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J. Phys. Chem. C 2007, 111, 10829-10835

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Modeling Silicon Carbide Synthesis on a Submicrometric Scale A. Lemarchand*,† and J. P. Bonnet‡ UniVersite´ Pierre et Marie Curie - Paris 6, UMR 7600 LPTMC, Paris, F-75005 France, CNRS, UMR 7600 LPTMC, Paris, F-75005 France, and Ecole Nationale Supe´ rieure de Ce´ ramique Industrielle, GEMH, E.A. 3178, 47-73, aVenue Albert Thomas, 87065 Limoges, France ReceiVed: February 23, 2007; In Final Form: May 21, 2007

We build a model, on a mesoscopic, submicrometric scale, describing the formation of silicon carbide during heating at a constant rate and holding at a temperature smaller than the eutectic temperature of carbon and silicon. Simulating a two-dimensional cut of the initial mixture of powders, we show that the mean number of contacts of a carbon plate with silicon increases as the typical size of the silicon grains decreases. We focus on the simulation of the dynamics of the reaction when the melting of silicon and the dissolution of carbon begin until some solid silicon remains. Precipitation of silicon carbide is assumed to obey a heterogeneous nucleation mechanism. The model sheds some light on the dynamics of dissolution-precipitation in self-heated synthesis and on the morphology of the resulting material. In agreement with the experiments, the simulations reproduce larger reaction speeds for smaller silicon grains. Larger silicon carbide grains are obtained when the typical sizes of silicon and carbon grains coincide.

Introduction Control of the submicrometric properties of advanced materials like ceramics or cermet is the main challenge of selfpropagating high-temperature synthesis (SHS).1-5 The mechanisms governing SHS have the complexity of condensed-phase reactions. Two different reaction modes can be obtained. If the reaction is initiated locally, at one end of the sample, a solid flame propagates through the sample. On the contrary, a homogeneous heating ignites a thermal explosion simultaneously in the whole sample. If one excepts the involved time-resolved diffraction experiments,6 experimental mechanistic investigations of self-sustained waves are usually indirect and require combustion front quenching methods. The quenched samples are then analyzed by scanning electron microscopy and X-ray diffraction.7-10 The extreme conditions and the speed of the reaction prevent one from easily obtaining reliable information on the dynamics. As an alternative, the formation of ceramics during homogeneous heating at a constant rate has been investigated,11-13 in order to study the mechanisms occurring during thermal explosion. During a sufficiently slow temperature ramp, the reactive medium can be efficiently cooled and the intermediate stages of the system can be analyzed in a more satisfactory way than in a violent SHS experiment. We choose the synthesis of silicon carbide (SiC) as a prototype of reaction following a dissolution-precipitation mechanism.14-16 We focus on the case of thermal explosion and, more particularly, on the stage of the reaction where carbon dissolves in liquid silicon and silicon carbide precipitates preferentially at a solid/liquid interface. Up to now, modeling of high-temperature synthesis has been based on an uncoupling between micro- and macrokinetic aspects.17-19 Our aim here is to reproduce the * Corresponding author. Address: Universite´ Pierre et Marie Curie Paris 6, CNRS, UMR 7600, Laboratoire de Physique The´orique de la Matie`re Condense´e, 4 place Jussieu, case courrier 121, 75252 Paris cedex 05, France. Tel: 33(0)144277290. Fax: 33(0)144277287. Electronic mail: [email protected]. † Universite ´ Pierre et Marie Curie - Paris 6 and CNRS. ‡ Ecole Nationale Supe ´ rieure de Ce´ramique Industrielle.

