Universality, nonuniversality, and the effects of anisotropy on diffusion-limited aggregation

Abstract

Large numbers (hundreds) of quite large [(4–8)×${10}^4$ lattice sites or particles] diffusion-limited-aggregation (DLA) clusters have been grown on two-dimensional square, hexagonal, and triangular lattices to investigate the effects of lattice anisotropy. Similar simulations have been carried out to explore other types of anisotropy, such as anisotropy in the diffusion process and sticking. Our results suggest that the DLA process is sensitive to anisotropy of low symmetry. DLA on the triangular and hexagonal lattices (both with sixfold effective symmetry) lead to structures which can be characterized by a radius of gyration exponent β of 0.585±0.003 (the same as that found for off-lattice DLA). For the square lattice an effective value of about 0.60 is found for the largest cluster sizes used and anisotropy in diffusion or sticking with threefold symmetry leads to even larger radius-of-gyration exponents. Our results suggest that off-lattice DLA and DLA in systems with about fivefold or higher symmetry belong to one universality class and that DLA in environments of lower symmetry lead to structures in a different universality class or classes. Our results for square-lattice DLA are consistent with the theoretical predictions of Turkevich and Sher, but our results for the hexagonal and triangular lattices do not agree with their theoretical prediction. However, in these cases, the discrepancy can be rationalized by using ideas contained in the work of Turkevich and Sher.