Emily S. NelsonGlenn Research Center, Cleveland, Ohio

Phillip ColellaLawrence Berkeley National Laboratory, Berkeley, California

Parametric Study of Reactive Melt Infiltration

NASA/TM—2000-209802

March 2000

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NASA/TM—2000-209802

March 2000

National Aeronautics andSpace Administration

Glenn Research Center

Prepared for the1999 International Mechanical Engineering Congress and Expositionsponsored by the American Society of Mechanical EngineersNashville, Tennessee, November 14–19, 1999

Emily S. NelsonGlenn Research Center, Cleveland, Ohio

Phillip ColellaLawrence Berkeley National Laboratory, Berkeley, California

Parametric Study of Reactive Melt Infiltration

Acknowledgments

The authors would like to acknowledge NASA Glenn Research Center and the Department of Energy for theirsponsorship of this work. The authors would also like to thank Jay Singh and Arnon Chait for their insightful

review of this paper.

Available from

NASA Center for Aerospace Information7121 Standard DriveHanover, MD 21076Price Code: A03

National Technical Information Service5285 Port Royal RoadSpringfield, VA 22100

Price Code: A03

NASA/TM—2000-209802 1

PARAMETRIC STUDY OF REACTIVE MELT INFILTRATION

Emily S. NelsonNational Aeronautics and Space Administration

Glenn Research CenterCleveland, Ohio 44135

Phillip ColellaLawrence Berkeley National Laboratory

Berkeley, California 94720

ABSTRACTReactive melt infiltration is viewed as a promising means

of achieving near-net shape manufacturing with quickprocessing time and at low cost. Since the reactants andproducts are, in general, of varying density, overallconservation of mass dictates that there is a force related tochemical conversion which can directly influence infiltrationbehavior. In effect, the driving pressure forces may competewith the forces from chemical conversion, affecting theadvancement of the front. We have developed a two-dimensional numerical code to examine these effects, usingreaction-formed silicon carbide as a model system for thisprocess. We have examined a range of initial porosities, poreradii, and reaction rates in order to investigate their effects oninfiltration dynamics.

NOMENCLATURERoman letterskr reaction rate constant

MW" species molecular weightp pressurer pore radiusR" rate of production/consumption of species "te i, time at which infiltration is complete

te r, time at which chemical conversion is complete

t3 time at which infiltration enters third, post-reaction-

dominated regime~uD macroscopic Darcy (front) velocity

v" species molar specific volumeV volumeyf location of the infiltration front

Greek lettersξ intrinsic infiltration rate

κ permeabilityη molar consumption rateφ " species volume fraction, V V"

Φ porosityΦs volume fraction of solid, Φ Φs = −1

σ surface tensionθ wetting angleµ fluid viscosity

Subscripts/superscriptsc capillaryf front" KPFGZ FGPQVKPI URGEKGU

BACKGROUNDIn reactive melt infiltration (RMI), a porous medium, or

“preform”, is first created, which is easy to form into a desiredshape. A liquid infiltrant is introduced into the preform througha pressure gradient caused by capillary action or by appliedpressure (see, e.g., Kochendoerfer, 1991; Hillig, 1994a, b;Haggerty and Chiang, 1990). The chemical conversionbetween the infiltrant and the preform results in a final product,which may be fully dense and is of the same dimensions asthe original preform (Singh and Behrendt, 1992).

In comparison to other methods of compositemanufacture, RMI is quick and efficient. Chemical vaporinfiltration requires days or weeks to complete the chemicalconversion process, but RMI needs only minutes or hours.Post-process machining is limited, making this processattractive for the manufacture of brittle and expensiveceramics.

Reaction-formed silicon carbide is chosen as a modelsystem for this study. The production of monolithic siliconcarbide system is appealing because the chemistry and

NASA/TM—2000-209802 2

preform geometry are reasonably well characterized. We willfocus on infiltration that is driven solely by capillary pressure.The relevant thermophysical properties of the chemicalconstituents are given in Table 1.

Table 1. Thermophysical properties of silicon, carbon,and silicon carbide.

Property Si C SiC

ρ (kg/m3) 2400 1500 3120

µ (kg/m⋅s) 5 x 10-4

MW (kg/kmol) 28.1 12.0 40.1

v" (m3/kmol) 0.0117 0.008 0.0128

Although the absolute viscosity of liquid silicon, µ, is aweak function of temperature (Turovskii and Ivanova, 1974),we will neglect any temperature dependence. Note that thespecific molar volume, v" , varies substantially among thespecies. In fact, the volume occupied by a mole of SiC isapproximately 35% smaller than that of a mole each of Si andC. Based on overall mass conversion, Singh and Behrendtcomputed that the critical porosity, Φc , for this process is0.358. For initial porosities below Φc , the flow will chokeprematurely because there simply is not enough volumeavailable to complete the conversion. If Φ Φ0 > c and all of thecarbon is converted in a fixed preform volume, the resultingceramic will be comprised of SiC and Si. In this study, we willinvestigate Φ0 in the range of 0.40-0.60, well above thecritical porosity.

