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Reaction-Bonded Silicon Carbide by Reactive Infiltration Prasert Sangsuwan, Joaquı´n A. Orejas,† Jorge E. Gatica,* Surendra N. Tewari, and Mrityunjay Singh‡ Chemical Engineering Department, Cleveland State University, 1960 E 24th Street-SH455, Cleveland, Ohio 44115-2425
Exothermic reactions between a porous matrix and an infiltrating melt provide a more economic alternative for synthesizing many ceramics, intermetallics, and composites. It has recently been demonstrated that infiltration of cast microporous carbon preforms by silicon melt can be used to fabricate high-density, nearly net-shaped silicon carbide components at significantly reduced cost. This paper describes the synthesis of reaction-bonded silicon carbide by reactive infiltration of microporous carbon preforms. The kinetics of unidirectional infiltration of silicon melt into microporous carbon preforms as a function of pore morphology and melt temperature is investigated in this paper. Qualitative agreement between experimental data and a mathematical model for capillarity-driven fluid flow through cylindrical pores is demonstrated. Experimental evidence of high parametric sensitivity is also presented. A simplified model relating fluid flow, transport, and reaction phenomena is formulated to interpret experimental evidence of pore closing and free silicon entrapment. A robust numerical formulation is also described. Introduction Chemical reactions between a porous matrix and an infiltrating phase provide a more economic alternative for synthesizing many ceramics, intermetallics, and composites. In reaction-bonded silicon carbide (RBSC), for instance, capillary infiltration of a carbon-containing body is carried out by molten silicon.1 The occurrence of a chemical reaction between silicon and carbon results in the formation of silicon carbide particles that bond existing SiC grains, which are usually added as inert fillers during the preform preparation step. In another approach, carbonaceous materials, such as carbon fiber tows, fiber cloths, or felts, are infiltrated by molten silicon in a vacuum to form Si/SiC (silicon reinforced by silicon carbide) composites.2 It has recently been demonstrated that infiltration of cast microporous carbon preforms by a silicon melt can be used to fabricate high-density, nearly net-shaped silicon carbide components at significantly reduced cost.3,4 Because molten silicon wets carbon, it wicks up into the microporous carbon network when brought into contact with the preform, thus converting carbon into silicon carbide. Components fabricated by this technique are expected to have a significant impact in gas turbine engine components, commercial combustion nozzles, and other advanced applications where their refractoriness (high strength at elevated temperatures), good oxidation resistance, high thermal conductivity, low density, and adequate toughness can be exploited. However, successful commercial exploitation of this technique requires a quantitative understanding of the various steps involved in the process, namely, (i) the fabrication of the microporous carbon preforms, (ii) their infiltration * Author to whom correspondence should be addressed. Phone: (216) 523-7274. Fax: (216) 687-9220. E-mail: j.gatica@ csuohio.edu. † Facultad de Ingenierı ´a, Universidad Nacional de Rı´o Cuarto, Ruta 36 Km 601, Co´rdoba, Argentina. ‡ NYMA, Inc., NASA John H. Glenn Research Center, 21000 Brookpart Road, Cleveland, OH 44135-3191, USA.
by the silicon melt, and (iii) the chemical reaction to convert carbon into silicon carbide. The process for fabricating the microporous carbon preforms is reasonably well understood. Porous carbon preforms with varying microstructures [pore size, pore volume fraction, carbon particle (strut) size] can be produced by controlling the chemistry of the organic mixture and the heat treatment.4 The process of silicon infiltration, however, is less understood. Unlike in other infiltration processes, the presence of a chemical reaction between the porous preforms and the infiltrating fluid adds complexity to the process. Indeed, nonlinear interactions between flow and transport phenomena complicate the scenario where thermomechanical and chemical phenomena determine the characteristics of the resulting composite. For example, the reaction between the infiltrating silicon melt and carbon results in reduced pore size (there is an approximately 58% volume increase when one mole of amorphous carbon reacts to form one mole of SiC). This affects the permeability of the preform reducing the infiltration velocity. Thus, for a preform with a small initial pore size, the silicon infiltration will stop prematurely, resulting in incomplete infiltration, a phenomenon frequently referred to as choking. Alternatively, if the pore constrictions in the preform are large, the preform will be fully infiltrated, but residual unconverted silicon will remain in the component, with deleterious effects for high-temperature applications. Therefore, it is important to understand (i) the kinetics of the chemical reaction, (ii) the kinetics of the melt infiltration, (iii) the coupling between these two phenomena, and (iv) their dependence on the preform morphology (volume fraction of carbon, pore size distribution, bed tortuosity, etc.) and the processing conditions (preform and melt temperatures, pressure, melt characteristics, etc.). A complete analysis of the process would be very complex because of the nonlinear interactions between microscale and macroscale phenomena. Microscale phenomena, such as the carbon-silicon reaction and mass
10.1021/ie001029e CCC: $20.00 © 2001 American Chemical Society Published on Web 07/31/2001
