Diffusion-limited Aggregation

The ow and deposition of polydisperse granular materials is simulated through the magnetic di usion limited aggregation (MDLA) model. The random walk undergone by an entity in the MDLA model is modiÿed such that the trajectories are ballistic in nature, leading to a magnetically controlled ballistic deposition (MBD) model. This allows to obtain important ingredients about a di cult problem that of the nonequilibrium segregation of polydisperse sandpiles and heterogeneous adsorption of a binary distribution of particles which can interact with each other and with an external ÿeld. Our detailed results from many simulations of MBD clusters on a two-dimensional triangular lattice above a at surface in a vertical ÿnite size box for binary systems indicates intriguing variations of the density, "magnetization", types of clusters and fractal dimensions. We derive the ÿeld and grain interaction-dependent susceptibility and compressibility. We deduce a completely new phase diagram for binary granular piles and discuss its complexity inherent to di erent grain competition and cluster growth probabilities.

By combining static and dynamic structure factor measurements under microgravity conditions, we obtain for the first time direct insight into the internal structure of colloidal aggregates formed over a wide range of particle attractions under ideal diffusion-limited conditions. By means of near-field scattering we measure the time-dependent density-density correlation function as the aggregation process evolves, and we determine the ratio of the hydrodynamic and gyration radius to elucidate the aggregate's internal structure as a function of its fractal dimension. Surprisingly, we find that despite the large variation of particle interactions, the mass is always evenly distributed in all objects with fractal dimension ranging from 2.55 for shallow potentials to 1.78 for deep ones.

An analysis of the Infrared Space Observatory (ISO) infrared spectra of comet C/1995 O1 Hale-Bopp has been conducted. The particles in the coma are assumed to be irregular aggregates that are built by a diffusionlimited aggregation (DLA) procedure and are composed of olivine, both amorphous and crystalline, and glassy carbon. To simulate the morphological structure of the interstellar dust, the silicate component is placed in the inner layers of the aggregate, while the carbon dipoles occupy the outermost layers. The particle emissivities are calculated using the discrete dipole approximation (DDA) method of Draine & Flatau. The dust size distribution functions at different heliocentric distances (2.8, 2.9, and 3.9 AU) have been determined as the solution to three overdetermined systems of equations for the infrared flux at each wavelength in the 8-40 lm region. These size distributions can be well fitted by a power law, n r ð Þ / r À , having power indexes that decrease with heliocentric distance as ¼ À3:6 AE 0:3 at 2.8 AU and À3:3 AE 0:3 at 3.9 AU.

Motivated by a wide-range of applications from ground water remediation to carbon dioxide sequestration and by di culties in reconciling experiments with previous modeling, we have developed a pore-level model of two-phase ow in porous media. We have attempted to make our model as physical and as reliable as possible, incorporating both capillary e ects and viscous e ects. After a detailed discussion of the model, we validate it in the very di erent limits of zero capillary number and zero-viscosity ratio. Invasion percolation (IP) models the ow in the limit of zero capillary number; results from our model show detailed agreement with results from IP, for small capillary numbers. Di usion limited aggregation (DLA) models the ow in the limit of zero-viscosity ratio; ow patterns from our model have the same fractal dimension as patterns from DLA for small viscosity ratios.

We present a model of a neural network that is based on the diffusion-limited-aggregation ͑DLA͒ structure from fractal physics. A single neuron is one DLA cluster, while a large number of clusters, in an interconnected fashion, make up the neural network. Using simulation techniques, a signal is randomly generated and traced through its transmission inside the neuron and from neuron to neuron through the synapses. The activity of the entire neural network is monitored as a function of time. The characteristics included in the model contain, among others, the threshold for firing, the excitatory or inhibitory character of the synapse, the synaptic delay, and the refractory period. The system activity results in ''noisy'' time series that exhibit an oscillatory character. Standard power spectra are evaluated and fractal analyses performed, showing that the system is not chaotic, but the varying parameters can be associated with specific values of fractal dimensions. It is found that the network activity is not linear with the system parameters, e.g., with the numbers of active synapses. The details of this behavior may have interesting repercussions from the neurological point of view.

