Comparative study of large-scale Laplacian growth patterns

Abstract

We investigate the scaling of cluster size with mass for our simulations of diffusion-limited aggregation (DLA) clusters and dielectric-breakdown (DB) clusters of 10 particles grown on a square lattice, and DLA clusters of 10 particles grown off lattice. We [2] are among the most widely studied models for generating fractal growth patterns (for recent reviews, see, e.g. , [3,4]). Each of these models is a Laplacian growth process in which difFusing particles released from a distant boundary attach to the perimeter of a growing cluster. The essential difFerence between the models is a boundary condition [5].In on-lattice versions the diffusing particle undergoes a random walk on a lattice and particles in the cluster are represented by occupied (aggregate) sites on the lattice.In the DLA model the diffusing particles are terminated at the first surface site (unoccupied site adjacent to an aggregate site) that they contact. This surface site is then converted to an aggregate site. In the DB model the diffusing particles can diffuse through surface sites but are terminated at the first aggregate site they contact. The surface site immediately preceding this termination site in the difFusion process is then converted to an aggregate site. The DB boundary condition introduces an inherent surface tension [6,7]. In this sense, the DB model is more physically appealing than the DLA model, which has zero surface tension [8].An "ofF-lattice" Laplacian growth model has also been introduced [9]. In this model, the diffusing particle slides over fixed step lengths in free space until it first intersects the cluster. The difFusing particle is then moved back and is attached at its first point of contact with the cluster. We will follow convention and refer to this growth process as "off-lattice DLA. " However, this offlattice growth process more closely realizes the boundary conditions for the DB model, since the difFusing particle can difFuse along the "surface" of the cluster in each of these models.Much of the theoretical interest in the above models has been directed at determining precise numerical and algebraic estimates for the &actal dimension D [3,4]. OfFlattice DLA clusters are homogeneous self-similar fractals, with fractal dimension D 1.7 in d = 2, independent of cluster size (see, e.g. , [10,11] DB clusters and the on-lattice DLA studies suggest that large clusters are necessary to reveal the full behavior of the model [14]. It is generally believed that the differences between large-scale on-lattice and off-lattice DLA patterns are due to the underlying lattice anisotropy. However, as noted above, the kinetics of attachment implied by the boundary conditions are also difFerent in these two models. The role played by this difFerence has not been fully explored.Large-scale studies of on-lattice DB patterns are also useful for this purpose because the kinetics of attachment are similar to off-lattice DLA, but the diffusion process is confined to the underlying lattice. The possibility of signatures of the micros...

Evidence of Discrete Scale Invariance in DLA and Time-to-Failure by Canonical Averaging

“…where M (r) is the "mass" of the cluster or, similarly, the number of particles within a disk of radius r centered on the initial cluster seed. The numerical value of the fractal dimension of off-lattice DLA-clusters have been reported to be D ≈ 1.71±0.02 [24,25] and D ≈ 1.60 ± 0.02 [26] depending on how D is estimated [26]. A reason for the surprisingly large deviation in the dimension reported by different authors may be the presence of log-periodic oscillations in the local fractal dimension D r as a function of r, or equivalently M, as shown in [15].…”

Exact solution for random walks on the triangular lattice with absorbing boundaries

“…By using the two matching conditions, Eqs. ( 9), (10), the four arbitrary constants, A, B, Ã, B, can be reduced to two arbitrary constants, A, B say. The solutions in Region II can then be written as…”

“…These include potential theory [6], electrical networks [7], surface diffusion [8] and Diffusion-Limited Aggregation (DLA) [9]. For example, the exact results for the square lattice problem [2] have been used to expedite the growth of large DLA clusters on the square lattice [10] and the formulae derived in this paper could similarly be used on the triangular lattice.…”

Simulation of electrochemical deposition process by a multiparticle diffusive aggregation model

“…1-3 Studies of pattern formations in a diffusive system include electrochemical deposition, 4 crystal growth, 5 viscous fingering, 6,7 dielectric breakdown, [8][9][10][11] chemical dissolution, 12 and bacterial colonies. 13,14 An approximation of these phenomena is provided by the Laplacian growth model that can be simulated by the diffusion-limited aggregation ͑DLA͒ model.…”