DLA Clusters Research Articles

Diffusion limited aggregation of particles with different sizes: Fractal dimension change by anisotropic growth

Clusters formation models have been extensively studied in literature, and one of the main task of this research area is the analysis of the particle aggregation processes. Some work support that the main characteristics of this processes are strictly correlated to the cluster morphology, for example in DLA. It is expected that in the DLA clusters formation with particles containing different sizes the modification of the aggregation processes can be responsible for changes in the DLA morphology. The present article is going to analyze the formation of DLA clusters of particles with different sizes and show that the aggregates obtained by this approach generate an angle selection mechanism on dendritic growth that influences the shielding effect of the DLA edge and affect the fractal dimension of the clusters.

The Arcsine law as the limit of the internal DLA cluster generated by Sinai’s walk

We identify the limit of the internal DLA cluster generated by Sinai's walk as the law of a functional of a Brownian motion which turns out to be a new interpretation of the Arcsine law.

Diffusion Limited Aggregation on a Cylinder

We consider the DLA process on a cylinder $$G \times {\mathbb{N}}$$ . It is shown that this process “grows arms”, provided that the base graph G has small enough mixing time. Specifically, if the mixing time of G is at most $$\log^{(2-\epsilon)}\left\vert G \right\vert$$ , the time it takes the cluster to reach the m th layer of the cylinder is at most of order $$m \cdot \frac{\left\vert G \right\vert}{\log\log\left\vert G \right\vert}$$ . In particular we get examples of infinite Cayley graphs of degree 5, for which the DLA cluster on these graphs has arbitrarily small density. In addition, we provide an upper bound on the rate at which the “arms” grow. This bound is valid for a large class of base graphs G, including discrete tori of dimension at least 3. It is also shown that for any base graph G, the density of the DLA process on a G-cylinder is related to the rate at which the arms of the cluster grow. This implies that for any vertex transitive G, the density of DLA on a G-cylinder is bounded by 2/3.

QUASI-FRACTALS: NEW METHOD OF DESCRIPTION OF A STRUCTURE OF DISORDERED MEDIA

The description of a self-similar structure with the help of fractal dimension represents an effective tool for understanding of the scaling properties of disordered media. In this paper we suggest a new type of fractals (they were defined as quasi-fractals with logarithmic asymptotics) that can be suitable for description of a wide class of clusters formed by random fractals. As independent parameter we chose a number of coordination sphere j (j = 1, 2 ...). With respect to this parameter the radius of coordination sphere can be expressed as R(j) = R0jα and number of particles located inside of the sphere R(j) is expressed by another power-low function N(j) = N0jβ. Different algorithms considered in this paper confirm the effectiveness of the quasifractals in description of DLA clusters. We suppose that these new geometrical approach will increase the applicability of the mathematics of the fractional calculus.

Diffusion-limited aggregation: A continuum mean field model

Mean field theory is used as a basis for a new approach to analyzing fractal pattern formation by diffusion-limited aggregation. A coarse time scale is introduced to take into account the discrete nature of DLA clusters. A system of equations is derived and solved numerically to determine the fractal dimension and density of a cluster as a function of distance from its center. The results obtained are in good agreement with direct numerical simulations.

Thermodynamically reversible generalization of diffusion limited aggregation

We introduce a lattice gas model of cluster growth via the diffusive aggregation of particles in a closed system obeying a local, deterministic, microscopically reversible dynamics. This model roughly corresponds to placing the irreversible diffusion limited aggregation model (DLA) in contact with a heat bath. Particles release latent heat when aggregating, while singly connected cluster members can absorb heat and evaporate. The heat bath is initially empty, hence we observe the flow of entropy from the aggregating gas of particles into the heat bath, which is being populated by diffusing heat tokens. Before the population of the heat bath stabilizes, the cluster morphology (quantified by the fractal dimension) is similar to a standard DLA cluster. The cluster then gradually anneals, becoming more tenuous, until reaching configurational equilibrium when the cluster morphology resembles a quenched branched random polymer. As the microscopic dynamics is invertible, we can reverse the evolution, observe the inverse flow of heat and entropy, and recover the initial condition. This simple system provides an explicit example of how macroscopic dissipation and self-organization can result from an underlying microscopically reversible dynamics. We present a detailed description of the dynamics for the model, discuss the macroscopic limit, and give predictions for the equilibrium particle densities obtained in the mean field limit. Empirical results for the growth are then presented, including the observed equilibrium particle densities, the temperature of the system, the fractal dimension of the growth clusters, scaling behavior, finite size effects, and the approach to equilibrium. We pay particular attention to the temporal behavior of the growth process and show that the relaxation to the maximum entropy state is initially a rapid nonequilibrium process, then subsequently it is a quasistatic process with a well defined temperature.

