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Modeling the ignition of self-propagating combustion synthesis of transition metal aluminides
Intermetallics 18 (2010) 2385e2393
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Modeling the ignition of self-propagating combustion synthesis of transition metal aluminides Silvia Gennari, Umberto Anselmi-Tamburini, Filippo Maglia*, Giorgio Spinolo Department of Physical Chemistry, University of Pavia, V.le Taramelli 16, 27100 Pavia, Italy
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 November 2009 Received in revised form 13 January 2010 Accepted 12 August 2010 Available online 9 September 2010
The ignition process in self-propagating high-temperature synthesis (SHS) reactions was modeled taking into account chemical and phase transformation processes. The model was applied to the synthesis of aluminides. The dependence of the ignition process on peak input energy density and peak temperature is demonstrated by ignition diagrams. The influence of the transition metal grain size and pellet porosity on the ignition conditions was demonstrated. With an increase in grain size, a significant increase in the input energy density is required and the region of SHS reactions in the ignition diagram is reduced. The experimentally observed difference in the ignition characteristics of the synthesis of different aluminides was attributed to differences in solubility limits and rate of dissolution. The effect of the ignition conditions on the velocity of the following self-sustaining reaction was analyzed. 2010 Elsevier Ltd. All rights reserved.
Keywords: C. Reaction synthesis A. Aluminides
1. Introduction Self-propagating high-temperature synthesis (SHS) is a well established method for the synthesis of various classes of inorganic materials that are characterized by a strongly exothermic formation reaction [1]. In the last two decades a considerable effort has been devoted to demonstrate how this method can be applied to the synthesis of a wide variety of materials. Despite that, the mechanisms of ignition and propagation of these complex reactions are still not well understood. In particular the ignition process has received a very limited attention, from either experimental or theoretical point of view. Information on the dependence of ignition temperatures and ignition energies from the experimental parameters such as heat of reaction, thermal conductivity, grain size and relative density are scarce, while the theoretical investigations are usually based on oversimplified chemical mechanisms derived from the chemistry of flames. This represents a major limitation in our understanding of solid-state combustion reactions, since the ignition process plays a pivotal role in defining their feasibility. In previous investigations [2,3] we developed a novel approach to the simulation of SHS processes, based on a detailed and realistic chemical mechanism that we utilized to investigate the propagation of SHS reactions bringing to the formation of some transition metals
* Corresponding author. Tel.: þ39 0382 987208; fax: þ39 0382 987575. E-mail address: fi[email protected] (F. Maglia). 0966-9795/$ e see front matter 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.intermet.2010.08.008
aluminide. Aim of the present paper is to apply the same approach to the study of the transient phenomena preceding the steady-state wave propagation in the same aluminide systems, with the explicit intent to investigate the role of the experimental parameters, and to compare the results with the available experimental data. Most of the presented results will be related to the most experimentally investigated processes, i.e.: the synthesis of NiAl and CoAl. These two systems, incidentally, are characterized by almost identical thermodynamic and kinetic characteristics and produce very similar results. For these reason in some sections the presented data will refer to NiAl and in other to CoAl. Simulations concerning the NbAl3 and TiAl synthesis reaction will also be presented in some sections to show as large differences in the thermodynamic and kinetics of the solideliquid interaction can produce dramatic differences in the SHS ignition process even for reactions characterized by similar reaction enthalpies. 2. Modeling the ignition process in the SHS of transition metal aluminides As reported before, literature accounts on the ignition of SHS processes are relatively few. Barzykin published the only comprehensive review on this topic, summarizing its experimental and theoretical aspects [4]. In general, the early theoretical works were based on a homogeneous pre-mixed flame model [5e7]. The first micro-mechanistic approach, including an heterogeneous chemical mechanism, was developed by Stangle and co-workers [8e11] for the SHS of NbeC. More recent experimental investigations have
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attempted to relate the initiation process to different chemical steps and reaction mechanisms, focusing primarily on the SHS of TiC [12e17]. For this process, Makino et al. [17] studied the effect of mixture composition, size of the carbon particles, degree of dilution, heating area, and heat flux on the ignition delay time. Few studies are available in the case of synthesis of intermetallic compounds. The most experimentally investigated process has been by far the synthesis of NiAl [18e23]. Monagheddu et al. [23] evaluated for this reaction the influence of laser energy on ignition delay time and temperature. Zhu et al. [24], on the other hand, provided for this reaction experimental evidences of a dissolutionprecipitation-controlled mechanism. Experimental information concerning the relationship between ignition delay time and reactants grain size (and hence heating rate) has been provided for the SHS of CoAl [25]. Determination of ignition temperatures and energies for different stoichiometries in the TieAl [26] and NbeAl systems [18] has also been made. On the modeling front, Fu et al. [27] proposed a mathematical approach that analyses the ignition process of these reactions, while more recently Makino, by applying a heterogeneous approach to the study of ignition of SHS in aluminides [28], has shown that ignition energy is not only related to heat losses of the sample, but also to other physico-chemical parameters (such as particle size of the transition metal) which are not taken into account in a homogeneous gas-phase like theory. The computational approach that we are proposing is based on a model that describes the ignition and propagation of the SHS of transition metal aluminides taking explicitly into account all the chemical and phase transformations involved in the combustion process. Details on the method have been provided in previous papers [2,29,30]. In this approach we describe the SHS reactions using a parabolic partial differential equation (PDE) system, which in the case of a one-dimensional problem, can be expressed as
(
2$s$3$ T 4 T 4 a vT v vT c þ C ¼ þ q_ chim a vt vx vx R q_ chim ¼ f ðx;T;tÞ b
(1)
where, in Fourier’s equation, Eq (1), s is the StefaneBoltzmann constant, 3 the emissivity, R the radius of the sample (cylindrical in this case with radius R and length L), T the temperature and Ta is the ambient temperature (typically 298 K), and each equation q_ chim ¼ f ðx;T;tÞ describes one chemical process and phase transformation involved in the process. In the case of the SHS of transition metal aluminides [2], we considered a chemical mechanism based on a dissolution-precipitation process [24,31,32]. This mechanism involves the following steps: a) melting of Al, b) melting of the transition metal, c) dissolution of the transition metal into liquid aluminum, d) precipitation/melting of the intermetallic compound, e) eutectic deposition. This list, however, does not necessarily represent a time sequence of events driving to the formation of the intermetallic phase, but a mere list of all processes that the algorithm is able to take into account. The actual sequence of events is decided by the algorithm itself depending on the actual conditions present at each time and space step during the simulation. As a result, one or more of the previous steps might be missing, depending on the reaction considered and on the actual reaction conditions. This approach allows enough flexibility to reproduce the actual reaction mechanism that is involved in the synthesis of the majority of the aluminides. In most cases, as we said before, the reaction is controlled by the solideliquid interaction between the liquid aluminum and the solid transition metal; the melting of the transition metals is possible only in few cases as well as the melting of the intermetallic phases, but these events do happen when the right combination of thermodynamic properties and experimental conditions is met. It must be further noticed that
our model implicitly implies that the phase diagram involved in the considered reacting system is represented by a simple eutectic. This, of course, is not the case for all aluminides. More complex phase equilibria, involving the formation of intermediate compounds or peritectic reaction, are often encountered. Strictly speaking our approach does not apply to these cases. However, it must be noted that, because of the very rapid kinetic of the SHS processes, some phase equilibria, when characterized by an intrinsically slow kinetics, are never attained and a simplified phase diagram is actually controlling the reaction mechanism. Peritectic reactions, for instance, rarely play an important role in SHS processes because of their slow kinetics. As a result, our approach can be applied also to systems characterized by a complex phase diagram, if the appropriate conditions are met. It must finally be noted that in the algorithm all phase transformations, steps (a), (b), (d) and (e), are described in terms of thermal balance only, without an explicit kinetic law; only the kinetics of dissolution of the transition metal spherical particle in the liquid Al are taken explicitly into account [33]. The ignition of an SHS process can in principle be reduced at two limiting cases, usually referred to as “heat flux-controlled” and “temperature-controlled” ignition. These two idealized situations have the experimental counterpart in the laser ignition (where heat flux is controlled) and chemical ignition (where an igniter characterized by a defined reaction temperature is used). In our approach both ignition conditions can be simulated using boundary conditions defined either as a function of temperature (corresponding to Cauchy boundary conditions) or of heat flux (corresponding to Von Neumann boundary conditions) [2,30]. Controlled temperature ignition (Cauchy boundary conditions) is expressed as:
8 < Tðx; t ¼ 0Þ ¼ Ta Tð0; tÞ ¼ f1 ðtÞ : TðL; tÞ ¼ f2 ðtÞ
cx˛f0; Lg (2)
where f2(t) is normally set to room temperature (Ta ¼ 298 K), while f1(t) represents the ignition function on the left-hand side of the sample. In order to avoid discontinuity and to reproduce the experimental situation, the ignition pulse in our simulations is not described by a square function, but by a Gaussian function, as:
2 2 f1 ðtÞ ¼ Ta þ Tp Ta eððtmÞ =2u Þ
(3)
with Tp being the maximum temperature value, u the time range, and m the time lag of ignition. On the other hand, controlled heat flux ignition (Von Neumann boundary conditions) is expressed as:
8 Tðx; t ¼ 0Þ ¼ Ta > < vT J0 ðtÞ vx jð0;tÞ ¼ c > JL ðtÞ : vT j ¼ vx ðL;tÞ
cx˛f0; Lg (4)
c
where the fluxes J0 and JL can assume a fixed value or can be a function of time. A convenient way to define these fluxes is through the use of a radiation term:
Jð0; tÞ ¼ s$3$ T04 Ts4
(5)
JðL; tÞ ¼ s$3$ TL4 Td4
(6)
with Td typically ¼ Ta, and Ts depends on time, as imposed by a suitable ignition function, and s and 3 are as defined earlier. Numerical solutions for the 1-D model have been obtained using the FD Crank Nicholson algorithm [34], and for the 3-D model [35],
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Fig. 1. Propagation diagram for the combustion synthesis of NiAl, with starting Ni grain size of 1 mm and bulk thermal conductivity. The boundaries have been calculated with the present model, while the references at the bottom refer to fluxes used experimentally for ignition, when available in literature [18,19,26,51].