In order to model the preform, we have drawn on theextensive work of Behrendt and Singh on the melt infiltration ofsilicon and silicon alloys into microporous carbon (seeBehrendt, 1986; Behrendt and Singh, 1993, 1994; Singh andBehrendt, 1992, 1994a, b, 1995). Their work was also thebasis for other experimental and analytical studies(Sangsuwan et al., 1998; Datta et al., 1996; Messner andChiang, 1990). The amorphous, glassy carbon preforms wereuniform and isotropic with good connectivity (Singh andBehrendt, 1992, 1994b; Sangsuwan et al., 1998). Moreover,they could be consistently reproduced by Behrendt andSingh’s process with pore diameters ranging from 0.1-3.0 µmand initial porosities from 0.33-0.64. We will examine poreradii ranging from 0.10-0.20 µm and initial porosities from0.40-0.60.

Infiltration takes place in an evacuated furnace. Withoutthe application of external pressure, the infiltration process isdriven by capillary action. Consequently, the infiltration rate isdependent upon the wetting characteristics of the infiltrantupon the preform surface, which is in turn sensitive to localtemperature and surface and melt impurities. At 1703 K, thecontact angle, θ, of liquid silicon on carbon is reported to be38° by Li and Hausner (1991). (See also Whalen andAnderson, 1974; Turovskii and Ivanova, 1974.) We will use avalue of 0.725 N/m for the surface tension, σ (Brandes, 1983),although other values have been reported (Minnear, 1982).

We will assume that the chemistry is reaction-controlledrather than diffusion-dominated, based on the work of Messnerand Chiang (1990) and Fitzer and Gadow (1984) so that thepore radii close by:

r t r k tr r( ) = −0 (1)

where r0 is the initial pore radius, kr is the reaction rate, andtr is the local reaction time, i.e., the amount of time that haselapsed since the reaction began in this region. Since thetemperature variation within the preform for this set ofparameters is on the order of 100°C (Nelson, 1998; Einset,1993), we will use a constant value for kr . Messner andChiang estimated the reaction rate to be 4x10–8 m/s, butbounded the estimate from 10–8 to 10–7 m/s. Alloyed melts maydecrease reaction rates as well as eliminate the residualsilicon phase (Singh and Behrendt, 1992, 1994a,b, 1995).Accordingly, we will investigate reaction rates in the range of10–9 to 10–7 m/s. In reality, the reaction rate is not a tunableparameter, and the effect on infiltration dynamics will vary ifthe chemistry is modified.

GOVERNING EQUATIONSA schematic of the two-dimensional numerical model of

reactive melt infiltration is sketched below in Fig. 1. Liquid Si isintroduced at the bottom of the preform and wicks up into thepreform due to capillary action.

Figure 1. Schematic of reactive melt infiltration.

The general form for the conservation of mass is given foreach of the species by:

∂∂

+ ∇ ⋅ = =t

u R Si C SiC( ) ~ , ,ρ φ ρ φ" " " " " "1 6 l (2)

NASA/TM—2000-209802 3

where R" is the rate of production/destruction of species " ,and φ " is the species volume fraction, i.e., the fraction of thevolume occupied by species " . The species volume fractionmust obey the constraints 0 1≤ ≤φ " and φ ""∑ ≤ 1. In anyregion that is completely filled, φ ""∑ = 1. Since overall massmust be conserved, the conservation equation for the totalmass in the system is subject to the constraint R""

=∑ 0 . Theliquid phase, comprised of a single component, i.e., Si, is theonly phase with a nonzero velocity.

We will model this process as a zeroth-order chemicalreaction. For stoichiometric conversion, this exothermicreaction is given by Si C SiC+ → , so that the local productionof species can be expressed as:

R MW" "= β η (3)

where β = +1 for the product and −1 for the reactants, η is thenumber of moles consumed per unit time, and MW" is thespecies molecular weight. We note that none of the speciescan be updated to a negative value. Since we are providedwith an abundant source of Si, the amount of C availablelocally will limit the chemical conversion. Therefore, when all ofthe C has been consumed locally, the chemical reactioncomes to a halt in that region.

We will make the critical assumption that the overallvolume of the preform remains constant (Singh and Behrendt,1992). Using the above assumptions, the overall conservationof mass can be written as:

∇ ⋅ = − −φ ηSi Siu SiC Si C~1 6 1 6v v v (4)

where v" " "= MW ρ is the specific molar volume.

The expression for η is found by relating the geometricdescription of the pore closure to that obtained from theconservation of mass. The volume fraction of the solid phaseis Φs C SiC= +φ φ . From mass conservation, the change in solidvolume fraction with respect to time is:

∂∂

=∂

∂+

∂∂

= −

Φ s C SiC

t t t

SiC C

φ φ

η v v1 6 (5)

If we envision the preform porosity to be comprised of abundle of parallel capillary tubes, then the permeability for thismodel is found by:

κ =Φr 2

8 (6)

where r is the pore radius and Φ is the porosity ( Φ Φs = −1 ).Experimental data (Datta et al., 1996) indicates that this is anadequate model for this preform. Comparing the cross-sectional area of pore space initially and at time n, we findthat:

