Self-Compaction or Expansion in Combustion Synthesis of Porous Materials

We propose a mathematical model for the combustion of porous deformable condensed materials, which we use to describe the deformation of the high temperature products, induced by the pressure difference of the gas outside and inside the sample, in the absence of any external forces. The deformation occurs as a result of pore compaction (expansion), resulting in a more (less) dense product material. To describe the evolution of porosity we derive an equation which allows us to define a characteristic time of deformation t_d. If t_dis sufficiently smaller than the characteristic time of combustion t_n the deformation process is sufficiently fast to compensate for pressure gradients, so that pressure is equalized almost instantaneously, and filtration is suppressed. If t_d t_r, deformation occurs solely in the product, and does not affect the propagation velocity. We determine various characteristics of a uniformly propagating combustion wave, and the materials produced by it, such as the propagation velocity, combustion temperature, final depth of conversion and final porosity of the product, as a function of the thermophysical parameters of the system. We also identify a regime of pulsating propagation, in which case the final porosity of the products is periodic in space. We show that both the uniformly propagating wave and the pulsating propagating wave solutions are stable, each corresponding to its own parameter regime. We also find that the deformation process can affect stability. In particular, the effect of viscosity is found to be stabilizing.