Site-bond percolation

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. 

Figure 5. Critical volume flow diagram for the site-bond percolation about a gradient of gel-sol sites.

The site-bond percolation can be given as an extension of the site percolation, which accounts for the sol-gel transition of the systems comprised of monomer and solvent. The result of the site-bond percolation over a lattice model is shown in Figure 2, where the probabihty of site occupation, and that of the bond formation, p were varied independently. The prediction of sol-gel transition like Figure 2 has been compered with the experimental results for the solution of pelystyrene and carbon disulfide. (Tan et al., 1983)... 

Fig. 2. Phase diagram of random site- bond percolation.

The percolation models represent many features of a gelling system, but there are obviously many differences. For example, in bond percolation, the lattice is assumed to have a monomer on every site, whereas gelation generally occurs in dilute systems. This deficiency is addressed by site-bond percolation, in which the sites are randomly populated with monomers and solvent molecules. As the concentration of solvent rises from zero, it is found [35] that Pc increases continuously from the bond-percolation to the site-percolation threshold. (See Fig. 13.) The remarkable (and convenient) fact... 

Tablet formation can be imagined as a combination of site and bond percolation phenomena. The formation of a tablet during compression can be described as a site-bond percolation phenomenon. The volume of the matrix is supposed to be spanned by a three-dimensional virtual lattice with lattice spacing of the order of a molecular diameter. This interpretation is more rigorous than an earlier one assuming a lattice spacing of approximately mean particle size, and takes into accoxmt a particle size distribution, i.e., a distribution of cluster sizes [76]. After pouring particles/granules to be compacted into the die, the lattice sites are either empty forming pores or occupied by molecules forming clusters.

Percolation transition of hydration water is intrinsically a site-bond percolation problem. At some temperature, percolation transition occurs upon increase in the surface coverage C, which is analogue of the occupancy variable p. At low coverages, only finite clusters are present in the system, whereas there is an infinite cluster above the percolation threshold. In Fig. 66, typical arrangement of water molecules, adsorbed at hydrophilic plane, is shown for three surface coverages. Visual inspection does not allow determination of the percolation threshold. This can be done by the analysis of various cluster properties for a system of a given dimensionality [396]. As hydration water is not a strict 2D system, the reliable estimation of a percolation threshold assumes an independent use of several criteria. 

A summary of these ten examples shows how the random percolation problem can be modified The critical exponents change only if the modification introduced can be seen on a scale which may become infinitely large, as in particular at the critical consolute point of phase separation. Otherwise, the modification concerns only inessential details and does not change the critical exponents. In some sense, the correlated site-bond percolation model described in Chapter D is only a further generalization of modifications 1 and 2 above providing similar results for the critical exponents. 

The theory of random-bond percolation in Sect. C.II. assumes that every site is occupied by a monomer, and bonds between monomers are formed randomly. In a real gel, besides the f-funtional monomers, also solvent molecules are usually present. In order to take this solvent into account in a first approximation, one can allow the sites to be oompied by a monomer with a probability

solvent molecule otherwise, with probability — nearest-neighbor monomers may form a bond with probability p whereas no bonds emanate fi-om or lead to the solvent molecules. The original random-bond percolation model is thus transformed into a random site-bond percolation in which the clusters consist of randomly distributed monomers connected by random bonds. 

If we assume that the monomers of the site-bond problem described above are no longer distributed randomly but instead are distributed as in a lattice gas (interactions between nearest neighbors) in thermal equilibrium at temperature T, then we obtain Ising-corre-lated site-bond percolation the most general percolation problem discussed... 

Fig. 6. Phase diagram of random site-bond percolation in the simple cubic lattice. (Monte Carlo simulation )...

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