An analysis of ﬂow, temperature, and chemical composition …swihart/Reprints/Swihart... · 2007....

PHYSICS OF FLUIDS VOLUME 11, NUMBER 4 APRIL 1999

An analysis of flow, temperature, and chemical composition distortionin gas sampling through an orifice during chemical vapor deposition

Mark T. Swiharta) and Steven L. GirshickDepartment of Mechanical Engineering, University of Minnesota, Minneapolis, Minnesota 55455

~Received 22 May 1998; accepted 17 December 1998!

Measurement of the chemical composition of gases sampled through a small hole in the subs-trate can be a useful diagnostic for investigations of the chemistry of chemical vapor deposition~CVD! processes. Ideally, one would measure the composition of the gas at the growth sur-face. However, the flow disturbance due to sampling causes the conditions at the mouth ofthe orifice to be different from those at the growth surface. Unless the orifice diameter is suffici-ently small, relative to the thickness of chemical and thermal boundary layers above the growthsurface, the sampled composition will differ from the composition at the growth surface. Inthis work, we present results of two-dimensional simulations of the flow, heat transfer, and chemicalreactions in an axisymmetric stagnation point flow with gas sampling through a small orifice inthe substrate on the symmetry axis of the flow field. Detailed results are given foratmospheric-pressure radio-frequency plasma CVD of diamond, corresponding to experimentsperformed in our laboratory. We also present more general results, approximate analyticalrepresentations of the flow field, and scaling rules for the size of the disturbance due to the samplingorifice. © 1999 American Institute of Physics.@S1070-6631~99!01204-0#

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I. INTRODUCTION

Gas sampling through an orifice in the substrate, flowed by gas chromatographic or mass spectrometric ansis of the sampled gases, has been used to investigatchemistry of a wide range of chemical vapor deposit~CVD! processes. Systems where this technique has bapplied include combustion CVD of diamond,1 hot filamentCVD of diamond,2 atmospheric-pressure radio-frequenplasma CVD of diamond,3–5 microwave plasma CVD ofdiamond,6 plasma-enhanced CVD of diamondlike carbo7

laser-assisted CVD of germanium,8 thermal CVD of AlNfrom ~~CH3!2AlNH2!3,

9 MOCVD of GaAs,10,11 photo-assisted CVD of Si3N4,

12 tungsten CVD,13 and laser-inducedCVD of amorphous hydrogenated silicon.14 In several ofthese studies,1,2,4 measured concentrations were explicicompared to one-dimensional simulations of a chemicreacting stagnation point flow computed using the SPcode15 from Sandia National Laboratories. In other casthough such explicit comparisons were not made, the cposition of the gas entering the sampling orifice was assuto be identical to the composition at the film surface. Tneglects any effects that the flow into the orifice has onflow, temperature, and concentration fields in its vicinity.

Our interest in exploring the effect of these samplidistortions arises from the experimental and modeling sties of atmospheric-pressure radio-frequency plasma CVDdiamond performed previously in our laboratory.4,5,16,17 In

a!Author to whom correspondence should be addressed. Fax:~716!645-3822.Current address: Department of Chemical Engineering, University at Bfalo ~SUNY!, Buffalo, New York 14260-4200. Electronic [email protected].

8211070-6631/99/11(4)/821/12/$15.00

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these experiments, diamond was deposited on a molybdesubstrate at atmospheric pressure using a radiofrequencduction plasma operating with argon as the main plasmaMethane and hydrogen were the reactants, with typical inCH4:H2:Ar ratios of 1:50:500. Gases were sampled throug70 mm orifice at the center of the substrate and analyzedgas chromatography. A two-dimensional model of tplasma17 predicted peak temperatures above 10 000 K.typical conditions, that model predicted that the 4000 K istherm is approximately 2 mm from the substrate. A ondimensional stagnation point flow model was used to mothe deposition and boundary layer chemistry, with tfreestream conditions obtained from the predictions oftwo-dimensional plasma model at the 4000 K isotherm. Tfreestream composition was taken as an equilibrium mixtof Ar, H, H2, and the C1 and C2 hydrocarbons at 4000 K an1 atm.17 A third model predicted composition changes in tsampling line to the gas chromatograph.4,18 The stagnationpoint flow model predicts chemical boundary layer thicnesses comparable to the orifice diameter for some speciwas also found that the predicted concentrations at a distaof 200 mm above the substrate surface were in much beagreement with the measured concentrations than werepredicted concentrations at the surface.4 This suggests thasampling distortions due to the flow into the orifice are important in this system.

