Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

AIAA 98-0162

NUMERICAL STUDY OF CHEMICALLY REACTING FLOWS WITHCATALYTIC SURFACE REACTIONS USING THE PRESSURE

CORRECTION ALGORITHM

Zhijian Chen, Lee Kania, Saif WarsiAdaptive Research Division

Pacific-Sierra Research CorporationHuntsville, AL 35805

AbstractThis paper describes a numerical method to

study a general chemically reacting flow withcatalytic surface reactions. Multi-step finite ratechemistry models are employed to model the physicsof the reaction process. A split-operator methodseparates the chemical kinetics terms from the fluiddynamical terms. The changes of speciesconcentrations on a catalytic surface are included asboundary conditions in the species transportationequations. The pressure-velocity correction methodin the CFD2000 package is used to enhancenumerical stability.

IntroductionNumerical simulation of chemically

reacting flows continues to become an importanttool in the design and analysis of modem reactorconcepts. Surface chemical reactions occur at a gas-solid, gas-liquid or liquid-solid interface. Thecomprehensive understanding of chemical vapordeposition (CVD) systems, which are usedthroughout the microelectronics industry today,requires this technology. Despite the importance ofCVD, many other chemical reaction devices, such assolid rocket propulsion systems, high efficiencyinternal combustion engines, high altitudehypersonic vehicles also need this technology. It iswell known that the detailed mechanism aboutsurface reaction procedures is very complicated.Numerically, a phenomenological model is required.

Chemical reactions are frequently coupledwith fluid flows and take place at extremely smalltime scales. Accordingly, chemically reacting flowsare divided into three general types: (1) Equilibrium;(2) Frozen and (3) Finite rate. The first two of theseare limiting cases which correspond to instantaneousreactions(equilibriura) and to no reactions(frozen).

Copyright © 1998 by the American Institute ofAeronautics and Astronautics, Inc. All rightsreserved.

Finite rate reactions occur on time scales which allow fornumerical modeling, and are those which are observed tooccur in most reacting flows. The numerical methodsused to simulate the flow physics for these "real" flowsface a major difficulty. The difficulty is the stiffness ofthe mathematical models which describe the transportcharacteristics for the species concentrations. For ageneral reacting flow, the gas phase chemical reactionsand surface reactions may exist simultaneously. Forsurface reaction, especially in CVD systems, thepioneering work of Moffat and Jensen pointed out thatthe thermal diffusion term sometimes may play animportant role in the study of several reactorconfigurations1. However, the method in their work islimited to boundary layer type flows. Currently, thispaper reports a new development combining the non-iterative pressure correction method2 with source termsplitting technique3 to simulate general non-equilibriumchemically reacting flows in which gas phase and surfacereactions occur simultaneously.

AnalysisGoverning Equations

The governing equationscompressible flow are as follows:Continuity:

Momentum:

in tensor form for

(1)

dpu,-*—*-dt

d , . dp—— (pu ,-«,-) = — —V J '+su

Energy:dph d-Z-+—dt dX

dp̂-dt

(2)

(3)

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Species:

dx

Equation of State:= pRT

(4)

(5)

Where p is the ensemble averaged density of themixture; u;, the 1th component of the velocity vector;Tij, the shear stress; s ,̂ the source for momentum; h,the enthalpy, defined by

Thus, the net reaction rate for species i is

jr1 NR ^fdt dt

The forward reaction rate is defined from the Arrheniuslaw expressed as

(9)

And the backward reaction rate is defined as

j = \,...,NR (10)

(6)

Y; (i = 1,...NS) mass fraction of the i* species; NS,total number of species; O, the viscous energydissipation term; Ft and FYi, .the effectivediffusivities; Cpi, the specific heat; a>;, the speciesproduction rate determined from the law of massaction; and h°fi heat of formation of the i* species.In equation (4), the second term on the right handside is called the thermal diffusion term whicheffects species concentrations by the temperaturegradient, ctj is the thermal diffusion coefficient of i*species. It is a function of temperature. It should bepointed out that the gas constant, R, in equation (5)is computed based on the gas mixture.

Finite Rate Chemical ReactionFor a general chemical reaction proceeding

in both the forward and backward directions, thestoichiometric equation can be written as

NS NS

Where,v-j andv-j are the stoichiometric coefficientsof reactants and products in the j& reaction; M;, thespecies formula; NR, the total reaction step; k§ andkbj, the forward and backward reaction rates for j*step respectively. Based on the law of mass action,the net reaction rate of species i at the j* reactionstep is4

dtNS

(7)

where the equilibrium coefficient is obtained fromkinetic theory which is given as

RJ

NS NS

i=l i=l

NS NS

1=1

(11)

(12)

(13)

where g; is the gjbbs free energy calculated from JANAFdata5 and R^ and R^ are the universal gas constants.

