Hybrid model of neutral diffusion, sheaths, and the to …doeplasma.eecs.umich.edu/files/PSC_Ding1.pdfHybrid model of neutral diffusion, sheaths, and the to transition in an atmospheric

This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 169.229.32.36

This content was downloaded on 03/09/2014 at 01:57

Please note that terms and conditions apply.

Hybrid model of neutral diffusion, sheaths, and the to transition in an atmospheric pressure

He/H2O bounded rf discharge

View the table of contents for this issue, or go to the journal homepage for more

2014 J. Phys. D: Appl. Phys. 47 305203

(http://iopscience.iop.org/0022-3727/47/30/305203)

Home Search Collections Journals About Contact us My IOPscience

iopscience.iop.org/page/termshttp://iopscience.iop.org/0022-3727/47/30http://iopscience.iop.org/0022-3727http://iopscience.iop.org/http://iopscience.iop.org/searchhttp://iopscience.iop.org/collectionshttp://iopscience.iop.org/journalshttp://iopscience.iop.org/page/aboutioppublishinghttp://iopscience.iop.org/contacthttp://iopscience.iop.org/myiopscience

Journal of Physics D: Applied Physics

J. Phys. D: Appl. Phys. 47 (2014) 305203 (17pp) doi:10.1088/0022-3727/47/30/305203

Hybrid model of neutral diffusion, sheaths,and the α to γ transition in an atmosphericpressure He/H2O bounded rf discharge

Ke Ding1,2, M A Lieberman1 and A J Lichtenberg1

1 Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA94720, USA2 College of Science, Donghua University, Shanghai, 201620 People’s Republic of China

Received 17 April 2014, revised 23 May 2014Accepted for publication 5 June 2014Published 3 July 2014

AbstractWater is a trace gas of interest for plasma-based medical applications. We use atwo-temperature hybrid global model to simulate a chemically complex, bounded, He/H2Oatmospheric pressure discharge, including 43 species with clusters up to H19O+9. The dischargeis embedded in a larger volume, in which the trace gas fraction is controlled, leading todepletion of water within the discharge and diffusive flows of reaction products to the walls.For a planar discharge with a 1 cm electrode radius and a 0.5 mm gap, driven at 13.56 MHz,we determine the depletion and diffusion effects and the α to γ transition, over a range of rfcurrents (100–1600 A m−2) and external H2O concentrations (500–10 000 ppm). The transitionfrom the low power α-mode to the high power γ -mode is accompanied by a collapse of thebulk electron temperature, an increase in the density and a decrease in the sheath width. At thehighest external H2O concentration studied, there are no low current (α-mode) solutionsbecause the sheath widths fill the device. The α-mode is recovered at larger gaps (e.g., 1 mm)or higher frequencies (e.g., 27.12 MHz). The higher mass cluster densities decrease rapidlywith increasing gas temperature. Each simulation takes about two minutes on a medium sizelaptop computer, allowing exploration of a large input parameter space.

Keywords: helium water discharge model, atmospheric pressure rf discharge, dischargesheaths

1. Introduction

Atmospheric pressure radio-frequency (rf) micro-dischargeshave applications in bio-medical and materials processing.Therefore they have been studied both experimentally [1–5]and by numerical modelling, using global models for theircomputational efficiency [6–9], or one-dimensional (1D) fluidor kinetic particle-in-cell (PIC) simulations [10–17], which aremore accurate, but the increased complexity leads to increasedcomputation times and limits the parameter range that canbe studied. A recent review of atmospheric pressure plasmamodelling and experiments has additional and extensivereferences [18].

In previous work [19, 20], we developed a fast hybridanalytical–numerical global model that included sheatheffects. In the model, the net ion density ni (total positiveion density minus total negative ion density) is assumed to

be uniform over the gap width l and the electron density neis uniform with ne = ni over a cloud of width d = l − sm,where sm is the maximum value of the oscillating sheath width.An analytical solution of this current-driven homogeneousdischarge yields the space- and time-varying electric fieldE(x, t) and the oscillating sheath width s(t). An ohmicpower balance is then used to find the time-varying electrontemperature Te(t) on the rf timescale, with averages over anrf period yielding rate coefficients used in the particle balanceequations, which are numerically integrated to find the globalspecies densities. The model results were compared to 1Dfluid calculations [15, 16] for atmospheric He/0.1%N2 andHe/0.5%O2 discharges with a 1 mm discharge gap and rfdriving frequency f = 13.56 MHz over a relatively low powerrange Stot = 0.33–3.3 W cm−2. The neutral and chargedparticle densities agreed within a factor of two but becameless accurate at the higher powers.

0022-3727/14/305203+17$33.00 1 © 2014 IOP Publishing Ltd Printed in the UK

http://dx.doi.org/10.1088/0022-3727/47/30/305203

J. Phys. D: Appl. Phys. 47 (2014) 305203 K Ding et al

At high powers, the sheath electric fields can be largeenough to produce avalanche multiplication of electronscreated either by ion-impact secondary emission from theelectrodes, or by metastable electron–ion pair productionwithin the sheaths. These processes produce a highertemperature ‘hot’ electron group that was neglected inthe previous hybrid global model [19, 20]. Other modelinaccuracies arise from spatial variations. For example, thehigh fields in the sheath regions lead to higher metastabledensities in the sheaths.

In a recent paper [21], we performed 1D PIC simulationsof an electropositive capacitive atmospheric pressure Hedischarge with trace amounts of N2, over a wide range of inputcurrents and frequencies. A simplified reaction set was usedwith only one kind of positive ion (N+2 ) and a single Penningionization process (He∗ + N2 → He + e + N+2 ). Both time-averaged as well as space- and time-varying PIC diagnosticsprovided a detailed description of the physics. We used theresults to guide the development of an expanded hybrid globalmodel applicable to the higher power regimes. The expandedmodel, which includes sheath multiplication, has two classes ofelectrons: higher temperature hot electrons associated with thesheaths, and cooler warm electrons associated with the bulk.A separate Child law (CL) sheath calculation determines theion wall loss flux �w. Reasonable agreement was obtainedbetween the model and the PIC simulation results for a1 mm gap atmospheric He/0.1%N2 discharge with appliedfrequencies of 27.12 and 13.56 MHz. For driving currentsvaried from 400 to 6000 A m−2, the discharge conditions variedfrom low power, with essentially no sheath multiplication,to high power, in which electron–ion pair production isdominated by sheath multiplication. As the frequency in themodel was raised from 13.56 to 40.68 MHz, the transitionoccurs at increasingly higher J , as is also seen experimentallyin a pure helium discharge [3]. The PIC simulations suppliedinformation that is important for understanding global modelaccuracy and its limitations, particularly the effects of spatialvariations.

Another trace gas of some importance is water. This isparticularly interesting as the water may arise from contactof a biomedical plasma with a subject who is being treated.An experiment investigating He/H2O mixtures has beenperformed [22] and analysed with a global model [8]. Thechemistry of water is quite complicated, giving rise to manyspecies and reactions. In addition, the discharge is weaklyelectronegative. The authors compared their extensive reactionset, over two ranges of water fraction, to two levels ofsimplification [8]. Their global model, however, used variousapproximations that may lead to significant errors, as will beseen in comparison to our analysis in this paper.

Building on our previous development of a hybridanalytic-numerical global model, which explicitly considersthe effect of sheaths and hot electron multiplication effects,we further expand the model to analyse configurations of thetype in [22], adding the following new physics:

1. In previous versions of the hybrid model, we usedspecified values of the trace gases. In the experiment,the discharge is embedded in a large chamber, in which

the fraction of water is controlled. However, withinthe discharge itself, the water can be depleted and isspatially varying. The amount of depletion and the spatialvariation depends on the strength of the discharge current,the external trace gas density, and on the boundaries.The boundary conditions require a diffusive analysis, inaddition to the volume sources and sinks.

2. Similarly, reaction products flow to the axial and radialwalls by diffusive processes, and are lost diffusively tothe side walls and with some reaction probability to theaxial end walls. For almost all species, the diffusion lossesare rate limiting and are typically much smaller than theusual gas kinetic losses used in global models, which canconsiderably alter the discharge chemistry.

3. In a He/H2O discharge, there are numerous types ofPenning ionization processes and numerous types ofpositive ions lost to the walls. The significant processesmust be incorporated into the particle and energy balancerelations for the warm and hot electron components.

4. The single mobility-limited N+2 ion flux to the walls inthe He/N2 discharge, previously treated [21], must bemodified to account for the multiple ion species and theelectronegative character of the He/H2O discharge.

In section 2.1, a brief summary is given of the choice ofspecies, reactions, and volume and surface rate coefficients.Section 2.2 describes the volume and surface physics andchemistry in the various models that we employ. In section 3.1,we compare results of the various models to the previous work[8] at a fixed electron density and external water concentration.In section 3.2, we use the two-temperature hybrid model at13.56 MHz to examine the discharge equilibrium over a rangeof external water concentrations from 500 to 10 000 ppm, andover a wide range of discharge currents, examining the waterdepletion in the discharge, the buildup of non-reactive speciessuch as H2 and O2, the limiting effects of a finite sheathwidth, and the α–γ transition. In section 3.3, we brieflyexamine frequency and gap width variations, and in section 3.4,we examine the gas temperature sensitivity of the water ionclusters. In section 4, we present the conclusions and furtherdiscussion.

2. Model

2.1. Species, reactions, and rate coefficients

We will first compare the results of our increasinglysophisticated models using the simplified M2 reaction set ofLiu et al [8], which contains 40 species and 128 reactions,to their results using their full reaction set of 46 species and477 reactions. With 1000 ppm of H2O, their own comparisonof M2 with the full reaction set in discharges containing30–3000 ppm H2O agreed to within about 20%. The M2 setcontains the following species: electrons e, He, He∗, He+, He∗2,He+2, HeH

+, H, H+, H−, H2, H+2 , O, O∗, O+, O−, O2, O2(a),

O2(b), O+2, O3, OH, OH∗, OH+, OH−, HO2, H2O, H2O+,

H2O2, H2O−2 , H2O

+3, H3O

+, H3O−2 , H4O

+2, H5O

+2, H5O

−3 , H7O

+3,

H9O+4, H11O+5, and H13O

+6. In the remainder of the work, using

the most sophisticated (two-temperature hybrid) model, we use

2


an extended M2 reaction set, adding the higher mass clustersH15O+7, H17O

+8, and H19O

+9, to examine the gas temperature

sensitivity of the cluster formation.The reactions and rate coefficients we use are given in

table C1. The Boltzmann solver Bolsig+ [23] with the Morgancross section set [24] is used to calculate the e-He and e-H2Orate coefficients, and the results are fit to the form AT Be e

−C/Teshown in the table. Reaction R1 for e–He elastic scatteringgives the electron energy loss for this dominant energy lossprocess.