Sampling distortions in molecular beam mass spectroetry ~MBMS! of reactive species in flames have been cosidered by several authors. Flame sampling is a similar,not identical problem to the one considered here. In flasampling, the goal is to obtain the composition of the unpturbed flame, the situation that would exist if no sampli

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© 1999 American Institute of Physics

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822 Phys. Fluids, Vol. 11, No. 4, April 1999 M. T. Swihart and S. L. Girshick

probe were present at all and, therefore, no surface reacoccurred. In sampling through a CVD substrate, the goato measure the concentrations that would exist at the surif it did not have a hole in it, with the surface reactionundisturbed. What the two situations have in common ishifting of the isotherms and composition profiles towardsampling orifice due to the flow into it. An early analysisMBMS sampling from flames was presented by HayhuKittelson, and Telford,19–21 who used simple analytical approximations to the flow field to estimate the disturbancethe flame due to flow into the probe and due to the boundlayer formed near the probe entrance. Later, Yi and Knu22

studied the shift of concentration profiles due to flow intosampling cone by performing two-dimensional calculatiofor inviscid flow, diffusion, and a single first order chemicreaction in a binary mixture. Knuth,23 and Smith24 have re-viewed these and many other studies of composition distions in MBMS sampling.

One goal of the present work was to quantitatively dtermine the effect of distortions due to flow into the sampliorifice on measured species concentrations duatmospheric-pressure radio-frequency plasma CVD ofmond under the conditions used in our laboratory. An adtional objective was to generalize these results to alsimple calculation of the approximate extent of the sampldisturbance under conditions relevant to other CVD pcesses. To do this, we have performed two-dimensiosimulations of the near-orifice region of a stagnation poflow in which there is critical flow through a small orificlocated on the axis of the flow. For conditions correspondto experiments from this laboratory that have been presepreviously,4,5 simulations that included detailed diamonCVD chemistry in both the gas phase and on the surfwere carried out. To generalize the results, we also pformed less computationally expensive simulations, incling only three species and a single surface reaction, ovrange of pressures, total flow rates, orifice diameters,freestream and substrate temperatures. These resultscompared to results based on simple, analytical streamftion representations similar to those used by Hayhurst,telson, and Telford.19 Approximate scaling rules for the sizof the sampling disturbance are obtained from the resuThese can be used to estimate the effects of sampling uother conditions.

II. SIMULATION METHODOLOGY

The numerical simulations presented in this work wecarried out using the commercial finite volume based coputational fluid dynamics codeCFD-ACE.25 The solution pro-cedure used in this code is essentially that presentedPatankar,26 with the additional capabilities of using multipldomains, body fitted coordinates, and higher-order differeing schemes. The equations solved were the mixture conuity equation, the axial and radial momentum conservaequations, the energy conservation equation, a continequation for each gas phase chemical species, and a suspecies conservation equation for all but one of the surfspecies. These equations are summarized in Table I. In


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work, first order upwinded differences were used for the dsity, second order upwinded differences were used forvelocity components, and central differences were usedthe enthalpy and species mass fractions. The flow was treas compressible, and variations of thermodynamic and traport properties with temperature, pressure, and composwere accounted for. Temperature and composition depenmixture viscosity, thermal conductivity, mixture-averagdiffusion coefficients, and thermal diffusion coefficienwere obtained from the kinetic theory of gases27 withLennard-Jones parameters taken from theCHEMKIN transportproperty library.28 Thermodynamic properties of the chemcal species were taken from theCHEMKIN thermodynamicdatabase.29 The CFD-ACE program was augmented with routines to allow calculation of the thermal diffusion coefficients using the rigorous methods of theCHEMKIN transportpackage,28 and with routines to solve for the fractional coverages of the surface species and to couple the rates offace reactions to the gas phase species continuity equathrough their boundary conditions at the growth surfaSURFACE CHEMKIN30 was used to manage the surface reactmechanism and to compute reaction rates of surface rtions, while the built-in capabilities ofCFD-ACE were used tohandle the gas phase chemistry. The computational domis sketched in Fig. 1, where we have indicated the boundconditions used. This geometry corresponds to the samporifice used in the experiments presented in previous wfrom this laboratory.4,5

For simulations used to obtain quantitative results corsponding to experiments performed in our laboratory~seeabove!, detailed reaction mechanisms were used in bothgas phase and on the surface. In the gas phase, a mechconsisting of 27 reactions among 16 species~H2, H, Ar, andthe C1 and C2 hydrocarbons! was used. This reduced mechnism was obtained from the larger set of reactions repoelsewhere31 by application of the mechanism reduction bason principal component analysis.32 The larger mechanismupon which this reduced mechanism is based containshydrocarbon reactions found in the GRI mechanismmethane combustion33 augmented by additional reactions involving atomic carbon, and with some rate parametersdated based on more recent results. The extended growthmethyl mechanism of Yu and Girshick,16 based on the Harrismechanism,34 was used at the diamond deposition surfaSimpler simulations were also performed, in which onlyH2, and Ar were included, and the only chemical reactiwas H atom recombination at the surface with a recombition coefficient of 0.16.

