top temperature Te (b), gas temperature Tg (c), and Ar2 density estimations for dNðÞ1s5 and for dR (assuming typical ICCD (d), calculated by the QN model (left column) and by the PS þ emission line measured error 10%), the final value for dTe model (right column). The main Ar ion density has a estimated in our case is about 5%. The relative error of ne is similar profile and values as the electron density, and estimated similar to the relative error of Te and is found to be therefore is not presented. The same color legend is used in also about 5%. The error of the Ar(4s) density is estimated the presentation of the plasma characteristics, calculated based on the average of different experimental measure- by the two models, to allow an easy comparison. ments which is in the range of 5–15%. The wave form in the profiles and the position of the local To estimate the gas temperature in an Ar plasma, the maxima can be explained as follows. Surface waves are rotational temperature derived from the first positive obtained only as slow waves[3] and the wave phase velocity

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is 1.5 108 [m s 1] (see Section Electromagnetic field equations, above). Thus the wavelength is calculated to be around 6 cm for the applied frequency of 2.45 GHz. The metal rings at the plasma tube ends (31 cm in length) confine the electromagnetic field and the surface waves can be reflected. In our experimental set up the tube length is shorter compared to the tube length of 60–80 cm used in the surfaguide experiments reported at pressures below 1000 Pa (cf. Refs.[26,37]). The local maximum values of the electron density (see Figure 4(a)) decrease twice from the center to the plasma tube end, which shows very weak wave damping because the tube length is comparatively short for the investigated pressure range. We found weak dissipation of the wave at the tube ends not only in the modeling results but also experimentally. Experimentally, we observed that the plasma column length is the same as the tube length for pressures between 80 and 3000 Pa, except for an applied power below 50 W at a pressure of 80 Pa in which case the plasma column contracted quickly with decreasing power and even the discharge extinguished below 40 W. Hence, we consider that standing waves are formed and local maxima along the tube length are formed at a distance close to a half wavelength (3 cm, see above). The local maximum values and the wavelength, and hence the distance between the maxima, along the tube, decrease away from the central position, where the wave is launched, due to plasma-wave interaction. In both models the local maximum values of the electron density (Figure 4(a)) have close to linear dependence with z, starting from the central position to the tube ends in agreement with previously reported 1D models in weakly collisional plasmas.[24–27] When the pressure increases, the electron-neutral collision frequency increases, which leads to significant contraction of the plasma column at atmospheric pressure.[40,41]

The maximum electron density, ne, is calculated to be 1.1 1020 and 6 1019 m 3 in the QN model and PS model, respectively. Both values are radially averaged in front of the waveguide gap. The electron density decreases in the radial direction, away from the center, and its profile has Bessel-type form of the first kind, in agreement with the radial profile found analytically as a solution of the electron density balance equation.[33]Although the models operate with the same basic input, such as the Ar chemistry and reaction rate coefficients (see the Supporting Information), the heat capacity at constant pressure or constant volume þ and the viscosity of Ar, and the Ar diffusion coefficient, the different diffusion regimes, and hence the different BCs for the particle fluxes at the wall might explain the difference Figure 6. Electron density (a) and temperature (b), gas in the electron density values calculated by the two models. þ temperature (c) and Ar2 density (d), calculated by the QN (left We observe that the profiles calculated by the QN model are side) and PS (right side) models. The same color legend is used for more pronounced, which is due possibly to the different both models to facilitate the comparison. mesh and numerical approach for solving the equations in the two modeling frameworks. In the QN model a

Plasma Process Polym 2016, DOI: 10.1002/ppap.201600185

14 ß 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim DOI: 10.1002/ppap.201600185 R Early View Publication; these are NOT the final page numbers, use DOI for citation !! Understanding Microwave Surface-Wave Sustained Plasmas ...

