Bond percolating networks

Fig. 2. The integrated DOSs per atom at p = Pc for d = 2, d = 3, and d = 4 bond-percolating networks. The symbols correspond to those in Fig. 1.

Arbabi, S Sahimi, M, Elastic Properties of Three-Dimensional Percolation Networks with Stretching and Bond-Bending Eorces, Physical Review B 38, 7173, 1988. 

In random bond percolation, which is most widely used to describe gelation, monomers, occupy sites of a periodic lattice. The network formation is simulated by the formation of bonds (with a certain probability, p) between nearest neighbors of lattice sites, Fig. 7b. Since these bonds are randomly placed between the lattice nodes, intramolecular reactions are allowed. Other types of percolation are, for example, random site percolation (sites on a regular lattice are randomly occupied with a probability p) or random random percolation (also known as continuum percolation the sites do not form a periodic lattice but are distributed randomly throughout the percolation space). While the... 

O2 diffusion through the membrane seems to be limited by the percolation network of the diffusion path, which is not only defined by the amount of water in the membrane, but also by the different chemical structure of the membranes. It is difficult to make comparisons of gaseous diffusion behavior among polymers with different structures because polymer morphology can change drastically without appreciable changes in density, and the presence of water and the hydrogen bonds formed between polymer-water moieties also has major effects on system properties. However, some points can be made from these particular studies. 

In addition to the amount of filler content, the shape, size and size distribution, surface wettability, interface bonding, and compatibility with the matrix resin of the filler can all influence electrical conductivity, mechanical properties, and other performance characteristics of the composite plates. As mentioned previously, to achieve higher electrical conductivity, the conductive graphite or carbon fillers must form an interconnected or percolated network in the dielectrical matrix like that in GrafTech plates. The interface bonding and compatibility between... 

The side chain separation varies in a range of 1 nm or slightly above. The network of aqueous domains exhibits a percolation threshold at a volume fraction of 10%, which is in line with the value determined from conductivity studies. This value is similar to the theoretical percolation threshold for bond percolation on a face-centered cubic lattice. It indicates a highly interconnected network of water nanochannels. Notably, this percolation threshold is markedly smaller, and thus more realistic, than those found in atomistic simulations, which were not able to reproduce experimental values. 

Network structures are still determined by nodes and strands when long chains are crosslinked at random, but the segmental spacing between two consecutive crosslinks, along one chain, is not uniform in these systems which are currently described within the framework of bond percolation, considered within the mean field approximation. The percolation process is supposed to be developed on a Cayley tree [15, 16]. Polymer chains are considered as percolation units that will be linked to one another to form a gel. Chains bear chemical functions that can react with functions located on crosslinkers. The functionality of percolation units is determined by the mean number f of chemical functions per chain and the gelation (percolation) threshold is given by pc = (f-1)"1. The... 

It is interesting to note that the (L-N-B)-model leads to similar expressions for the moduli like the VTG-model apart from the first summand of Eq. (38). However, contrary to the semi-empirical weighting functions W(y6) of the VTG-model, the corresponding density distribution function/la(y) in the (L-N-B)-model is related to the morphological structure of the filler network, i.e., the distribution of singly connected bonds in a percolation network. Unfortunately, this distribution function is not known, exactly. Therefore, a simple exponential... 

The formation of spanning H-bonded water networks on the surface of biomolecules has been connected with the widely accepted view that a certain amount of hydration water is necessary for the dynamics and function of proteins. Its percolative nature had been suggested first by Careri et al. (59) on the basis of proton conductivity measurements on lysozyme this hypothesis was later supported by extensive computer simulations on the hydration of proteins like lysozyme and SNase, elastine like peptides, and DNA fragments (53). The extremely interesting... 

Most of the pore structures (e.g., spongy structures) consist of extensive three-dimensional networks in which there is a profusion of interconnections between voids within the structure. The latter interconnections affect considerably the kinetics of various processes in porous solids. This effect can adequately be described by employing the ideas developed in percolation theory 7-13). In the framework of this theory, the medium is defined as an infinite set of sites interconnected by bonds. Percolation theory can be applied to porous solids via identification of network sites with voids, and bonds with necks. Thus, the theory is applicable primarily to spongy porous structures but in some cases also to corpuscular structures. 

Gelation is a connectivity transition that can be described by a bond percolation model. Imagine that we start with a container full of monomers, which occupy the sites of a lattice (as sketched in Fig. 6.14). In a simple bond percolation model, all sites of the lattice are assumed to be occupied by monomers. The chemical reaction between monomers is modelled by randomly connecting monomers on neighbouring sites by bonds. The fraction of all possible bonds that are formed at any point in the reaction is called the extent of reaction p, which increases from zero to unity as the reaction proceeds. A polymer in this model is represented by a cluster of monomers (sites) connected by bonds. When all possible bonds are formed (all monomers are connected into one macroscopic polymer) the reaction is completed (/> = 1) and the polymer is a fully developed network. Such fully developed networks will be the subject of Chapter 7, while in this chapter we focus on the gelation transition. 

Figure 23.5. In both cases, the stress prior to expansion is negative since the volume exclusion outweighs the positive contribution to the stress from the elastic bonds. During expansion, the system with very-easy-to-break bonds (curve (a)) relaxes slowly. In this case, the set of isolated clusters of particles of Type 2 prevents the elastic bonds from pulling-in from the imaginary moving barrier. The system with long-lasting bonds (curve (b)) seems to relax more rapidly initially. This is due to the initial stretching of the bonds of the percolated network of particles of Type 2, which adds an increasingly positive contribution to the stress. Since the bonds eventually break, the rapid increase in stress is halted at the point at which the percolated network breaks into two disconnected parts.

Extensive empirical studies of percolation networks have revealed universal critical percolation thresholds (pc). That is, for percolation to occur across a network, some critical proportion of the squares (or blocks in three-dimensions) must be pores. In two dimensions, //, 0.59 for conductive bonds at all shared edges... 

On the basis of random walk arguments, Koplik et al. (1988) show the relations K Pn, K Pn In Pn, and K Pi for (i) a bundle of uniform stream tubes that meet at perfect mixing chambers separated by a characteristic length /, (ii) nonuniform bond transit times, and (iii) the presence of dead-end pores, respectively. The relationship K oc pi is also obtained for percolation networks near the percolation threshold when the characteristic length is the percolation correlation length and the molecular diffusion coefficient is adjusted for the presence of the percolation network. 

Yanuka (1992) considered convection-dominated flows on two-dimensional percolation networks. A flow realization was obtained by randomly assigning conductivities to each bond in the network, imposing a pressure gradient across the network, and solving for the flow throughout the network. Two cases were considered in the first, all bonds between sites were considered open while in the second, the probability of an open bond was near the percolation threshold. Random walks through the networks were weighted by the flow in each bond. In the first case, K is constant and in the second, K °c 12 as observed previously. 

Figure 18. A bond percolation model of gas flow in a pore network containing foam lamellae blocking pore throats. Line intersections represent pore bodies. (Reproduced with permission from reference 14. Copyright 1993.)...

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