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and energy transport phenomena, will determine the flow characteristics within an individual pore. At the macroscale, the interactions among neighboring pores will determine the stability of the infiltration front and, thus, the composite microstructure and overall degrees of infiltration and silicon carbide conversion. Stresses due to the severe thermal gradients resulting from the exothermic reaction could cause cracking, and residual stresses due to volume change/cooling mechanisms could have deleterious effects on the component brittleness. Therefore, when faced with the task of designing the process leading to a final component, a number of questions are clearly without a definite answer. In other words, how can we predict the maximum extent of infiltration for a given initial melt or preform temperature? What will be the stability characteristics of the infiltration front, and what will be its effects on the composite microstructure (morphology)? How does the kinetics of infiltration depend on the structural properties of the preform, i.e., preform porosity and tortuosity, pore size distribution, and pore orientation and interconnection? Can the resulting temperature and stress distributions in a component be predicted? Can the overall extent of conversion be predicted by combining knowledge about the chemical reaction and infiltration kinetics? A combined experimental and numerical modeling approach is followed in this paper to answer these and other related questions. The main purpose of this paper is to develop a theoretical understanding of the reactive infiltration process by investigating the infiltration process, the two phases (kinetics- and diffusion-limited stages) of the chemical reaction between the infiltrating melt and the carbonaceous preform, and the associated thermal phenomena. The present work addresses the following components: (i) Investigation of the kinetics in simple geometries, (ii) microstructural characterization of porous preforms, (iii) experimental analysis of reactive infiltration using two different configurations (microporous preforms infiltrated by molten silicon and silicon/silicon carbide/carbon preforms prepared from particulate materials), and (iv) phenomenological modeling and numerical analysis of the infiltration process. Preliminary proof-of-concept experimental results are presented to lend support to the model formulated. Experimental Characterization The microporous amorphous carbon preforms used in this study were made from a mixture of furfuryl alcohol resin, diethylene and triethylene glycols, and p-toluene sulfonic acid. This mixture was polymerized to form a porous solid polymer. The solid polymer was then heated slowly in a flowing argon atmosphere, which resulted in the production of microporous carbon preforms. Preforms with a range of pore sizes were obtained by varying the composition of the organic mixture. Further details on preform fabrication can be found elsewhere.4 The microstructure of the preforms was examined by scanning electron microscopy and characterized by mercury porosimetry and permeability determinations. The experimental setup used to analyze this process consisted of an alumina assembly heated via a graphite susceptor using induction coils. The carbon specimens were inserted in an alumina tube, and the silicon melt was contained in a quartz crucible. The preform and melt were fitted with tungsten-rhenium thermocouples interfaced with an HP data logger. The whole assembly
Figure 1. Permeability determinations for preforms of type A (inset shows results from mercury porosimetry).
was kept under a vacuum-controlled atmosphere in a bell jar. Details on the experimental setup and protocol can be found elsewhere.5 Infiltrated samples were cooled to room temperature, sectioned, and examined by optical and scanning electron metallography and electron microprobe analysis. Microstructural Characterization of Preforms. The inset in Figure 1 presents the results of mercury porosimetry determinations carried out on type A preforms. Both preforms, types A and B, have narrow pore size distributions. Their median pore diameters, based either on the pore volume or on the pore surface area, are nearly identical. A comparison between their skeletal densities, 1.48 and 1.51 g cm-3, and the literature-reported density of glassy carbon, 1.5 g cm-3, suggests that the pore networks in these preforms are interconnected. In the presence of isolated pores, the skeletal density calculated from mercury porosimetry would be smaller than that corresponding to glassy carbon. However, the B type preform is more porous than the A type, as shown by their fractional porosities, 0.53 versus 0.48. Permeability Analysis of Uninfiltrated Specimens. The porous medium permeability, κ, is an overall macroscopic parameter that is widely used to describe fluid flow through porous structures in many engineering applications. This parameter can easily be extracted from experimental data. For steady-state unidirectional infiltration of a Boussinesq’s fluid, Darcy’s relationship reduces to the wellknown permeability expression
u)-
κ dp µf dz
(1)
where u is the fluid velocity, p is the pressure, µ denotes the dynamic viscosity of the infiltrating fluid, and κ represents the permeability of the porous medium. This relationship can be used to extract the permeability of the medium from a linear correlation between fluid flow velocity, u, and the applied pressure gradient. Figure 1 shows the correlation between the water flow rates and the applied pressure in a standard permeameter6 for type A performs. The goodness of fit of these determinations supports the applicability of Darcy’s law for the preforms used in this study. For a room-temperature viscosity of water of 9.6 × 10-10 MPa s, the permeabilities of the two preforms, A and B, can be estimated as 2.29 × 10-14 and 2.70 × 10-14 m2, respectively.