a commercial engine for the solution of PDEs [36] has been interfaced to the chemical routines in order to solve the system. Unless otherwise specified, the simulation results reported here have been obtained with the 1-D approach, whose features are more than sufficient for the study of the ignition process.
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Fig. 2. Ignition energy as a function of starting high-melting metal grain size for the SHS of NiAl with bulk thermal conductivity.
suggesting that an energy threshold exists, with a value that is almost independent from peak temperature. On the other hand, no combustion can be initiated, regardless of the total energy supplied to the sample, if the temperature does not reach at least the melting point of Al. This is a consequence of the absence of any solidesolid interaction in our model and agrees well with the experimental evidence [24,31,32]. Finally, in the case of low ignition temperatures and very large energies no real propagation front can be
3. Results and discussion 3.1. Ignition diagrams We indicated before the distinction between temperaturecontrolled and heat flux-controlled ignition conditions [4]. This distinction has been largely used in traditional combustion studies because the use of simplified boundary conditions allows an easier analytical solutions of the systems of Eq (1). As a result, the ignition conditions for a certain reacting system are usually defined on the basis of a one parameter such as the temperature of ignition or the heat of ignition. Sometimes, less direct parameters such as the ignition delay time, are used, although they all refer ultimately to a temperature or energy condition. The actual situation, however, is far more complex and the ignition of an SHS process can hardly be defined by a single parameter. The best way to visualize this situation is through the use of ignition diagrams, such as the one reported in Fig. 1 for the synthesis of NiAl, using powders of 1 mm in grain size and a sample 1 cm in diameter and 2 cm long. This diagram shows the outcome of a Gaussian ignition pulse for all possible combinations of peak temperature and peak energy. It is evident as a self-propagating combustion front can be initiated only if a certain combination of temperature and energy density is used. It should be here clarified that in this figure, the ignition time is implicitly involved moving in both directions of the diagram. By moving vertically along the y axis, for instance, the same total energy amount is released to the system slowly, in the case of low peak temperature, or very quickly, in the case of high peak temperatures. On the other hand, moving horizontally, keeps the peak temperature constant, but involves releasing different energies by means of maintaining the ignition source on for different times. Fig. 1 tells us that if the ignition impulse is very short, as happens on the left side of the diagram, no self-propagating process can ever be initiated even if very high peak temperatures are reached. The boundary between extinction and propagation in this region (on the left-hand side of the diagram) is almost vertical,
Fig. 3. Solid Al (continuous line, left axis), solid Me (dots, left axis), solid compound (dashes, left axis) and temperature (dashedot, right axis) as a function of time, for the SHS of NiAl with starting Ni grain size of 1 mm (a), and 70 mm (b).
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Fig. 4. Space profiles (at different time steps) of temperature (a,b), amount of solid metal (a1,b1), and enthalpy released by chemical/phase transformations (a2,b2) for the SHS of CoAl with starting grain size of 10 mm (a,a1,a2) and 100 mm (b,b1,b2).
initiated, but the entire sample ends up reacting at once; a reaction mode that is usually referred to as thermal explosion or volume combustion. The high thermal conductivity produces in this case an homogeneous increase in temperature in the whole sample, as a result of the slow energy supply. It is important to note that the actual shape of these ignition diagrams is not uniquely defined by the intrinsic (thermodynamic and kinetic) nature of the reacting systems, but it is, for a given a system, very dependent on the values of other experimental parameters, in particular on the grain size of the transition metal and on the sample porosity. We will discuss this influence in the following sections.
3.2. Effect of grain size An example of the influence of the transition metal grain size on the ignition conditions is shown in Fig. 2, as obtained by using our model. The figure shows the energy density required to ignite the reaction for a fixed ignition peak temperature of 2000 K when the grain size is changed in the range between 101 to 50 mm. The figure shows an increase of three orders of magnitude in the ignition energy when the grain size is varied in this range. When compared with the ignition diagram of Fig. 1, which was calculated
for a grain size of 1 mm, it is evident as the region of SHS ignition shrinks dramatically as a result of the larger grain size. The reason for such a strong influence of grain size on the ignition conditions can be better understood by analyzing the details of the reaction mechanism. This will be illustrated at first in the case of a steady propagating front (Fig. 3) and then in the more complex transient leading to the ignition (Fig. 4). In Fig. 3 the temperature profile (dashedot), the amount of solid Al (continuous line), solid Me (dots) and solid intermetallic compound (dashes) for the SHS of NiAl with starting Ni grain sizes of 1 mm (a) and 70 mm (b) are reported. The simulations have been performed using a thermal conductivity corresponding to the bulk value, i.e. in a zero porosity situation. The effect of thermal conductivity (and porosity) will be explicitly considered in the next section. When small grain sizes are used (Fig. 3a), the dissolution of the transition metal proceeds extremely fast due to the large interface area between the reactants, and comes to completion within few time steps. Since Ni dissolves in large amounts at each time step, a considerable amount of heat of solution is released, producing a sharp increase in the temperature profile. In contrast, when r0 ¼ 70 mm (Fig. 3b), Ni dissolution starts and proceeds at a much slower rate, taking z0.01 s to come to completion. At each time step, Ni dissolves into the molten pool in small quantities and no temperature spike is
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Fig. 5. Ignition energy as a function of (a) effective thermal conductivity, and (b) sample porosity for the SHS of NiAl, with starting nickel grain size of 1 mm.
