Simulation Study of Wave Propagation Instabilities for the Combustion

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Simulation Study of Wave Propagation Instabilities for the Combustion Synthesis of Transition Metals Aluminides Silvia Gennari,*,† Umberto Anselmi-Tamburini,†,‡ Filippo Maglia,† Giorgio Spinolo,† and Zuhair A. Munir‡ INSTM, IENI/CNR, and Dipartimento di Chimica Fisica, UniVersita` di PaVia, Vle Taramelli, 27100 PaVia, Italy, and Department of Chemical Engineering and Materials Science, UniVersity of California, DaVis, California 95616 ReceiVed: NoVember 11, 2005; In Final Form: January 26, 2006

Interest in the mode of propagation of self-sustaining reactions has been motivated by the influence of the mode on the microstructure and composition of the final product. However, comprehensive studies relating the onset of the various propagation modes to the chemical and phase transformations taking place in the sample are still lacking. In the present work propagation instabilities in self-propagating high-temperature synthesis (SHS) of transition metal aluminides are studied using a computer simulation approach. The results are presented for the SHS of NiAl, CoAl, TiAl, and NbAl3. Particular emphasis is made with respect to the influence of process variables and system parameters on the onset of propagation instabilities, in relation to the physicochemical processes taking place during the propagation of the combustion front.

1. Introduction That the propagating of combustion waves in self-propagation high-temperature synthesis (SHS) can take place in a variety of modes has been known for several decades. Investigations on these modes (and their transitions) have been motivated by both basic and practical considerations. Since non-steady-state propagation (e.g., pulsating and spin) often results in different microstructures than steady-state propagation, understanding theoretical parameters (and experimental conditions) that influence mode transitions is of significant practical value. Investigations on the propagation mode in the SHS of intermetallic compounds have been carried out during the past 2 decades.1-18 In addition to the steady-state mode, several of these compounds exhibit oscillating or micropulsating propagation modes.19 Using a simple approach based on homogeneous combustion, Hardt et al.20 presented a justification for the occurrence of pulsating behavior in these reactions, confirming the experimental observations of Belyaev and Komkova.21 Considerable effort has been devoted by many on the question of wave stability using different theoretical approaches for the study of the numerical stability and the identification of the existence and uniqueness of the solutions of the partial differential equation (PDE) system describing the combustion processes. Mercer and Weber,7-12 for example, developed a numerical and analytical approach that demonstrates how heat loss can be responsible for the onset of oscillatory combustion modes, while Bayliss and Matkowsky13,14,16 and Bayliss and co-workers15 have demonstrated the existence of two different routes to chaos by numerically solving two models characterized by the presence or absence of the melting of the solid reactants. Raymond and Volpert22 investigated the stability of SHS waves * Corresponding author. Tel.: +39 0382 987208. Fax: +39 0382 987575. E-mail: silvia.gennari@ unipv.it. † Universita ` di Pavia. ‡ University of California.

in systems in which melting occurs and showed that the flow of the liquid affects the one-dimensional stability of the wave. Experimental investigations on the combustion modes in SHS are relatively few and generally qualitative in nature. Using a digital high-speed microscopic video recording, Mukasyan et al.23 have recently been able to investigate experimentally the wave oscillations during the SHS synthesis of NiAl. Using the same experimental approach, Varma et al.24 showed the existence of microscopic fluctuations in the shape and instantaneous velocity in macroscopically steady reaction waves and explained these observations in terms of initial heterogeneity in the reactant powders. Much of the theoretical work on the mode of SHS waves has been based on chemical models derived from the gas-phase homogeneous combustion, including the assumption of a single chemical step. For this reason no information on the influence of the elementary steps involved in the reaction mechanism on the propagation modes has been provided. In the present work we investigated the propagation instabilities using a simulation approach for the SHS of transition metal aluminides that we developed recently and that is specifically oriented toward the mechanistic aspects of the combustion process.25,26 In this paper we present the results on instabilities, with particular attention to the influence of process variables and system parameters on the onset of propagation instabilities. A detailed explanation of the causes of the appearance of instabilities in relation to the physicochemical processes taking place during the propagation of the combustion front will be provided. 2. Modeling the SHS Process of Transition Metal Aluminides The simulation method used in this paper has been described in detail in a previous paper26 and it will be only briefly summarized here. The code we developed is of a general nature so as to be applicable for the simulation of combustion reactions