∂Φ∂

=∂∂

−���

���

srt t r

r r tΦ0

02 0

2 2( )4 9 (7)

For details, see Nelson, 1998; Nelson and Colella, 1999a,b.Recall that r t r k tr r( ) = −0 , where r0 is the initial pore radiusand tr is the relevant reaction time. Combining Eqs. (5) and(7) gives the molar consumption rate as:

η = −���

���−

21

00

0v vSiC C

k

r

k

rtr rrΦ (8)

At the pore level, the convective time scale is much longerthan the diffusive time scale. Therefore, we can use d’Arcy’slaw to describe the motion of the front:

~~

u pD = − ∇κµ

(9)

where ~uD is the front velocity, κ is the permeability of theporous medium, µ is the absolute viscosity of the fluid, and pis the pressure. For this system, κ is a function of space andtime. Furthermore, any body force term is negligible incomparison to the pressure force provided by capillary action.

We can relate the fluid velocity, ~uSi , to the front velocity,~uD , by the Dupuit-Forcheimer relation, i.e., ~ ~u uD Si Si= φ . We donot explicitly know ~uD or the pressure field; however, ouranalysis of continuity provided a relationship for thedivergence of velocity in Eq. (4). Combining the divergence ofd’Arcy’s law, Eq. (9), with Eq. (4), we arrive at a steadyPoisson problem for pressure that stems from theconservation of momentum and of mass:

∇ − ∇���

���

= − −κµ

ηp SiC Si Cv v v1 6 (10)

Equation (10) identifies the mechanism by which capillaryforces and chemical conversion can enter into competition inthe infiltration process. For this system, the right-hand side ofEq. (10) is at minimum zero. When it is nonzero, the pressureprofiles within the preform exhibit some curvature, resulting ina nonuniform velocity field.

For this system, the ambient pressure is negligible. Weapply a homogeneous Dirichlet boundary condition at thelower boundary; at the infiltration front, the pressure is equal tothe negative of the capillary pressure, − pc . Using the Young-Laplace equation, we can estimate pc at the front as a functionof surface tension, σ, pore radius, r , and the contact angle, θ:

prc =

2σ θcos (11)

Depending on the initial pore radius, pc varied from about5.94x106 to 1.88x107 Pa. As in the experimental setup ofSangsuwan et al. (1998), we will assume that the preform is

NASA/TM—2000-209802 4

coated with, e.g., boron nitride, to inhibit mass transport acrossthe sidewalls. This ensures that the liquid silicon from the bathwill not wick up the sidewalls, and that no fluid will escape fromthe system through the sidewalls. Therefore, homogeneousNeumann boundary conditions are applied to the pressure atthe side boundaries.

After the pressure solve, we can directly calculate thevelocity from the pressure field by using d’Arcy’s law, Eq. (9).The normal component of velocity at the sidewalls is zero, dueto the homogenous Neumann boundary condition used on thepressure field. Although we would expect some tangentialvelocity slip in the immediate vicinity of the sidewalls, theseeffects are confined to distances on the order of a porediameter (Kaviany, 1991). Therefore, we will assume that, atthe left and right boundaries, ∂ ∂ =v x 0 . In this set ofnumerical simulations, the infiltration process is essentiallyone-dimensional, as in experimental work of a similar process(Einset, 1993). This can be expected due to uniform initialporosity and pore size, the choice of boundary conditions onpressure and velocity, and the simplicity of the chemistry andpermeability models.

The derivation of the energy equation, as well as the fulldetails of the above derivations, can be found in Nelson (1998)or Nelson and Colella (2000b).

NUMERICAL PROCEDUREThe equations derived above are used to develop a finite-

volume algorithm for reactive melt infiltration. The discreteformulation in two dimensions is discussed in Nelson (1998)and Nelson and Colella (2000a). Briefly, the macroscopicvelocity is used to advance the front on a fixed mesh using afront-tracking technique developed by Chern and Colella(1988), coupled with a volume-of-fluid method and frontreconstruction. Chemical reaction is allowed to occur based onthe local species volume fractions and local reaction time. Theupdated species volume fractions are used to form a newpermeability field. Next, a Poisson problem is solved on theinfiltrated region only to find the new pressure field, using thetechnique of Johansen and Colella (1998). Finally, d’Arcy’s lawis used to find the new velocity field. The process is repeateduntil all of the C in the preform is consumed or the finalsimulation time is reached.

The mesh size was 32x128 on a 0.01x0.04 m domain sothat the mesh spacing ∆ ∆x y= = 0.0003125 m. A descriptionof the discrete equations, as well as the effects of gridresolution and other sources of numerical error are discussedin Nelson and Colella (2000a) and Nelson (1998).

RESULTSIn this section, we will explore the effects of varying

reaction rate, initial porosity and initial pore radius.

