PHYSICAL REVIEW A VOLUME 36, NUMBER 7 OCTOBER 1, 1987 Scaling theory for the anisotropic behavior of generalized diffusion-limited aggregation clusters in two dimensions Mitsugu Matsushita Department of Physics, Chuo University, Kasuga, Bunkyo ku, -Tokyo ii2, Japan Fereydoon Family Department of Physics, Emory University, Atlanta, Georgia 30322 Katsuya Honda Department of Applied Physics, Faculty of Engineering, Nagoya University, Nagoya 464, Japan (Received 29 June 1987) A scaling description of the crossover from isotropic to anisotropic cluster growth for ordinary diffusion-limited aggregation (DLA) in two dimensions developed recently by Family and Hentschel is extended to the generalized DLA or g model. The dependence of various exponents necessary to characterize the anisotropic growth on the local-growth probability exponent g of the generalized DLA is obtained explicitly. The rt dependence of the exponent P describing the varia- tion of the crossover mass N, on the degree of symmetry m, N, — m, is derived. The results indi- cate that the anisotropic star-shaped clusters can be easily observed for g ) 1, while their appear- ance is much more difFicult for g & 1. All our results are consistent with those of computer simu- lations reported so far. Formation of large aggregates from small particles is a topic of considerable recent interest. ' " Among many nonequilibrium growth models' proposed so far to de- scribe aggregation processes the diffusion-limited aggre- gation (DLA) model has received the most attention. Much of the interest in DLA was stimulated by the as- sumption that DLA clusters are self-similar fractals characterized by a unique fractal dimension D. The value D was thought to be dependent only on the spatial dimen- sion d but independent of, e.g. , the underlying lattice de- tails. In fact, a number of theoretical attempts have been made to obtain an analytical expression for D for ordinary DLA, extended DLA' " including DLA on fractal patterns such as percolation clusters and/or with general random walks such as Levy Aight, and generalized DLA (or the ri model' ). ' ' Recent simulations, however, in- dicate that asymptotically large DLA clusters grown in two dimensions on lattices with m-fold symmetry or even those of moderate size but grown through an anisotropic growth mechanism exhibit strong m-fold star-shaped an- isotropy ' in their global structure. Recently, some progress has been made in under- standing the crossover from isotropic to anisotropic cluster shapes and in calculating the exponents for simplified models of two-dimensional DLA. However, generalized DLA clusters with g) 1 show even stronger anisotro- py. On the other hand, Eden clusters never show star-shaped anisotropy even for the particle number N = 1. 7 & 10 . Unfortunately, little progress has been made in a quantitative understanding for this unusual but interesting phenomenon. We believe, however, that this problem is very important in that it is intimately related to the role of anisotropy in dendritic solidification, viscous fingering, ' and dielectric breakdown in elong- ated plastics. The exponent p is to be determined later. g is, therefore, regarded as the crossover radius from the isotropic to an- isotropic cluster growth. For the star-shaped cluster, i.e. , Rii)) g one must write down, instead of (1), Ria — I ""N ' and Ri — m (3) This means that the anisotropic cluster is not self-similar but self-affine. Combining (1)-(3) we obtain the follow- In this Rapid Communication we extend the scaling theory for the appearance of anisotropy in ordinary DLA developed recently by Meakin and Family and, in par- ticular, by Family and Hentschel " to the generalized DLA or g model. We present the scaling description for the structure of generalized DLA patterns and the cross- over from an initially isotropic to asymptotically aniso- tropic growth. We then obtain the dependence of the cluster mass N, at the crossover on m and g. The results, in fact, explain prominent anisotropic behaviors in the generalized DLA described before. Let us presuppose that in the presence of an rn-fold an- isotropy, originating either from the symmetry of back- ground lattice or the anisotropy of growth mechanism, an asymptotically large cluster resembles a star-shaped pat- tern with m spokes;' for example, it would be a cross for m =4. Each spoke is of length R ii and width R &. ' ' Initially, we expect the cluster to grow isotrop- ically, and R(i and R& to be scaled as Ri(-R~ -N', where N is the cluster mass and v is the inverse of the fractal dimension D of an isotropic DLA. Suppose that this isotropic growth continues until Rii — R~ — (, where 3518 1987 The American Physical Society