10.1021/jp0565249 CCC: $33.50 © 2006 American Chemical Society Published on Web 03/23/2006

Combustion Synthesis of Transition Metals Aluminides TABLE 1: Experimentally Determined Dissolution Kinetic Parameters30

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system

Ea (kJ‚mol-1)

Ω0 × 108 (m2‚s-1)

compound

∆Hform (kJ‚mol-1)

Tad (K)

Ni-Al Ti-Al Co-Al Nb-Al

45.4 19.9 68.9 26.6

30 41 264 0.19

NiAl TiAl CoAl NbAl3

-118.407 -75.312 -110.458 -45/-32

1910 1518 1900 1953

TABLE 2: Thermal Conductivities of Solid Metals Utilized in the Simulation χ (W‚m-1‚K-1) solid Al Ni Ti

237 90.7 22

χ (W‚m-1‚K-1) solid Co Nb

100 54

in a large group of transition metal aluminides sharing the same reaction mechanism. In general terms the method uncouples the solution of the heat transfer from the description of the chemical reactions/ phase transformations to provide a solution at the level of the single reactant particle. The reaction mechanism on this scale follows the experimental observations27-29 and includes the following general steps: (a) melting of Al, (b) diffusion-controlled dissolution of the high-melting metal (indicated as Me in subsequent discussion) into the molten pool using dissolution kinetic parameters that have been obtained experimentally under isothermal conditions30), (c) melting of Me if Ti > TmMe, (d) precipitation and possible melting of the resulting compound, and (e) deposition of the eutectic mixture. Only the irreversible step of dissolution (step b) is taken into account with an explicit kinetic law, while phase transformations (steps (a), (c), (d), and (e)) are simply considered in terms of energy balance. The experimentally obtained values for the dissolution rate of the high-melting metal into the aluminum pool are shown in Table 1, while the literature values of thermal conductivity for the solid metals utilized in this analysis are shown in Table 2. The thermodynamics of the system involved are described using the SGTE (Scientific Group Thermodata Europe) standard and hence the underlying data for the binary systems Ni-Al,31 Co-Al,32 Ti-Al,33 and Nb-Al34 have been obtained from the literature using the CALPHAD (Calculation of Phase Diagrams) approach. Computationally, the PDE system is solved by using the finite difference Crank Nicolson algorithm. The accuracy of the results has been checked with short runs with many different space and time steps. In this work we will limit our investigation to one-dimensional instabilities. Our main goal is to show the connection between propagation instabilities and the details of the reaction mechanism. The investigation of more complex three-dimensional propagation modes, briefly introduced in a previous paper,35 although interesting from the theoretical and numerical point of view, does not add any new information. Propagation wave velocity is calculated separately at the end of the simulation run from temperature space profiles. For each time step, position data for a given interval of temperatures (typically Ti - 50 < Ti < Ti + 50 K and Ti g Tm(Al)) are stored by the program. After the simulation run, the file that has been written is read and T values at several space/time points are used to evaluate velocity by using spline functions of velocity vs time and of velocity vs space. It is important to indicate that some different ways of calculating propagation velocity have been tried in order to ensure that the oscillations observed are not an artifact of the numerical algorithm. In the present work, instabilities will be mainly identified as a

