The garnet-biotite geothermometry is widely used by metamorphic petrologists to estimate the temperature condition of metamorphism [Spear 1993 and references therein]. The basic assumption for the geothermometry is temperature-dependent equilibrium distribution of Fe and Mg ions in biotite and garnet. During equilibration, diffusive mass transfer along the diffusion pathways (i.e. grain or phase boundaries) as well as volume diffusion within crystals of biotite and garnet occurs. The effect of temperature-dependent diffusion has been successfully demonstrated with finite difference calculations [e.g. Spear 1991; Ehlers et al. 1994; Powell & White 1995]. These models usually assume that the rate of grain boundary diffusion is much faster than the rate of volume diffusion in crystals. More specifically, these models assume that the rate-limiting step for equilibration is volume diffusion in garnet because the rate of diffusion in garnet is much slower than both diffusion in biotite [Hoffman & Giletti 1974; Cygan & Lasaga 1985] and diffusion along grain boundaries. Geologically meaningful results have been produced by these models and applied for the interpretation of zoning pattern in garnets [e.g. Spear 1991; O'Brien 1997].
Although the aforementioned models [e.g. Spear 1991; Ehlers et al. 1994] generated useful results, these models fail to produce thin-section scale textural features associated with the zoning patterns in garnet. For example, the effect of distribution geometry in phases participating in the reaction cannot be studied with the conventional model because of the simple assumption that volume diffusion in garnet is the rate-limiting step. Furthermore, these models cannot be used for multiple geologic processes, for example, deformation and metamorphism. In this paper, we present a two dimensional model where the ion exchange reaction occurs in a grain network created by digitally mapping a thin section. Since calculation of diffusion is enormously different for different dimensions (2D in our model vs 3D in nature), it is difficult to apply the model results for rocks. However, we believe that this type of approach can lead to further understanding of the ion exchange reaction since we can infer the effect of various textural parameters of rocks (e.g. phase distribution and grain size) on the development of zoning in garnet and we can also model the situation where textures change during metamorphism (e.g. grain growth and foliation development). We adopted the Elle scheme [Jessell et al. 2001] for the exchange reaction because the preexisting Elle code allows us to model multiple processes by integrating incremental textural changes. We will present three examples of our model results; (1) zonation development at a constant temperature with different phase distribution of garnet and biotite, (2) zonation development at constant temperature with different textures of non-reacting matrix phases, and (3) zonation development during cooling.