Chemical and Catalytic Reactor Modeling - ACS Publications

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Modeling of Chemical Vapor Deposition Reactors for the Fabrication of Microelectronic Devices KLAVS F. JENSEN Department of Chemical Engineering and Materials Science, UniversityofMinnesota, Minneapolis, MN 55455

The modelling of reactors for chemical vapor deposition (CVD) of thin solid films is reviewed. Both production and experimental reactor systems are considered. The discussion of production equipment centers on conventional horizontal, barrel, and tubular reactors while the treatment of experimental systems focuses on rotating disk and stagnation point flow reactors. The analogies between CVD and heterogeneous catalysis are pointed out and also illustrated through a modelling study of the multiple-wafer-in-tube low pressure CVD reactor. The reactor model is shown to be equivalent to a fixed bed reactor model. The deposition of polycrystalline Si from SiH4 is considered as a specific example. The model predicts experimental observations and provides quantitative comparison with experimental data from reactor studies. Chemical vapor deposition (CVD) is one of the fundamental processes in the microelectronics industry where it is used to deposit stable, thin solid films. The terms "chemical" and "vapor" derive from the fact that the solid film is deposited by chemical reactions of gaseous components. This distinguishes CVD from physical deposition processes such as sputtering and gives the process its flexibility. In the microelectronics industry CVD is used to grow a wide range of thin solid films which serve as dielectronics, conductors, passivation layers, dopant sources, and oxidation barriers. These films in turn form the basis for many different electronic devices including semiconductor memories, semiconductor lasers, light detectors, power transistors, and microprocessors. CVD is well established in the growth of the basic silicon derived electronic materials: polycrystalline Si, epitaxial (doped) Si, Si02, and S13N4. In addition, the process is gaining importance in the formation of the next generation of materials based on III-V compounds (e.g. GaAs, InP, and AlGaAs) which currently find use in optoelectronic devices. 0097-6156/83/0237-0197$06.00/0 © 1984 American Chemical Society In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

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CHEMICAL AND CATALYTIC REACTOR MODELING

Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on October 30, 2015 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch011

Chemical vapor d e p o s i t i o n i s not r e s t r i c t e d to the m i c r o e l e c t r o n i c s i n d u s t r y . I t i s the key process i n the f a b r i c a t i o n of o p t i c a l f i b e r s where i t enables grading of the r e f r a c t i v e index as a f u n c t i o n of the r a d i a l p o s i t i o n i n the f i b e r (JL). In the manu f a c t u r i n g i n d u s t r y the technique provides coatings with s p e c i a l p r o p e r t i e s such as high hardness, low f r i c t i o n , and high c o r r o s i o n r e s i s t a n c e . Examples of CVD r e a c t i o n s and processes are given i n Table 1.

T a b l e I . T y p i c a l C h e m i c a l V a p o r D e p o s i t i o n Systems

System „

. . .

O v e r a l l Reaction _

_

H a l i d e transport

WF5

^

+ Ho ^

W +

s

±

SiH4 + S i Low pressure CVD

SiH4 + N2O -> S1O2

S1H4 + NH3 ->· S13N4 Plasma enhanced CVD

J«* + ξ ±

Λ

Έ

Μ

}™ **° >

SiH4 + NH3 > Si3N4 S1H4 + N2O ~ > S1O2 Photochemical

vapor d e p o s i t i o n

^

+

^

The widespread use of CVD i s due to i t s many advantages, ( i ) Gaseous reactants are e a s i e r to handle and keep pure than are l i q u i d s or melts, ( i i ) The substrate can be cleaned before deposi t i o n by a s l i g h t etch with a r e a c t i v e gas. ( i i i ) The f i l m i s formed at temperatures w e l l below i t s melting p o i n t , e s p e c i a l l y i n plasma or photo-enhanced CVD, where f i l m temperatures t y p i c a l l y are a few hundred degrees, ( i v ) The growth of an e p i t a x i a l l a y e r on top of another s i n g l e c r y s t a l l i n e m a t e r i a l with d i f f e r i n g l a t t i c e parameters ( i . e . heteroepitaxy) can be r e a l i z e d by varying the composition of the growing f i l m so that the l a t t i c e parameters change smoothly across the i n t e r f a c e between the two m a t e r i a l s , (v) S i m i l a r l y , donor/acceptor concentrations i n the f i l m may be c o n t r o l l e d by varying the gas phase composition, ( v i ) In c o n t r a s t to s p u t t e r i n g methods, CVD provides good step coverage, which i s p a r t i c u l a r l y important since patterns are etched and subsequently covered by new l a y e r s during the processing of a device.