Effect of reaction rateWith the initial porosity fixed at 0.50 and the initial pore

radius at 0.17 µm, the infiltration curves for reaction rates of 0,10–9, 10–8, and 10–7 m/s are shown below in Fig. 2. Thecapillary pressure corresponding to this case is 6.99x106 Pa.Fig. 2(a) presents the location of the front at the centerline,y tf ( ) , while Fig. 2(b) shows y tf

2 ( ) .In the absence of chemical reaction, y tf ( ) exhibits a

square-root dependence, and the slope of y tf2 ( ) , called the

intrinsic infiltration rate (Hillig, 1994b) and denoted as ξ, isconstant. The front reaches the upper boundary of thepreform, located at 0.04 m, at ti =15.51 s.

time( )s0 5 10 15

time( )s0 5 10

(a) (b)

12

3

41

23

4

0000.

0001.

0002.

yf ( )m22yf ( )m

000.

001.

020.

003.

004.

15

Figure 2. Centerline infiltration profiles for Φ0 =0.50;r0 =0.17 µm; kr =0 (1), 10-9 (2), 10-8 (3), and 10 -7 (4) m/s.

(a) y tf ( ) ; (b) y tf2 ( )

For the chemical system of Si/C/SiC, the infiltrationprocess is retarded by the chemical reaction. For the lowreaction rate, kr =10–9 m/s, the infiltration curves closely matchthe nonreacting case for the first couple of seconds. After thatpoint, the front begins to lag behind the nonreacting case.Fig. 2(b) shows that that ξ is not constant over this timeinterval. In contrast, yf

2for kr =10–7 m/s appears quite linearafter half a second. For kr =10–8 m/s, there is a short period ofunder one second during which the curves closely follow thenonreacting profiles. Up to about 8 seconds, the slope of yf

2

is continuously changing. After that time, ξ appears constant,nearly matching that of kr =10–7 m/s.

What explains this behavior? The infiltration movesthrough a sequence of flow regimes in which the forcesinduced by capillary action and by chemical reaction are incompetition (Nelson, 1998; Nelson and Colella, 2000a, b).Initially, the pressure forces dominate as the infiltrantimpulsively moves into the preform, indicated through theclose correspondence of yf with the nonreacting case. Thiscan be seen in approximately the first two seconds of kr =10–9

m/s and the first second of kr =10–8 m/s.In the second regime, the forcing term provided by the

chemical reaction flares up and wanes. Within the preform,

NASA/TM—2000-209802 5

(b)(a)

(d)(c)

Figure 3. Centerline profiles of (a) pressure; (b) velocity; (c) permeability; and (d) φC for Φ0 =0.50, r0 =0.17 µm, andkr =10-8 m/s. Profiles are presented in approximately one-second increments for t =1-5 seconds, and in ten-second

increments thereafter.

the centerline pressure profiles become progressively morecurved (see Eq. (13)), leading to decreased flow velocities(see Eq. (9)), as shown in Figs. 3(a) and (b) for kr =10–8 m/s.Behind the front, the permeability, κ, Fig. 3(c), and φC ,Fig. 3(d), steadily decrease. Previous work (Nelson, 1998,Nelson and Colella, 2000a, b) indicates that the centerlinethermal maximum lags well behind the infiltration front in thisregime. Furthermore, the maximum operating temperatureswere observed in this regime, particularly with insulatedboundaries nearby.

When all of the available C has been consumed near theentry to the preform, Fig. 3(d) at t = 7.99 s, the pressurerelaxes to a more linear profile except in the immediate vicinityof the front, Fig. 3(a). Correspondingly, the velocity profilebecomes constant over part of the infiltrated region, Fig. 3(b).Consequently, y tf ( ) exhibits the characteristic bump in theinfiltration profiles of Fig. 2, e.g., in kr =10–8 m/s between 4and 8 seconds. This behavior can be observed in experimentalwork by Einset (1996).

For kr =10–9 m/s, the infiltration is still early in thereaction-dominated regime at 15 seconds, while for kr =10–7

m/s, the infiltration has passed through the reaction-dominated

regime by about one-half second. As will be seen throughoutthis work, the temporal and physical extent of the reaction-dominated regime decreases with increasing reaction rate.

After 10 seconds, the infiltration process for kr =10–8 m/senters a third regime, in which the reaction-generated forcesare confined to a small region behind the front and theinfiltration dynamics are largely set by the permeability in thefully reacted zone and the capillary pressure. From10 seconds to complete infiltration at 63 seconds, y tf ( )resembles a square-root profile, and ξ can be approximated bya constant. The reaction zone becomes progressively morelocalized to the vicinity of the front. For example, the reactingzone at 63 seconds is about 0.00296 m, appearing over eightcomputational cells. As a result, the thermal maximum is veryclose to the infiltration front (Nelson, 1998).

Although the curve for the highest reaction rate inFig. 2(b) appears linear after one second, notice that, unlikethe nonreacting case, the curve does not intersect the origin.Examination of the initial time interval more closely indicatesthat the infiltration simply proceeds through the reaction-dominated regime more rapidly than the other cases, resultingin the offset. If we were to extend the temporal window for

NASA/TM—2000-209802 6

(b)(a)

Figure 4. Effect of Φ0 on infiltration profiles for r0 = 0.17 µm and Φ0 =0.40 (1), 0.45 (2), 0.50 (3), 0.55 (4) and 0.60 (5):kr =0 (black), 10 -9 (blue), 10 -8 (green), and 10 -7 m/s (red): (a) yf vs. t ; (b) y tf

2 ( ) vs. t .

kr =10–9 m/s to later times (and with a longer preform length, inmost cases), we would also see the characteristic bumpsignaling the ascendance and conclusion of the reaction-dominated regime, but occurring at a retarded pace.