structured mesh in the form of an ordered rectangular grid particle energy balance equations by the addition of the is used. The computational domain consists of 482 cells in term neglected in the heat flux vector when the gas flow is the axial direction and 202 in the radial direction. In axial not 0 (the last term in Equation (32)). Only a few models direction the cells are stretched towards the ends, so that consider these terms till now, however, they are developed the size gradually changes from 2.073 mm at the edges to for localized MW plasma created in resonant cavity 0.833 mm in the center. In the radial direction the cells are reactors[54,82] and therefore, a comparison is difficult to stretched beyond a radius of 7 mm, so that up to 7 mm their make. þ size is 0.167 mm and for higher radii the size gradually The density of the molecular ions Ar2 exhibits increases up to a size of 0.635 mm. The portion of the maximum values close to the tube wall and at the plasma computational domain occupied by the plasma is therefore column ends, that is, where Tg is low. Analysis of the 42 cells in radial direction over the entire length. In the PS reaction rates for formation and loss of the molecular ion model, a non-uniform triangular grid in the plasma volume shows that high Tg reduces the population of the molecular and a finer non-uniform quadrilateral grid next to the ion.[42] The maximum calculated value is 1.2 1017 and plasma tube wall is constructed, based on algorithms 0.8 1017 m 3 in the QN and PS model, respectively. described in the manual of COMSOL.[64] The number of the The maximum values of the Ar(4s) density (not shown triangular cells in the computational domein is 25,810 and here) calculated by the two models in the order of the number of the quadrilateral cells is 7428. Regarding 1–2 1018 m 3 are found at the plasma column ends, the numerical algorithms used for solving the equations, which is in agreement with experimental measurements information can be found in the documentation of along the z-axis in surfatron sustained Ar discharges.[49,52] PLASIMO[63] and COMSOL.[64] The radially averaged density in the discharge center is in 17 3 The radially averaged Te (Figure 4(b)) in the discharge the order of 5–6 10 m . The density of Ar(4p), again center is calculated to be 1.1 and 1.27 eV by the QN and PS radially averaged in the discharge center, is 4 1017 m 3 17 3 models, respectively. The profile of Te does not change and 2 10 m in the QN and PS model, respectively. Due considerably in the volume, and is in the order of 1 eV. to the higher electron density calculated in the QN model, Therefore, the temperature gradients in the plasma can be all other species have up to twice as high values compared neglected compared to the density gradients (see for to the corresponding densities calculated in the PS model. instance the profile of the electron density presented in Figure 7 presents the electron density (a) and tempera- [70] Figure 4(a)). The difference in the values of Te calculated ture (b) and the gas temperature (c) calculated by both by both models is comparatively small (in the order of 15%). models for an applied power of 50–200 W at a pressure of A reason might be neglecting the additional terms in the 1000 Pa and flow rate of 125 sccm. Figure 7(a) shows the electron heat flux vector in the QN model, which requires linear dependence of the electron density on applied power, further investigation, as discussed in Section Specific in agreement with theoretical and experimental studies Energy Balance Equations. previously reported.[1–3,42]

The spatial profile of Tg is similar to the profile of the Experimental measurements for similar pressure electron density since the elastic collisions are the main gas (1000 Pa) and similar quartz tube radius are scarce in heating mechanism. There is a good agreement both in the literature. A similar surfaguide reactor sustains Ar plasma profile and the values of the gas temperature calculated by in a quartz tube with a radius of 7.5 mm.[26] The electron the two models. Still, we consider to improve the heavy density is measured at a pressure of 133 Pa (1 Torr). Based on

Figure 7. Radially averaged electron density (a) and temperature (b), and gas temperature (c) calculated by the QN and PS models as a function of the applied power at 1000 Pa and 125 sccm. The legend in (a) also applies for (b) and (c). The measurements available in the literature at 1333 Pa in surfaguide[26] and at 1000 Pa in surfatron[50] Ar discharges are shown.

Plasma Process Polym 2016, DOI: 10.1002/ppap.201600185 ß 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.plasma-polymers.org 15 R Early View Publication; these are NOT the final page numbers, use DOI for citation !! V. Georgieva et al.

the experimental measurements, laws of similarity have performed some simulations without gas flow, and the been developed and used to calculate the electron density at results show that the gas temperature in the discharge [26] different operating conditions. For example, ne in front of centre is calculated to be about 660 K, which is 25% the waveguide gap was calculated to be 1.4 1020 m 3 at a higher compared to the results for a gas flow of 125 sccm pressure of 1333 Pa and applied power of 170 W.[26] The (i.e., 530 K). At a pressure of 1000 Pa, the gas flow velocity is other relevant reference we have found considers measure- 3 m.s 1. The difference in the gas temperature with or ments of the electron density in a surfatron Ar discharge without gas flow is then less pronounced and is about 7%. sustained in a quartz tube with a radius of 3 mm.[50] The The gas temperature without flow is calculated to be electron density at the position where the wave is launched 1030 K, while it is 960 K at a flow rate of 125 sccm. We is measured to be around 4 1019 m 3 at a pressure of investigated the influence of the flow rate from 125 till [50] 1000 Pa and applied power of 32 W. The measured Te at 500 sccm and found that the plasma characteristics do not the same operating condition is 1.3 eV. The experimental depend on the flow rate in the considered power regime, measurements listed above are shown in Figure 7(a) and (b). only the profile changes slightly in the direction of the flow. We could not find simulation results or experimental However, when we increase further the flow rate till measurements of the gas temperature in SW sustained Ar 1000 sccm, and at a power of 500 W, we observe highly discharges at a pressure of 1000 Pa in the literature. We find, asymmetric profiles and extension of the plasma outside of however, good agreement with measurements at lower the EM region in the direction of the flow,[21] which cannot pressure (see the next section). be observed when the gas flow is 0. In the experiment in The developed simulation models also have limitations. ref.[21] a catalytic reactor is placed 3 cm below the bottom When the pressure decreases, the mean free path of the EM confinement ring positioned at 45.5 cm. The simulation electrons increases and at 100 Pa, it is calculated to be predicts a gas temperature in the order of 600–800 K in the 4.5 mm (see Section Validation of the assumptions in the catalytic reactor. The extension of the plasma is detected QN model), which is comparable with the radius of the experimentally by measuring a high gas temperature of plasma tube (7 mm). The minimum pressures (indepen- about 600 K in the catalytic reactor (note that the cooling dently of the applied power) at which the developed QN and system keeps the plasma tube wall temperature at 5 8C).[21] PS models produce stable solutions are 200 and 600 Pa, For a comparison, at a low flow rate used in the respectively. The lower pressure limit for the QN model can present work, the gas temperature measured and predicted be explained by the calculated higher electron density by the simulations (see Figure 6(c)) at the same position compared to the values calculated by the PS model. In both (z ¼ 48.5–50 cm) is close to room temperature. models at pressures below the critical value for each model, Figure 8(a)presents themeasured and simulated electron we observed that the charged particles are lost at the wall temperature as a function ofpressure at an applied power of