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Formulation of a Mathematical Model Infiltration of Microporous Structures by Reactive Fluids. The siliconizing process has been addressed by a large number of authors, with most of the investigations via experiments performed between molten silicon and isolated carbon fibers or plates. However, there has been very little effort to study the most important step in the conversion process: the process of silicon infiltration through the porous carbon preforms. Because the infiltration of the silicon melt precedes the chemical reaction itself, it is a crucial link in determining the shape, size, and characteristics of the carbon preforms that can be successfully converted into silicon carbide. The infiltration of porous structures by a fluid is usually described by Darcy’s model, which for a Boussinesq’s fluid can be written as
F°f
µf ∂u ) -∆p - u - F°f[βT(T - T°) + βC(C - C°)]g ∂t κ (2)
where u is the fluid velocity, p is the pressure, µ is the fluid dynamic viscosity, F is the fluid density, β is the expansion coefficient (with C and T standing for concentration and temperature, respectively), g is the constant of gravitational acceleration, and κ is the (porous) medium permeability. This latter parameter, which allows one to treat the fluid-solid system as a quasicontinuum medium, is intimately related to the structure properties (void fraction, pore size distribution, etc.). A number of empirical relationships have been derived for packed7 and fibrous beds and bundles of aligned fibers,8 including correction factors for nonspherical shapes and misalignment of the fibers, as well. There is, however, no reliable equation9 that can predict the permeability for a given microporous structure. The approach, therefore, has been to extract this value from experimental measurements as outlined above. A different approach can alternatively be followed. Instead of characterizing the medium by a single parameter, κ, a description of the capillarity-driven flow in a single pore can be made using Newton’s law.10 A force balance for the isothermal infiltration of a vertical cylindrical pore (with a constant cross-sectional area) yields
1 πrp2Ff
∑Fz ) dt(z dt ) ) - 4( dt ) d
dz
1 dz 2
-
8µf
dz - gz + z rp2Ff dt 2 γ cos(θ) (3) rpFf LV
where r is the pore radius; µ and F are the fluid dynamic viscosity and density, respectively; γ is the surface tension; θ stands for the contact angle; and z and t represent the infiltration height and time, respectively. The subscripts f and LV stand for fluid and liquidvapor interface, respectively. For highly viscous fluids and for short infiltration lengths (neglecting end effects and body forces) this equation can be solved analytically. For most cases, where the flow occurs as a result of capillary pressure, the fluid weight is negligible when compared to the surface tension during the early stages of the infiltration (i.e., short infiltration lengths) and the above equation yields
z2 ) Φt or u )
Φ1 2 z
with Φ )
DpγLV cos θ (4) 4µf
The parameter Φ, often considered11 an intrinsic infiltration rate parameter, can be obtained from the experimental data by plotting infiltration length or infiltration velocity versus infiltration time. However, the above equation has been derived for single-pore flow and is strictly valid for a porous structure of parallel pores of diameter Dp. In contrast, the experimental data yield an overall Φ value that is representative of the pore network (pore size, shape, distribution and microstructure) being infiltrated. The Dp in the equation for Φ thus represents an effective pore diameter, i.e., Dpe. It is interesting to note here that, assuming a linear dependence for the pressure in the permeability definition, a relationship between the intrinsic infiltration parameter and the medium permeability can easily be derived. Most of the attempts to analyze reactive infiltration, therefore, have naturally followed one of these two routes. Thus, for instance, Messner and Chiang12 attempted to develop an analytical model to predict the infiltration phenomenon: based on a varying permeability correlated to the pore size reduction with time, the infiltration behavior of silicon through graphite preforms was predicted via Darcy’s law. However, no measurement of the infiltration kinetics was carried out. An early study reported in the literature13 attempted to measure the infiltration front velocity as a function of temperature. To examine the infiltration kinetics, Einset exposed thin tapes of carbon preforms containing silicon carbide particles to silicon melt. Thermocouples were inserted at different points along these tapes to record the onset of the temperature rise due to the reaction at the infiltrating front. These experiments, however, are not an accurate representation of infiltration phenomena through a porous preform. Indeed, the silicon melt would preferentially wet and traverse along the outer surface of the tapes where there is no resistance to the melt flow and not through the inner tortuous channels of the intricate porous network. Preliminary experiments are described next. Hillig’s simple expression is used to assess the rate-controlling mechanism for the reactive infiltration process. Temperature Dynamics of Reactive Infiltration Experiments. Typical temperature-time plots recorded by the thermocouple assembly described earlier are shown in Figure 2. The thermal response of the thermocouples located at 0.3, 0.5, 0.8, and 1.1 cm from the (bottom) surface of a type A preform exposed to the silicon melt are indicated as TC1, TC2, TC3, and TC4, respectively. The thermal response corresponding to the silicon melt is indicated as Si. The uncertainty in the temperature values is approximately (6 K. Figure 3a shows the different stages involved in the experiment. The arrows at the bottom of the figure indicate the different stages of the infiltration experiments. The melt-carbon preform contact occurred at 1853 s, with 1.7 cm of the preform being submerged into the melt. The exothermic reaction between the infiltrating silicon melt and the carbon preform resulted in a steep temperature rise, recorded by all thermocouples. It is worth noticing, however, that, despite the highly exothermic reaction, the temperatures recorded by all preform thermocouples return to the silicon melt temperature within 60 s after the initial temperature rise. The
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Figure 2. Thermal responses in a preform of type A.
