- Email: [email protected]
Chemical vapor infiltration: modelling solid matrix deposition in ceramic—ceramic composites
Chemwof Engineermg Science. Vol 46, No. 3, pp. 723 733, 1991. Prmted ID Circa1 Bntam
ooop-2so9)91 s3.w + 0.00 Q 1991 Pergamon Press plc
CHEMICAL VAPOR INFILTRATION: MODELLING SOLID MATRIX DEPOSITION IN CERAMIC-CERAMIC COMPOSITES
Department
G. Y. CHUNG, B. J. MCCOY and J. M. SMITH’ of Chemical Engineering, University of California, Davis, CA 95616, U.S.A and D. E. CAGLIOSTRO
and M. CARSWELL
NASA, Ames Research Center, Moffett Field, CA 94035, USA
(First receined 14 December
19X9; accepted in revisedform
15 June 1990)
model is developed to predict the effect of diffusion, the rate of deposition, and spatial ufa solid rnatrin irr a woven fabric forming a ceramic-r;eramic composite. To the knowledge of the authors there has been no prior treatment of chemical vapor infiltration in a system of plies consisting of an assembly of woven tows containing bundles of filaments. The model predicts the times to fill the gaps around the filaments of a tow, the space between plies, and the holes between the tows. It also predicts the porosity of the composite as a function of position and processing time. Predictions are made for an illustrative case of deposition of silicon carbide matrix within a woven carbon fabric. Abstract-A
diskibulion
INTRODUCTION
Ceramic fiber-reinforced composites, formed by the deposition of a solid within the void spaces in layers of woven fabrics, are resistant to oxidation and high temperature and have a low density compared to metals. This process of chemical vapor infiltration (CVI) requires different deposition conditions from those of conventional chemical vapor deposition (CVD). It is desirable for the deposit (e.g. Sic) to fill the void spaces as uniformly as possible to maintain good mechanical properties. The main difficulty in CVI is to avoid premature closing of the access to interior plies. Generally CVI requires deposition temperatures and total pressures much lower than those used in conventional CVD (Naslain et a[., 1983). van den Brekel et al. (1981) developed a model for the deposition of Ni, TiN and TIC on the inner wall of steel and ceramic tubes assuming a first-order deposition reaction. Later, Rossignol et (II. (1984) studied the formation of TiC from titanium chloride and methane in porous carbon media using the van den Brekel model and assuming that the porous region is equivalent to a cylindrical pore. Starr (1987) modelled short-fiber preforms with the assumption of a constant deposition thickness throughout a unit cell. The goal was to predict changes in composite structure with deposition. Jensen and Melkote (1989) also modelled CVI of short-fiber preforms. The influence on the final composite properties of operating conditions and structural parameters was studied. Tai and Chou (1989) studied the growth of alumina within one Sic-fiber bundle in an isothermal, hot-wall reactor
‘Author
to whom correspondence
should be addressed. 723
where the direction of gas diffusion is parallel to the axis of the fiber. They also simulated the pore space between the fibers as cylindrical capillary tubes. Recently, Sotirchos and Tomadakis (1989) studied the densification of fibrous substrates using Monte Carlo simulation methods. Monte Carlo simulation methods were employed to determine the variation of the structural properties of the fibrous structure and of the effective diffusivity. They proposed optimal reaction temperature and pressure histories for obtaining uniformIy densified composites. To our knowledge there has been no treatment of CVI for a substrate consisting of fabric or plies where each ply is an assembly of woven tows containing bundles of filaments. Sic is relatively inert and stable, and has good oxidation resistance and mechanical properties at high temperature. In conjunction with high-strength fibers it can be used to make high-temperature structural composites. Our objective was to develop a model for the in-depth CVD of SIC from dichlorodimethylsilane (DDS) into layers of woven carbon fabrics. It is regarded that DDS first decomposes to chlorosilanes and methane in the presence of an excess of hydrogen. The chlorosilanes are subsequently reduced to silicon metal, which reacts with methane to form Sic (Choudhury and Formigoni, 1969; Graul and Wagner, 1972; Sherman, 1972; DudukoviC, (1983). The model given here assumes that the overall deposition kinetics are first-order in DDS. The specific objective is to predict the overall rate of deposition as well as how the dimensions in the sample change with time, including amount of void space left around the filaments and between plies as a function of geometry and processing conditions.
724
G. Y. MODEL
CHUNG
DEVELOPMENT
Ceramic-ceramic composites can be fabricated in various types of CVI reactors. The model proposed here is for infiltration of Sic in a woven fabric preform in the reactor configuration shown in Fig. 1. The concepts could be applied to other geometries involving mass transfer in various subsections of the preform. For the arrangement in Fig. 1, boundary conditions at locations 1 and 2 are required as input data. These may be estimated from appropriate internal reacting flow codes or experimental measurements (Jensen, 1984; Moffdt and Jensen, 1988; Gokoglo er al., 1989). The substrate geometry assumes a multi-layer construction. Figures 2 and 3 show a sample consisting of 12 layers (plies) of woven fabric. In the plies each tow is assumed rectangular in cross-section and is a bundle of nonporous filaments woven to form approximately square holes. The plies are spaced so that the holes in every ply are aligned above each other. This forms a continuous straight hole through the plies as indicated in Fig. 3. The filaments within a tow are assumed to be arranged on triangular centers. Diffusion of gaseous DDS occurs from both ends to the center of the sample; through the holes, through the space between plies, and through the gaps around the filaments. The terms hole, space and gap are used to identify the different void regions of the sample (Figs 2 and 3). Because of symmetry, the model is considered for only half of the sample. Numbering of spaces and plies begins from the middle space and the middle ply (Fig.
3). Calculations
show
that
b
flow
b
flow
reactor wall Fig. 1. Reactor
configuration.
,tow
h
b P Y (b)
Details
plies
f,larnent
d,ff;s,an (a) Sample
aYer
of two
(C) Part of one ply
of 12 pies
considered
Fig. 2. Schematic
drawing
in the
of the woven fabric
modelhng
preform.
significant
deposition in the holes does not occur on the hole walls (edge of ply) until the gaps and spaces are filled. Hence, deposition can be considered sequential, occurring first on the outside lateral surface of the cylindrical filaments of the tow, and later on both sides of the flat surface of the ply and on the edge of the ply after the gaps around the outer filaments are filled. For the development of the model, it is assumed that the sample is isothermal and diffusion is onedimensional. Here, one-dimensional diffusion implies that there is no concentration gradient in the x- and y-directions in the hole, no concentration gradient in the y- and z-directions in the space between plies, and no concentration gradient in the x-direction in the gaps around filaments. It is also assumed that the
I
et al.
Fig 3. Notation numbering
diffusion velocity in the void regions is large with respect to the rate of change of the thickness of the deposited SIC. This assumption is generally valid when the density of the solid deposit is large with respect to the density of the diffusing gas (Bischoff, 1963). With this assumption, gaseous diffusion in the voids can be treated as a steady-state process. As deposition occurs, the concentration profiles in the void regions change as the porosity and surface area change. In addition, it is assumed that there is no diffusion from the holes into the gaps around the filaments along the edge of the ply. Rather, this region of the hole wall is considered to be a flat surface on which deposition occurs, reducing the dimension of the hole. The model is developed by solving mass balance equations for DDS for the three parts of the system (holes, spaces and gaps) and equations for the change with time in the dimensions of the system. Mass
-I
for dimensions and dimensionless variables, of spaces and plies, and coordinates.
balance
in the holes between
tows
The hole can be considered as two parts; one part is the hole wail along the edge of the ply through the distance c,, where deposition occurs. The other part is the open space between plies of distance h,, where
725
Chemical vapor infiltration diffusion in the x-direction between plies takes place (Fig. 3). Deposition starts on the hole wall, along distance c,, after filling the gaps around the outer filaments. It is assumed that these gaps are filled only by diffusion in the z-direction from the space between plies. Since the width of a tow is much greater than its thickness, this assumption should not introduce much error. Calculations indicate that at a given time the amount of deposition from the holes is only 9% of that into the filaments from the spaces. Therefore, the mass balance equation for the hole wall along the edge of the ply can be written as follows. Before filling gaps:
and Nz~,n,mlz~=~ = diffusion flux into the gaps on the lower side of ply n
.
aG*,.,nl
eg
af
In eq. (7), 9 is the width of diffusion as shown in Fig. 3. This dimension with x as follows: g=
(9)
*‘=O
area in the space changes linearly
-2x+a,+d,.
(10)
After filling the gaps around the outer filaments, deposition occurs on the flat surfaces of the upper and lower plies. The mass balance equation then becomes
LnDes
, 1 agac,,,,, _ gax ax
a2~,~ dX'
2kC
X.“,lfl
=
(11)
After filling gaps: & bc
Boundary
a2c, ,ky-4kC,=0 at z = h/2
C, = C,
a2C. aDmk L -
conditions
4Nx,nix=d,/Z
=
C3Z2
(4)
0 ‘
(ac,/az) = 0
&= ax
(3)
atz=O
0
c x,n,m = CZIr=(n-l)@,+c,)
Mass
balance
d2C i’,“.rn.l
In the hole, either eq. (1) or (2) and eq. (4) are applied alternately along the z-direction through the multiple-ply system. As deposition goes on, the open space between plies of distance bn,M,+l in the hole becomes smaller and the hole wall, through the distance c,, becomes larger. Hence, eq. (2) is substituted for eq. (4) for an increasing distance c, as the space between plies decreases. Mass balance in the void space between plies Deposition on the flat surface of one ply starts after the gaps around the outer filaments are filled with deposit. Before these gaps are filled, the mass balance in the spaces between plies includes diffusion on both sides of a ply. Hcncc, the mass balance before the gaps are filled is
~2CXnm
1agaC,., + --A g6x ax + N,,.n.Ji,=o)
where N r’.n--l.m z,=c* = diffusion the upper side of ply n - 1
(13)
= 0
(14)
G.n,m.~ = C,,,,,
at z’ = 0
(15)
C L .“.!?I.,- C,,,,,.,
at zi = c,
(16)
D,
- (N,~,n-l,ml,~=c,
at x = dJ2.