faster than their creation in the plasma bulk. We extend the 100 W. The decrease of Te with pressure is attributed to more pressure range by the experimental measurements done in frequent electron-neutral collisions when the pressure the modeled set-up at lower pressure. The results are increases. The experimental data are fitted linearly and the presented in the next section. fit is extrapolated to a pressure of 200 Pa, which corresponds to the first simulation point. The extrapolated value of 1.4 eV is in good agreement with the calculated value of 1.37 eV at 200 Pa by the QN model. The measurements in a 5.2. Pressure Range Extension by Experimental surfatron sustained Ar discharge available in the literature Measurements [50] show also a decrease of Te with pressure and are in good The plasma characteristics were measured for a pressure agreement with our simulation results, especially with the range of 80–170 Pa, which corresponds to a flow rate range results from the PS model. of 25–175 sccm, since a change in pressure is connected to a The radially averaged electron density at 4 cm above the change of the flow rate at which the gas is supplied in the wave launcher (or 26 cm along the z-axis) and in front of experimental set-up. The flow rate in thesimulations iskept the wave launcher calculated by the QN and PS models are fixed at 125 sccm, or 2.08 10 6 m3 s 1, which is not a shown in Figures 8(b) and (c), respectively. The estimation very large value at standard temperature and pressure. of the electron density based on the OES measured plasma However, at a pressure of 200 Pa, it is 4.22 10 3 m3.s 1. emission at 4 cm above the wave launcher is also shown in Taking into account the plasma tube cross-section of Figure 8(b). The calculated electron density at 200 Pa is 1.54 10 4 m2 and a parabolic velocity profile, the about six times higher than the measured one at 170 Pa (see maximum gas flow velocity is along the axis of symmetry below in Table 2, which presents the experimental and and is calculated to be 14 m.s 1. Therefore, neglecting the simulation results at their pressure limits). The difference in gas flow velocity will affect the plasma transport, especially the calculated and measured electron densities might be for the heavy particles, in the axial direction. Indeed, we explained by the reaction rate coefficients used in the

Plasma Process Polym 2016, DOI: 10.1002/ppap.201600185

16 ß 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim DOI: 10.1002/ppap.201600185 R Early View Publication; these are NOT the final page numbers, use DOI for citation !! Understanding Microwave Surface-Wave Sustained Plasmas ...

Figure 8. Comparison of electron temperature (a) and density (b and c), experimentally estimated based on OES measured plasma emission and calculated by the QN and PS models, as a function of pressure, for an applied power of 100W: (b) electron density at 4 cm above the waveguide gap, that is, at 26 cm along the z-axis; (c) electron density calculated in the discharge center, that is, at 30 cm along the z-axis. The measured electron temperature[50] and electron density[49,50] in front of the wave launcher in a surfatron Ar discharge are presented in (a) and (c), respectively. The measured electron density in the discharge center of a surfaguide Ar discharge[37] is presented in (c). analysis of the optical emission spectra, considering each 600 Pa, and therefore the calculated values of the PS model [59] 1sy level separately, which are different from the could not be compared with the experimental data, which reaction rate coefficient in the simulation, where a single explains why they are not included in Table 2. The lumped excitation level 4s is considered. The measured (see simulated plasma characteristics are radially averaged in 18 3 Table 2) ne of 4 10 m (160 Pa, 100 W, 175 sccm) is in a cylindrical volume of 1 cm height, at 4 cm above the good agreement with the electron density of 2 1018 m 3 discharge center, where the optical fiber is positioned. Good measured in a surfaguide Ar discharge at similar operating agreement in Te, Tg, and Ar(4s) density is observed. Our data [37] conditions (133 Pa, 100 W, 100 sccm). for Tg are in good agreement also with the measured Tg of The calculated electron density in the discharge center is around 480 K in a surfaguide Ar discharge at similar compared with measurements available in the literature operating conditions (133 Pa, 100 W, 100 sccm).[37] in surfatron[49,50] and surfaguide[37] Ar discharges (see Finally, we present the results for the plasma potential Figure 8c). Investigation of the electron density as a calculated in the PS model by solving Poisson’s equation. function of pressure by the QN and PS models shows that When resolving the sheath region in the PS model, a special the electron density starts to decrease at a pressure above boundary rectangular mesh is used. At the boundary, the 1333 Pa. In that region, the volume recombination and the length of the cell in the r-direction is 0.01 mm and it is molecular assisted recombination start to play a role and 0.5 mm in the z-direction. We observed no large variations therefore the charged particle density decreases, although of the calculated plasma potential when decreasing the the electron-Ar collision frequency increases with pressure. mesh size. The equation is discretized linearly. The radial This observation is in agreement with the classification[3] distribution of the plasma potential at z ¼ 0 for a pressure according to the pR product (see the Introduction). range 600–2667 Pa is shown in Figure 9. The sheath width, Table 2 presents the measured plasma characteristics defined as the distance from the wall where considerable and the calculated values by the QN model at 170 and deviation from quasineutrality is observed was found to be 200 Pa, respectively. As explained above, the critical in the order of, or less than, 0.1 mm. The plasma potential minimum pressure at which the PS model is reliable is and the sheath drop decrease from 15 to 13 V and from 12 to

Table 2. Plasma characteristics calculated by the QN model and measured by OES or FPS at 4 cm above the wave launcher.

