On Step‐Function Reaction Kinetics Model in the Absence of Material Diffusion

Abstract

We propose a precise definition of the step-function kinetics suitable for approximating diffuse propagating reaction fronts in one-dimensional gasless-combustion-type models when a Lewis number is large. We investigate this kinetics in the context of free-radical frontal polymerization (FP) in which a monomer-initiator mixture is converted into a polymer via a propagating self-sustaining reaction front. The notion of step-function kinetics has been extensively used in studies of the frontal dynamics both in FP and in combustion problems when the material diffusion is negligible. However, the models have always been effectively reduced to their point-source approximations without defining exactly what the step-function kinetics is for diffuse reaction fronts. We demonstrate numerically that dynamics of diffuse fronts in systems modeled with step-function kinetics and in systems modeled with Arrhenius kinetics are qualitatively the same at time scales at which the bulk reaction ahead of the front can be ignored. We perform stability analysis for the traveling reaction wave and show that the stability threshold is in close agreement with numerical simulations as well as with other existing kinetics approximations. The benefits of using step-function kinetics are two-fold. The reaction dynamics predicted by the step-function kinetics approximates the dynamics predicted by the Arrhenius kinetics over a wider range of system parameters than the point-source approximation. Second, the systems governed by the step-function kinetics can be analyzed both analytically and numerically within the framework of a single model. Downloaded 11/26/14 to 131.193.242.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php ON STEP-FUNCTION KINETICS 793 rate coupled with the exothermicity of the reaction must be sufficient to overcome heat losses into the reactants and product zones [5].Note that an alternative to frontal polymerization is bulk polymerization, in which a mixture of reagents is heated uniformly and polymer formation occurs simultaneously throughout the mixture.A more extensively studied chemical process with a frontal reaction mechanism is self-propagating high-temperature synthesis (SHS)-a combustion process characterized by a heat release large enough to propagate a combustion front through a powder compact while consuming the reactant powders [6], [13]. The simplest models and front propagation mechanisms for FP and SHS are essentially the same, except for the magnitudes of the model parameters.Both steady and unsteady front propagation have been observed in FP [18] as well as in SHS [14]. Unsteady front propagation is usually undesirable, as it leads to nonuniform "layered" structure of the final product. One of the goals of our modeling is to determine the range of material parameters within which the stability of a uniformly propagating polymerization front is guaranteed. The analysis of the full model is, however, too complicated because it requires solving a system of...

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