Figure 3. Correlation between infiltration velocity and infiltration height for reactive infiltration.
preform was raised out of the melt at about 2090 s, at which time the induction power was switched off. This resulted in a rapid cooling, at a rate of approximately 50 K min-1, recorded by all five thermocouples. The isothermal plateau, starting at 2240 s, registered by the thermocouple in the silicon melt corresponds to the solidification of silicon (1687 K). It is interesting to note that the isothermal trend was also indicated by the thermocouples located in the preform, especially TC1, TC2, and to some extent TC3 (Figure 3c). This indicates
the presence of free silicon in the preform, as was confirmed by microstructural analysis. It is interesting to note that the peak temperatures show a decrease with increasing distance from the exposed surface of the preform. This is due to the continuous decrease in the infiltration front velocity as it travels into the preform, as predicted by the capillary flow analysis. Such thermal profiles can be used to estimate the infiltration front velocity for a quantitative comparison with theoretical predictions. Detailed metallographic analyses and additional runs for different preforms and conditions can be found elsewhere.5 Dynamics of Reactive Infiltration. The front velocity was estimated by following the displacement of the inflection point in the temperature profile. The time responses of the different thermocouples were differentiated, and estimates of the front velocity were thus obtained. Figure 3 shows infiltration data and the least-squares results (linear correlation and the corresponding 95% confidence intervals) corresponding to three infiltration experiments. As predicted by the capillary flow analysis, the infiltration velocity decreased with increasing infiltration distance. An analysis of capillarity-driven infiltration suggests that the infiltration behavior can be predicted considering a single-pore representation of the preform structure and using eq 4. Using the pore geometry characterization from permeability measurements and literature values for physical properties, eq 4 was used to calculate theoretical infiltration dynamics (Hillig’s parameter or slope in Figure 3). However, the discrepancies between experimental (slope in Figure 3) and theoretical (eq 4) slopes can reach up to 2 orders of magnitude. This difference, despite the many simplifying assumptions, suggests that the occurrence of the exothermic chemical reaction has an accelerating effect on the infiltration process. A summary of the possible kinetic mechanisms for this reaction and their possible integration into a mathematical model, which would explain the disagreement between experiments and theory, is presented next. Reaction Mechanism and Kinetics. The siliconizing process has been addressed by a large number of authors, with most of the investigations via experiments performed between molten silicon and isolated carbon fibers or plates. However, there has been very little effort to study this process in conjunction with the most important step in the conversion process: silicon infiltration through the porous carbon preforms. The infiltration of the silicon melt precedes the chemical reaction itself and is, therefore, a crucial link in determining the shape, size, and characteristics of the carbon preforms that can be successfully converted into silicon carbide. In a reactive infiltration process, the chemical reaction competes with the infiltration process in becoming the rate-controlling step. Understanding and characterizing the infiltration dynamics and the chemical kinetics will therefore be crucial for the design and fabrication of actual components by this process. Significant controversy regarding the exact mechanism for the formation of reaction-bonded silicon carbide (RBSC) has been raised in the literature. Here, the two most feasible basic mechanisms for the RBSC process are examined. It was initially believed that, after a thin film of silicon carbide formed on the carbon surface, diffusion of silicon or carbon through solid SiC (the
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melting point of silicon carbide, 2100 K, is much higher than that of silicon, 1687 K) is responsible for further conversion.14,15 Diffusion through a grain boundary was invoked to explain the fast reaction generally observed. It was later proposed that the initial silicon carbide layer spalls off because of the volume increase caused by the reaction, exposing a fresh unreacted carbon surface to liquid silicon. Further conversion was then postulated to occur by dissolution of carbon into molten silicon at higher temperatures, followed by the precipitation of silicon carbide in regions of lower temperatures.16-18 Diffusion-Controlled Mechanisms. The “shrinkingcore” model assumes that silicon diffuses through the solid SiC and reacts with carbon at the C-SiC interface. This mechanism has been associated with some of the early work on RBSC in Germany14,15,19 where SiC was observed to grow as a continuous layer on graphite plates. In this model, the silicon carbide layer is assumed to form via a mechanism similar to that involved in the oxidation of p-type metals.20,21 This model is simple, and a closed form solution can be obtained for the first-order kinetics.22 The “bulging-core” model, on the other hand, assumes that carbon diffuses through SiC and reacts with Si at the SiC-melt interface. Liquid-Phase Reaction Mechanism. This mechanism was initially proposed by Pampuch et al.18 and later supported by Chiang et al.23 It assumes that carbon first dissolves in the molten silicon by a self-accelerated (autocatalytic) mechanism. The dissolved carbon then diffuses to the cooler-temperature region of the melt, where, because of reduced solubility, it precipitates as SiC particles. This process can also lead to the growth of larger SiC particles at the expense of smaller ones. In an attempt to verify this mechanism, Chiang et al.23 carried out experiments similar to those reported by Fitzer and Gadow19 but focusing on the onset of the process (short times, i.e., first-order-dynamics phase). A correlation between the temperature dependence of carbon solubility in silicon melt and that of the initial SiC growth rate was used to support this mechanism. Analysis of Literature Data. Fitzer and Gadow19 measured silicon carbide growth rates in carbon plates in contact with molten silicon at 1873 and 2073 K. More recently, similar experiments have been reported24 at lower temperatures, 1703-1783 K. These data can be analyzed through the bulging- and/or shrinking-core models. Indeed, these models yield the following nonlinear model for siliconizing growth rates