For the simulations including detailed chemistry, the epanding portion of the domain downstream of the samplorifice was omitted. The pressure profile at the orifice ecalculated without detailed chemistry was imposed asboundary condition for the simulations that included detaichemistry. The velocity profile at the orifice outlet obtainusing this pressure boundary condition was identical to tobtained with the larger domain. Solutions of the quasi-odimensional stagnation point flow problem, for comparisto the two-dimensional results, were obtained using the Scode from theCHEMKIN package.15 These included the sam


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823Phys. Fluids, Vol. 11, No. 4, April 1999 M. T. Swihart and S. L. Girshick

TABLE I. Governing equations.a

Mixture continuity

1

r

]

]r~rrv!1

]

]z~ru!50

Radial momentum

rSv ]v]r 1u]v]zD1 ]p]r 1 1r ]]r ~rtrr !1 ]trz]z 2 tuur 50Axial momentum

rSv ]u]r 1u]u]zD1 ]p]z 1 1r ]]r ~rtrz!1 ]tzz]z 50Energy

rCpSv ]T]r 1u]T]zD2 1r ]]r Srl ]T]r D2 ]]zSl ]T]zD1(

k51

Kg SrCpkYkSUk ]T]z 1Vk ]T]r D1v̇khkD1v

]p

]r1u

]p

]z50

Species continuity

rSv ]Yk]r 1u]Yk]z D1 1r ]]r ~rrYkVk!1 ]]z~rYkUk!1Mkv̇k50, k51, Kg21

YKg512(k51Kg21Yk

Surface site conservation

ṡk50, k51, Ks21

uKs512(k51Ks21uk

where

t rr 52mS 2]v]r 2 23 S 1r ]]r ~rv !1 ]u]zD Dtuu52mS 2vr 2 23 S 1r ]]r ~rv !1 ]u]zD Dtzz52mS 2]u]z 2 23 S 1r ]]r ~rv !1 ]u]zD Dt rz52mS ]u]r 1 ]v]zDVk52

1

XkDkm

dXkdr

2Dk

T

rYk

1

T

]T

]r, k51, Kg21

Uk521

XkDkm

dXkdz

2Dk

T

rYk

1

T

]T

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VKg51

YKgS 12 (

k51

Kg21

YkVkD UKg5 1YKg S 12 (k51Kg21

YkUkDar5radial coordinate,z5axial coordinate,u5axial velocity,v5radial veloc-ity, T5temperature,r5density,p5pressure,Yk5mass fraction of speciesk, Xk5mole fraction of speciesk, uk5fractional coverage of surface specie

k, hk5molar enthalpy of speciesk, t5stress tensor,v̇k5molar productionrate of speciesk by gas phase reactions,Kg5number of gas phase specie

ṡk5molar production rate of speciesk by surface reactions,Ks5number ofsurface species,Cp5mixture specific heat,Cpk5specific heat of speciesk,l5mixture thermal conductivity,Uk5axial diffusion velocity of speciesk,Vk5radial diffusion velocity of speciesk, Dkm5mixture-averaged diffusioncoefficient of speciesk in the mixture,Mk5molecular weight of speciesk,Dk

T5thermal diffusion coefficient for speciesk.


chemistry as the two-dimensional simulations. The pressprofile boundary condition applied along the radial outfloboundary of the domain@p1~z! in Fig. 1# was obtained byintegration of the axial momentum equation for the quaone-dimensional stagnation point flow using the velocidensity, and viscosity profiles predicted by the SPIN coTherefore the imposed pressure boundary condition waspressure profile that would be present in an ideal stagnapoint flow. This boundary condition was necessaryachieve quantitative agreement between the quasi-odimensional solution and the two-dimensional simulationspoints radially distant from the orifice.

III. RESULTS AND DISCUSSION

A. Typical results

Typical isotherms and streamlines are shown in Fig.These are for a freestream velocity (u0) of 10 m/s, an up-stream pressure@p05p1~z50!# of 1 atm, a freestream temperature (T0) of 4000 K, a wall temperature (Tw) of 1200 K,an orifice diameter of 70mm, and a downstream pressu(p2) of 20 Torr. These are conditions corresponding toexperiments and calculations presented by Girshick, Li, Yand Han.17 Only the streamlines near the orifice are showThe streamlines are those of an ideal stagnation point flexcept near the sampling orifice. Likewise, the isother~and concentration isolines, not shown! are parallel to thesubstrate far from the orifice, as they would be in an idstagnation point flow. Compressibility effects are not sign

FIG. 1. Sketch of computational domain showing the boundary conditiapplied. Variables are defined in Table I.


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cant in the region upstream of the orifice. For the conditioof Fig. 2, the Mach number is 0.38 at the center of the orifimouth and the pressure is 0.89 atm.