Catalytic Surface ReactionSurface chemical reactions occur at a gas-solid,

gas-liquid or liquid-solid interfaces. The detailedmechanism about the reaction procedure is even morecomplicated than a typical gas phase reaction.Numerically, a phenomenological model is required.

Considering a general surface reaction of theform4

A + Wall <-??-> B + WallV

where A and B are general chemical species. The wallacts as a catalyst which induces the reaction. The rate ofproduction of the reactant A per unit area is

(14)

where nA and nB are the orders of the forward andbackward reactions, respectively. At equilibrium, onehas

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

(15)

and equation (14) gives

(16)

For catalytic reactions, kw f is usually modeled byintroducing the catalytic efficiency, yw

1>6, thus

(17)

where R is universal gas constant, T, temperature,mA, molecule weight of species A. yw is a modelconstant In some situations it is a function oftemperature.

In order to solve the species transportationequation (4), the catalytic wall boundary conditionfor species i is proposed as4

8Y-= -(pDt —*-) (18)

at (19)

where, C() and D() denote the convection and diffusionoperators respectively. At the predictor step, only thesource term is taken into account, thus

dY—l-dt

(20)

since eoi is a function of species mass fraction Yi? theequation (20) can be written in a vector form:

dY_dt

= F(Y) (21)

where Y=(Yl5 Y2, ..., YNS)T. The ODE equation (21) canbe integrated using the fully implicit scheme with the

. n+lJacobian 5F/3Y from time tn to t""' to get species massfractions Y i. An effectivecalculated from

source term then can be

p(YJ"-Yi")— — — — (22)

where, D;is the species diffusivity.

Numerical procedureThe governing equations are transformed to

an arbitrary curvilinear coordinate and are thenintegrated over an arbitrary control volume using anon-staggered grid arrangement with all dependentvariables stored in the cell center. The discretizationin the spatial domain is achieved by centraldifferencing scheme for the diffusion terms and by ageneralized upwind scheme based on Chakravarthyand Osher7 for the convection terms. The non-iterative operator-splitting algorithm PISO2 is usedto solve equations (1) through (3). In order to avoidthe strong stiffness from the production rate ofchemical species, A splitting method8' 9 or pointimplicit method3 for source terms of the speciesequations has been used in this study. This methodseparates the computation of the effect between flowfield and chemical source terms into two steps:predictor and corrector steps. The resulting effectivechemical source terms are then coupled into flowfield computation. This method is successful inavoiding the stiffness difficulties arising fromchemical reaction kinetics models.

Using the operator representation, thespecies governing equation is written as thefollowing form

Following this, a correction step involving theconvection and diffusion parts of the species utilizing themost updated flow field variables at the current time isimplicitly integrated to get the species mass fraction attime t n+1. The current method and catalytic reactionmodel have been implemented in the general commercialCFD software system CFD2000 of Adaptive Research.Predictions made for a 2-D CVD reactor are presentedbelow.

Numerical SolutionsSilicon Chemical Vapor Deposition

Figure 1 shows a two dimensional CVD reactorwhich is 2 cm in height and 20 cm in length. The bottomwall, the susceptor, is isothermally heated to a constanttemperature, 1323 K, and the cold top wall is kept at theconstant temperature, 300 K. The incoming mixture is

Y(cm)'2.0

0.

= 300K

20.0 X(cm)TH=1323K

Figure 1. The geometry of SfflL, CVD reactor.

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composed of SiBLt and H2 flowing through thereactor at average velocity 17.5 cm/sec, andtemperature of 300 K. The initial partial pressure ofeach component is 0.76 torr for SiH, and 759.24 torrfor H2. The gas phase chemical reaction is assumedto be a two step, 4 species finite rate reaction1 givenas

kjXSi2H6 SiH + SiH

where the forward reaction rates are

fr/1=6.10*1028r-5-°°exp(-^)

k =2.1

(23)

(24)

where E1=58.73kcalAnolandE2=56.40kcal/moLFor the surface reactions, following the

work of Coltrin10 and Moffat1, the catalyticefficiency for species SiHt is set as:

„„,,_ ,= 0.0537exp( -18680ca//wo/.——— — ——— )K.1

(25)

withys;H2= 1.0 and ySiH6=1.0.The computational grid employed is

uniform in the X direction and stretched toward thehot wall in the Y direction. Several different gridcombinations were used to the solution griddependency. Figure 2 shows the comparisons for theX direction velocity profiles at the reactor outlet.The velocity profiles shift slightly toward the coldupper wall This is due to the variance in laminarviscosity which is a function of temperature.11

0.-4-.

0.3.