∆ ) ∆o - β + xβ2 + 2Rt with
β ) DSi/SiCek˜ or β ) DC/SiCek˜
(6)
and
R ) FSi/FC or R ) FC/FSi where ∆ denotes the SiC layer thickness, t represents time, kr is the reaction rate constant, km is the mass transfer coefficient, and De stands for the effective diffusion coefficient. Although the model breaks down for ∆o ) 0, it provides a useful tool for examining the controlling mechanism(s) in the RBSC process. The parameters R
Figure 4. Experimental siliconizing data adapted from Fitzer and Gadow.19
and β can readily be associated with well-known kinetic and transport criteria. Namely, β is the inverse of the Damkohler number, and R is associated with the diffusion characteristic time. A critical examination of the data from Fitzer and Gadow19 and Zhou and Singh24 can then be provided in light of this model. Figure 4 shows the data and the linear correlation corresponding to the bulging- and shrinking-core models. Messner and Chiang12 proposed the following model for pore shrinkage
r(t) ) Ro - ∆(t) with ∆(t) ) k1t or ∆(t) ) k2xt
(7)
where the constants k1 ) R/β and k2 ) (2R)1/2 can be obtained from experimental data as shown above. These experimental data do not support the assumption of kinetically controlled growth (linear with time), as was originally assumed by Messner and Chiang:12 the Damkohler number, 1/β, cannot be statistically distinguished from infinity, which suggests a diffusioncontrolled mechanism. However, it might be that this absence of a kinetically controlled stage is due to the lack of data at short times or the existence of a condensed-phase reaction during the preheating/melting stages. Experiments at different temperatures can be used to correlate the effective diffusion coefficient of carbon through silicon carbide via an Arrhenius dependence on temperature. Support for the Si-diffusion-based mechanism came from the observation that the siliconized carbon approximately retained its original dimensions as it was converted to SiC. This model seems to correlate well with results reported in Russian literature.25 However, it is in conflict with more recent experimental observations. For example, it is inconsistent with the phenomenon of pore closure (choking) observed experimentally during siliconization of porous carbon preforms. It is also contradictory to the observation of carbon dissolution during siliconization of carbon fibers. The mechanism based on the diffusion of C, on the other hand, would be consistent with (a) the phenomenon of preform choking during silicon infiltration,12 (b) the growth of a continuous silicon carbide layer observed on graphite plates in contact with the silicon melt,14,15 and (c) the observation of carbon fiber dissolution.18,23 The molar volume increase associated with conversion of a carbon “particle” into silicon carbide (about 58% in volume) reduces the size of the “interparticle” region, containing the silicon melt. The resulting decrease in the permeability will ultimately choke further infiltra-
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tion of silicon melt. At first, the observation of carbon dissolution might appear inconsistent with the formation of a continuous layer of silicon carbide. However, as discussed below, this conclusion is misleading because these observations were made under two very different sets of experimental conditions. The carbon dissolution was observed for carbon fibers in contact with silicon melt, whereas formation of the continuous SiC layer was observed for carbon plates. A thin layer of silicon carbide instantaneously forms as soon as carbon is brought in contact with silicon. This SiC acts as a barrier for further reaction, and the process becomes diffusion-controlled. It is generally believed that this layer is subject to increasing tensile stresses with its further growth. The tensile strength of silicon carbide is between 50 and 200 MPa. Experiments in which dissolution of carbon and spalling off of the silicon carbide layer were reported were on carbon fibers 10100 µm in diameter (or porous preforms). It is obvious that the radial stresses in a ring-like silicon carbide layer will be far larger than those in a thin (0-50 µm) silicon carbide layer growing on a flat graphite plate (0.1 cm thick). The spalling, therefore, is likely to occur during the early stages of siliconizing of fibers or porous structures, whereas the process of siliconizing graphite plates will not exhibit the spalling phenomena. Messner and Chiang12 identified a transition in the growth dynamics. An apparent linear correlation consistent with a dissolution mechanism was confirmed in a later paper.3 A closer inspection, however, of the experimental data reported by Chiang et al.3 clearly shows nonzero intercepts for the linear correlation. This is clear evidence of the existence of a protective layer and, therefore, the presence of a diffusion mechanism. For reasons similar to those explained above, this protective layer is not likely to spall in the siliconizing of small-sized pores. Moreover, growth data corresponding to diffusion-controlled mechanisms will appear distorted if plotted as degree of carbon conversion as a function of time and will then be misleading. The square-root dynamics typical of diffusion-limited processes will thus be masked. To compound the problem, the dissolution process is highly exothermic and, therefore, should accelerate itself until mass transport becomes the controlling mechanism or heat conduction promotes an equilibrium. This calls for a more detailed analysis of the correlation between deposition dynamics and carbon dissolution. Moreover, the important question of how the cycle of dissolution and precipitation is terminated remains unanswered. These observations suggest a “hybrid” mechanism, which would be consistent with the arguments supporting the above-described theories. A Hybrid Stress-Controlled Mechanism. The diffusion mechanism is consistent with many of the experimental observations. However, a realistic RBSC model has to include the stress-induced “spalling off” feature discussed earlier. The diffusive growth of SiC will introduce tensile stresses in the SiC layer. As explained above, these stresses will rapidly exceed the tensile strength of SiC, and the film will crack and/or spall. This will then be followed by a rapid reaction stage due to the exposure of fresh carbon surface to molten silicon. A new diffusion layer will be formed and the diffusioncontrolled stage will follow until the stresses again expose fresh carbon. The SiC fragments will precipitate