Under all conditions considered here, the flow throuthe orifice is choked and reaches a Mach number ofM51just before the orifice outlet. For the conditions of Fig. 2, ttotal mass flow through the orifice is about 2.131027 kg/s.The maximum gas velocity is about 1100 m/s~M52.1!, onthe centerline, about 0.1 mm past the orifice exit. Thispansion is followed by a recompression with slight overcopression, then relaxation to the final pressure. This isexpected behavior downstream of the orifice. For the contions of Fig. 2, the first shock~expansion! was located abou0.03 mm before the orifice outlet, and the second shock~re-compression! was about 0.35 mm past the orifice outlet,the centerline. Due to boundary layers on the walls, thshocks were not perpendicular to the walls. Since our fowas on the behavior upstream of the orifice, we did nottempt to sharply resolve these shocks. The length of themain downstream of the orifice did not affect the flow, prvided that it was sufficiently long that radial pressugradients could be neglected and a constant pressure boary condition could be specified at the outlet. The dowstream pressure did not affect the flow upstream of thefice. Changing the downstream pressure by a factor ofeither direction changed the mass flowrate into the orificeless than 0.1%. The downstream flow will not be considefurther, since it does not significantly affect the flow ustream of the orifice.

FIG. 2. Typical flow and temperature fields from two-dimensional simutions. Boundary conditions areT054000 K, u0510 m/s, p051 atm,Ts51200 K, p2520 Torr.


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For the conditions of Fig. 2, the separation streamlbetween flow that enters the orifice and flow that exits raally meets the substrate at about 0.4 mm from the symmaxis. The radial position of this stagnation ring~hereafterdenotedRst) is one measure of the size of the disturbandue to sampling. Significant distortion of the isothermsobserved near the orifice. The radial extent of this distortis comparable to the stagnation ring location,Rst. Anothermeasure of the size of the disturbance is the axial distancwhich the isotherms are shifted at the center of the orifiWe define this shift~hereafter denoteddT) as the distancefrom the substrate, at points radially distant from the orififor which the temperature is equal to the temperature atcenter of the orifice. For the case shown in Fig. 2, this shifabout 70mm. There are corresponding shifts of the concetration isolines for each chemical species. The distortionsthe concentration profiles are coupled to each other anthe distortions in the temperature profile by gas phase retions, since the rates of these reactions are highly tempture dependent. These shifts in the isotherms and concetion isolines will significantly affect the concentration ofspecies entering the orifice if they are comparable tothickness of the chemical boundary layer for that species

B. Results from simulations with detailed chemistry

As described above, simulations including detailchemistry corresponding to the diamond deposition expments described by Lindsay, Larson, and Girshick5 were per-formed to determine quantitatively the effect of the sampldisturbance in those experiments. Figure 3 shows resultcalculations corresponding to deposition at a substrate tperature of 1200 K, a pressure of 1 atm, an orifice diameof 70 mm, and an input methane to hydrogen ratio of 2Figure 4 shows corresponding axial species profiles onaxis of symmetry and 2 mm from the axis of symmetry fseveral compounds of interest in diamond deposition.acetylene, shown on the left-hand side of Fig. 3, the conctration boundary layer is thick compared to the size ofsampling disturbance, and the acetylene mole fraction atorifice is only slightly different from its value at the growtsurface away from the orifice. However, methane is formin a boundary layer whose thickness is comparable todimensions of the sampling disturbance, and thereforemole fraction at the orifice is substantially different from ivalue at the growth surface. Similar observations canmade for the other species shown in Fig. 4. Mole fractionsall of the species at the orifice center, at the growth surfaand averaged over the flow through the orifice are givenTable II. The difference between the mole fractions of spcies at the growth surface and at the orifice ranges frabout 2% for argon, H2, and C2H2, to several orders of magnitude for trace species that are formed or destroyed in a vthin region near the surface.

For each species we can define an effective sampdistance (dk) as the distance from the substrate, at poiradially distant from the orifice, where its mole fractionequal to its mole fraction at the center of the orifice. Tablepresents these effective sampling distances. These r

-


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FIG. 3. Acetylene and methane concentration profiles from two-dimensional simulations including detailed gas phase and surface chemistry. Conthe same as in Fig. 2. The freestream composition is an equilibrium mixture of the sixteen species considered at 4000 K and 1 atm with atom f40.3% H, 59.1% Ar, and 0.54% C.

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from 31 to 160mm, depending on the diffusion coefficient othe species, its unperturbed concentration profile, and hois coupled to the other species by reaction. The correspoing shift in the isotherms (dT as defined above! is 60 mm.This is comparable to the shifts in the concentration isolifor most of the species. Yi and Knuth22 found, in their simu-lations of flow into an orifice at the tip of a sampling conthat the effective sampling distance for species whose cacteristic reaction time was larger than the characteristic fltime was approximately given by

dkd

5a Pem,k0.5 , ~1!

where d5orifice diameter, Pem,k5Peclet number for mastransfer5Re Sck5uord/Dkm , uor5average velocity at the

FIG. 4. Concentration profiles for several important species in diamdeposition from the same simulation presented in Fig. 3. Solid lines areaxial profiles at radial positions far from the orifice, and the dashed linesthe profiles along the symmetry axis.