To find the time- and space-averaged hot electrontemperature and rate coefficients, we also use the sameBolsig+ calculation to determine the metastable excitation ratecoefficient Khm, the mean electron energy E , and the firstTownsend coefficient α as a function of E/ng, with ng the gasdensity. The metastable rate coefficient Khm, which appears inthe hot electron balance, is fit to a power law function of E/ngover the range of 10–300 Td (1 Td = 10−21 V-m2), Khm =Beff(E/ng)

q , and is similarly fit to an exponential functionof Th = 23E , Khm = Aeff exp(−Ceff/Th). Equating these twoforms and using a space- and time-averaged electric field in thehigh field sheath region allows the determination of the averagehot electron temperature Th. The details of the calculation aregiven in [21]. The fits yield Beff = 8.735 × 10−20 m3 s−1and q = 1.944, and Aeff = 4.279 × 10−14 m3 s−1 andCeff = 34.45 V. For the first Townsend coefficient, we usea fitting function that works over a wide E/ng range [25,p 56, section 4.1.5], α = Ang e−B(ng/E)1/2 , obtaining A =2.54 × 10−20 m2 and B = 31.60 Td1/2.

Table C2 gives the surface reactions, diffusion coefficientsD, mobilities µ, and reaction probabilities γ used in our model.The reaction probabilities are the same as in [8]. The gas-temperature dependent diffusion coefficients are determinedusing the procedure given in [26]. The mobilities are evaluatedat E/ng ≈ 20 Td from the data in [27] where available, and,where unavailable, are rescaled from a near-mass equivalention according to the theoretical scaling [28] that µ ∝ 1/√mR,with mR being the reduced mass of the ion–neutral pair.

2.2. Volume and surface model physics

A detailed description of the two-temperature hybrid model,with electron multiplication in the sheath, is given in [21]. Inthe model, the rf current density and frequency are specifiedas the independent electrical quantities.

2.2.1. Hybrid model volume physics. The model assumptionsare:

(a) The net ion density ni is uniform over the gap width l,except when determining the wall flux in (f).

(b) The electron density is equal to ni over a ‘cloud’ widthd = l − sm, and the maximum sheath width in thehomogeneous model is sm = 2J/(eωne) with ω as theradian frequency.

(c) There are two electron populations: a uniform warmcomponent (density ne, temperature Te), produced in thelow-field region of the discharge within the oscillatingelectron cloud, and a small hot component (temperature

Th > Te), produced in the high-field sheath regionsexterior to the oscillating cloud. The hot componentdensity nh (averaged over the width of the cloud) isassumed (and found) to be much smaller than ne. Allsecondary electrons and half the Penning ionization occursin the high field sheath region [21].

(d) The fields and sheath oscillation are determined using thehomogeneous model [19].

(e) The metastable density is approximated to be zero withinthe bulk plasma, is equal tonm0 at the sheath edges (x = smand x = l − sm), and falls linearly to zero at the walls.The average metastable density (over the gap width l) isthen nm = nm0 · (sm/l).

(f) The ion flux �w lost to a wall is determined from a constantmobility rf CL that includes the triangular ion generationprofile within the sheath; a collisional Bohm criterion [29]is used to join the sheath to the plasma.

(g) The oscillating temperature Te(t) of the warm componentis determined by equating the ohmic power to thecollisional losses, using the Maxwellian elastic andinelastic rate coefficients for the warm electrons [19].

(h) The oscillating hot component temperature Th(t) is foundfrom a kinetic calculation [23], using the space- and time-varying sheath electric field.

(i) The time-average rate coefficients for the warm andhot activated electron reactions are found by averagingover the oscillating temperatures. The electron-activatedreaction rates from the two classes of electrons aresummed to determine the total reaction rates used in theparticle balance equations.

(j) Multiplication factors Mγ for ionization by secondaryelectrons and MP for Penning ionization in the sheath arecalculated using the homogeneous model expressions forthe time-varying sheath motion and the space- and time-varying electric fields.

2.2.2. Hybrid model surface physics. We use the theorydescribed in appendix A1 for the diffusion of H2O from theexterior region into the reaction region across the cylindricalside wall, and for the diffusion of reaction products to the sideand end walls. The flow of H2O into the reaction region canbe written as

dn̄H2Odt

= KH2On̄H2O, (1)

where the rate coefficient (A14) is

KH2O =4DH2O

R2

(n2ext

n̄2H2O− 1

), (2)

with DH2O as the diffusion coefficient and next as the fixed,external H2O density. The H2O average density n̄H2O in (2)is obtained self-consistently in the complete solution to thehybrid model.

Reactive neutral species a such as H, O, O∗, O2(a), O2(b),OH∗, He∗, and He∗2 are assumed to be lost to the axial endwalls with some finite probability γa, through recombination

3


Table C1. Reactions and rate coefficients.

n Reaction Rate Coefficienta,b,c,d Reference

Electron elastic scattering(R1) e + He → e + He 4.937 × 10−14T 0.2579e [23, 24](R2) e + H2O → e + H2O 5.886 × 10−14T 0.2829e [23, 24]Electron impact ionization(R3) e + He → He+ + e + e 1.5942 × 10−14T 1.6767e e−49.2949/Te [23, 24](R4) e + He∗ → He+ + e + e 2.254 × 10−13T −0.1241e e−5.725/Te [23, 24](R5) e + H2O → H2O+ + e + e 9.9988 × 10−15T 0.74193e e−21.5733/Te [23, 24](R6) e + H → H+ + e + e 5.08 × 10−15T 0.6e e−13.6/Te [34](R7) e + H2 → H+2 + e + e 9.1 × 10−15T 0.5e e−15.4/Te [34](R8) e + O → O+ + e + e 9.0 × 10−15T 0.7e e−13.6/Te [35](R9) e + O2 → O+2 + e + e 9.0 × 10−16T 2.0e e−12.6/Te [35](R10) e + O2(a) → O+2 + e + e 9.0 × 10−16T 2.0e e−11.6/Te [35](R11) e + OH → OH+ + e + e 2.0 × 10−16T 1.78e e−13.8/Te [35]Electron impact excitation(R12) e + He → e + He∗ 1.0022 × 10−13T −0.96989e e−33.8044/Te

+3.8066 × 10−15T 0.74604e e−32.1884/Te [23, 24](R13) e + H2O → e + H2O 1.0817 × 10−14T −0.23465e e−1.3388/Te

+1.0896 × 10−14T −0.2369e e−1.3657/Te [23, 24](R14) e + H2O → H + OH + e 5.5258 × 10−14T −0.34295e e−14.0641/Te [23, 24](R15) e + H2O → H2 + O∗ + e 2.4162 × 10−14T −0.062054e e−22.4206/Te [23, 24](R16) e + H2O → H + OH∗ + e 1.0307 × 10−13T −1.1304e e−17.879/Te [23, 24](R17) e + H2 → H + H + e 8.73 × 10−14T −0.5e e−11.7/Te [34](R18) e + O → O∗ + e 4.5 × 10−15e−2.29/Te [35](R19) e + O2 → O + O + e 7.1 × 10−15e−8.6/Te [35](R20) e + O2 → O∗ + O + e 4.0 × 10−14e−8.4/Te [36](R21) e + O2 → O2(b) + e 3.24 × 10−16e−2.218/Te [37](R22) e + O2 → O2(a) + e 1.7 × 10−15e−3.1/Te [35](R23) e + O2(a) → O2 + e 5.6 × 10−15e−2.2/Te [35](R24) e + O2(b) → O∗ + O + e 3.49 × 10−14e−4.29/Te [37](R25) e + OH → O + H + e 2.08 × 10−13T −0.76e e−6.9/Te [38](R26) e + HO2 → H + O2 + e 3.1 × 10−15 [39](R27) e + H2O2 → OH + OH + e 2.36 × 10−15 [39](R28) e + H2O2 → H + HO2 + e 3.1 × 10−17 [40](R29) e + O3 → O + O2 + e 5.88 × 10−15 [39]Electron impact attachment and dissociative attachment(R30) e + H2O → OH + H− 8.6515 × 10−15T −1.66e e−8.5394/Te [23, 24](R31) e + H2O → H2 + O− 2.2818 × 10−15T −1.5914e e−11.2067/Te [23, 24](R32) e + H2O → OH− + H 1.0384 × 10−16T −0.78038e e−8.0822/Te [23, 24](R33) e + H2O2 → H2O + O− 1.57 × 10−16T −0.55e [41](R34) e + H2O2 → OH + OH− 2.7 × 10−16T −0.5e [41]Dissociative recombination(R35) e + H4O+2 → H2O + OH + H 9.6 × 10−13T −0.2e [42](R36) e + H7O+3 → H + 3H2O 2.24 × 10−12T −0.08e [43](R37) e + H9O+4 → H + 4H2O 3.6 × 10−12 [43](R38) e + H11O+5 → H + 5H2O 4.0 × 10−12 [44](R39) e + H13O+6 → H + 6H2O 4.0 × 10−12 [44](R40) e + H15O+7 → H + 7H2O 4.0 × 10−12 [44](R41) e + H17O+8 → H + 8H2O 4.0 × 10−12 [44](R42) e + H19O+9 → H + 9H2O 4.0 × 10−12 [44]Ion-molecule reactions: collisional detachment(R43) H− + He → He + H + e 8.0 × 10−18(Tg/300)0.5 [45]Ion-molecule reactions: helium ions(R44) He+ + 2He → He+2 + He 1.4 × 10−43(Tg/300)−0.6 [46](R45) He+ + H2O → H+ + OH + He 2.04 × 10−16 [47](R46) He+ + H2O → H + OH+ + He 2.86 × 10−16 [47](R47) He+ + H2O → H2O+ + He 6.05 × 10−17 [47](R48) HeH+ + H → H+2 + He 9.1 × 10−16 [47](R49) HeH+ + H2O → H3O+ + He 4.3 × 10−16 [48](R50) He+2 + H2O → HeH+ + He + OH∗ 1.3 × 10−16 [49](R51) He+2 + H2O → O+ + H2 + 2He 2.1 × 10−16 [49](R52) He+2 + H2O → OH+ + H + 2He 2.1 × 10−16 [49](R53) He+2 + H2O → H+ + OH + 2He 2.1 × 10−16 [49](R54) He+2 + H2O → HeH+ + OH + He 2.1 × 10−16 [49](R55) He+2 + H2O → H+2 + O + 2He 2.1 × 10−16 [49]