in the bulk phase of the silicon melt and act as nucleation sites for further SiC precipitation/growth. This mechanism, although “simplistic” in appearance, would add the displacement equations and precipitation kinetics to the above diffusion mechanism. However, it presents a more realistic representation of the reaction during silicon infiltration of large-pore carbon preforms. Preforms with a small pore size will most likely not manifest this mechanism. Indeed, the growth of cylindrical silicon carbide shells is accompanied by stresses that can exceed the material tensile strength. The siliconizing of fibers creates silicon carbide layers subject to tensile stresses against a free surface, which therefore can exhibit spalling. For the silicon carbide growth on the inside of carbon pores, on the other hand, silicon carbide layer breakage is contained by the unconverted carbon, and therefore, the layer can grow to larger thicknesses before the hollow SiC cylinder collapses. A Modified Model and its Numerical Solution. In summary, diffusion-limited SiC growth appears to be the most feasible mechanism in the siliconizing of microporous carbon preforms. The infiltration analysis presented earlier dealt with a single-pore representation of a structure consisting of a randomly oriented array of multisize pores. The pores were considered to remain unaltered during the infiltration process. However, infiltration of silicon through microporous carbon is accompanied by structural changes in the pores, which calls for a dynamic percolation model. Any predictive tool, therefore, cannot be based on static measurements alone (permeability experiments) but must also include the dynamics of the structure changes. Messner and Chiang12 proposed a simple model for pore shrinkage
r(t) ) Ro - ∆(t) with ∆(t) ) k1t or ∆(t) ) k2xt where the constants k1 ) R/β and k2 ) (2R)1/2 can be obtained from experimental data as described above. However, in addition to the difficulty of justifying the linear assumption (i.e., a kinetically controlled siliconizing process), this model has a major drawback: the above equations, valid for flat plates, cannot be used for cylindrical fibers without a proper reformulation. For a cylindrical geometry, they should be
r(t) ) Ro - ∆(t) ) Ro(1 - δ(t))
(8)
with δ(t) being the solution to
(δ - 1)2 [1 - 2 ln(1 - δ)] ) 1 -
4R t Ro2
(9)
This simple modification predicts substantially different dynamics compared with that predicted by assuming a simple extension of the plate results to fiber bundles. Changes in the infiltrated structure as the infiltration proceeds have been treated at several levels of complexity: from infiltration of a porous preform by a pure metal or an alloy where reactions at the fluid-preform interface are neglected26,27 to the mechanisms where nucleation, thermophoresis, and kinetics interact to deposit films on the pore walls.28,29 However, reactive infiltration has received limited attention. The infiltration of a channel with a time-dependent cross-sectional area is governed by equations far more complex than
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Figure 6. Infiltration dynamics vs infiltration height. Numerical results for infiltration of type B preforms.
Figure 5. Schematic of an infiltrated pore.
eq 3, and the simplifications leading to simple permeability-structure relationships can no longer be made (i.e., the simple geometric model for permeability in which the square of the infiltration length is proportional to time can not be recovered). The dynamics of the infiltration process is then described by a modified formulation of Newton’s law
∑F(z,h) ) F ∂t(πr2h ∂t ) ∂
∂h
(10)
or, in detail
FV(t)
d2h ) A(h)γLV cos(θ) - FV(t)g dt2 1 A(0,t)F v(0,t)2 - FD(t) (11) 4
where z is the infiltration direction, r is the radius of the pore, and h is the position of the silicon melt meniscus. A and V represent the cross-sectional area and volume of the silicon melt, respectively; the crosssectional area will depend on both the position along the infiltration path and the time. FD represents the viscous drag effects. Here, surface tension, gravitational, viscous-drag, and end-effect forces will be time-dependent as they are interrelated to structural changes. Moreover, the infiltration of a single pore will then necessitate two additional governing equations, a mass balance and an energy balance, which would govern the dynamics of the infiltration geometry. To compound the difficulty introduced by these two equations, the description of the infiltration domain requires the redefinition of a dynamic three-region domain. Figure 5 presents an schematic of the different regions identifiable in the reactive infiltration of a single pore, i.e., a carbon layer, a silicon carbide wedge and flowing liquid silicon. A detailed formulation of the corresponding mathematical model and boundary conditions is beyond the scope and extent allotted for this paper, and it will be published elsewhere. The numerical solution of the governing momentum, mass, and energy transport governing equations is better suited by a method that treats the different regions illustrated in Figure 5 separately. The authors have adopted orthogonal collocation on finite elements (OCFEM),30 a method that has been reported to be successful in many diffusion-reaction problems. This method is particularly suited to represent many of the qualitative features of the problem, it results in a robust and simple formulation, and it is generally more accurate than finite difference (FD) approaches while being simpler than more traditional finite element