fluctuation of the propagation wave speed as a function of either time or space. 3. Results and Discussion A summary of the characteristics of the SHS reactions considered in this work is reported in Table 3, which contains the heats of formation and the adiabatic temperatures (always close to the melting point of the reaction product) of the intermetallic compounds examined. Figure 1 shows typical experimental evidence of propagation instabilities in the SHS of aluminides; in this case the sample represents the synthesis of NiAl using large Ni particles. The lamellar microstructure is typical of unstable wave propagation. The mechanism responsible for the appearance of these oscillations is understood only in general terms. It is well-known that highly nonlinear terms, such as an exponential dependence on temperature, can induce instabilities in macroscopic dynamic systems.13-16 But the role played by the parameters controlling the microscopic mechanism in starting and maintaining the oscillations remains obscure. With our modeling approach it is possible to investigate both the macroscopic and microscopic features related to the propagation of the combustion front, as we will demonstrate in this paper. 3.1. Role of Particle Size. A parameter that plays a major role in defining the onset of oscillating behavior in wave propagation is particle size.36 Despite intuitive anticipation of an effect of particle size, this consideration has been examined in relatively few experimental investigations. A typical example of the calculated dependence of the mode of propagation on particle size is seen in Figure 2. The figure shows the variation of the wave velocity, V, with time for the SHS synthesis of CoAl for four different starting Co particle sizes, r0. The thermal conductivity used in these calculations is that corresponding to a fully dense sample (i.e., bulk conductivity). As will be discussed later, high thermal conductivity enhances the appearance of unstable propagation modes. For large particle sizes

Figure 1. Microscope image of the final product for the SHS of NiAl. Coarse starting Ni grain size (r0 > 40 µm) was used.

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Figure 3. Propagation velocity as a function of transition metal grain size for four different aluminides. Black dots refer to oscillating propagation modes, while white squares outline steady propagation.

Figure 2. Wave velocity as a function of time for the SHS of CoAl with Co starting grain size 10 µm (a), 7 µm (b), 5 µm (c), and 0.7 µm (d). Bulk thermal conductivity and xCo ) 0.49 are used in these simulations.

(Figure 2a, with r0 ) 10 µm) chaotic propagation behavior is found with Vmax ) 0.8755 m‚s-1, Vmin ) 0.0509 m‚s-1, and Vmean ) 0.2459 m‚s-1. As the particle size decreases, different oscillating propagation modes develop, passing through doubleperiod oscillation for r0 ) 7 µm (Vmean ) 0.3536 m‚s-1) in Figure 2b and single-period oscillation for r0 ) 5 µm (Vmean ) 0.4678 m‚s-1) in Figure 2c and finally reaching steady propagation mode for r0 ) 0.7 µm with an average velocity of 0.97 m‚s-1 (Figure 2d). The transition from chaotic behavior to steady-state propagation for all systems investigated is shown

in Figure 3, plotted as velocity vs particle size. In the figure, filled circles denote oscillatory propagation while open squares denote steady-state propagation. For the case of NiAl synthesis (Figure 3a), steady-state propagation is only seen when the particle size is smaller than 1 µm. In contrast, in the case of TiAl synthesis (Figure 3c), steady-state propagation is calculated for all sizes up to 50 µm, albeit with lower velocity absolute values than in the cases of the synthesis of CoAl and NiAl. The only case where the onset of oscillating behavior is encountered corresponds to a particle size of 50 µm (Figure 3c). In this case the SHS of TiAl propagates with a single-period oscillation; the use of Ti particle sizes bigger than 50 µm corresponds to extinction, unless the sample is strongly preheated. The existence of lamellar microstructure in TiAl synthesized by SHS has been reported for starting Ti particle size > 40 µm, suggesting the occurrence of pulsed propagation.37,38 It should be noted that the system TiAl is generally identified as difficult to ignite and thus requires a significant amount of preheating before the reaction is able to self-propagate.38 However, in these studies the Ti particles are rather large, around 80 µm, and hence unstable propagation conditions are expected. From our analysis, the use of smaller

