DLA Model Research Articles

Formation of silver dendrites under microwave irradiation

Well-defined silver dendrites were successfully prepared in DMF solution containing poly( N-vinly-2-pyrrolidone) (PVP) as the stabilizer under microwave irradiation. Morphology similar to those of the DLA model fractal dimensions around 1.7 has been observed in our systems. It was found that microwave power and the existence of PVP played important roles in the formation of well-defined silver dendrites. The products were characterized by X-ray diffraction (XRD) and transmission electron microscopy (TEM). The proposed formation mechanism of dendrites was also investigated.

DLA Model and Hitting Distribution of Random Walk

DLA Model and Hitting Distribution of Random Walk

Aggregation in the Plane and Loewner's Equation

We study an aggregation process which can be viewed as a deterministic analogue of the DLA model in the plane, or as a regularized version of the Hele-Shaw problem. The process is defined in terms of the Loewner differential equation. Using complex analytic methods, we establish a Kesten-type estimate for the growth of the cluster. We also indicate a real-variable approach based on a certain martingale structure in the phase space of the inverse Loewner chain.

Precipitation selectivity of boron compounds from slags

The effects of slag composition, additive agents and heat treatment on the crystal morphology, the precipitation behavior and the efficiency of extraction of boron (EEB) from slags were investigated. The EEB varied with the slag composition; the further the slag composition deviated from the line between the borate 2MgO·B 2O 3 and the silicate 2MgO·SiO 2 in the MgO–B 2O 3–SiO 2 system, the lower the EEB. The EEB varied with the temperature of heat treatment, the highest EEB appeared at 1100°C. During the solidification of molten slags the behavior of the boron compounds involved two steps: phase separation and crystal precipitation. The micrographs of the phase separation were described using the fractal dimension which was 1.87–1.92. The result of computer simulation indicated that the mechanism of the phase separation was Spinodal decomposition. The fractal dimension of the crystalline phase 2MgO·B 2O 3 whose micrographs showed fractal characteristics was 1.62. The result of computer simulation showed that the growth of the borate 2MgO·B 2O 3 was dependent on the DLA model. The additives TiO 2 and MO x decrease the activation energy of crystallization and make the boron compounds precipitate in the form of the crystalline phase more easily, thus the EEB was increased. The structure of slag was modified by addition of TiO 2. A lot of [B 2O 5] and [BO 3] clusters, which were favorable to make crystals of 2MgO·B 2O 3 and 3MgO·B 2O 3 precipitate, were formed in the slags.

99/00198 Study of the relationship between fractal dimension and viscosity ratio for viscous fingering with a modified DLA model

99/00198 Study of the relationship between fractal dimension and viscosity ratio for viscous fingering with a modified DLA model

Study of the relationship between fractal dimension and viscosity ratio for viscous fingering with a modified DLA model

The morphological evolution characters of viscous fingering, which occurred in two-dimensional Hele-Shaw cells with various viscosity ratio of the displaced-to-driving fluids from 1 to 10 000, were simulated by a modified DLA model on off-lattices. The model can reproduce experimental patterns conveniently and generate various finger growth aggregations from skeletal patterns to fleshy patterns. Both the shapes and the fractal dimension of fractal clusters obtained by controlling the parameters in the present model are in good agreement with experiments. The present work indicates that the effective fractal dimension of viscous fingering decreases and the corresponding morphologies of finger growth vary from fleshy patterns to skeletal patterns when viscosity ratio V R is increased in laboratory scale. Low fractal dimension corresponds to very ramified viscous fingering fractal structure and short breakthrough time. In small scale, the effective fractal dimension can be reasonably regarded as a useful parameter for the fluid displacement process. In the scale of laboratory experiments, the unfavorable viscous fingering can be controlled by reducing the viscosity ratio, thus, increasing the effective fractal dimension of viscous fingering.

Interface structure in colored DLA model

We propose a model for the simultaneous diffusion-limited growth of two clusters A and B, where the growth of one cluster screens the growth of the other one. We consider the possibility that the A and B clusters can penetrate into each other in course of their growth in different spatial dimensions and express the conjecture that the A-B boundary is flat in all dimensions. Using an electrostatic analogy, we compute some spatial characteristics of the clusters.