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orifice, Dkm5mixture-averaged diffusion coefficient for species k, a5a constant of proportionality, 0.19 in Yi anKnuth’s work.22

The final column of Table II gives the value ofdk fromthis expression for each species, with the constant of protionality set ata50.163, which gave the best agreement wthe detailed calculations when H2, Ar, CH2, CH3, and C2H2were excluded~see footnotes to Table II!. For these speciesthere is reasonable agreement between the predictionEq. ~1! and the observed value of (dk). Thus, Eq.~1! cangive a useful, semiquantitative estimate of the sampling dtance for species that have a significant concentration grent near the surface and for which the concentrationcreases or decreases monotonically over a distance of aorifice diameters from the surface. The concentrationsspecies without a significant concentration gradient nearsurface will be unaffected.

We would expect the effective sampling distance to dpend only on the Peclet number if the only processes gerning the species concentrations were convection andlecular diffusion. In this reactive flow, gas phase reactiosurface reactions, and thermal diffusion lead to dependeon other dimensionless quantities in addition to the Penumber. In principle, the effective sampling distanceeach component should depend on its thermal diffusion r@which can be defined askT5Dk

T/(rYkDkm) in a multicom-ponent system#, its gas phase Damko¨hler number@defined as

Dag5(v̇kd2)/(ckDkm)] and its surface Damko¨hler number

@defined as Das5( ṡkd)/(ckDkm)]. These numbers are defined based on reaction rates and diffusion coefficientsparticular location. It may not be possible to define a rep

sentative production rate (v̇k) for a species, since that quantity may change by orders of magnitude over distances cparable to the orifice diameter. We evaluated these thdimensionless groups using the concentrations, properand production rates at the substrate surface and examthe dependence of the effective sampling distance on th

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TABLE II. Species concentrations.

Species

Mole fractionat

orificecenter

Mole fractionaveraged

overoutflow

Molefraction

atsurface

Effectivesampling

distance (dk)~mm!

Estimateddk from Eq. ~1!

~mm!

H2 2.3131021 2.3431021 2.3731021 a 38

CH4 2.7131025 3.9331025 9.8731025 88 74

C2H4 5.9031026 1.1331025 3.8331025 82 86

Ar 7.4431021 7.4331021 7.5431021 a 160CH3 3.12310

25 2.8431025 1.8431025 43b 74H 2.1231022 1.9731022 5.3131023 46 31C2H6 5.72310

29 2.3031028 1.7931027 81 90C 5.0531025 4.3931025 1.0731025 89 64CH 1.7631026 1.3231026 1.6931027 88 603CH2 5.98310

26 4.9531026 2.6331026 153c 73C2 1.96310

28 1.2431028 5.18310210 81 76C2H 4.74310

27 2.7731027 6.3431029 61 85C2H2 3.57310

23 3.5431023 3.4931023 a 85C2H3 5.38310

26 8.5431026 1.6231025 85 86C2H5 6.98310

29 2.2431028 1.5031027 80 901CH2 1.12310

27 7.8131028 9.2231029 58 73

aThe effect of the sampling disturbance on the concentrations of these species is negligible.bCH3 behaves anomolously because its unperturbed concentration profile goes through a maximum amm from the surface.

cCH2 behaves anomolously because its unperturbed concentration profile goes through a minimum aboummfrom the surface.

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The effective sampling distance appeared to correlate wDag and withkT , but we do not have a sufficient numberresults over a large enough range of these parametemake any general statements about this dependence. Thfective sampling distance did not appear to depend sigcantly on Das , but again there is insufficient information tgeneralize.

In previous modeling of the diamond deposition expements to which these calculations correspond,4 it was foundthat there was significant disagreement between the contrations measured in a sample withdrawn through the 70mmorifice and the predictions of the gas phase concentrationthe surface obtained from a one-dimensional boundary lamodel that did not account for perturbations due to the flinto the sampling orifice. It was speculated that this diffence might be due to sampling effects. This hypothesisbased on the observation that the measured concentraagreed reasonably with the gas phase concentrationsdicted by the model at a position 200mm above the growthsurface. The simulations presented here, and others notsented, showed that sampling effects could not fully accofor the discrepancies. However, they also showed that spling effects were significant, and that inclusion of thesefects moves the model predictions closer to agreementthe experimental observations. For example, as seen in TII, sampling effects can cause the sampled CH4 concentra-tion to differ from the concentration at the surface by mothan a factor of 2. The previous calculations used the retion mechanisms presented by Yu and Girshick16 and byLindsay.18 More recent calculations,31 that used the samreaction mechanism as the present work, showed thatbulk of the discrepancy between the model and experimwas due to inadequacy of the gas phase reaction mecha