When fully reacted, the final value for φ Si is 0.197 and forφ SiC , it is 0.803.

Effect of initial porosityTo examine the effect of varying porosity, infiltration

profiles are shown in Fig. 4 for kr =0, 10–9, 10–8, and 10–7 m/swith Φ0 =0.40, 0.45, 0.50, 0.55 and 0.60, holding r0 fixed at0.17 µm. The capillary pressure corresponding to this r0 is6.99x106 Pa. In this time interval, the infiltration profiles forkr =0 and 10–9 m/s collapse into two curves. The increasedpermeability caused by increasing the porosity by 50% doesnot significantly impact the infiltration in the pressure-dominated regime and the early stages of the reaction-dominated regime.

At increased reaction rate, we can still see an early periodduring which the curves for the varying porosity areindistinguishable. However, in the subsequent reaction-dominated regime, Fig. 4 indicates that the reaction-generatedforces can greatly slow down the infiltration, as seen mostdramatically for kr = 10–8 m/s at the lowest porosities. In fact,the reaction-generated forces can theoretically stop the flowfor a time. The fanning effect seen for kr = 10–8 and 10–7 m/scan be explained as follows: Since carbon is the only speciespresent in the preform initially, Φ0 is related to the amount ofC present, i.e., φC0

= 1 – Φ0. This in effect places an upperbound on the amount of chemical conversion that is possibleover a given volume. Once the carbon in the initial reaction-dominated region is consumed, the dynamics of the infiltrationproceed at a different pace. At constant kr , examination of

the centerline profiles reveal that the fanning occurs soon afterthe pressure profile relaxes back from its most distortedshape. These trends indicate that the reaction-dominatedregime becomes larger in spatial and temporal extent withdecreasing porosity and reaction rate. For the fully reactedzone, the final φ SiC was 0.96, 0.88, 0.80, 0.72, and 0.64 forΦ0 =0.40, 0.45, 0.50, 0.55 and 0.60, respectively.

The times required for complete infiltration and completeconversion, te i, and te r, , are presented in Fig. 5. We did notcarry out the infiltration process to complete conversion forkr =10–7 m/s due to numerical instability; this is being

Figure 5. Influence of Φ0 on required time for completeinfiltration (unfilled symbols) and conversion (filledsymbols) for r0 = 0.17 µm and kr =0, ��; 10-9 , zz; and 10 -8

m/s, ��.

NASA/TM—2000-209802 7

addressed in a modified algorithm (Nelson and Colella,2000a).

Notice that the most efficient reaction rate for infiltratingthe preform, i.e., the lowest kr , does not imply the shortestconversion time, particularly at low initial porosities. Also notethat, as the reaction rate is increased at fixed porosity, thedifference between te c, and te i, decreases. It is zero forΦ0 =0.40, kr =10–8 m/s, indicating that the carbon behind thefront is essentially consumed as fast as the front can move.

(a)

(b)Figure 6. Influence of Φ0 in the post-reaction-dominated

regime: (a) time of entry into this regime; and (b) intrinsicinfiltration rate.

The time at which the infiltration moved from the reaction-dominated regime to the third post-reaction-dominated regime,t3 , shown in Fig. 6(a), was estimated from infiltration profiles.The intrinsic infiltration rate, ξ, in this regime is presented inFig. 6(b).

For kr =0, ξ=1.0x10–4 m/s over the range of porosity. Forkr =10–8 m/s and 10–7 m/s, we can see that ξ increaseslinearly with porosity. We might expect this, because thepermeability is proportional to the porosity, while capillarypressure remains fixed. When chemical reaction occurs, theentry into the post-reaction-dominated regime is retarded asporosity decreases, reflecting the increased influence of thereaction-generated forces on the infiltration dynamics.However, once in the post-reaction-dominated regime, thepermeability in the fully reacted zone behind the front is moreinfluential on the infiltration dynamics than the reaction-generated forces.

Effect of initial pore radiusNext, we will examine the influence of the initial pore

radius on the infiltration process, holding the porosity fixed atΦ0 =0.50 and varying the reaction rate, kr , to 0, 10–9, 10–8,10–7 m/s and the pore radius to 0.10, 0.14, 0.17 and 0.20 µm.The capillary pressure corresponding to these pore radiiranges from 5.94x106 Pa to 1.88x107 Pa (see Eq. (11)). Forclarity, the infiltration curves are presented in sets by reactionrate in Fig. 7 for the first 30 seconds of the infiltration.

Decreasing pore radius retards the infiltration, asobserved by Einset (1996) experimentally. Ignoring thechemical reaction for the moment, this would be expectedsince the capillary pressure is inversely proportional to r0 ,while the permeability is proportional to r0

2. Unlike theparameterization in porosity, the curves for kr =0 and 10–9 m/sdo not collapse onto one another in Figs. 7(a) and (b). We alsosee the effects of chemistry in the curves for kr =10–-8 and 10–7

m/s in Figs. 7(c) and (d).The curves for kr =10–9 m/s (Fig. 7(b)) are nearly

equivalent to the corresponding nonreacting case for the firstsecond or two. All of the preforms at this reaction rate are fullysaturated by 30 seconds except for r0 =0.10 µm. The intrinsicinfiltration rate, ξ, shown in Fig. 7(f) is not constant for any ofthe pore radii over this time interval.

