Critical properties of cluster size distribution in an asymmetric diffusion-aggregation model (original) (raw)
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Journal of Physics A: Mathematical and Theoretical, 2020
We consider the stochastic dynamics of a system of diffusing clusters of particles on a finite periodic chain. A given cluster of particles can diffuse to the right or left as a whole and merge with other clusters; this process continues until all the clusters coalesce. We examine the distribution of the cluster numbers evolving in time, by means of a general time-dependent master equation based on a Smoluchowski equation for local coagulation and diffusion processes. Further, the limit distribution of the coalescence times is evaluated when only one cluster survives.
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 three-dimensional 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.
Diffusion, fragmentation and merging: Rate equations, distributions and critical points
Physica D: Nonlinear Phenomena, 2006
The fundamental processes of diffusion, fragmentation and merging are very common in many physical systems. We study situations where either two or all three of these processes are present in the dynamical evolution of the system. Specifically, we formulate rate equations in terms of the distribution N (x, t) of fragments of linear size x at time t which include a combination of diffusive growth, size fragmentation and fragment coagulation. Our goal is to obtain analytical solutions for N (x, t) in varying situations and in specific limits of x and t. (J. Ferkinghoff-Borg), mhjensen@nbi.dk (M.H. Jensen), mathies@nbi.dk (J. Mathiesen), polesen@nbi.dk (P. Olesen).
Diffusion Limited Aggregation: A Paradigm of Disorderly Cluster Growth
The purpose of this talk is to present a brief overview of our group's recent research into dynamic mechanisms of disorderly growth, an exciting new branch of condensed matter physics in which the methods and concepts of modern statistical mechanics are proving to be useful. Our strategy has been to focus on attempting to understand a single model system -diffusion limited aggregation (DLA). This philosophy was the guiding principle for years of research in phase transitions and critical phenomena. For example, by focusing on the Ising model, steady progress was made over a period of six decades and eventually led to understanding a wide range of critical point phenomena, since even systems for which the Ising model was not appropriate turned out to be described by variants of the Ising model (such as the XY and Heisenberg models). So also, we are optimistic that whatever we may learn in trying to "understand" DLA will lead to generic information helpful in understanding general aspects of dynamic mechanisms underlying disorderly growth.
Probability Theory and Related Fields, 1993
The purpose of this paper is to explore the connection between multiple space-time scale behaviour for block averages and phase transitions, respectively formation of clusters, in infinite systems with locally interacting components. The essential object is the associated Markov chain which describes the joint distribution of the block averages at different time scales. A fixed-point and stability property of a particular dynamical system under a renormalisation procedure is used to explain this pattern of cluster formation and the fact that the longtime behaviour is universal in entire classes of evolutions.
Scaling in steady-state cluster-cluster aggregation
Physical Review A, 1985
The diffusion-limited cluster-cluster aggregation model is investigated under conditions which for long times lead to steady-state coagulation. Single particles are added to the system at a constant rate and the larger clusters appearing as a result of the aggregation ...
Diffusion-limited aggregation without branching
Physical Review A, 1986
Diffusion limited aggregation without branching is a model of irreversible growth where the accreting particles move diffusively and the rules determining the growth of the cluster force a very simple structure on the resulting aggregates. The simplicity of the model makes it possible to give a more complete theoretical treatment than in other diffusive growth processes. Here a brief overview is given of the available simulation results and of the theoretical work: these results are compared with those obtained for ordinary DLA.
Self-consistent rate equation theory of cluster size distribution in aggregation phenomena
Physica A: Statistical Mechanics and its Applications, 2002
Cluster nucleation and growth by aggregation is the central feature of many physical processes, from polymerization and gelation in polymer science, occulation and coagulation in aerosol and colloidal chemistry, percolation and coarsening in phase transitions and critical phenomena, agglutination and cell adhesion in biology, to island nucleation and thin-ÿlm growth in materials science. Detailed information about the kinetics of aggregation is provided by the time dependent cluster size-distribution, a quantity which can be measured experimentally. While the standard Smoluchowski rate-equation approach has been in general successful in predicting average quantities like the total cluster density, it fails to account for spatial uctuations and correlations and thus predicts size distributions that are in signiÿcant disagreement with both experiments and kinetic Monte Carlo simulations. In this work we outline a new method which takes into account such correlations. We show that by coupling a set of evolution equations for the capture-zone distributions with a set of rate-equations for the island densities one may obtain accurate predictions for the time-and size-dependent rates of monomer capture. In particular, by using this method we obtain excellent results for the capture numbers and island-size distributions in irreversible growth on both one-and two-dimensional substrates.
Scaling laws in the diffusion limited aggregation of persistent random walkers
Physica A: Statistical Mechanics and its Applications, 2011
We investigate the diffusion limited aggregation of particles executing persistent random walks. The scaling properties of both random walks and large aggregates are presented. The aggregates exhibit a crossover between ballistic and diffusion limited aggregation models. A non-trivial scaling relation ξ ∼ 1.25 between the characteristic size ξ, in which the cluster undergoes a morphological transition, and the persistence length , between ballistic and diffusive regimes of the random walk, is observed.