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used. The revised reaction mechanism, when samplingfects are also properly accounted for, gives good agreemwith the experimentally measured concentrations ofstable hydrocarbons.31

C. Comparison of detailed simulations to simplifiedstreamfunction expressions

The flow field from the two-dimensional simulationpresented here can be reasonably approximated by anlytical streamfunction. In their analysis of sampling effectsmolecular beam mass spectrometry from flames, HayhuKittelson, and Telford19 approximated the flow field usingthe superposition of the streamfunctions for inviscid, incopressible flow against a flat plate and into a point sink. Ifadopt the same approach, the streamfunction and cosponding velocity components are

c~r ,z!52kr2z2mz

Ar 21z2, ~2!

ru521

r

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]r524kz2

mz

~r 21z2!3/2, ~3!

rv51

r

]c

]z52kr2

mr

~r 21z2!3/2, ~4!

where c(r ,z)5streamfunction ~lines of constantc arestreamlines!; r5radial coordinate;z5axial coordinate, withthe orifice in thez50 plane;r5density;u5axial velocity;



FIG. 5. Comparison of streamlines from a complete two-dimensional simulation to the simplified streamfunction given by Eq.~2! in the text. Boundaryconditions areT054000 K,u0510 m/s,p051 atm,Ts51200 K, p2520 Torr.

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v5radial velocity; m5ṁ/(2p)5~total mass flow rate intosink!/~2p); k52(r0u0)/(4z0); and r0 , u05density andaxial velocity in a reference plane atz5z0.

The first term on the right-hand side of each expressdescribes the flow against an infinite plane, and the secterm describes flow into a point sink. The streamfunctformulation guarantees satisfaction of the overall continuequation, but the velocity components are not requiredsatisfy any momentum balances. Figure 5 comparesstreamlines from the above streamfunction to those fromdetailed simulation for the same conditions described inprevious section. The sink strength~m! was set to match theflow rate through the orifice from the two-dimensional nmerical simulations, and the reference plane~z5z0) wastaken to be 2 mm from the substrate. This was the inflboundary location for the detailed simulations. This simpfied streamfunction gives a reasonable approximation toflow field. A useful result from the simplified streamfunctiois the simple scaling rule that it provides for the size of tdisturbance in the flow field. The location of the stagnatring (Rst) can be obtained by solving for the radial positioof the stagnation streamline (c50! and taking the limit aszgoes to zero. This gives

Rst5S m2kD1/3

, ~5!

which upon substitution of expressions form andk gives thedependence of the radial extent of the flow disturbanceoperating parameters such as temperatures, freestream vity, pressure, orifice diameter, etc.

The above streamfunction may be improved to give bter agreement with the flow field from the detailed simu


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tions by making modifications that require it to satisfy mophysically realistic boundary conditions. For predicting taxial extent of the disturbance, it seems particularly imptant that the streamfunction result in a velocity field that hzero radial velocity at the substrate surface so that it satisthe no-slip boundary condition there. Several modificatioto Eq. ~2! that satisfy appropriate boundary conditions fthis problem (c5u5v50 at z50, c52m andv50 at r50,and c52kr2z at z5z0) were investigated. The simplest expression that was found to give significantly improved agrment with the numerical simulations was

c~r ,z!52kr2zS ~a22!S zz0

D 31~322a!S zz0

D 21a zz0

D2

mz2

A~r 21z2!~~br !21z2!, ~6!

wherea andb are dimensionless constants.We treat the dimensionless numbersa and b as free

parameters that may be used to match this streamfunctiothose from the detailed simulations. Constant values ofa59andb51 gave good results for the range of conditions stuied in this work~see below!. Figure 6 compares this streamfunction ~with a59 and b51! to the detailed simulationresults for the same conditions shown in Fig. 5. The agrment is much improved, particularly for the location of thstagnation streamline. Similar agreement was observed frange of other conditions, using the same values ofa andb.This expression predicts that the radial position of the stnation ring will be given by

Rst5S mz02abkD1/4

~7!



FIG. 6. Comparison of streamlines from a complete two-dimensional simulation to the simplified streamfunction given by Eq.~6! in the text. Boundaryconditions areT054000 K,u0510 m/s,p051 atm,Ts51200 K, p2520 Torr. Parameter values in Eq.~6! area59, b51.

e

m

eao

n

qre

ield

.

thet

usowed,the

heill

ber.

te

which has a slightly weaker dependence ofRst on the oper-ating parameters than expression~5!.