The characteristic bump in the infiltration profile due to thepresence of the reaction-dominated regime is seen mostclearly in Fig. 7(c) for kr =10–-8 m/s under 10 seconds. Thecorresponding slope of yf

2 (t) in Fig. 7(g) is not linear duringthis time. The infiltration has moved beyond the reaction-dominated regime by 10 seconds in Fig. 7(c). This occursmore quickly for the higher reaction rate in less than a secondin Fig. 7(d). From Figs. 7(g) and (h), we can see that ξappears constant once the infiltration process has left thereaction-dominated regime.

NASA/TM—2000-209802 8

yf ( )m

time( )s0 10 20 30000.

001.

020.

003.

004.

kr =0

yf ( )m

time( )s0 10 20 30

000.

001.

020.

003.

004.

kr = −10 9

(a)

yf ( )m

time( )s0 10 20 30000.

001.

020.

003.

004.

yf ( )m

time( )s0 10 20000.

001.

020.

003.

004.

(b) (c) (d)

123

4

kr = −10 8 kr = −10 7

1234

12

34

12

34

time( )s0 10 20 300000.

0001.

0002.kr =0

time( )s0 10 20 30

kr = −10 9

(e)time( )s

0 10 20 30time( )s

0 10 20

(f) (g) (h)

123

4

kr = −10 8 kr = −10 7

1234

12

3

4

12

34

yf ( )m22

0000.

0001.

0002.

yf ( )m22

0000.

0001.

0002.

yf ( )m22

0000.

0001.

0002.

yf ( )m22

Figure 7. Effect of r0 on infiltration profiles for Φ0 =0.50; r0 =0.10 (1), 0.14 (2), 0.17 (3) and 0.20 (4) µm: (a) y t kf r( ), =0.0; (b)y t kf r( ), =10-9 m/s; (c) y t kf r( ), =10-8 m/s; and (d) y t kf r( ), =10-7 m/s; (e) y t kf r

2 ( ), =0.0; (f) y t kf r2 ( ), =10-9 m/s; (g)

y t kf r2 ( ), =10-8 m/s; and (h) y t kf r

2 ( ), =10-7 m/s.

The dependence of the total infiltration and conversiontimes on pore radius is shown in Fig. 8. A wide gap is onceagain apparent between te i, and te r, for kr =10–9 m/s (unfilledand filled circles). For the lowest pore radius and reaction rate(filled circle, left of arrow), the infiltration process had movedwell beyond the reaction-dominated regime when the upperboundary was encountered. On the other hand, much of theconversion took place in a fully saturated preform for the largerpore radii (filled circles, right of arrow). Recall that, for a fullysaturated preform, the pore radius decreases linearly withtime, Eq. (1), to complete conversion. The trend of increasingte r, with increasing radius at high pore radii therefore indicatesthat, under these conditions, the absence of the front reducesthe effectiveness of the conversion.

Figure 9 shows the influence of pore radius on the start ofthe post-reaction-dominated regime and on the intrinsicinfiltration rate. Fig. 9(a) indicates that decreasing pore radiusreduces the temporal extent of the reaction-dominated regime,an effect which is most pronounced at the lower reaction rate.Unlike the porosity-dependent parameterization, ξ is notconstant for kr =0 (squares), but rather increases withincreasing pore radius, due to increasing permeability. Asbefore, we should expect this because the capillary pressure isinversely proportional to r0 , while the permeability isproportional to r0

2. Fig. 9(b) indicates that ξ is linearly

proportional to r0 and is not affected greatly by the reactionrate. These results imply that the infiltration in the post-reaction-dominated regime is governed primarily by thepermeability of the preform in the reacted zone behind thefront and by the capillary pressure.

Figure 8. Influence of r0 on required time for completeinfiltration (unfilled symbols) and complete conversion(filled symbols) for kr =0, ��; 10-9 , zz; and 10 -8 m/s, ��.

NASA/TM—2000-209802 9

(a)

(b)

Figure 9. Influence of r0 in the post-reaction-dominatedregime: (a) time of entry into this regime; and (b) intrinsic

infiltration rate.

CONCLUSIONSWe have examined the infiltration of liquid Si into a

constant-volume microporous C preform to form a monolithicSiC ceramic composite for the range of Φ0 =0.40-0.60,r0 =0.10-0.20 µm, and kr =0, 10–9, 10–8, 10–7 m/s. The initialpore radius and porosity were uniform over the entire preform.

We have seen that the infiltration dynamics are dependenton reaction rate, initial porosity and initial pore radius.Significantly, it is also a function of the flow regime arising fromthe competition between pressure and reaction forces. We didnot find a simple correlation for the time and distanceassociated with the reaction-dominated regime. However,numerical simulation can be used to predict these details.

