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The heterogeneity maps for micropyretic synthesis of Ni–Al composite
Materials Science and Engineering A 392 (2005) 262–268
The heterogeneity maps for micropyretic synthesis of Ni–Al composite H.P. Li∗ Jin-Wen Institute of Technology, 99 An-Chung Road, Taipei County, Hsintien 231, Taiwan Received 21 June 2004; received in revised form 13 September 2004; accepted 15 September 2004
Abstract Heterogeneities in initial composition and porosity are common during micropyretic synthesis when powders are pressed or mixed, and the conventional modeling treatments so far have only considered uniform systems. Heterogeneities in composition and porosity are thought to result in local variations in reaction yield, or such thermophysical/chemical parameters for the reactants as density, heat capacity, and thermal conductivity; further changing the propagation velocity of a combustion front. This study, thus, investigates the impact of heterogeneities in initial composition and porosity on micropyretic synthesis with Ni + Al by a numerical simulation. The heterogeneity maps for the thermophysical/chemical reactant parameters (such as density, heat capacity, thermal conductivity, and reaction yield) and corresponding micropyretic parameters (such as pre-heat zone thickness and propagation velocity) with the heterogeneities in composition and porosity are generated. The calculation results show that the heterogeneity in porosity has a stronger effect on thermal conductivity and density as compared with the heterogeneity in Ni composition. However, the results suggest that the reactivity of the micropyretic reaction is significantly decreased only when the heterogeneity in Ni composition increases. From the knowledge of heterogeneity maps, the effects of heterogeneities in initial composition and porosity on micropyretic reaction can be acquired. © 2004 Elsevier B.V. All rights reserved. Keywords: Composition heterogeneity; Porosity heterogeneity; Micropyretic synthesis; Combustion synthesis; SHS
1. Introduction Many exothermic non-catalytic solid–solid or solid–gas reactions, after being ignited locally, can release enough heat so as to sustain the self-propagating combustion front throughout the specimen without additional energy [1–9]. Since the 1970s, this kind of exothermic reaction has been used in the synthesis of refractory compounds in the former Soviet Union [5]. This novel technique, so-called micropyretic synthesis or combustion synthesis, has been intensively studied for process implication [1–9]. This technique employs exothermic reaction processing, which circumvents difficulties associated with conventional methods of time and energy-intensive sintering processing. The advantages of micropyretic synthesis also include the rapid ∗
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net shape processing and clean products. In addition, the micropyretic-synthesized products have been reported to possess better mechanical and physical properties. Micropyretic-synthesized products have been reported to have better mechanical and physical properties [10,11]. An example is the formation of shape-memory alloys of nickel and titanium [10]. It has been reported that those prepared by micropyretic synthesis possess greater shaperecovery force than corresponding alloys produced by conventional methods [10]. On account of the high thermal gradients encountered in micropyretic synthesis, it has been speculated that the products of such a process may contain a high-defect concentration. The presence of high levels of defects has led to expectation of higher reactivity, namely higher sinterability [11]. The micropyretic synthesis technique also provides rapid net shape processing. When compared with conventional powder metallurgy operations, micropyretic synthesis not only offers shorter processing
H.P. Li / Materials Science and Engineering A 392 (2005) 262–268
Nomenclature Cp
heat capacity of product (general form) (kJ/(kg K)) d diameter of the specimen (m) E activation energy (kJ/kg) fR (j) random number at node j h surface heat transfer coefficient (J/(m2 K s)) Heterocomp heterogeneity in composition (%) Heteroporosity heterogeneity in porosity (%) K0 pre-exponential constant (s−1 for zero-order reaction) Po original porosity (%) Pj porosity at node j (%) Q heat of reaction (kJ/kg) R gas constant (kJ/(kg K)) Ryield,j reaction yield at node j (%) t time (s) T temperature (K) T0 initial temperature (K) Vs volume fraction of component (species) s (%) Xi,j molar fraction of component (species) i at node j (%) z dimensional coordinate (m) Greek letters η fraction reacted κ thermal conductivity (general form) (kJ/(ms K)) ρ density (kg/m3 ) Φ (T, η) reaction rate (s−1 )
time, but also excludes the requirement for high-temperature sintering. Several numerical and analytical models of micropyretic synthesis in a composite system have been well developed [12–19]. Lakshmikantha and Sekhar first explored the numerical model that includes the effects of dilution and porosity and melting of each constituent of the reactants and products [13]. The analytical modeling of the propagation of the combustion front in solid–solid reaction systems has also been reported [15]. The analytical model gives good results when compared with the experimentally determined numbers and the numerically calculated values. In addition, a multidimensional numerical model and dynamic modeling of the gas and solid reaction have also been carried out to illustrate the effects of various parameters on the micropyretic synthesis [14,15]. These numerical and analytical analyses provide a better understanding of the reaction sequence during micropyretic synthesis reactions. However, heterogeneities in initial composition and porosity are common during micropyretic synthesis when powders are pressed or mixed, and the conventional modeling
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treatments [12–19] so far have only considered uniform systems. Heterogeneities in initial composition and porosity result in local variations in reaction yield and the thermophysical/chemical reactant parameters, such as density, heat capacity, and thermal conductivity. This changes the propagation velocity and combustion temperature of a combustion front. Such a change may further lead to the non-homogeneous microstructures. In this study, a numerical simulation is used to characterize the effects of heterogeneities in initial composition and porosity on NiAl micropyretic synthesis. The heterogeneity maps, considering the heterogeneities in initial composition and porosity concurrently, are also generated. From the knowledge of heterogeneity maps, the effects of heterogeneities in initial composition and porosity on micropyretic reaction can be acquired.
