Witten-Sander model

Simulations may be grouped according to the general uses for which they were developed. As such, the first to be developed, and the simplest simulation techniques, are those investigating the structure of aggregates. These models first appeared in 1963 [18], and after Mandelbrot s seminal work on fractal geometry [45], development of these types of simulations increased dramatically, spurred by the well-known Witten-Sander model [46]. 

Dolbin, I. V Kozlov, G. V The polymer films structure formation Witten-Sander model. Bulletin of KBSC RAS, 2004, 2, 40. ... 

The authors of Ref [19] used the stated above treatment of polymers cold flow with application of Witten-Sander model of diffusion-limited aggregation [20] on the example of PC. As it has been shown in Refs. [21, 22], PC structure can be simulated as totality of Witten-Sander clusters (WS clusters) large number. These clusters have compact central part, which in the model [18, 23] is associated with notion cluster. Further to prevent misunderstandings the term cluster will be understood exactly as a compact local order region. At translational motion of such compact region in viscous medium molecular friction coefficient of each cluster, a particle, having radius a, is determined as follows [24] ... 

The mechanism of growth of columnar structure ZnO layers assumes the presence of nanosized clusters in the reagent flow and that the formation of ZnO layers can be described by a model close to the one of Witten-Sanders. Increase of target-to-substrate distance leads to the decrease of the number of aggregates in the flow near the substrate and to the improvement of the structural perfection of ZnO layers. 

Let us consider the parameter estimation within the framework of Witten-Sander irreversible aggregation model [ 194]. As it has been shown in Refs. [195, 196], the amorphous glassy polymers stmcture can be simulated as totality of Witten-Sander clusters (WS clusters) large munber, having radius R., which is determined as follows [197] ... 

The last model is a hierarchical model where the fractal dimension is tuned by selecting the orientation and the sticking point of clusters of size 2 [53], where p is the iteration number. Four values of the fractal dimension were studied 1.6, 1.9. 2.2, and 2.5. Two of these values were chosen close to the values of the Witten-Sander aggregate and to the standard hierarchical model, for reasons that will be clear in the next section. 

Monomer-cluster or cluster-cluster growth can be limited by diffusion or by reaction. In diffusion-limited monomer-cluster aggregation (DLMCA), simulated by the Witten and Sander model (25) in Fig. 5.11, it is assumed that monomers are released one by one from sites arbitrarily far from a central cluster. The monomers travel by a random walk diffusion mechanism and stick irreversibly at first contact with the growing cluster. Because of this trajectory, the monomers cannot penetrate deeply into a cluster without intercepting a cluster arm and the arms effectively screen the interior of the cluster from incoming monomers. Growth occurs preferentially at exterior sites, resulting in objects in which the density decreases radially from the center of mass (in three dimensions dm = 2.45). 

For interactions nanoclusters - loosely packed matrix estimation within the range o T = 293- -373K the authors of Ref. [48] used the model of Witten-Sander clusters friction, stated in Ref. [46]. This model application is due to the circumstance, that amorphous glassy polymer structure can be presented as an indicated clusters large number set [47]. According to this model, Witten-Sander clusters generalized friction coefficient t can be written as follows [46] ... 

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... 

We discuss below the DLA model in detail which was proposed by Witten and Sander [16, 18]. The DLA is a particular model of a random irreversible growth. The growth process starts from a seed particle. A second particle is launched far enough from the seed and makes a random walk. If it visits a position next to the seed, it is stuck to it and both form a two-particle cluster extending the initial seed. Then a third particle is launched and moves randomly around this cluster. It may join the two-particle... 

The DLA model has been investigated in 1981, by Witten and Sander [112]. This was catching the attention of researchers due to property of various levels of crystal patterns especially in non-equilibrium state can be simulated instantly. The standard DLA model includes some basic terminologies which are conceptually helpful to describe complexity of the growth process. 

In 1981 the diffusion limited aggregation (DLA) model was introduced by Witten and Sanders [96]. In this model particles are added, one at a time, to a cluster or aggregate of particles via random walk trajectories. According to this model, there is competing growth of polymer chains from a surface, which leads to the formation of independent clusters. 

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