Structure formation during precipitation reaction in gels: New liesegang patterns

AbstractThe paper reports on an investigation carried out into one‐ to three‐dimensional Liesegang systems (test tube arrangement) composed of different concentrations of lead nitrate in Agar gel with either potassium iodide or potassium dichromate as the precipitation reagent. A rich variety of different structure formations, including screw formations of the weakly soluble reaction product is documented.Measurements of the progression of the precipitation front show a dependence of the diffusion path on time that which deviates from the Einstein‐Smoluchowski root mean square displacement. The measured exponent n of the displacement xn =f(t) is discussed in detail and interpreted as the dynamic dimension dw of the diffusion through the fractal medium of the gel, the backbone dimension of which could be estimated. A cellular automaton model is used to describe the DLA cluster formation of the precipitate as well as the geometrical shape of the Liesegang patterns.

Dynamical Characterization of Electroless Deposition in the Diffusion-Limited Regime

Electroless deposition of metallic aggregates in thin gap geometry has recently been recognized as a good candidate for producing DLA-like morphology. In the continuation of these studies, which were mainly focused on the fractal geometry of the growing aggregate, we address the issue of characterizing the interfacial dynamics via some local measurement of the growth velocity during an electroless deposition process. Direct computations of the harmonic measures of DLA and electroless clusters are compared with experimental estimations of the local growth velocity of these same clusters. This test is shown to be more relevant than the computation of statistical quantities like the generalized fractal dimensions Dq and the f(α) singularity spectrum, as advocated by the multifractal formalism. On the one hand, it does not require working with clusters of very large size. On the other, it provides a direct comparison of the theoretical and experimental local distributions of the growth velocity along the interface of the growing cluster. This experimental approach is therefore adapted to a real time determination of the regime during which the growth process can be reasonably approximated by a purely diffusion-limited process.

THE MOST PROBABLE APPROACH TO THE DLA MODEL

The most probable approach is used to treat the problem of the DLA model.The most probable distribution of the particles,the fractal dimension and the screening exponent of the DLA cluster are obtained.The results agree with the computer simulation results.

Growth and form of zero-noise diffusion-limited-aggregation on the cubic lattice

The growth and form of zero-noise DLA clusters on a cubic lattice is investigated. Numerical simulations reveal a stable cluster consisting of six fourfold symmetric (almost planar) arms with a staircase structure along each arm. Empirical expressions are obtained for; the shape of the arms in the vicinity of the tips, the scaling of the arm lengths and the scaling of the tip growth rates. The empirical expressions are used to derive results for the growth rates along the arms of the cluster and for the step lengths in a needle staircase. These results are in good agreement with the numerical simulations.

Construction of a DLA cluster model

On the occasion of the 50th birthday of a distinguished colleague, a three-dimensional wood model of a computer-generated DLA cluster was built. In this paper the design of the model and the construction process are described. The experiment may be carried out in the framework of a classroom experiment to demonstrate some of the fundamental concepts used in current research on growth phenomena. It is suitable as a first introduction to fractal geometry.Zusammenfassung. Anläßlich des 50sten Geburtstag eines bedeutenden Kollegen wurde ein dreidimensionales Holzmodell eines computererzeugten DLA-Clusters gebaut. Hier beschreiben wir die Entwickelung des Modells und den Konstruktionsprozess. Das Experiment kann im Rahmen eines Klassenzimmerexperiments durchgeführt werden um einige der fundamentalen Konzepte innerhalb der aktuellen Forschung auf dem Gebiet der Wachstumsphänomene zu demonstrieren. Es eignet sich als eine erste Einführung in die fraktale Geometrie.

True self-avoiding walks on fractal lattices above the upper marginal dimension

We study by Monte Carlo simulations the critical behaviour and the cross-over scaling of the true self-avoiding walks (TSAW) on fractal lattices above the upper marginal dimension. We estimate the Flory exponent nu which characterizes the RMS end-to-end distances of SAW on a percolation cluster at percolation thresholds both in three and four dimensions and on a DLA cluster in three dimensions. Results were in good agreement with the predictions of the known Flory formulae. We also discuss the fractal-to-Euclidean and the RW-to-TSAW cross-over scaling. Monte Carlo data appear to collapse in the scaling regions for both cases; however, we found that the scaling function for the latter is different from that on the regular lattices.

Kinetics of the first-order reaction on fractals and Monte Carlo simulations

For diffusion controlled reactions on fractals, the kinetic expression of the first-order reaction takes the form: C A = C A0 exp(−k′ t 2 d w ) , where k′ is a constant and d W is the fractal dimension of the trajectory. Monte Carlo simulations show that the expression can effectively describe the kinetics of the first-order reaction on DLA clusters.