Percolation in nanoporous gold can be achieved with as little as 8% by volume of gold. Samples of nanoporous gold of various morphologies are analysed with a combination of electrical and optical data. Growing thin films and complex multiply-connected 3-dimensional networks both display non-universal character. Growing films have 2-dimensional morphology but a 3dimensional percolation threshold and non-universal critical coefficients, yet similar silver films percolate as expected with universal coefficients. Growing gold however regresses to 2dimensional resistive behaviour between 65% to 100% gold, and this regime lies along a single power law curve shared by the hyper-dimensional networks of gold, suggesting underlying symmetry governed by diffusion limited aggregation. Models of data imply either hyperdimensionality or major internal property changes as density shifts. The distinctive flat spectral signature found near the percolation threshold is common to all highly porous samples and is explained quantitatively in terms of effective plasmonic response. Parameters from fits of effective medium models to optical and resistivity data are in close agreement, especially at the highest porosities. They imply an effective dimension which increases continuously as porosity grows via the increased branching needed for structural integrity.

We study the possibility of using numerical modelling in the process of design a membrane of prescribed morphology and transport properties. We started from a real example of the cross-section of alginate membrane cross-linked by glutaraldehyde containing 25 wt% magnetite particles and searched for a numerical model that will resemble its morphological properties like amount of polymer matrix, sizes of polymer matrix domains, fractal dimension and others. Two different methods of generating models of such were proposed. After choosing the best models based on its morphological similarities to the real membrane, we study their transport properties in terms of Brownian diffusion. We showed that there is a good agreement of diffusion type and diffusion constant between the models and the real membrane.

We prove that the harmonic measure is stationary, unique and invariant on the interface of DLA growing on a cylinder surface. We provide a detailed theoretical analysis puzzling together multiscaling, multifractality and conformal invariance, supported by extensive numerical simulations of clusters built using conformal mappings and on lattice. The growth properties of the active and frozen zones are clearly elucidated. We show that the unique scaling exponent characterizing the stationary growth is the DLA fractal dimension.

We extend the conformal mapping approach elaborated for the radial Diffusion Limited Aggregation model (DLA) to the cylindrical geometry. We introduce in particular a complex function which allows to grow a cylindrical cluster using as intermediate step a radial aggregate. The grown aggregate exhibits the same selfaffine features of the original cylindrical DLA. The specific choice of the transformation allows us to study the relationship between the radial and the cylindrical geometry. In particular the cylindrical aggregate can be seen as a radial aggregate with particles of size increasing with the radius. On the other hand the radial aggregate can be seen as a cylindrical aggregate with particles of size decreasing with the height. This framework, which shifts the point of view from the geometry to the size of the particles, can open the way to more quantitative studies on the relationship between radial and cylindrical DLA.

Analogies between critical phenomena and the continuous spectrum of scaling exponents associated with fractal measures are pointed out. The analogies are based first on the Hausdorff-Bernstein reconstruction theorem, which states that the positive integer moments suf5ce to characterize a probability distribution function with finite support, and second on the joint probability distribution for the positive integer moments. This joint probability distribution, which can be considered as a fixed point, is universal and exhibits both gap scaling and the infinite set of exponents. Monte Carlo simulations of the electrical properties of percolation clusters on the square and triangular lattices support this general result. Extensions to other fields where infinite sets of exponents have arisen, such as diAusion-limited aggregation and localization, should be straightforward.

We performed extensive numerical simulation of diffusion-limited aggregation in two dimensional channel geometry. Contrary to earlier claims, the measured fractal dimension D = 1.712 ± 0.002 and its leading correction to scaling are the same as in the radial case. The average cluster, defined as the average conformal map, is similar but not identical to Saffman-Taylor fingers.