initial distribution of the reactant grains in the reactive compact, the dynamics of the explosive reaction, and the morphology of the ceramic on a mesoscopic, submicrometric scale. A comparison of the predictions of the theoretical approach with the experimental results11-13 will offer a good test of the model. We recently proposed a model to simulate the initial mixture of silicon and carbon grains.20 For a given chemical nature, the grains considered were all of the same size, and we could not reach the desired porosity for the resulting mixture of powders. Here we introduce distributions of the grain size in order to bring us as close as possible to the experimental conditions. The paper is organized as follows: In the second section we recall experimental results on the formation of silicon carbide during heating of the sample at a constant rate. In the third section we present a method for simulating the mixture of carbon and silicon powders and determine the initial number of contacts between the reactants as a function of the typical size of silicon grains. In the fourth section we propose an algorithm to simulate the dynamics of formation of silicon carbide, and we study the influence of silicon granulometry on the reaction speed and the size of the silicon carbide grains. The fifth section is devoted to conclusions. Experimental Results and Hypotheses We consider the exothermic formation of silicon carbide from a mixture of powders of silicon and graphite according to

Si + C f SiC + heat

(1)

We first recall experimental results11 obtained during heating at a rate of 15 °C per minute and holding at a temperature smaller than the eutectic temperature (Te ) 1404 °C) of carbon and silicon. The experiments are carried out for a mixture of carbon and silicon powders in a molar ratio of 1:1 and a porosity of 50%. Note that a carbon excess is used in order to compensate the effect of carbon combustion due to the presence of oxygen traces in the surrounding argon. Unless otherwise stated, the

10.1021/jp071520k CCC: $37.00 © 2007 American Chemical Society Published on Web 07/04/2007

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Lemarchand and Bonnet

same type of natural graphite powder and two silicon powders differing by the size of their grains are used. Electronic microscopy images of the powders reveal that the carbon grains look like hexagonal plates, whereas the silicon grains can be approximated by spheres. Instead of the mean value and the variance that would be sufficient to define a symmetrical Gaussian distribution function, the experimentalists use three quantities to characterize possibly nonsymmetrical distribution functions of the grain size. The two different mass-distribution functions of the silicon powders used in the experiments obey

r10 ) 1.15 µm, r50 ) 2.85 µm, r90 ) 6.8 µm for the small grains (2) r10 ) 11.65 µm, r50 ) 16.95 µm, r90 ) 27.1 µm for the large grains (3) where the value of rx expresses that x% of the mass of silicon consists of grains of radius smaller than rx. In other words, rx obeys

x ∫0r m(r)H(r)dr ) 100 ∫0∞ m(r)H(r)dr x

(4)

where m(r) is the mass of the grains of radius between r and r + dr, and H(r) is the number-distribution function, that is, the number of grains of radius between r and r + dr. By definition, r50 is the median value of the radius of the silicon grains according to the mass-distribution function. The mass-distribution function of the carbon plates used in the experiments has the following characteristics:

l10 ) 4.7 µm, l50 ) 10.5 µm, l90 ) 20.8 µm

(5)

where the value of lx expresses that x% of the mass of carbon consists of grains of typical length smaller than lx. Figure 1a gives the experimental results observed when the samples of powder mixtures are brought at a constant rate of 15 °C min-1 up to a maximal temperature Tmax ranging from 1325 to 1400 °C. The maximal temperature remains smaller than the eutectic temperature. When Tmax is reached, the sample is immediately cooled, and the conversion rate in SiC is deduced from X-ray diffraction patterns.11 Figure 1b represents the evolution of the conversion rate during holding at 1400 °C. The maximum temperature reached during a ramp at a constant rate can be identified with time. Using Figure 1a,b, we give in Figure 1c a schematic representation of the SiC conversion rate versus time during holding at Tmax ) 1400 °C. We can define three domains associated with qualitatively different behaviors. Domains I and III, at the beginning and end of the reaction, respectively, are associated with slow reaction regimes that are nearly independent of the silicon grain size. These domains would certainly be shorter during SHS than during a slow temperature ramp or holding at 1400 °C. In the following, we discard these two slow regimes.11,20 The most interesting behavior is observed in domain II, which is associated with the fast reaction regime. The slope of the curves in domain II would probably be larger in an actual SHS experiment, since the conditions would be more violent and the formation of SiC would be faster. Nevertheless, the qualitative behavior of the conversion rate and, in particular, the dependence on the size of the silicon grains are supposed to be the same during thermal explosion as during the model experiments reported in Figure 1a,b. With regard to dynamics, the main experimental result is that the speed of the reaction is sensitively influenced by the