2. Numerical calculation procedure During the passage of the combustion front in the reaction, the energy equation for transient heat conduction, including the source term, containing heat release due to the exothermic micropyretic reaction is given as [5,13,15]: ∂T ∂ ∂T 4h(T − T0 ) ρ Cp = κ − + ρ Q Φ (T, η) ∂t ∂z ∂z d (1) Each symbol in the equation is explained in the Nomenclature section. The reaction rate, Φ (T, η), in Eq. (1) is given as: ∂η E Φ(T, η) = = K0 (1 − η) exp − (2) ∂t RT In this study, a numerical calculation for Eq. (1) is carried out with the assumption of the first-order kinetics. In the Eq. (1), the energy required for heating the synthesized product from the initial temperature to the adiabatic combustion temperature is shown on the left-hand side. The terms on the righthand side are the conduction heat transfer term, the surface heat loss parameter, and the heat release due to the exothermic micropyretic reaction, respectively. The surface heat loss is assumed radically Newtonian in this study. The previous studies [3,13] have shown that the surface heat loss is much less than the exothermic heat of the reaction; thus, the surface heat loss is taken to be zero in the numerical calculation. In the computational simulation, a one-dimensional sample of 1 cm length is divided into 1201 nodes (regions) to calculate the local temperature using an enthalpy–temperature method coupled with the Guass–Seidel iteration procedure. The choice of 1 cm sample length is only for computational purpose. It has been found that the length of sample does not affect the numerical results in this study. Thus, the simulation results are applicable to practical experimental condition. First, the proper initial and boundary conditions are used to initialize the temperatures and enthalpies at all nodes. The initial conditions in the simulation are taken as follows: (1)
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at the ignition node, at time t ≥ 0, the temperature is taken to be the adiabatic combustion temperature (T = Tc and η = 1). It has been reported that the chosen temperature value at the ignition node only slightly influences the temperature profiles at the initial stage [13,15]. (2) At the other nodes, at time t = 0, the temperatures are taken to be the same as the substrate temperature (T = T0 and η = 0). Depending on the values of the temperature and enthalpy occurred in the reaction, the proper thermophysical/chemical parameters are considered and the proper limits of the reaction zone are determined for each node in the numerical calculation. At any given time, the fraction reacted and enthalpy of the current iteration are calculated from the previous fraction reacted and enthalpy of the earlier iteration. The range of the enthalpy as well as the molar ratio for each material and node is thus determined, and the values of temperature, density, and thermal conductivity at each node can be further calculated in appropriate zone. Initial composition at each node is first calculated from the random number (fR (j) at node j) and the assigned heterogeneity (Heterocomp ) that determines the magnitude of the variation. The sequence of the random numbers (−0.5 to +0.5) generated from the computation is repeatedly used in the specimens with different heterogeneities to compare the magnitude of heterogeneity effect. In this study, Ni + 50 at.% Al composition which corresponds to NiAl compound is chosen. The initial composition before the beginning of the reaction can be expressed as follows.
ingly determined, as shown in Table 1. After the molar fractions of reactants are determined, the reaction yield at node j can be further determined from the molar fractions of reactants and diluents: XNi,j XAl,j (4) , Ryield,j = min 0 0 XNi XAl In addition, the porosity effects of the reactants and product that influence the density (ρ) and thermal conductivity (κ) profiles are also considered in the numerical calculation. During the numerical calculation, the average global porosities (Po ) of the reactants and product are both taken to be 25%. The initial porosity of the reactants at each node is calculated from the random number (fR (j) at node j) and the assigned heterogeneity (Heteroporosity ) that determines the magnitude of the porosity variation: porosity at node j : Pj = P o (1 + Heteroporosity fR (j))
(5)
where −0.5 ≤ fR (j) ≤ +0.5, Po = 25%, 0% < Heteroporosity < 100%, and j = 1, 2, . . ., 1201. The studied compositions with the different heterogeneities in porosity are also shown in Table 1. Once the initial composition and porosity at each node are set to given heterogeneities, the thermophysical/chemical parameters at node j can be thus calculated as: [ρs Vs,j (1 − Pj )] (6) density at node j : ρj = s
Ni molar fraction at node j : XNi,j 0 = XNi (1 + Heterocomp fR (j))
(3a)
Al molar fraction at node j : XAl,j = 1 − XAl,j
thermal conductivity at node j : κj 1 − Pj κs Vs,j = 1 + Pj /2 s
(3b)
0 = 50.0 at.%, where XNi −0.5 ≤ fR (j) ≤ +0.5 0% < Heterocomp < 100%, and j = 1, 2, . . ., 1201. In order to assure the sum of the initial compositions for all 1201 nodes equal to the stoichiometric values, the calculated Ni and Al compositions of each node are adjusted so that the average values of each initial composition are equal to the ideal homogeneous 0 ¯ Ni : X ¯ Al = 1 : 1, i.e., (1/n) n=1201 values (X XNi,j = XNi j=1 n=1201 0 ). For example, as 30% and (1/n) j=1 XAl,j = XAl heterogeneity in initial Ni composition has occurred, Ni composition is correspondingly varied within 15 at.% (=50.0 at.% × 30%). Thus, Ni composition is noted to vary from 42.5 to 57.5 at.% and Al composition is correspond-
heat capacity : Cpj =
(Cps Xs,j )
(7)
(8)
s
where s denotes the component involved in the reaction, including Ni and Al in this study. The effect of melting of reactants and product is also included in the calculation. The various thermophysical/chemical parameters, such as thermal conductivity, density and heat capacity of the reactants and product are assumed to be independent of temperature, but they are different in each state. The average values of these parameters vary as the reaction proceeds, depending upon the degree of reaction. The parameter values used in the computational calculation are shown in the Table 2 [20–22]
Table 1 The examples of the studied Ni + 50 at.% Al compositions with different heterogeneities in initial composition and porosity Heterogeneity in composition (%)
Heterogeneity in porosity (%)
Ni composition (at.%)
Al composition (at.%)
Porosity
0 0 30 30
0 60 0 60
50.0 50.0 42.5–57.5 42.5–57.5
50.0 50.0 57.5–42.5 57.5–42.5
25.0 17.5–32.5 25.0 17.5–32.5
H.P. Li / Materials Science and Engineering A 392 (2005) 262–268
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Table 2 The thermophysical/chemical parameters for the reactants (Ni and Al) and product (NiAl) at 300 K and liquid state [20–22] Thermophysical/chemical parameters
Al
Ni
NiAl
Heat capacity (at 300 K) (J/(kg K)) Heat capacity (liquid state) (J/(kg K)) Thermal conductivity (at 300 K) (J/(ms K)) Thermal conductivity (liquid state) (J/(ms K)) Density (at 300 K) (kg/m3 ) Density (liquid state) (kg/m3 )
902 [20] 1178 [20] 238 [22] 100 [22] 2700 [22] 2385 [22]
445 [20] 735 [20] 88.5 [22] 53 [21] 8900 [22] 7905 [22]
537 [20] 831 [20] 75 [21] 55 [21] 6050 [21] 5950 [22]
and Table 3 [20,23]. In this study, the combustion temperature is defined as the highest reaction temperature during micropyretic synthesis, and the propagation velocity is the velocity of the combustion front propagation. In addition, the higher pre-exponential factor (K0 ) value 4 × 108 s−1 is used to be capable of illustrating the variation of the propagation velocity for the NiAl combustion reaction. The criterion used to ascertain whether the fraction reacted and enthalpies at each time level converge or not is determined from the relative error criterion. Once the convergence criterion for every node is met, the enthalpy and fraction reacted of the last iteration in a time step are considered to be the corresponding final values. The calculations are normally performed from 500 to 2000 times, depending upon the calculated thermal parameters; to make all 1201 sets (nodes) meet the criterion for each time step. At least 600 time steps are calculated to allow the combustion front propagate the 1-cm long specimen completely.