Combustion Synthesis of Transition Metals Aluminides Ti particle size should lead to steady propagation, without the need for prior heating of the reactants. Regarding the SHS of NbAl3 on the other hand (Figure 3d), the exothermicity of the process is markedly lower (-45 < ∆H < -32 kJ‚mol-1), and the possibility of self-propagation of the reaction wave is confined to a much smaller grain size range, in good agreement with experimental findings.39 As a matter of fact, the largest grain size that can produce a self-propagating front is around 10 µm for this system, with the propagation mode being always steady state. The general trend of decreasing velocity with increasing particle size, and the transition from steady-state to non-steady-state wave propagation has also been observed in other intermetallic reactions. In the case of the synthesis of CoTi, the transition occurs at a particle size of about 140 µm, while in the case of the synthesis of NiTi steady state is observed even for sizes as large as 400 µm.40-42 The results of these experimental studies are in general agreement with theoretically calculated velocity dependence on particle radius by Makino.36 In comparing the results of Figure 3 to each other, we note a decrease of the highest velocity (corresponding to a particle size of 0.1 µm) in the following order of phase formation intended: NiAl, CoAl, TiAl, and NbAl3. This trend is consistent with the values of the enthalpy of formation for the aluminides, as reported in Table 3. 3.2. Effect of Thermal Conductivity. As already indicated, thermal conductivity plays an important role in defining the propagation mode in SHS processes.43-45 Literature results show that the effective thermal conductivity of a compact of powders can be as much as 2 orders of magnitude lower than the bulk value.45 This parameter is strongly related to the porosity of the sample and is not easy to obtain experimentally. However, changes in porosity produce not only a modification of the effective thermal conductivity of the sample but also a marked change in the amount of heat released per unit volume, giving rise to a more complex modification of the combustion process. From an experimental point of view, thermal conductivity (as well as heat per unit volume produced by the SHS process) can be manipulated by the use of a diluent (normally the product itself), added to the reacting mixture of the two metals or, as stated above, by changing the porosity. We will discuss this point in more detail in a following paper,44 where a closer comparison with experimental data will be made. In this paper, thermal conductivity will be treated as an independent parameter since our main focus is on the role that the various parameters play on propagation stability, independent of each other. In other words, the present simulation approach allows us to study the effect of different process variables independently, which is not always possible under experimental conditions. In this case, a reduction factor is applied to the bulk value of thermal conductivity, and the range of conductivity values spanned has been explored in previous papers in order to reproduce experimental propagation wave speed values.43 In Figure 4 we show the effect of thermal conductivity on the velocity and mode of propagation for the SHS synthesis of CoAl for a constant Co particle size of 10 µm. When the thermal conductivity is reduced by half, i.e., χ ) χbulk/2, the wave propagates by a strong double-period oscillation, Figure 4a. It should be recalled that when the bulk conductivity is used for the same particle size, a chaotic behavior is seen in the propagation, Figure 2a. When the conductivity is further reduced, χ ) χbulk/5, a unimodal oscillation with a markedly lower peak amplitude is seen, Figure 4b. As the thermal conductivity is reduced further, the amplitude continues to decrease, Figure 4c, and when a very low thermal conductivity

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Figure 4. Wave speed as a function of time for the SHS of CoAl with r0 ) 10 µm, xCo ) 0.49, and χ ) χbulk/2 (a), χ ) χbulk/5 (b), χ ) χbulk/9 (c), and χ ) χbulk/30 (d).

is utilized, χbulk/30, the wave now propagates in steady-state mode, Figure 4d. A similar behavior is observed also in the synthesis of NiAl, differing only in the absolute values of wave speed. The influence of thermal conductivity on the absolute values of the propagation rates, together with the propagation mode for selected aluminides, is shown in Figure 5. It should be pointed out here that if the change in conductivity is solely based on porosity, the porosities corresponding to 1/2, 1/5, and 1/30 of the thermal conductivity value of a dense sample are 33, 53, and 64%, respectively, as calculated from the derivation of Maxwell.46 A porosity of 64% corresponds to a nonconsolidated (poured) powder.

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Figure 6. Mode diagram of combustion behaviors while varying thermal conductivity and high melting metal grain size for the SHS of CoAl/NiAl (a), TiAl (b), and NbAl3 (c). Black dots indicate oscillating or chaotic propagation modes, while white squares show steady propagation. Continuous line shows the limits to extinction.

