Chemical Attack in Free Boundary Domains

Abstract

Abstract. This paper presents a mathematical model for a chemical process used to machine cristal as glass or silica. A short physical description is presented from which we draw the mathematical model. We obtain a coupled parabolic equations system on a free boundary domain with a non-linear condition on the boundary. The existence and the uniqueness is proved in the one-dimensional case.

Local classical solutions for a chemical system with free boundary

“…The parabolic system (1.1) describes the Fickian diffusion of the multicompound concentration c that reacts chemically at the interface S T following a nonlinear law (1.2) and creating a dissolution-growth process, in which the boundary evolves according to the kinetic condition (1.3). In a model for the chemical attack of crystals proposed in [4] the different ionic concentrations may be written on the reactive boundary by a conservation law of the type…”

Capillarity–dissolution system for a two-dimensional geometry

“…For example, recent developments in micro-optics [1,2] consist in manufacturing small glass tips from a dissolution process [3]. Fluoride acid is used in order to dissolve the glass thereby leading to a reaction-diffusion problem in the fluid domain which is delimited by a reactive solid-liquid interface and a second free boundary separating the reactive liquid from its neutral environment and where surface tension acts [4,5]. Oil is generally superposed to the acid in order to protect the etched part of the tip [3], hence affecting the capillary properties of the latter interface.…”

“…Our analysis is based, on the one hand, on the Young-Dupré criterion in order to determine the potential triple-junction points between the two fluids and the solid and, on the other hand, on the new concept of metastable curve which governs the process static stability. Since little theoretical and technological information exists in the literature about microtip forming by chemical erosion [1][2][3][4][5] while the process stability has not been addressed yet, we refer the reader to the above cited papers and to classical textbooks [6,7] for more detail. In addition, the thorough stability analysis of shaped crystal growth performed by Tatarchenko [8] could be used as a basis to further investigate the dynamic stability of microtip forming.…”