D. Predicted and observed scaling of the flowdisturbance

To predict the size of the flow disturbance, we first neto calculate the mass flow rate through the orifice (ṁ). Forisentropic, adiabatic choked flow through an orifice of diaeterd, the mass flow rate is given by35

ṁ5S 2g11D @~g11!/2~g21!#AgM

RT0~pd2!p0 , ~8!

whereg is the specific heat ratio,M is the molecular weightof the gas,R is the universal gas constant,T0 is the upstreamtemperature, andp0 is the upstream pressure. Since a rflow is neither isentropic nor adiabatic, the actual mass flwill be lower. Equation~8! is generally multiplied by a dis-charge coefficient (CD), which is an empirical factor relatingthe observed flow rate to the theoretical flow rate from~8!.CD depends on the orifice geometry and the flow conditioMeasurements and correlations forCD for different orificegeometries and conditions are available in the literature.36,37

Comparing results of our detailed flow simulations to E~8!, with T0 and g evaluated at the substrate temperatugave discharge coefficients in the rangeCD50.08 toCD50.56. As can be seen in Fig. 7, the discharge coefficwas found to scale with the square root of the Reynonumber. In this figure, both the Reynolds number and Eq.~8!were evaluated using conditions at the substrate surface


d

-

lw

s.

.,

nts

Of

course, this scaling cannot be obeyed for large values ofReynolds number, since as Re→`, the discharge coefficienmust go to one~frictional losses will be negligible!. For theorifice geometry considered here, this correlation allowsto estimate the discharge coefficient, and therefore the flthrough the orifice. Note that an iterative process is requirsince the Reynolds number depends on the flow throughorifice. It is also easy to measure the total flow into torifice experimentally. In the remainder of this work, we w

FIG. 7. Dependence of the discharge coefficient on the Reynolds num

The Reynolds number for the flow into the orifice is defined as (4ṁ/pdm),

whereṁ is the mass flow rate into the orifice,d is the orifice diameter, andm is the dynamic viscosity~evaluated at the conditions at the substrasurface!.


or

ura

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sizu

tioal

on

efptitini-

th

stecote-gr

prae

to

in

seioetodly

te

na-

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is

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aregas, soas


assume that the mass flow rate through the orifice (ṁ) isknown, either from a correlation like the one in Fig. 7from experiment.

To test the expected variation of the sampling distbance with reactor conditions, a set of fifty simulations wconducted with only three chemical species~H, H2, and Ar!,no gas phase reactions, and only hydrogen recombinatiothe substrate surface. The base conditions for these simtions were those given above for the simulations thatcluded detailed chemistry. These correspond to our atspheric pressure radio-frequency plasma CVD system, wthe boundary layer above the substrate is quite thin and tsampling effects are more likely to be important thanlower temperature systems. Five parameters (p0 , d, u0 , T0 ,andTs) were independently varied, and measures of theof the sampling disturbance predicted by the numerical simlations were compared to simple scaling rules. The locaof the stagnation ring (Rst) is a good measure of the radiextent of the disturbance, and can be predicted from Eq.~5!or ~7!. Figure 8 compares those predictions to the predictiof the detailed numerical simulations for variation ofRst withpressure (p0), orifice diameter~d!, freestream velocity (u0),substrate temperature (Ts), and freestream temperature (T0).The solid lines are the predictions from~5!, and the dashedlines are those from~7!. For this range of conditions, thesexpressions capture essentially the correct dependence oradial extent of the sampling disturbance on these fiverameters. This allows us to predict, at least semiquantively, the radial size of the sampling disturbance and, usEq. ~4! or ~6!, to obtain an approximate flow field for arbtrary conditions.

As can be seen from Eqs.~5! and~7!, the location of thestagnation ring depends on the ratio of the strength offlow into the orifice~of which m is a measure! to the strengthof the overall stagnation point flow~of which k is a mea-sure!. The change in the stagnation ring position with presure@Fig. 8~a!# arises from the increased volumetric flow rainto the orifice with increasing pressure. If the dischargeefficient (CD) were constant, then the volumetric flow rainto the orifice would be constant~mass flow rate proportional to pressure!, and the location of the stagnation rinwould be independent of pressure. However, the dischacoefficient increases with increasing pressure~roughly as thesquare root of pressure, since the Reynolds number isportional to pressure! and therefore the radius of the stagntion streamline increases with increasing pressure as wThis is a kinematic effect. The higher volumetric flow inthe orifice at higher pressures leads to a larger regionstreamlines entering the orifice. Likewise, the stagnation rradius increases with increasing orifice diameter@Fig. 8~b!#because the volumetric flow rate into the orifice increawith the orifice diameter while the strength of the stagnatflow is unchanged. The volumetric flow rate increasroughly asd2.5, since the Reynolds number is proportionald ~giving CD}d

0.5), and the flow rate for an ideal chokeflow is proportional tod2. The discharge coefficient is nearindependent of the inlet velocity (u0) so both the mass flowrate and volumetric flow rate into the orifice are unaffec


-s

atla--o-rese

e-n

s

thea-a-g

e

-

-

ge

o--ll.