Figure 5. Propagation speed as a function of thermal conductivity of the transition metal for compounds with different reaction enthalpies. Black dots indicate oscillating propagation modes and white squares show steady propagation, while the arrow remarks the bulk value of thermal conductivity of the transition metal of the compound considered. Transition metal particle size is in all cases 10 µm.

As before, the filled circles in Figure 5 denote oscillatory propagation while the open squares denote steady-state propagation. The arrow refers to the bulk thermal conductivity of the transition metal. The figure shows that velocity for any given conductivity value is related to the enthalpy of formation of the aluminide. Moreover, the figure shows that, at low and high conductivities, the wave does not propagate. This result is in qualitative agreement with observation on the dependence of wave velocity on relative density.47,48

The combined effect of particle size and thermal conductivity on the dynamic behavior of the SHS reactions for selected systems is shown in Figure 6. The figure shows regions of chaotic and oscillatory propagations (filled circles) and steadystate propagation (open squares) for the SHS of CoAl (and, in this respect, also for NiAl), TiAl, and NbAl3. In the case of CoAl (and also NiAl) the field of oscillating and chaotic modes is relatively large and occurs under the combination of large thermal conductivity and large particle size (Figure 6a). In this and all other cases (Figures 6b and 6c), the line boundary surrounding the points signifies the boundary of extinction. For TiAl, only a very limited region of instability can be found corresponding to the maximum particle size. No oscillations could be found in the case of the SHS of NbAl3, whose field of existence of SHS is smaller with respect to particle size in comparison with the other systems. It is also evident from these diagrams that propagation instabilities are induced by high values of thermal conductivities and large particle size. We will return to this point when we discuss the mechanism responsible for the oscillations.

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Figure 8. Mode diagram of combustion behaviors while varying effective thermal conductivity and intrinsic dissolution coefficient. The reference values of dissolution coefficient and thermal conductivity, as well as the thermodynamic functions used in these simulations, refer to the SHS of NiAl.

dissolution depends to large extent on the shape of the phase diagram, in particular on the temperature dependence of solubility. ∆H and phase diagram details are not independent since they are connected by thermodynamic relationships, and for this reason in our treatment they cannot be varied independently. The role of the dissolution kinetics in the propagation mode is subtle and to our knowledge has never been investigated before. The dissolution rate is the key microscopic parameter controlling the kinetics of the combustion process. As we have indicated earlier,26 it depends in part on the solubility limit. For a spherical particle,

r2 ) r02 - Kdisst where r is the radius of the Me (transition metal) particle at a given time, t, and r0 is its initial value. Kdiss is the rate of dissolution of the transition metal into aluminum. This in turn is expressed as

Kdiss ) ksΩ

Figure 7. Wave propagation speed as a function of intrinsic dissolution coefficient for different values of effective thermal conductivity. The dissolution coefficient Ω is given in an Arrhenius form. Black dots indicate oscillating propagation modes and white squares show steady propagation while the dashed line remarks the true value of dissolution coefficient, as obtained from experiments in isothermal conditions performed by our group.30 In all these simulations, the SHS of NiAl was taken as the reference situation.

3.3. Effect of ∆H and Ω. In a previous paper,35 we showed that the presence of oscillatory propagation can be related to the thermodynamic characteristics of the system. In this regard, it should be emphasized that thermodynamic properties relevant to SHS reactions include not only the ∆H of the reaction but also phase equilibria, particularly in the case of the synthesis of aluminides. In these reactions, the dissolution of the transition metal in molten aluminum is the most relevant step. This process is thermodynamically controlled by the solubility limit and kinetically controlled by the rate of dissolution. The kinetics of

