Abstract :
A model of deterministic Laplacian growth with variable anisotropy is simulated on a square lattice. Upon changing a parameter α that controls the amount of anisotropy, three clearly distinct morphologies appear, separated by sharp transitions. The model is also shown to describe a problem of fluid invasion in a regular array of chambers connected by pores, in which case α is related to the relative volume of pores and chambers. Particular cases of this model include the usual bond- and site noise reduction cases of the infinite noise reduction models of Laplacian growth, whose relation to fluid invasion is discussed.
The transitions found here can be described as splitting-merging of dendrites, a phenomenon that has been qualitatively observed in many systems. Two different mechanisms are responsible for the transitions in this model: One of them (splitting transition) is due to the fact that the dendrites originally growing along the main axes become unstable and split. The other transition (screening transition) occurs because two stable morphologies compete due to long-range screening. Each transition has an associated characteristic length, which is found to diverge at the critical point. The noiseless character and large lattice sizes of the simulations here presented allow the determination of critical indices associated with these morphological phase transitions, in a similar way to what is done in the case of thermodynamical phase transitions.
The growth velocity is found to behave continuously across one of the transitions and discontinuously across the other. The role of the long range screening in producing this difference is discussed. A phenomenological model is proposed to describe the role of screening, which predicts both the existence of a divergent length and a discontinuity in the growth velocity
