Model of layered pattern formation in binary igneous systems

Centimeter to meter-scale repetitive patterns in composition and texture are sometimes observed in igneous systems. Examples are found in layered intrusions and multi-shelled orbicular granites. These patterns may result from the action of nonlinear self-organization processes in which the interplay between crystallization dynamics, diffusion and thermal conduction causes mineral and crystal size segregations. These mechanisms are analogous to the ones underlying the formation of Liesegang bands and lead to comparable features, such as a geometric progression of the band positions and the presence of doublets. We present here a comprehensive one-dimensional numerical model of Liesegang pattern formation process from binary eutectic melts in igneous systems. The model incorporates nucleation, growth and Ostwald ripening and is applied to both the layered intrusion and the orbicular granite configuration with appropriate simple geometries and cooling boundary conditions. The emergence of cyclic layering is described in terms of two key parameters that control the pattern formation: the scaled latent heat of crystallization (Stefan number) and the ratio of the thermal diffusivity to a characteristic diffusion coefficient (Lewis number). It is found that, in intrusions, a banding pattern compatible with the Liesegang spacing law is generated when the Stefan number is large and the Lewis number small, in agreement with previous studies. For orbicular granites with low Lewis number, we show that the band thickness and the crystal size increase with distance from the rim to the core, in agreement with the observations of Zhang and Lee (2020). This suggests that the pattern progresses inwards from the outer boundary, rather than from a colder core, thus supporting the conceptual model reported in Zhang and Lee (2020). Moreover, the results for both geometries indicate that ripening plays an important role in the formation of realistic patterns.