Figure 1. Experimental percentage of silicon carbide formed11 versus maximal temperature reached during heating at constant rate (15 °C min-1) followed by immediate cooling (a) and versus time during holding at 1400 °C (b). Carbon and silicon were in a molar ratio of 1:1. The porosity of the initial mixture was 50%. The powder of carbon used was natural graphite (Alfa Aesar, ref 14736) with plates of median length l50 ) 10.5 µm. The triangles give the results obtained for small silicon grains (Fluka) of median radius r50 ) 2.85 µm. The squares represent the results for larger silicon grains (Fluka) of median radius r50 ) 16.95 µm. (c) Schematic representation of the SiC conversion rate versus time for an equimolar mixture of silicon and carbon during holding at 1400 °C. The solid line corresponds to smaller Si grains relative to those represented by the dashed line for identical carbon grains. A fast regime (II) between two slow regimes (I and III) is identified.

granulometry of the silicon powder and that the reaction is faster for smaller silicon grains. The experimental observations11-16 incite us to formulate the following hypotheses: The local heat release that accompanies reaction 1 results in the local melting of silicon. Carbon then dissolves in liquid silicon, and the formation of silicon carbide is assumed to follow a dissolution-precipitation mechanism. We admit that dissolution is fast compared to precipitation so that the speed of the reaction is imposed by precipitation. Heterogeneous nucleation of SiC is observed at the solid-liquid interfaces, that is, at the interface between the eutectic liquid and carbon or solid silicon. During a sufficiently slow temperature ramp and holding, we can assume that the temperature of the sample remains constant: we admit that the reaction is sufficiently slow to ensure the relaxation toward a partial and local equilibrium. Until some solid silicon remains, we admit that the temperature is fixed at the eutectic temperature Te. This last hypothesis is probably the less reliable one in the case of SHS, where nonequilibrium conditions are imposed by the violence of the heat release. Simulation of an Initial Mixture of Powders We aim to reproduce an initial mixture of powders of carbon and silicon on a mesoscopic scale. We simulate a bidimensional (2D) cut of the mixture. In the 2D cut, carbon is represented by sticks of width ∆l ) 200 nm fixed by the typical width of the graphite plates used in the experiments. In the simulation, a given stick is defined by four real numbers: the two coordinates of one of its tips, its length l, and its angle with the horizontal line. A silicon grain is represented by a disk defined by three real numbers: the two coordinates of its center and its radius.

Modeling SiC Synthesis

Figure 2. Number-distribution function H(l) of the length l of the simulated carbon sticks leading to a mass-distribution function with l10 ) 4 µm, l50 ) 10 µm, l90 ) 21 µm.

A given configuration of a 2D cut of the initial mixture of powders is generated as follows: A number ns of silicon disks are generated without overlap in a square box whose side is adjusted to obtain a porosity of 50% for the final mixture. Periodic boundary conditions are applied. We choose ns close to 20 for all the configurations. We introduce the two parameters rS and σS, which respectively control the mean value and the width of the radius distribution of the silicon disks. The disks are added one by one: the three real numbers that characterize one disk are randomly chosen and accepted only if they lead to a disk that does not overlap the previously generated disks. Note that the mass distribution obtained can only be characterized after the total number, ns, of disks has been created; the values of r10, r50, and r90 are then determined. If the mass-distribution function obtained differs from the expected shape, the procedure is repeated for a different value of the parameters rS and σS until the desired mass-distribution function, with r10, r50, and r90 close to the experimental values, is obtained. Then a number nC of carbon sticks are generated without overlap, with nC chosen to obtain an equimolar mixture. The procedure followed to create the carbon sticks is similar to the one adopted to generate the disks. The sticks are created one by one according to a number-distribution function controlled a priori by the mean value lC and the variance σC. The four real numbers that characterize one stick are randomly chosen according to this distribution function but are finally accepted only if they lead to a stick that does not overlap the previously generated disks and sticks. This condition induces a bias. We give in Figure 2 the number-distribution function H(l) of the length l of the carbon sticks deduced by the simulations. It is to be noted that the number-distribution function is not symmetric due to the constraint of nonoverlapping, that favors the creation of small sticks. The number-distribution function obtained is a posteriori examined to check whether it agrees with the experimental conditions. The corresponding massdistribution function obeys l10 ) 4 µm, l50 ) 10 µm, and l90 ) 21 µm, close to the characteristics of the graphite powder used in the experiments and given in eq 5. We generate configurations