formulations. The time integration is carried over using a standard adaptive fourth-order Runge-Kutta method, with the domain being updated at each integration step. Typical results are presented in Figure 6. These results intend to mimic the infiltration of type B preforms. The dynamics corresponding to the infiltration height (h), pore radius (r), and reaction-diffusion interfaces (C/SiC/Si) are presented in a format depicting the predicted structure at the time that pore closure occurs. To avoid dealing with the singularity of the reaction at time t ) 0, the initial infiltration stage was correlated to experimental data extracted from Fitzer and Gadow19 and the analytical solution to the momentum equation. Despite these assumptions, the numerical solution shows a striking resemblance to the experimental results. Indeed, the infiltration lengths and times are on the order of magnitude of experimental observations (i.e., infiltration lengths of 1-2 cm are predicted, with pore closure occurring in the first 10 s of infiltration, cf. Figure 3). Moreover, the pore-closing and free silicon entrapment predictions (consistent with the evidence of free silicon found for both types of preforms) are also clearly demonstrated by the numerical solution. Issues related to the initial reactioninfiltration stages and thermal interaction between the different regions, physical properties, and diffusionreaction parameters remain unresolved by the present model. These phenomena might be responsible for the instabilities and high parametric sensitivity observed experimentally but not captured by the formulated mathematical model. The above analysis, however, is based on isothermal observations. For the small fluid (melt) volume contained inside the pores, a strong interaction with flow and transport phenomena will occur. The analysis of such an infiltration process would most likely require additional dynamic measurements in order to further understand the siliconizing mechanism, and it will be the subject of future contributions. The macrodynamics of the infiltration-reaction process has received very limited attention. Some efforts in this direction have recently been reported in the literature.31 These studies involved the use of commercial CFD software (FIDAP) to model mass and energy transport phenomena in a regular arrangement of parallel graphite fibers in contact with a silicon melt. Flow was assumed to occur parallel to the fiber axes, and a two-dimensional concentration and temperature distribution was formulated. The reaction was assumed to be under kinetic control and the initial (only the dynamic analysis for the first 0.1 µs was reported) stages of the Si/C counterdiffusion process were simu-
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lated. Although the permeability was related to the cross-sectional area change due to the reaction, the dynamics of the reactive infiltration process was not analyzed. The authors concluded that a phenomenological model (similar to the one proposed above) where carbon dissolution and breakage of the SiC layer combine to complete the siliconizing process was needed. Nevertheless, the unidirectional flow assumption and the ordered geometry reduce the applicability of these results to the problem analyzed in this paper. Conclusions The results indicate the need for a dynamic percolation model that includes structural changes in the pores. The fluid contained within the microchannels (1-5 µm) interacts thermally with the solid skeleton, and the local fluid properties change drastically with the sharp temperature rise. The species distribution is also influenced. Structural changes are local phenomena, strongly influenced by the temperature and species distribution. Therefore, any predictive tool must include the dynamics of the flow environment, thermal interactions between the melt and the solid, temperature effects on the siliconizing process, and residual reaction/thermal effects resulting from of the free silicon entrapped ahead of the choking front. In other words, a model based purely on flow analysis considerations, even when the pore structure is adapted dynamically to account for the occurrence of a chemical reaction, might not be adequate for predicting all features of a reactive infiltration dynamics. Acknowledgment Alppreciation is expressed to Pat Dickerson, Dave Epperly, Jim Barker, and Christopher Palda for their technical assistance and to Thomas K. Glasgow, Chief, Processing Science and Technology Branch, at the NASA Glenn Research Center for partial support of this research. One of the authors (J.E.G.) gratefully acknowledges support from the Established Full-time Faculty Research Development (EFFRD) program at Cleveland State University. Dr. Orejas acknowledges the support given by the Secretarı´a de Ciencia y Te´cnica Universidad Nacional de Rı´o Cuarto (UNRC) and by the Consejo de Investigaciones Cientı´ficas y Te´cnicas de la provincia de Co´rdoba (CONICOR), Argentina. Literature Cited (1) Popper, P. The Reaction of Dense Self-bonded Silicon Carbide. In Special Ceramics; Heywood: London, 1960; p 209. (2) Mehan, R. L. Effect of Silicon Carbide Content and Orientation on the Properties of Si/SiC Ceramic Composite. J. Am. Ceram. Soc. 1977, 60 (3-4), 177. (3) Chiang, Y. M.; Messner, R. P.; Terwilliger, C. D.; Behrendt, D. R. Reaction Formed Silicon Carbide. Mater. Sci. Eng. 1991, A144, 63. (4) Singh, M.; Behrendt, D. R. Studies on the Reactive Melt Infiltration of Silicon and Silicon-Molybdenum Alloys in Porous Carbon; NASA Report TM-105860; National Aeronautics and Space Administration: Washington, D.C., 1992. (5) Sangsuwan, P.; Tewari, S. N.; Gatica, J. E.; Singh, M.; Dickerson, R. Reactive Infiltration of Silicon Melt Through Microporous Carbon Preforms. Metall. Trans. 1999, 30B, 933. (6) Simhai, N.; Datta, S. K.; Singh, M.; Dickerson, R.; Smith, J.; Gatica, J. E.; Tewari, S. N. Characterization of Microporous Preforms: Analysis of Permeability. Metall. Mater. Trans. A 1996, 27, 3669. (7) Ergun, S. Fluid Flow through Packed Columns. Chem. Eng. Prog. 1952, 48, 89.