Initially, the pressure forces dominated as the infiltrantimpulsively entered the preform. Later, the presence of thechemical reaction retarded the infiltration due to a competitionbetween the pressure forces and the volumetric forces due tothe chemical conversion. When the reactants in the entry tothe preform were consumed, the reaction zone was confinedto a small region behind the front. Furthermore, the infiltration

rate increased, although it was significantly smaller than that ofthe corresponding nonreacting case.

In the first pressure-dominated regime and the earlystages of the second reaction-dominated regime, a porosityincrease of 50% had a negligible effect on the infiltrationprofiles, as seen in the data for kr =0 and 10–9 m/s.

The reaction-generated forces may impede or enhancethe flow, depending on the density variation among theproducts and reactants. For the Si/C/SiC system with a denseproduct, the conversion was shown to retard the flow. Thetemporal and physical extent of the reaction-dominated regimedecreased with increasing reaction rate, decreasing poreradius and increasing porosity. In this regime, the velocitywithin the preform was retarded significantly due to thecurvature of the pressure profiles, causing a characteristicbump in the infiltration profiles. Theoretically, the flow canbecome stationary at the front; this is to be distinguished fromchoking in that this is a temporary phenomenon. For kr = 10–8

m/s, the reaction-dominated regime was in evidence fromabout 2 to 10 seconds, while it was completed by one secondfor kr = 10–7 m/s. The intrinsic infiltration rate was not constantin this regime, indicating that the location of the front did nothave a square-root dependence on time. Aside from theimplications on infiltration dynamics, the maximum processtemperature was found, in previous work, to occur in thisregime. The reaction-dominated regime ended soon after thecarbon in the initial entry zone was completely consumed.

Beyond the reaction-dominated regime, the infiltrationsettled into a simpler pattern. The reaction zone was localizedto an area just behind the front. The intrinsic infiltration ratewas constant and proportional to the initial porosity and to theinitial pore radius. Although the permeability is proportional tothe square of the pore radius, the capillary pressure isinversely proportional to the pore radius, accounting for thisbehavior. The intrinsic infiltration rate was not greatlyinfluenced by the magnitude of the reaction rate in this regime.We therefore deduce that the infiltration was primarily affectedby the capillary pressure at the front and the permeability inthe fully reacted zone behind the front. Consequently, theintrinsic infiltration rate can be characterized by a constantbeyond the reaction-dominated regime, although the linerepresenting ξt must be offset from the origin to coincide withthe data.

For this system, numerical simulation of the parameterspace provides the best prediction of optimal parameters forminimum conversion time. For an actual process, we shouldalso include preform length as a parameter, as that will affectwhether or not the process leaves the reaction-dominatedregime. For kr = 10–9 m/s, the infiltration was completed duringthe early stages of the reaction-dominated regime, except atthe smallest pore radii. If the entire preform reached saturationduring this time, the overall conversion time was seen toincrease with increasing pore radius, since the evolving poreradius decreased strictly linearly with time, as set by thechemistry of the system. In contrast, if the infiltration movedinto the post-reaction-dominated regime, as seen at lower pore

NASA/TM—2000-209802 10

radius, the conversion time decreased with increasing poreradius. The presence of the advancing front, with itsaccompanying complex infiltration dynamics, can eitherimprove or reduce the efficiency of the conversion process,depending on the initial parameters. If the infiltration wasallowed to proceed through the sequence of infiltrationregimes, the overall conversion time decreased withincreasing porosity, pore radius and reaction rate.

This sort of analysis could provide insight into the designof functionally graded materials. For example, localmodification of the pore radius or porosity in the post-reaction-dominated regime could be used to speed up the infiltrationrate. Such a modification in the reaction-dominated regimecould have major implications on the efficiency of the process.The advantage of a numerical simulation is that it could rangeover the desired parameter space with minimal effort, allowingfor more precise optimization.

ReferencesBehrendt, D.R. “Porous silicon carbide as a matrix for

ceramic composites.” NASA TM 88837 (1986).Behrendt, D.R. and M. Singh. “Effect of carbon preform

pore volume and infiltrants on the composition of reaction-formed silicon carbide materials.” J Mat Synthesis andProcessing 2:117-123 (1994).

Behrendt, D.R. and M. Singh. “Theoretical considerationsfor reaction-formed silicon carbide.” NASA TM 106414 (1993).

Brandes, E.A., ed. Smithells Metals Reference Book, 6th

ed. Butterworths (1983).Chern, I-L and P. Colella. “A conservative front tracking

method for hyperbolic conservation laws”. Unpublished paper(1988).

Datta, S.K., N. Simhai, S.N. Tewari, J.E. Gatica, and M.Singh. “Permeability of microporous carbon preforms.” MetTrans A 27A:3669-74 (1996).

Einset, E. O. Private communication (1993).Einset, E. O. “Capillary infiltration rates into porous media

with applications to Silcomp processing.” J Am Ceram Soc79:333-338 (1996).

Fitzer, E. and R. Gadow. "Investigations of the reactivity ofdifferent carbons with liquid silicon." In Proceedings ofInternational symposium on ceramic components for engines.Japan, 1983. KTK Scientific Publishers, Tokyo, Japan (1984).