3. Results and discussion To investigate the influences caused by the heterogeneities in initial porosity and composition, the heterogeneity maps for the thermophysical/chemical reactant parameters (such as density, heat capacity, thermal conductivity, and reaction yield) and the corresponding micropyretic parameters (such as propagation velocity and thickness of pre-heat zone) are generated. Figs. 1 and 2 show the percentage variations in density and thermal conductivity for the specimens with different heterogeneities in Ni composition and porosity, respectively. Since the density and the thermal conductivity at each node are calculated from the composition and the porosity, the variations in these parameters are found to correlate strongly with the changes in composition and porosity. In the ideal homogeneous specimen (0% heterogeneities in Ni composition and porosity), the thermal con-
Fig. 1. A map for the variations in thermal conductivity with the different heterogeneities in initial composition and porosity. The thermal conductivity for the homogeneous specimen is 178.2 J/(ms K).
ductivity and density remain as constants at all nodes, and variations in these parameters are zero. Density and thermal conductivity start to vary with the distance for the heterogeneous specimens. Fig. 1 shows that the variation in thermal conductivity along the specimen is increased to 26.7% with the increase in the heterogeneity in porosity to 60%.
Table 3 The values of various parameters used in the numerical calculation [20,23] Parameters
NiAl
Combustion temperature (K) Activation energy (kJ/mol) Exothermic heat (kJ/mol) Pre-exponential factor (s−1 ) Time step (s)
1912 139 [20] 118.5 [23] 4 × 108 0.0005
Fig. 2. A map for the variations in density with the different heterogeneities in initial composition and porosity. The density for the homogeneous specimen is 3.87 g/cm3 .
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Fig. 5. Time variations of the combustion front temperature along the Ni + Al specimens. The interval time between two consecutive time steps (profiles) is 0.0005 s. The number 20 denotes the 20th time step (0.010 s) after ignition. The heterogeneities in composition and porosity are, respectively, 0 and 0%. Fig. 3. A map for the variations in heat capacity with the different heterogeneities in initial composition and porosity. The heat capacity for the homogeneous specimen is 673.5 J/(kg K).
Under such a variation, the thermal conductivity is varied between 145.5 and 210.9 J/(ms K) [=178.2 (homogeneous value) × (1 ± 26.7%/2)]. On the other hand, the variation in thermal conductivity is only increased to 15.5% upon increasing the heterogeneity in Ni composition to 30%. A 60% heterogeneity in porosity (=25% × 60% = 15%) and a 30% heterogeneity in Ni composition (=50 at.% × 30% = 15 at.%) both lead to 15% change in porosity or composition, as shown in Table 1. The heterogeneity map in Fig. 1 reveals that the heterogeneity in porosity has stronger effects on changing thermal conductivity than the heterogeneity in Ni composition for a given change percentage. A similar phenomenon is also found in the variation in density. Fig. 2 shows that the variations in density are enhanced to 20.0 and 17.2% as the heterogeneities are increased to 60% for porosity and 30% for composition, respectively.
Fig. 4. A map for the maximum decrements in reaction yield with the different heterogeneities in initial composition and porosity. The reaction yield for the homogeneous specimen is 100%.
In addition, Eqs. (4) and (8) also show that the reaction yield and the heat capacity at each node are only influenced by the heterogeneity in initial Ni composition. Thus, Figs. 3 and 4 show that the variations in heat capacity and reaction yield are correlated with the changes in heterogeneity in Ni composition, but independent of the changes in porosity. Fig. 3 shows that an increase in 30% heterogeneity in Ni composition enhances the variation in heat capacity to 20.4% as compared with the homogeneous specimen. As also seen in Eq. (4) and Fig. 4, the maximum decrease in the reaction yield is calculated to be 15% for the specimen with 30% heterogeneity in Ni composition. A decrease in the reaction yield correspondingly reduces the exothermic heat of the micropyretic synthesis and propagation velocity of the combustion front. The heterogeneity maps in Figs. 1–4 show that the variations in composition and porosity change density, heat capacity, thermal conductivity and exothermic heat, further influencing the reactivity at each node. The corresponding micropyretic parameters (e.g. propagation velocity and thickness of pre-heat zone) and reaction temperature profiles are expected to change with the heterogeneities during micropyretic synthesis.
Fig. 6. Time variations of the combustion front temperature along the Ni + Al specimens. The interval time between two consecutive time steps (profiles) is 0.0005 s. The number 20 denotes the 20th time step (0.010 s) after ignition. The heterogeneities in composition and porosity are, respectively, 0 and 60%.
H.P. Li / Materials Science and Engineering A 392 (2005) 262–268
Figs. 5–7 show the temperature profiles of combustion fronts at different time points along the specimens for different heterogeneities in composition and porosity. The micropyretic reaction is ignited at an initial position of 0 cm and the combustion front propagates from left to right. The number 20 shown in the figures denotes the 20th time step (0.010 s) after ignition. For the homogeneous specimen, the combustion front propagates at the velocity of 45.4 cm/s (Fig. 5). The value is higher than the experimental result because a high pre-exponential constant is used in the numerical calculation to illustrate the heterogeneous effects. As the heterogeneity in porosity is increased to 60%, it is noted from Fig. 6 that the propagation velocity is slightly increased to 46.3 cm/s. However, as the heterogeneity in initial Ni composition is increased to 30% and the heterogeneity in porosity is kept at 0%, the propagation velocity has been significantly reduced to 38.1 cm/s, as shown in Fig. 7. In addition, the combustion temperature and propagation velocity are noted to dramatically change with the distance. The heterogeneity maps for the pre-heat zone thickness and propagation velocity with the heterogeneities in composition and porosity are further generated in this study. Fig. 8 shows the heterogeneity maps for the pre-heat zone thickness. The average zone thickness is noted to increase with the heterogeneity in Ni composition, whereas the average zone thickness is only slightly changed as the heterogeneity in porosity is increased. A previous study [17] has indicated that a narrow pre-heat zone results is normally referred to a higher oscillatory frequency of the combustion front, further increasing the reaction temperature and propagation velocity. Thus, the stability and propagation velocity of the combustion front are expected to increase when the pre-heat zone becomes narrower. The heterogeneity map for the propagation velocity (Fig. 9) shows a continuous decrease in the propagation velocity upon increasing the heterogeneity in Ni composition. As compared with the homogeneous specimen, the propagation velocity is decreased within 16.2% for the specimen
Fig. 7. Time variations of the combustion front temperature along the Ni + Al specimens. The interval time between two consecutive time steps (profiles) is 0.0005 s. The number 20 denotes the 20th time step (0.010 s) after ignition. The heterogeneities in composition and porosity are, respectively, 30 and 0%.