Percolation-like transition in DLA with two species

We introduce a DLA model with two species in order to simulate the growth of alternate clusters. We perform intensive numerical simulations in two dimensions in which the proportions of the two species are varied. A critical point, analogous to a percolation threshold, is found and a new class of critical exponents is obtained for this transition. Within numerical accuracy, the fractal dimension of the clusters is found to be the same as in the usual DLA model, independent of the species concentrations. Possible connections with the growth of two-dimensional ionic crystals are discussed.

Theory of the kinetics of nucleation in adsorbing layer: the approach based on the relaxation of order parameter field

We use a Ginzburg-Landau type equation for the order parameter field to study the growth kinetics of two-dimensional new phase nuclei, as well as the phenomenon of instability of the azimuthal form of growing nuclei. A connection between the developed approach and phenomenological Zeldovich-Volmer (ZV) theory is established. The onset of various types of conditions imposed in ZV theory on the nucleus boundary is considered. We obtain instability diagrams of the angular perturbations and show that the instability occurs when the main growth mechanism is a diffusive one. Also, we determine the parameters region where the criteria obtained in this paper are the same as the results of phenomenological ZV theory and the continuum DLA model. Additionally, we study the criteria for the growth of angular perturbations in the case of a nonquasistationary growth regime.

Modelling urban growth patterns

CITIES grow in a way that might be expected to resemble the growth of two-dimensional aggregates of particles, and this has led to recent attempts1a¤-3 to model urban growth using ideas from the statistical physics of clusters. In particular, the model of diffusion-limited aggregation4,5 (DLA) has been invoked to rationalize the apparently fractal nature of urban morphologies1. The DLA model predicts that there should exist only one large fractal cluster, which is almost perfectly screened from incoming a¤˜development unitsa¤™ (representing, for example, people, capital or resources), so that almost all of the cluster growth takes place at the tips of the clustera¤™s branches. Here we show that an alternative model, in which development units are correlated rather than being added to the cluster at random, is better able to reproduce the observed morphology of cities and the area distribution of sub-clusters (a¤˜towns') in an urban system, and can also describe urban growth dynamics. Our physical model, which corresponds to the correlated percolation model6a¤-8 in the presence of a density gradient9, is motivated by the fact that in urban areas development attracts further development. The model offers the possibility of predicting the global properties (such as scaling behaviour) of urban morphologies.

Enhanced Visualization of Acid Carbonate Rock Interaction

Visualisation of the reaction patterns between acid and carbonate rocks is the subject of this note.

The Diffusive Instability in Growth: DLA, Dendritic Growth, and Solidification

AbstractWe give an overview of the influence of the diffusive instability in forming various growth morphologies. We review dendritic growth of crystals, and discuss the DLA model as an approach to the disordered, fractal case. We consider various systems which are reasonably well described by DLA including island shapes in surface growth.

Real and simulated fractal aggregates

We have obtained interesting metallic fractal aggregates through an evaporation-condensation method. In order to interpret these results, computer simulation programs based on the DLA model have been developed, as well as methods to approximate the determination of the fractal dimension and another to compare two aggregates.

超微粒子凝集体の充填シミュレーション 充填構造に及ぼすシミュレーション条件の影響

The porous structure of ultra-fine powder beds, the porosity ratio of which is more than 90%, was constructed by the simulation of aggregate packing. The various aggregates of ultra-fine powders were obtained by the extended DLA model. The aggregating process was analyzed by Smoluchowski's equation, and the aggregate structure was discussed quantitatively with fractal dimensionality. The simulation had some adverse effects on the aggregation, when the length of the side of the simulated space was smaller than the critical value. The critical values of the length were 180 and 50 times the particle diameters, respectively for two and three dimensional cases. As the mean number of particles in a single aggregate increased, the width of the packing simulation space must be increased to reproduce the actual shape of pore size distribution in the packed bed. If the mean number of particles in a single aggregate is more than 100, for example, the width required is more than 200 times the particle diameter both for two and three dimensional cases. The effect of adhesion between aggregates on packed bed structure decreased as the size of aggregate increased, because the bridging among several aggregates created large voids.

