A conductivity-dependent phase transition from closed-loop to open-loop dendritic networks

Motivated by a principle of minimum dissipation per channel length, we introduce a model for branching, hierarchical networks in an open, dissipative system. Global properties of the resulting structures are observed to scale with the ratio of conductivity in the dendrite material to conductivity in the lattice material. Beyond a critical conductivity ratio, the resulting structures are naturally self-avoiding and possess scale-independent branching ratios. Our findings suggest that the conductivity ratio determines the geometric properties of naturally arising dendritic structures. We discuss empirical verification in the context of a system of self-organizing agglomerates of metal particles in castor oil.