observed. Once the solid intermetallic compound starts to precipitate, the heat released by this process raises the temperature further, increasing also the dissolution rate of the transition metal into the molten pool. The precipitation of the compound can be observed also in the case of small grain size, but it is not as visible on the temperature profile. The phase evolution during the transient accompanying the ignition is shown in Fig. 4 in the case of a reaction that can reach the self-sustaining mode and of another which undergoes extinction as a result of a larger grain size (synthesis of CoAl, with starting grain sizes of 10 and 100 mm, respectively). Fig. 4a1 and b1 reports the amount of solid transition metal along the propagation direction, while, Fig. 4a2 and b2 shows profiles of enthalpy variation related to chemical and phase transition processes. In both cases when the ignition source is turned on a limited number of space points close to the ignition source reach the melting point of aluminum and the dissolution of Co starts taking place while the ignition source is still on. Since dissolution is faster with smaller Co grain sizes (Fig. 4a1), the heat produced at every time step and at every space point in this case is fairly high (Fig. 4a2). As a result, the reaction front moves from its initial position and becomes self-sustaining. When larger grain size of Co is considered, the heat released by the chemical processes (at every space step and for every time step) is an order of magnitude smaller (Fig. 4b2). This means that after the first space points have reacted, the combustion wave is not able to produce enough heat to compensate for the heat losses and does not
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Fig. 6. Temperature distribution along the sample when the ignition coil reaches the melting temperature of Al while heating (a) and while cooling (b). Black circles refer to bulk thermal conductivity while white circles to simulations performed with c ¼ cbulk/ 70. All simulations were run with Ni starting grain size of 10 mm.
propagate (Fig. 4b1). More space points “warm up” due to heat transfer, but the dissolution process does not produce enough heat for the process to become self-propagating. 3.3. The effect of thermal conductivity The role of thermal conductivity on the ignition and propagation of SHS processes is intrinsically more complex. Although in the modeling it is possible to consider thermal conductivity as an independent parameter, this can lead to unrealistic conclusions, since it cannot be controlled at will in real experiments. The experimental parameter that more closely controls the thermal conductivity of a sample is its porosity. A variation in porosity, however, produces a complex result, as it does not only modify the thermal conductivity of the mixture, but it also influences the amount of energy for unit volume that can be released by the chemical process. An example of the influence of thermal conductivity and porosity on the ignition energy of NieAl mixtures for an ignition temperature of 2000 K and a grain size of 1 mm is shown in Fig. 5a and b, respectively. A reduction in thermal conductivity, starting from the thermal conductivity of a fully dense pellet, (Fig. 5a) produces initially a reduction in the amount of energy required for ignition followed by a sharp increase. A somewhat similar behavior is observed when the porosity is considered as a free parameter (Fig. 5b): an increase in porosity initially produces
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a decrease in the energy required to ignite the reaction, followed by a steep increase for porosity values above 30%. Some general conclusions can be drawn on the basis of these results. When the thermal conductivity is very high, a large fraction of the supplied heat is drained by the cold part of the sample, requiring a high energy input in order to bring a considerable fraction of the sample above the ignition temperature (Fig. 6). A reduction in the thermal conductivity reduces at first the energy required, as the temperature gradient inside the sample becomes steeper and gets easier to increase the temperature of the first part of the sample. For very low values of thermal conductivity, however, the temperature gradients become so steep that only a very small fraction of the sample is brought above the ignition temperature. The total amount of heat released by that region becomes, at that point, too small to sustain
Fig. 7. Solid Al (continuous line, left axis), solid Me (dots, left axis), amount of liquid phase (dashedotedot, left axis), solid compound (dashes, left axis) and temperature (dashedot, right axis) as a function of time, for the SHS of CoAl (a), NbAl3 (b) and TiAl (c).
the propagation of the reaction, unless a very large amount of heat is supplied from the outside. It is interesting to note that the trends reported in Fig. 5a reproduce closely experimental observations reported by Barzykin in the case of the ignition of TiC [4]. 3.4. Temperature profiles and phase evolution As we pointed out earlier, in most cases experimental investigation of the ignition processes involves the determination of a single parameter, such as the ignition temperature or the ignition power. Only in very few cases more detailed information on the dynamics of the ignition process are presented. Milanese et al. [25] reported recently an investigation on the ignition of the SHS of CoeAl and NbeAl for different transition metal grain size using laser powers sources and determining the temperature profiles with micro-thermocouples. In both systems the temperature profiles showed a slow initial rise, due to the laser heating, followed by a distinct increase in the heating rate when the reaction ignites. The two systems, however, showed quite different behavior. In the case of CoAl, a very sharp ignition is observed right after the melting of Al for all grain sizes, while in the case of the NbAl3 a much more complex behavior is observed, involving an induction period between the melting of Al and the actual ignition. As a result, ignition temperatures between 800 and 1000 C are reported, depending on the grain size. This behavior contrasts with the commonly accepted idea that SHS processes in intermetallic systems are always triggered by the melting of one of the components. It must also be noted that the heat of reaction, and the corresponding adiabatic temperatures, are quite similar for the compounds present in the two systems, so the different ignition behavior cannot be explained on the bases of a difference in the thermochemical properties. We investigated the ignition processes in these two systems using our approach, attempting to provide justification for the experimental evidence through the analysis of phase evolution during the ignition process. In Fig. 7, profiles of temperature (dashedot), amount of solid Al (continuous line), solid Me (dots), liquid phase (dashedotedot), and solid intermetallic compound (dashes) are reported as a function of time for the SHS of CoAl (a), NbAl3 (b), and TiAl (c), on the surface of a pellet that is being ignited by a radiating source. In all cases, the simulations were performed with r0 ¼ 50 mm and standard ignition conditions (as reported in the Appendix), in order to make a reliable comparison with the experimental results [25,37]. For the same reason pre-heating of the starting mixture at 400 C was simulated for the NbeAl system. In the case of the SHS of CoAl (Fig. 7a), after the initial heating by conduction and melting of Al, the dissolution process starts and the heat released by this highly exothermic process immediately causes a sharp increase in the temperature profile. A different situation can be observed for the SHS of NbAl3 (Fig. 7b). In this case, after heating and Al melting an intermediate region (between points 1 and 2 in the figure), prior to the sharp temperature increase, can be observed. In this period the temperature increases more slowly in comparison to what was observed in the case of the SHS of CoAl, where such a region was not observed. Such a result reproduces quite well what observed by Milanese et al. [25]. These differences in behavior have the basis in a different kinetics of the solideliquid interactions between these two systems, that produces a different sequence of events during the ignition process, as evidenced by our simulations. In the case of CoAl, in fact, the dissolution of Co into liquid Al is a very fast process due to the high intrinsic dissolution coefficient (U0) and high solubility of Co into Al, that in the range between 1100 and 1500 K, typical of these processes, spans the range 8% < S < 20%. The dissolution of Co into liquid Al is a strongly exothermic process and hence at every time step, a considerable
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amount of heat is produced by dissolution. The fast dissolution and strong exothermicity cause a strong release of heat resulting in a spike in the temperature profile. In contrast, the dissolution of Nb in liquid Al is a much slower process, due to the low solubility of Nb
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in Al (S < 1% at 1100 K and z3% at 1500 K). The exothermicity of this process is also considerably lower. The heat released by the dissolution process (starting at point 1) is thus not sufficient to cause a sharp increase in the temperature, but only a change in the
Fig. 8. Wave speed as a function of ignition energy E for the SHS of NiAl (a,e), CoAl (b,f), TiAl (c,g), and NbAl3 (d,h) with c ¼ cbulk and c ¼ cbulk/70 respectively. Black dots refer oscillating propagation modes, and white squares to steady propagation. The dashed lines show the lowest E value before extinction.
