A Free Boundary Problem Modelling Zoning in Rocks

Abstract

<p>Oscillatory zoning in rocks can be explained by a kinetic mathematical model of crystal growth. In this model, zoning is an autonomously occurring phenomenon resulting from the interaction of crystal growth dynamics and diffusion of solutes within the solution. Here the rates of crystal formation have a positive feedback dependency such that these rates depend on the composition of the crystal surface.</p> <p>A moving free boundary problem is presented describing the growth of two essential crystal end-members that are formed from two solutes on a solid-solute interface. The simplest possible case is presented in which there are two first order crystal formation reactions, and all the variation of concentration is confined to one solute. Bifurcation analysis is used as a criteria for the local existence of oscillatory zoning. Under certain physical conditions, we can show, using rigorous analysis, that planar constant composition front solutions lose their stability to oscillatory solutions through a Hopf bifurcation when important parameter values exceed some critical value. The analysis is very sensitive to the precise stoichiometry of the crystal formation reactions and to the initial conditions of the state.</p>