The creation of fractal clusters by diffusion limited aggregation (DLA) is studied by using iterated stochastic conformal maps following the method proposed recently by Hastings and Levitov. The object of interest is the function Φ (n) which conformally maps the exterior of the unit circle to the exterior of an n-particle DLA. The map Φ (n) is obtained from n stochastic iterations of a function φ that maps the unit circle to the unit circle with a bump. The scaling properties usually studied in the literature on DLA appear in a new light using this language. The dimension of the cluster is determined by the linear coefficient in the Laurent expansion of Φ (n) , which asymptotically becomes a deterministic function of n. We find new relationships between the generalized dimensions of the harmonic measure and the scaling behavior of the Laurent coefficients.

In analogy to recent results on nonuniversal roughening in surface growth [Lam and Sander, Phys. Rev. Lett. 69, 3338 (1992)],we propose a variant of difFusion-limited aggregation (DLA) in which the radii of the particles are chosen from a power-law distribution. For very broad distributions, the huge particles dominate and the fractal dimension is calculated exactly using a scaling theory. For narrower distributions, it crosses back to DLA. We simulated 1200 clusters containing up to 200000 particles. The &actal dimensions obtained are in reasonable agreement with our theory. This variant of DLA might have relevance to the cluster-cluster aggregation model. PACS number(s): 64.60.Ak Diffusion-limited aggregation (DLA) [1] is an important prototype of fractal growth [2]. In two dimensions,

It is shown that screening greatly diversifies the type of patterns that can grow in an electrostatic field. Screening introduces a new length scale and a nontrivial dependence on the boundary conditions. Growing patterns can either have a fractal character (dilTusion-limited-aggregation-like) at scales shorter than the screening length, be similar to the Eden model, or even be dense. A transition from dense to multibranched growth occurs at a point which depends on the potentials at the boundaries, the distance between them, and the screening length.

This paper deals with the synchronous implementation of situated Multi-Agent Systems (MAS) in order to have no execution bias and to allow their programming on massively parallel computing devices. For this purpose we investigate the translation of discrete MAS into Cellular Automata (CA). Contrarily to the sequential scheduling generally used in MAS simulations, CA is a model for massively parallel computing where the updating of the components is synchronous. However, CA expressivity is limited and not adapted to build models where independent entities may move and act on neighbor cells. After illustrating these issues on a simple example, we propose a generic method to translate discrete MAS into CA, called transactional CA. Our approach consists in translating MAS specified with the influence-reaction model into a transactional CA.

Binary diffusion-limited cluster-cluster aggregation processes are studied as a function of the relative concentration of the two species. Both, short and long time behaviors are investigated by means of threedimensional off-lattice Brownian Dynamics simulations. At short aggregation times, the validity of the Hogg-Healy-Fuerstenau approximation is shown. At long times, a single large cluster containing all initial particles is found to be formed when the relative concentration of the minority particles lies above a critical value. Below that value, stable aggregates remain in the system. These stable aggregates are composed by a few minority particles that are highly covered by majority ones. Our off-lattice simulations reveal a value of approximately 0.15 for the critical relative concentration. A qualitative explanation scheme for the formation and growth of the stable aggregates is developed. The simulations also explain the phenomenon of monomer discrimination that was observed recently in single cluster light scattering experiments.

A sticking probability model based on the average cluster lifetime is employed for deducing a kernel capable to describe the kinetics of computer simulated irreversible aggregation processes in two dimensions. The deduced kernel describes not only the time evolution of the cluster size distribution for diffusion limited aggregation (DLCA) and reaction limited aggregation (RLCA) but also for the entire transition region between both regimes. The model predicts a crossover to diffusion limited cluster aggregation for all sticking probabilities at long aggregation times. The time needed for reaching the DLCA limit increases for decreasing sticking probability.

... Since this value is smaller than the Bjerrum length, the electrical double layer may be considered to be completely suppressed. ... 1. Bandini P, Prestidge CA, Ralston J (2001) Miner Eng 14:487 2. Luna-Xavier JL, Guyot A, Bourgeat-Lami E (2002) J Colloid Interface Sci 250:82 3 ...