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Table 1 Propagation wave speed as a function of ignition energy, as obtained in Fig. 7. System
c
v/cm s1
E/J cm2
NieAl CoeAl TieAl NbeAl NieAl CoeAl TieAl NbeAl
c ¼ cbulk c ¼ cbulk c ¼ cbulk c ¼ cbulk c ¼ cbulk/70 c ¼ cbulk/70 c ¼ cbulk/70 c ¼ cbulk/70
24 15 17 15 2.6 2 2.1 2
120 120 200 200 1800 1800 2000 2000
slope with the temperature increasing quite slowly until the liquid phase reaches saturation and the compound starts precipitating (point 2). The heat released by this process, the precipitation of the intermetallic compound, is in this case responsible for the rapid increases of the temperature to values at which the solubility of Nb into Al is strongly increased. The dissolution process hence becomes faster and a temperature spike is produced due to both the heat produced by dissolution and to the heat released by the precipitation of the compound. Fig. 7c shows the result for the third reaction considered, SHS of TiAl, whose thermodynamic and kinetic characteristics can roughly be defined as “intermediate” between NbAl3 and CoAl (DH z 75 kJ mol1, S z 3% at 1100 K and z18% at 1500 K). As a result, the time profiles obtained in this case can be defined as intermediate, since it is not as sharp as the one of CoAl, but not as slow as the one of NbAl3, due to the intermediate thermodynamic and kinetic characteristics of this system in comparison to the two other systems. As a matter of fact, after melting of Al, the dissolution process starts fairly slow. As a consequence, it is not possible to obtain an immediate ignition, but the slope of the intermediate region of the T profile (between points 1 and 2) is higher than that for the SHS of NbAl3, since more heat is released at every time step, for both kinetic and for thermodynamic reasons. 3.5. Effect of ignition conditions on propagation parameters The ignition conditions have a strong influence not only on the dynamic of the ignition process itself, but also on the behavior of the subsequent steadily propagating wave. In several cases the energy supplied during the ignition produces an increase in
temperature in a fairly large portion of the sample or even in all of it, modifying the combustion temperature and propagating rate. Limited attention has been given to this effect, since experimentally the ignition conditions are mostly uncontrolled. As a result, most of the data relative to reaction temperature and propagation rates in SHS processes are altered, particularly when intermetallic phases are concerned, since these systems are characterized by high thermal conductivities. Ideally, the intrinsic value of an SHS propagating wave velocity should be determined at an infinitely long distance from the ignition surface. However, this is very difficult to do experimentally, as the production of very long green pellets is problematic. In alternative, different energies of ignition should be used, identifying for each reaction the limiting conditions. Such approach, however, is complex and time consuming and has not been used in real experimental investigations. The influence of the ignition conditions on wave propagation, however, can be easily investigated through our modeling. In our approach we analyzed the propagation rate in the case of the synthesis of several transition metal aluminides varying the heat flux given to the sample during the ignition process. Fig. 8 shows the dependence of wave velocity on ignition energy for the SHS of NiAl (a,e), CoAl (b,f), TiAl (c,g), and NbAl3 (d,h). For all systems, two sets of simulations were made; one with bulk thermal conductivity (a,b,c,d) and the other with a strongly reduced value of thermal conductivity (e,f,g,h). As a general observation, it can be seen that all four systems exhibit a decrease in wave velocity when the energy of ignition is lowered. Propagation wave velocities obtained with a reduced thermal conductivity, c, are always lower than those observed with bulk thermal conductivities and the value of E required to ignite the reaction becomes larger as the effective thermal conductivity is decreased. The minimum values of wave velocity and the corresponding ignition energies for each system and each thermal conductivity are presented in Table 1. The results shown in Fig. 8 suggest that an intrinsic limit in the value of the propagation wave velocity actually exists, although such a value is dependent on several experimental factors (grain size, compaction degree, etc.). It also shows that very energetic ignition conditions can alter significantly the propagation conditions. A comparison of these results with experimental data is very difficult, as the problem of the influence of ignition conditions on the propagation conditions has never been investigated systematically. The experimental data that are more closely related to this analysis focus on the effect of homogeneous pre-heating on the propagation rate. Such an effect has been studied experimentally for the SHS of NiAl by Maslov et al. [37], Hardt and Holsinger [38], and Naiborodenko and Itin [39e41]. Simulation results (1-D, continuous line and 3-D dashed line) are compared with the experimental results of Maslov et al. (black dots) in Fig. 9. Both the 1-D and 3-D simulation results are in good agreement with the experimental values for pre-heating temperatures, Ti, below 600 K. But a marked difference is encountered for higher pre-heating temperatures. At higher temperatures, only the 3-D simulation results are in agreement with experiment. Makino [28] suggested that such a discrepancy is due to the experimental difficulty in identifying the combustion wave in samples that undergo a high level of pre-heating. Another possible cause of the discrepancy might be related to the fact that solid-state processes have not been considered in the present simulation approach. Such processes are expected to be significant as the preheating temperature is increased. 4. Summary
Fig. 9. Propagation speed as a function of pre-heating temperature for the SHS of NiAl. Black dots report experimental results by Maslov et al. [37], the continuous line shows 1-D simulation results and the dashed line refers to 3-D simulations results, obtained with r0 (Ni) ¼ 10 mm and c ¼ cbulk/70.
Modeling of the ignition process in self-propagating hightemperature synthesis (SHS) reactions of transition metal aluminides was made taking explicitly into account chemical changes and phase
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transformations. Ignition diagrams were developed, showing the effect of input ignition energy density and peak temperature on the occurrence of different modes of combustion. The boundary between wave extinction and propagation was shown to be nearly independent of peak temperature. In contrast, regardless of the ignition energy supplied, no combustion can be initiated if the temperature does not reach the melting point of Al, in agreement with experimental observations. Volume combustion (thermal explosion) is predicted when the peak ignition temperature is low and the input energy is high. The influence of the transition metal grain size on ignition was investigated. It was shown that with an increase in grain size, a significant increase in the input energy density is required with a reduction of the stable SHS region in the ignition diagram. The influence of thermal conductivity and porosity on the ignition energy was determined for the NieAl system. A reduction in the thermal conductivity results initially in a reduction in the amount of energy required for ignition. Further decrease in thermal conductivity causes a sharp increase in the required energy. Similar behavior is observed with changes in porosity. Initially an increase in porosity results in a decrease in the energy required to ignite the reaction, followed by a steep increase in the energy when the porosity exceeds 30%. The influence of the ignition conditions on the wave velocity of the subsequent SHS front in several aluminides was also examined. Simulations were made and the results were compared with the published experimental results and a general good agreement was found.
Appendix Numerical values of all parameters used in the present work are reported in this section. All phase diagrams have been calculated following the CALPHAD approach [42] and the underlying thermodynamic data for NieAl [43], TieAl [44], CoeAl [45], NbeAl [46] have been obtained from literature. The dissolution of the transition metal into the molten aluminum pool is represented by a dissolution constant Kdiss given as Kdiss ¼ ks U, with ks taking into account the solubility of the transition metal into aluminum (depending on the phase diagram) and U is the intrinsic dissolution coefficient of Me into Al. In all present simulations U is given in an Arrhenius form, where both pre-exponential factor and activation energy have been obtained from experimental measurements [47]. Specifically, for NieAl, U0 ¼ 30 108 m2 s1 and Ea ¼ 45.4 kJ mol1, for TieAl, U0 ¼ 41 108 m2 s1 and Ea ¼ 19.9 kJ mol1, for CoeAl, U0 ¼ 68.9 108 m2 s1 and Ea ¼ 264 kJ mol1 and for NbeAl, U0 ¼ 0.19 108 m2 s1 and Ea ¼ 19.2 kJ mol1. Emissivity is always set at 0.99. Thermal conductivity has been modeled in two different ways. In both cases, at each time step and for each space slice, thermal conductivity is calculated as the amount-weighted summation of thermal conductivities of substances present in a phase. Since previous works and literature evidence [2,3,48] demonstrated that bulk values of thermal conductivity are far too high to simulate SHS of a compact of powders, in a first approach, the calculated (bulk) value of thermal conductivity is reduced simply applying a reduction factor homogeneously on the whole sample. In this case, thermal conductivity values explored span the range cbulk/70 < c < cbulk. In the second approach, non-linear dependence of thermal conductivity on porosity has been modeled using the approach of Cunningham
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and Peddicord [49]. In all cases, thermal conductivity values for pure elements have been obtained from literature [50]. The high-melting metal grain size values used in the present work (where not differently specified) are 105 m. The pellet is 0.02 m long with dx ¼ 106 m and dt ¼ 107 s. Parameters used for the Gaussian ignition function are 104 s
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Intermetallics journal homepage: www.elsevier.com/locate/intermet
Modeling the ignition of self-propagating combustion synthesis of transition metal aluminides Silvia Gennari, Umberto Anselmi-Tamburini, Filippo Maglia*, Giorgio Spinolo Department of Physical Chemistry, University of Pavia, V.le Taramelli 16, 27100 Pavia, Italy
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 November 2009 Received in revised form 13 January 2010 Accepted 12 August 2010 Available online 9 September 2010
The ignition process in self-propagating high-temperature synthesis (SHS) reactions was modeled taking into account chemical and phase transformation processes. The model was applied to the synthesis of aluminides. The dependence of the ignition process on peak input energy density and peak temperature is demonstrated by ignition diagrams. The influence of the transition metal grain size and pellet porosity on the ignition conditions was demonstrated. With an increase in grain size, a significant increase in the input energy density is required and the region of SHS reactions in the ignition diagram is reduced. The experimentally observed difference in the ignition characteristics of the synthesis of different aluminides was attributed to differences in solubility limits and rate of dissolution. The effect of the ignition conditions on the velocity of the following self-sustaining reaction was analyzed. 2010 Elsevier Ltd. All rights reserved.
