Aggregation diffusion-limited, computer

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. 

Transport control of flocculation is realized in an especially direct way in the process known as diffusion-limited cluster-cluster aggregation5 (aggregation as used in this term means flocculation in the present chapter). In this process, which is straightforward to simulate and visualize on a computer, particles undergo Brownian motion (i.e., diffusion) until they come together in close proximity, after which they coalesce instantaneously and irreversibly to form floccules (or clusters ). The clusters then diffuse until they contact one another and combine to form larger clusters, and so on, until gravitational... 

The basic theoretical approach for the analysis of diffusion controlled reactions is due to Smoluchowski [9] who developed it for the analysis of diffusion limited aggregation of colloidal particles. We discuss the generalization of this approach to the case of rodlike molecules here. The computational method best suited for the simulation of the polymerization of rodlike molecules is Brownian dynamics. We discuss in this review both multiparticle Brownian dynamics and pairwise Brownian dynamics the latter is a hybrid method combining Smoluchowski s [9] theory and Brownian... 

Figure 8. Transmission electron micrographs of typical clusters of gold, silica, and polystyrene colloids, prepared by both diffusion-limited and reaction-limited cluster aggregation and by computer simulation. There is a striking similarity in the structure of the clusters of different colloids in each regime.

Figure 3.9 Box-counting analysis of a projection of a computer-generated diffusion-limited cluster aggregate of 10 000 particles with mass fractal dimension of 1.88. The images have box sizes L of (top left to bottom right) 1, 2,4,8,16, 32 and 64 pixels and require 205 245,59519, 17 062, 4895,1436, 462 and 135 squares respectively to cover the image.

Computer simulations have shown that the value of fractal dimension largely depends on whether the aggregation process is controlled by the diffusion rate of the clusters and single particles or by their chemical reactivity at the time of collision, the latter being mainly controlled by the DLVO forces. This observation, in agreement with experimental work on aerosols and colloids, has led to a new classification of aggregation processes the reaction-limited and diffusion-limited cluster aggregation (RLCA and DLCA respectively) processes. 

The patterns produced by the diffusion-limited aggregation (DLA) processes are characterized by the open random and tree-type structures and can be well described as fractals. Computer simulations of fractal growth have been shown to produce structures... 

Pierce F, Sorensen CM, Chakrabarti A (2006) Computer simulation of diffusion-limited cluster-cluster aggregation with an epstein drag force. Phys Rev E 74 021411. 

Figure la. Fractal aggregate, constructed by computer simulated diffusion limited aggregation. Fractal dimension D = 1.44... 

Self-similar behavior has also been observed in computer simulation of aggregation processes. Thus aggregates of colloidal particles in diffusion-limited aggregation processes have been found to display self-similar behavior (Meakin, 1983). 

These procedures not only indicate the kinetics of aggregation, but also allow study of the size distribution and the fractal geometry of the clusters. Typical results for the fractal dimension of clusters grown in computer simulations are summarized in Table 5. Note that the monomer-cluster aggregation models produce dense nonfractal clusters, except for the diffusion-limited case, whereas cluster-cluster models yield fractal dimensions in good agreement with experiments on real colloids. 

Fig. 6. Domain shape obtained by computer simulation with the diffusion limited aggregation model (a), in comparison with experimental observation following a pressure jump (b).

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