(8) Jackson, G. W.; James, D. F. The Permeability of Fibrous Porous Media. Can. J. Chem. Eng. 1986, 64, 364. (9) Dullien, F. A. L. Porous Media. Fluid Transport and Pore Structure; Academic Press: New York, 1979. (10) Martins, G. P.; Olson, D. L.; Edwards, G. R. Modeling of Infiltration Kinetics for Liquid Metal Processing of Composites. Metall. Trans. 1988, 19B, 95. (11) Hillig, W. B. Melt Infiltration Process for Making Ceramic Composites. In Fiber Reinforced Ceramic Composites. Materials, Processing and Technology; Mazdiyasni, K. S., Ed.; Noyes Publications: Park Ridge, NJ, 1990; pp 260-277. (12) Messner, R. P.; Chiang, Y. M. Liquid-Phase ReactionBonding of Silicon Carbide Using Alloyed Silicon-Molybdenum Melts. J. Am. Ceram. Soc. 1990, 73 (5), 1193. (13) Einset, E. O. Transport and Reaction in Silicon Melt Infiltration of Porous Carbon Preforms. J. Am. Ceram. Soc. 1996, 79 (2), 333. (14) Fitzer, E.; Gadow, R. Fiber Reinforced Silicon Carbide. Ceram. Bull. 1986, 65 (2), 326. (15) Minnear, W. P. Interfacial Energies in the Si/SiC System and the Si + C Reaction. Am. Ceram. Soc. Commun. 1982, Jan, C-10-11. (16) Ness, J. N.; Page, T. F. Microstructural Evolution in Reaction Bonded Silicon Carbide. J. Mater. Sci. 1986, 21, 1372. (17) Hase, T.; Suzuki, H.; Isekei, T. Formation Process of β-SiC During Reaction Sintering. J. Nucl. Mater. 1976, 59, 42. (18) Pampuch, P.; Walasek. E.; Bialoskorski, J. Reaction Mechanism in Carbon-Liquid Silicon Systems at Elevated Temperatures. Ceram. Int. 1986, 12, 99. (19) Fitzer, E.; Gadow, R. Investigation of the Reactivity of Different Carbons with Liquid Silicon. In Proceedings of the International Symposium on Ceramics for Engines; Technoplaza Co. Ltd.: Tokyo, Japan, 1983; pp 561-572. (20) Rode, H.; Hlavacek, V.; Viljoen, H. J.; Gatica, J. E. Combustion of Metallic Powders: A Phenomenological Model for the Initiation of Combustion. Combust. Sci. Technol. 1993, 88, 153. (21) Zhou, H.; Singh, R. N. Reaction Kinetics and Mechanism in Si-C and Si-C-Mo Systems. In Advances in Ceramic-Matrix Composites; American Ceramic Society: Westerville, OH, 1993; pp 91-102. (22) Levenspiel, O. Chemical Reaction Engineering; John Wiley & Sons: New York, 1972; Chapter 12. (23) Chiang, Y. M.; Haggerty, J. S.; Messner, R. P.; Demetry, C. Reaction-Based Processing Methods for Ceramic-Matrix Composites. Ceram. Bull. 1989, 68 (2), 420. (24) Zhou, H.; Singh, R. N. Kinetics Model for the Growth of Silicon Carbide by the Reaction of Liquid Silicon with Carbon. J. Am. Ceram. Soc. 1995, 78 (9), 2456. (25) Lukin, B. V. Determination of the Linear Velocity of the formation of β-SiC. Izv. Akad. Nauk SSSR 1971, 7 (7), 1169. (26) Mortensen, A.; Masur, L. J.; Cornie, J. A.; Flemmings, M. C. Metall. Trans. A. 1989, 23, 2535. (27) Biswas, D. K.; Gatica, J. E.; Tewari, S. N. Dynamic Analysis of Pressure Infiltration Processes. Ceram. Trans. 1995, 58, 105-110. (28) Jensen, K. F. Micro-Reaction Engineering: Applications of Reaction Engineering to Processing of Electronic and Photonic Materials. Chem. Eng. Sci. 1987, 42, 923. (29) Gomez, A.; Rosner, D. E. Combust. Sci. Technol. 1993, 89, 335 (see also Koylu, O. U.; Tandon, P.; Xing, Y.; Rosner, D. E. Experimental and Theoretical Studies of the Structure of Inorganic Particle-Producing Seeded Laminar Counterflow Diffusion Flames. Presented at the Annual AIChE Meeting, Miami, FL, 1995; Paper 59p. (30) Finlayson, B. A. Nonlinear Analysis in Chemical Engineering; McGraw-Hill, New York, 1980. (31) Rajesh, G.; Bhagat, R. B.; Nelson, E. Modeling the flow of molten silicon in porous carbon preforms and the subsequent formation of silicon carbide. In Ceramic Matrix Compositess Advanced High-Temperature Structural Materials; Lowden, R. A., Ferbere, M. K., Hollmann, J. R.,Chawla, K. K.,DiPietroand, S. G., Eds.; MRS Symposium Proceedings 365; Materials Research Society: Pittsburgh, PA, 1995; pp 147-152.
Received for review December 5, 2000 Revised manuscript received June 6, 2001 Accepted June 6, 2001 IE001029E