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Fig. 8. A map for the changes in thickness of pre-heat zone with the different heterogeneities in initial composition and porosity. The average thickness of pre-heat zone for the homogeneous specimen is 0.714 mm.
with 30% heterogeneity in composition. However, the propagation velocity is only changed in <2%, even though the heterogeneity in porosity has been enhanced to 60%. It has been shown that the variations in thermal conductivity and density caused by the heterogeneity in composition are smaller as compared with those caused by the same heterogeneity in porosity. However, the variation in the propagation velocity is larger for the specimen with heterogeneous composition distribution. As already discussed, this is because a change in the composition at each node directly changes the ratio of reactants far from the stoichiometric ratio. The reaction yield is correspondingly decreased, further reducing the exothermic heat of reaction. The combustion temperature has thus been
Fig. 9. A map for the changes in propagation velocity of combustion front with the different heterogeneities in initial composition and porosity. The propagation velocity for the homogeneous specimen is 50.4 cm/s.
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effect on thermal conductivity and density as compared with the heterogeneity in initial Ni composition. However, the heterogeneity maps for the corresponding micropyretic parameters suggest that an increment in the pre-heat zone thickness occurs only when the heterogeneity in initial Ni composition increases. An increment in the pre-heat zone thickness has been reported to decrease the oscillatory frequency of the unstable combustion front, further decreasing the reactivity of the micropyretic reaction. Therefore, a heterogeneity map also reveals that the propagation velocity is significantly decreased with the heterogeneities in initial composition. From the knowledge of heterogeneity maps, the effects of heterogeneities in initial composition and porosity on micropyretic reaction can be acquired. Fig. 10. A map for the standard deviations of propagation velocity of combustion front with the different heterogeneities in composition and porosity.
significantly reduced with the distance and the variations in the propagation velocity are correspondingly decreased, as shown in Fig. 9. On account of the variations in the reaction yield and exothermic heat of reaction along the specimen, the combustion front is found to propagate in a succession of rapid and slow movements. Therefore, the standard deviation of propagation velocity is also found to increase with the heterogeneity in composition and porosity, as shown in Fig. 10. Again, the heterogeneity in composition has stronger effects on increasing the standard deviation of the velocity than the effects caused by the heterogeneity in porosity. The maximum standard deviation of propagation velocity is found when the 30% heterogeneity in composition and 40% heterogeneity in porosity occurs.
4. Summary and conclusions This study investigates the effects of heterogeneities in initial composition and porosity on micropyretic synthesis of Ni + Al using a numerical simulation. The heterogeneity maps for the thermophysical/chemical reactant parameters (such as density, heat capacity, thermal conductivity, and reaction yield) and the corresponding micropyretic parameters (such as pre-heat zone thickness and propagation velocity) with the heterogeneities in composition and porosity have been explored in this study. These heterogeneity maps for the thermophysical/chemical reactant parameters have shown that the heterogeneities in initial composition and porosity influence the thermal conductivity and density, whereas the heat capacity and reaction yield are only influenced by the heterogeneity in initial composition. Such variations change the nature of propagation of a combustion front. The calculations show that the heterogeneity in initial porosity has a stronger
Acknowledgments The supports from the National Center for HighPerformance Computing (account no.: y48hpl00) and National Science Council in Taiwan (NSC93-2216-E-228-1) are acknowledged.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
H.P. Li, J. Mater. Res. 17 (12) (2002) 3213–3221. H.P. Li, Mater. Sci. Eng. A 345 (1–2) (2002) 336–344. H.P. Li, Mater. Chem. Phy. 80 (3) (2003) 758–767. Z.A. Munir, U. Anselmi-Tamburini, Mater. Sci. Rep. 3 (1989) 277–365. A.G. Merzhanov, B.I. Khaikin, Prog. Energy Combust. Sci. 14 (1988) 1–98. H.P. Li, J. Mater. Sci. 39 (2) (2004) 707–710. H.P. Li, Mater. Res. Bull. 39 (12) (2004) 1881–1894. H.P. Li, Mater. Chem. Phy. 89 (1) (2005) 130–137. Z.A. Munir, Am. Ceram. Bull. 67 (2) (1988) 249–342. F. Booth, Trans. Farad. Soc. 49 (1953) 272–281. J.D. Walton, N.E. Poulos, J. Am. Ceram. Soc. 42 (1) (1959) 40–49. H.P. Li, Acta Mater. 51 (2003) 3213–3224. M.G. Lakshmikantha, A. Bhattacharys, J.A. Sekhar, Metall. Trans. A 23A (1992) 23. M.G. Lakshmikantha, J.A. Sekhar, J. Mater. Sci. 28 (1993) 6403. M.G. Lakshmikantha, J.A. Sekhar, J. Am. Ceram. Soc. 77 (1994) 202–210. V. Subramanian, M.G. Lakshmikantha, J.A. Sekhar, J. Mater. Res. 10 (1995) 1235. H.P. Li, Scripta Mater. 50 (7) (2004) 999–1002. H.P. Li, Metall. Mater. Trans. A 34 (9) (2003) 1968–1978. G.K. Dey, A. Arya, J.A. Sekhar, J. Mater. Res. 15 (2000) 63. I. Brain, O. Knacke, O. Kubaschewski, Thermochemical Properties of Inorganic Substances, Springer–Verlag, New York, NY, 1973. D.R. Lide, CRC Handbook of Chemistry and Physics, CRC, Boca Raton, FL, 1990. E.A. Brandes, G.B. Brook, Smithells Metals Reference Book, Butterworth–Heinemann Ltd., 1992. Y.S. Naiborodenko, V.I. Itin, Combust. Explos. Shock Waves 11 (3) (1975) 293–300.