where ks is the solubility at a given temperature and Ω is the intrinsic dissolution coefficient in m2‚s-1 of a specific metal into liquid aluminum. In other words, the dissolution kinetics depends on two parameters: the solubility of the transition metal at a given temperature, which is related to the shape of the phase diagram, and the intrinsic solubility rate, which is related to the kinetics of the interfacial process. Since with the approach we used we could not change ks without varying the shape of the phase diagram and, ultimately, the value of ∆H, we investigated the role of the dissolution rate by varying in each system the value of the intrinsic dissolution constant Ω. The results are reported in Figures 7 and 8. These results refer only to NiAl, which together with CoAl appears to be more prone to propagation instabilities. Figure 7 shows what would be anticipated intuitively, that an increase in the intrinsic dissolution rate results in an increase in the propagation rate for different thermal conductivities relative to the bulk value in the NiAl case. In all cases, the increase in propagation rate appears to reach a plateau asymptotically. The onset of the plateau corresponds to a process where the dissolution is no longer ratedetermining. Quite surprisingly, propagation instabilities appear only for lower values of the dissolution rate. High values of the parameter seem to stabilize the propagation. However, Figure

7150 J. Phys. Chem. B, Vol. 110, No. 14, 2006 7 shows also how dissolution rate and thermal conductivity are closely related in defining the propagation mode. This relationship is better seen in Figure 8 where both parameters are varied in order to produce a propagation mode diagram. The diagram shows that low dissolution rate and low thermal conductivity generally lead to extinction while low thermal conductivity and high Ω lead to steady-state propagation. Oscillatory propagation occurs generally when both parameters are high. 3.4. Microscopic Mechanism for Oscillation. The results discussed above show that the parameters of particle size, enthalpy of the reaction, dissolution rate, and thermal conductivity are strongly interrelated and the relative role of each in defining the conditions of stable, oscillatory, or chaotic propagation cannot be defined. This implies that the appearance of oscillation is not based on a unique mechanism. Different contributions corresponding to the different combinations of the experimental parameters can lead to the immergence of oscillatory propagation. Aside from macroscopic thermodynamic and kinetic considerations, little attention has been paid thus far to the investigation of the physical mechanism that initiates and sustains the oscillations. In Figure 9 the oscillatory wave pattern during the SHS of NiAl (Figure 9a) is related to the phase evolution corresponding to the minimum (Figure 9b) and maximum (Figure 9c) of the pulses. The phase evolutions depicted in Figures 9b and 9c show the temperature profile as a function of distance in the region across the combustion front together with the relative amounts of the various phases involved in the process. In both figures, the combustion front propagates from left to right. At the extreme right end of each figure, the temperature of the reactants increases in the region ahead of the combustion front. When the temperature reaches the melting point of Al, the amount of solid aluminum drops to zero (as seen by the line labeled Al(s)). At this point, the dissolution of the solid transition metal (Me(s)) in liquid Al commences. When the liquid becomes saturated, the intermetallic phase starts to precipitate, as seen by the line labeled MeAl(s). When the relative amount of this phase reaches 1, the chemical process is finished. Marked differences can be observed between the two situations representing the low and high velocities (of Figure 9a) and depicted in Figures 9b and 9c, respectively. For the case of the minimum velocity (Figure 9b), the temperature increases with a low gradient and the maximum value is reached well after the completion of the chemical process. In contrast, for the maximum velocity (Figure 9c), the initial sharp increase in temperature corresponds to the near completion of the chemical process (MeAl(s) is close to 1). This is followed by a second broad maximum, which is the consequence of the partial melting of the intermetallic phase. In this case, the much higher propagation rate reduces the time available for the chemical steps (dissolution and precipitation). It also produces a temperature profile that is much steeper, reducing the width of the preheated zone ahead of the combustion front and pushing the front itself toward reactants that are still relatively cold. An interesting insight can be gained from the analysis of the modifications that the spatial temperature profile undergoes during a single oscillation cycle (Figure 10). Particularly interesting is the comparison of the temperature profiles for points placed symmetrically with respect to the velocity pulse (points A and E or points B and D in Figure 10a). Although these respective points have the same propagation velocity, their temperature profiles are very different during the speedup and slowdown phases of the combustion front. The drastic slowdown in the propagation rate, observed after point C, appears to be