4


.Ion-molecule reactions: small ions

(R56) H+ + H2O → H2O+ + H 6.9 × 10−15 [47](R57) H− + H2O → OH− + H2 3.8 × 10−15 [47](R58) H+2 + He → HeH+ + H 1.3 × 10−16 [47](R59) H+2 + H2O → H2O+ + H2 3.9 × 10−15 [47](R60) H+2 + H2O → H3O+ + H 3.4 × 10−15 [47](R61) O+ + H2O → H2O+ + O 2.6 × 10−15 [36](R62) O− + H2O → OH− + OH 1.4 × 10−15 [50](R63) OH+ + O → O+2 + H 7.1 × 10−16 [47](R64) OH+ + H2O → H2O+ + OH 1.5 × 10−15 [51](R65) OH+ + H2O → H3O+ + O 1.3 × 10−15 [52](R66) H2O+ + O2 → H2O + O+2 3.3 × 10−16 [36](R67) H2O+ + H2O → H3O+ + OH 1.85 × 10−15 [36]Ion-molecule reactions: clusters(R68) O+2 + H2O + He → H2O+3 + He 2.6 × 10−40(Tg/300)−4.0 [53](R69) H2O+3 + H2O → H4O+2 + O2 1.0 × 10−15 [42](R70) H2O+3 + H2O → H3O+ + OH + O2 3.0 × 10−16 [50](R71) H4O+2 + H2O → H5O+2 + OH 1.4 × 10−15 [42](R72) H3O+ + H2O + He → H5O+2 + He 3.2 × 10−39(Tg/300)−4.0 [53](R73) H5O+2 + H2O + He → H7O+3 + He 7.4 × 10−39(Tg/300)−7.5 [53](R74) H7O+3 + H2O + He → H9O+4 + He 2.5 × 10−39(Tg/300)−8.1 [53](R75) H9O+4 + He → H7O+3 + H2O + He 2.0 × 1012T −8.1g e−8360/Tg [53](R76) H9O+4 + H2O + He → H11O+5 + He 3.3 × 10−40(Tg/300)−14.0 [53](R77) H11O+5 + He → H9O+4 + H2O + He 6.3 × 1024T −14g e−5750/Tg [53](R78) H11O+5 + H2O + He → H13O+6 + He 4.0 × 10−41(Tg/300)−15.3 [53](R79) H13O+6 + He → H11O+5 + H2O + He 2.62 × 1027T −15.3g e−5000/Tg [53](R80) H13O+6 + H2O + He → H15O+7 + He 4.5 × 10−42(Tg/300)−16.0 [53](R81) H15O+7 + He → H13O+6 + H2O + He 1.98 × 1029T −16.0g e−5000/Tg [53](R82) H15O+7 + H2O + He → H17O+8 + He 4.5 × 10−42(Tg/300)−16.0 [53](R83) H17O+8 + He → H15O+7 + H2O + He 1.98 × 1029T −16.0g e−5000/Tg [53](R84) H17O+8 + H2O + He → H19O+9 + He 4.5 × 10−42(Tg/300)−16.0 [53](R85) H19O+9 + He → H17O+8 + H2O + He 1.98 × 1029T −16.0g e−5000/Tg [53](R86) O− + H2O + He → H2O−2 + He 1.3 × 10−40 [54](R87) OH− + H2O + He → H3O−2 + He 2.5 × 10−40 [54](R88) H2O

−2 + H2O → H3O−2 + OH 1.0 × 10−17 [54]

(R89) H3O−2 + H2O + He → H5O−3 + He 3.5 × 10−40 [54]

Ion–ion recombination(R90) OH+ + H2O

−2 + He → O + OH + H2O + He 2.0 × 10−37(Tg/300)−2.5 [55, 56]

(R91) OH+ + H5O−3 + He → 2OH + 2H2O + He 2.0 × 10−37(Tg/300)−2.5 [55, 56]

(R92) H2O+ + O− + He → O + H2O + He 2.0 × 10−37(Tg/300)−2.5 [55, 56](R93) H2O+ + OH

− + He → OH + H2O + He 2.0 × 10−37(Tg/300)−2.5 [55, 56](R94) H2O+ + H2O

−2 + He → O + 2H2O + He 2.0 × 10−37(Tg/300)−2.5 [55, 56]

(R95) H2O+ + H3O−2 + He → OH + 2H2O + He 2.0 × 10−37(Tg/300)−2.5 [55, 56]

(R96) H2O+ + H5O−3 + He → OH + 3H2O + He 2.0 × 10−37(Tg/300)−2.5 [55, 56]

(R97) H2O+3 + H5O−3 + He → OH + 3H2O + O2 + He 2.0 × 10−37(Tg/300)−2.5 [55, 56]

(R98) H9O+4 + H2O−2 + He → OH + 5H2O + He 2.0 × 10−37(Tg/300)−2.5 [55, 56]

(R99) H9O+4 + H5O−3 + He → 7H2O + He 2.0 × 10−37(Tg/300)−2.5 [55, 56]


(R101) H11O+5 + H5O−3 + He → 8H2O + He 2.0 × 10−37(Tg/300)−2.5 [55, 56]


(R103) H13O+6 + H5O−3 + He → 9H2O + He 2.0 × 10−37(Tg/300)−2.5 [55, 56]


(R105) H15O+7 + H5O−3 + He → 10H2O + He 2.0 × 10−37(Tg/300)−2.5 [55, 56]


(R107) H17O+8 + H5O−3 + He → 11H2O + He 2.0 × 10−37(Tg/300)−2.5 [55, 56]


(R109) H19O+9 + H5O−3 + He → 12H2O + He 2.0 × 10−37(Tg/300)−2.5 [55, 56]

Neutral reactions: Penning ionization(R110) He∗ + He∗ → He+2 + e 2.03 × 10−15(Tg/300)0.5 [46](R111) He∗ + H → H+ + He + e 1.1 × 10−15 [57](R112) He∗ + H2 → H+2 + He + e 2.9 × 10−17 [6, 58](R113) He∗ + O → O+ + He + e 3.96 × 10−16(Tg/300)0.17 [59](R114) He∗ + O2 → O+2 + He + e 2.54 × 10−16(Tg/300)0.5 [10](R115) He∗ + OH → OH+ + He + e 7.8 × 10−16 Estimated same as H2O(R116) He∗ + H2O → He + H2O+ + e 6.6 × 10−16 [48]

5


Table C1. (continued).

(R117) He∗ + H2O → He + OH+ + H + e 1.5 × 10−16 [48](R118) He∗ + H2O → He + OH + H+ + e 2.6 × 10−17 [48](R119) He∗ + H2O → HeH+ + OH + e 8.5 × 10−18 [48](R120) He∗ + H2O2 → He + OH+ + OH + e 7.8 × 10−16 Estimated same as H2O(R121) He∗2 + H2O → 2He + H2O+ + e 6.0 × 10−16 [60]Neutral reactions: helium species(R122) He∗ + He + He → He∗2 + He 2.0 × 10−46 [61](R123) He∗2 + He → 3He 1.5 × 10−21 [47](R124) He + O∗ → O + He 1.0 × 10−19 [10](R125) He + OH∗ → OH + He 1.5 × 10−20 [10](R126) He + H + O2 → He + HO2 2.0 × 10−44(Tg/300)−0.8 [62](R127) He + H + OH → He + H2O 1.56 × 10−43(Tg/300)−2.6 [63](R128) He + O + O2 → He + O3 3.4 × 10−46(Tg/300)−1.2 [10]Neutral reactions: H atom(R129) H + O3 → OH + O2 2.71 × 10−17(Tg/300)0.75 [64](R130) H + HO2 → O2 + H2 1.1 × 10−18T 0.56g e−346/Tg [36](R131) H + HO2 → 2OH 2.35 × 10−16e−373.7/Tg [36]Neutral reactions: O atom(R132) O∗ + O2 → O + O2(b) 2.56 × 10−17e67/Tg [36](R133) O + OH → H + O2 6.0 × 10−17T −0.186g e−154/Tg [36](R134) O + HO2 → OH + O2 2.9 × 10−17e200/Tg [36](R135) O∗ + H2O2 → H2O + O2 5.2 × 10−16 [36](R136) O∗ + H2O → OH + OH 1.62 × 10−16e64.95/Tg [64](R137) O∗ + H2O → O + H2O 1.2 × 10−17 [65]Neutral reactions: Others(R138) O2(b) + H2O → O2(a) + H2O 4.52 × 10−18e89/Tg [66](R139) OH + OH → H2O + O 2.5 × 10−21T 1.14g e−50/Tg [64](R140) OH + OH → H2O2 1.5 × 10−17(Tg/300)−0.37 [62](R141) OH + HO2 → O2 + H2O 4.38 × 10−17e110.9/Tg [36](R142) OH + H2O2 → HO2 + H2O 4.53 × 10−18e−288.9/Tg [36](R143) OH∗ + H2O → H2O + OH 4.9 × 10−16(Tg/300)0.5 [67](R144) OH∗ + H2O2 → H2O + HO2 2.93 × 10−16 [68]

Radiation(R145) OH∗ → OH 1.25 × 106 [49]a Rate coefficients for 1, 2 and 3 reactants in s−1, m3 s−1, and m6 s−1, respectively.b Temperatures in roman and italic typeface in volts and kelvins, respectively.c A factor of e−0.01/Te was added to (R26)–(R29) for numerical convergence.d A factor of T −0.008e was added to (R37)–(R42) for numerical convergence.

or de-excitation processes. Although γa may be small, the rate-limiting step for loss to the end walls is usually not reaction-limited, but diffusion-limited. The axial diffusive loss of areactive species is written as

dn̄adt

= −Kaxialn̄a, (3)

where n̄a is the average density, and, from (A22) and (A20),

Kaxial = k2zDa, (4)with

k2z =(

l2

12+

Dal

v̄a

2 − γaγa

)−1(5)

and v̄a = (8kTg/πMa)1/2 as the mean speed. For γa �12Da/v̄al, the first term in the parentheses in (5) is much largerthan the second term, and the diffusive loss is rate-limiting;the loss does not depend on the value of γa. As described inappendix A2, this condition is met for all reactive species inthe simulations except O2(a).

