Dynamic bond percolation model

In addition to specific applications, the dynamic bond percolation model has been extended to focus on the importance of lattice considerations. The role of correlations among different renewal processes and the... 

The brief discussion above shows that the structure of a polymer electrolyte and the ion conduction mechanism are complex. Furthermore, the polymer is a weak electrolyte, whose ions form ion pairs, triple ions, and multidentate ions after its ionic dissociation. Currently, there are several important models that attempt to describe the ion conduction mechanisms in polymer electrolytes Arrhenius theory, the Vogel-Tammann-Fulcher (VTF) equation, the Williams-Landel-Ferry (WLF) equation, free volume model, dynamic bond percolation model (DBPM), the Meyer-Neldel (MN) law, effective medium theory (EMT), and the Nernst-Einstein equation [1]. 

More detailed theoretical approaches which have merit are the configurational entropy model of Gibbs et al. [65, 66] and dynamic bond percolation (DBP) theory [67], a microscopic model specifically adapted by Ratner and co-workers to describe long-range ion transport in polymer electrolytes. 

To describe the change in reptation dynamics of the chains as a function of nanoparticle volume fraction, a percolation model was used. At the percolation threshold, a physical network formed by interconnection of immobilized chains on individual nanoparticles penetrates the entire sample volume. In this case, only physical cross-links are considered and the terminal relaxation time reaches the value characteristic for the life time of the physical filler-polymer bond. Thus, the relaxation time near the percolation threshold is expressed in the form [44] ... 

In a recent paper [9], the present author has considered a model for the dynamics of percolation clusters with the aim of harmonizing the two independent results obtained in references [4] and [5] for the connection-disconnection probability Ngv(s) of a finite cluster of size s. These two results are in mutual agreement in d=2 but not in d > 2 and the only way to obtain coherent results is to consider that the scaling of the anti-red bonds (or reconnecting bonds) is different from that of the red bonds (or disconnecting bonds). The general expression, supposed valid in any case is,... 

Cates [56] introduced the notion polymer fractal by replacing the rigid bonds in a percolation cluster by flexible (phantom) links. This model can be used to describe the gelation process (Figure 11.2) [57]. Within the framework of this approach, Cates [56] described variations of the structure and the dynamics of dilute solutions of polymers. 

Figure 3.58. Problem of bond and node percolation a model of gel formation during polymerization of a polyfunctional monomer in solution (Stanley et al., 1980 Efros, 1982 ) [Reprinted with permis.sion from Dynamics of Synergetic Systems Proc. Int. Symp. on Synergetics, Butlfeld, Germany, September 2f-29, 1979 Ed. M.Haken. Copyright by Springer-VerlagJ...

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