Structure of large two-dimensional square-lattice diffusion-limited aggregates: Approach to asymptotic behavior

Meakin, Paul; Ball, Robin C.; Ramanlal, P.; Sander, Leonard M.

doi:10.1103/physreva.35.5233

Cited by 87 publications

(41 citation statements)

References 23 publications

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“…Several improvements to this original algorithm were developed in order to speed up the simulations [15,16]. Later it was found that large cluster exhibit anisotropy which reflects the symmetry of the lattice [17].…”

Section: Simulations a Dla Algorithmmentioning

confidence: 99%

Finite size effect of harmonic measure estimation in a DLA model: Variable size of probe particles

Menshutin

Shchur

Vinokour

2008

Physica A: Statistical Mechanics and its Applications

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A finite size effect in the probing of the harmonic measure in simulation of diffusion-limited aggregation (DLA) growth is investigated. We introduce variable size of the probe particles to estimate harmonic measure and extract fractal dimension of DLA clusters taking two limits, of vanishingly small probe particle size and of infinitely large size of a DLA cluster. We generate 1000 DLA clusters consisting of 50 million particles each using off-lattice killing-free algorithm developed in the early work. The introduced method leads to an unprecedented accuracy in the estimation of the fractal dimension. We discuss the variation of the probability distribution function with the size of probing particles.

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Section: Simulations a Dla Algorithmmentioning

confidence: 99%

Finite size effect of harmonic measure estimation in a DLA model: Variable size of probe particles

Menshutin

Shchur

Vinokour

2008

Physica A: Statistical Mechanics and its Applications

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“…In contrast, the morphology of on-lattice DLA is size dependent [12 -14]. Small clusters appear to be self-similar fractals with D 1.7 [1,8] [14].…”

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confidence: 99%

“…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.…”

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confidence: 99%

Comparative study of large-scale Laplacian growth patterns

1995

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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...

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“…Thus one can reasonably suspect that the screening effects induced by the fractal geometry irreversibly influence the dynamics of the growth. Because the complexity of the geometry is intricately connected with the dynamical evolution, most attention has been initially focused on the fractal structure of Laplacian aggregates [1][2][3][17][18][19][20][21][22]. But only very recently [23,24], DLA clusters were shown to be statistically self-similar as generally believed.…”

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confidence: 99%

“…Extensive computer investigations [1][2][3] [lc, 3-7] or to mimic anisotropic growth [3, [8][9][10][11][12][13][14][15][16]. The fractal geometry of these aggregates has been analyzed by computational [2,3,11,[17][18][19][20][21][22][23][24] and analytical methods [15,[25][26][27][28]. But [3] or the result of a proliferation of deterministic tip-splitting instabilities [29].…”

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confidence: 99%

Experimental evidence for deterministic chaos in electrochemical deposition

Argoul

Arnéodo

1990

J. Phys. France

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We report an experimental study of the statics and dynamics of electrodeposition of zinc in an aqueous medium, in the intermediate regime between dendritic and DLA-like metallic clusters. During the growth process, periodic and nonperiodic oscillations of the voltage are recorded under constant applied current intensity. Period-doubling bifurcations are identified. Analysis of the data in terms of phase portraits, Poincaré maps and 1-D maps shows that the fractal geometry of these electrochemical deposits is the signature of a low-dimensional deterministic chaotic dynamics which displays sensitivity to initial conditions. The study of the apparently more complex dynamics recorded in the DLA limit is a very promising experimental challenge

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Structure of large two-dimensional square-lattice diffusion-limited aggregates: Approach to asymptotic behavior

Cited by 87 publications

References 23 publications

Finite size effect of harmonic measure estimation in a DLA model: Variable size of probe particles

Finite size effect of harmonic measure estimation in a DLA model: Variable size of probe particles

Comparative study of large-scale Laplacian growth patterns

Experimental evidence for deterministic chaos in electrochemical deposition

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