L. K. Gross

Weakly Nonlinear Stability Analysis of Frontal Polymerization

A description of frontal polymerization is given via a free boundary model with nonlinear kinetic and kinematic conditions at the free boundary. We perform a weakly nonlinear analysis for the development of pulsating instabilities on the cylinder, building on the linear stability analysis of [1]. We take as a bifurcation parameter an experimentally measurable combination of material and kinetic parameters. The asymptotic analysis leads to the derivation of an ordinary differential equation of Landau-Stuart type for the slowly varying amplitude of a linearly unstable mode. We classify nonlinear dynamics of the polymerization front by doing a parameter sensitivity study of the amplitude equation.

Thermo-kinetically controlled pattern selection

Through a combination of asymptotic and numerical approaches we investigate bifurcation and pattern formation for a free boundary model related to a rapid crystallization of amorphous films and to the self-propagating high-temperature synthesis (solid combustion). The unifying feature of these diverse physical phenomena is the existence of a uniformly propagating wave of phase transition whose stability is controlled by the balance between the energy production at the interface and the energy dissipation into the medium. For the propagation on a two-dimensional strip with thermally insulated edges, we develop a multi-scale weakly-nonlinear analysis that results in a system of ordinary differential equations for the slowly varying amplitudes. We identify a nonlinear parameter which is responsible for the pattern selection, and utilize the amplitude system for predicting the evolving patterns. The pattern selection is confirmed by direct numerical simulations on the free boundary problem. Some numerical results on strongly nonlinear regimes are also presented.

Weakly nonlinear dynamics of interface propagation

A simple conceptual description of condensed-phase combustion, explosive solidification, and certain other exothermic phenomena can be given via a free-boundary model with a nonlinear kinetic condition at the free boundary. For a wide range of parametric regimes, the reaction front exhibits a great variety of spatial patterns and instabilities. In [1], we did a linear stability analysis of interfaces that move along a two-dimensional semi-infinite strip with thermally insulated edges. Here we use the normal-mode method to perform a weakly nonlinear analysis for the development of transverse instabilities in the strip. The asymptotic analysis leads to the derivation of ordinary differential equations of Landau-Stuart type for the slowly varying amplitudes of linearly unstable modes. We focus on a strip in which two eigenmodes lose stability at the same value of a parameter related to the activation energy. Such a case gives rise to nontrivial couplings between the amplitude equations, and the two unstable modes compete for dominance. Based on the bifurcation analysis of the amplitude equations, we classify the front configurations that will emerge for any given choice of the kinetics parameter.

Snell’s law of refraction observed in thermal frontal polymerization

We demonstrate that Snells law of refraction can be applied to thermal fronts propagating through a boundary between regions that support distinct frontal velocities. We use the free-radical frontal polymerization of a triacrylate with clay filler that allows for two domains containing two different concentrations of a peroxide initiator to be molded together. Because the polymerization reaction rates depend on the initiator concentration, the propagation speed is different in each domain. We study fronts propagating in two parallel strips in which the incident angle is 90 degrees. Our data fit Snells law v(r)/v(i)=sin theta(r)/sin theta(i), where v(r) is the refracted velocity, v(i) is the incident velocity, theta(r) is the angle of refraction, and theta(i) is the incident angle. Further, we study circular fronts propagating radially from an initiation point in a high-velocity region into a low-velocity region (and vice versa). We demonstrate the close resemblance between the numerically simulated and experimentally observed thermal reaction fronts. By measuring the normal velocity and the angle of refraction of both simulated and experimental fronts, we establish that Snells law holds for thermal frontal polymerization in our experimental system. Finally we discuss the regimes in which Snells law may not be valid.

The onset of linear instabilities in a solid combustion model

This article concerns the onset of linear instability in a simple model of solid combustion in a semi-infinite two-dimensional strip of width l. The free boundary problem that describes the model involves initial and boundary conditions, including a nonlinear kinetic condition at the interface. The linear problem governing perturbations to a basic solution is solved by the method of images with the reaction front perturbation satisfying an integro-differential equation. This equation is then solved using Laplace transforms. Finally, we perform a stability analysis for the model by studying the solution of the reaction front perturbation. The inclusion of initial conditions enables us to show the development of linear instability from arbitrary initial small disturbances.

WEAKLY NONLINEAR AND NUMERICAL ANALYSES OF DYNAMICS IN A SOLID COMBUSTION MODEL

This paper contains qualitative and quantitative comparisons between a weakly nonlinear analysis and direct numerical simulations of a free-boundary problem. The former involves modulating the most linearly unstable mode, taking a small perturbation of the neutrally stable value

Strengthening information literacy in a writing-designated course in the mathematics major

\nu_c

Comparison Study of Dynamics in One-Sided and Two-Sided Solid-Combustion Models

of a parameter

Frontal reaction in a layered polymerizing medium

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A numerical study of one-step models of polymerization: Frontal versus bulk mode

related to the activation energy. Analogously, we perform the direct numerical computations near the marginally unstable value, namely,