3 3 Te [eV] ne[m ] Tg [K] Ar(4s) density [m ]

Modeling 200 Pa, 50 W 1.3 1.5 1019 396 2.2 1017 Experimenta) 170 Pa, 50 W 1.65 0.2 1019 400 90 2.8 1017 Modeling 200 Pa, 100 W 1.37 2.5 1019 463 2.3 1017 Experiment 160 Pa, 100 W 1.7 0.4 1019 450 60 2.4 1017 a)Incident power. The reflected power in the experiment has always a non-zero value, and for all cases it is measured to be in the range 3–6% from the incident power.

Plasma Process Polym 2016, DOI: 10.1002/ppap.201600185 ß 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.plasma-polymers.org 17 R Early View Publication; these are NOT the final page numbers, use DOI for citation !! V. Georgieva et al.

The models are benchmarked at a pressure of 1000 Pa. The similarities and differences are discussed based on the set of equations and corresponding BCs solved in each model, the applied transport coefficients and diffusion approximations. We found good agreement in the calculated power deposition density and electric field, and in general in all plasma characteristics. The influence of power and pressure on the plasma characteristics is presented and discussed. Both simula- tion models show that the electron temperature Figure 9. Plasma potential radial distribution at z ¼ 0, calculated decreases with pressure till 1333 Pa and is more or less in the PS model for different pressures. constant close to 1 eV when the pressure is increased further. The electron density increases till 1333 Pa, and then starts to decrease due to loss of charged particles 10 V, respectively, when the pressure increases from 600 to via volume and molecular-assisted recombination. The 1333 Pa. Both parameters almost do not change at 2667 Pa. change in the plasma behavior is in agreement with a The same behavior is observed in the electron temperature previously developed classification[3] based on the dependence on pressure (see Figure 8(a)). The sheath pressure-radius pR product. The electron density and potential difference in ambipolar electric field approxima- gas temperature have a dependence on the applied tion (QN model) is given by Equation (38). It is proportional power close to linear. to the electron temperature and is calculated to be 5–6 V The pressure limitation of the models is clarified. ¼ for Te 1.0–1.2 eV in the QN model in the considered Therefore, the modeling results were extended in the pressure range of 600–2667 Pa. Hence, in both models the pressure range by experimental measurements carried potential drop across the sheath is proportional to the out in the modeled set-up. Comparison with experimen- electron temperature. tally measured plasma characteristics available from literature for similar operating conditions is also shown 6. Conclusion and discussed. A good agreement is found, taking into account a number of different operating conditions and reactors. We present and benchmark two 2D self-consistent In conclusion, both diffusion approximations can be models of a surface-wave sustained Ar discharge operat- applied successfully to simulate the SW sustained ing at intermediate pressure in the range of 200–2667 Pa, plasmas at the intermediate pressure range. The modeled and extended down to 80 Pa by experimental data Ar plasma characteristics are benchmarked with previ- measured in the simulated set-up. The plasma is ously developed analytical solutions or parametrizations, sustained by electromagnetic waves launched by a as well as with experimental measurements available in surfaguide reactor operating at 2.45 GHz in continuous literature. Hence, the developed 2D models can be regime. One of the models considers quasi-neutral plasma adapted in future to simulate complex chemical plasmas, in the complete plasma region and neglects sheath which are of interest for environmental or industrial formation (QN model) and the other model resolves the applications. formation of the sheath (PS model). Both models solve the electromagnetic field equations describing the surface wave propagation along the plasma-dielectric border. The complete general set of fluid (transport) equations is Acknowledgements: This research is carried out in the framework presented in local and laboratory reference frames. The of the network on Physical Chemistry of Plasma Surface Interactions Interuniversity Attraction Poles phase VII project additional terms in the heat flux vector and hence in the (http://psi-iap7.ulb.ac.be/), supported by BELSPO. N. Britun is a energy balance equations due to the transformation from post-doctoral researcher of the FNRS, Belgium. A. Berthelot’s a local to a laboratory reference frame are derived and research was supported by the European Marie Curie RAPID discussed. The specific density and energy balance project, which has received funding from the European Union’s equations along with the corresponding BCs are devel- Seventh Framework Programme for research, technological development and demonstration under grant agreement no oped and compared for the two models. The transport 606889. coefficients used in the models are given. The gas flow is taken into account through solving the plasma continuity Supporting Information is available from the Wiley Online and momentum equations. Library or from the author.

Plasma Process Polym 2016, DOI: 10.1002/ppap.201600185

18 ß 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim DOI: 10.1002/ppap.201600185 R Early View Publication; these are NOT the final page numbers, use DOI for citation !! Understanding Microwave Surface-Wave Sustained Plasmas ...