A sticking probability model for irreversible aggregation processes is developed. It allows a kernel capable of describing not only the diffusion-limited and reaction-limited aggregation regimes but also the whole transition region to be deduced. According to the definition given by van Dongen and Ernst, the kernel establishes λ = 0 for the entire range of sticking probabilities. The model presented

Smoluchowski’s equation is widely applied to describe the time evolution of the cluster-size distribution during aggregation processes. Analytical solutions for this equation, however, are known only for a very limited number of kernels. Therefore, numerical methods have to be used for describing the time evolution of the cluster-size distribution. In this work, we present a novel self-consistent method for solving Smoluchowski’s equation for any homogeneous kernel. The method considers dynamic scaling to be valid but does not need to assume a given form for the scaling distribution Φ(x). Moreover, the scaling distribution Φ(x) is obtained as a natural result of the algorithm. Due to the implementation of dynamic scaling concepts, the algorithm converges almost immediately with a minimal calculation effort. Comparing calculated size distributions with the corresponding analytical solutions shows the validity of the method. The method is then used to fit experimental data for diffusi...

In the past Monte-Carlo simulations have been used to study the influence of morphological properties of fractal surfaces on the selectivity between competing three-step catalytic reactions. These simulations were conducted using abstract two-dimensional diffusion-limited aggregate models. In an attempt to identify the appropriate properties of a real material that should be represented in a model this previous work has been extended to consider three-dimensional models (cluster-cluster aggregates and the Menger sponge), in order to correlate selectivity with model characteristics which may be related to input from experimental characterisation data, such as that from magnetic resonance imaging. The simulations on three-dimensional models have shown that both the porosity and the mass fractal dimension of the models are important to accurately represent a real porous material; a finding not explicit in previous work in two dimensions.

The kinetics of particle growth in lead zirconate titanate sol gel precursor solutions has been investigated. It was found that chemical reaction limited aggregation was responsible for most of the sol aging, followed by diffusion limited aggregation in the later stage. At the beginning, particles grow by reaction between initial polymers and particles. As the particle number-density increases, a particle-particle aggregation characterised by an exponential growth law becomes the predominant mechanism. When particles grow larger, towards gel formation, the aggregation changes to diffusion limited. Mathematical models derived according to the above mechanisms agreed well with experimentally measured particle growth profiles.

Using Monte Carlo simulations we have modeled the diffusion of a single particle in twoand three-dimensional lattices with different crowding conditions given by distinct obstacles size and density. All registered data emphasize that diffusion process is anomalous and diffusing particle describes fractal trajectories. We have introduced a new time-scale fractal dimension, dm, which is related to the anomalous diffusion exponent, α. This allows us to relate the well-known length-scale fractal dimension of the random walk, dw, to the new one introduced here as a time-scale fractal dimension. Moreover, the 3D simulations consider similar conditions to those used in our previous FRAP experiments in order to reveal the relationship between the length and time-scale fractal dimensions.

We have studied the aggregation of the porphyrin trans-bis(N-methylpyridinium-4-yl)-diphenylporphyrine (t−H 2Pagg) in aqueous solutions by means of light scattering (elastic and quasielastic) and UV-visible absorption measurements. Elastic and dynamic light scattering indicate that the aggregation produces large monodisperse clusters having a fractal structure driven by di usion limited aggregation (DLA). UV-visible absorption measurements have been applied to study the corresponding kinetics, obtaining the time dependence of the monomer concentration and of the mean cluster size. The obtained ÿndings are in agreement with the picture proposed by current theoretical models and molecular dynamics (MD) simulations on DLA.

