Cluster spanning

At the beginning of the desorption process, some condensate is removed from the wider pores (i.e. unoccupied bonds) near the surface. As the pressure is reduced, the vapour-filled pores (occupied bonds) form clusters, which eventually extend across the particle. The stage at which a spanning cluster is formed across the particle corresponds to the percolation threshold, when the pore emptying becomes rapid. This stage corresponds to the knee of an H2 hysteresis loop. 

Miranda et al. demonstrated experimentally the influence of the particle size of the components on the percolation threshold in hydrophilic matrices as well as the importance of the initial porosity in the formation of the gel layer (sample-spanning cluster of excipient) [74],... 

In hydrophilic matrices the drug threshold is less evident than the excipient threshold, which is responsible for the release control [73], In order to estimate the percolation threshold of HPMC K4M, different kinetic parameters were studied Higuchi rate constant, normalized Higuchi rate constant, and relaxation rate constant. The evolution of these release parameters has been studied as a function of the sum of the excipient volumetric percentage plus initial porosity. Recent studies of our research group have found the existence of a sample-spanning cluster of excipient plus pores in the hydrophilic matrix before the matrix is placed in contact with the liquid, clearly influences the release kinetics of the drug [73]. 

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]... 

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).

There are a number of useful points to be drawn from this analysis of the water cluster distribution. First, it appears that the cumulative water cluster distribution with a properly chosen cutoff distance is a metric that allows one to see clearly differences in connectivity of the aqueous domain as a function of both water content and polymer architecture. Clearly, Fig. 9(b) shows the connectivity of the aqueous domain moving from many small disconnected clusters to a single sample-spanning cluster as a func-... 

The probability that a given site (bond) is occupied and is a part of an s-size cluster is sns p). Let us denote by P(p), the probability that any occupied site (bond) belongs to the infinite (lattice spanning) cluster. Then we have the obvious relation. 

Below the percolation line, there is predicted to be a sample-spanning cluster of contacting spheres. Woutersen et al. (1994) found that the gel point for 47-nm octadecyl-grafted silica spheres in benzene is in reasonable agreement with the predicted percolation transition. However, Grant and Russel (1993) found that the gelation line is below the percolation ... 

Percolation is the process of network formation by random filling of bonds between sites on a lattice. If one increases the fraction of bonds (p) formed, then larger clusters of bonds form until an infinite lattice-spanning cluster (at the percolation threshold, p=Pc) is formed. Figure 2.15 (Stanley, 1985) shows the growth of the network corresponding to values of the fraction p of (a) 0.2, (b) 0.4, (c) 0.6 (p=p ) and (d) 0.8. 

Luxmoore and Ferrand (1993) pointed out that pores that belong to the sample spanning cluster but not the backbone can be thought of as containing stagnant backwater zones. Thus, empirical determination of the proportions of backbone and backwater porosity in random and nonrandom pore percolation networks could be quite useful. They anticipated that percolation modeling would play an important role in understanding the effects of transient pore scale processes on solute transport. 

In materials of infinite extent, the above definitions remain valid. As noted previously (Chapter 4), for pore space topologies with a given coordination number, there exists a critical filling probability (porosity). In materials with filling probabilities above this critical value, the size of the largest cluster is comparable to the size of the lattice. The presence of this lattice spanning cluster does not require that the material be finite in extent in fact, most analytical results in percolation theory assume that the lattice is infinite. For... 

As noted previously, the critical probability for the Bethe lattice is (equivalent to Pc defined in Chapter 4) = /(( — 1). For lattices below this critical probability (e < Cc), the root of Equation 9-26 is e = e. The accessible porosity, from Equation 9-25, is zero, which indicates that a lattice spanning cluster is not present. For lattices above the critical probability (e > e ) Equation 9-26 can be solved to find e. Results for coordination numbers 3 and 4 are ... 

In Figure 9.14b, the fraction accessible porosity is plotted versus the total porosity for Bethe lattices of coordination numbers 3 and 7. For all coordination numbers, a is zero for porosities less than the critical value. The critical porosity is indicated for each coordination number by the intercept of the curve with the x-axis. Above the critical porosity, rises sharply. In this transition region, the infinite, lattice-spanning cluster is growing and incorporating pores and smaller pore clusters that are isolated at lower porosities. At high porosities, 0a becomes equal to unity, indicating that all the pores are members of the infinite cluster. 

Figure 16.5. Snapshots of flie simulation of different binary mixtures - water-DMSO in the top panel, water-eflianol in die middle, and water-TBA in the bottom panel. Water moleeules are shown in silver. Co-solvents (DMSO, edianol, and TBA) are represented in blue. The snapshot is shown at two different coneentrations - one before the onset of percolation to show the microheterogeneity in the system, and one after the onset of percolation to show the spanning cluster of the cosolvent. Figure adapted with permission from J. Phys. Chem. B, 115 (2011), 685. Copyright (2011) American Chemical Society.

Like the SAW, the problem of percolation can also be treated as geometrical critical phenomenon [74]. To introduce the percolation problem, let us consider a regular hyper-cubic d-dimensional lattice, where either the sites or bonds are occupied with probability p. In percolation one asks questions concerning the connectivity of occupied bonds. Sets of mutually connected bonds form cluster. One can then ask what is the probability that there is a cluster spanning from the one end of the lattice to the opposite end in the thermodynamic limit (the number of sites goes to infinity) this spanning cluster becomes the infinite cluster. 

Figure 5.1, we see a cluster that connects opposite edges of the system. The latter is called the spanning cluster. In percolation theory, we are concerned mainly with infinite systems and thus we call the spanning duster the percolation or the infinite cluster. The illustration given in Figure 5.1 reflects here a finite (small) portion of the infinite system but it has to be remembered that quantitative characterizations of the various properties are considered meaningful only at the infinite system limit... 

From Figure 5.1 it is easy to appreciate that the numbers and sizes of the clusters will grow with the site occupation probability, p 1]. The central consequence, however, is that there will be a value of p that is known in the literature as the critical probability, p, such that only for p>Pc there is a spanning cluster. This pc is known as the percdation threshdd of the lattice l-3]. The behavior of the various geometrical or physical properties around p as a function of p — is known as the critkal behav-... 

The question that is expected to be relevant to systems (such as the composites) in the continuum is What happens if the resistors in the system do not all have the same Tq value Since the resistors in the LNB model are connected in series, the fink s resistance Rx can be represented by (ro)Ii, where (ro) is the average of the individual resistors in the system and thus in the link. TTiis simple average has a meaning only if we assume (as we will do throughout) that there is no correlation between the position of the resistors in the system and their values. Now we note that each group of pc resistors that is chosen randomly provides a percolation spanning cluster and this applies in particular to the Pc resistors of the lowest values in... 

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