The asymptotic theory of gasless combustion synthesis
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INTRODUCTION
THE propagation of reaction fronts in condensed phase combustion, often referred to as "self-propagating hightemperature synthesis" or simply as "combustion synthesis," has a broad range of pyrotechnical and metallurgical applications, particularly in the synthesis of metal alloys and refractory materials, t~-4} Examples of simple "gasless" systems include Ti/A1, T i / B , N b / B , F e / Z r , and Cr/Zr, to name but a few, while an example of a more complex system is T i O E / A 1 / C , which forms TiC and A1203. Although the reactants are initially in the solid phase, melting of at least one of them is frequently observed and is generally required for the reaction to sustain itself and to propagate. The explanation for this is that microscopic interdiffusion of the reacting powders, which determines the local reaction rate, is limited by surface-to-surface contact between individual particles. When one of the reactants melts, the melted species can coat the unmelted particles, thereby greatly increasing the local reaction rate. Subsequent to melting, the reaction may be regarded on the microscopic scale as being either kinetically limited or diffusion limited, [5,6~ although one may also consider the intermediate case in which both processes assume equal importance at elevated temperatures, as discussed below. In the kinetically limited case, which is referred to as homogeneous combustion, it is assumed that the reaction is essentially limited by the temperature T and by the unreacted fraction Y of the mixture in a manner similar to that of gaseous combustion. The reaction rate ~ is then usually taken to be of the form ff~ = .Y,Y"e-~/f
[ 1]
STEPHEN B. MARGOLIS, Senior Member of Technical Staff, is with the Combustion Research Facility, Sandia National Laboratories, Livermore, CA 94551-0969. This paper is based on a presentation made in the symposium "Reaction Synthesis of Materials" presented during the TMS Annual Meeting, New Orleans, LA, February 17-21, 1991, under the auspices of the TMS Powder Metallurgy Committee. METALLURGICAL TRANSACTIONS A
where n > 0 is a kinetic parameter, often taken to be unity, ,( is the rate coefficient (assumed constant for simplicity), a n d / ~ is the activation temperature (activation energy divided by the gas constant) of the reaction. This rate law is most valid in the limit of small particle size or when all components of the mixture melt during the reaction process. In contrast, the diffusion-limited assumption (heterogeneous combustion) explicitly takes into account the fact that there is a (solid) product layer across which atoms must diffuse in order to react with one another. In this case, the factor Y" in Eq. [1] is replaced by a function f ( Y ) of the form
f(Y)
= e-m~
- Y)-"
[2]
where m and n are parameters.J71 The general rationale behind Eq. [2] lies in the nature of the transfer mechanism across the product layer, the thickness of which is directly related to the level of conversion (1 - Y). The parameters m and n are usually determined fr
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