The heterogeneity maps for micropyretic synthesis of Ni–Al composite H.P. Li∗ Jin-Wen Institute of Technology, 99 An-Chung Road, Taipei County, Hsintien 231, Taiwan Received 21 June 2004; received in revised form 13 September 2004; accepted 15 September 2004
Abstract Heterogeneities in initial composition and porosity are common during micropyretic synthesis when powders are pressed or mixed, and the conventional modeling treatments so far have only considered uniform systems. Heterogeneities in composition and porosity are thought to result in local variations in reaction yield, or such thermophysical/chemical parameters for the reactants as density, heat capacity, and thermal conductivity; further changing the propagation velocity of a combustion front. This study, thus, investigates the impact of heterogeneities in initial composition and porosity on micropyretic synthesis with Ni + Al by a numerical simulation. The heterogeneity maps for the thermophysical/chemical reactant parameters (such as density, heat capacity, thermal conductivity, and reaction yield) and corresponding micropyretic parameters (such as pre-heat zone thickness and propagation velocity) with the heterogeneities in composition and porosity are generated. The calculation results show that the heterogeneity in porosity has a stronger effect on thermal conductivity and density as compared with the heterogeneity in Ni composition. However, the results suggest that the reactivity of the micropyretic reaction is significantly decreased only when the heterogeneity in Ni composition increases. From the knowledge of heterogeneity maps, the effects of heterogeneities in initial composition and porosity on micropyretic reaction can be acquired. © 2004 Elsevier B.V. All rights reserved. Keywords: Composition heterogeneity; Porosity heterogeneity; Micropyretic synthesis; Combustion synthesis; SHS
1. Introduction Many exothermic non-catalytic solid–solid or solid–gas reactions, after being ignited locally, can release enough heat so as to sustain the self-propagating combustion front throughout the specimen without additional energy [1–9]. Since the 1970s, this kind of exothermic reaction has been used in the synthesis of refractory compounds in the former Soviet Union [5]. This novel technique, so-called micropyretic synthesis or combustion synthesis, has been intensively studied for process implication [1–9]. This technique employs exothermic reaction processing, which circumvents difficulties associated with conventional methods of time and energy-intensive sintering processing. The advantages of micropyretic synthesis also include the rapid ∗
Tel.: +886 2 8212208; fax: +886 2 23813621. E-mail address: [email protected].
0921-5093/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2004.09.031
net shape processing and clean products. In addition, the micropyretic-synthesized products have been reported to possess better mechanical and physical properties. Micropyretic-synthesized products have been reported to have better mechanical and physical properties [10,11]. An example is the formation of shape-memory alloys of nickel and titanium [10]. It has been reported that those prepared by micropyretic synthesis possess greater shaperecovery force than corresponding alloys produced by conventional methods [10]. On account of the high thermal gradients encountered in micropyretic synthesis, it has been speculated that the products of such a process may contain a high-defect concentration. The presence of high levels of defects has led to expectation of higher reactivity, namely higher sinterability [11]. The micropyretic synthesis technique also provides rapid net shape processing. When compared with conventional powder metallurgy operations, micropyretic synthesis not only offers shorter processing
H.P. Li / Materials Science and Engineering A 392 (2005) 262–268
Nomenclature Cp
heat capacity of product (general form) (kJ/(kg K)) d diameter of the specimen (m) E activation energy (kJ/kg) fR (j) random number at node j h surface heat transfer coefficient (J/(m2 K s)) Heterocomp heterogeneity in composition (%) Heteroporosity heterogeneity in porosity (%) K0 pre-exponential constant (s−1 for zero-order reaction) Po original porosity (%) Pj porosity at node j (%) Q heat of reaction (kJ/kg) R gas constant (kJ/(kg K)) Ryield,j reaction yield at node j (%) t time (s) T temperature (K) T0 initial temperature (K) Vs volume fraction of component (species) s (%) Xi,j molar fraction of component (species) i at node j (%) z dimensional coordinate (m) Greek letters η fraction reacted κ thermal conductivity (general form) (kJ/(ms K)) ρ density (kg/m3 ) Φ (T, η) reaction rate (s−1 )
time, but also excludes the requirement for high-temperature sintering. Several numerical and analytical models of micropyretic synthesis in a composite system have been well developed [12–19]. Lakshmikantha and Sekhar first explored the numerical model that includes the effects of dilution and porosity and melting of each constituent of the reactants and products [13]. The analytical modeling of the propagation of the combustion front in solid–solid reaction systems has also been reported [15]. The analytical model gives good results when compared with the experimentally determined numbers and the numerically calculated values. In addition, a multidimensional numerical model and dynamic modeling of the gas and solid reaction have also been carried out to illustrate the effects of various parameters on the micropyretic synthesis [14,15]. These numerical and analytical analyses provide a better understanding of the reaction sequence during micropyretic synthesis reactions. However, heterogeneities in initial composition and porosity are common during micropyretic synthesis when powders are pressed or mixed, and the conventional modeling
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treatments [12–19] so far have only considered uniform systems. Heterogeneities in initial composition and porosity result in local variations in reaction yield and the thermophysical/chemical reactant parameters, such as density, heat capacity, and thermal conductivity. This changes the propagation velocity and combustion temperature of a combustion front. Such a change may further lead to the non-homogeneous microstructures. In this study, a numerical simulation is used to characterize the effects of heterogeneities in initial composition and porosity on NiAl micropyretic synthesis. The heterogeneity maps, considering the heterogeneities in initial composition and porosity concurrently, are also generated. From the knowledge of heterogeneity maps, the effects of heterogeneities in initial composition and porosity on micropyretic reaction can be acquired.