= 0
(7)
flux into the gaps on
around the filaments occurs on the lateral outside filaments. The mass balance
in the gaps
In the gaps, deposition surface of the cylindrical equation is
x --d,,z
i
(12)
Initially, eq. (7) is applied everywhere in the space. Then, it is replaced with eq. (11) for the element of distance x where the gaps around the filaments at the outer sides of the plies are filled.
(6)
NX,.L
T
atx=O
(5)
is the diffusion flux into the space where ~x,nlx=d,~2 between plies at the entrance from the hole and can be written as
b,,,D,,
for eqs (7) and (11) are
(2)
aZ
where a is the dimension of the square hole. The mass balance equation in the hole along the distance b, is
bc
0.
>
bcs
SZ’2
- kG,,.,,fs
where s is the surface area for deposition of SIC which is expressed as the sum of the lateral surface areas of all cylindrical filaments in unit volume (W = number of filaments per unit cross-sectional area in the tow): s = 27cr.,,,,[ W. Equation
(14) is applied
(17)
until the gaps are filled.
Changes of dimensions with time The DDS concentration profiles in the voids change with dimensions of the system, i.e. the sidelength of the hole, distance between plies, thickness of the ply, and radius of the filament. The dimensions change with time due to deposition, and are expressed by the equations that follow. The rate of change of the dimension of the hole is aa
-
at’
ic
a = a,
ZYM”AC, pm
at t < t,.
(18) (19)
G.Y.
726
CHUNG
For the hole wall along the edge of the ply, t, is the time required to plug the gaps around the outer filaments at the edge of a ply. For the open space of the hole along the distance b,, t, is the time required to plug the space between plies at the entrance from the hole. The equation for the change of distance between plies is
abn.m _ _ 2qM”kC at
Pm
b,,, = b,
ic
(21)
ic
ah+ ,,m at
(22) Ix=d,,2
at t < t tl,n.hf,+1.
c, = c,
arm,,= Lx
ic
qM
~kC,~,,,,,r
w=
r,
number
of filaments
(27)
The first-order reaction rate constant, k, was estimated from the experimental deposition data of Carswell (1989). The walls of the plies are not flat surfaces since the tows are woven to form the plies. The effective diffusion coefficient in the uneven space between plies was estimated
using the tortuosity
factor $:
The effective diffusion coefficient in the gaps around the filaments was estimated from the porosity and tortuosity factor, K, according to the equation (Smith, 1970; Froment and Bischoff, 1979)
(24) D,,
at t = 0.
in a tow
0,
(23)
(25)
The method of solution of these equations is first to calculate the initial concentration profiles in holes, spaces between plies, and gaps around the filaments from eqs (i)-( 17) using the initial dimensions of the system. Then, new dimensions are calculated using eqs (18)-(25) with the calculated concentration profiles. This procedure is repeated for successive time increments to yield concentration profiles and dimensions of the system as a function of time. From these results, deposition amounts and rates and porosities can also be evaluated. The dimensionless forms of eqs (l)-(25) are in the Appendix. They were solved numerically in finitedifference form with 400 nodal points in the q-direction, 10 nodal points in the c-direction, and eight nodal points in the c-direction. The previous time-step values of Y were used in eq. (A4). Similarly, the previous time step values of X were used in eq. (A6). A time increment of 2 s was used up to 10 h processing time and 2 min thereafter. All the calculations were done with a Vax 11-785 computer and took about 20 mincpu time for a typical run. RESULTS
of filaments in a unit cross-sectional to the axis of the filaments, W, is estimated using the total number of filaments in a tow and the dimensions d, and c,:
Pm
T,,,,~ =
(26)
)
Again tb,n,M,+1 is the time required to plug the gaps around the filaments at the outer surface of a ply. The rate of change in the radius of a filament is given by the mass balance: 2 *
h-2Nc b, = 0 2N - 1
(20) area perpendicular
at t G tb.“+
+
distance between holes, d,; the radius of one fiIament, r,; the thickness of one ply, c,,; and the thickness of the whole system, h. Also the number of plies in the system, 2N, must be known. The distance between plies, b,, can be calculated using h, c, and N:
The number
x,n,m
to plug the gaps where tb,n,m is the time required around the filaments at the outer surface of a bundle of filaments (tow) at position x. Note that these gaps at the outer surface may be plugged while voids still exist around the inner filaments in the hundle. The equation for the change in thickness of a ply can be expressed in terms of the change in distance between plies: ab, m dc,_ ___ 1 2 at 2 ( at Ix=d,/l
et al.
AND DISCUSSION
Parameter values The following initial dimensions are needed for the calculations: the dimension of the square hole, a,; the
=
:Dmk. K
(29)
Since the porosity and tortuosity are changing with time due to deposition, D,, is a function of time. According to the random pore model of Wakao and Smith (1962), the tortuosity factor is inversely proportional to Lhe porosity. Using this proportionality:
where K, is the initial tortuosity eq. (30) into eq. (29) gives
factor.
Substituting
(31) With the assumption that, on average, a pore makes an angle of 4.5” with the radial direction of the cylindrical filament, $ was used as the initial value of the tortuosity factor. The local porosity in the ply is given by E = 1 - 71l.2w.
(32)
Since the dimensions of the voids may be very small, the temperature high, and the pressure low, Knudsen diffusion can be significant. Hence, Dmk is taken as a composite diffusivity given by the equation 1
-=-$D mk
1 D,
1 4
(33)
Chemical vapor infiltration
727
where D,, the molecular diffusion coefficient, is estimated from the Chapman-Enskog formula (Smith, 1970) and D,, the Knudsen diffusion coefficient in cm2/min, is calculated from the equation (34) where Tis temperature in K, MA the molecular weight of DDS in g/mol and R the average void opening in cm. R was taken as half the side-length (a) of the hole and as half the distance (b) for the space between plies. Within a tow the filaments initially are close together but not in contact. Hence R for the gaps was evaluated in the following way. First there will be some parts of the gaps around filaments that never can be filled with deposit. These regions are due to isolated volumes formed when there is sufficient deposit to bring the filaments in contact with each other. This porosity due to geometry is 0.093. With this porosity, the radius r,, of the filaments at the time that the gaps are plugged is obtained from eq. (32). Then, half of the least distance between filaments is the difference between ry and r, i.e. (rl - r). Then, an approximate value of R is taken as an average of r,, and (rP - r). Results for a specific set of initial dimensions and a range of diffusivities and rate constants are shown in Figs 4-l 1. There are three dimensionless variables, 4, i and cp. corresponding to Thiele modulus numbers. However, these three variables all change with a change in D, or k. Corresponding changes in D, or k that leave these variables unchanged yield different deposition curves. The parameter values listed in Table 1 are based upon the geometries derived for a composite sample fabricated at the NASA Ames Research Center. Comparison
of calculation
t
Concentration vs time in the gaps around the filain the first ply at point (A) (Fig. 3; x = 0, z’ = 0.5~~) for diffusion coefficients and reaction rate constants. D, = 17,300 cm’/min, k = 10 cm/min
1.c
0.8
0” ._0 E 0
04
oz0
20
time
methods
As mentioned earlier, the prior time step values of Y in eq. (A4) and X in eq. (A6) were used for the finite-difference calculations. This reduces the calcu-
10,
’ 60
40
.
80
[hr]
Fig. 6. Concentration YS time in the second space between plies at point (B) (Fig. 3; x = 0) for different diffusion coefficients and reaction rate constants. D, = 17,300 cm’/min, k = 10 cm/min.
,
1 .’O\ \
0.84 \
u” 0. 6‘N
0
0 4’1
0.31 0.0
0.2
I 04
’ 06
08
’ 1.0
“/(d&2)
Fig. 4. Concentration profiles in the first space between plies (Fig. 3) at 5 h for different diffusion coefficients and reaction rate constants. D, = 17,300 cm*/min, k = 10 cm/min.
0. 2’
0
10
20
30
time
330
630
do.2 930
[hr]
Fig. 7. Concentration vs time in a hole at point (C) (Fig. 3; z = 0) for different difTusion coefficients and reaction rate constants. D, = 17,300 cm’/min, k = 10 cm/min.
G. Y.
728
$ ::
-
0.6-
CHUNG
et cd.
0.6
v)
$ L
= c" E o 0
_ _.
_
_
- 0.9 0.4r..._..:::::::~~~~~~----..... Dm.Ik --. --Q..___ --.._.___ --._ ..I__ *x_ Ddk‘.... -1.2 0.2*\.. .\,
0
d
'... o,. 00
I
I
0.2
0.4
0.6
,
"-,,5
0.8
1.0
x/(do/2)
Fig. 8. Distance between plies and deposition along the x-direction in the first space (Fig. 3) at 20 h for different diffusion coefficients and reaction rate constants. D, = 17,300 cm2/min, k = 10 cm/min.
8
0.6
z g I
'Om.2k‘ --__._;
;
1.2
time
Fig. 11. Rate of deposition per unit cross-sectional area (csa) of sample vs time for different diffusion coefficients and reaction rate constants. D, = 17,300 cm’/min, k = 10 cm/min.
lations to solving a tridiagonal set of equations. This simplification was checked with the Gauss-Seidel iteration method avoiding the use of values for the prior time step. No significant differences between the two computation methods were found. The difference became smaller for larger diffusion coefficients. In calculating the concentration profile in the space between plies, it was assumed that concentration gradients
04
18
02
2.4
[hr)
in the y- and
z-directions
are
independent.
steady-state calculations in the space between plies were carried out for one- and two-dimensional geometries: One-dimensional: To check
this assumption,
Dnlg
(35)
z/(h/2)
with
Fig. 9. Dimension of a hole and deposition thickness along the z-direction at 200 h for different diffusion coefficients and reaction rate constants. D_ = 17.300 cm”/min,
y = x + a,/2
Two-dimensional:
bcs
C, = C, at the side line of the hole
(37)
ac,jax=o or aC,/ay
= 0 at the center
of the space.
(33)
The dimensionless forms of these equations were solved separately by the iteration method. Even if the shape of the concentration contour lines is different in the region far from the hole, the concentration profiles along the y-axis show small differences (less than I %). The rate of deposition results in Table 2 also showed little difference, which implies that the one-dimensional approximation is reasonable.
time
[hr]
Fig. 10. Porosity of the whole sample of 12 plies vs time for different diffusion coefficients and reaction rate constants. The porosity at time zero is 0.706. D, = 17,300cm2/min, k = 10 cm/min.