Appendix @ rm ~ þ r:ðr ~uÞ¼0: ðA4Þ Transport Equations in Local and Laboratory @t m Reference Frames The specific momentum equation is obtained from ~ Equation (A1) by setting x mav: The formulation Information for the macroscopic variables can be derived with respect to the particle velocity~v presentation relative from the BE, Equation (12) from the main paper, without ~ [69] to ua is solving it. A macroscopic variable xðÞ~r;~v; t is related to the moment of the distribution function as shown by @ ~ ~ ~ ~ ðÞþr ua r:ðÞr uaua Equation (11). The transport equation of that variable can @t a a ~ ~ ~ be obtained by taking the corresponding moment of the þ r:Pa na < F>a ¼ Aa; ðA5Þ BE, which means multiplying the BE by x and integrating over all the velocity space. When x is dependent only on ~ where Aa is the exchange of momentum between different position ~r; and time t, the transport equation is given species by means of collisions, P ¼ r ~c ~c is the kinetic by[69] a a a a pressure tensor and is expressed further by the scalar pressure @ ÀÁÀÁ p and the viscosity tensor t (see Equation (A14) below). The ~ a a nahix þr: nahixv ~ @t * a + a external force F in plasma is electromagnetic in nature and is ~ Z   F df a 3 ~ ¼ ~ þ ~ ~ ; na :rvx ¼ x d v: ðA1Þ given by the Lorentz force: F Zae E v B where Za is ma dt coll a v the charge number of the species a and e is the elementary ~ charge; B ¼ m H~ is the magnetic induction vector. The Тhe specific density, momentum, and energy balance 0 gravitational force is neglected. Hence, for the chargedDE equations are derived from the zero-th, (i.e., x 1), 2 ~ ¼ ðÞ ~ ; mav particles the averaged external force is F first x mav and second x 2 moments of the  a ~ ~ ~ Boltzmann equation, respectively. Combinations of Zae E þ ua B ; and for the neutral particles the external these equations form the continuity equation force is 0. Equation (A5) in two velocity presentations is and the momentum equation for the plasma as a @ whole. ~ ~ ~ ~ ðÞþr ua r:ðÞþrr uaua p The specific density equation is obtained from Equation @t a a a  ð ~ Þ x ~ ~ ~ ~ d naua (A1) by setting 1 and applying Equation (16) from þ r:ta naZae E þ ua B ¼ ma [69] the main paper. In addition, the particle flux defined dt coll ~ ~ ~ ~ ð Þ as Ga ¼ naua ¼ naðÞu þ wa ; is a sum of two components: A6 theparticlefluxduetothecollectivemotionofall ~ particles n ~u and the diffusion particle flux G ; ¼ a a dif @ ~ ¼ ~ : ~ ~ Du 0 ~ 0 nawa Hence, the specific density equation in particle ðÞþr w~ r:ðÞþr ~uw~ r ðw~ :rÞ~u þ r þrp þ r:t @t a a a a a a a Dt a a velocity representations with respect to ~u or to ~u is a  given by ðÞ~ ~0 ~ ~ d nawa naZaeðE þ wa BÞ¼¼ma ðA7Þ  dt @n dðn Þ coll a þ r~:ðn ~u Þ¼ a ; ðA2Þ @ a a ¼0 t dt coll The expressions for kinetic pressure tensor P a,the 0 ¼ 0 scalar pressure p a, viscosity tensor ta and electric field ~0  E in the particle velocity representation relative to ~u are @ ð Þ na ~ ~ ~ ~ d na þ r:ðnauÞþr:Ga;dif ¼ : ðA3Þ given below in Section Definition of quantities in two @ d [69] t t coll velocity representations. In absence of an external magnetic field, neglecting The continuity equation is obtained by multiplying viscosity and time dependence of ~u, applying the relations Equation (A2) or (A3) by ma, and summing over all species. (A29), Equation (A7) is transformed to Having the total mass density rm, the average plasma velocity expressed by Equation (13) from the main paper, @ ~ ~ ðÞþr w~ r:ðÞþr ~uw~ r ðw~ :rÞ~u þrp taking into account Equation (17) and the fact that the @t a a a a a a a collision term vanishes when summing over all species, as a  dðÞn w~ consequence of the total mass conservation of the system, n Z e~E ¼ m a a : ðA7aÞ a a a d we receive t coll