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Figure 3. Initial configuration of an equimolar mixture of ns ) 21 silicon disks (in white) and R ) 652 carbon sticks (in black) of width ∆l ) 200 nm. The size of the box is adjusted to obtain a porosity of 50%. The mass-distribution of the silicon disks obeys r10 ) 6 µm, r50 ) 7 µm, r90 ) 8 µm. The distribution function of the carbon sticks is the same as in Figure 2.

for different values for the median radius r50 of the silicon disks and the same values for the other parameters. The generation of a configuration does not require the introduction of a length cutoff. However we wish to determine the mean number of contacts NC/Si between a carbon stick and silicon for different median values of the silicon radius in the range 1.3 µm e r50 e17 µm. To this aim it is convenient to discretize space. We choose the width ∆l ) 200 nm of the carbon sticks as the site length. Figure 3 gives an example of a discretized configuration obtained for r50 ) 7µm. A site is considered a contact site between carbon and silicon when the distance between a stick and a disk is smaller than or equal to ∆l. For a given configuration with nS disks of silicon, NC/Si is defined as the ratio of the number of contact sites and the number nC of carbon sticks: NC/Si is the mean number of contacts of a carbon stick with silicon. In the range of the values of r50 used in the experiments, we find that the mean number of contacts NC/Si monotonically decreases when the median value r50 of the silicon disk radius increases. Quantitatively, 1/NC/Si linearly increases with r50, as shown in Figure 4. We checked the robustness of this result by changing the value of l10, that is, the proportion of small carbon sticks, which sensitively modifies the number of contacts. As expected, the number of contacts decreases as l10 increases, but the same linear variation of 1/NC/Si versus r50 is observed with an identical slope. The main result of this section is the following: the simulation performed for a stoichiometric mixture of reactants reveals that the mean number of contacts of a carbon plate with silicon increases as the typical size of the silicon grains decreases. Simulation of the Dynamics of the Reaction To simulate the formation of silicon carbide on a mesoscopic scale, we start from a discretized configuration of the mixture of carbon and silicon powders obtained according to the procedure given in the previous section. Initially, four different

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Figure 4. Reciprocal 1/NC/Si of the number of contacts between carbon and silicon referred to one graphite stick versus the median value r50 of the silicon disk radius. The simulations are performed for equimolar mixtures of C and Si of porosity equal to 50%. Solid squares represent the distribution function of carbon given in Figure 2 (l10 ) 4µm). Crosses correspond to a mass-distribution function of carbon characterized by l10 ) 7 µm with l50 and l90 unchanged.

natures of sites can be defined: the pores, the carbon sites, the solid silicon sites, and the contact sites between carbon and solid silicon. When the sample is brought to a temperature smaller than the eutectic temperature, we assume that the exothermic reaction (eq 1) is initiated at these contact sites that are considered nucleation sites. Then silicon carbide sites are formed and heat is released locally, leading to the formation of liquid silicon in which carbon slowly dissolves. The eutectic liquid is assumed to wet the skeleton of carbon sticks. We introduce three different types of liquid sites: (i) the reactive sites RC with a side in common with a carbon site, (ii) the reactive sites RS with a side in common with a solid silicon site or a silicon carbide site, and (iii) the other liquid sites. Since the molar volume of silicon and silicon carbide are very close,20 the consumption of reactive sites leads to the creation of the same number of silicon carbide sites. We neglect the consumption of the carbon sites. Considering the standard enthalpy of reaction at the eutectic temperature (∆RH° ) -3.424 × 106 J kg-1) and the enthalpy of fusion of silicon (∆fH° ) 1.787 × 106 J kg-1), we find that the mean number of liquid sites created for each formation of a SiC site is xj ) -∆RH°FSiC/ ∆fH°FSi ) 2.65, where the densities of SiC and Si are respectively given by FSiC ) 3220 kg m-3 and FSi ) 2330 kg m-3. Following a Monte Carlo procedure, we transform the reactive sites RC and RS into SiC sites with probabilities pC and pS during the time step ∆t. The reactive sites RC and the sites RS with a side in common with a solid silicon site are nucleation sites. The reactive sites RS with a side in common with a silicon carbide site are growth sites. For the sake of simplicity, we assume that the initial contact sites between solid silicon and carbon react with the same probability pC as the liquid silicon sites with a side in common with carbon: all these contact sites are considered reactive sites RC. We also admit that the two