Subdiffusive Fluctuations for Internal Diffusion Limited Aggregation

Internal diffusion limited aggregation (internal DLA) is a cluster model in $\mathbb{Z}^d$ where new points are added by starting random walkers at the origin and letting them run until they have found a new point to add to the cluster. It has been shown that the limiting shape of internal DLA clusters is spherical. Here we show that for $d \geq 2$ the fluctuations are subdiffusive; in fact, that they are of order at most $n^{1/3}$, at least up to logarithmic corrections. More precisely, we show that for all sufficiently large $n$ the cluster after $m = \lbrack\omega_dn^d\rbrack$ steps covers all points in the ball of radius $n - n^{1/3}(\ln n)^2$ and is contained in the ball of radius $n + n^{1/3}(\ln n)^4$.

Fractal and Dendritic Growth of Surface Aggregates

ABSTRACTThe similarity of patterns formed in non-equilibrium growth processes in physics, chemistry and biology is conspicuous and many attempts have been made to discover common mechanisms underlying their growth. The central question in this context is what causes some patterns to be dendritic, as e.g. snowflakes, while others grow fractal (randomly ramified). Here we report a crossover from fractal to dendritic patterns for growth in two dimensions: the diffusion limited aggregation of Ag atoms on a Pt(111) surface as observed by means of variable temperature STM. The microscopic mechanism of dendritic growth can be analyzed for the present system. It originates from the anisotropy of the diffusion of adatoms at corner sites which is linked to the trigonal symmetry of the substrate. This corner diffusion is observed to be active as soon as islands form, therefore, the classical DLA clusters with the hit and stick mechanism do not form. The ideas on the mechanism for dendritic growth have been verified by kinetic Monte-Carlo simulations which are in excellent agreement with experiment.

Self-similarity of the branching structure in very large DLA clusters and other branching fractals

We study the branching structure of very large DLA clusters, of up to 100 million particles. The Horton-Strahler ordering of the branches in these clusters shows a relaxation towards a state with the stream numbers forming a geometric series. This behaviour is compared with those of several self-similar trees. It indicates that DLA clusters converge to topologically self-similar objects.

EVIDENCE FOR MULTISCALING IN LARGE DLA CLUSTERS

We have measured the multiscaling dimension D(x) on a set of large off-lattice DLA clusters of one million particles, finding results which strongly support multiscaling behavior. The multiscaling picture is also confirmed by the slow logarithmic decay in the fluctuations of the radius (relative to the cluster radius) as the size of clusters grow.

Is diffusion limited aggregation scale invariant?

Recent work showing evidence for multiscaling in large two dimensional DLA clusters is reviewed. A connection between multiscaling and multifractality is also shown resulting in a weak localization of the growth sites (δ-sites) in shells around the cluster seed.

BEYOND CLASSICAL MULTIFRACTAL ANALYSIS USING WAVELETS: UNCOVERING A MULTIPLICATIVE PROCESS HIDDEN IN THE GEOMETRICAL COMPLEXITY OF DIFFUSION LIMITED AGGREGATES

We emphasize the wavelet transform as a very promising tool for solving the inverse fractal problem. We show that a dynamical system which leaves invariant a fractal object can be Uncovered from the space-scale arrangement of its wavelet transform modulus maxima. We illustrate our theoretical considerations on pedagogical examples including Bernoulli invariant measures of linear and nonlinear expanding Markov maps as well as the invariant measure of period-doubling dynamical systems at the onset of chaos. We apply this wavelet based technique to analyze the fractal properties of DLA azimuthal Cantor sets defined by intersecting the inner frozen region of large mass off-lattice DLA clusters with a circle. This study clearly reveals the existence of an underlying multiplicative process that is likely to account for the Fibonacci structural ordering recently discovered in the apparently disordered arborescent DLA morphology. The statistical relevance of the golden mean arithmetic to the fractal hierarchy of the DLA azimuthal Cantor sets is demonstrated.

A Monte Carlo simulation for the catalytic oxidation of CO on DLA clusters

A Monte Carlo simulation for the heterogeneously catalyzed oxidation of CO on diffusion limited aggregation clusters is introduced. Kinetic phase transition points which appear in this reaction system are studied as a function of the system parameters. Both the O poisoning transition at y 1 and the CO poisoning transition at y 2 are found at very low values and they are second-order transitions. This holds even in the presence of CO diffusion. The results will be compared with simulations which were performed on regular and other fractal surfaces. The strong influence of the lattice structure is discussed in detail.