The neutral radial diffusive losses for both reactive andnon-reactive species to the cylindrical wall are determined in

appendix A3 for a net generation having the H2O radial profile:

dn̄adt

= −Kradialn̄a, (6)

where for the reactive species, from (A33)

Kradial(reactive) = 2kzR

next

n̄H2ODa, (7)

and for the non-reactive species, from (A34),

Kradial(non-reactive) = 4R2

(next

n̄H2O+ 1

)Da. (8)

For the positive ion losses, we use a mobility-limited ionloss flux [15, 16], �+ = µ+n+wEw, with n+w and Ew as theion density and time-average electric field at the axial end wal,respectively. The ion density at the wall is obtained from a CLcalculation, as described in appendix B. For each kind of ion,the positive ion loss is

dn̄+dt

= −K+n̄+, (9)

6


Table C2. Surface reactions, diffusion coefficients D, mobilities µ, and reaction probabilities γ .

n Reaction D(m2/s) µ(m2/V-s) Reaction probability γ Reference

(R146) H → 12 H2 1.558 × 10−8T 1.75g — 0.03 [26](R147) O∗ → O 5.508 × 10−9T 1.75g — 1.00 [26](R148) O2(a) → O2 3.359 × 10−9T 1.75g — 0.0004 [26](R149) O2(b) → O2 3.359 × 10−9T 1.75g — 0.02 [26](R150) O → 12 O2 5.508 × 10−9T 1.75g — 0.02 [26](R151) OH∗ → OH 4.874 × 10−9T 1.75g — 1.00 [26](R152) He∗ → He 8.733 × 10−9T 1.75g — 1.00 [26](R153) He∗2 → He 5.923 × 10−9T 1.75g — 1.00 [26](R154) He+ → He — 3.624 × 10−6Tg 1.00 [27](R155) He+2 → He + He — 6.48 × 10−6Tg 1.00 [27](R156) HeH+ → He + H — 3.438 × 10−6Tg 1.00 [27](R157) H+ → H — 1.076 × 10−5Tg 1.00 [27](R158) H+2 → H2 — 1.106 × 10−5Tg 1.00 [27](R159) O+ → O — 9.116 × 10−6Tg 1.00 [27](R160) O+2 → O2 — 7.907 × 10−6Tg 1.00 [27](R161) OH+ → OH — 9.062 × 10−6Tg 1.00 [27](R162) H2O+ → H2O — 9.014 × 10−6Tg 1.00 [27](R163) H2O+3 → H2O + O2 — 7.748 × 10−6Tg 1.00 [27](R164) H3O+ → H2O + H — 8.97 × 10−6Tg 1.00 [27](R165) H4O+2 → 2H2O — 7.858 × 10−6Tg 1.00 [27](R166) H5O+2 → 2H2O + H — 7.848 × 10−6Tg 1.00 [27](R167) H7O+3 → 3H2O + H — 7.722 × 10−6Tg 1.00 [27](R168) H9O+4 → 4H2O + H — 7.657 × 10−6Tg 1.00 [27](R169) H11O+5 → 5H2O + H — 7.617 × 10−6Tg 1.00 [27](R170) H13O+6 → 6H2O + H — 7.591 × 10−6Tg 1.00 [27](R171) H15O+7 → 7H2O + H — 7.572 × 10−6Tg 1.00 [27](R172) H17O+8 → 8H2O + H — 7.557 × 10−6Tg 1.00 [27](R173) H19O+9 → 9H2O + H — 7.546 × 10−6Tg 1.00 [27]

where n̄+ is the average ion density, and the rate coefficient is,from (B5)

K+ = 98

µ+

l

(eTen̄+

25�0

)1/12 (J

ω�0

)5/6 (ne

n+,tot

)13/12, (10)

with n+,tot as the average total positive ion density.

2.2.3. Comparison models. We have used three additionalmodels for comparisons, in the calculations of section 3.1: asingle-temperature hybrid model and two global models. Inglobal models, the time-average electron power absorbed isspecified as the independent electrical quantity, and the particleand energy balances are solved to determine the dischargeequilibrium. There is no specified driving frequency and nocalculation of the underlying discharge dynamics. This meansthat there is no calculation of the sheath width, electric field,discharge voltage, rf current and ion power. There is no wayto determine a mobility-driven ion flux in these global models.

1. The single-temperature hybrid model is the same as thetwo-temperature hybrid model, except that the generationof hot electrons in the high-field sheath regions isneglected. This model is useful within the α-mode, atlow rf currents, where the electron multiplication is small,but it cannot capture the higher current (higher power)regime, including the α–γ transition physics [21].

2. The global diffusion model has a single electrontemperature and the sheath width is zero. It uses the same

diffusion fluxes for H2O and the reaction products as thehybrid models. Because the gas pressure is high, a gaskinetic axial ion loss flux �+ = n+v̄+/4 is used, yieldingan ion loss rate coefficient K+ = v̄+/2l.

3. The global model in [8] also has a single electrontemperature and a zero sheath width. There is no neutraldiffusion. The H2O density within the discharge regionis taken to be the fixed external density next, and theneutral fluxes lost to the end and side walls are takento be �a = γanav̄a/4 and �a = nav̄a/4, respectively.The positive ion flux to the end wall is taken to be�+ = n+uB, where uB = (eTe/M+)1/2 is the low pressure(collisionless) Bohm speed. One should note that thisspeed is significantly higher than a collisional Bohm speedat atmospheric pressures [29].

A summary of the volume and surface loss physics for thefour models used in the present work is given in table 1.

3. Results

The simulations are done using the Matlab stiff integratorode15s and are started with low densities of all dissociated andcharged species, and the simulation time is tf = 0.5 s, abouttwice the time for all the densities in the system to reach a steadystate. A typical simulation takes less than two minutes on amedium-speed laptop computer. As a check on the integrationaccuracy, we determined the electron density by two methods:(1) by direct integration of the electron balance equation; (2) by

7


Table 1. Volume and surface model physics.

Reference [8] global Global diffusion Hybrid Hybrid with multiplication

nH2O const Inward diffusion determines nH2O (appendix A1)Axial neutral loss flux γ · 14 nv̄ Outward diffusion (appendix A3)Radial neutral loss flux 14 nv̄ Outward diffusion (appendix A3)Axial ion loss flux n+uB 14 n+v̄+ CL flux µ+n+wE (appendix B)Maximum sheath width 0 0 Homogeneous model 2s̄EEDF Single temperature Te Two temperatures Te, Th

Figure 1. Principal neutral densities (a) and ion densities (b), at ne = 1017 m−3 for a 1 cm radius 0.5 mm gap cylindrical helium discharge,with 1000 ppm external H2O concentration, driven at 13.56 MHz, for five calculations: reference [8] (1st bars), the global M2 set (2nd bars),the global diffusion model (3rd bars), the single temperature hybrid model (4th bars), and the two-temperature hybrid model (5th bars).

summing the heavy particle charge densities. The two methodsagree to within 1% for all results in this work.

3.1. Comparison to previous global results

In this subsection, simulations are done using the four differentdischarge models indicated in table 1, and the results arecompared to those in [8] at 1000 ppm H2O and an electrondensity of 1017 m−3. For all four simulations, the discharge gapis 0.5 mm, the electrode radius is 1 cm, and the gas temperatureis 300 K. The time-average power (in the global models) or therf current density (in the hybrid models) is chosen to obtain therequired electron density. A driving frequency of 13.56 MHzis used in the single- and two-temperature hybrid models, witha secondary emission coefficient, based on [16], of 0.25.

Figures 1(a) and (b) give bar graphs with five bars foreach selected species density. The first two bars for eachspecies compare the [8] results using the full reaction setto our calculations with the reduced M2 set. We see aclose correspondence, particularly for the dominant products,indicating that the M2 set captures most of the importantreactions. For the neutral species shown in figure 1(a), thelarger differences occur for the lower density species, and maydo so for a variety of reasons. Probably the most importanteffect is that small differences in production and loss of thehigher density species can result in larger fractional differencesin species with much lower density. For the positive ion speciesshown in figure 1(b), we also see a close correspondence,except for some high mass clusters. The volume generation andloss reactions for these clusters have highly gas-temperaturedependent rate coefficients. Although the simulation in [8]

was stated to be performed at a gas temperature of 300 K,we find that their data best fits the M2 simulation better at atemperature of 350 K. In section 3.4, we further explore thevariations of cluster formation with gas temperature.

The addition of neutral H2O depletion, reaction productdiffusion, and reduced positive ion losses in the global model,shown as the third bar in the figures, strongly affects thedensities of most dominant species. For example, the dominantnon-reactive species H2, O2 and H2O2, resulting from thedissociation of H2O and subsequent wall reactions, becomeorders of magnitude larger. These effects can be furtherenhanced in the metastable states, e.g., O2(a), and for the lowmass positive ions. Water depletion effects alone are minimalunder these low power conditions (≈0.5 W cm−2), but becomemore important at the higher powers, as will be seen in the nextsubsection. The addition of the sheath physics in the fourthbar, and both the sheath and hot electron multiplication in thefifth bar, do not make very significant changes in the densities.The reason is that the sheath widths do not dominate the bulkplasma, and at these low powers, the multiplication effects aresmall. In other operating ranges, explored below, importantdifferences may arise.

3.2. Variations with discharge current and external waterconcentration

In this subsection, simulations are done using the two-temperature hybrid model, which incorporates the sheathmultiplication phenomena. We use the same configurationas in section 3.1, with the extended M2 reaction model,including the higher mass clusters H15O+7, H17O

+8, and H19O

+9.

8


The discharge current J is varied from 100 to 1600 A m−2,with four external H2O concentrations: 500, 1000, 3000 and10 000 ppm. The lowest concentration is chosen such thatthe depletion of H2O at the higher rf currents leads to anaverage H2O concentration within the validity of the M2 model(30 ppm). The 500–10 000 range is probably appropriate foractual applications.

