PHYSICAL REVIE% A VOLUME 31, NUMBER 2 Rapid Communications FEBRUARY 1985 corrections unless requested by the author. New class of screened growth aggregates with a continuously tunable fractal dimension Paul Meakin Experimental Station, Central Research and Development Department, E. I. DuPont de ¹mours and Company, Inc. , 8 ilmington, Delaware 19898 Francois Leyvraz' Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109 H. Eugene Stanley Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215 (Received 22 October 1984) A new family of fractals is investigated. The fractal dimension Df is found to be equal to a variable parameter of the model characterizing the strength of the screening. Thus we can make fractals with arbi- trary Df, and study anomalous diffusion as a function of D&. Our data support a generalization we propose of the recent Aharony-Stauffer conjecture based on the spatial distribution of "growth sites" of a fractal. p(x) =K(x) y E perimeter E(y) (la) lt is of considerable general interest to discover how the familiar laws of physics are modified for fractals, in part be- cause of the numerous examples of fractal structures in nature. '~ Some studies have focused on regular fractals — such as the Sierpinski gasket — for which the fractal dimension D~ is usually known exactly. ' Recently it has become increasingly apparent that the physical systems of interest are not describable by regular fractals, and hence many studies of random fractals have been undertaken. A major problem that plagues these studies is that Df is not generally known exactly, even for simple d = 2 systems. Here we develop a family of random fractal structures for which Df is known exactly. Moreover, one can continuous- ly tune Df in order to test laws that may not be readily test- ed using the discrete values of D~ available from the above-mentioned fractals; these fractals thereby provide an ideal testing ground for properties of random fractals in gen- eral. More important, perhaps, is the conceptual rationale for this model. It bears the same relation to the Rikvold model7 '(or any other model with a discrete value of Df) that the Fisher Ma Nickel model o-f sp-in spin intera-ctions that de cay as a power law bears to the ordinary Ising model with short-range forces. Vfe are concerned with clusters generated by starting from a seed and successively adding new sites at the perimeter. The probability for adding a new site at the vacant perimeter site x is given by7 where &(x)= g exp( — lx —. yl ') . yE cluster (lb) Here e is a free parameter. Thus we grow a cluster by suc- cessive addition of new sites on the perimeter with a long- range screening effect as a result of the nature of the depen- dence of p(x) on the existing cluster sites. We have three main objectives: (i) to give a compelling argument that Df=e, (ii) to put this prediction to a searching test by means of very large scale numerical simulations, and (iii) to investigate the properties of random walks on these clusters, thereby testing the relative validity of two competing theories of fractal dynamics. 9' Fractal dimension. To find Df it is more convenient to visualize the cluster being generated by growing sites at a rate K(x) given by Etl. (lb), so that X„E(y) is the aver- age number of sites created per unit time. Now consider E(x) as a function of Df. Changing from sums to integrals and going over to polar coordinates, I R lx — yl '= ~ drr I r '=O(R I )+O(1) y 6 cluster (2) where a is a short-distance cutoff on the order of the lattice length. First suppose that Df & e, so that K(x) =exp( — AR r ), (3a) K(y) & R fexp( — AR f ) « 1 . (3b) y E perimeter 31 1195 1985 The American Physical Society