Eflects of d$iision coeficients and constants Concentration profiles. Concentration space between plies are displayed in specific parameter values given in Table are shown concentration changes
reaction
rate
profiles in the Fig. 4 for the 1. In Figs 5-7 with time in
729
Chemical vapor infiltration Table 1. Parameter values used to illustrate the model a, = 0.06 cm,
b, = 0.003 cm, i-, = 0.0004 cm, h = 0.321 cm, Number of filaments in a tow = 3000 Deposition
conditions:
c, = 0.024 cm, N = 6,
10 tarr total pressure,
C, = 5 x LOP mol/cm3, p, = 3.217 g/cm3
d, = 0.18 cm W= 695,000cm-2
9OOT, and 3.75% DDS in hydrogen k = 10 cm/min
D, = 17,300 cm2/min,
Table 2. Rate of deposition pet unit cross-sectional area of sample in the space between plies in the two- and one-dimensional calculations+ Rate of deposition [ x lo6 g/(h cm’)] D,,k
0.1 D,,k
O.O5D,,k
D,,2k
D,,0.5k
Two-dimensional calculation
3.935
3.851
3.761
7.732
1.985
One-dimensional calculation
3.95
3.877
3.804
7.777
1.989
+D, = I7,300 cm2/min, k = 10 cm/min.
the gaps around the filaments at point (A) (z’ = O.k,, x = 0), in the space between plies at point (B) (x = 0), and in the hole at point (C) in Fig. 3 (z = 0) for different diffusion coefficients and reaction rate constants. While the concentration gradients in the gaps around the filaments are small, gradients in the space between plies (Fig. 4) are relatively large. These large gradients are due to the large flux into the gaps around the filaments before they are filled. For instance, for the parameter values of AI,,,and k given in Table 1 and at 5 h, the concentration difference (0.87 - 0.61) between the entrance and the middle of the space between plies is about one-quarter of the initial concentration (i.e. 0.26C,). This concentration difference is reduced to O.O4C, at 20 h when the gaps around the outer filaments are filled. The same phenomenon occurs in the holes, i.e. the concentration gradient is large when the spaces and the gaps are not plugged and is small after these regions are plugged. The concentration in the gaps around the filaments (Fig. 5) changes with time due to the concentration change in the spaces between plies (Fig. 6). The curves in Fig. 6 end when the gaps in the outer filaments are plugged. After this time the gaseous silane concentration in the very small voids around the remaining filaments will decrease to zero. The corresponding amount of deposition is negligible. As the flux into the plies becomes larger due to the increased surface areas of the filaments for deposition (as the radius of the filaments grows), the concentration in the void spaces between plies decreases. As a result, the concentration in the gaps around the filaments decreases. Later, as the flux into the plies is reduced due to plugging of some of the gaps around the outer filaments, the concentration in the spaces between plies increases. Hence, the concentration in the remaining gaps increases. After plugging of all the gaps around the
outer filaments, the concentration in the spaces between plies decreases monotonically with time due to plugging. Since the concentration in the holes is already high (see Fig. 7) when the gaps around the filaments are plugged, the decrease in flux into the spaces between plies does not significantly affect the concentration profiles in the holes and spaces. Therefore, the concentration between spaces decreases monotonically due to the decrease in the distance between plies. When the diffusion coefficient decreases, the concentrations throughout the system are smaller, as expected (Fig. 4). It is also expected that higher reaction rate constants would lead to lower concentrations. However, the concentrations in the spaces between plies at 4 times the rate constant (4k) are higher than those at 0.5k, k and 2k because the gaps are plugged. It is shown in Fig. 5 that the gaps around the outer filaments at 5 h are already closed for 4k. The steeper concentration profile for 2k in Fig. 4 is due to the greater flux into the plies. This increased flux is due to the larger surface area for deposition around the filaments. Thus, the model demonstrates that the process at the first ply (near the center of the sample) is affected by what happens at the last ply. Plugging occurs first in the gaps around the filaments, next in the spaces between plies, and finally in the holes. For instance, for the specific conditions of k and D, in Table 1, Figs 5-7 show that plugging of the gaps occurs at 10 h, plugging of the spaces between plies occurs at 48 h, and plugging of the holes occurs at 808 h. For lower diffusion coefficients and lower reaction rate constants longer times are required to fill the gaps and spaces. As mentioned earlier, as the gaps around the filaments and the spaces between plies are closed, concentrations in the hole increase. Since closing of the spaces between all
730
G.Y.
CHUNG
the 12 plies occurs almost at the same time for a high diffusion coefficient, the concentration in the holes increases sharply as shown in Fig. 7. However, for a low diffusion coeficient, the increase is slow due to the step-by-step closing of the space between plies. The variation in slope for the concentrations in the gaps for O.lD, and O.O5D,, as shown in Fig. 5, is due to plugging of the gaps around the outer filaments in the spaces between plies. Amounts of deposition and porosities. The model predicts the changes in dimensions of the sample as a result of deposition. Results are illustrated in Figs 8 and 9. The deposition thickness between plies is expressed as (b, - b)/2, where h, and 6 are the distances between plies initially and at any time, respectively. The deposition thickness in the holes is expressed as (a, - a)/2, where a, and a are the dimensions of the hole initially and at any time. The deposition thickness in the spaces (Fig. 8) is nonuniform due to the large concentration gradients shown in Fig. 4. On the other hand, the deposition profile through the holes at 200 h (Fig. 9) is essentially uniform except for a high reaction rate constant, 4k. The amount of deposition was calculated for each part of the system as follows: In the gaps around the filaments:
47~~~ WTAx
AZ’ 5 F i=l j=l
In the spaces 4~mTAx
between
F gj(b,/2 [ j=j
plies:
2’
(r&
- r2)gj
(39)
k=l
j=, 1
- b1.j) + ; y ;=2
gj(b, - bi.i)
(40)
In the holes: 2p,HAz
5 (a,’ - a?) (41) i=l The porosities of the whole system and each ply were calculated using the amounts of deposition for each part of the system. The porosities of the whole system are shown in Fig. IO. The discontinuous points for the rate of change in porosity in Fig. 10 show the times that each part is plugged. As expected, a more dense deposition (lower porosity) is obtained for a low reaction rate constant and a large diffusion coefficient. This is because the concentration gradients are smaller. Rate of deposition curves calculated from the amounts of deposition at each time step are displayed in Fig. 11. The section of the curves at low times gives the deposition rate before the gaps around the outer filaments are fiIled at any position (x) between plies. At O.lD, and O.OlD, the increase in deposition rate due to surface area increase is less than the decrease due to concentration decrease. The concentration decreases because the area for diffusion decreases. Hence, the resuItant rate of deposition decreases. However, at D,, the rate of deposition increases because the increase in rate due to surface area
et ul.
increase is larger than the decrease due to concentration decrease. When the gaps around the outer filaments are plugged at some positions x, the total rate of deposition decreases due to the decrease in the total deposition area. After plugging the gaps along all the distance x, the deposition rate decreases slowly with the monotonic decrease of concentration in the spaces between plies, as shown in Fig. 6. To check the importance of diffusion resistance in the sample, calculations were made for the condition that the concentration in all the void regions is a constant
equal
to the boundary
value,
C,.
The
results
showed that for the reference values, D, and k, the porosity was about 7% less when diffusion resistance was neglected. An empirical relation has been proposed to describe deposition (Rossignol et al., 1984). The equation representing this relation is log [(M, - M,)/M,]
= - Wt
(42)
where M, is the initial pore volume multiplied by the density of deposited material, M, is the weight of deposited material, and w is a constant which is a function of temperature and initial porosity. As shown in Fig. 12, plotting log(M, - M,)/M, vs time shows linear segments, except at long processing times when the holes are becoming plugged. The resulting approximate values of w from our data are given in Table 3. Compared to the values of 2 x 1O-3 and 4.5 x 10-j h- ’ for carbon
of dimensions
of sample
The effects of initial distance between plies and size of the holes are shown in Figs 13 and 14. Changing the size of the holes resulted in different times for the holes to be plugged. Decreasing the size of the holes also resulted in more area for deposition to occur between plies, where the deposition is nonuniform. Furthermore, while the dimension of the hole does not signi-
-0 2
T z. -0.4 % I i =p) -0.6 0
-0 a
50
100
150
200
time
400
600
800
[hr]
Fig. 12. Log [(M, - M,)/M,] vs time for different diffusion coefficients reaction and rate D, = constants. 17,300 cm2/min, k = 10 cm/min.
Chemical 3. w values in eq. (42) for different
Table
vapor
infiltration
731
diffusion coefficients and reaction rate constants’ W( x lo3 h-‘)
O.lD,,k
D,,k In the gaps
42
O.O5D,,k
O.OlD,,k
2.5
15
30
In the spaces
4
3.55
3.01
1.59
In the holes
0.2
0.18
0.17
0.13
‘D,
D,,4k
D,,2k
117
70
17 0.55
D,,0.5k 27.5
8.02
2.29
0.36
0.17
= 17,300 cm2/min, k = IO cm/min.
0.5
0.4
.g cn
0.3
2 g
02
0.1
I 300
600
lime
900
0.0
1200
L
J
0
200
[hr]
600
400
time
600
[hr]
Fig. 13. Porosity of the entire sample of 12 plies YS time for different dimensions of the holes. The porosity at time zero for a, is 0.706 (a, = 0.06 cm, b, = 0.003 cm).
