Topological properties of diffusion limited aggregation and cluster - cluster aggregation

The detailed topological or ‘connectivity’ properties of the clusters formed in diffusion limited aggregation (DLA) and cluster-cluster aggregation (CCA) are considered for spatial dimensions d = 2,3 and 4. Specifically, for both aggregation phenomena we calculate the fractal dimension d,,, = i-’ defined by e R d m l n where e is the shortest path between two points separated by a Pythagorean distance R For CCA, we find that dmin increases monotonically with d, presumably tending toward a limiting value dmi, = 2 at the upper critical gimensionality d, as found previously for lattice animals and percolation. For DLA, on the other hand, we find that dmin = 1 within the accuracy of our calculations for d = 2, 3 and 4; suggesting the absence of an upper critical dimension. We also discuss some of the subtle features encountered in calculating dmin for DLA. Considerable recent attention has focussed on models of aggregation, in large part due to their potential promise in providing tractable models for a range of flocculation phenomena. The diffusion limited aggregation (DLA) model of Witten and Sander (1981) is the prototype of modern models of aggregation: a seed particle is placed at the origin and a random walker is released from a large circle encompassing the origin. This particle is assumed to undergo a random walk until it sticks to the seed particle. Another random walker is then released, and this process is continued until typically a large aggregate containing N = O( lo4) particles has been formed. DLA describes a range of natural phenomena in which identifiable ‘seed sites’ exist, but for the flocculation of particles ranging from soot to colloids no such stationary seed exists. Accordingly, the cluster-cluster aggregation (CCA) model (Meakin 1983d, Kolb et a1 1983) assumes that a large number of particles randomly diffuse at the same time: when they touch one another they stick, forming clusters, which themselves diffuse randomly touching other particles or clusters until at a large time a ramified fractal aggregate has been formed. Both DLA and CCA clusters qualitatively resemble aggregates found in nature, and have been the subject of intensive recent study. However, there is thus far only a single quantitative parameter that has been used to characterise these aggregates. This is the fractal dimension df that is a direct measure of how the density approaches zero as /I Supported in part by grants from ONR and NSF. 0305-44701841 180975 + 07$02.25 @ 1984 The Institute of Physics L975 L976 Letter to the Editor the length scale over which it is measured increases. If M ( R ) is the cluster mass within a Pythagorean distance R of a cluster point, and p ( R ) = M ( R ) / R d is the density, then one writes M ( R ) ~~f p ( R ) Rdf-d. ( 1 ) The fractal dimension concept has permitted extensive comparisons between large-scale computer simulations (e.g. Meakin 1983a, b, c, e) and several mean-field type theories (e.g. Muthukumar 1983, Tokuyama and Kawasaki 1984, Hentschel 1984, Hentschel and Deutch 1984). More recently, it has become possible to actually measure df for naturally-occurring aggregates and to compare the experimental values with results from simulations and from theory (see e.g. Forrest and Witten 1979, Nittmann et a1 1984, Weitz and Oliveira 1984, Niemeyer et a1 1984, Schaefer et a1 1984, Schaefer and Keefer 1984, Bale and Schmidt 1984, Laibowitz and Gefen 1984, Matsushita et a1 1984). Although df has proved extremely useful as a quantitative parameter with which to characterise clusters, it is by no means sufficient. For example, in d = 3 both DLA and percolation clusters have df 2.5, yet even the most casual visual inspection reveals that they are quire different (see e.g. figure 4 of Stanley et a1 (1984a)). Accordingly, one motivation for the present study is to investigate the utility of a second parameter in the quantitative characterisation of DLA and CCA. This is the exponent dmin that governs the dependence of the minimum path length between two points, e, on the Pythagorean distance R between them,