Keywords: C. Reaction synthesis A. Aluminides
1. Introduction Self-propagating high-temperature synthesis (SHS) is a well established method for the synthesis of various classes of inorganic materials that are characterized by a strongly exothermic formation reaction [1]. In the last two decades a considerable effort has been devoted to demonstrate how this method can be applied to the synthesis of a wide variety of materials. Despite that, the mechanisms of ignition and propagation of these complex reactions are still not well understood. In particular the ignition process has received a very limited attention, from either experimental or theoretical point of view. Information on the dependence of ignition temperatures and ignition energies from the experimental parameters such as heat of reaction, thermal conductivity, grain size and relative density are scarce, while the theoretical investigations are usually based on oversimplified chemical mechanisms derived from the chemistry of flames. This represents a major limitation in our understanding of solid-state combustion reactions, since the ignition process plays a pivotal role in defining their feasibility. In previous investigations [2,3] we developed a novel approach to the simulation of SHS processes, based on a detailed and realistic chemical mechanism that we utilized to investigate the propagation of SHS reactions bringing to the formation of some transition metals
* Corresponding author. Tel.: þ39 0382 987208; fax: þ39 0382 987575. E-mail address: fi[email protected] (F. Maglia). 0966-9795/$ e see front matter 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.intermet.2010.08.008
aluminide. Aim of the present paper is to apply the same approach to the study of the transient phenomena preceding the steady-state wave propagation in the same aluminide systems, with the explicit intent to investigate the role of the experimental parameters, and to compare the results with the available experimental data. Most of the presented results will be related to the most experimentally investigated processes, i.e.: the synthesis of NiAl and CoAl. These two systems, incidentally, are characterized by almost identical thermodynamic and kinetic characteristics and produce very similar results. For these reason in some sections the presented data will refer to NiAl and in other to CoAl. Simulations concerning the NbAl3 and TiAl synthesis reaction will also be presented in some sections to show as large differences in the thermodynamic and kinetics of the solideliquid interaction can produce dramatic differences in the SHS ignition process even for reactions characterized by similar reaction enthalpies. 2. Modeling the ignition process in the SHS of transition metal aluminides As reported before, literature accounts on the ignition of SHS processes are relatively few. Barzykin published the only comprehensive review on this topic, summarizing its experimental and theoretical aspects [4]. In general, the early theoretical works were based on a homogeneous pre-mixed flame model [5e7]. The first micro-mechanistic approach, including an heterogeneous chemical mechanism, was developed by Stangle and co-workers [8e11] for the SHS of NbeC. More recent experimental investigations have
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attempted to relate the initiation process to different chemical steps and reaction mechanisms, focusing primarily on the SHS of TiC [12e17]. For this process, Makino et al. [17] studied the effect of mixture composition, size of the carbon particles, degree of dilution, heating area, and heat flux on the ignition delay time. Few studies are available in the case of synthesis of intermetallic compounds. The most experimentally investigated process has been by far the synthesis of NiAl [18e23]. Monagheddu et al. [23] evaluated for this reaction the influence of laser energy on ignition delay time and temperature. Zhu et al. [24], on the other hand, provided for this reaction experimental evidences of a dissolutionprecipitation-controlled mechanism. Experimental information concerning the relationship between ignition delay time and reactants grain size (and hence heating rate) has been provided for the SHS of CoAl [25]. Determination of ignition temperatures and energies for different stoichiometries in the TieAl [26] and NbeAl systems [18] has also been made. On the modeling front, Fu et al. [27] proposed a mathematical approach that analyses the ignition process of these reactions, while more recently Makino, by applying a heterogeneous approach to the study of ignition of SHS in aluminides [28], has shown that ignition energy is not only related to heat losses of the sample, but also to other physico-chemical parameters (such as particle size of the transition metal) which are not taken into account in a homogeneous gas-phase like theory. The computational approach that we are proposing is based on a model that describes the ignition and propagation of the SHS of transition metal aluminides taking explicitly into account all the chemical and phase transformations involved in the combustion process. Details on the method have been provided in previous papers [2,29,30]. In this approach we describe the SHS reactions using a parabolic partial differential equation (PDE) system, which in the case of a one-dimensional problem, can be expressed as
(
2$s$3$ T 4 T 4 a vT v vT c þ C ¼ þ q_ chim a vt vx vx R q_ chim ¼ f ðx;T;tÞ b
(1)
where, in Fourier’s equation, Eq (1), s is the StefaneBoltzmann constant, 3 the emissivity, R the radius of the sample (cylindrical in this case with radius R and length L), T the temperature and Ta is the ambient temperature (typically 298 K), and each equation q_ chim ¼ f ðx;T;tÞ describes one chemical process and phase transformation involved in the process. In the case of the SHS of transition metal aluminides [2], we considered a chemical mechanism based on a dissolution-precipitation process [24,31,32]. This mechanism involves the following steps: a) melting of Al, b) melting of the transition metal, c) dissolution of the transition metal into liquid aluminum, d) precipitation/melting of the intermetallic compound, e) eutectic deposition. This list, however, does not necessarily represent a time sequence of events driving to the formation of the intermetallic phase, but a mere list of all processes that the algorithm is able to take into account. The actual sequence of events is decided by the algorithm itself depending on the actual conditions present at each time and space step during the simulation. As a result, one or more of the previous steps might be missing, depending on the reaction considered and on the actual reaction conditions. This approach allows enough flexibility to reproduce the actual reaction mechanism that is involved in the synthesis of the majority of the aluminides. In most cases, as we said before, the reaction is controlled by the solideliquid interaction between the liquid aluminum and the solid transition metal; the melting of the transition metals is possible only in few cases as well as the melting of the intermetallic phases, but these events do happen when the right combination of thermodynamic properties and experimental conditions is met. It must be further noticed that
our model implicitly implies that the phase diagram involved in the considered reacting system is represented by a simple eutectic. This, of course, is not the case for all aluminides. More complex phase equilibria, involving the formation of intermediate compounds or peritectic reaction, are often encountered. Strictly speaking our approach does not apply to these cases. However, it must be noted that, because of the very rapid kinetic of the SHS processes, some phase equilibria, when characterized by an intrinsically slow kinetics, are never attained and a simplified phase diagram is actually controlling the reaction mechanism. Peritectic reactions, for instance, rarely play an important role in SHS processes because of their slow kinetics. As a result, our approach can be applied also to systems characterized by a complex phase diagram, if the appropriate conditions are met. It must finally be noted that in the algorithm all phase transformations, steps (a), (b), (d) and (e), are described in terms of thermal balance only, without an explicit kinetic law; only the kinetics of dissolution of the transition metal spherical particle in the liquid Al are taken explicitly into account [33]. The ignition of an SHS process can in principle be reduced at two limiting cases, usually referred to as “heat flux-controlled” and “temperature-controlled” ignition. These two idealized situations have the experimental counterpart in the laser ignition (where heat flux is controlled) and chemical ignition (where an igniter characterized by a defined reaction temperature is used). In our approach both ignition conditions can be simulated using boundary conditions defined either as a function of temperature (corresponding to Cauchy boundary conditions) or of heat flux (corresponding to Von Neumann boundary conditions) [2,30]. Controlled temperature ignition (Cauchy boundary conditions) is expressed as:
8 < Tðx; t ¼ 0Þ ¼ Ta Tð0; tÞ ¼ f1 ðtÞ : TðL; tÞ ¼ f2 ðtÞ
cx˛f0; Lg (2)
where f2(t) is normally set to room temperature (Ta ¼ 298 K), while f1(t) represents the ignition function on the left-hand side of the sample. In order to avoid discontinuity and to reproduce the experimental situation, the ignition pulse in our simulations is not described by a square function, but by a Gaussian function, as:
2 2 f1 ðtÞ ¼ Ta þ Tp Ta eððtmÞ =2u Þ
(3)
with Tp being the maximum temperature value, u the time range, and m the time lag of ignition. On the other hand, controlled heat flux ignition (Von Neumann boundary conditions) is expressed as:
8 Tðx; t ¼ 0Þ ¼ Ta > < vT J0 ðtÞ vx jð0;tÞ ¼ c > JL ðtÞ : vT j ¼ vx ðL;tÞ
cx˛f0; Lg (4)
c
where the fluxes J0 and JL can assume a fixed value or can be a function of time. A convenient way to define these fluxes is through the use of a radiation term:
Jð0; tÞ ¼ s$3$ T04 Ts4
(5)
JðL; tÞ ¼ s$3$ TL4 Td4
(6)
with Td typically ¼ Ta, and Ts depends on time, as imposed by a suitable ignition function, and s and 3 are as defined earlier. Numerical solutions for the 1-D model have been obtained using the FD Crank Nicholson algorithm [34], and for the 3-D model [35],
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Fig. 1. Propagation diagram for the combustion synthesis of NiAl, with starting Ni grain size of 1 mm and bulk thermal conductivity. The boundaries have been calculated with the present model, while the references at the bottom refer to fluxes used experimentally for ignition, when available in literature [18,19,26,51].