We study invasion percolation in the presence of viscous forces, as a model of the drainage of a wetting fluid from a porous medium. Using concepts from gradient percolation, we consider two different cases, depending on the magnitude of the mobility ratio M. When M is sufficiently small, the displacement can be modeled by a form of gradient percolation in a stabilizing gradient, involving a particular percolation probability profile. We develop the scaling of the front width and the saturation profile, in terms of the capillary number. In the opposite case, the displacement is described by gradient percolation in a destabilizing gradient and leads to capillary-viscous fingering. This regime is identified in the context of viscous displacements and in general differs from diffusion-limited aggregation, which also describes displacements at large M. Constraints for the validity of the two regimes are developed. Limited experimental and numerical results support the theory of stabilized displacement. The effect of heterogeneity is also discussed. ͓S1063-651X͑97͒08812-0͔

We study the morphology of the chain-like aggregates formed when a external constant and uniaxial magnetic field is applied to a magneto-rheological (MR) fluid. In order to characterize the conformation of the aggregates, we study the evolution of various fractal dimensions during aggregation and disaggregation processes (i.e., when the applied field is switched on and off), using video-microscopy and image analysis. Experiments have been performed by varying the values of two external parameters: the magnetic field amplitude and particle concentration. We found that the box-counting dimension, related with how the aggregates occupy the surrounding space, depends on the ratio R 1 /R 0 . During the first stage of the disaggregation process, when the particles are moving by Brownian motion inside the aggregate, Family-Vicsek scaling function is verified.

The understanding of architecture as an ecology of interactive systems moves past limitations and restricted tendencies toward spatial environments that are adaptive, perceptual, and behavioral. In this framework, the environment seeks to build interaction scenarios to activate relationships between components. In this case, architecture moves away from well-known models that always lead to disciplined responses and toward understanding adaptive ecologies that include active particles for communication and exploration. The current research investigated and discussed the design of a system that can replace the current methods of planning the construction of infrastructure units in the future. The result is the simulation of a cellular self-assembly system that can produce and rebuild its structure when needed. Therefore, there is a revolution in construction and a complete revision of architectural standards and construction planning, disremembering demolitions and laborious construction costs. This study aims to explore the function of quorum sensing, a mechanism of intra-species communication that serves as the central regulatory system in the formation and concurrent response to environmental changes. The algorithmic design features of the system's constituent units were also examined. During the assembly process, extracellular matrices act as the milieu in which individual cells interact with one another and the desired environment as constructed building blocks. This allows for parallel assembly and error correction at a general level, both automatically, and has significant implications for the field. The proposed model uses a two-level control system (micro-macro) to ensure that the interactions among the components, on the one hand, and between the components and their environment, on the other hand, are in line with the objectives of the plan and the balance of the whole system. This two-level approach primarily follows a bottom-up process at the scale of the interaction of its constituent particles and subsequently utilizes a top-down process during the overall regulation of the system through the extracellular matrix. Ultimately, the simulation results obtained using Grasshopper3d software were shown in three scales: small (chair), medium (shelter), and large. This approach has been demonstrated to be effective in previous research and offers a promising framework for further investigation in this field. This research can take essential steps in developing simulator machines to build a construction self-assembly system based on sequential configurations and numbers.
Keywords: Adaptive Ecologies  Self-assembly  Living Architecture  Diffusion Limited Aggregation  Reversible Fabrication

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Detonation nanodiamonds (DND) form fractal-like aggregates composed of polydisperse DND particles. We present a novel methodology for the visualisation and characterisation of fractal clusters of DND from one-dimensional small-angle X-ray scattering (SAXS) intensity. The fractal nature and polydispersity of DND are modelled by combining a diffusion-limited aggregation (DLA) process implemented in Monte Carlo simulations with the distribution of DND sizes measured from high-resolution transmission electron microscopy. The radius of gyration (42e44 nm), aggregation number (850e1150), and the maximum dimension (226e242 nm) of DND fractal clusters were obtained from the fitting of the synchrotron-based SAXS data (q~0.11e4.75 1/nm) measured for two samples of commercialized DND powders by the developed theoretical model.

Mass-transfer driven growth of a single gas cluster in a porous medium under the application of a supersaturation in the far field is examined. We discuss the growth pattern and its growth rate. Contrary to compact (spherical) growth in the bulk, growth patterns in porous media are disordered and vary from percolation to di6'usion-limited aggregation (DLA) as the cluster size increases. At conditions of low supersaturation, scaling laws for the boundaries that delineate these patterns and of the corresponding growth rates are derived. In three dimensions {3D), it is found that the cluster grows as 1/(D-1) R~-t,wh ere D& is the pattern fractal dimension (=2.50 for percolation or DLA). A similar result involving logarithmic corrections is found for 2D. These results generalize the classical scaling R~-I, " to fractal clusters.