Fig. 14. Porosity of the entire sample of 12 plies vs time for different distances between plies. The porosity at time zero for b, is 0.706 (a, = 0.06 cm, b, = 0.003 cm)
ficantly affect the concentration in the hole, decreasing the hole size decreases the area for diffusion into
plies determines the total height of system. The local porosities when all areas are plugged are given in Table 4. The porosity due to geometry (= 0.093) is the lower bound that is obtained in case of uniform deposition. The porosity due to nonuniform deposition is obtained by subtracting the porosity due to geometry from the total porosity in the gaps. From Table 4, as the distance (b,) between plies increases, the local
the space between plies. Hence, a large concentration gradient results and even greater nonuniformity of deposition in the space between plies and in the gaps around the filaments. This leads to a higher porosity at the time of plugging the holes, even if this time is shorter. The distance between plies and number of
Table 4. Porosities
of each part of the sample at the time of plugging of holes for different distances between plies’
dimensions
of holes and
Porosity
aoh, In the gaps
Due to nonuniform deposition Total*
.%,,b,
0.5a,,b,
a,,%,
a,,2b,
a,,0.5b,
0.042
0.038
0.05 1
0.032
0.033
0.056
0.135
0.131
0.144
0.125
0.126
0.149
In the spaces
0.142
0.133
0.161
0.047
0.07 1
0.287
In the holes
O.OC06
0.0012
0.0035
0.0027
0.0011
O.OCKM
In Ihe whole sample
0.127
0.117
0.143
0.098
0.109
0.147
i
‘The individual porosities are defined as the void volume in each part divided by the total volume of that part. The results are for reference conditions (D,,k) and where a, = 0.06 cm and 5, = 0.003 cm. The initial porosities (based upon the total sample volume) for the reference case (a,,b,) are 0.547 in the gaps, 0.096 in the spaces and 0.063 m the holes. ‘Porosity due to geometry is 0.093.
732
G. Y.
CHUNG
porosities in the space between plies decrease in almost the same ratio. This means that the final pore volume in the spaces does not change much even if the total volume of the system increases. Hence, the porosity in the spaces between plies based on the total volume decreases. This decrease of porosity due to increase of total volume is greater than the increase of porosity due to the increase of nonuniform deposition in the spaces between plies. Therefore an increase in the distance between plies reduces the final porosities (Fig. 14).
et
ai.
k M, M, M, M,* N N, N,,
CONCLUSlONS
A model is proposed for CVD within layers of woven carbon fabrics. The model predicts concentration profiles in the system, amount and rate of deposition, and changes of dimensions with time. The results suggest sequential plugging of the gaps around the outer filaments, the spaces between plies, and finally the holes. The concentration in the gaps around the filaments first decreases due to the increased surface area for deposition and then increases due to the gradual plugging of the gaps around the filaments. Predicted final porosities are lower if the diffusion coefficient is large and the reaction rate constant is low. A comparison of the final porosities with and without diffusion resistance also shows, as expected, that porosity increases substantially if diffusion resistance is significant. For the diffusion coefficient, the reaction rate constant and geometries tested, increasing the size of holes and the distance between plies gives lower final porosities (higher sample densities). Acknowledgement-This work was supported Cooperative Agreement NCC 2-590.
by NASA
4s W
X
Y
(CYlC,) coordinate in the direction of diffusion in the space between plies, cm reduced concentration along the x-direction
Z
(C,/C,) reduced
x
NOTATION
a
side-length of the square hole, cm b distance between plies, cm concentration of DDS at the outside of the co sample, mol/cm3 concentration of DDS along the x-axis in the C, nth space between plies, mol/cm3 concentration of DDS along the z-axis in the c, hole, mol/cm’ concentration of DDS along the z-axis in the gap, mol/cm3 C thickness of one ply, cm effective diffusion coefficient in the gap around De, the filaments, cm’/min effective diffusion coefficient in the space beD, tween plies, cm’/min Knudsen diffusion coefficient, cm2/min 4 molecular diffusion coefficient, cm’/min D, D rnL composite diffusion coefficient, cm’/min d distance between holes, i.e. width of tow, cm width of diffusion area in the space between 9 plies, cm H number of holes in unit cross-sectional area, cm-’ h height of the sample, cm
first-order reaction (deposition) rate constant, cm/min molecular weight of deposited Sic, g/mol total number of divisions in the x-direction in the space (d,/2Ax) total number of divisions in the z-direction in the hole (h/2Az) total number of divisions in the z’-direction in the gap (c,/Az’) half of the total number of plies flux ofdiffusion in the x-direction, mol/cm’min flux of diffusion in the z’-direction, mol/ cm2 min number of deposited Sic molecule per molecule of DDS radius of average void opening between filaments, cm radius of a filament, cm surface area of the filaments per unit volume of the ply, cmz/cm3 number of spaces or plies of length d, in unit cross-sectional area, cm - 2 time, min time required to plug the gaps around the outer filaments at the hole-edge of a ply, or time required to plug the space between plies at the entrance from the hole at position z; f, = [f’(z)], min time required to plug the gaps around the outer filaments at position x; t, = [f(x)], min number of filaments per unit cross-sectional area perpendicualr to the axis of the filaments, cm-’ reduced concentration along the z’-direction
z z’
Greek
concentration
along
the
z-direction
(C,/CJ coordinate in the direction of diffusion hole, cm coordinate in the direction of diffusion gaps around the filaments, cm
in the in the
letters
reduced side-length of the square hole (a/a,) reduced distance between plies (b/b,) reduced thickness of the nth ply (c/c,) reduced width of the diffusion area in the space between plies (g/d,) local porosity of a ply reduced coordinate in the z’-direction (z’/c,) reduced coordinate in the z-direction [z/(/1/2)] tortuosity in the gaps around the filaments dimensionless parameter [d,(k/2b,D,)‘~2] dimcnsionlcss parameter [h(2/a,d,)1’Z] dimensionless parameter [(d,/2)( l/b,c,)“*] reduced coordinate in the x-direction [x,$&/Z)] density of the deposited material, g/cm3
Chemical
vapor
reduced radius of a filament (r/t-J dimensionless time [(2qM,kC,/a,p,)t] dimensionless time required to plug the gaps around the outer filaments at the hole-edge of a ply, or time required to plug the space between plies at the entrance from the hole at position 4; z, = C_f(a)l CGWfmkCo/wmb.l dimensionless time required to plug the gaps around the outer filaments at position <; tb = ES(<)] [(%~,&/%L’,&~ dimensionless parameter [h(k/u,D,)1’2] dimensionless parameter [c,(2ar,‘Wk/D,)1i2] defined by eq. (42) Subscripts the Ith segment of a ply in z’-direction 1 the mth segment of a space or a ply in x-direction m the nth ply or space n zero time 0 plugging time P
733
infiltration
Tai, N. H. and Chou, T. W., 1989, Analytical modeling of chemical vapor infiltration in fabrication of ceramic composites. J. Am. &ram. Sot. 72, 414-420. van den Brekel, C. H. J., Fonville, R. M. M., van der Straten, P. J. M. and Verspul, G., 1981, CVD of Ni, TIN, and TIC on complex shapes. Proc. elecrrockem. Sot. 81, 142-156. Wakao, N. and Smith, J. M., 1962, Diffusion in catalyst pellets. Chem. Engng Sci. 17, 825-834. APPENDIX
dimensionless form eqs (1)-(25) are as follows: In the holes between tows: Hole walls: Before filling gaps: In
d2Z -_=O a?+
After filling gaps:
WI bc Open spaces
atq=l
Z=l
between
plies:
REFERENCES
BischotT, K. B., 1963, Accuracy of the pseudo steady state approximation for moving boundary diffusion problems. CRem. Engng Sri. 18, 711-713. Carswell, M., Personal communication. Choudhury, R. and Formigoni, N. P., 1969, Beta-Sic films. J. electrochem. Sot. 116, 1441-1443. Dudukovi?, M. P., 1983, Reactor models for CVD of silicon, in Flat Plate Soiar Array Workshop on the Science of Silicon Marerial Prepararion, J.P.L. Publication 83(13), 199-226. Froment, G. F. and Bischoff, K. B., 1979, Chemical Reactor Analysis and Design, pp. 167-173. John Wiley, New York. Gokoglo, S. A., Kuczmarski, M. A., Tsui, P. and Chait, A., 1989, Convection and chemistry effects in CVD: a 3-D analysis for silicon deposition, Paper presented at the EURO 7 Conference, Perpignan, pp. 19-23. Graul, J. and Wagner, E., 1972, Growth mechanism of polycrystalline beta-Sic layers on silicon substrate. Appl. Phys. Lett. 21(2), 67-69. Jensen, K F., 1984, Modelling of CVD reactors. Proc. ektrochem. Sm. 84(6), 3-20. Jensen, K. F. and Melkote, R. R., 1989, Chemical vapor infiltration of short fiber preforms. Extended abstract presented at the 1989 AlChE Annual Meeting, San
bc
aZj@
In the spaces between Before filling gaps:
atq=O
= 0
(A5)
plies:
and 6, = a,/d, +
I- 5
(A7)
After filling gaps:
(A81
ar.,,iat =0
bcs
at5=0 at 5 = 1
Yn.I = Z 1~= z(n - I)(b, + c.)/h In the gaps around
the filaments: (All)
Francisco, p. 54. Moffat, H. K. and Jensen, K. F., 1988, 3-D flow effects in silicon CVD in horizontal reactors. J. elecrrochem. Sot.
bcs Xn,,., = Yn.m
135,459-471. Naslain, R., Hannache,
H., Heraud, L., Rossignol, J. Y., Christin, F. and Bernard, C., 1983, Chemical vapor infiltration technique, Proc. 4th European Conf: on Chemical Vapor Deposition, pp. 293-304. Rossignol, J. Y., Langlais, F. and Naslain, R., 1984, A tentative modelization of titanium carbide C.V.I. within the pore network of two-dimensional carbon-carbon composite preforms. Proc. electrochem. Sot. l&4(6), 596-614. Sherman, A., 1972, Chemical Vapor Depositionfor Microelectronics, Principles, Technology, and Applications, pp. 67-69. Noyes, Park Ridge, N.J. Smith, J. M., 1970, Chemical Engineering Kinetics, pp. 414-419. McGraw-Hill, New York. Sotirchos, S V. and Tomadakis, M., 1989, Modelling the densification of fibrous structures, Extended abstract presented at the 1989 AlChE Annual Meeting, San Francisco, p. 135. Starr, T. L., 1987, Model for CVl of short fiber preforms. Ceram. Engng Sci. Proc. 8.951-957.
(A9) (Al01
at[=O
(Al2)
at c = 1
(Al31
X n.n.1 - Y.+,,, Changes
of dimensions
with time:
d”=_Z
(Al4)
d+
ic
a=1
at T $ z,
aS..,=_a”y_ dr
b,
’
(Al51 (Al61 (Al71
ah
_=
--
(Al8)
ar
(‘419)
ic
atr n.m.l
a,
dr
2r,
~“,rn.l = I
n,m*i at t = 0.