In figure 2 we plot important discharge quantities versusJ , with H2O external concentration as a parameter. Infigure 2(a) we see the decay of the average water density withincreasing current, as the water is depleted by the interactions,observing that for the highest current at 500 ppm, the averageconcentration has been reduced by a factor of ten. Thiseffect is reduced as the external water concentration increases,as is physically reasonable. The main consequence of thedepletion is the buildup of non-reactive neutral species, as seenin figure 1.

The maxima of the oscillating hot and warm temperaturesare plotted in figure 2(b) (the time-average temperatures arehalf of the maximum values). The rising temperature of thehot sheath electrons becomes sufficiently large that it begins totake over the ionization from the higher density warm electronspecies, causing the temperature of that species to decline. Thissignals the transition to the γ -mode, as was also observed inprevious work [21] for the He/0.1%N2 discharge.

Figure 2(c) gives the normalized sheath widths sm/l.The sheath widths are fairly large in the simulation, andat 10 000 ppm the sheaths become too large to sustainthe discharge, demonstrating that finite sheath widths cansignificantly limit operating regimes in atmospheric pressuredischarges. This phenomenon is not captured in the usualglobal models, in which the sheath width is essentially takento be zero [20]. For the lower concentrations in the figure, wesee the α–γ transition, i.e., the nearly constant sheath width inthe α-mode shrinks at the onset of the γ -mode. These resultscan be understood from the homogenous model. From thetime-average electron power balance and the oscillating sheathwidth, one obtains [20]

ωsm =(

24 ζeTe

MHe

)1/2, (11)

where ζ is the ratio of total (elastic + inelastic) to elasticelectron energy losses. Equation (11) predicts that the sheathwidth within the α-mode increases with increasing H2Odensity, and that the nearly constant sheath width in the α-mode shrinks at the onset of the transition, due to the collapseof the warm Te, as is seen in the figure. The inelastic lossesincrease with the H2O concentration, with the simulationsyielding (at 400 A m−2) ζ = 1.2, 1.28, and 1.56 for externalconcentrations of 500, 1000, and 3000 ppm, respectively. Arequired condition to have an equilibrium solution is that theelectron cloud width d = l−sm be greater than zero; i.e., sm <l. (The actual condition used in the calculations is d > 0.15 l.)This restrictive condition is encountered at 10 000 ppm, as seenin figure 2(c), where there is no equilibrium solution within theα-mode.

The rf voltage Vrf in figure 2(d) also shows the α–γtransition. The homogeneous model capacitive sheath gives

the result

Vrf = Jω�0

sm. (12)

Figure 2(d) shows that Vrf rises linearly with J when smis constant, flattens, and then falls slowly as sm decreasesfaster than J increases, in agreement with (12). Near themaximum seen in the figure, J is a double-valued functionof Vrf , indicating that a voltage-driven discharge can exhibithysteresis and sudden jumps from the α to the γ mode, andvice versa, as the voltage is varied. This behaviour, also seenin fluid simulations of atmospheric pressure discharges [15],is characteristic of the α–γ transition [30, 31].

The space- and time-averaged factors Mγ and MP forsecondary and Penning multiplication are plotted in figure 2(e).Note that Mγ − 1 and MP − 1 are the number of additionalelectron–ion pairs created in the high-field sheath region by anemitted secondary electron and a high-field Penning ionization,respectively. When the multiplication is significant, we see thatMγ is significantly larger than MP. This is due, in part, to theassumed triangular profile of the metastable density within thesheath. Because the profile falls to zero at the electrode, thereare few Penning pairs created within the high-field region nearthe electrode, while all the secondaries, emitted at the electrodesurface, see the high fields.

Before the onset of the multiplication, the largerproportion of the total ionization is due to Penning ionization,rather than direct ionization by the warm electrons. Thisreverses with the onset of multiplication, producing a highertemperature hot species, as illustrated in figure 2(f ). Inthe α-mode, Mγ ≈ MP ≈ 1, such that the hot electroncreation by multiplication is negligible, and the Penning pairproduction is about a factor of 5–10 larger than the directionization. However, in the γ -mode, when the multiplication issignificant, the direct ionization becomes 2–4 times larger thanthe Penning; the direct production from the hot secondaries andtheir multiplication products becomes the dominant process.We have confirmed that this is true even if the secondaryemission coefficient is reduced to a significantly smaller value;e.g., γse = 0.05.

In figure 2(g), we give the electron power Se and the totalpower (electrons + ions) Stot. In the α-mode, for all but thelowest rf currents (J � 200 A m−2), the homogeneous modelelectron power is [21]

Se = mνmde

(ζ

3

2

eTe

MHe

)1/2J, (13)

where νm is the electron–neutral collision frequency. Since Teand the electron cloud width d are constant in the α-mode, Sescales linearly with J , as seen in the figure. For the mobility-dominated flow of ions across the sheath, the ion power is [21,equation (B21)]

Si = 5µ̄6ω3�20

J 3, (14)

where µ̄ is a density-weighted average ion mobility. Hence Siscales more strongly with J than Se, becoming the dominantdissipation at the higher currents. (One should note that theapplied powers in typical experiments are generally much

9


γ

Figure 2. Discharge quantities versus rf current density J , for H2O concentrations of 500, 1000, 3000 and 10 000 ppm; (a) nH2O/next,H2O,(b) Maximum hot Thmax and warm Temax electron temperatures, (c) normalized sheath width sm/l, (d) rf discharge voltage. (e) secondaryMγ and Penning MP multiplication factors, (f ) total pair creation rates for Penning and direct ionization processes, (g) total power Stot andelectron power Se, (h) electronegativity n−/ne.

10


Figure 3. (a) Electron and positive ion densities, (b) positive ion cluster densities and (c) principal neutral densities, versus J at 1000 ppmH2O.

higher than the powers dissipated in the discharge, due tohigh external losses in the matching networks and externalcircuits used, a feature commonly seen in atmospheric pressurerf plasmas [15, page 76].)

Finally, in figure 2(h), we give the electronegativityn−,tot/ne, the ratio of total negative ion to electron density.Except at the lowest currents, the electronegativities arerelatively small, and they decrease with increasing current.A simple global negative ion balance, equating an averagedissociative attachment rate to an average positive–negativeion recombination rate, gives the relation n−,tot/ne ≈Kattng/Krecn+,tot. This indicates that the decrease inelectronegativity with increasing current (increasing n+,tot)seen in the figure is a consequence of the negative ion balance.In the simulations, the two most important negative ion speciesdensities (not shown in figure 3) are the clusters H5O

−3 and

H3O−2 .In figure 3 at 1000 ppm external H2O density, we plot

ne, the hot component density nh, the positive ion and clusterdensities, and the neutral densities, versus J . The electrondensity, shown along with the ion densities in figures 3(a)and (b), increases roughly linearly with J in the α-mode, inagreement with the electron power balance scaling [21]

n2eTe =MHe

6ζe3J 2 −

(�0mωνme2

)Te, (15)

as Te is roughly constant in the α-mode. The second term onthe right is important only at the lowest currents and gives alower value than the linear scaling with J , seen in the figure at

100 A m−2. After the α–γ transition, the figure shows a faster-than-linear scaling of ne with J , again in accordance with (15),as Te collapses with increasing J .

The density nh of the hot component shown in figure 3(a)(dashed line) is much smaller than ne and is roughly constant,decreasing slightly from approximately 2 × 1015 m−3 at100 A m−2 to 1×1015 m−3 at 1600 A m−2. We can understandthe near-constancy of nh from the hot electron and positiveion balance relations, as follows: In the α-mode, half of thePenning ionization goes to producing hot electrons, which arelost producing metastables

1

2nm

∑j

KPjnj = KhmnhnHe, (16)

where the j -sum is over the neutral species participating inthe Penning processes. The positive ions created by all thePenning processes are lost to the walls

nm∑

j

KPjnj = 2l

∑k

�+k. (17)

Eliminating the Penning sum from these equations and solvingfor nh, we obtain

nh = 1KhmnHel

∑k

�+k. (18)

Both Khm and �+k are functions of the rf current density J .We noted in section 2.1 that Khm is well-fitted to a power law

11


function of the electric field E ∝ J , Khm = Beff(E/ng)q ,with q = 1.944. This implies the scaling that Khm ∝ J 1.944.The ion flux scales as �+k ∝ J 5/6n13/12+s from (B4). Asn+s ≈ ne ∝ J , we obtain the scaling �+k ∝ J 1.917. BecauseKhm and �+k scale with J in nearly the same way, we see from(18) that the hot component density nh is roughly a constant,independent of J . In the transition to the γ -mode, an additionalhot electron and ion generation due to secondary and Penningmultiplication appears in the left hand sides of (16) and (17)respectively, leading to a decrease in nh, as seen in figure 3(a).

Within the α-mode, the positive ion cluster H11O+5dominates the total positive ion density, with little contributionfrom the light ions, demonstrating the importance of the clusterdynamics in He/H2O discharges. However, at the highestcurrents within the γ -mode, we see that the lighter ionsdominate, particularly HeH+, as the discharge becomes highlydissociated.

Reflecting the drop in nH2O/next that was seen infigures 2(a), 3(c) shows that the H2O density is nearly theexternal density next at 100 A m−2 and gradually drops bynearly a factor of ten at 1600 A m−2, as ne, and hence thedissociation processes increase. The dissociation at the highestcurrents within the γ -mode produces copious densities of H2and O2, exceeding the H2O density by factors of 2–4. Thedominant biologically-reactive species produced is H2O2, and,to a lesser extent, OH and O2(a). There is a smaller productionof O-atoms, and very little O3 is produced (not shown).

3.3. Sheath width and minimum current variations with gapsize and frequency

We noted in figure 2 that at 10 000 ppm H2O there are nosolutions for lower currents because the sheath widths becometoo large. We expect from the scaling in (11) that for largergaps or higher frequencies, the range of J will extend to lowervalues. We explore this in figure 4 which plots sm at 10 000 ppmfor the three different cases of 13.56 MHz at 0.5 mm gap,13.56 MHz at 1 mm gap, and 27.12 MHz at 0.5 cm gap. Asshown in the figure, the α mode is recovered by increasingeither the gap width or the frequency. The scaling of sm withfrequency is seen to obey the relation (11).