In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.


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Chemical Vapor Deposition

Reactors

199

To e f f e c t i v e l y e x p l o i t the above advantages i t i s necessary to have a d e t a i l e d understanding of the chemical and p h y s i c a l rate mechanisms underlying CVD. These encompass mass transport i n the gas phase by convection and d i f f u s i o n , homogeneous as w e l l as heterogeneous r e a c t i o n s , and heat t r a n s f e r by both convection and r a d i a t i o n . The s i t u a t i o n i s f u r t h e r confounded by complex flow f i e l d s and boundary c o n d i t i o n s . The formation of Si02 from S1CI4 exemplifies the importance of knowing the r o l e of each rate proc e s s . The heterogeneous r e a c t i o n s are necessary to grow t h i n S i f i l m s f o r b i p o l a r devices, while homogeneous n u c l e a t i o n of Si02 i s e s s e n t i a l i n the production of o p t i c a l f i b e r s . There e x i s t s a considerable l i t e r a t u r e on CVD (2) but r e l a t i v e l y few attempts have been made to combine chemical and p h y s i c a l r a t e processes to give a complete representation of the d e p o s i t i o n process. Most CVD studies have focused on demonstrating the growth of a p a r t i c u l a r m a t e r i a l or c r y s t a l s t r u c t u r e . However, the combined a n a l y s i s i s necessary i n order to design CVD reactors where i t i s p o s s i b l e to deposit t h i n f i l m s of constant thickness and u n i f o r m i t y across an e n t i r e wafer. This i s p a r t i c u l a r l y important i n the r e a l i z a t i o n of submicron feature s i z e s f o r Very Large Scale Integrated C i r c u i t s . The f u r t h e r development of devices based on III-V compounds also depends on CVD reactor design improvements s i n c e the composition and thus the e l e c t r o n i c p r o p e r t i e s of these m a t e r i a l s vary considerably with process c o n d i t i o n s . Chemical vapor d e p o s i t i o n and heterogeneous c a t a l y s i s share many k i n e t i c and transport features, but CVD reactor design lags the corresponding c a t a l y t i c reactor a n a l y s i s both i n l e v e l of sop h i s t i c a t i o n and i n scope. In the f o l l o w i n g we review the s t a t e of CVD reactor modelling and demonstrate how c a t a l y t i c reactor design concepts may be a p p l i e d to CVD processes. This i s i l l u s t r a t e d with an example where f i x e d bed reactor concepts are used to describe a commercial "multiple-wafers-in-tube" low pressure CVD r e a c t o r . CVD

Reactors

F i g u r e 1 i l l u s t r a t e s conventional CVD r e a c t o r s . These reactors may be c l a s s i f i e d according to the w a l l temperature and the depos i t i o n pressure. The h o r i z o n t a l , pancake, and b a r r e l r e a c t o r s are u s u a l l y c o l d - w a l l r e a c t o r s where the wall temperatures are conside r a b l y cooler than the d e p o s i t i o n s u r f a c e s . This i s accomplished by heating the susceptor by e x t e r n a l r f induction c o i l s or quartz r a d i a n t heaters. The h o r i z o n t a l multiple-wafer-in-tube (or boat) r e a c t o r i s a hot-wall reactor i n which the w a l l temperature i s the same as that of the d e p o s i t i o n s u r f a c e . Therefore, i n t h i s type of r e a c t o r , the d e p o s i t i o n also occurs on the reactor walls which presents a p o t e n t i a l problem since f l a k e s from the wall deposit cause defects i n the f i l m s grown on the wafers. This i s avoided i n the c o l d - w a l l r e a c t o r s , but the large temperature gradients i n those reactors may induce convection c e l l s with associated problems i n maintaining uniform f i l m thickness and composition.