(A20) (‘421)
ooop-2so9)91 s3.w + 0.00 Q 1991 Pergamon Press plc
CHEMICAL VAPOR INFILTRATION: MODELLING SOLID MATRIX DEPOSITION IN CERAMIC-CERAMIC COMPOSITES
Department
G. Y. CHUNG, B. J. MCCOY and J. M. SMITH’ of Chemical Engineering, University of California, Davis, CA 95616, U.S.A and D. E. CAGLIOSTRO
and M. CARSWELL
NASA, Ames Research Center, Moffett Field, CA 94035, USA
(First receined 14 December
19X9; accepted in revisedform
15 June 1990)
model is developed to predict the effect of diffusion, the rate of deposition, and spatial ufa solid rnatrin irr a woven fabric forming a ceramic-r;eramic composite. To the knowledge of the authors there has been no prior treatment of chemical vapor infiltration in a system of plies consisting of an assembly of woven tows containing bundles of filaments. The model predicts the times to fill the gaps around the filaments of a tow, the space between plies, and the holes between the tows. It also predicts the porosity of the composite as a function of position and processing time. Predictions are made for an illustrative case of deposition of silicon carbide matrix within a woven carbon fabric. Abstract-A
diskibulion
INTRODUCTION
Ceramic fiber-reinforced composites, formed by the deposition of a solid within the void spaces in layers of woven fabrics, are resistant to oxidation and high temperature and have a low density compared to metals. This process of chemical vapor infiltration (CVI) requires different deposition conditions from those of conventional chemical vapor deposition (CVD). It is desirable for the deposit (e.g. Sic) to fill the void spaces as uniformly as possible to maintain good mechanical properties. The main difficulty in CVI is to avoid premature closing of the access to interior plies. Generally CVI requires deposition temperatures and total pressures much lower than those used in conventional CVD (Naslain et a[., 1983). van den Brekel et al. (1981) developed a model for the deposition of Ni, TiN and TIC on the inner wall of steel and ceramic tubes assuming a first-order deposition reaction. Later, Rossignol et (II. (1984) studied the formation of TiC from titanium chloride and methane in porous carbon media using the van den Brekel model and assuming that the porous region is equivalent to a cylindrical pore. Starr (1987) modelled short-fiber preforms with the assumption of a constant deposition thickness throughout a unit cell. The goal was to predict changes in composite structure with deposition. Jensen and Melkote (1989) also modelled CVI of short-fiber preforms. The influence on the final composite properties of operating conditions and structural parameters was studied. Tai and Chou (1989) studied the growth of alumina within one Sic-fiber bundle in an isothermal, hot-wall reactor
‘Author
to whom correspondence
should be addressed. 723
where the direction of gas diffusion is parallel to the axis of the fiber. They also simulated the pore space between the fibers as cylindrical capillary tubes. Recently, Sotirchos and Tomadakis (1989) studied the densification of fibrous substrates using Monte Carlo simulation methods. Monte Carlo simulation methods were employed to determine the variation of the structural properties of the fibrous structure and of the effective diffusivity. They proposed optimal reaction temperature and pressure histories for obtaining uniformIy densified composites. To our knowledge there has been no treatment of CVI for a substrate consisting of fabric or plies where each ply is an assembly of woven tows containing bundles of filaments. Sic is relatively inert and stable, and has good oxidation resistance and mechanical properties at high temperature. In conjunction with high-strength fibers it can be used to make high-temperature structural composites. Our objective was to develop a model for the in-depth CVD of SIC from dichlorodimethylsilane (DDS) into layers of woven carbon fabrics. It is regarded that DDS first decomposes to chlorosilanes and methane in the presence of an excess of hydrogen. The chlorosilanes are subsequently reduced to silicon metal, which reacts with methane to form Sic (Choudhury and Formigoni, 1969; Graul and Wagner, 1972; Sherman, 1972; DudukoviC, (1983). The model given here assumes that the overall deposition kinetics are first-order in DDS. The specific objective is to predict the overall rate of deposition as well as how the dimensions in the sample change with time, including amount of void space left around the filaments and between plies as a function of geometry and processing conditions.
724
G. Y. MODEL
CHUNG
DEVELOPMENT
Ceramic-ceramic composites can be fabricated in various types of CVI reactors. The model proposed here is for infiltration of Sic in a woven fabric preform in the reactor configuration shown in Fig. 1. The concepts could be applied to other geometries involving mass transfer in various subsections of the preform. For the arrangement in Fig. 1, boundary conditions at locations 1 and 2 are required as input data. These may be estimated from appropriate internal reacting flow codes or experimental measurements (Jensen, 1984; Moffdt and Jensen, 1988; Gokoglo er al., 1989). The substrate geometry assumes a multi-layer construction. Figures 2 and 3 show a sample consisting of 12 layers (plies) of woven fabric. In the plies each tow is assumed rectangular in cross-section and is a bundle of nonporous filaments woven to form approximately square holes. The plies are spaced so that the holes in every ply are aligned above each other. This forms a continuous straight hole through the plies as indicated in Fig. 3. The filaments within a tow are assumed to be arranged on triangular centers. Diffusion of gaseous DDS occurs from both ends to the center of the sample; through the holes, through the space between plies, and through the gaps around the filaments. The terms hole, space and gap are used to identify the different void regions of the sample (Figs 2 and 3). Because of symmetry, the model is considered for only half of the sample. Numbering of spaces and plies begins from the middle space and the middle ply (Fig.
3). Calculations
show
that
b
flow
b
flow
reactor wall Fig. 1. Reactor
configuration.
,tow
h
b P Y (b)
Details
plies
f,larnent
d,ff;s,an (a) Sample
aYer
of two
(C) Part of one ply
of 12 pies
considered
Fig. 2. Schematic
drawing
in the
of the woven fabric
modelhng
preform.
significant
deposition in the holes does not occur on the hole walls (edge of ply) until the gaps and spaces are filled. Hence, deposition can be considered sequential, occurring first on the outside lateral surface of the cylindrical filaments of the tow, and later on both sides of the flat surface of the ply and on the edge of the ply after the gaps around the outer filaments are filled. For the development of the model, it is assumed that the sample is isothermal and diffusion is onedimensional. Here, one-dimensional diffusion implies that there is no concentration gradient in the x- and y-directions in the hole, no concentration gradient in the y- and z-directions in the space between plies, and no concentration gradient in the x-direction in the gaps around filaments. It is also assumed that the
I
et al.
Fig 3. Notation numbering
diffusion velocity in the void regions is large with respect to the rate of change of the thickness of the deposited SIC. This assumption is generally valid when the density of the solid deposit is large with respect to the density of the diffusing gas (Bischoff, 1963). With this assumption, gaseous diffusion in the voids can be treated as a steady-state process. As deposition occurs, the concentration profiles in the void regions change as the porosity and surface area change. In addition, it is assumed that there is no diffusion from the holes into the gaps around the filaments along the edge of the ply. Rather, this region of the hole wall is considered to be a flat surface on which deposition occurs, reducing the dimension of the hole. The model is developed by solving mass balance equations for DDS for the three parts of the system (holes, spaces and gaps) and equations for the change with time in the dimensions of the system. Mass
-I
for dimensions and dimensionless variables, of spaces and plies, and coordinates.
balance
in the holes between
tows
The hole can be considered as two parts; one part is the hole wail along the edge of the ply through the distance c,, where deposition occurs. The other part is the open space between plies of distance h,, where
725
Chemical vapor infiltration diffusion in the x-direction between plies takes place (Fig. 3). Deposition starts on the hole wall, along distance c,, after filling the gaps around the outer filaments. It is assumed that these gaps are filled only by diffusion in the z-direction from the space between plies. Since the width of a tow is much greater than its thickness, this assumption should not introduce much error. Calculations indicate that at a given time the amount of deposition from the holes is only 9% of that into the filaments from the spaces. Therefore, the mass balance equation for the hole wall along the edge of the ply can be written as follows. Before filling gaps:
and Nz~,n,mlz~=~ = diffusion flux into the gaps on the lower side of ply n
.
aG*,.,nl
eg
af
In eq. (7), 9 is the width of diffusion as shown in Fig. 3. This dimension with x as follows: g=
(9)
*‘=O
area in the space changes linearly
-2x+a,+d,.
(10)
After filling the gaps around the outer filaments, deposition occurs on the flat surfaces of the upper and lower plies. The mass balance equation then becomes
LnDes
, 1 agac,,,,, _ gax ax
a2~,~ dX'
2kC
X.“,lfl
=
(11)
After filling gaps: & bc
Boundary
a2c, ,ky-4kC,=0 at z = h/2
C, = C,
a2C. aDmk L -
conditions
4Nx,nix=d,/Z
=
C3Z2
(4)
0 ‘
(ac,/az) = 0
&= ax
(3)
atz=O
0
c x,n,m = CZIr=(n-l)@,+c,)
Mass
balance
d2C i’,“.rn.l
In the hole, either eq. (1) or (2) and eq. (4) are applied alternately along the z-direction through the multiple-ply system. As deposition goes on, the open space between plies of distance bn,M,+l in the hole becomes smaller and the hole wall, through the distance c,, becomes larger. Hence, eq. (2) is substituted for eq. (4) for an increasing distance c, as the space between plies decreases. Mass balance in the void space between plies Deposition on the flat surface of one ply starts after the gaps around the outer filaments are filled with deposit. Before these gaps are filled, the mass balance in the spaces between plies includes diffusion on both sides of a ply. Hcncc, the mass balance before the gaps are filled is
~2CXnm
1agaC,., + --A g6x ax + N,,.n.Ji,=o)
where N r’.n--l.m z,=c* = diffusion the upper side of ply n - 1
(13)
= 0
(14)
G.n,m.~ = C,,,,,
at z’ = 0
(15)
C L .“.!?I.,- C,,,,,.,
at zi = c,
(16)
D,
- (N,~,n-l,ml,~=c,
at x = dJ2.
= 0
(7)
flux into the gaps on
around the filaments occurs on the lateral outside filaments. The mass balance
in the gaps
In the gaps, deposition surface of the cylindrical equation is
x --d,,z
i
(12)
Initially, eq. (7) is applied everywhere in the space. Then, it is replaced with eq. (11) for the element of distance x where the gaps around the filaments at the outer sides of the plies are filled.