2. Numerical calculation procedure During the passage of the combustion front in the reaction, the energy equation for transient heat conduction, including the source term, containing heat release due to the exothermic micropyretic reaction is given as [5,13,15]: ∂T ∂ ∂T 4h(T − T0 ) ρ Cp = κ − + ρ Q Φ (T, η) ∂t ∂z ∂z d (1) Each symbol in the equation is explained in the Nomenclature section. The reaction rate, Φ (T, η), in Eq. (1) is given as: ∂η E Φ(T, η) = = K0 (1 − η) exp − (2) ∂t RT In this study, a numerical calculation for Eq. (1) is carried out with the assumption of the first-order kinetics. In the Eq. (1), the energy required for heating the synthesized product from the initial temperature to the adiabatic combustion temperature is shown on the left-hand side. The terms on the righthand side are the conduction heat transfer term, the surface heat loss parameter, and the heat release due to the exothermic micropyretic reaction, respectively. The surface heat loss is assumed radically Newtonian in this study. The previous studies [3,13] have shown that the surface heat loss is much less than the exothermic heat of the reaction; thus, the surface heat loss is taken to be zero in the numerical calculation. In the computational simulation, a one-dimensional sample of 1 cm length is divided into 1201 nodes (regions) to calculate the local temperature using an enthalpy–temperature method coupled with the Guass–Seidel iteration procedure. The choice of 1 cm sample length is only for computational purpose. It has been found that the length of sample does not affect the numerical results in this study. Thus, the simulation results are applicable to practical experimental condition. First, the proper initial and boundary conditions are used to initialize the temperatures and enthalpies at all nodes. The initial conditions in the simulation are taken as follows: (1)
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at the ignition node, at time t ≥ 0, the temperature is taken to be the adiabatic combustion temperature (T = Tc and η = 1). It has been reported that the chosen temperature value at the ignition node only slightly influences the temperature profiles at the initial stage [13,15]. (2) At the other nodes, at time t = 0, the temperatures are taken to be the same as the substrate temperature (T = T0 and η = 0). Depending on the values of the temperature and enthalpy occurred in the reaction, the proper thermophysical/chemical parameters are considered and the proper limits of the reaction zone are determined for each node in the numerical calculation. At any given time, the fraction reacted and enthalpy of the current iteration are calculated from the previous fraction reacted and enthalpy of the earlier iteration. The range of the enthalpy as well as the molar ratio for each material and node is thus determined, and the values of temperature, density, and thermal conductivity at each node can be further calculated in appropriate zone. Initial composition at each node is first calculated from the random number (fR (j) at node j) and the assigned heterogeneity (Heterocomp ) that determines the magnitude of the variation. The sequence of the random numbers (−0.5 to +0.5) generated from the computation is repeatedly used in the specimens with different heterogeneities to compare the magnitude of heterogeneity effect. In this study, Ni + 50 at.% Al composition which corresponds to NiAl compound is chosen. The initial composition before the beginning of the reaction can be expressed as follows.
ingly determined, as shown in Table 1. After the molar fractions of reactants are determined, the reaction yield at node j can be further determined from the molar fractions of reactants and diluents: XNi,j XAl,j (4) , Ryield,j = min 0 0 XNi XAl In addition, the porosity effects of the reactants and product that influence the density (ρ) and thermal conductivity (κ) profiles are also considered in the numerical calculation. During the numerical calculation, the average global porosities (Po ) of the reactants and product are both taken to be 25%. The initial porosity of the reactants at each node is calculated from the random number (fR (j) at node j) and the assigned heterogeneity (Heteroporosity ) that determines the magnitude of the porosity variation: porosity at node j : Pj = P o (1 + Heteroporosity fR (j))
(5)
where −0.5 ≤ fR (j) ≤ +0.5, Po = 25%, 0% < Heteroporosity < 100%, and j = 1, 2, . . ., 1201. The studied compositions with the different heterogeneities in porosity are also shown in Table 1. Once the initial composition and porosity at each node are set to given heterogeneities, the thermophysical/chemical parameters at node j can be thus calculated as: [ρs Vs,j (1 − Pj )] (6) density at node j : ρj = s
Ni molar fraction at node j : XNi,j 0 = XNi (1 + Heterocomp fR (j))
(3a)
Al molar fraction at node j : XAl,j = 1 − XAl,j
thermal conductivity at node j : κj 1 − Pj κs Vs,j = 1 + Pj /2 s
(3b)
0 = 50.0 at.%, where XNi −0.5 ≤ fR (j) ≤ +0.5 0% < Heterocomp < 100%, and j = 1, 2, . . ., 1201. In order to assure the sum of the initial compositions for all 1201 nodes equal to the stoichiometric values, the calculated Ni and Al compositions of each node are adjusted so that the average values of each initial composition are equal to the ideal homogeneous 0 ¯ Ni : X ¯ Al = 1 : 1, i.e., (1/n) n=1201 values (X XNi,j = XNi j=1 n=1201 0 ). For example, as 30% and (1/n) j=1 XAl,j = XAl heterogeneity in initial Ni composition has occurred, Ni composition is correspondingly varied within 15 at.% (=50.0 at.% × 30%). Thus, Ni composition is noted to vary from 42.5 to 57.5 at.% and Al composition is correspond-
heat capacity : Cpj =
(Cps Xs,j )
(7)
(8)
s
where s denotes the component involved in the reaction, including Ni and Al in this study. The effect of melting of reactants and product is also included in the calculation. The various thermophysical/chemical parameters, such as thermal conductivity, density and heat capacity of the reactants and product are assumed to be independent of temperature, but they are different in each state. The average values of these parameters vary as the reaction proceeds, depending upon the degree of reaction. The parameter values used in the computational calculation are shown in the Table 2 [20–22]
Table 1 The examples of the studied Ni + 50 at.% Al compositions with different heterogeneities in initial composition and porosity Heterogeneity in composition (%)
Heterogeneity in porosity (%)
Ni composition (at.%)
Al composition (at.%)
Porosity
0 0 30 30
0 60 0 60
50.0 50.0 42.5–57.5 42.5–57.5
50.0 50.0 57.5–42.5 57.5–42.5
25.0 17.5–32.5 25.0 17.5–32.5
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Table 2 The thermophysical/chemical parameters for the reactants (Ni and Al) and product (NiAl) at 300 K and liquid state [20–22] Thermophysical/chemical parameters
Al
Ni
NiAl
Heat capacity (at 300 K) (J/(kg K)) Heat capacity (liquid state) (J/(kg K)) Thermal conductivity (at 300 K) (J/(ms K)) Thermal conductivity (liquid state) (J/(ms K)) Density (at 300 K) (kg/m3 ) Density (liquid state) (kg/m3 )
902 [20] 1178 [20] 238 [22] 100 [22] 2700 [22] 2385 [22]
445 [20] 735 [20] 88.5 [22] 53 [21] 8900 [22] 7905 [22]
537 [20] 831 [20] 75 [21] 55 [21] 6050 [21] 5950 [22]
and Table 3 [20,23]. In this study, the combustion temperature is defined as the highest reaction temperature during micropyretic synthesis, and the propagation velocity is the velocity of the combustion front propagation. In addition, the higher pre-exponential factor (K0 ) value 4 × 108 s−1 is used to be capable of illustrating the variation of the propagation velocity for the NiAl combustion reaction. The criterion used to ascertain whether the fraction reacted and enthalpies at each time level converge or not is determined from the relative error criterion. Once the convergence criterion for every node is met, the enthalpy and fraction reacted of the last iteration in a time step are considered to be the corresponding final values. The calculations are normally performed from 500 to 2000 times, depending upon the calculated thermal parameters; to make all 1201 sets (nodes) meet the criterion for each time step. At least 600 time steps are calculated to allow the combustion front propagate the 1-cm long specimen completely.