CHEMICAL AND CATALYTIC REACTOR MODEl ING

ι-GAS INLET-,

QUARTZ BELL JAR

SILICON WAFERS

000 —Ν EXHAUST

J

INDUCTION COIL OR HEATERS

EXHAUST

QUARTZ BELL JAR RADIANT HEATERS

EXHAUST

BARREL REACTOR

GAS INLET

PANCAKE REACTOR INDUCTION COIL OR RADIANT HEATERS

OOO OOO OOO OOO OOCiXw

3-ZONE RESISTANCE HEATER TILT ANGLE

HORIZONTAL REACTOR

LPCVD REACTOR

Figure 1· T y p i c a l CVD reactors,



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There have been a number of modelling s t u d i e s ; i n p a r t i c u l a r h o r i z o n t a l and b a r r e l r e a c t o r s have been considered. However, the usefulness of these models i n the o p t i m i z a t i o n of reactor design and operating c o n d i t i o n s i s l i m i t e d by many s i m p l i f y i n g assumpt i o n s which bring e m p i r i c a l constants i n t o the modelling equations. Shepherd (3) described the d e p o s i t i o n of S i from SiCl4 i n a h o r i z o n t a l r e a c t o r assuming a p a r a b o l i c v e l o c i t y p r o f i l e , l i n e a r temperature v a r i a t i o n , and d i f f u s i o n to the susceptor surface while n e g l e c t i n g the change i n gas phase concentration i n the flow d i r e c t i o n due to the d e p l e t i o n of r e a c t a n t s . Rundle (4,5) included t h i s a x i a l d e p l e t i o n of r e a c t a n t , but assumed a plug flow and neg l e c t e d temperature v a r i a t i o n s . Bradshaw (6_) and Eversteyn et a l . (7,8) considered a stagnant l a y e r of f l u i d adjacent to the susceptor coupled with a well-mixed main flow region between the upper end of t h i s layer and the react o r w a l l . Eversteyn et a l . combined the d i f f u s i o n equation f o r the stagnant layer with a plug flow model f o r the main flow r e g i o n . An e m p i r i c a l r e l a t i o n was used to p r e d i c t the thickness of the l a y e r as a f u n c t i o n of the gas v e l o c i t y . The major r e s u l t of t h i s soc a l l e d stagnant layer model was the p r e d i c t i o n that i f the suscept o r were t i l t e d at a small angle to the h o r i z o n t a l , the u n i f o r m i t y of s i l i c o n growth rates through the reactor would be s u b s t a n t i a l l y improved. T h i s improvement i s due to a s t a b i l i z a t i o n of the flow and a thinning of the boundary l a y e r . The l a t t e r e f f e c t only ent e r s into the modelling equations through the e m p i r i c a l c o r r e l a t i o n of layer thickness and gas v e l o c i t y . In f a c t , the stagnant l a y e r model i n c o r r e c t l y p r e d i c t s that the reactor e f f i c i e n c y passes through a minimum and then increases with i n c r e a s i n g gas v e l o c i t y while the e f f i c i e n c y should approach zero i n the l i m i t of very l a r g e flow r a t e s . The stagnant layer concept o r i g i n a t e d from flow v i s u a l i z a t i o n experiments with Ti02 p a r t i c l e s showing an almost p a r t i c l e free l a y e r c l o s e to the susceptor and n a t u r a l convection c e l l s i n the main gas stream. This was i n t e r p r e t e d as evidence f o r a stagnant l a y e r , but because of the l a r g e temperature gradients i n the react o r , the p a r t i c l e free l a y e r more l i k e l y a r i s e s from thermodiffus i o n of Ti02 p a r t i c l e s away from susceptor ( 9 ) . Takahashi et a l . (10) have a l s o observed n a t u r a l convection s t r u c t u r e s i n a h o r i z o n t a l CVD r e a c t o r . They present f i n i t e d i f f e r e n c e simulations of v e l o c i t y and gas phase c o n c e n t r a t i o n p r o f i l e s f o r pure laminar flow and f o r the s p i r a l l i n g flow caused by n a t u r a l convection e f f e c t . U n f o r t u n a t e l y , as was the case with previous i n v e s t i g a t o r s , they l i m i t the a n a l y s i s to cases where the surface r e a c t i o n rate i s very l a r g e so that mass t r a n s f e r to the surface c o n t r o l s the depos i t i o n process. However, both k i n e t i c and transport e f f e c t s are important i n the m a j o r i t y of CVD processes. The two major modelling approaches based on e i t h e r boundary l a y e r approximations or w e l l developed laminar flow have a l s o been a p p l i e d to the b a r r e l r e a c t o r . Dittman (11) used a C h i l t o n Colburn analogy f o r flow over a f l a t p l a t e to p r e d i c t S i growth



