Bond percolation

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. 

Percolation theory describes [32] the random growth of molecular clusters on a d-dimensional lattice. It was suggested to possibly give a better description of gelation than the classical statistical methods (which in fact are equivalent to percolation on a Bethe lattice or Caley tree, Fig. 7a) since the mean-field assumptions (unlimited mobility and accessibility of all groups) are avoided [16,33]. In contrast, immobility of all clusters is implied, which is unrealistic because of the translational diffusion of small clusters. An important fundamental feature of percolation is the existence of a critical value pc of p (bond formation probability in random bond percolation) beyond which the probability of finding a percolating cluster, i.e. a cluster which spans the whole sample, is non-zero. 

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... 

This results in a value of d = 2.5 for bond percolation on a 3-dimensional lattice. The fractal dimension of the Bethe lattice (Flory-Stockmayer theory) is... 

Around 1970 computer simulations of the branching processes on a lattice started to become a common technique. In bond percolation the following assessment is made [7] whenever two units come to lie on adjacent lattice sites a bond between the two units is formed. The simulation was made by throwing at random n units on a lattice with ISP lattice sites. Clusters of various size and shape were obtained from which, among others, the weight fraction distribution could be derived. The results could be cast in a form of [7]... 

Bloch functions, 25 7, 8 Bloch state, stationary, 34 237, 246 Blow-out phenomenon, 27 82, 84 Bohr magneton, 22 267 number, 27 37 Boltzmann law, 22 280 Bonding energy, BOC-MP, 37 106-107 Bondouard disproportionation reaction, 30 196 Bond percolation, 39 6-8 Bonds activation... 

Percolation models, 39 5-11 AB percolation, 39 10-11 bond percolation on regular lattices, 39 6-... 

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. 

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... 

Experimental e"(t) curves exhibit an inflection point that corresponds to the maximum rate of change of conductivity, associated to the maximum decrease of mobility. For this reason this inflection point has been attributed to gelation (Wasylyshyn and Johari, 1997). In some cases good agreement between the inflection time and the gel time obtained by rheological methods was observed. A very complex explanation taking into account inter-molecular H bonds, protonic conduction, and a bond percolation law has been proposed to explain this mobility change (Johari and Wasylyshyn, 2000). 

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... 

Richard Zallen, The Physics of Amorphous Materials (New York Wiley-Interscience, 1998), chap. 4. The percolation model distinguishes between bond percolation, in which a pairwise connection exists between sites with bonds either connected or unconnected (percolation threshold is reached at 18 percent), and site percolation, in which all sites are connected and the sites, rather than the bonds between sites, establish the character of connectivity (percolation threshold is reached at 15 percent). If the analogy holds, and the number of connected bonds is the critical component for women in an academic department, then the critical mass threshold is reached at 18 percent. 

In the model of bond percolation on the square lattice, the elements are the bonds formed between the monomers and not the sites, i.e., the elements of the clusters are the connected bonds. The extent of a polymerization reaction corresponds to the fraction of reacted bonds. Mathematically, this is expressed by the probability p for the presence of bonds. These concepts can allow someone to create randomly connected bonds (clusters) assigning different values for the probability p. Accordingly, the size of the clusters of connected bonds increases as the probability p increases. It has been found that above a critical value of pc = 0.5 the various bond configurations that can be formed randomly share a common characteristic a cluster percolates through the lattice. A more realistic case of a percolating cluster can be obtained if the site model of a square lattice is used with probability p = 0.6, Figure 1.5. Notice that the critical value of pc is 0.593 for the 2-dimensional site model. Also, the percolation thresholds vary according to the type of model (site or bond) as well as with the dimensionality of the lattice (2 or 3). 

Both problems share the common property that P(p) is of measure zero for p < Pc, with the critical threshold value pc as a function of the type of lattice considered. Few exact formulas exist for P(p) or even pc. There are, however, a number of empirical rules. For instance, for the bond percolation problem,... 

Nevertheless, in a previous study dealing with inert matrices of naltrexone-HCl [74], two different excipient percolation thresholds pc2 were found for the matrixforming excipient Eudragit RS-PM the site percolation threshold related to a change in the release kinetics and the site-bond percolation threshold derived from the mechanical properties of the tablet, where the excipient failed to maintain tablet integrity after the release assay. 

Basically, the process of tablet compression starts with the rearrangement of particles within the die cavity and initial elimination of voids. As tablet formulation is a multicomponent system, its ability to form a good compact is dictated by the compressibility and compactibility characteristics of each component. Compressibility of a powder is defined as its ability to decrease in volume under pressure, and compactibility is the ability of the powdered material to be compressed into a tablet of specific tensile strength [1,2], One emerging approach to understand the mechanism of powder consolidation and compression is known as percolation theory. In a simple way, the process of compaction can be considered a combination of site and bond percolation phenomena [5]. Percolation theory is based on the formation of clusters and the existence of a site or bond percolation phenomenon. It is possible to apply percolation theory if a system can be sufficiently well described by a lattice in which the spaces are occupied at random or all sites are already occupied and bonds between neighboring sites are formed at random. 

TABLE 10.11 Percolation Liinites ° 4>c, for Site and Bond Percolation... 

Pc, at which a spanning cluster occurs is called the percolation threshold. There are two versions of percolation, site and bond. With bond percolation, the sites are initially filled and the bonds are added to connect the sites. With site percolation, a grid placed over a region is gradually filled with spheres. The percolation threshold is lower for bond percolation than for site percolation because a bond is attached to two sites while a site is connected to a maximum of z bonds. Taking the coordination number, z, around the sites, the threshold for bond percolation is seen to be close to that of the classical theory [21]... 

The percolation probability has different values based on the classical theory site or bond percolation for different structures, as shown in Table 12.2. This critical percolation volume fraction, <, is calculated from the percolation threshold and the space filling factor. The volume fraction for site percolation for various structures is essentially the same as follows. In three dimensions, the site percolation threshold occurs at —16% volume. Near the percolation threshold the average cluster size diverges as does the spanning length of clusters. 

Structure Coordination z Probaldlify of percolation 1 z - 1 Probability of site percolation P - Probability of bond percolation Space fUUng factor V I rcolation volume fraction 0 = opf"... 

Site and bond percolation can be shown to be equivalent, and for simplicity from this point we discuss only site percolation. 

The numerical results reviewed above were obtained for infinite lattices. How do the various quantities of interest behave near the percolation threshold in a large but finite lattice This problem has been studied by renormalization methods, which are essentially equivalent to finite-size scaling. For finite lattices the percolation transition is smeared out over a range of p, and one must expect a similar trend in other functions, including the conductivity. Computer simulations by the Monte Carlo method have been carried out for bond percolation on a three-dimensional simple cubic lattice by Kirkpatrick (1979). Five such experimental curves are shown in Fig. 40, each of which corresponds to a cube of size b, containing bonds. In Fig. 40 the vertical axis gives the fraction p of such samples that percolate (i.e., have opposite faces con-... 

Fig. 40. Scaling for finite-sized lattices. Computer calculations of scaling properties for bond percolation on the three-dimensional simple cubic lattice. When p is the fraction of connected bonds, p = p (p,b) is the fraction of cubic samples of edge length b that contain a continuous path of connected bonds (a spanning cluster) which links opposite faces of the sample. From Kirkpatrick (1979).

Table 1.1 The site and bond percolation thresholds for different lattice types...

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