a commercial engine for the solution of PDEs [36] has been interfaced to the chemical routines in order to solve the system. Unless otherwise specified, the simulation results reported here have been obtained with the 1-D approach, whose features are more than sufficient for the study of the ignition process.
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Fig. 2. Ignition energy as a function of starting high-melting metal grain size for the SHS of NiAl with bulk thermal conductivity.
suggesting that an energy threshold exists, with a value that is almost independent from peak temperature. On the other hand, no combustion can be initiated, regardless of the total energy supplied to the sample, if the temperature does not reach at least the melting point of Al. This is a consequence of the absence of any solidesolid interaction in our model and agrees well with the experimental evidence [24,31,32]. Finally, in the case of low ignition temperatures and very large energies no real propagation front can be
3. Results and discussion 3.1. Ignition diagrams We indicated before the distinction between temperaturecontrolled and heat flux-controlled ignition conditions [4]. This distinction has been largely used in traditional combustion studies because the use of simplified boundary conditions allows an easier analytical solutions of the systems of Eq (1). As a result, the ignition conditions for a certain reacting system are usually defined on the basis of a one parameter such as the temperature of ignition or the heat of ignition. Sometimes, less direct parameters such as the ignition delay time, are used, although they all refer ultimately to a temperature or energy condition. The actual situation, however, is far more complex and the ignition of an SHS process can hardly be defined by a single parameter. The best way to visualize this situation is through the use of ignition diagrams, such as the one reported in Fig. 1 for the synthesis of NiAl, using powders of 1 mm in grain size and a sample 1 cm in diameter and 2 cm long. This diagram shows the outcome of a Gaussian ignition pulse for all possible combinations of peak temperature and peak energy. It is evident as a self-propagating combustion front can be initiated only if a certain combination of temperature and energy density is used. It should be here clarified that in this figure, the ignition time is implicitly involved moving in both directions of the diagram. By moving vertically along the y axis, for instance, the same total energy amount is released to the system slowly, in the case of low peak temperature, or very quickly, in the case of high peak temperatures. On the other hand, moving horizontally, keeps the peak temperature constant, but involves releasing different energies by means of maintaining the ignition source on for different times. Fig. 1 tells us that if the ignition impulse is very short, as happens on the left side of the diagram, no self-propagating process can ever be initiated even if very high peak temperatures are reached. The boundary between extinction and propagation in this region (on the left-hand side of the diagram) is almost vertical,
Fig. 3. Solid Al (continuous line, left axis), solid Me (dots, left axis), solid compound (dashes, left axis) and temperature (dashedot, right axis) as a function of time, for the SHS of NiAl with starting Ni grain size of 1 mm (a), and 70 mm (b).
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Fig. 4. Space profiles (at different time steps) of temperature (a,b), amount of solid metal (a1,b1), and enthalpy released by chemical/phase transformations (a2,b2) for the SHS of CoAl with starting grain size of 10 mm (a,a1,a2) and 100 mm (b,b1,b2).
initiated, but the entire sample ends up reacting at once; a reaction mode that is usually referred to as thermal explosion or volume combustion. The high thermal conductivity produces in this case an homogeneous increase in temperature in the whole sample, as a result of the slow energy supply. It is important to note that the actual shape of these ignition diagrams is not uniquely defined by the intrinsic (thermodynamic and kinetic) nature of the reacting systems, but it is, for a given a system, very dependent on the values of other experimental parameters, in particular on the grain size of the transition metal and on the sample porosity. We will discuss this influence in the following sections.
3.2. Effect of grain size An example of the influence of the transition metal grain size on the ignition conditions is shown in Fig. 2, as obtained by using our model. The figure shows the energy density required to ignite the reaction for a fixed ignition peak temperature of 2000 K when the grain size is changed in the range between 101 to 50 mm. The figure shows an increase of three orders of magnitude in the ignition energy when the grain size is varied in this range. When compared with the ignition diagram of Fig. 1, which was calculated
for a grain size of 1 mm, it is evident as the region of SHS ignition shrinks dramatically as a result of the larger grain size. The reason for such a strong influence of grain size on the ignition conditions can be better understood by analyzing the details of the reaction mechanism. This will be illustrated at first in the case of a steady propagating front (Fig. 3) and then in the more complex transient leading to the ignition (Fig. 4). In Fig. 3 the temperature profile (dashedot), the amount of solid Al (continuous line), solid Me (dots) and solid intermetallic compound (dashes) for the SHS of NiAl with starting Ni grain sizes of 1 mm (a) and 70 mm (b) are reported. The simulations have been performed using a thermal conductivity corresponding to the bulk value, i.e. in a zero porosity situation. The effect of thermal conductivity (and porosity) will be explicitly considered in the next section. When small grain sizes are used (Fig. 3a), the dissolution of the transition metal proceeds extremely fast due to the large interface area between the reactants, and comes to completion within few time steps. Since Ni dissolves in large amounts at each time step, a considerable amount of heat of solution is released, producing a sharp increase in the temperature profile. In contrast, when r0 ¼ 70 mm (Fig. 3b), Ni dissolution starts and proceeds at a much slower rate, taking z0.01 s to come to completion. At each time step, Ni dissolves into the molten pool in small quantities and no temperature spike is
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Fig. 5. Ignition energy as a function of (a) effective thermal conductivity, and (b) sample porosity for the SHS of NiAl, with starting nickel grain size of 1 mm.