Because the first Townsend coefficient α, whichdetermines the multiplication factors Mγ and MP, dependsstrongly (exponentially) on the electric field near the wall,the condition for the α–γ transition and collapse of the warmTe occurs at a particular value of the dc field Ew = J/ω�0.For the previous He/0.1%N2 simulations, we obtained Ew ≈1 × 106 V m−1 at the transition [21]. Figure 4 for He/H2O at13.56 MHz and 10 000 ppm indicates that the transition occursat J ≈ 800 A m−2, which gives the same transition field as inthe He/N2 simulations. The 27.12 MHz simulation also showsthe expected frequency scaling of J ∝ ω at the transition.

In addition to the sheath width, the electron power balancesets a lower limit on the rf current density for an equilibriumsolution to exist; the minimum current density is found tobe [19]

Jmin = ω�0νm me

(6 ζ

eTe

MHe

)1/2. (19)

Figure 4. Sheath widths sm versus J , at 10 000 ppm H2O, forvarious combinations of gap width and frequency.

The minimum current is proportional to the frequency. For the13.56 MHz case with a 1 mm gap, the simulation values areνm ≈ 1.35×1012 s−1, ζ ≈ 1.9 and T e ≈ 1.6 V (at 200 A m−2).Using these values in (19), we obtain Jmin ≈ 120 A m−2. Thesimulations indicate that the discharge equilibrium is lost at156 A m−2, in reasonable agreement with the calculated value.For the 27.12 MHz case with a 0.5 mm gap, the equilibrium islost at 312 A m−2, confirming the linear scaling of Jmin with ω.

3.4. Cluster size variations with gas temperature

The rate coefficients in table C1 for the volume creation andloss of the high-mass clusters are strongly gas-temperaturedependent. In figure 5 we plot a set of 9 bar graphs for theregular cluster densities; each set has four densities at 275, 300,325, and 350 K, which is a reasonable range for biomedicalapplications. The calculations are done using the extendedM2 reaction set at an external H2O concentration of 1000 ppmfor l = 0.5 mm, f = 13.56 MHz and a fixed J = 300 A m−2,corresponding to a power of approximately 0.43 W cm−2. Weobserve a buildup of water clusters at all temperatures, reachinga maximum density for H11O+5. The notable difference is thatthe higher mass clusters maintain significant densities at thecoolest gas temperatures, while dropping off more rapidly athigher Tg.

The gas heating can limit the maximum power used inapplications. To estimate the heating, we assume that a powerper unit volume ptot is deposited uniformly into helium gas atatmospheric pressure within two parallel plates, each held ata fixed temperature Tw, and separated by a gap width l. Theheat flow equation

− κT d2Tg

dz2= ptot (20)

has the parabolic solution

Tg = Tw + ptot2κT

(l2

4− z2

), (21)

12


Figure 5. Water cluster densities versus gas temperature, atJ = 300 A m−2 and 1000 ppm H2O.

where κT ≈ 0.156 W/m-K is the thermal conductivity ofthe helium gas and Tg(z) is the gas temperature. Averagingover the parabolic profile gives T g = Tw + T g, with anaverage temperature rise T g = ptotl2/12κT . Putting ptot =Stot/l, and for a 0.5 mm gap, we obtain the practical formula

T g ≈ 2.67 Stot, with Stot in W cm−2. For the typical α-mode powers of 0.1–1 W cm−2, the average temperature rise issmall. At the highest power of 20 W cm−2, after the transitionto the γ -mode, the average temperature rise is significant,

T g ∼ 50 K.

4. Conclusions and discussion

Most modelling of the complex chemistry of atmosphericpressure plasmas has been done with simple global modelsthat do not consider diffusive flows, sheaths, or the α–γtransition. In this work, we extend a previous hybrid globalmodel of atmospheric pressure noble gas/trace gas rf capacitivedischarges that includes the important sheath and transitioneffects [19–21] to bounded discharges and to more complexchemistries, by adding new physics of (1) trace gas depletiondue to diffusion into and reactions within the discharge,(2) reaction product diffusion to the discharge boundaries, and(3) multiple Penning reactions and multiple positive ion walllosses. The complete model determines the limits to operationin the low current α-mode and the electron multiplicationeffects at higher currents, both arising from finite-size sheaths.The complete model has two classes of electrons: a lowdensity class of hot electrons associated with the high-fieldoscillating sheath regions, and a high density class of warmelectrons associated with the bulk, allowing exploration ofthe transition from a low power, bulk-dominated α-modeto a high power, sheath-dominated γ -mode. The dischargedynamics on the rf timescale is determined using a current-driven homogeneous model in which the net ion density (totalpositive ion density minus total negative ion density) is uniformover a gap width l and the electron density is uniform overan oscillating cloud of width d = l − sm, where sm is the

maximum value of the oscillating sheath width. An analyticalsolution of the homogeneous model determines the space-and time-varying electric fields and the oscillating sheathwidth, and an analytic ohmic power balance determines thetime-varying warm electron temperature on the rf timescale.A Child law sheath calculation determines the positive ionwall flux. Analytic averages over an rf period determinethe rate coefficients used in the particle balance equations,which are then numerically integrated to find the global speciesdensities. The combination of the analytic solutions of thedynamics and power balance, and the numerical solutions ofthe particle balances, gives a fast solution of the dischargeequilibrium, typically of order two minutes on a medium-speed laptop. The model physics is described in detail insection 2.2.

We apply the model to determine the dischargeequilibrium properties in a bounded cylindrical He/H2Odischarge with a 0.5 mm gap and 1 cm radius electrodes,at an rf frequency of 13.56 MHz. The simulations include40–43 species. We compare the two-temperature hybridmodel, as well as less complete hybrid and global models,with a previous global calculation that used a larger reactionset [8] at an external H2O concentration of 1000 ppm. Thereis reasonable agreement using their model assumptions.Significant differences appear if the H2O depletion andreaction product diffusive losses are incorporated into themodel. We use the model, including additional clusters upto H19O+9, to determine the equilibrium over a wide rangeof rf currents and for external H2O concentrations varyingfrom 500 to 10 000 ppm. At high currents, the average H2Oconcentration within the discharge is found to be depletedby as much as a factor of ten. Within the α-mode, thepositive ion cluster H11O+5 dominates the total positive iondensity, demonstrating the importance of the cluster dynamicsin He/H2O discharges. As in previous calculations of a He/N2system with simpler chemistry [21], the α–γ transition is foundto occur for a dc electric field at the electrode ofEw = J/ω�0 ≈1 × 106 V m−1 (Ew/ng ≈ 40 Td). At 10 000 ppm, the sheathbecomes too large to sustain the discharge in the α-mode. Atlarger gaps (1 mm) or higher frequencies (27.12 MHz), thislimitation is overcome, in agreement with the sheath widthscaling in (11). The high mass H2O cluster reactions arestrongly sensitive to the gas temperature Tg. Varying Tg from275 to 350 K, we find that the higher mass clusters maintainsignificant densities only at the cooler temperatures.

The M2 model that we use in this paper did not includeH+3 . As pointed out by a referee, the reaction of the primary H

+2

with H2 can be fast, resulting in H+3 having larger equilibriumdensities than H+2 . To investigate this, we added the 20reactions of H+3 from the complete Liu et al model [8] into ourbasic case (1000 ppm H2O, 13.56 MHz) and did simulationsat low and high currents. For low currents, the simulationresults show that the density of H+3 is a little higher than theH+2 density. In the case of high currents, the density of H

+3 is

dominant. But neither the H+3 or H+2 species are important in the

water reactions or in determining the total positive ion density.The species of primary concern did not change significantlyeven at high currents. Consequently we did not change ourmany calculations in the paper to include H+3 .

13


Acknowledgments

This work was partially supported by the Department of EnergyOffice of Fusion Energy Science Contract DE-SC000193and by the Natural Science Foundation of China Contract11375042. We thank Felipe Iza for very useful comments onhis previous simulations of the He/H2O atmospheric pressuredischarge.

Appendix A. Neutral diffusion

We determine the diffusive loss fluxes of various neutralspecies in a cylindrical (r, z) atmospheric pressure He/H2Odischarge, consisting of two solid plates with radius Rseparated in z by a gap of width l, embedded in a largerchamber. In an experiment [22], R = 1 cm and l = 0.5 mm.The H2O fraction is variable, with a typical external valueof 0.001 in the larger chamber. The cylindrical side wall isassumed to be open, such that neutral species can freely diffuseacross it into the discharge region.

A.1. Water molecule diffusion

The H2O enters through the cylindrical side wall with a fixedradial edge density. Assuming that the volume loss of H2O isproportional to its density, we obtain the diffusion equation

− D∇2n = −νn, (A1)where D and n are the H2O diffusion coefficient and density,and the eigenvalue ν is a constant volume loss frequency. Weassume no losses of H2O to the axial end walls. For thisboundary condition, the diffusion is independent of the axialcoordinate z, and (A1) reduces to

d2n

dr2+

1

r

dn

dr− ν

Dn = 0, (A2)

with the solutionn = n0I0(κr), (A3)

where I0 is the modified zero-order Bessel function of the firstkind, and κ = (ν/D)1/2 is the radial decay constant. Theaverage density is

n̄ = 1πR2

∫ R0

2πrdr n0I0(κr) = 2κR

I1(κR)n0, (A4)

where I1 is the modified first-order Bessel function of the firstkind. The net inward flux at r = R is

�H2O = Ddn

dr

∣∣∣∣R

= n0κDI1(κR). (A5)

Substituting (A4) into (A5) to eliminate n0 in favor of n̄ gives

�H2O =D

2Rκ2R2n̄. (A6)

To determine κR, a mixed boundary condition is used atthe radial wall,

�H2O(R) =1

4v̄(next − n(R)

), (A7)

where v̄ = (8kTg/πM)1/2 is the mean speed and next = f0nHeis the external density of H2O molecules. Substituting (A3)–(A5) into (A7), we obtain

n̄

(I0(κR) +

4κD

v̄I1(κR)

)= next 2I1(κR)

κR, (A8)

which can be solved numerically for κR as a function of next/n̄.However, a very good analytic solution can be found in the limitthat 4κD/v̄ 1, which is the regime of interest at atmosphericpressure. Neglecting the second term in parentheses on the lefthand side in (A8), we obtain

κRI0(κR)

I1(κR)= 2next

n̄. (A9)

To a very good approximation (within 4%),

κRI0(κR)

I1(κR)≈ (4 + κ2R2)1/2 (A10)

for all κR. Using this in (A9), we obtain

κ2R2 ≈ 4(

n2ext

n̄2− 1

)(A11)

Substituting (A11) into (A6), we obtain the flux of H2Odiffusing into the cylindrical plasma through the radial wall as

�H2O =2D

R

(n2ext

n̄2− 1

)n̄. (A12)

Multiplying this flux by 2πRl and dividing by the dischargevolume πR2l, we obtain the volume rate of diffusive flow ofH2O into the discharge,

dn̄

dt= KH2On̄, (A13)

where the rate coefficient is

KH2O =4D

R2

(n2ext

n̄2− 1

). (A14)

The H2O average density n̄ is obtained self-consistently in thecomplete solution to the global or hybrid model.