(6)
NX,.L
T
atx=O
(5)
is the diffusion flux into the space where ~x,nlx=d,~2 between plies at the entrance from the hole and can be written as
b,,,D,,
for eqs (7) and (11) are
(2)
aZ
where a is the dimension of the square hole. The mass balance equation in the hole along the distance b, is
bc
0.
>
bcs
SZ’2
- kG,,.,,fs
where s is the surface area for deposition of SIC which is expressed as the sum of the lateral surface areas of all cylindrical filaments in unit volume (W = number of filaments per unit cross-sectional area in the tow): s = 27cr.,,,,[ W. Equation
(14) is applied
(17)
until the gaps are filled.
Changes of dimensions with time The DDS concentration profiles in the voids change with dimensions of the system, i.e. the sidelength of the hole, distance between plies, thickness of the ply, and radius of the filament. The dimensions change with time due to deposition, and are expressed by the equations that follow. The rate of change of the dimension of the hole is aa
-
at’
ic
a = a,
ZYM”AC, pm
at t < t,.
(18) (19)
G.Y.
726
CHUNG
For the hole wall along the edge of the ply, t, is the time required to plug the gaps around the outer filaments at the edge of a ply. For the open space of the hole along the distance b,, t, is the time required to plug the space between plies at the entrance from the hole. The equation for the change of distance between plies is
abn.m _ _ 2qM”kC at
Pm
b,,, = b,
ic
(21)
ic
ah+ ,,m at
(22) Ix=d,,2
at t < t tl,n.hf,+1.
c, = c,
arm,,= Lx
ic
qM
~kC,~,,,,,r
w=
r,
number
of filaments
(27)
The first-order reaction rate constant, k, was estimated from the experimental deposition data of Carswell (1989). The walls of the plies are not flat surfaces since the tows are woven to form the plies. The effective diffusion coefficient in the uneven space between plies was estimated
using the tortuosity
factor $:
The effective diffusion coefficient in the gaps around the filaments was estimated from the porosity and tortuosity factor, K, according to the equation (Smith, 1970; Froment and Bischoff, 1979)
(24) D,,
at t = 0.
in a tow
0,
(23)
(25)
The method of solution of these equations is first to calculate the initial concentration profiles in holes, spaces between plies, and gaps around the filaments from eqs (i)-( 17) using the initial dimensions of the system. Then, new dimensions are calculated using eqs (18)-(25) with the calculated concentration profiles. This procedure is repeated for successive time increments to yield concentration profiles and dimensions of the system as a function of time. From these results, deposition amounts and rates and porosities can also be evaluated. The dimensionless forms of eqs (l)-(25) are in the Appendix. They were solved numerically in finitedifference form with 400 nodal points in the q-direction, 10 nodal points in the c-direction, and eight nodal points in the c-direction. The previous time-step values of Y were used in eq. (A4). Similarly, the previous time step values of X were used in eq. (A6). A time increment of 2 s was used up to 10 h processing time and 2 min thereafter. All the calculations were done with a Vax 11-785 computer and took about 20 mincpu time for a typical run. RESULTS
of filaments in a unit cross-sectional to the axis of the filaments, W, is estimated using the total number of filaments in a tow and the dimensions d, and c,:
Pm
T,,,,~ =
(26)
)
Again tb,n,M,+1 is the time required to plug the gaps around the filaments at the outer surface of a ply. The rate of change in the radius of a filament is given by the mass balance: 2 *
h-2Nc b, = 0 2N - 1
(20) area perpendicular
at t G tb.“+
+
distance between holes, d,; the radius of one fiIament, r,; the thickness of one ply, c,,; and the thickness of the whole system, h. Also the number of plies in the system, 2N, must be known. The distance between plies, b,, can be calculated using h, c, and N:
The number
x,n,m
to plug the gaps where tb,n,m is the time required around the filaments at the outer surface of a bundle of filaments (tow) at position x. Note that these gaps at the outer surface may be plugged while voids still exist around the inner filaments in the hundle. The equation for the change in thickness of a ply can be expressed in terms of the change in distance between plies: ab, m dc,_ ___ 1 2 at 2 ( at Ix=d,/l
et al.
AND DISCUSSION
Parameter values The following initial dimensions are needed for the calculations: the dimension of the square hole, a,; the
=
:Dmk. K
(29)
Since the porosity and tortuosity are changing with time due to deposition, D,, is a function of time. According to the random pore model of Wakao and Smith (1962), the tortuosity factor is inversely proportional to Lhe porosity. Using this proportionality:
where K, is the initial tortuosity eq. (30) into eq. (29) gives
factor.
Substituting
(31) With the assumption that, on average, a pore makes an angle of 4.5” with the radial direction of the cylindrical filament, $ was used as the initial value of the tortuosity factor. The local porosity in the ply is given by E = 1 - 71l.2w.
(32)
Since the dimensions of the voids may be very small, the temperature high, and the pressure low, Knudsen diffusion can be significant. Hence, Dmk is taken as a composite diffusivity given by the equation 1
-=-$D mk
1 D,
1 4
(33)
Chemical vapor infiltration
727
where D,, the molecular diffusion coefficient, is estimated from the Chapman-Enskog formula (Smith, 1970) and D,, the Knudsen diffusion coefficient in cm2/min, is calculated from the equation (34) where Tis temperature in K, MA the molecular weight of DDS in g/mol and R the average void opening in cm. R was taken as half the side-length (a) of the hole and as half the distance (b) for the space between plies. Within a tow the filaments initially are close together but not in contact. Hence R for the gaps was evaluated in the following way. First there will be some parts of the gaps around filaments that never can be filled with deposit. These regions are due to isolated volumes formed when there is sufficient deposit to bring the filaments in contact with each other. This porosity due to geometry is 0.093. With this porosity, the radius r,, of the filaments at the time that the gaps are plugged is obtained from eq. (32). Then, half of the least distance between filaments is the difference between ry and r, i.e. (rl - r). Then, an approximate value of R is taken as an average of r,, and (rP - r). Results for a specific set of initial dimensions and a range of diffusivities and rate constants are shown in Figs 4-l 1. There are three dimensionless variables, 4, i and cp. corresponding to Thiele modulus numbers. However, these three variables all change with a change in D, or k. Corresponding changes in D, or k that leave these variables unchanged yield different deposition curves. The parameter values listed in Table 1 are based upon the geometries derived for a composite sample fabricated at the NASA Ames Research Center. Comparison
of calculation
t
Concentration vs time in the gaps around the filain the first ply at point (A) (Fig. 3; x = 0, z’ = 0.5~~) for diffusion coefficients and reaction rate constants. D, = 17,300 cm’/min, k = 10 cm/min
1.c
0.8
0” ._0 E 0
04
oz0
20
time
methods
As mentioned earlier, the prior time step values of Y in eq. (A4) and X in eq. (A6) were used for the finite-difference calculations. This reduces the calcu-
10,
’ 60
40
.
80
[hr]
Fig. 6. Concentration YS time in the second space between plies at point (B) (Fig. 3; x = 0) for different diffusion coefficients and reaction rate constants. D, = 17,300 cm’/min, k = 10 cm/min.
,
1 .’O\ \
0.84 \
u” 0. 6‘N
0
0 4’1
0.31 0.0
0.2
I 04
’ 06
08
’ 1.0
“/(d&2)
Fig. 4. Concentration profiles in the first space between plies (Fig. 3) at 5 h for different diffusion coefficients and reaction rate constants. D, = 17,300 cm*/min, k = 10 cm/min.
0. 2’
0
10
20
30
time
330
630
do.2 930
[hr]
Fig. 7. Concentration vs time in a hole at point (C) (Fig. 3; z = 0) for different difTusion coefficients and reaction rate constants. D, = 17,300 cm’/min, k = 10 cm/min.
G. Y.
728
$ ::
-
0.6-
CHUNG
et cd.
0.6
v)
$ L
= c" E o 0
_ _.
_
_
- 0.9 0.4r..._..:::::::~~~~~~----..... Dm.Ik --. --Q..___ --.._.___ --._ ..I__ *x_ Ddk‘.... -1.2 0.2*\.. .\,
0
d
'... o,. 00
I
I
0.2
0.4
0.6
,
"-,,5
0.8
1.0
x/(do/2)
Fig. 8. Distance between plies and deposition along the x-direction in the first space (Fig. 3) at 20 h for different diffusion coefficients and reaction rate constants. D, = 17,300 cm2/min, k = 10 cm/min.
8
0.6
z g I
'Om.2k‘ --__._;
;
1.2
time
Fig. 11. Rate of deposition per unit cross-sectional area (csa) of sample vs time for different diffusion coefficients and reaction rate constants. D, = 17,300 cm’/min, k = 10 cm/min.
lations to solving a tridiagonal set of equations. This simplification was checked with the Gauss-Seidel iteration method avoiding the use of values for the prior time step. No significant differences between the two computation methods were found. The difference became smaller for larger diffusion coefficients. In calculating the concentration profile in the space between plies, it was assumed that concentration gradients
04
18
02
2.4
[hr)
in the y- and
z-directions
are
independent.
steady-state calculations in the space between plies were carried out for one- and two-dimensional geometries: One-dimensional: To check
this assumption,
Dnlg
(35)
z/(h/2)
with
Fig. 9. Dimension of a hole and deposition thickness along the z-direction at 200 h for different diffusion coefficients and reaction rate constants. D_ = 17.300 cm”/min,
y = x + a,/2
Two-dimensional:
bcs
C, = C, at the side line of the hole
(37)
ac,jax=o or aC,/ay
= 0 at the center
of the space.
(33)
The dimensionless forms of these equations were solved separately by the iteration method. Even if the shape of the concentration contour lines is different in the region far from the hole, the concentration profiles along the y-axis show small differences (less than I %). The rate of deposition results in Table 2 also showed little difference, which implies that the one-dimensional approximation is reasonable.
time
[hr]
Fig. 10. Porosity of the whole sample of 12 plies vs time for different diffusion coefficients and reaction rate constants. The porosity at time zero is 0.706. D, = 17,300cm2/min, k = 10 cm/min.