observed. Once the solid intermetallic compound starts to precipitate, the heat released by this process raises the temperature further, increasing also the dissolution rate of the transition metal into the molten pool. The precipitation of the compound can be observed also in the case of small grain size, but it is not as visible on the temperature profile. The phase evolution during the transient accompanying the ignition is shown in Fig. 4 in the case of a reaction that can reach the self-sustaining mode and of another which undergoes extinction as a result of a larger grain size (synthesis of CoAl, with starting grain sizes of 10 and 100 mm, respectively). Fig. 4a1 and b1 reports the amount of solid transition metal along the propagation direction, while, Fig. 4a2 and b2 shows profiles of enthalpy variation related to chemical and phase transition processes. In both cases when the ignition source is turned on a limited number of space points close to the ignition source reach the melting point of aluminum and the dissolution of Co starts taking place while the ignition source is still on. Since dissolution is faster with smaller Co grain sizes (Fig. 4a1), the heat produced at every time step and at every space point in this case is fairly high (Fig. 4a2). As a result, the reaction front moves from its initial position and becomes self-sustaining. When larger grain size of Co is considered, the heat released by the chemical processes (at every space step and for every time step) is an order of magnitude smaller (Fig. 4b2). This means that after the first space points have reacted, the combustion wave is not able to produce enough heat to compensate for the heat losses and does not
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Fig. 6. Temperature distribution along the sample when the ignition coil reaches the melting temperature of Al while heating (a) and while cooling (b). Black circles refer to bulk thermal conductivity while white circles to simulations performed with c ¼ cbulk/ 70. All simulations were run with Ni starting grain size of 10 mm.
propagate (Fig. 4b1). More space points “warm up” due to heat transfer, but the dissolution process does not produce enough heat for the process to become self-propagating. 3.3. The effect of thermal conductivity The role of thermal conductivity on the ignition and propagation of SHS processes is intrinsically more complex. Although in the modeling it is possible to consider thermal conductivity as an independent parameter, this can lead to unrealistic conclusions, since it cannot be controlled at will in real experiments. The experimental parameter that more closely controls the thermal conductivity of a sample is its porosity. A variation in porosity, however, produces a complex result, as it does not only modify the thermal conductivity of the mixture, but it also influences the amount of energy for unit volume that can be released by the chemical process. An example of the influence of thermal conductivity and porosity on the ignition energy of NieAl mixtures for an ignition temperature of 2000 K and a grain size of 1 mm is shown in Fig. 5a and b, respectively. A reduction in thermal conductivity, starting from the thermal conductivity of a fully dense pellet, (Fig. 5a) produces initially a reduction in the amount of energy required for ignition followed by a sharp increase. A somewhat similar behavior is observed when the porosity is considered as a free parameter (Fig. 5b): an increase in porosity initially produces
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a decrease in the energy required to ignite the reaction, followed by a steep increase for porosity values above 30%. Some general conclusions can be drawn on the basis of these results. When the thermal conductivity is very high, a large fraction of the supplied heat is drained by the cold part of the sample, requiring a high energy input in order to bring a considerable fraction of the sample above the ignition temperature (Fig. 6). A reduction in the thermal conductivity reduces at first the energy required, as the temperature gradient inside the sample becomes steeper and gets easier to increase the temperature of the first part of the sample. For very low values of thermal conductivity, however, the temperature gradients become so steep that only a very small fraction of the sample is brought above the ignition temperature. The total amount of heat released by that region becomes, at that point, too small to sustain
Fig. 7. Solid Al (continuous line, left axis), solid Me (dots, left axis), amount of liquid phase (dashedotedot, left axis), solid compound (dashes, left axis) and temperature (dashedot, right axis) as a function of time, for the SHS of CoAl (a), NbAl3 (b) and TiAl (c).
the propagation of the reaction, unless a very large amount of heat is supplied from the outside. It is interesting to note that the trends reported in Fig. 5a reproduce closely experimental observations reported by Barzykin in the case of the ignition of TiC [4]. 3.4. Temperature profiles and phase evolution As we pointed out earlier, in most cases experimental investigation of the ignition processes involves the determination of a single parameter, such as the ignition temperature or the ignition power. Only in very few cases more detailed information on the dynamics of the ignition process are presented. Milanese et al. [25] reported recently an investigation on the ignition of the SHS of CoeAl and NbeAl for different transition metal grain size using laser powers sources and determining the temperature profiles with micro-thermocouples. In both systems the temperature profiles showed a slow initial rise, due to the laser heating, followed by a distinct increase in the heating rate when the reaction ignites. The two systems, however, showed quite different behavior. In the case of CoAl, a very sharp ignition is observed right after the melting of Al for all grain sizes, while in the case of the NbAl3 a much more complex behavior is observed, involving an induction period between the melting of Al and the actual ignition. As a result, ignition temperatures between 800 and 1000 C are reported, depending on the grain size. This behavior contrasts with the commonly accepted idea that SHS processes in intermetallic systems are always triggered by the melting of one of the components. It must also be noted that the heat of reaction, and the corresponding adiabatic temperatures, are quite similar for the compounds present in the two systems, so the different ignition behavior cannot be explained on the bases of a difference in the thermochemical properties. We investigated the ignition processes in these two systems using our approach, attempting to provide justification for the experimental evidence through the analysis of phase evolution during the ignition process. In Fig. 7, profiles of temperature (dashedot), amount of solid Al (continuous line), solid Me (dots), liquid phase (dashedotedot), and solid intermetallic compound (dashes) are reported as a function of time for the SHS of CoAl (a), NbAl3 (b), and TiAl (c), on the surface of a pellet that is being ignited by a radiating source. In all cases, the simulations were performed with r0 ¼ 50 mm and standard ignition conditions (as reported in the Appendix), in order to make a reliable comparison with the experimental results [25,37]. For the same reason pre-heating of the starting mixture at 400 C was simulated for the NbeAl system. In the case of the SHS of CoAl (Fig. 7a), after the initial heating by conduction and melting of Al, the dissolution process starts and the heat released by this highly exothermic process immediately causes a sharp increase in the temperature profile. A different situation can be observed for the SHS of NbAl3 (Fig. 7b). In this case, after heating and Al melting an intermediate region (between points 1 and 2 in the figure), prior to the sharp temperature increase, can be observed. In this period the temperature increases more slowly in comparison to what was observed in the case of the SHS of CoAl, where such a region was not observed. Such a result reproduces quite well what observed by Milanese et al. [25]. These differences in behavior have the basis in a different kinetics of the solideliquid interactions between these two systems, that produces a different sequence of events during the ignition process, as evidenced by our simulations. In the case of CoAl, in fact, the dissolution of Co into liquid Al is a very fast process due to the high intrinsic dissolution coefficient (U0) and high solubility of Co into Al, that in the range between 1100 and 1500 K, typical of these processes, spans the range 8% < S < 20%. The dissolution of Co into liquid Al is a strongly exothermic process and hence at every time step, a considerable
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amount of heat is produced by dissolution. The fast dissolution and strong exothermicity cause a strong release of heat resulting in a spike in the temperature profile. In contrast, the dissolution of Nb in liquid Al is a much slower process, due to the low solubility of Nb
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in Al (S < 1% at 1100 K and z3% at 1500 K). The exothermicity of this process is also considerably lower. The heat released by the dissolution process (starting at point 1) is thus not sufficient to cause a sharp increase in the temperature, but only a change in the
Fig. 8. Wave speed as a function of ignition energy E for the SHS of NiAl (a,e), CoAl (b,f), TiAl (c,g), and NbAl3 (d,h) with c ¼ cbulk and c ¼ cbulk/70 respectively. Black dots refer oscillating propagation modes, and white squares to steady propagation. The dashed lines show the lowest E value before extinction.