The growth of dendrites of fractal pattern on a conducting polymer

Dendrite, of fractal pattern, has been observed on the surface of the conducting polymer polypyrrole, after an 'undoping' process. Metal ions have been deposited on the surface of the conducting polymer by applying a voltage for undoping. The fractal dimension has been calculated from the self-correlation function. The growth mechanism of the dendrite has been explained on the basis of the DLA model and the undoping process.

Diffusion-limited growth in bacterial colony formation

Colonies of bacterial species called Bacillus subtilis have been found to grow two-dimensionally and self-similarly on agar plates through diffusion-limited processes in a nutrient concentration field. We obtained a fractal dimension of the colony patterns of D=1.73±0.02, very close to that of the two-dimensional DLA model, and confirmed the existence of the screening effect of protruding main branches against inner ones in a colony, the repulsion between two neighboring colonies and the tendency to grow toward nutrient. These effects are all characteristic of the pattern formation in a Laplacian field. This finding implies the importance of physical properties of the environment for the morphology of bacterial colonies in general.

Urban Growth and Form: Scaling, Fractal Geometry, and Diffusion-Limited Aggregation

In this paper, we propose a model of growth and form in which the processes of growth are intimately linked to the resulting geometry of the system. The model, first developed by Witten and Sander and referred to as the diffusion-limited aggregation or DLA model, generates highly ramified tree-like clusters of particles, or populations, with evident self-similarity about a fixed point. The extent to which such clusters fill space is measured by their fractal dimension which is estimated from scaling relationships linking population and density to distances within the cluster. We suggest that this model provides a suitable baseline for the development of models of urban structure and density which manifest similar scaling properties. A typical DLA simulation is presented and a variety of measures of its structure and dynamics are developed. These same measures are then applied to the urban growth and form of Taunton, a small market town in South West England, and important similarities and differences with the DLA simulation are discussed. We suggest there is much potential in extending analogies between DLA and urban form, and we also suggest future research directions involving variants of DLA and better measures of urban density.

Hierarchy of effective sizes of DLA clusters

The authors discuss the possibility of having an infinite hierarchy of length scales characterising the moments of the Laplacian field outside DLA clusters, in the entire space. This hierarchy constitutes a multifractality for DLA which is different from the usual one associated with the near field of these clusters. The radius of gyration and the hydrodynamic radius are both members of this new family of length scales. They present numerical simulations on the usual DLA model, and on the fracture growth model (i.e. 'elastic' DLA), testing this hypothesis.

Fractal Aggregation Near the Threshold11Projects supported by the National Natural Science Foundation of China.

In this paper, we present two fractal aggregation models, line pattern seed model (model 1) and point pattern seed model (model 2), which are particle-cluster models. Using the current models, we investigate the critical transition in fractal aggregation processes in two dimensions, and suggest a method for finding the critical transition point. The computer simulation results show that the critical concentration is for model 1 and for model 2, critical fractal dimension. for model 1 and for model 2, which are in good agreement with those of DLA model and experimental data. The results also show that the critical transition point in two dimensions seems to be independent of the size of lattices and the initial seed patterns. The results seem to belong to the same universality class.

Multiparticle diffusion-limited aggregation with strip geometry

Two finite density versions of the multiparticle DLA model have been investigated on strips of width L lattice units with periodic boundary conditions. In model I single particles are added to the deposit at the position in which they first attempt to step onto the deposit and in model II particles directly and indirectly connected to the deposit (via nearest neighbor occupancy) are added when they first contact the growing deposit. For model I the growth of the deposit mass crosses over from t 1 2 dependence in the low density/short time limit to M ∼ t in the high density/long time limit. For model II a similar crossover is found but as the density approaches the percolation threshold the deposit mass is found to grow according to M ∼ t δ with δ having a value of about 4/3. These results help to resolve some apparent discrepancies between the results of earlier work on closely related models by Voss and by Meakin and Deutch.