3. Results and discussion To investigate the influences caused by the heterogeneities in initial porosity and composition, the heterogeneity maps for the thermophysical/chemical reactant parameters (such as density, heat capacity, thermal conductivity, and reaction yield) and the corresponding micropyretic parameters (such as propagation velocity and thickness of pre-heat zone) are generated. Figs. 1 and 2 show the percentage variations in density and thermal conductivity for the specimens with different heterogeneities in Ni composition and porosity, respectively. Since the density and the thermal conductivity at each node are calculated from the composition and the porosity, the variations in these parameters are found to correlate strongly with the changes in composition and porosity. In the ideal homogeneous specimen (0% heterogeneities in Ni composition and porosity), the thermal con-
Fig. 1. A map for the variations in thermal conductivity with the different heterogeneities in initial composition and porosity. The thermal conductivity for the homogeneous specimen is 178.2 J/(ms K).
ductivity and density remain as constants at all nodes, and variations in these parameters are zero. Density and thermal conductivity start to vary with the distance for the heterogeneous specimens. Fig. 1 shows that the variation in thermal conductivity along the specimen is increased to 26.7% with the increase in the heterogeneity in porosity to 60%.
Table 3 The values of various parameters used in the numerical calculation [20,23] Parameters
NiAl
Combustion temperature (K) Activation energy (kJ/mol) Exothermic heat (kJ/mol) Pre-exponential factor (s−1 ) Time step (s)
1912 139 [20] 118.5 [23] 4 × 108 0.0005
Fig. 2. A map for the variations in density with the different heterogeneities in initial composition and porosity. The density for the homogeneous specimen is 3.87 g/cm3 .
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Fig. 5. Time variations of the combustion front temperature along the Ni + Al specimens. The interval time between two consecutive time steps (profiles) is 0.0005 s. The number 20 denotes the 20th time step (0.010 s) after ignition. The heterogeneities in composition and porosity are, respectively, 0 and 0%. Fig. 3. A map for the variations in heat capacity with the different heterogeneities in initial composition and porosity. The heat capacity for the homogeneous specimen is 673.5 J/(kg K).
Under such a variation, the thermal conductivity is varied between 145.5 and 210.9 J/(ms K) [=178.2 (homogeneous value) × (1 ± 26.7%/2)]. On the other hand, the variation in thermal conductivity is only increased to 15.5% upon increasing the heterogeneity in Ni composition to 30%. A 60% heterogeneity in porosity (=25% × 60% = 15%) and a 30% heterogeneity in Ni composition (=50 at.% × 30% = 15 at.%) both lead to 15% change in porosity or composition, as shown in Table 1. The heterogeneity map in Fig. 1 reveals that the heterogeneity in porosity has stronger effects on changing thermal conductivity than the heterogeneity in Ni composition for a given change percentage. A similar phenomenon is also found in the variation in density. Fig. 2 shows that the variations in density are enhanced to 20.0 and 17.2% as the heterogeneities are increased to 60% for porosity and 30% for composition, respectively.
Fig. 4. A map for the maximum decrements in reaction yield with the different heterogeneities in initial composition and porosity. The reaction yield for the homogeneous specimen is 100%.
In addition, Eqs. (4) and (8) also show that the reaction yield and the heat capacity at each node are only influenced by the heterogeneity in initial Ni composition. Thus, Figs. 3 and 4 show that the variations in heat capacity and reaction yield are correlated with the changes in heterogeneity in Ni composition, but independent of the changes in porosity. Fig. 3 shows that an increase in 30% heterogeneity in Ni composition enhances the variation in heat capacity to 20.4% as compared with the homogeneous specimen. As also seen in Eq. (4) and Fig. 4, the maximum decrease in the reaction yield is calculated to be 15% for the specimen with 30% heterogeneity in Ni composition. A decrease in the reaction yield correspondingly reduces the exothermic heat of the micropyretic synthesis and propagation velocity of the combustion front. The heterogeneity maps in Figs. 1–4 show that the variations in composition and porosity change density, heat capacity, thermal conductivity and exothermic heat, further influencing the reactivity at each node. The corresponding micropyretic parameters (e.g. propagation velocity and thickness of pre-heat zone) and reaction temperature profiles are expected to change with the heterogeneities during micropyretic synthesis.
Fig. 6. Time variations of the combustion front temperature along the Ni + Al specimens. The interval time between two consecutive time steps (profiles) is 0.0005 s. The number 20 denotes the 20th time step (0.010 s) after ignition. The heterogeneities in composition and porosity are, respectively, 0 and 60%.
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Figs. 5–7 show the temperature profiles of combustion fronts at different time points along the specimens for different heterogeneities in composition and porosity. The micropyretic reaction is ignited at an initial position of 0 cm and the combustion front propagates from left to right. The number 20 shown in the figures denotes the 20th time step (0.010 s) after ignition. For the homogeneous specimen, the combustion front propagates at the velocity of 45.4 cm/s (Fig. 5). The value is higher than the experimental result because a high pre-exponential constant is used in the numerical calculation to illustrate the heterogeneous effects. As the heterogeneity in porosity is increased to 60%, it is noted from Fig. 6 that the propagation velocity is slightly increased to 46.3 cm/s. However, as the heterogeneity in initial Ni composition is increased to 30% and the heterogeneity in porosity is kept at 0%, the propagation velocity has been significantly reduced to 38.1 cm/s, as shown in Fig. 7. In addition, the combustion temperature and propagation velocity are noted to dramatically change with the distance. The heterogeneity maps for the pre-heat zone thickness and propagation velocity with the heterogeneities in composition and porosity are further generated in this study. Fig. 8 shows the heterogeneity maps for the pre-heat zone thickness. The average zone thickness is noted to increase with the heterogeneity in Ni composition, whereas the average zone thickness is only slightly changed as the heterogeneity in porosity is increased. A previous study [17] has indicated that a narrow pre-heat zone results is normally referred to a higher oscillatory frequency of the combustion front, further increasing the reaction temperature and propagation velocity. Thus, the stability and propagation velocity of the combustion front are expected to increase when the pre-heat zone becomes narrower. The heterogeneity map for the propagation velocity (Fig. 9) shows a continuous decrease in the propagation velocity upon increasing the heterogeneity in Ni composition. As compared with the homogeneous specimen, the propagation velocity is decreased within 16.2% for the specimen
Fig. 7. Time variations of the combustion front temperature along the Ni + Al specimens. The interval time between two consecutive time steps (profiles) is 0.0005 s. The number 20 denotes the 20th time step (0.010 s) after ignition. The heterogeneities in composition and porosity are, respectively, 30 and 0%.