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The momentum balance equation for the plasma or the @ ÀÁ ~ 0 ~ 0 Du equation of motion for the plasma as a whole is obtained by ðÞþn e r: n e ~u þ r w~ @ a a a a a a  summing Equation (A.6) or (A.7) over all species in the t Dt 0 0 0 ð e Þ [69] ~ ~ ~ ~0 ~ ~ d na a plasma. Taking into account that the exchange of þðP :rÞ:u þ r:q naZaewa:E ¼ a a dt momentum due to collisions for the plasma as a whole is coll ðA13Þ 0, and introducingX the total electric charge density per unit volume r ¼ n Z e; we obtain[69] a a The term in the right-hand side of Equation (A12) and a @  (A13) represents the rate of change in the thermal energy ~ ~ ¼ ~ ~ ~ ðÞþr ~u rðÞþr ~u~u rP rE þ J B ¼ 0; ðA8Þ @t m m density due to collisions and radiation. In plasma, the viscosity effect in energy transport can [69,70] where the total kinetic pressure tensor P and the total usually be neglected. In both models presented here electric current density~J are developed in Section Definition we neglect the viscosity terms in the specific energy balance of quantities in two velocity representations.[69] If the equations. Applying Equation (A14) for the kinetic pressure plasma is neutral, that is, r is 0, no external magnetic field is tensor, and the relations (A29) and (A32), neglecting the applied and using Equation (A30), Equation (A8) is trans- time dependence of ~u, and in absence of an external formed to magnetic field (see (A27)), Equation (A13) becomes ÀÁ @ @ ~ ~ ~ ðÞþ~ r~:ðÞþr~~ þ r~: ¼ : ð Þ ðÞþn e r:ðÞþn e ~u p r:~u þ r: ~q þ p w~ þ n e w~ rmu rmuu p t 0 A9 @ a a a a a a a a a a a @t t  ðÞe ~ ~ d na a naZaew a:E ¼ : ðA13aÞ Equation (A4) and (A9) are known as Navier-Stokes dt equations. coll The specific energy balance is obtained from Equation (A1) The last term in the left-hand side is the Ohmic (Joule) 2 by setting x mav : In the presentation relative to~u : v2 ¼ ¼ ~ :~ ¼ ~}:~; 2 a heating, Qohm neewe E J E when a denotes elec- 2 þ 2 ; ua ca and the total average particle energy is a sum of trons; and it is 0 when a denotes ions since the ion current is ~} the kinetic energy of directed motion and kinetic energy of neglected (see also Equation (A28), showing that J and~J are random, thermal, motion equivalent when the plasma is neutral).

2 2 m u ma c x K þ e ¼ a a þ a : ðA10Þ ua a 2 2 Kinetic pressure tensor For an isotropic distribution of the random velocity The kinetic pressure tensor can be presented as a sum of the

2 scalar pressure of particles type a, pa, and the viscous stress ma c 3 e ¼ a ¼ : ð Þ [69–71] a kBTa A11 tensor, t 2 2 a ¼ d þ t : ð Þ Introducing the heat flux vector (which is the random Paij pa ij aij A14 ~ ¼ 1 2~ ; flux of the thermal energy) qa ra caca and applying 2 ¼ ¼ Equation (A2) and (A6), the specific energy balance equation Here dij is the Kroenecker delta defined as dij 1 for i j, ~ d ¼ 6¼ ; ¼ ; : in velocity presentation relative to ua is ij 0 for i j, for i j 1 3 The scalar pressure pa is defined as one-third the trace of the pressure tensor[69] @ ðÞþe r~:ðÞþðe ~ :rÞ~ :~ þ r~:~ na a na aua Pa ua qa X @t  1 1 2 p ¼ Pa ¼ r c : ðA15Þ dðn e Þ a i¼1;3 ii a a ¼ a a : ðA12Þ 3 3 dt coll According to the thermodynamic definition of the [69] A derivation of the equation can be found in ref. Note that absolute temperature Ta, there is a mean thermal energy the external force term vanishes in that velocity presentation. associated with each translational degree of freedom ðÞ¼ ; : 1 ¼ 1 < 2 >: The specific energy balance equation can be written i 1 3 2 kBTai 2 ma cai When the distribution 2 ¼ 2= with particle velocity representation with respect to the of the random velocity is isotropic cai ca 3 (which is the average plasma velocity.[65,69] The derivation is too case of the Maxwell-Boltzmann distribution) the scalar tedious to be shown here. It is based on the specific pressure is given by the equation of state of an ideal gas[69] continuity equation (A3), the specific momentum transfer ¼ < 2 >¼ : ð Þ equation (A7), and the transformations (A18)–(A28). The pa ra cai nakBTa A16 final general form is[65,69]

Plasma Process Polym 2016, DOI: 10.1002/ppap.201600185

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The viscous-stress tensor elements are defined as[69,70] 0 ¼ þ ~ ~ : ð Þ Pa Pa rawawa A19 ¼ < ~ ~ 1 2 >: ð Þ taij ra caicaj c dij A17 3 a Partial scalar pressure When the viscosity effect is relatively unimportant X 0 ¼ 1 0 ¼ 1 2 ¼ þ 1 2: ð Þ (typically this is valid for the electrons), the non-diagonal pa Paii ra ca0 pa rawa A20 3 i¼1;3 3 3 elements of Paij can be neglected. Hence, the force per unit 0 r~: ; volume inside the plasma, Pa is reduced to the negative Pa can be written in the form of Equation (A14), that is, a 0 0 r~: ¼r : sum of the scalar pressure, p , and viscosity tensor, t , gradient of the scalar pressure: Pa pa a a relative to the plasma flow velocity