Lemarchand and Bonnet types of reactive sites RSsthe nucleation sites in contact with solid silicon and the growth sites in contact with SiCsreact with the same probability pS. The allocation of physical values to pC + pS or equivalently to the time scale would require knowledge of the activation energy of the reaction. We impose pS > pC to reproduce the preferential precipitation of silicon carbide on Si and SiC surfaces rather than on carbon surfaces.11 In the frame of the hypotheses of the model, we have checked20 that the simulation results are not sensitive to the value of pC/ pS in the range 0.1 e pC/pS e 10. Here we have arbitrarily fixed pC ) 0.2 and pS ) 0.5. To follow the reaction kinetics, we do not need to simulate the displacement of the eutectic liquid and the shrinking of the silicon disks. Determining the total initial number of solid silicon sites, we create a priori an identical number of potentially liquid sites around the carbon skeleton. Such a liquid layer would cover the carbon sticks if all silicon would be melted. As the reaction proceeds, potentially liquid sites are first transformed into actual liquid sites and then into silicon carbide sites. According to the experimental results, the pore distribution of the resulting ceramics is close to the initial distribution of the silicon grains.11 In the simulation, the place occupied by the initial solid silicon disks is eventually replaced by pores of smaller radius, which are surrounded by silicon carbide. Consequently, during the dynamics, the pores take on the role of the solid silicon sites. To determine if a liquid site can be considered a reactive site RS of contact with solid silicon, we therefore check whether a neighboring site is a pore. The simulation procedure can be summarized as follows: At time t, we assume that NC reactive sites RC and NS reactive sites RS have been formed. During the time step ∆t, (i) pCNC and pSNS reactive sites are transformed into SiC sites, (ii) two or three potentially liquid sites are transformed into actual liquid sites in the neighborhood of each SiC site formed at step (i), so that the mean number of liquefied sites is xj ) 2.65, and (iii) the type of each liquid site formed at step (ii) is determined according to the nature of its neighbors, and the values of NC and NS are updated accordingly. Time is incremented by ∆t, and the same procedure is repeated until reactive sites remain. Details of the configuration obtained at the beginning of the melting of silicon is given in Figure 5. Small isolated grains of SiC are first formed around the nucleation sites located at the initial points of contact between solid silicon and carbon. Silicon progressively melts, and new nucleation sites appear at the contact between liquid silicon and carbon or solid silicon. These new nucleation sites give rise to new grains. Consequently, the number of SiC grains rapidly increases. The number of growth sites, that is, the reactive sites with a side in common with silicon carbide, also increases. Consequently, the grains grow and then coalesce. Due to this phenomenon, the number of grains finally decreases. These antagonistic processes lead to the nonmonotonous time evolution of the number of SiC grains given in Figure 6. The Monte Carlo simulation intrinsically includes the description of the fluctuations of the number of grains. As observed in Figure 6 for r50 ) 2.9 µm, the number of grains can be small at some time due to a phenomenon of percolation. The scale of the ordinates reinforces this impression. However, the number of grains tends to a well-defined limit, independent of the fluctuations observed during the reaction. We use the method proposed by Hoshen and Kopelman21 used in the frame of percolation theory22 or fractals23 to determine the number of SiC grains in a given configuration.