Compact layers of uracil films grown on the mercury electrode-aqueous solution interface were studted by means of time-resolved FFT impedance spectroscopy. The ftlms are charactertzed by the fractal dimension which evolves with time. The results are discussed m terms of the cluster-cluster aggregation model and the hole formation mechanism proceeding in the compact layer.

ABSTRACTPreliminary results of an aggregation model that takes into account both the Brownian motion as well as the gravitational drift experienced by the colloidal particles and clusters is presented. It is shown that for high strengths of the drift the system crosses over to a regime different from diffusion-limited colloid aggregation, for which there is an increase of the fractal dimension, a speeding up of the aggregation rate and a widening of the cluster size distribution, becoming algebraically decaying with an exponent τ.

Large-cluster growth by diffusive processes has been made practical by a diffusion-enhancement technique introduced by Meakin [Phys. Rev. A 27, 604 (1985); 27, 1495 (1985)]. This technique has been questioned on the grounds that a variable diffusivity will change the flux of walkers at the surface of the cluster given any particular boundary conditions. We demonstrate that, carried out properly, diffusion enhancement does not modify the flux. Thus, enhanced diffusion yields the same growth equations that nonenhanced diffusion does. Diffusion-limited growth processes occur in many dis

Using stochastic conformal mappings, we study the effects of anisotropic perturbations on diffusion-limited aggregation (DLA) in two dimensions. The harmonic measure of the growth probability for DLA can be conformally mapped onto a constant measure on a unit circle. Here we map m preferred directions for growth to a distribution on the unit circle, which is a periodic function with

The potential of the original m-spoke model, conceived by Family and Hentschel (FH) [Faraday Discuss. Chem. Soc. 83, 139 (1987)],does not satisfy the correct boundary conditions for diffusionlimited aggregation (DLA) or dielectric breakdown. Moreover, while the suggested scaling forms hold, if the growth probability exponent g of the dielectric breakdown model is larger than 2, they are not generally valid for g(2. The correct potential of an m-spoke structure is given, and the scaling behavior of this new model is calculated. Since the scaling dependence on the cluster size X is determined by the local-field behavior, the exponent vll found by FH is recovered with the new potential. For a careful evaluation of the m dependence, however, a nonlocal quantity must be considered, leading to novel results. We conclude that only if the fractal dimension is m dependent can the asymptotic shape of DLA clusters be m-spoke-like. Large-scale clusters obtained by simulation of diffusion-limited aggregation (DLA) on a lattice rellect, in their global structure, the anisotropy of the lattice. '

The model of cluster growth by diffusion-limited aggregation of clusters is studied in two dimensions and the cluster size distribution n, (t) is determined as a function of the cluster size s and the time t Ad.ynamic scaling function of the form n, (t)t "s 'f (s/t*) is proposed and is shown to lead to the scaling relation w = (2r)z among the critical exponents. The simulation results support this scaling law.

An off-lattice version of the diffusion-limited aggregation model is used to simulate pattern formation in viscous flows. The patterns generated are very similar to those obtained in the experiments on fingering in the radial Hele Shaw cell. We find a crossover from compact to fractal structure as the length scale is increased. The effective fmctal dimension of dusters obtained for low values of a parameter corresponding to the surface tension is about 1.74, which is close to that of the diffusion-limited aggregates.