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Figure 9. Solid Al (black line, left axis), solid Me (red line, left axis), solid intermetallic compound (yellow line, left axis), liquid composition (green line, left axis), and temperature (blue line, right axis) as a function of space on the reaction wave front for the SHS of NiAl (r0 ) 5 µm, χ ) χbulk) for the time steps corresponding to the minimum (b) and maximum (c) in propagation wave speed (a).

related to the presence of a steep temperature profile and the slowdown continues until the temperature profile becomes less steep. The results shown in Figures 9 and 10 suggest a mechanism based on a “decoupling” between the propagation of the “heat wave” and the propagation of the wave associated with the chemical process (the “chemical wave”). If the appropriate conditions are met, the chemical front tends to “run away”, producing regions of localized high temperature that propagate much faster than the thermal wave. These spikes eventually die off when they reach the part of the sample that is still too cold to sustain any further propagation. The evolution of this process is visualized in Figure 11, where the temperature profiles as a function of space for a time interval of ≈10-4 s are shown for the SHS of NiAl with a starting Ni particle size of 10 µm and with use of the bulk thermal conductivity. The identification of the conditions that lead to these instabilities is not an easy task since all the parameters involved appear to be interconnected. However, the results shown in Figures 5, 6, and 8 suggest that the combination of high exothermicity, high

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J. Phys. Chem. B, Vol. 110, No. 14, 2006 7151 8 show that reducing the kinetics of the dissolution process (the chemical step) always induces an enhancement of the instabilities. This could be done either by reducing the intrinsic kinetics of the process (reducing Ω) or by increasing the particle size (this reduces the surface of contact). The same figures show that high thermal conductivities produce the same result. It is interesting to note that the various aluminides differ significantly not only in their exothermicity but also in their thermal conductivity (Figure 5 and Table 2). TiAl, characterized by stable propagation, shows an Me thermal conductivity that is one-fourth of the thermal conductivity of NiAl and CoAl that are characterized by propagation instabilities in almost every condition. This shows the importance of thermal conductivity in defining the stability of the propagation. 4. Concluding Remarks

Figure 10. Wave speed as a function of time for an oscillating NiAl SHS process (a) and space profiles of temperature corresponding to different points of the wave speed cycle (b).

The present simulation approach, focused on the heterogeneous aspects of SHS processes, has made it possible to relate the onset of the various propagation modes for the synthesis of some intermetallic compounds to the chemical and phase transformations taking place in the sample. The onset of oscillations has been determined to be due to the separation of the thermal and chemical fronts. Such information remains lacking in the literature and is extremely difficult to obtain from an experimental point of view (particularly in terms of detaching the effect of different process parameters) and hence shows the utility of a modelistic approach in the study of complex processes such as these SHS reactions. Acknowledgment. Cosan Unuvar and Daniela Fredrik are kindly acknowledged for the sample of Figure 1. Dr. Johan Bratberg is gratefully thanked for helpful discussion regarding the CALPHAD approach to the calculation of phase diagrams. The financial support of the National Science Foundation to one of us (Z.A.M.) is also acknowledged. References and Notes

Figure 11. Temperature profiles as a function of space for the SHS of NiAl with starting nickel grain size of 10 µm and bulk thermal conductivity for the powder mixture.

thermal conductivity, and relatively slow kinetics of the chemical process seems to be the more favorable combination for the onset of oscillating propagation modes. The presence of a chemical step that is intrinsically slow in comparison with the thermal conductivity seems to be particularly important. In such a situation the chemical front tends to lag behind the heat front. Such a situation would lead to extinction if it is continued for a sufficient time. However, it produces also a fairly large region of unreacted but hot sample ahead of the “chemical” front. When in this region the melting point of Al is reached, a large amount of heat is released, increasing the temperature and producing a burst that eventually dies off when this preheated region is totally converted. At that point the advancement of the combustion front slows down drastically and the cycle starts again. This mechanism is supported by several observations. Figures 6 and

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