Diffusion-limited aggregates

Meakin P 1983 Formation of fractal clusters and networks by irreversible diffusion-limited aggregation Phys. Rev. Lett. 51 1119-22... 

Actually, this is not really diffusion-XimiiQd, but rather Laplacian growth, since the macroscopic equation describing the process, apart from fluctuations, is not a diffusion equation but a Laplacian equation. There are some crucial differences, which will become clearer below. In some sense DLA is diffusion-limited aggregation in the limit of zero concentration of the concentration field at infinity. 

More than 20 years ago, Matsushita et al. observed macroscopic patterns of electrodeposit at a liquid/air interface [46,47]. Since the morphology of the deposit was quite similar to those generated by a computer model known as diffusion-limited aggregation (D LA) [48], this finding has attracted a lot of attention from the point of view of morphogenesis in Laplacian fields. Normally, thin cells with quasi 2D geometries are used in experiments, instead of the use of liquid/air or liquid/liquid interfaces, in order to reduce the effect of convection. 

Witten, T. A. and Sander, L. M. (1981) Diffusion-limited aggregation, A kinetic critical phenomenon. Phys. Rev. Lett., 47, 1400-1403. 

Let us illustrate first how different (idealized) aggregation processes may result in different structures. There is extensive literature on diffusion-limited aggregation (DLA) (for a comprehensive review, see Meakin, 1988). Three methods of simulation are common (standard) diffusion-limited aggregation (DLA), reaction-limited aggregation (RLA), and linear trajectory aggregation (LTA). DLA structures are generated by placing a seed particle in the middle of a lattice. Other particles are placed in the lattice... 

Fig. 37. Typical clusters obtained by diffusion-limited aggregation (DLA). Top Two-dimensional diffusion-limited aggregation. Bottom Reaction-limited hierarchical cluster-cluster aggregation (HCCA) (Meakin, 1988 with permission, from the Annual Review of Physical Chemistry, Vol. 39. by Annual Reviews www.Annual/Reviews.org).

CA in which many filled cells execute a random walk but never interact with one another, cannot give rise to stable pattern formation since the cells will move at random forever. However, if cells can interact when they meet, so that one diffusing cell is allowed to stick to another, stable structures can be created. These structures illustrate the modeling of diffusion-limited aggregation (DLA), which is of interest in studies of crystal formation, precipitation, and the electrochemical formation of solids. 

For example, in the case of PS and applying the Smoluchowski equation [333], it is possible to estimate the precipitation time, fpr, of globules of radius R and translation diffusion coefficient D in solutions of polymer concentration cp (the number of chains per unit volume) [334]. Assuming a standard diffusion-limited aggregation process, two globules merge every time they collide in the course of Brownian motion. Thus, one can write Eq. 2 ... 

In contrast, the three- or two-dimensional morphologies of colloidal aggregates via Brownian particle trajectories show a fractal-like structure. One of the most prominent features of the surface deposits formed by the diffusion-limited aggregation mechanism is the formation of isolated treelike clusters (9). In our experiments, the surface morphology of the silica-coated polyethylene composite prepared by... 

An extension of the coupled-cluster approximation to the non-equilibrium classical systems [43-45] has allowed to study asymptotics of bimolecular reactions. It resulted in a rather unexpected conclusion that now the generally-accepted time dependence of the A+B —> 0 reaction for d = 3, n(t) oc f-3/4, is only the pre-asymptotic stage, with the true asymptotics n(t) oc f 1 Similar technique was used also for the study of diffusion-limited aggregation and structure formation processes [47],... 

Parkinson, J., Kadler, K. E., and Brass, A. (1995). Simple physical model of collagen fibrillogenesis based on diffusion-limited aggregation./. Mol. Biol. 247, 823-831. 

Evidence of the fraction of free monomer micelles at the clotting time would help to determine the appropriate form of the reaction kernel and whether growth is limited by diffusion or by the reaction itself. The growth of polymers from polyfunctional monomers, the formation of diffusion-limiting aggregates, and many other natural phenomena can all be scale invariant fractals with a similar fractal... 

Fig. 6. (a) Diffusion limited aggregation of 4-trans-2-(pyrid-4-yl-vinyl) benzoic acid... 

Avnir et al. llbl have examined the classical definitions and terminology of chirality and subsequently determined that they are too restrictive to describe complex objects such as large random supermolecular structures and spiral diffusion-limited aggregates (DLAs). Architecturally, these structures resemble chiral (and fractal) dendrimers therefore, new insights into chiral concepts and nomenclature are introduced that have a direct bearing on the nature of dendritic macromolecular assemblies, for example, continuous chirality measure44 and virtual enantiomers. ... 

The mathematical model called diffusion-limited aggregation (DLA) was introduced by Witten and Sander in 1981 [46]. The model starts with a particle at the origin of a lattice. Another particle is allowed to walk at random (simulating Brownian motion) until it arrives at a site adjacent to the seed particle. At each time step, the traveling particle moves from one site to... 

This structure is generated via the modified diffusion-limited aggregation (DLA) algorithm of [205] using the law p = a (m/N). Here, N = 2, 000 (the number of particles of the DLA clusters), a = 10 and ft = 0.5 are constants that determine the shape of the cluster, p is the radius of the circle in which the cluster is embedded, pc = 0.1 is the lower limit of p (always pc < p), and to is the number of particles sticking to the downstream portion of the cluster. This example corresponds to a radial Hele-Shaw cell where water has been injected radially from the central hole. Due to heterogeneity a sample cannot be used to calculate the dissolved amount at any time, i.e., an average value for the percent dissolved amount at any time does not exist. This property is characteristic of fractal objects and processes. 

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