A dynamical model for fractal and compact growth in supercooled systems

A dynamical model that can exhibit both fractal percolation growth and compact circular growth is presented. At any given cluster size, the dimension of a cluster growing on a two-dimensional square lattice depends on the ratio between the rates of two probabilistic processes, namely (i) the aggregation of lattice sites into the growing cluster and (ii) the relaxation of lattice sites into those available for potential aggregation. The proposed model approaches the limit of two-dimensional invasion percolation if the aggregation process is much faster than the relaxation process, and it approaches Eden's model for compact circular growth if the relaxation process is much faster than the aggregation process. Experimental examples of the fractal-growth regime include the percolation-like growth of bent-core smectics and calamitic smectics, where such fractal growth is attributed to the slow relaxation of molecules in a viscous supercooled medium.