Modeling SiC Synthesis

Figure 5. Detail of a discretized configuration obtained at the beginning of the melting of silicon at time t ) 5 (arbitrary units) for the parameters of Figure 3. The probability of reaction of a (solid or liquid) silicon site in contact with carbon is pc ) 0.2, and the probability of reaction of a liquid silicon site in contact with silicon carbide or solid silicon (whose role is played by pores) is ps ) 0.5. Black squares represent carbon sites, small gray dots are potentially liquid silicon sites, blue squares are reactive sites in contact with carbon, green squares are reactive sites in contact with silicon carbide or solid silicon, yellow squares are nonreactive liquid silicon sites, and red squares are silicon carbide sites.

We consider that two sites of SiC belong to the same grain if they have a side in common. We use the same algorithm to obtain the mass-distribution function of the simulated SiC grains in the final configurations for different granulometries of silicon. In the following, rSiC 50 is the median radius of the simulated SiC grains according to the mass-distribution function. As already mentioned, r50 is the corresponding quantity for the silicon disks. As shown in Figure 7, rSiC 50 versus r50 presents an interesting maximum for 7 µm < r50 < 13 µm, that is, when the silicon disks have a median radius of the same order of magnitude as the median length of the carbon sticks, fixed here at l50 ) 10µm. For r50 ) 7µm, close to the optimum properties, the median radius of the simulated SiC grains is equal to rSiC 50 ) 5.4 µm. Note that the final numbers of simulated SiC grains given in Figure 6 and their dependence on the median radius r50 of silicon cannot be directly related to the median radii rSiC 50 according to the mass-distribution function and their dependence on r50 given in Figure 7. Actually, the mass-distribution function of the SiC grains is not trivial. It possesses a long tail and even a bimodal character for r50 ) 7 µm. In this last case, it means that many small grains coexist with a few large grains. Under these conditions, the mass-distribution functions are different from the number-distribution functions. In the computation of the median radius rSiC 50 according to the mass-distribution function, the weight of the large grains is larger than that in the corresponding computation according to the number-distribution function, which is influenced more by the many small grains. The median radius of the SiC grains according to the numberdistribution function is found to be on the order of 0.9 µm. The predictions of the model agree quite well with the experimental results. Figures 8 and 9 give electronic microscopy images of the silicon carbide grains obtained for different

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Figure 6. Time evolution of the number of silicon carbide grains for different median radii r50 of the silicon disks. Open triangles correspond to r50 ) 1.3 µm, solid triangles correspond to r50 ) 2.9 µm, crosses correspond to r50 ) 7µm, open squares correspond to r50 ) 13µm, and solid squares correspond to r50 ) 17µm. The distribution function of the carbon sticks is the same as in Figure 2. The simulations are performed for equimolar mixtures of carbon sticks and silicon disks with a porosity of 50% in square boxes with sides equal to 103 sites (i.e., 200 µm). The probability of reaction of a (solid or liquid) silicon site in contact with carbon is pc ) 0.2, and the probability of reaction of a liquid silicon site in contact with silicon carbide or solid silicon is ps ) 0.5.

Figure 7. Median radius rSiC 50 of the silicon carbide grains according to the mass-distribution function obtained when the reaction is complete versus the median radius r50 of the silicon disks according to the massdistribution function. Same parameters as in Figure 6.

granulometries of silicon and two carbon powders.11 As shown in Figure 8c,d, for a graphite powder with l50 ) 10.5 µm, the experiments11 lead to a maximal size of the SiC grains for 7.0 µm < r50 e 12.85 µm. Hence, according to the qualitative analysis of the experimental results, we deduce that the silicon carbide grains reach a maximal size when the median radius r50 of the silicon disks and the median length l50 of the carbon plates are of the same order of magnitude. The bimodal character of the mass-distribution function of the SiC grains obtained in the experiments is clear in Figure 8d for r50 ) 12.85 µm, where

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Figure 8. Scanning electronic microscopy image of silicon carbide obtained for different median values r50 of the silicon grain radius and the same graphite powder (Alfa Aesar, ref 14736) with median length l50 ) 10.5 µm. The five images all have the same scale.