ofg

sns

d

by changes in the inlet velocity. The decrease in the stagtion ring radius with increasing inlet velocity@Fig. 8~c!# re-flects the fact that for an increasing inlet flow rate~increasingstrength of the stagnation point flow!, the constant flow ratethrough the orifice is a decreasing fraction of the total floChanging the substrate temperature at fixed inlet temperaaffects the flow into the orifice but does not affect the stanation point flow. The flow rate into the orifice decreasroughly asTs

21/2, so the stagnation ring radius decreasessomething likeTs

21/8, consistent with the weak dependenobserved in Fig. 8~d!. Increasing the inlet temperature dcreases the mass flow rate of the stagnation point flow,to the decrease in the inlet density~with fixed inlet velocity!.It has little effect on the flow into the orifice, since the desity near the orifice is set by the substrate temperature. Asinlet temperature is increased, the flow into the orifice isincreasing fraction of the total flow, and the radius of tstagnation ring increases@Fig. 8~e!#, roughly asT0

1/4.The axial extent of the sampling disturbance does

correlate directly with the radial extent. A good measurethe axial extent is the shift of the isotherm correspondingthe temperature at the center of the orifice. The isotherm s(dT) is defined as the distance from the surface, at poradially far from the orifice, where the temperature is eqto the temperature at the center of the orifice entrance. Ifenergy balance equation~see Table I! is made dimensionlessand the terms corresponding to compressibility, reaction,species diffusion are dropped, the Peclet number remainthe only dimensionless parameter in the equation. Therefwhen convection and conduction dominate the effectscompressibility and reaction, the solution will depend onon the Peclet number. In analogy to Eq.~1! and the work ofYi and Knuth,22 we expect that the isotherm shift (dT) willbe proportional to the square root of the Peclet number,

dTd

5a Peh0.5, ~9!

where Peh5Peclet number for heat transfer5Re Pr5(ṁCp)/(pdl), Cp5 specific heat,l5thermal conductiv-ity, a5a constant of proportionality, taken to be 0.22 in thwork.

Figure 9 compares the predictions of Eq.~9! to the val-ues ofdT obtained from examination of the detailed calcutions. The scaling predicted by Eq.~9! is followed closely,except when the freestream temperature and the substemperature are close to each other, so that the systenearly isothermal@see Fig. 9~e!#. If the temperature profilewas determined solely by convection and conduction, thwe would expectdT to depend only on the Peclet numbeHowever, for nearly isothermal conditions, this is not tcase. When the system is nearly isothermal, then the tperature gradients in the system are small, and thereforeconductive and diffusive terms in the energy equationsmall. The terms in the energy equation corresponding toexpansion do not depend on the temperature gradientthey become relatively important under these conditions. G


f theults


FIG. 8. Comparisons of the predicted location of the stagnation ring~Rst! predicted by scaling rules based on simplified streamfunctions to results odetailed calculations. The solid line is from Eq.~5! and the dashed line is from Eq.~7! in the text. The filled circles connected by a solid line are the resof individual runs of the detailed simulations.

iso-nlficeig

eth

ver-tecon-

ss-ur-tive

expansion cools the gas near the orifice, shifting thetherms away from the orifice~less net shift toward the orifice! when the substrate temperature is lower than the itemperature and shifting the isotherms toward the ori~more net shift! when the substrate temperature is highthan the inlet temperature. This is what is observed in F9~e!.

Taken together, Eqs.~7! and~9! allow us to obtain semi-quantitative estimates of the size of the disturbance duthe flow field. These may be compared to the thickness of


-

eter.

toe

concentration and temperature boundary layers and the oall dimensions of the flow system to qualitatively evaluathe effect of sampling disturbance on measured speciescentrations.

IV. SUMMARY AND CONCLUSIONS

Results of detailed two-dimensional reacting compreible flow calculations of gas sampling through an orifice ding chemical vapor deposition were presented. Quantita



FIG. 9. Comparisons of the predicted shift in the isotherms from Eq.~9! in the text to results of the detailed calculations. The dashed line is from Eq.~9!. Thefilled circles connected by a solid line are the results of individual runs of the detailed simulations.

smsee

ses

heong

ceRe-the

ullyndto

ne-

results corresponding to atmospheric radiofrequency plaCVD of diamond showed that these disturbances can benificant and can partially account for discrepancies betwexperimental concentration measurements and modelingsults. Simplified streamfunctions were presented that cloapproximate the flow field from the detailed calculationFinally, scaling rules for the radial and axial extent of tsampling disturbance were presented and evaluated by cparison to the results of the detailed simulations over a raof operating conditions.


aig-n

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m-e

ACKNOWLEDGMENTS

This work was partially supported by National ScienFoundation Grant No. CTS-9424271, by the Engineeringsearch Center on Plasma-Aided Manufacturing, and byMinnesota Supercomputer Institute. The authors gratefacknowledge contributions to this work by John Lindsay aJohn Larson in the form of helpful discussions, accessdetails of previous work, and assistance with the quasi-odimensional modeling.


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