Applying finite-size scaling analysis to diffusion-limited aggregation (DLA) clusters grown in finite width strips on a square lattice we find that Iii and Ii, the cluster lengths along and perpendicular, respectively, to the direction of growth, diverge as N " and N ', respectively, where N is the number of particles in the cluster. %e find numerically that vii-', and v&-, '. From the finite-size scaling analysis we derive the expression D 1+(Ivt)/vi for the fractal dimension D

Irreversible fractal-growth models like diffusion-limited aggregation (DLA) and the dielectric breakdown model (DBM) have confronted us with theoretical problems of a new type for which standard concepts like field theory and renormalization group do not seem to be suitable. The fixed-scale transformation (FST) is a theoretical scheme of a novel type that can deal with such problems in a reasonably systematic way. The main idea is to focus on the irreversible dynamics at a given scale and to compute accurately the nearest-neighbor correlations at this scale by suitable lattice path integrals. The next basic step is to identify the scale-invariant dynamics that refers to coarse-grained variables of arbitrary scale. The use of scaleinvariant growth rules allows us to generalize these correlations' to coarse-grained cells of any size and therefore to compute the fractal dimension. The basic point is to split the long-time limit (t-+ 00) for the dynamical process at a given scale that produces the asymptotically frozen structure, from the large-scale limit (r-+ 00) which defines the scale-invariant dynamics. In addition, by working at a fixed scale with respect to dynamical evolution, it is possible to include the fluctuations of boundary conditions and to reach a remarkable level of accuracy for a real-space method. This new framework is able to explain the self-orgauized critical nature and the origin of fractal structures in irreversible-fractal-growth models. It also provides a rather systematic procedure for the analytical calculation of the fractal dimension and other critical exponents. The FST method can be naturally extended to a variety of equilibrium and nonequilibrium models that generate fractal structures. CONTENTS I. Introduction: 'Physics of Fractals II. Properties of Fractal-Growth Models A.' The basic models: Diffusion-limited aggregation and dielectric breakdown model B. The fractal dimension C. Essential features of the problem D. Why usual methods are problematic III. The Fixed-Scale Transformation Strategy and its Basic Ideas IV. Basic Concepts A. Pair-correlation configurations and the fractal dimension 1. Characterization of fractal structures with a fine-graining procedure 2. Void distribution B. The fixed-scale transformation C. Fixed-scale transformation application to the Eden model D. Fluctuations of boundary conditions E. Empty configurations: The extended fixed-scale transformation F. Asymptotic scale-invariant dynamics V. Applications of Fixed-Scale Transformation to Hamiltonian Equilibrium Problems A. Critical fluctuations and fractal growth B. The percolating cluster 1. Fixed-scale transformation application to the percolation problem 2. Alternative connectivity conditions 3. Percolation on the triangular lattice

The size-specific coagulation frequencies of fractal aggregates formed by the simulation of di8'usionlimited cluster-cluster aggregation in two and three dimensions are determined from dynamic-scaling spectra by an inverse-problem approach. The four cases considered are twoand three-dimensional aggregation in which the cluster difFusivity is independent and dependent upon the mass of the cluster. The frequencies derived from this approach describe the transient simulation results very well, the predictions being better than those with the Smoluchowski Brownian-coagulation frequency [Phys. Z. 17, 557 (1916)]modified by scaling arguments in three dimensions. Furthermore, frequencies are obtained for two-dimensional aggregation for which physical models have been evasive. The validity of the mean-field equation with a time-independent homogeneous frequency is examined. It is found that the mean-Geld approximation is valid in the range of simulation results but progressively deteriorates with time. The power of the inverse-problem approach here lies in its yielding size-specific aggregation rates of particles whether or not physical models are available.

A novel method for obtaining 2-d diffusion limited aggregates of copperfrom copper sulphate solution using a simple displacement reaction is reported. No external field is applied for accelerating the aggregation process. The aggregates evince scale invariance over a range ofsolution concentrations.

A theoretical mode1 of the electronic current instability observed experimentally by Bredikhin et al. in the cell Ag') /RbA&I,/ C'+' at various constant voltages is proposed. The model is based on the concept of fractal growth of silver dendritic clusters from the Ag'-'/RbAgJ, interface. It is shown that some of the clusters make connections between opposite electrodes at sufficiently large voltages, which leads to stochastic current oscillations. We show that the number of connecting clusters increases as the voltage is raised and therefore current oscillations become more frequent. All experimental dependences on current instability are well described within the framework of a mean-field approach to the theory of reversible diffusion-limited aggregation without branching.