0 0 0 Definition of quantities in two velocity P ¼ p I þ t : ðA21Þ [69] a a a representations 0 The relationship between ta and ta is given by  ~ ~ ~ 0 1 1) Relative to the average particle velocity: v ¼ ua þ ca, ð Þ ð Þ ¼ 2 : ð Þ ta ij ta ij ra waiwaj wadij A22 that is, in a reference frame moving with the plasma 3 flow velocity, or in a laboratory reference frame if the Particle kinetic energy of thermal motion. The relationship plasma flow velocity is 0. 0 m hic2 m hic2 ~ ¼ ~ þ~ e ¼ a a e0 ¼ a a0 2) Relative to the average plasma velocity: v u ca0, between a 2 and a 2 that is, in a laboratory reference frame. The correspond- 2 0 0 m w ing quantities are denoted with a superscript e ¼ e þ a a : ðA23Þ a a 2 ~ ~ ~ ~ < v>a ¼ ua ¼ u þ wa; ðA18aÞ 1 Particle heat f lux vector : ~q ¼ r c2~c ; a 2 a a a ~ ~ ~ ~ ~ < ca>a ¼ 0; < ua>a ¼ ua; < ca0>a ¼ wa; ~0 ¼ 1 2 ~ : < ~> ¼ ~ ðA18bÞ q a ra ca0ca0 u a u 2 After certain mathematical transformations: ~ ~ ~ ca0 ¼ ca þ wa; ðA18cÞ DE c2 ~c ¼ð~c þ w~ Þ2ðÞ~c þ w~ a0 a0 a a a a ¼ c2~c þ 2w~ :hi~c ~c þ c2 w~ þ w2w~ : < 2> ¼ 2 þ < 2 > ; ðA18dÞ a a a a a a a a a v a ua ca a ¼ hi~ ~ Hence, having in mind that Pa ra caca and e ¼ 1 2 ~ ~ 0 na a ra c , the relationship between q and q is < 2> ¼ 2þ < 2 > þ 2~:~ : ðA18eÞ 2 a a a v a u ca0 a u wa 0 1 ~q ¼ ~q þ w~ :P þ n e w~ þ r w2w~ : ðA24Þ Below the subscript a in the expression for average a a a a a a a 2 a a a values is omitted. Total (plasma) kinetic pressure tensor P is connected to the : ¼ hi~ ~ ; partial kinetic pressure tensor Pa by Particle pressure tensor Pa ra caca 0 X X P ¼ r hi~c ~c : ¼ þ ~ ~ : ð Þ a a a0 a0 P Pa rawaw a A25 a a Applying that Total scalar pressure p is connected to the partial scalar hi~ ~ ¼ hiðÞ~ þ ~ ðÞ~ þ ~ ca0ca0 ca wa ca wa pressure pa by ~ ~ ~ ~ ~ ~ ¼ hicaca þ 2wahica þ hiwawa X X ¼ hi~ ~ þ ~ ~ ; ¼ þ 1 2: ð Þ caca wawa p pa rawa A26 a 3 a the relationship between the two presentations of the The electric field seen by the charged particles when kinetic pressure tensor is: plasma moves with ~u is

Plasma Process Polym 2016, DOI: 10.1002/ppap.201600185 ß 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.plasma-polymers.org 21 R Early View Publication; these are NOT the final page numbers, use DOI for citation !! V. Georgieva et al.

~0 ~ ~ ~ ~ E ¼ E þ ~u B: ðA27Þ r:P ¼rp þ r:t: ðA30Þ

The total electric current density or the charge flux,~J; and The components of the plasma viscous stress tensor can 0 [70] the conduction charge current density,~J ; are related by the be found from ~  convection current, ru, where r is the total electric charge @ @  ¼ ui uj þ r~:~ ; ð Þ density tij m m2dij u A31 @xj @xi X X ~ ~ ~ ~ J ¼ naZaeua ¼ ru þ naZaewa a a where m is the viscosity coefficient, and the coefficient m2 0 ¼ r~u þ~J : ðA28Þ depends on the nature of the gas. For a mono-atomic gas ¼2 :[70] m2 3 m The mobility of ions is low compared to the electron Finally, when the viscosity can be neglected the pressure mobility and therefore, their contribution to the total tensor is connected to the partial scalar pressure by ~ current can be neglected. Hence, J is equivalent to the ¼ : ~ : ¼ ~ Pa paI Hence w a Pa pawa and we receive for the electron current density ~J ¼n e~u : Similarly, the total 0e e e particle heat flux vector electric current density ~J in the velocity representation 0 ~ ~ ~ 0 relative to the plasma flow velocity u,isJ ¼neewe: The ~ ¼ ~ þ ~ þ e ~ : ð Þ q a qa pawa na aw a A32 definitions of the total electric current density in both 0 velocity representations,~J and~J , are equivalent when the plasma is neutral, i.e., r ¼ 0. In case of surface waves~J is the electron current density, Diffusion calculated by Equation (4). ~ Simplifications: The diffusion velocities wa are small ~ compared to the thermal velocities ca,andthe Drift-Diffusion Approximation for Charged quadratic terms in Equation (A19), (A20), and Particles (A22–A26) can be neglected.[69] We receive the follow- ing relationships for the quantities in both velocity To analyze the transport of electrons we consider the representations specific density equation (A3) and specific momentum 0 equation (A7a). To obtain the diffusion velocity we set ¼ ¼ þ ; ð Þ Pa Pa paI ta A29a the plasma flow velocity to 0 and neglect the time- variation.[70,71] Next, the electron diffusion velocity is much 0 greater than the neutral particle diffusion velocity, and 0 ¼ ; t ¼ t ; ð Þ pa pa a a A29b therefore the latter is neglected in the collision term. Hence, the momentum balance equation for the electrons [70,71] 0 becomes e a ¼ ea; ðA29cÞ 1 rðn k T Þe~E ¼m n w~ ; ðA33Þ n e B e e m e ~ 0 ¼ ~ þ ~ þ e ~ þ ~ : ; ð Þ e qa qa pawa na aw a wa ta A29d which is the equation of motion of a single electron in a X reference frame moving with the plasma or when the total P ¼ P ; ðA29eÞ plasma (gas) flow velocity is 0. From this equation the a a electron diffusion velocity in the so-called drift-diffusion [70,71] X approximation is obtained as ¼ ; ð Þ p pa A29f a e k T rn k T rT w~ ¼ ~E B e e B e e : ðA34Þ e m n m n n m n T X e m e m e e m e t ¼ t : ðA29gÞ a a The first term determines the directed (drift) velocity due to the acceleration in the external electric field, the second For isotropic distribution of the thermal velocity ~c ; a term describes the diffusion due to the density gradient, and the plasma kinetic pressure tensor has the following the third term describes the (thermal) diffusion due to the form, taking into account Equation (A29e), (A29f), and temperature gradient. The proportionality factors are called (A29g): electron mobility me, and electron diffusion coefficient De