0.2-

0.1-

2O*30 o JtO*3O

Figure 3 shows the temperature contours. Figure 4, 5 and6 show mass fraction contours for SiHt, SiH2 and Si2H«.The deposition of silylene appears in a narrow boundarylayer above the susceptor. Figure 7 shows thecomparison of the growth rate of silicon12 in units ofHm/min.

Figure 3 Temperature contours.

1.500E-021.Z17E-023.014E-028.114E-036.086E-034.057E-032.029E-03

Figure 4 Contours of mass fraction of SiEU.

1.935E-063.871E-065.606E-067.742E-OS

1.1S1E-05

Figure 5 Contours of mass fraction of Si2H6.80*30

Figure 2. Comparison of velocity profiles.

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

2.649E-055.Z96S-05

A

BC

nT.

F

ABCDEF

1.135E-052.270E-053.306E-054.451E-OS5.6761-056.B11E-05

comparison of the mass fraction of SiH) at the outlet ofsusceptor along the vertical direction in units of cm.

5

0.014.

CL012.

0.008.

0.006.

0.004.

0,002.

0

A: without thermal diffusion

B: with thermal diffusion

0 05 1 15 2Y

Figure 8 Comparison of mass fraction of SfflU at outlet.Figure 6 Contours of mass fraction of SiH2.

0.5

0.4.

0.3.

0.2.

0 5

Numericol o Experimented

Dcc

20

Figure 7 Comparison of growth rale betweennumerical solution and experimental data. The unitof X is in (cm) and growth rate in um/min.

Figure 7 shows that the difference between thenumerical solution and experimental growth ratedata is about 10-15%. This difference comes mainlyfrom the chemical reaction model. In gas phase, asimplified two-step model is used. In the surfacereaction region, there are some assumptions on thereaction model. The growth rate is a function ofsurface reaction rate and species concentrations. It isexpected that the more complete reaction modelwould give a better numerical solution. As statedbefore, the thermal diffusion will affect the masstransportation of species. Figure 8 shows a

SummaryWe have combined a numerical technique

involving multi-step chemical reactions and catalyticsurface reactions with a general all-speed flow solver tostudy the chemical vapor deposition process. Thesplitting of the source term is effective in avoiding thestiffness within the chemical kinetics model. The studypredicts the formation of silicon from silane and theresults show that the difference is within an acceptablemargin. The study also shows that the thermal diffusionmay be included in the calculation of speciestransportation in high temperature gradient flows. In thefuture more applications, including the full chemicalreaction mechanism, will be performed to verify thecurrent methodology.

Reference1. H.K. Moffat and K.F. Jensen, "Three-

Dimensional Flow Effects in Silicon CVD inHorizontal Reactors," J. Electrochem. Soc., V.135, No. 2, p. 459, 1988.

2. R.I. Issa, "Solution of the Implicitly DiscretisedFluid Flow Equations by Operator-Splitting," J.Comp. Phys., V. 62, p. 40, 1985.

3. J.B. Greengerg, "A New Reliable Family ofSplit-Operator Methods for ComputingReacting Flows," Int J. Num. Meth. Fluid, V. 4,p. 653, 1984.

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4. K.K. Guo, Principles of Combustion, JohnWilly&Sons, 1986.

5. M.W. Chase, et aL, JANAF Thermodyna-mics Tables, 3rd ed., Americal ChemicalSociety and American Institute of Physicsfor the National Bureau of Standards, 1986.

6. R.B. Pope, "Stagnation Point ConvectiveHeat Transfer in Frozen Boundary Layer,"AIAA J., V. 6, No. 4, p. 169,1968.

7. S.R. Chakravathy and S. Osher, "A NewClass of High Accuracy TVD Schemes forHyperbolic Conservation Laws," AIAA-85-0363, 1985.

8. CM. Rhie, S.T. Steven and H.B. Ebabimi,"Numerical Analysis of Reacting FlowsUsing Finite Rate Chemistry Models," J.PropuL And Power, V. 9, No. 1, p. 119,1993.

9. Z.J. Chen, C.P. Chen and Y.S. Chen, "APressure Correction Method for theCalculation of Compressible ChemicallyReacting Flows," AIAA-92-3032, 1992.

10. M.E. Coltrin, RJ. Kee and J.A. Mffler, "AMathematical Model of Silicon ChemicalVapor Deposition," J. Electrochem. Soc.,V.I 33, No. 6, P. 1206,1986.

11. J. Ouazzani, K. Chru and F. Rosenberger,"On the 2D Modeling of Horizontal CVDReactors and its Limitations," J. CrystalGrowth, V. 9l,p.497, 1988.

12. F.C. Eversteyn, et aL, "A Stagnant LayerModel for the Epitaxial Growth of Siliconfrom Silane in a Horizontal Reactor," J.Electrochem. Soc., V.I 17, No. 7, p. 925,1970.