very small grains coexist with much larger grains. Quantitatively, the analysis of the experimental samples11 gives the following range for the median radius of the silicon carbide grains according to the mass-distribution function: 3 µm < rSiC 50,exp < 10 µm, where the subscript exp stands for experiments. This experimental result is also in agreement with the prediction of the model. The existence of a maximum size for the silicon carbide grains when the reactants have similar granulometries is confirmed by an other series of experiments for a graphite powder with l50 ) 5.7 µm. As shown in Figure 9, the silicon carbide grains have a maximum size for silicon disks obeying 2.85 µm < r50 e 7.0 µm. Once again, the experiments lead to the conclusion that the grains of the resulting ceramics have a maximal size when the value of l50 is close to the value of r50, that is, when the typical sizes of the reactants coincide. The silicon carbide grains observed in the vicinity of the optimum conditions in Figure 9c, for r50 ) 7.0µm, are smaller than those observed in analogous conditions in Figure 8d, for r50 ) 12.85µm. According to the experiments performed for r50 = l50, we conjecture that, for reactants of identical typical size, the size of the silicon carbide grains increases when the reactant size increases. We finish the analysis of the results with the dependence of the dynamics on the granulometry of the silicon powder. As shown in Figure 6, the time variation of the number of SiC grains changes with the typical size of the silicon disks: the

Lemarchand and Bonnet

Figure 9. Same caption as Figure 8 for a carbon powder (Prolabo, ref 97063) with median length l50 ) 5.7 µm.

Figure 10. Same caption as in Figure 6 for the silicon carbide conversion rate.

maximum of the curve is reached sooner for smaller Si disks, indicating that the reaction is faster. For r50 g 7 µm, an initial plateau associated with a small number of SiC grains is reached before the formation of the peak and the final relaxation toward a second higher plateau. The influence of the size of the silicon disks on the reaction speed is confirmed in Figure 10, which gives the time evolution of the percentage of SiC formed for

Modeling SiC Synthesis different granulometries of silicon. The model correctly reproduces the nonlinear character of thermal explosion: after an induction period, the reaction suddenly accelerates. The comparison of the simulation results given in Figure 10 with the experimental results given in Figure 1 is very satisfying: the fast regime II observed experimentally identifies with the time evolution deduced from the simulations. First of all, this result validates the model, and second, it confirms that smaller silicon grains speed up the reaction. It is to be noted that the initial number of contacts between carbon and silicon also increases as the size of the silicon grains decreases: this property of the initial mixture can be considered a good parameter to control the kinetics of the reaction. Conclusions We propose a simulation model on a submicrometric scale in order to reproduce the formation of silicon carbide during heating at a constant rate and holding at a temperature smaller than the eutectic temperature of the mixture of carbon and silicon. We simulate a 2D cut of the initial mixture of powders and show that the number of contacts between carbon and silicon decreases as the typical size of the silicon grains increases. The simulation of the dynamics of formation of silicon carbide from these initial contact sites shows that the reaction speed decreases when the typical size of the silicon grains increases. The analysis of the morphology of the obtained ceramic reveals that the typical size of the silicon carbide grains formed is maximal for silicon grains and carbon plates of similar typical sizes. Starting from hypotheses on the micrometric properties of the reactants, we deduce micrometric properties of the resulting materials, such as the distribution of size of the silicon carbide grains. Moreover, the simulation of the processes occurring on a submicrometric scale, such as the local melting of the silicon sites, the dissolution of carbon, and the local precipitation of silicon carbide sites, gives us access to macroscopic properties of the dynamics and, in particular, the speed of the reaction. The results of the simulation model are in good quantitative agreement with the experimental results. Our model proposes a bridge between the microscopic and macroscopic descriptions of the dissolution-precipitation mechanism observed in many high-temperature syntheses. In order to extend the results obtained here for a temperature ramp, that is, for thermal explosion, to the case of the propagation mode of SHS, we need to reexamine the different hypotheses of the model and introduce the description of a selfsustained wave. The melting of silicon, the dissolution of carbon, and the heterogeneous precipitation of silicon carbide is also expected. The main difference is that temperature is not