Aggregation kinetics and aggregate structure are studied for chemical interactions, aggregate hydrodynamic behavior, monodisperse polystyrene latex particles of diameter 60 and 140 and restructuring, the picture of aggregation kinetics and nm. The experimental part consists of measurements over a rather structure formation is far from unified and complete. broad range of electrolyte concentrations (0.1 to 0.8 M NaCl) and In perikinetic aggregation, which is the subject of the for particle volume fractions 10 06 to 10 05 , using dynamic and present study, Brownian motion is the driving mechanism, static light scattering techniques. The aggregation kinetics data as opposed to orthokinetic aggregation by which particles obtained can be fitted into a single curve with a proper scaling of are brought together by fluid shear. The currently accepted time and size for both particle sizes. For relatively long times, we framework, against which theories and experiments are observe power-law type kinetics, characteristic of diffusion limited aggregation (DLA). Fractal dimensions in all cases are close to tested, suggests that there exist two regimes of perikinetic 1.7 (as expected for DLA). The light scattering behavior of aggreaggregation (6): diffusion-limited aggregation (DLA), in gates of arbitrary size is also studied numerically by employing which the rate-limiting step is particle diffusion, and reaccomputer simulations, and existing models (Lin et al., 1990. J. tion-limited aggregation (RLA), in which the collision prob-Colloid Interface Sci. 137, 263) are tested and improved for certain ability between particles is small and becomes the controlconditions (large angles and/or large particles). The shape of the ling step. The main differences refer to the structure, size experimental kinetics curves is successfully described with numeridistribution and growth kinetics; RLA aggregates tend to be cal simulations, further suggesting that (under the conditions more compact, having a fractal dimension of Ç2.1 as optested) DLA aggregation prevails for short times as well. ᭧ 1997 posed to Ç1.8 for DLA (6-9).

Introduction Urban growth modeling has evolved over recent years to capture increasingly well the details of urban morphology and structure on a qualitative as well as a quantitative level. In this paper we are concerned mainly with demonstrating the ability of the proposed framework to produce realistic urban growth patterns. Urban growth is a complex dynamical process in which many of the patterns which are observed at a macroscopic scale emerge as a result of microscopic dynamics. It is also often macroscopic properties that can be directly observed and measured in real systems. Therefore, comparative analysis of patterns observed in simulation and reality can be helpful for validating a model and thereby gaining knowledge about microscopic properties. Fractal analysis has been used extensively in the analysis of urban growth modeling and can capture morphological properties that elude other means of measurement. In addition to fractal dimension analysis we also introduce complementary methods and concepts for spatial analysis that are borrowed from statistical physics. Broadly speaking, urban modeling is directed at two objectives. On the one hand, modeling is aimed at providing a deeper theoretical understanding of urban dynamics.

General and mathematically transparent models of urban growth have so far suffered from a lack in microscopic realism. Physical models that have been used for this purpose, i.e. Diffusionlimited Aggregation (DLA), Dielectric Breakdown Models (DBM) and Correlated Percolation all have microscopic dynamics for which analogies with urban growth appear stretched. Based on a Markov Random Field (MRF) formulation we have developed a model that is capable of reproducing a variety of important characteristic urban morphologies and that has realistic microscopic dynamics. The results presented in this paper are particularly important in relation to "urban sprawl", an important aspect of which is aggressively spreading low-density land uses. This type of growth is increasingly causing environmental, social and economical problems around the world. The micro dynamics of our model, or its "first principles", can be mapped to human decisions and motivations and thus potentially also to policies and regulations. We measure statistical properties of macrostates generated by the urban growth mechanism that we propose, and we compare these to empirical measurements as well as to results from other models. To showcase the open-ended-ness of the model and to thereby relate our work to applied urban planning we have also included a simulated city consisting of a large number of land use classes in which also topographical data have been used.