202


r a t e s , but such an approach i s l i m i t e d to the mass t r a n s f e r cont r o l l e d regime and the s p e c i f i c reactor f o r which the c o r r e l a t i o n i s developed. F u j i i et a l . (12) s p l i t the annular flow region into three concentric l a y e r s where convection dominates i n the c e n t r a l one and d i f f u s i o n governs the transport i n both the l a y e r s next to the w a l l and the susceptor. By a d j u s t i n g the r e l a t i v e thickness of the l a y e r s the authors could match experimental data. Again, t h i s type of approach only a p p l i e s to the reactor at hand and prov i d e s no new i n s i g h t into the d e p o s i t i o n process. Manke and Donaghey (13) assumed f u l l y developed laminar flow and treated the mass t r a n s f e r i n the annular region between reactor w a l l and susceptor as an extended Graetz problem n e g l e c t i n g a l l k i n e t i c s . The so f a r most complete a n a l y s i s of the b a r r e l reactor i s given by Juza and Cermak (9,14) making use of the 2D momentum, mass, and energy balances to demonstrate the development of the transverse v e l o c i t y , concentration, and temperature p r o f i l e s along the susceptor. They include o v e r a l l surface k i n e t i c s f o r the surface r e a c t i o n : S i C l 4 + 2H2 > S i + 4HC1, and by comparison with the previous studies they demonstrate the importance of k i n e t i c e f f e c t s . They a l s o show that thermodiffusion should be included because of the steep thermal gradients and the large d i f f e r e n c e i n molecular weight between H2 and S i - s p e c i e s . The gas phase react i o n s associated with S i d e p o s i t i o n from SiCl4 ( c f . (15)) are not included i n the a n a l y s i s although they i n f l u e n c e the growth rate predictions· Experimental Reactor Systems There have been s e v e r a l experimental studies using the h o r i z o n t a l r e a c t o r , but most of these have been l i m i t e d to simple measurements of f i l m thickness. Only a few studies have been performed on the gas phase, notably by Sedgwick, Ban, and t h e i r coworkers. Sedgwick et a l . (16) used l a s e r Raman s c a t t e r i n g techniques to measure the transverse concentration and temperature p r o f i l e s i n a h o r i z o n t a l r e a c t o r . Ban et a l . (15,17) determined gas phase c o n t r a c t i o n s by mass spectrometry. However, the h o r i z o n t a l reactor i s not s u i t a b l e f o r mechanistic studies of CVD since i t i s v i r t u a l l y impossible to separate p h y s i c a l and k i n e t i c rate processes f o r t h i s complex r e a c t o r geometry. On the other hand, r o t a t i n g disk or stagnation point flow reactors are a t t r a c t i v e f o r d e t a i l e d i n v e s t i g a t i o n s s i n c e the hydrodynamics are c h a r a c t e r i z e d . Thus, the use of these c o n f i g u r a t i o n s makes i t p o s s i b l e to decouple the transport e f f e c t s more a c c u r a t e l y from the chemical k i n e t i c s . The r o t a t i n g d i s k has long been used to study e l e c t r o c h e m i c a l r e a c t i o n s and has a l s o found e a r l y use i n CVD (18). Sugawara (19) has presented an a n a l y s i s of e p i t a x i a l growth of S i from SiCl4« His treatment includes n a t u r a l convection but i s l i m i t e d to mass t r a n s f e r c o n t r o l l e d d e p o s i t i o n and e q u i l i b r i u m d i s t r i b u t i o n s i n the gas phase. P o l l a r d and Newman (20) d e t a i l S i d e p o s i t i o n on a r o t a t i n g disk t r e a t i n g the multicomponent mass and heat t r a n s f e r