A.2. Reactive and nonreactive species

Reactive neutral species a such as H, O, O∗, O2(a), O2(b),OH∗, He∗, and He∗2 are assumed to be lost to the end wallswith some finite probability γa, through recombination orde-excitation processes. Although γa may be small, therate-limiting step for loss to the end walls is usually notreaction-limited, but diffusion-limited. The one exception,discussed below, is the oxygen metastable O2(a).

To understand the interplay between diffusion to andreaction at a surface, we examine a 1D model having diffusiveloss to the end walls with a reaction probability γa and aconstant volume generation Ga [32, section 9.4]. The diffusionequation is then

− d2na

dz2= Ga

Da, (A15)

14


with the solution

na = Gal2

8Da

(1 − 4z

2

l2

)+ naS, (A16)

where naS is the density at the end walls. The flux is � =−Dadna/dz = Gaz. The boundary condition at the endwallz = l/2 is [33]

�(l/2) = 12na(l/2) v̄a

γa

2 − γa . (A17)

Applying the boundary condition, we obtain

naS = 2 − γaγa

Gal

v̄a. (A18)

Integrating (A16) over the volume yields the average density

n̄a = Gak2zDa

(A19)

with

k2z =(

l2

12+

Dal

v̄a

2 − γaγa

)−1. (A20)

The volume loss to both end walls is then

dn̄adt

= −Kaxialn̄a, (A21)

where

Kaxial = 2�(l/2)n̄al

= k2zDa. (A22)

For γa � 12Da/v̄al, the first term in the parentheses in(A20) is much larger than the second term, and the diffusiveloss is rate-limiting; the loss does not depend on the value ofγa. (In this case, the simplified boundary condition naS ≈ 0can be used to determine the loss.) For typical values Da ∼10−4 m2 s−1, v̄a ∼ 500 m s−1, and l = 0.5 mm, we findγa � 4.8×10−3 for rate-limiting diffusive loss. This conditionis met for all reactive species listed above except O2(a), forwhich γa = 0.0002.

A.3. Neutral diffusion with the radial generation profile of water

We now consider the diffusion of a reactive or non-reactiveneutral species a having a volume generation with the radialprofile of the H2O density. From (A3) and (A4), the H2Odensity is

n(r) = n̄ κR2I1(κR)

I0(κr). (A23)

where n̄ is the volume-averaged H2O density. For a reactivespecies, there are diffusive losses to the axial walls. We assumethat l R, such that we can model the axial losses as anequivalent volume loss term −Dak2z na(r), with k2z given by(A20) and with na the radially-varying axial-averaged density.(For a non-reactive species, we set kz = 0.) The radialdiffusion equation is then

− d2na

dr2− 1

r

dnadr

= GaDa

I0(κr) − k2z na. (A24)

where

Ga = Kan0n̄ κR2I1(κR)

, (A25)

with Ka the rate coefficient for generation of a by a specieswith uniform radial density n0 and water. (The uniform densityspecies could be electrons or helium metastables, for example.)The solution to (A24) with the radial boundary condition thatna = 0 at r = R is

na = Ga(k2z − κ2)Da

[I0(κr) − I0(κR)

I0(kzR)I0(kzr)

]. (A26)

The volume-averaged density is found by averaging (A26) overthe radial variation

n̄a = 2Ga I0(κR)(k2z − κ2)R2Da

[I1(κR)

κR I0(κR)− I1(kzR)

kzR I0(kzR)

]. (A27)

Substituting (A9) into (A27) gives

n̄a = 2Ga I0(κR)(k2z − κ2)R2Da

[n̄

2next− I1(kzR)

kzR I0(kzR)

]. (A28)

The axial loss coefficient is found by averaging the axialloss flux k2zDal na(r) over the radius and dividing by thedischarge volume, yielding

dn̄adt

= −Kaxialn̄a, (A29)

with Kaxial given by (A22) and with k2z given by (A20). Theradial loss flux is

�radial = −Da dnadr

∣∣∣∣R

= −GaI0(κR)k2z − κ2

[κI1(κR)

I0(κR)− kzI1(kzR)

I0(kzR)

]. (A30)

Substituting (A9) and (A11) into (A30), we obtain

�radial = − GaI0(κR)(k2z − κ2)R

[2

(next

n̄− n̄

next

)− kzR I1(kzR)

I0(kzR)

].

(A31)

Substituting n̄a for Ga from (A28) into (A31), we obtain therate coefficient for radial loss

Kradial = −DaR2

[2

(nextn̄

− n̄next

)− kzR I1(kzR)I0(kzR)

][

n̄2next

− I1(kzR)kzR I0(kzR)

] . (A32)For reactive species with kzR � 2next/n̄, this reduces to

Kradial(reactive) = 2kzR

next

n̄Da. (A33)

For the non-reactive species with kzR 1, we find

Kradial(non-reactive) = 4R2

(nextn̄

+ 1)

Da. (A34)

15


Appendix B. Mobility-driven ion losses

To determine the positive ion losses to the end walls, we usethe mobility-driven CL theory developed in [21] for a He/N2atmospheric pressure discharge, extending it to the multipleions and electronegative character of the He/H2O discharge.The positive ion flux to an end wall is

�+ = µ+n+wEw, (B1)

where µ+ is the mobility, n+w is the ion density at the wall,and Ew = J/ω�0 is the dc electric field at the wall, with Jthe rf current amplitude driving the discharge. A constantmobility collisional CL with ion generation within the sheathwas used to relate the wall density to the density n+s at theplasma-sheath edge. Particle-in-cell simulations for He/N2were roughly consistent with a triangular ion generation profile(maximum at the sheath edge and zero at the wall). For thisprofile, one finds [21]

n+w = 916

(Es

Ew

)1/6n+s , (B2)

where

Es = TeNλDs

, (B3)

with λDs = (�0Te/en+s)1/2 the electron Debye length at thesheath edge. The PIC simulations also indicated that N ≈ 5,which we use for the He/H2O discharge. Substituting (B2) and(B3) into (B1), we obtain

�+ = 916

µ+

(eTe

N2�0

)1/12 (J

ω�0

)5/6n

13/12+s , (B4)

which depends only weakly on N .In an electronegative discharge, the total global positive

ion density n+,tot =∑

n̄+ is larger than the total densityn+s,tot = ne at the sheath edge, because the negative ions areconfined to the core of the plasma and do not penetrate into thehigher field sheath region. Therefore, for each type of positiveion in the He/H2O discharge, we use n+s = (n̄+/n+,tot)ne in(B4) to determine the ion flux. This yields the volume rate ofloss dn̄+/dt = −2�+/l = −K+n̄+ for each kind of ion, where

K+ = 98

µ+

l

(eTen̄+

N2�0

)1/12 (J

ω�0

)5/6 (ne

n+,tot

)13/12(B5)

is the rate coefficient.

References

[1] Knake N, Reuter S, Niemi K, Schulz-von der Gathen V andWinter J 2008 J. Phys. D: Appl. Phys 41 194006

[2] Schulz-von der Gathen V, Schaper L, Knake N, Reuter S,Niemi K, Gans T and Winter J 2008 J. Phys. D: Appl. Phys.41 194004

[3] Liu D W, Iza F and Kong M G 2009 Plasma Process. Polym.6 446

[4] Ellerweg D, Benedikt J, von Keudell A, Knake N andSchulz-von der Gathen V 2010 New J. Phys. 12 013021

[5] Bibinov N, Knake N, Bahre H, Awakowicz P andSchulz-von der Gathen V 2011 J. Phys. D: Appl. Phys.44 345204

[6] Park G, Lee H, Kim G and Lee J K 2008 Plasma Process.Polym. 5 569

[7] Park G, Hong Y, Lee H, Sim J and Lee J K 2010 PlasmaProcess. Polym. 7 281

[8] Liu D, Bruggeman P, Iza F, Rong M and Kong M 2010 PlasmaSources Sci. Technol. 19 025018

[9] Murakami T, Niemi K, Gans T, O’Connell D and Graham W G2013 Plasma Sources Sci. Technol. 22 015003

[10] Stafford D S and Kushner M J 2004 J. Appl. Phys. 96 2451[11] Iza F, Lee J K and Kong M G 2007 Phys. Rev. Lett. 99 075004[12] Liu D W, Iza F and Kong M G 2008 Appl. Phys. Lett.

93 261503[13] Martens T, Bogaerts A, Brok W J M and Dijk P V 2008 Appl.

Phys. Lett. 92 041504[14] Hong Y J, Moon M, Iza F, Kim G C and Lee J K 2008 J. Phys.

D: Appl. Phys. 41 245208[15] Waskoenig J 2010 Thesis Queen’s University Belfast[16] Waskoenig J, Niemi K, Knake N, Graham L M, Reuter S,

Schulz-von der Gathen V and Gans T 2010 Plasma SourcesSci. Technol. 19 045018

[17] Hemke T, Eremin D, Mussenbrock T, Derzsi A, Donko Z,Dittmann K, Meichsner J and Schulze J 2013 PlasmaSources Sci. Technol. 22 015012

[18] Lee H W, Park G W, Seo Y S, Im Y H, Shim S B and Lee H J2011 J. Phys. D: Appl. Phys. 44 053001

[19] Lazzaroni C, Chabert P, Lieberman M A, Lichtenberg A J andLeblanc A 2012 Plasma Sources Sci. Technol. 21 035013

[20] Lazzaroni C, Lieberman M A, Lichtenberg A J and Chabert P2012 J. Phys. D: Appl. Phys. 45 495204

[21] Kawamura E, Lieberman M and Lichtenberg A 2014 PlasmaSources Sci. Technol. 23 035014

[22] Bruggeman P, Iza F, Lauwers D and Gonzalvo Y A 2010J. Phys. D: Appl. Phys. 43 012003

[23] Hagelaar G J M and Pitchford L C 2005 Plasma Sources Sci.Technol. 14 722

[24] Morgan Database 2013 Retrieved for He/H2O (www.lxcat.net)[25] Raizer Y P 1991 Gas Discharge Physics (Berlin: Springer)[26] Fuller E, Schettler P and Giddings J 1966 Ind. Eng. Chem.