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Table 1 Propagation wave speed as a function of ignition energy, as obtained in Fig. 7. System
c
v/cm s1
E/J cm2
NieAl CoeAl TieAl NbeAl NieAl CoeAl TieAl NbeAl
c ¼ cbulk c ¼ cbulk c ¼ cbulk c ¼ cbulk c ¼ cbulk/70 c ¼ cbulk/70 c ¼ cbulk/70 c ¼ cbulk/70
24 15 17 15 2.6 2 2.1 2
120 120 200 200 1800 1800 2000 2000
slope with the temperature increasing quite slowly until the liquid phase reaches saturation and the compound starts precipitating (point 2). The heat released by this process, the precipitation of the intermetallic compound, is in this case responsible for the rapid increases of the temperature to values at which the solubility of Nb into Al is strongly increased. The dissolution process hence becomes faster and a temperature spike is produced due to both the heat produced by dissolution and to the heat released by the precipitation of the compound. Fig. 7c shows the result for the third reaction considered, SHS of TiAl, whose thermodynamic and kinetic characteristics can roughly be defined as “intermediate” between NbAl3 and CoAl (DH z 75 kJ mol1, S z 3% at 1100 K and z18% at 1500 K). As a result, the time profiles obtained in this case can be defined as intermediate, since it is not as sharp as the one of CoAl, but not as slow as the one of NbAl3, due to the intermediate thermodynamic and kinetic characteristics of this system in comparison to the two other systems. As a matter of fact, after melting of Al, the dissolution process starts fairly slow. As a consequence, it is not possible to obtain an immediate ignition, but the slope of the intermediate region of the T profile (between points 1 and 2) is higher than that for the SHS of NbAl3, since more heat is released at every time step, for both kinetic and for thermodynamic reasons. 3.5. Effect of ignition conditions on propagation parameters The ignition conditions have a strong influence not only on the dynamic of the ignition process itself, but also on the behavior of the subsequent steadily propagating wave. In several cases the energy supplied during the ignition produces an increase in
temperature in a fairly large portion of the sample or even in all of it, modifying the combustion temperature and propagating rate. Limited attention has been given to this effect, since experimentally the ignition conditions are mostly uncontrolled. As a result, most of the data relative to reaction temperature and propagation rates in SHS processes are altered, particularly when intermetallic phases are concerned, since these systems are characterized by high thermal conductivities. Ideally, the intrinsic value of an SHS propagating wave velocity should be determined at an infinitely long distance from the ignition surface. However, this is very difficult to do experimentally, as the production of very long green pellets is problematic. In alternative, different energies of ignition should be used, identifying for each reaction the limiting conditions. Such approach, however, is complex and time consuming and has not been used in real experimental investigations. The influence of the ignition conditions on wave propagation, however, can be easily investigated through our modeling. In our approach we analyzed the propagation rate in the case of the synthesis of several transition metal aluminides varying the heat flux given to the sample during the ignition process. Fig. 8 shows the dependence of wave velocity on ignition energy for the SHS of NiAl (a,e), CoAl (b,f), TiAl (c,g), and NbAl3 (d,h). For all systems, two sets of simulations were made; one with bulk thermal conductivity (a,b,c,d) and the other with a strongly reduced value of thermal conductivity (e,f,g,h). As a general observation, it can be seen that all four systems exhibit a decrease in wave velocity when the energy of ignition is lowered. Propagation wave velocities obtained with a reduced thermal conductivity, c, are always lower than those observed with bulk thermal conductivities and the value of E required to ignite the reaction becomes larger as the effective thermal conductivity is decreased. The minimum values of wave velocity and the corresponding ignition energies for each system and each thermal conductivity are presented in Table 1. The results shown in Fig. 8 suggest that an intrinsic limit in the value of the propagation wave velocity actually exists, although such a value is dependent on several experimental factors (grain size, compaction degree, etc.). It also shows that very energetic ignition conditions can alter significantly the propagation conditions. A comparison of these results with experimental data is very difficult, as the problem of the influence of ignition conditions on the propagation conditions has never been investigated systematically. The experimental data that are more closely related to this analysis focus on the effect of homogeneous pre-heating on the propagation rate. Such an effect has been studied experimentally for the SHS of NiAl by Maslov et al. [37], Hardt and Holsinger [38], and Naiborodenko and Itin [39e41]. Simulation results (1-D, continuous line and 3-D dashed line) are compared with the experimental results of Maslov et al. (black dots) in Fig. 9. Both the 1-D and 3-D simulation results are in good agreement with the experimental values for pre-heating temperatures, Ti, below 600 K. But a marked difference is encountered for higher pre-heating temperatures. At higher temperatures, only the 3-D simulation results are in agreement with experiment. Makino [28] suggested that such a discrepancy is due to the experimental difficulty in identifying the combustion wave in samples that undergo a high level of pre-heating. Another possible cause of the discrepancy might be related to the fact that solid-state processes have not been considered in the present simulation approach. Such processes are expected to be significant as the preheating temperature is increased. 4. Summary
Fig. 9. Propagation speed as a function of pre-heating temperature for the SHS of NiAl. Black dots report experimental results by Maslov et al. [37], the continuous line shows 1-D simulation results and the dashed line refers to 3-D simulations results, obtained with r0 (Ni) ¼ 10 mm and c ¼ cbulk/70.
Modeling of the ignition process in self-propagating hightemperature synthesis (SHS) reactions of transition metal aluminides was made taking explicitly into account chemical changes and phase
S. Gennari et al. / Intermetallics 18 (2010) 2385e2393
transformations. Ignition diagrams were developed, showing the effect of input ignition energy density and peak temperature on the occurrence of different modes of combustion. The boundary between wave extinction and propagation was shown to be nearly independent of peak temperature. In contrast, regardless of the ignition energy supplied, no combustion can be initiated if the temperature does not reach the melting point of Al, in agreement with experimental observations. Volume combustion (thermal explosion) is predicted when the peak ignition temperature is low and the input energy is high. The influence of the transition metal grain size on ignition was investigated. It was shown that with an increase in grain size, a significant increase in the input energy density is required with a reduction of the stable SHS region in the ignition diagram. The influence of thermal conductivity and porosity on the ignition energy was determined for the NieAl system. A reduction in the thermal conductivity results initially in a reduction in the amount of energy required for ignition. Further decrease in thermal conductivity causes a sharp increase in the required energy. Similar behavior is observed with changes in porosity. Initially an increase in porosity results in a decrease in the energy required to ignite the reaction, followed by a steep increase in the energy when the porosity exceeds 30%. The influence of the ignition conditions on the wave velocity of the subsequent SHS front in several aluminides was also examined. Simulations were made and the results were compared with the published experimental results and a general good agreement was found.
Appendix Numerical values of all parameters used in the present work are reported in this section. All phase diagrams have been calculated following the CALPHAD approach [42] and the underlying thermodynamic data for NieAl [43], TieAl [44], CoeAl [45], NbeAl [46] have been obtained from literature. The dissolution of the transition metal into the molten aluminum pool is represented by a dissolution constant Kdiss given as Kdiss ¼ ks U, with ks taking into account the solubility of the transition metal into aluminum (depending on the phase diagram) and U is the intrinsic dissolution coefficient of Me into Al. In all present simulations U is given in an Arrhenius form, where both pre-exponential factor and activation energy have been obtained from experimental measurements [47]. Specifically, for NieAl, U0 ¼ 30 108 m2 s1 and Ea ¼ 45.4 kJ mol1, for TieAl, U0 ¼ 41 108 m2 s1 and Ea ¼ 19.9 kJ mol1, for CoeAl, U0 ¼ 68.9 108 m2 s1 and Ea ¼ 264 kJ mol1 and for NbeAl, U0 ¼ 0.19 108 m2 s1 and Ea ¼ 19.2 kJ mol1. Emissivity is always set at 0.99. Thermal conductivity has been modeled in two different ways. In both cases, at each time step and for each space slice, thermal conductivity is calculated as the amount-weighted summation of thermal conductivities of substances present in a phase. Since previous works and literature evidence [2,3,48] demonstrated that bulk values of thermal conductivity are far too high to simulate SHS of a compact of powders, in a first approach, the calculated (bulk) value of thermal conductivity is reduced simply applying a reduction factor homogeneously on the whole sample. In this case, thermal conductivity values explored span the range cbulk/70 < c < cbulk. In the second approach, non-linear dependence of thermal conductivity on porosity has been modeled using the approach of Cunningham
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and Peddicord [49]. In all cases, thermal conductivity values for pure elements have been obtained from literature [50]. The high-melting metal grain size values used in the present work (where not differently specified) are 105 m. The pellet is 0.02 m long with dx ¼ 106 m and dt ¼ 107 s. Parameters used for the Gaussian ignition function are 104 s