Witten-Sander clusters

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] ... 

And at last, in a gelation point the static geometrical characteristics of a system are changed. Theoretically [26] and experimentally [27] it is shown, that the value Dj. changes from values about approx. 1.5-2.1 typical for a macromolecular coil in solution [45], up to approx. 2.5, characterizing Witten-Sander cluster [26]. 

In Fig. 6.1, the images of the studied nanocomposites, obtained in the force modulation regime, and corresponding to them nanoparticles aggregates fractal dimension distributions are adduced. As it follows from the adduced values particles aggregates in the studied nanocomposites are formed by a mechanism particle-cluster (P-Cl), that is, they are Witten-Sander clusters [7]. The variant A, was chosen which according to mobile particles are added to the lattice, consisting of... 

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] ... 

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] ... 

Dependences of type 5.1 are typical for fractal structures and are an experimental confirmation of the fractality of the structure of polymers [16] and the value of corresponds well to the known earlier values of the fractal dimension of the structure of polymers [17]. The numeric magnitude of gives the possibility to suppose the cluster structure formation type. As it is known [18], Witten-Sander clusters, formed by limited aggregation of diffusion of particles, have dimension d =2.5 0.26, which is close to the result obtained above. This allows the cluster structure forming in EP to be attributed to the type mentioned above. 

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. 

A modified Witten-Sander scheme was used to accelerate the convergence to the asymptotic regime [51]. The random particle does not stick to the cluster when it is on a site adjacent to occupied sites, but it is moved until it is on an occupied site then its final position is assumed to be the previous one. Such a scheme yields a fractal dimension of 2.6, which is slightly larger than the 2.5 obtained with the original Witten-Sander scheme. 

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. 

Eor Witten-Sander (WS) clusters, from the set of which the structure of crosslinked polymers is simulated [100], it can be written [60] ... 

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... 

Diffusion Limited Cluster Aggregation (DLCA) Process whereby clusters of particles undergoing a random walk due to Brownian motion, aggregate together. This theory, proposed by Witten and Sander in 1981... 

In 1981, Witten and Sander used the fractal dimension to describe their numerically designed aggregates of a difiusion-limited particle-cluster aggregation scheme. They derived the fractal dimension on to ways (i) via flie density-correlation function (Eq. (4.8)) and (ii) via the correlation between the aggregate radii of gyration Rg and the dimensionless mass, i.e. the number of primary particles N ... 

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). 

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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