Eflects of d$iision coeficients and constants Concentration profiles. Concentration space between plies are displayed in specific parameter values given in Table are shown concentration changes
reaction
rate
profiles in the Fig. 4 for the 1. In Figs 5-7 with time in
729
Chemical vapor infiltration Table 1. Parameter values used to illustrate the model a, = 0.06 cm,
b, = 0.003 cm, i-, = 0.0004 cm, h = 0.321 cm, Number of filaments in a tow = 3000 Deposition
conditions:
c, = 0.024 cm, N = 6,
10 tarr total pressure,
C, = 5 x LOP mol/cm3, p, = 3.217 g/cm3
d, = 0.18 cm W= 695,000cm-2
9OOT, and 3.75% DDS in hydrogen k = 10 cm/min
D, = 17,300 cm2/min,
Table 2. Rate of deposition pet unit cross-sectional area of sample in the space between plies in the two- and one-dimensional calculations+ Rate of deposition [ x lo6 g/(h cm’)] D,,k
0.1 D,,k
O.O5D,,k
D,,2k
D,,0.5k
Two-dimensional calculation
3.935
3.851
3.761
7.732
1.985
One-dimensional calculation
3.95
3.877
3.804
7.777
1.989
+D, = I7,300 cm2/min, k = 10 cm/min.
the gaps around the filaments at point (A) (z’ = O.k,, x = 0), in the space between plies at point (B) (x = 0), and in the hole at point (C) in Fig. 3 (z = 0) for different diffusion coefficients and reaction rate constants. While the concentration gradients in the gaps around the filaments are small, gradients in the space between plies (Fig. 4) are relatively large. These large gradients are due to the large flux into the gaps around the filaments before they are filled. For instance, for the parameter values of AI,,,and k given in Table 1 and at 5 h, the concentration difference (0.87 - 0.61) between the entrance and the middle of the space between plies is about one-quarter of the initial concentration (i.e. 0.26C,). This concentration difference is reduced to O.O4C, at 20 h when the gaps around the outer filaments are filled. The same phenomenon occurs in the holes, i.e. the concentration gradient is large when the spaces and the gaps are not plugged and is small after these regions are plugged. The concentration in the gaps around the filaments (Fig. 5) changes with time due to the concentration change in the spaces between plies (Fig. 6). The curves in Fig. 6 end when the gaps in the outer filaments are plugged. After this time the gaseous silane concentration in the very small voids around the remaining filaments will decrease to zero. The corresponding amount of deposition is negligible. As the flux into the plies becomes larger due to the increased surface areas of the filaments for deposition (as the radius of the filaments grows), the concentration in the void spaces between plies decreases. As a result, the concentration in the gaps around the filaments decreases. Later, as the flux into the plies is reduced due to plugging of some of the gaps around the outer filaments, the concentration in the spaces between plies increases. Hence, the concentration in the remaining gaps increases. After plugging of all the gaps around the
outer filaments, the concentration in the spaces between plies decreases monotonically with time due to plugging. Since the concentration in the holes is already high (see Fig. 7) when the gaps around the filaments are plugged, the decrease in flux into the spaces between plies does not significantly affect the concentration profiles in the holes and spaces. Therefore, the concentration between spaces decreases monotonically due to the decrease in the distance between plies. When the diffusion coefficient decreases, the concentrations throughout the system are smaller, as expected (Fig. 4). It is also expected that higher reaction rate constants would lead to lower concentrations. However, the concentrations in the spaces between plies at 4 times the rate constant (4k) are higher than those at 0.5k, k and 2k because the gaps are plugged. It is shown in Fig. 5 that the gaps around the outer filaments at 5 h are already closed for 4k. The steeper concentration profile for 2k in Fig. 4 is due to the greater flux into the plies. This increased flux is due to the larger surface area for deposition around the filaments. Thus, the model demonstrates that the process at the first ply (near the center of the sample) is affected by what happens at the last ply. Plugging occurs first in the gaps around the filaments, next in the spaces between plies, and finally in the holes. For instance, for the specific conditions of k and D, in Table 1, Figs 5-7 show that plugging of the gaps occurs at 10 h, plugging of the spaces between plies occurs at 48 h, and plugging of the holes occurs at 808 h. For lower diffusion coefficients and lower reaction rate constants longer times are required to fill the gaps and spaces. As mentioned earlier, as the gaps around the filaments and the spaces between plies are closed, concentrations in the hole increase. Since closing of the spaces between all
730
G.Y.
CHUNG
the 12 plies occurs almost at the same time for a high diffusion coefficient, the concentration in the holes increases sharply as shown in Fig. 7. However, for a low diffusion coeficient, the increase is slow due to the step-by-step closing of the space between plies. The variation in slope for the concentrations in the gaps for O.lD, and O.O5D,, as shown in Fig. 5, is due to plugging of the gaps around the outer filaments in the spaces between plies. Amounts of deposition and porosities. The model predicts the changes in dimensions of the sample as a result of deposition. Results are illustrated in Figs 8 and 9. The deposition thickness between plies is expressed as (b, - b)/2, where h, and 6 are the distances between plies initially and at any time, respectively. The deposition thickness in the holes is expressed as (a, - a)/2, where a, and a are the dimensions of the hole initially and at any time. The deposition thickness in the spaces (Fig. 8) is nonuniform due to the large concentration gradients shown in Fig. 4. On the other hand, the deposition profile through the holes at 200 h (Fig. 9) is essentially uniform except for a high reaction rate constant, 4k. The amount of deposition was calculated for each part of the system as follows: In the gaps around the filaments:
47~~~ WTAx
AZ’ 5 F i=l j=l
In the spaces 4~mTAx
between
F gj(b,/2 [ j=j
plies:
2’
(r&
- r2)gj
(39)
k=l
j=, 1
- b1.j) + ; y ;=2
gj(b, - bi.i)
(40)
In the holes: 2p,HAz
5 (a,’ - a?) (41) i=l The porosities of the whole system and each ply were calculated using the amounts of deposition for each part of the system. The porosities of the whole system are shown in Fig. IO. The discontinuous points for the rate of change in porosity in Fig. 10 show the times that each part is plugged. As expected, a more dense deposition (lower porosity) is obtained for a low reaction rate constant and a large diffusion coefficient. This is because the concentration gradients are smaller. Rate of deposition curves calculated from the amounts of deposition at each time step are displayed in Fig. 11. The section of the curves at low times gives the deposition rate before the gaps around the outer filaments are fiIled at any position (x) between plies. At O.lD, and O.OlD, the increase in deposition rate due to surface area increase is less than the decrease due to concentration decrease. The concentration decreases because the area for diffusion decreases. Hence, the resuItant rate of deposition decreases. However, at D,, the rate of deposition increases because the increase in rate due to surface area
et ul.
increase is larger than the decrease due to concentration decrease. When the gaps around the outer filaments are plugged at some positions x, the total rate of deposition decreases due to the decrease in the total deposition area. After plugging the gaps along all the distance x, the deposition rate decreases slowly with the monotonic decrease of concentration in the spaces between plies, as shown in Fig. 6. To check the importance of diffusion resistance in the sample, calculations were made for the condition that the concentration in all the void regions is a constant
equal
to the boundary
value,
C,.
The
results
showed that for the reference values, D, and k, the porosity was about 7% less when diffusion resistance was neglected. An empirical relation has been proposed to describe deposition (Rossignol et al., 1984). The equation representing this relation is log [(M, - M,)/M,]
= - Wt
(42)
where M, is the initial pore volume multiplied by the density of deposited material, M, is the weight of deposited material, and w is a constant which is a function of temperature and initial porosity. As shown in Fig. 12, plotting log(M, - M,)/M, vs time shows linear segments, except at long processing times when the holes are becoming plugged. The resulting approximate values of w from our data are given in Table 3. Compared to the values of 2 x 1O-3 and 4.5 x 10-j h- ’ for carbon
of dimensions
of sample
The effects of initial distance between plies and size of the holes are shown in Figs 13 and 14. Changing the size of the holes resulted in different times for the holes to be plugged. Decreasing the size of the holes also resulted in more area for deposition to occur between plies, where the deposition is nonuniform. Furthermore, while the dimension of the hole does not signi-
-0 2
T z. -0.4 % I i =p) -0.6 0
-0 a
50
100
150
200
time
400
600
800
[hr]
Fig. 12. Log [(M, - M,)/M,] vs time for different diffusion coefficients reaction and rate D, = constants. 17,300 cm2/min, k = 10 cm/min.
Chemical 3. w values in eq. (42) for different
Table
vapor
infiltration
731
diffusion coefficients and reaction rate constants’ W( x lo3 h-‘)
O.lD,,k
D,,k In the gaps
42
O.O5D,,k
O.OlD,,k
2.5
15
30
In the spaces
4
3.55
3.01
1.59
In the holes
0.2
0.18
0.17
0.13
‘D,
D,,4k
D,,2k
117
70
17 0.55
D,,0.5k 27.5
8.02
2.29
0.36
0.17
= 17,300 cm2/min, k = IO cm/min.
0.5
0.4
.g cn
0.3
2 g
02
0.1
I 300
600
lime
900
0.0
1200
L
J
0
200
[hr]
600
400
time
600
[hr]
Fig. 13. Porosity of the entire sample of 12 plies YS time for different dimensions of the holes. The porosity at time zero for a, is 0.706 (a, = 0.06 cm, b, = 0.003 cm).