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203

problem and i n c l u d i n g simultaneous homogeneous and heterogeneous r e a c t i o n s . In a d d i t i o n , the p h y s i c a l parameters vary across the d e p o s i t i o n zone. Their treatment c l e a r l y demonstrates that both homogeneous and heterogeneous r e a c t i o n s are important i n CVD of S i , but they observe some discrepancy between model p r e d i c t i o n s and experiments presumably because of poorly known k i n e t i c constants and n a t u r a l convection e f f e c t s . Hitchman et a l . (21,22) also consider the r o t a t i n g d i s k reactor and f i n d that the i n l e t gas flow perturb the i d e a l f l u i d flow p a t t e r n . They hypothesize that the d i f f i c u l t y of r o t a t i n g d i s k CVD experiments to confirm to t h e o r e t i c a l l y pred i c t e d flow patterns could be caused by t h i s i n l e t e f f e c t which i s not apparent i n l i q u i d phase systems. The stagnation point flow r e a c t o r , where the flow impinges on the heated substrate, appears to be an a t t r a c t i v e a l t e r n a t i v e to the r o t a t i n g d i s k c o n f i g u r a t i o n . There i s no ambiguity i n the way the gas i s introduced i n t o the d e p o s i t i o n r e g i o n . Moreover, natu r a l convection driven i n s t a b i l i t i e s may be avoided by i n v e r t i n g the reactor so that the buoyancy force and the i n l e t gas v e l o c i t y point i n the same d i r e c t i o n . Donaghey (23) discusses the use of the stagnation point flow i n c r y s t a l growth and reviews s e v e r a l CVD a p p l i c a t i o n s . The c o n t r i b u t i o n s by Wahl (24,25) seem to be the most s i g n i f i c a n t . He solves the two-dimensional momentum, s p e c i e s , and energy balances f o r an incompressible gas flow i n an enclosed stagnation point flow reactor by using f i n i t e d i f f e r e n c e s . The model p r e d i c t i o n s compare w e l l with flow v i s u a l i z a t i o n experiments. The conventional stagnation point c o n f i g u r a t i o n , where the gas flows downward, has been used i n s e v e r a l CVD s t u d i e s . However, none of these researchers seem to have accounted f o r the f a c t that i n a region c l o s e to the flow a x i s , the surface of the stagnation p l a t e i s "equally a c c e s s i b l e " to transport implying that the f i l m growth rate i s independent of the r a d i a l p o s i t i o n . Graves and Jensen (26,27) have r e c e n t l y analyzed t h i s case d e t a i l i n g the transformation of the general p a r t i a l d i f f e r e n t i a l modelling equat i o n s to a set of ordinary d i f f e r e n t i a l equations. These are i n turn solved by a G a l e r k i n f i n i t e element technique. The authors consider various general chemical mechanisms f o r f i l m growth to demonstrate how the stagnation point flow c o n f i g u r a t i o n may be used to d i s t i n g u i s h whether homogeneous or heterogeneous r e a c t i o n s dominate the o v e r a l l d e p o s i t i o n process. Low

Pressure CVD

Reactors

Low pressure CVD (LPCVD) has become a dominant process i n the growth of t h i n f i l m s of m i c r o e l e c t r o n i c m a t e r i a l s . I t i s widely used to deposit t h i n f i l m s of p o l y c r y s t a l l i n e S i , Si02> and Si3N4* In a d d i t i o n i t has been demonstrated f o r d e p o s i t i o n of metals, s p e c i f i c a l l y A l and W. The process i s c a r r i e d out i n tubular, hot w a l l r e a c t o r s where the wafers are placed perpendicular to the flow d i r e c t i o n as i l l u s t r a t e d i n Figure 2. The very large packing d e n s i t i e s that can be r e a l i z e d i n LPCVD r e a c t o r s without adverse




F i g u r e 2.

LPCVD reactor system.