Haggerty, J.S. and Y.M. Chiang. "Reaction-basedprocessing methods for ceramics and composites." CeramEng and Sci Proc 11:757-781 (1990).

Hillig, W.B. “Making ceramic composites by meltinfiltration.” Am Ceram Soc Bull 73:56-62 (1994).

Hillig, W.B. “Melt infiltration approach to ceramic matrixcomposites.” Am Ceram Soc, Comm 71:C-96 to C-99 (1994).

Hillig, W.B., R.L. Mehan, C.R. Morelock, V.J. DeCarlo andW. Laskow. “Silicon/silicon carbide composites.” Amer CeramSoc Bull 54:1054-56 (1975).

Johansen, H. and Colella, P. “A Cartesian grid embeddedboundary method for Poisson’s equation on irregulardomains.” J Comp Phys 147:60-85 (1998).

Kaviany, M. “Principles of heat transfer in porous media.”Springer-Verlag (1991).

Kochendoerfer, R. "Liquid silicon infiltration: a fast andlow-cost CMC manufacturing process." In Composites:Proceedings of the 8th International Conference on CompositeMaterials. Held in Honolulu, HI, July 15-19, 1991, pp. 23-F-1 to23-F-9.3 (1991).

Li, J.G. and H. Hausner. “Wettability of silicon carbide bygold, germanium and silicon.” J Mat Sci Ltrs 10:1275-76(1991).

Messner, R. and Y.M. Chiang. "Liquid-phase reaction-bonding of silicon carbide using alloyed Si-Mo melts." J AmerCeram Soc 73:1193 (1990).

Minnear, W.P. "Interfacial energies in the Si/SiC systemand the Si+C reaction." J Amer Ceram Soc 65:C10-C11(1982).

Nelson, E.S. “A numerical algorithm for the simulation ofreactive melt infiltration.” Ph.D. Thesis. University of Californiaat Berkeley (1998).

Nelson, E.S. and P. Colella. “A numerical algorithm forreactive melt infiltration.” To be submitted to J Comp Phys(2000a).

Nelson, E.S. and P. Colella. “Infiltration dynamics inreactive melt infiltration.” To be submitted to J Amer CeramSoc (2000b).

Nelson, E.S. and P. Colella. Unpublished work (2000c).Sangsuwan, P., S.N. Tewari, J.E. Gatica, M. Singh and R.

Dickerson. “Reactive infiltration of silicon melt throughmicroporous amorphous carbon preforms.” Met Trans30B:933-944 (1998).

Singh, M. and D.R. Behrendt. “Reactive melt infiltration ofsilicon-molybdenum alloys into microporous carbon preforms.”Mat Sci Eng A 194:193-200 (1995).

Singh. M. and D.R. Behrendt. “Microstructure andmechanical properties of reaction-formed silicon carbide(RSFC) ceramics.” Mat Sci Eng A 187:183-187 (1994).

Singh, M. and D.R. Behrendt. “Reactive melt infiltration ofsilicon-niobium alloys in microporous carbons.” J Mat Res7:1701-1708 (1994).

Singh. M. and D.R. Behrendt. "Studies on the reactivemelt infiltration of silicon and silicon-molybdenum alloys inporous carbon." NASA TM 105860 (1992).

Turovskii, B.M. and I.I. Ivanova. "Temperaturedependence of the viscosity of fused silicon." InorganicMaterials 10:2108-2111 (1974).

Whalen, T.J. and A.T. Anderson. “Wetting of SiC, Si3N4,and carbon by Si and binary Si alloys.” J Am Ceram Soc58:396-399 (1974).

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NASA TM—2000-209802

E–12107

WU–101–43–0C–00

16

A03

Parametric Study of Reactive Melt Infiltration

Emily S. Nelson and Phillip Colella

Infiltration; Porosity; Composite materials; Silicon carbides; Porous materials; Reactingflow; Fluid dynamics; Manufacturing; Production engineering; Permeability

Unclassified -UnlimitedSubject Categories: 24, 27 and 34 Distribution: Nonstandard

Prepared for the 1999 International Mechanical Engineering Congress and Exposition sponsored by the AmericanSociety of Mechanical Engineers, Nashville, Tennessee, November 14–19, 1999. Emily S. Nelson, NASA Glenn Re-search Center, and Phillip Colella, Lawrence Berkeley National Laboratory, M-S 50D-133, Berkeley, California 94720.Responsible person, Emily S. Nelson, organization code 6727, (216) 433–3268.

Reactive melt infiltration is viewed as a promising means of achieving near-net shape manufacturing with quick process-ing time and at low cost. Since the reactants and products are, in general, of varying density, overall conservation of massdictates that there is a force related to chemical conversion which can directly influence infiltration behavior. In effect, thedriving pressure forces may compete with the forces from chemical conversion, affecting the advancement of the front.We have developed a two-dimensional numerical code to examine these effects, using reaction-formed silicon carbide as amodel system for this process. We have examined a range of initial porosities, pore radii, and reaction rates in order toinvestigate their effects on infiltration dynamics.