Fig. 14. Porosity of the entire sample of 12 plies vs time for different distances between plies. The porosity at time zero for b, is 0.706 (a, = 0.06 cm, b, = 0.003 cm)
ficantly affect the concentration in the hole, decreasing the hole size decreases the area for diffusion into
plies determines the total height of system. The local porosities when all areas are plugged are given in Table 4. The porosity due to geometry (= 0.093) is the lower bound that is obtained in case of uniform deposition. The porosity due to nonuniform deposition is obtained by subtracting the porosity due to geometry from the total porosity in the gaps. From Table 4, as the distance (b,) between plies increases, the local
the space between plies. Hence, a large concentration gradient results and even greater nonuniformity of deposition in the space between plies and in the gaps around the filaments. This leads to a higher porosity at the time of plugging the holes, even if this time is shorter. The distance between plies and number of
Table 4. Porosities
of each part of the sample at the time of plugging of holes for different distances between plies’
dimensions
of holes and
Porosity
aoh, In the gaps
Due to nonuniform deposition Total*
.%,,b,
0.5a,,b,
a,,%,
a,,2b,
a,,0.5b,
0.042
0.038
0.05 1
0.032
0.033
0.056
0.135
0.131
0.144
0.125
0.126
0.149
In the spaces
0.142
0.133
0.161
0.047
0.07 1
0.287
In the holes
O.OC06
0.0012
0.0035
0.0027
0.0011
O.OCKM
In Ihe whole sample
0.127
0.117
0.143
0.098
0.109
0.147
i
‘The individual porosities are defined as the void volume in each part divided by the total volume of that part. The results are for reference conditions (D,,k) and where a, = 0.06 cm and 5, = 0.003 cm. The initial porosities (based upon the total sample volume) for the reference case (a,,b,) are 0.547 in the gaps, 0.096 in the spaces and 0.063 m the holes. ‘Porosity due to geometry is 0.093.
732
G. Y.
CHUNG
porosities in the space between plies decrease in almost the same ratio. This means that the final pore volume in the spaces does not change much even if the total volume of the system increases. Hence, the porosity in the spaces between plies based on the total volume decreases. This decrease of porosity due to increase of total volume is greater than the increase of porosity due to the increase of nonuniform deposition in the spaces between plies. Therefore an increase in the distance between plies reduces the final porosities (Fig. 14).
et
ai.
k M, M, M, M,* N N, N,,
CONCLUSlONS
A model is proposed for CVD within layers of woven carbon fabrics. The model predicts concentration profiles in the system, amount and rate of deposition, and changes of dimensions with time. The results suggest sequential plugging of the gaps around the outer filaments, the spaces between plies, and finally the holes. The concentration in the gaps around the filaments first decreases due to the increased surface area for deposition and then increases due to the gradual plugging of the gaps around the filaments. Predicted final porosities are lower if the diffusion coefficient is large and the reaction rate constant is low. A comparison of the final porosities with and without diffusion resistance also shows, as expected, that porosity increases substantially if diffusion resistance is significant. For the diffusion coefficient, the reaction rate constant and geometries tested, increasing the size of holes and the distance between plies gives lower final porosities (higher sample densities). Acknowledgement-This work was supported Cooperative Agreement NCC 2-590.
by NASA
4s W
X
Y
(CYlC,) coordinate in the direction of diffusion in the space between plies, cm reduced concentration along the x-direction
Z
(C,/C,) reduced
x
NOTATION
a
side-length of the square hole, cm b distance between plies, cm concentration of DDS at the outside of the co sample, mol/cm3 concentration of DDS along the x-axis in the C, nth space between plies, mol/cm3 concentration of DDS along the z-axis in the c, hole, mol/cm’ concentration of DDS along the z-axis in the gap, mol/cm3 C thickness of one ply, cm effective diffusion coefficient in the gap around De, the filaments, cm’/min effective diffusion coefficient in the space beD, tween plies, cm’/min Knudsen diffusion coefficient, cm2/min 4 molecular diffusion coefficient, cm’/min D, D rnL composite diffusion coefficient, cm’/min d distance between holes, i.e. width of tow, cm width of diffusion area in the space between 9 plies, cm H number of holes in unit cross-sectional area, cm-’ h height of the sample, cm
first-order reaction (deposition) rate constant, cm/min molecular weight of deposited Sic, g/mol total number of divisions in the x-direction in the space (d,/2Ax) total number of divisions in the z-direction in the hole (h/2Az) total number of divisions in the z’-direction in the gap (c,/Az’) half of the total number of plies flux ofdiffusion in the x-direction, mol/cm’min flux of diffusion in the z’-direction, mol/ cm2 min number of deposited Sic molecule per molecule of DDS radius of average void opening between filaments, cm radius of a filament, cm surface area of the filaments per unit volume of the ply, cmz/cm3 number of spaces or plies of length d, in unit cross-sectional area, cm - 2 time, min time required to plug the gaps around the outer filaments at the hole-edge of a ply, or time required to plug the space between plies at the entrance from the hole at position z; f, = [f’(z)], min time required to plug the gaps around the outer filaments at position x; t, = [f(x)], min number of filaments per unit cross-sectional area perpendicualr to the axis of the filaments, cm-’ reduced concentration along the z’-direction
z z’
Greek
concentration
along
the
z-direction
(C,/CJ coordinate in the direction of diffusion hole, cm coordinate in the direction of diffusion gaps around the filaments, cm
in the in the
letters
reduced side-length of the square hole (a/a,) reduced distance between plies (b/b,) reduced thickness of the nth ply (c/c,) reduced width of the diffusion area in the space between plies (g/d,) local porosity of a ply reduced coordinate in the z’-direction (z’/c,) reduced coordinate in the z-direction [z/(/1/2)] tortuosity in the gaps around the filaments dimensionless parameter [d,(k/2b,D,)‘~2] dimcnsionlcss parameter [h(2/a,d,)1’Z] dimensionless parameter [(d,/2)( l/b,c,)“*] reduced coordinate in the x-direction [x,$&/Z)] density of the deposited material, g/cm3
Chemical
vapor
reduced radius of a filament (r/t-J dimensionless time [(2qM,kC,/a,p,)t] dimensionless time required to plug the gaps around the outer filaments at the hole-edge of a ply, or time required to plug the space between plies at the entrance from the hole at position 4; z, = C_f(a)l CGWfmkCo/wmb.l dimensionless time required to plug the gaps around the outer filaments at position <; tb = ES(<)] [(%~,&/%L’,&~ dimensionless parameter [h(k/u,D,)1’2] dimensionless parameter [c,(2ar,‘Wk/D,)1i2] defined by eq. (42) Subscripts the Ith segment of a ply in z’-direction 1 the mth segment of a space or a ply in x-direction m the nth ply or space n zero time 0 plugging time P
733
infiltration
Tai, N. H. and Chou, T. W., 1989, Analytical modeling of chemical vapor infiltration in fabrication of ceramic composites. J. Am. &ram. Sot. 72, 414-420. van den Brekel, C. H. J., Fonville, R. M. M., van der Straten, P. J. M. and Verspul, G., 1981, CVD of Ni, TIN, and TIC on complex shapes. Proc. elecrrockem. Sot. 81, 142-156. Wakao, N. and Smith, J. M., 1962, Diffusion in catalyst pellets. Chem. Engng Sci. 17, 825-834. APPENDIX
dimensionless form eqs (1)-(25) are as follows: In the holes between tows: Hole walls: Before filling gaps: In
d2Z -_=O a?+
After filling gaps:
WI bc Open spaces
atq=l
Z=l
between
plies:
REFERENCES
BischotT, K. B., 1963, Accuracy of the pseudo steady state approximation for moving boundary diffusion problems. CRem. Engng Sri. 18, 711-713. Carswell, M., Personal communication. Choudhury, R. and Formigoni, N. P., 1969, Beta-Sic films. J. electrochem. Sot. 116, 1441-1443. Dudukovi?, M. P., 1983, Reactor models for CVD of silicon, in Flat Plate Soiar Array Workshop on the Science of Silicon Marerial Prepararion, J.P.L. Publication 83(13), 199-226. Froment, G. F. and Bischoff, K. B., 1979, Chemical Reactor Analysis and Design, pp. 167-173. John Wiley, New York. Gokoglo, S. A., Kuczmarski, M. A., Tsui, P. and Chait, A., 1989, Convection and chemistry effects in CVD: a 3-D analysis for silicon deposition, Paper presented at the EURO 7 Conference, Perpignan, pp. 19-23. Graul, J. and Wagner, E., 1972, Growth mechanism of polycrystalline beta-Sic layers on silicon substrate. Appl. Phys. Lett. 21(2), 67-69. Jensen, K F., 1984, Modelling of CVD reactors. Proc. ektrochem. Sm. 84(6), 3-20. Jensen, K. F. and Melkote, R. R., 1989, Chemical vapor infiltration of short fiber preforms. Extended abstract presented at the 1989 AlChE Annual Meeting, San
bc
aZj@
In the spaces between Before filling gaps:
atq=O
= 0
(A5)
plies:
and 6, = a,/d, +
I- 5
(A7)
After filling gaps:
(A81
ar.,,iat =0
bcs
at5=0 at 5 = 1
Yn.I = Z 1~= z(n - I)(b, + c.)/h In the gaps around
the filaments: (All)
Francisco, p. 54. Moffat, H. K. and Jensen, K. F., 1988, 3-D flow effects in silicon CVD in horizontal reactors. J. elecrrochem. Sot.
bcs Xn,,., = Yn.m
135,459-471. Naslain, R., Hannache,
H., Heraud, L., Rossignol, J. Y., Christin, F. and Bernard, C., 1983, Chemical vapor infiltration technique, Proc. 4th European Conf: on Chemical Vapor Deposition, pp. 293-304. Rossignol, J. Y., Langlais, F. and Naslain, R., 1984, A tentative modelization of titanium carbide C.V.I. within the pore network of two-dimensional carbon-carbon composite preforms. Proc. electrochem. Sot. l&4(6), 596-614. Sherman, A., 1972, Chemical Vapor Depositionfor Microelectronics, Principles, Technology, and Applications, pp. 67-69. Noyes, Park Ridge, N.J. Smith, J. M., 1970, Chemical Engineering Kinetics, pp. 414-419. McGraw-Hill, New York. Sotirchos, S V. and Tomadakis, M., 1989, Modelling the densification of fibrous structures, Extended abstract presented at the 1989 AlChE Annual Meeting, San Francisco, p. 135. Starr, T. L., 1987, Model for CVl of short fiber preforms. Ceram. Engng Sci. Proc. 8.951-957.
(A9) (Al01
at[=O
(Al2)
at c = 1
(Al31
X n.n.1 - Y.+,,, Changes
of dimensions
with time:
d”=_Z
(Al4)
d+
ic
a=1
at T $ z,
aS..,=_a”y_ dr
b,
’
(Al51 (Al61 (Al71
ah
_=
--
(Al8)
ar
(‘419)
ic
atr n.m.l
a,
dr
2r,
~“,rn.l = I
n,m*i at t = 0.
(A20) (‘421)