Plasma Process Polym 2016, DOI: 10.1002/ppap.201600185

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r ¼ e ; ¼ kBTe : ð Þ ~ ¼ ~ ¼ ~ ¼ n ; ð Þ me De A35 we w i w A DA A38 menm menm n

The relation between them is given by the Einstein where the ambipolar diffusion coefficient DA is: relation: De ¼ kBTe : In a similar way the ion diffusion velocity me e þ ¼ miDe meDi miDe þ can be defined with corresponding properties: ion mobility DA þ Di me mi me m , and ion diffusion coefficient D . Hence, the electron i i ¼ þ Te : ð Þ ¼ Di 1 A39 (Ze 1) and ion diffusion velocities are Ti

~ ¼ ~ 1 rðÞ; ð Þ The expression for DA shows that the ambipolar diffusion we meE Dene A36 ne coefficient is much smaller than the coefficient of free electron diffusion and greater than the free ion diffusion coefficient: D < D D : Thus the ambipolar field reduces 1 i A e w~ ¼ Z m ~E rðÞD n : ðA37Þ substantially the electron diffusion velocity. i i i n i i i The expression for the ambipolar diffusion coefficient is In plasma typically rT rn and the thermal diffusion valid for a system with one dominant ionic species. For the T n þ can be neglected,[70] hence the second term is transformed Ar plasma under study, the density of the Ar ion is 3–4 r orders of magnitude larger than the density of the other to D na : a na þ positive, molecular ion Ar2 . Therefore, Equation (A39) can be applied in the developed QN model. In case of a complex mixture and electronegative plasmas, another method is Ambipolar Diffusion Approximation for needed.[83] Charged Particles

The plasma is quasi-neutral in nature and only in a region Diffusion of Neutral Particles, Fick’s Model adjacent to a solid surface the neutrality is not fulfilled. This region is referred to as a sheath. Its width is in the order of The diffusion velocity of neutral particles can be found from ~ G ; ¼ ~ the Debye length lD, which is defined as the distance at Fick’s law, which states that the diffusion flux a dif nawa which the plasma confines the electric field of a point is proportional to the density gradient, that is, charge.[71] ~ ¼ r ; ð Þ Letusassumethatthequasi-neutralityintheplasma naw a Da na A41 ¼ ¼ bulk is fulfilled at a given moment ne ni n.Since where Da is the diffusion coefficient of a species a.Fick’s me mi; the electron mobility and diffusion coefficient are much greater than the ion mobility and diffusion model for diffusion assumes that there is one dominant : species and one type of particle moving through the coefficient, me mi and De Di Therefore, the electron flux exceeds the ion flux and the charge separation leads background species. The atomic neutral gas in a weakly to the formation of an electric field, called ambipolar ionized plasma can be assumed to be a dominant species. electric field, which speeds up the ions and retards the In case of multiple types of particles Fick’s model can be electrons. To maintain the quasi-neutrality, the changes applied if all species have much lower number densities intheelectronandionconcentrationineachvolume than the background gas. Indeed, the number densities of @ @ the electrons, ions and excited species are several orders element must be equal, that is, ne ¼ ni :[70] Using @t @t of magnitude lower than the background neutral gas in Equation (A3) and the fact that an electron and ion the surface wave sustained plasma and in case of Ar there appear or disappear simultaneously in a collision is one dominant species, which is the Ar atom in the process, we find that independently of the plasma ground state. velocity ~u, the following condition applies ~ ~ ~ ~ [70] r:ðneweÞ¼r:ðniwiÞ: This leads to the equality of the diffusion velocities of the electrons and ions. Hence, Heat Flux the ambipolar electric field in absence of an external electric field source can be found by applying Equation The heat flux is found from the third moment of the BE. The ¼ [69] (A36) and (A37) for Zi 1. Using the obtained expression general form can be found in and the equation is difficult fortheambipolarelectricfield,thevelocityofjoint to solve without assumptions. For a stationary case, in (ambipolar) motion of the charged particles is due to the absence of a magnetic field and for plasma flow velocity 0, density gradient[70] the heat flow equation is simplified so that the heat flux is

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