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Fig. 8. A map for the changes in thickness of pre-heat zone with the different heterogeneities in initial composition and porosity. The average thickness of pre-heat zone for the homogeneous specimen is 0.714 mm.
with 30% heterogeneity in composition. However, the propagation velocity is only changed in <2%, even though the heterogeneity in porosity has been enhanced to 60%. It has been shown that the variations in thermal conductivity and density caused by the heterogeneity in composition are smaller as compared with those caused by the same heterogeneity in porosity. However, the variation in the propagation velocity is larger for the specimen with heterogeneous composition distribution. As already discussed, this is because a change in the composition at each node directly changes the ratio of reactants far from the stoichiometric ratio. The reaction yield is correspondingly decreased, further reducing the exothermic heat of reaction. The combustion temperature has thus been
Fig. 9. A map for the changes in propagation velocity of combustion front with the different heterogeneities in initial composition and porosity. The propagation velocity for the homogeneous specimen is 50.4 cm/s.
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effect on thermal conductivity and density as compared with the heterogeneity in initial Ni composition. However, the heterogeneity maps for the corresponding micropyretic parameters suggest that an increment in the pre-heat zone thickness occurs only when the heterogeneity in initial Ni composition increases. An increment in the pre-heat zone thickness has been reported to decrease the oscillatory frequency of the unstable combustion front, further decreasing the reactivity of the micropyretic reaction. Therefore, a heterogeneity map also reveals that the propagation velocity is significantly decreased with the heterogeneities in initial composition. From the knowledge of heterogeneity maps, the effects of heterogeneities in initial composition and porosity on micropyretic reaction can be acquired. Fig. 10. A map for the standard deviations of propagation velocity of combustion front with the different heterogeneities in composition and porosity.
significantly reduced with the distance and the variations in the propagation velocity are correspondingly decreased, as shown in Fig. 9. On account of the variations in the reaction yield and exothermic heat of reaction along the specimen, the combustion front is found to propagate in a succession of rapid and slow movements. Therefore, the standard deviation of propagation velocity is also found to increase with the heterogeneity in composition and porosity, as shown in Fig. 10. Again, the heterogeneity in composition has stronger effects on increasing the standard deviation of the velocity than the effects caused by the heterogeneity in porosity. The maximum standard deviation of propagation velocity is found when the 30% heterogeneity in composition and 40% heterogeneity in porosity occurs.
4. Summary and conclusions This study investigates the effects of heterogeneities in initial composition and porosity on micropyretic synthesis of Ni + Al using a numerical simulation. The heterogeneity maps for the thermophysical/chemical reactant parameters (such as density, heat capacity, thermal conductivity, and reaction yield) and the corresponding micropyretic parameters (such as pre-heat zone thickness and propagation velocity) with the heterogeneities in composition and porosity have been explored in this study. These heterogeneity maps for the thermophysical/chemical reactant parameters have shown that the heterogeneities in initial composition and porosity influence the thermal conductivity and density, whereas the heat capacity and reaction yield are only influenced by the heterogeneity in initial composition. Such variations change the nature of propagation of a combustion front. The calculations show that the heterogeneity in initial porosity has a stronger
Acknowledgments The supports from the National Center for HighPerformance Computing (account no.: y48hpl00) and National Science Council in Taiwan (NSC93-2216-E-228-1) are acknowledged.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
H.P. Li, J. Mater. Res. 17 (12) (2002) 3213–3221. H.P. Li, Mater. Sci. Eng. A 345 (1–2) (2002) 336–344. H.P. Li, Mater. Chem. Phy. 80 (3) (2003) 758–767. Z.A. Munir, U. Anselmi-Tamburini, Mater. Sci. Rep. 3 (1989) 277–365. A.G. Merzhanov, B.I. Khaikin, Prog. Energy Combust. Sci. 14 (1988) 1–98. H.P. Li, J. Mater. Sci. 39 (2) (2004) 707–710. H.P. Li, Mater. Res. Bull. 39 (12) (2004) 1881–1894. H.P. Li, Mater. Chem. Phy. 89 (1) (2005) 130–137. Z.A. Munir, Am. Ceram. Bull. 67 (2) (1988) 249–342. F. Booth, Trans. Farad. Soc. 49 (1953) 272–281. J.D. Walton, N.E. Poulos, J. Am. Ceram. Soc. 42 (1) (1959) 40–49. H.P. Li, Acta Mater. 51 (2003) 3213–3224. M.G. Lakshmikantha, A. Bhattacharys, J.A. Sekhar, Metall. Trans. A 23A (1992) 23. M.G. Lakshmikantha, J.A. Sekhar, J. Mater. Sci. 28 (1993) 6403. M.G. Lakshmikantha, J.A. Sekhar, J. Am. Ceram. Soc. 77 (1994) 202–210. V. Subramanian, M.G. Lakshmikantha, J.A. Sekhar, J. Mater. Res. 10 (1995) 1235. H.P. Li, Scripta Mater. 50 (7) (2004) 999–1002. H.P. Li, Metall. Mater. Trans. A 34 (9) (2003) 1968–1978. G.K. Dey, A. Arya, J.A. Sekhar, J. Mater. Res. 15 (2000) 63. I. Brain, O. Knacke, O. Kubaschewski, Thermochemical Properties of Inorganic Substances, Springer–Verlag, New York, NY, 1973. D.R. Lide, CRC Handbook of Chemistry and Physics, CRC, Boca Raton, FL, 1990. E.A. Brandes, G.B. Brook, Smithells Metals Reference Book, Butterworth–Heinemann Ltd., 1992. Y.S. Naiborodenko, V.I. Itin, Combust. Explos. Shock Waves 11 (3) (1975) 293–300.