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e f f e c t s i n f i l m u n i f o r m i t y are p o s s i b l e because of the reduced pressure (~ 1 t o r r ) where the d i f f u s i o n c o e f f i c i e n t s are three orders of magnitude l a r g e r than at atmospheric pressure. This implies that the chemical r e a c t i o n s at the surface of the wafers are rate c o n t r o l l i n g rather than mass t r a n s f e r processes. Moreover, i n s p i t e of the low pressures, rates of d e p o s i t i o n i n LPCVD r e a c t o r s are only an order of magnitude l e s s than those obtained i n atmospheric CVD since the reactants are used with l i t t l e or no d i l u t i o n i n LPCVD whereas they are s t r o n g l y d i l u t e d i n the convent i o n a l cold w a l l processes reviewed above. There appear to have been few modelling e f f o r t s f o r hot-wall LPCVD r e a c t o r s . Gieske et a l . (28) and Hitchman et a l . (29) present experimental data and discuss flow f i e l d s , mass t r a n s f e r e f f e c t s , and p o s s i b l e k i n e t i c s i n rather general terms. A recent model by Kuiper et a l . (30) cannot account f o r d i f f u s i o n i n the spaces between the wafers and the s i g n i f i c a n t volume expansion commonly associated with LPCVD processes. Furthermore, i t i s r e s t r i c t e d to isothermal c o n d i t i o n s and plug flow i n the main flow region i n s p i t e of the l a r g e d i f f u s i v i t i e s associated with LPCVD. In the f o l l o w i n g we present a d e t a i l e d model of the commercial, multiple-wafer-in-tube reactor i l l u s t r a t e d i n Figure 2. We have s e l e c t e d the LPCVD as an example because of i t s c e n t r a l r o l e i n the m i c r o e l e c t r o n i c s i n d u s t r y and because i t n i c e l y demonstrates the analogies to heterogeneous c a t a l y t i c r e a c t o r s , i n p a r t i c u l a r the f i x e d bed r e a c t o r . LPCVD Reactor Model The modelling approach behind the LPCVD reactor model i s not r e s t r i c t e d to any s p e c i f i c d e p o s i t i o n k i n e t i c s . However, to l i m i t the a l g e b r a i c complexity and to be able to compare model p r e d i c t i o n s with experiments we consider the simplest major d e p o s i t i o n process, the d e p o s i t i o n of p o l y c r y s t a l l i n e S i from SiH4. The model i s based on the f o l l o w i n g k i n e t i c mechanism: SiH (g) * SiH (g) + 4

2

H (g) 2

SiH (g) * SiH (ads) 2

2

SiH (ads) > Si(s) + 2

and and

H (g) 2

i t i s assumed that the surface r e a c t i o n i s rate c o n t r o l l i n g follows the rate expression:

(1) L

2 4 This p a r t i c u l a r form may be j u s t i f i e d by the experimental observations: the rate i s i n h i b i t e d by H , f i r s t order i n SiH4 at low SiH4 p a r t i a l pressures and approaches zero order i n S1H4 at high 2



206


p a r t i a l pressures (31) . The d e t a i l s of the development of the model are given i n r e f . (32) and f o l l o w the same approach as com monly used i n modelling f i x e d bed r e a c t o r s . The complete d e s c r i p t i o n of the physicochemical processes i n the tubular LPCVD reactor e n t a i l s p a r t i a l d i f f e r e n t i a l equations i n the r a d i a l and a x i a l coordinates. However, at low pressures the time s c a l e associated with d e p o s i t i o n i s of the same order or l a r g e r than that f o r d i f f u s i o n so that one may assume p e r f e c t r a d i a l mixing i n the narrow annular flow r e g i o n . S i m i l a r l y , since the wafer spacing i s small compared to the radius of the wafer, we may neglect v a r i a t i o n s i n the a x i a l d i r e c t i o n w i t h i n each c e l l formed by adjacent wafers. Under those conditions the reactor equation takes the form:

+

ο dz

where

and

ψ - -

V

entrance mole f r a c t i o n of SiH4. The expansion c o e f f i c i e n t , ε i s introduced as o r i g i n a l l y described by Levenspiel (33). The two r e a c t i o n terms r e f e r to the d e p o s i t i o n on the reactor w a l l and wafer c a r r i e r and that on the wafers, r e s p e c t i v e l y . The remaining q u a n t i t i e s i n these equations and the f o l l o w i n g ones are defined at the end of the paper. The boundary conditions are equivalent to the w e l l known Danckwerts' boundary conditions f o r f i x e d bed r e a c t o r models. The r e a c t i o n and d i f f u s i o n of SiH4 between the wafers i s governed by the c o n t i n u i t y equation: 0

0

t

Ai^-"*fe

n

e

(4)

dis»

with the boundary c o n d i t i o n s : X(r = R ) = X ( z ) w

ix

b

= 0

dr


(5)

11.

JENSEN


207

Reactors

where the center boundary c o n d i t i o n i s the usual symmetry condi t i o n . The f a c t o r 1/(1+εχ) comes about because of the increase i n the number of moles i n the d e p o s i t i o n r e a c t i o n . The quantity η, connecting the d e s c r i p t i o n s f o r the flow region (2) and the wafers (4) i s defined as:


R

R'

(

Chemical and Catalytic Reactor Modeling - ACS Publications

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