58 18[27] Ellis H, Pai R, McDaniel E, Mason E and Viehland L 1976

At. Data Nucl. Data Tables 17 177[28] Viehland L and Mason E 1975 Ann. Phys. 91 499[29] Godyak V A and Sternberg N 1990 IEEE Trans. Plasma Sci.

18 159[30] Godyak V A and Khanneh A S 1986 IEEE Trans. Plasma Sci.

14 112[31] Raizer Y, Shneider M and Yatsenko N 1995 Radio-Frequency

Capacitive Discharges (Boca Raton, FL: CRC Press)[32] Lieberman M and Lichtenberg A 2005 Principles of Plasma

Discharges and Materials Processing 2nd edn (New York:Wiley)

[33] Chantry P 1987 J. Appl. Phys. 62 1141[34] Janev R K, Langer W D, Evans J K and Post D E 1987

Elementary Processes in Hydrogen–Helium Plasma: CrossSections and Reaction Rate Coefficients (Berlin: Springer)

[35] Gudmundsson J T, Kouznetsov I G, Patel K K andLieberman M A 2001 J. Phys. D: Appl. Phys. 34 1100

[36] Gordillo-Vazquez F J 2008 J. Phys. D: Appl. Phys.41 234016

[37] Gudmundsson J T and Thorsteinsson E G 2007 PlasmaSources Sci. Technol. 16 399

[38] Riahi R, Teulet P, Lakhdar Z B and Gleizes A 2006 Eur. Phys.J. D 40 223

[39] Soloshenko I A, Tsiolko V V, Pogulay S S, Terent’yeva A G,Bazhenov V Y, Schedrin A I, Ryabtsev A V andKuzmichev A I 2007 Plasma Sources Sci. Technol. 16 56

16

http://dx.doi.org/10.1088/0022-3727/41/19/194006http://dx.doi.org/10.1088/0022-3727/41/19/194004http://dx.doi.org/10.1002/ppap.200930009http://dx.doi.org/10.1088/1367-2630/12/1/013021http://dx.doi.org/10.1088/0022-3727/44/34/345204http://dx.doi.org/10.1002/ppap.200800019http://dx.doi.org/10.1002/ppap.200900084http://dx.doi.org/10.1088/0963-0252/19/2/025018http://dx.doi.org/10.1088/0963-0252/22/1/015003http://dx.doi.org/10.1063/1.1768615http://dx.doi.org/10.1103/PhysRevLett.99.075004http://dx.doi.org/10.1063/1.3058686http://dx.doi.org/10.1063/1.2839613http://dx.doi.org/10.1088/0022-3727/41/24/245208http://dx.doi.org/10.1088/0963-0252/19/4/045018http://dx.doi.org/10.1088/0963-0252/22/1/015012http://dx.doi.org/10.1088/0022-3727/44/5/053001http://dx.doi.org/10.1088/0963-0252/21/3/035013http://dx.doi.org/10.1088/0022-3727/45/49/495204http://dx.doi.org/10.1088/0963-0252/23/3/035014http://dx.doi.org/10.1088/0022-3727/43/1/012003http://dx.doi.org/10.1088/0963-0252/14/4/011http://www.lxcat.nethttp://dx.doi.org/10.1021/ie50677a007http://dx.doi.org/10.1016/0092-640X(76)90001-2http://dx.doi.org/10.1016/0003-4916(75)90233-Xhttp://dx.doi.org/10.1109/27.45519http://dx.doi.org/10.1109/TPS.1986.4316513http://dx.doi.org/10.1063/1.339662http://dx.doi.org/10.1088/0022-3727/34/7/312http://dx.doi.org/10.1088/0022-3727/41/23/234016http://dx.doi.org/10.1088/0963-0252/16/2/025http://dx.doi.org/10.1140/epjd/e2006-00159-2http://dx.doi.org/10.1088/0963-0252/16/1/008


[40] Soloshenko I A, Tsiolko V V, Khomich V A, Bazhenov V Y,Ryabtsev A V, Schedrin A I and Mikhno I L 2002 IEEETrans. Plasma Sci. 30 1440

[41] Nandi D, Krishnakumar E, Rosa A, Skimidt W F andIllenberger E 2003 Chem. Phys. Lett. 373 454

[42] Bortner M H and Baurer T 1979 Defense Nuclear AgencyReaction Rate Handbook, Section 24 revison No.7 2nd edn(NTIS) AD-763699

[43] Huang C M, Whitaker M, Biondi M and Johnsen R 1978 Phys.Rev. A 18 64

[44] Mitchell J B A 1990 Phys. Rep. 186 215[45] Champion R L, Doverspike L D and Lam S K 1976 Phys.

Rev. A 13 617[46] Wang Q, Economou D J and Donnelly V M 2006 J. Appl.

Phys. 100 023301[47] Miller T J, Farquhar P R and Willacy K 1997 Astrom.

Astrophys. Suppl. Ser. 121 139[48] Sanders R A and Muschlitz E E 1977 Int. J. Mass Spectrosc.

Ion Phys. 23 99[49] Binns W R and Ahl J L 1978 J. Chem. Phys. 68 538[50] Ferguson E E 1973 At. Data Nucl. Tables 12 159[51] Eichwald O, Yousfi M, Hennad A and Benabdessadok M D

1997 J. Appl. Phys. 82 4781[52] Zhao J, Su J, Guo H and Wang X 2009 Plasma Sci. Technol.

11 181[53] Sieck L W, Herron J T and Green D S 2000 Plasma Chem.

Plasma Process. 20 235

[54] Fritzenwallner J and Kopp E 1988 Adv. Space Res. 21 891[55] Kushner M J 1999 Bull. Am. Phys. Soc. 44 63[56] Kossyi I A, Kostinsky A Y, Matveyev A A and Silakov V P

1992 Plasma Sources Sci. Technol. 1 207[57] Hagelaar G J M and Kroesen G M W 2000 J. Appl. Phys.

88 2252[58] Ivanov V A and Skoblo Y E 2005 Opt. Spectrosc. 98 811[59] Cardoso R P, Belmonte T, Henrion G and Sadeghi N 2006

J. Phys. D: Appl. Phys. 39 4178[60] Ricard A, Decomps P and Massines F 1999 Surf. Coat.

Technol. 112 1[61] Golubovskii Y B, Maiorov V A, Behnke J and Behnke J F

2003 J. Phys. D: Appl. Phys. 36 39[62] GRI-MECH 3 Reaction Rate Database

(http://www.me.berkeley.edu/gri-mech/)[63] Tsang W and Hampson R F 1986 J. Phys. Chem. Ref. Data

15 1087[64] NIST chemical kinetic database

(http://kinetics.nist.gov/kinetics/index.jsp)[65] Atkinson R, Baulch D L, Cox R A, Hampson R F, Kerr J A

and Troe J 1997 J. Phys. Chem. Ref. Data 26 1329[66] Dunlea E J, Talukdar R K and Ravishankara A R 2005 J. Phys.

Chem. A 109 3912[67] Koike T and Morinaga K 1982 Bull. Chem. Soc. Japan 55 52[68] Mckendrick C B, Kerr E A and Wilkinson J P T 1984 J. Phys.

Chem. 88 3930

17

http://dx.doi.org/10.1109/TPS.2002.804192http://dx.doi.org/10.1016/S0009-2614(03)00622-5http://dx.doi.org/10.1103/PhysRevA.18.64http://dx.doi.org/10.1016/0370-1573(90)90159-Yhttp://dx.doi.org/10.1103/PhysRevA.13.617http://dx.doi.org/10.1063/1.2214591http://dx.doi.org/10.1051/aas:1997118http://dx.doi.org/10.1016/0020-7381(77)80092-2http://dx.doi.org/10.1063/1.435763http://dx.doi.org/10.1016/0092-640X(73)90017-Xhttp://dx.doi.org/10.1063/1.366336http://dx.doi.org/10.1088/1009-0630/11/2/10http://dx.doi.org/10.1023/A:1007021207704http://dx.doi.org/10.1016/S0273-1177(97)00649-2http://dx.doi.org/10.1088/0963-0252/1/3/011http://dx.doi.org/10.1063/1.1287529http://dx.doi.org/10.1134/1.1953971http://dx.doi.org/10.1088/0022-3727/39/19/009http://dx.doi.org/10.1016/S0257-8972(98)00797-Xhttp://dx.doi.org/10.1088/0022-3727/36/1/306http://www.me.berkeley.edu/gri-mech/http://dx.doi.org/10.1063/1.555759http://kinetics.nist.gov/kinetics/index.jsphttp://dx.doi.org/10.1063/1.556010http://dx.doi.org/10.1021/jp044129xhttp://dx.doi.org/10.1246/bcsj.55.52http://dx.doi.org/10.1021/j150662a007

1. Introduction2. Model2.1. Species, reactions, and rate coefficients2.2. Volume and surface model physics

3. Results 3.1. Comparison to previous global results3.2. Variations with discharge current and external water concentration3.3. Sheath width and minimum current variations with gap size and frequency3.4. Cluster size variations with gas temperature

4. Conclusions and discussion AcknowledgmentsAppendix A. Neutral diffusionA.1. Water molecule diffusionA.2. Reactive and nonreactive speciesA.3. Neutral diffusion with the radial generation profile of water

Appendix B. Mobility-driven ion losses References

Documents

Hybrid model of neutral diffusion, sheaths, and the to …doeplasma.eecs.umich.edu/files/PSC_Ding1.pdfHybrid model of neutral diffusion, sheaths, and the to transition in an atmospheric