J. Phys. Chem. C, Vol. 111, No. 29, 2007 10835 supposed to remain homogeneous and constant: a temperature wave is observed in the direction perpendicular to the 2D cut modeled here. A three-dimensional description of the sample is necessary. Moreover, the probabilities of reaction pC and pS and, consequently, the speed of the reaction increase with the temperature. Due to this acceleration, the slope of the curves representing the evolution of the conversion rate is expected to be larger in SHS conditions. However, we believe that the qualitative results about the relative speeds of the reactions and the typical sizes of the SiC grains obtained for different silicon granulometries remain valid in the case of SHS. References and Notes (1) Merzhanov, A. G.; Borovinskaya, I. P. Dokl. Akad. Nauk SSSR 1972, 204, 366-369. (2) Merzhanov, A. G. In Combustion and Plasma Synthesis of HighTemperature Materials; Munir, Z. A., Holt, J. B., Eds.; VCH Publishers: New York, 1990; pp 1-53. (3) Munir, Z. A.; Anselmi-Tamburini, U. Mater. Sci. Rep. 1989, 3, 277-365. (4) Makino, A. Prog. Energy Combust. Sci. 2001, 27, 1-74. (5) Varma, A.; Rogachev, A. S.; Mukasyan, A. S.; Hwang, S. Proc. Natl. Acad. Sci. U.S.A. 1998, 95, 11053-11058. (6) Bernard, F.; Paris, S.; Vrel, D.; Gailhanou, M.; Gachon, J. C.; Gaffet, E.; Int. J. Self-Propag. High-Temp. Synth. 2002, 11, 181-190. (7) Narayan, J.; Raghunathan, R.; Chowdhury, R.; Jagannadham, K. J. Appl. Phys. 1994, 75, 7252-7257. (8) Thiers, L.; Mukasyan, A. S.; Varma, A. Combust. Flame 2002, 131, 198-209. (9) Fan, Q.; Chai, H.; Jin, Z. Intermetallics 2001, 9, 609-619. (10) Xiao, G.; Fan, Q.; Gu, M.; Wang, Z.; Jin, Z. Mater. Sci. Eng., A 2004, 382, 132-140. (11) Moranc¸ ais, A. Porous SiC Ceramics Obtained by Reaction Between Silicon and Graphite: Influence of Self-propagating Reactions on Porosity. Ph.D. Thesis, Universite´ de Limoges, France, 2002 (unpublished). (12) Moranc¸ ais, A.; Balhoul-Hourlier, D.; Bonnet, J. P. Int. J. SelfPropag. High-Temp. Synth. 2003, 12, 99-106. (13) Moranc¸ ais, A.; Louvet, F.; Smith, D. S.; Bonnet, J. P. J. Eur. Ceram. Soc. 2003, 23, 1949-1956. (14) Rudnik, T.; Lis, J.; Pampuch, R.; Lihrmann, J. M.; Stobierski, L. Arch. Combust. 1996, 16, 3-12. (15) Pampuch, R.; Bialoskorski, J.; Walasek, E. Ceram. Int. 1987, 13, 63-68. (16) Pampuch, R.; Stobierski, L. Mater. Res. Bull. 1987, 22, 12251231. (17) Arimondi, M.; Anselmi-Tamburini, U.; Gobetti, A.; Munir, Z. A.; Spinolo, G. J. Phys. Chem. B 1997, 101, 8059-8068. (18) Maglia, F.; Anselmi-Tamburini, U.; Gennari, S.; Spinolo, G. Phys. Chem. Chem. Phys. 2001, 3, 489-496. (19) Gennari, S.; Maglia, F.; Anselmi-Tamburini, U.; Spinolo, G. J. Phys. Chem. B 2003, 107, 732-738. (20) Lemarchand, A.; Bonnet, J. P. J. Eur. Ceram. Soc. 2006, 26, 23892396. (21) Hoshen, J.; Kopelman, R. Phys. ReV. B 1976, 14, 3438-3445. (22) Bunde, A.; Havlin, S. In Fractals and Disordered Systems; Bunde, A., Havlin, S., Eds.; Springer: Berlin, 1991; pp 51-149. (23) Lemarchand, A.; Nainville, I.; Mareschal, M. Europhys. Lett. 1996, 36, 227-231.