Percolation theory, determination

The main notion of the percolation theory is the so-called percolation threshold Cp — minimal concentration of conducting particles C at which a continuous conducting chain of macroscopic length appears in the system. To determine this magnitude the Monte-Carlo method or the calculation of expansion coefficients of Cp by powers of C is used for different lattices in the knots of which the conducting par-... 

It took a long time before the percolation theory could be proved to be the better one in most cases. The reason for this delay resulted in part from the fact that until ten years ago the size exclusion chromatography (SEC) with on-hne light scattering was not sufficiently well developed. A direct molar mass determination is, however, imperative, since the separation in SEC is due to the hydrodynamic volume of the particles. A branched macromolecule has, however, a significantly higher molar mass than a linear one of the same hydrodynamic volume. Since 1989 a number of results have been reported which all strongly supported the percolation theory [109,111-116]. 

As discussed above, hysteresis loops can appear in sorption isotherms as result of different adsorption and desorption mechanisms arising in single pores. A porous material is usually built up of interconnected pores of irregular size and geometry. Even if the adsorption mechanism is reversible, hysteresis can still occur because of network effects which are now widely accepted as being a percolation problem [21, 81] associated with specific pore connectivities. Percolation theory for the description of connectivity-related phenomena was first introduced by Broad-bent et al. [88]. Following this approach, Seaton [89] has proposed a method for the determination of connectivity parameters from nitrogen sorption measurements. 

G. C. Wall and R. J. C. Brown,/. Colloid Interface Sci., 82,141 (1981). The Determination of Pore-Size Distributions from Sorption Isotherms and Mercury Penetration in Interconnected Pores The Application of Percolation Theory. 

Computer modelling of physisorption hysteresis is simplified if it is assumed that pore filling occurs reversibly (i.e. in accordance with the Kelvin equation) along the adsorption branch of the loop. Percolation theory has been applied by Mason (1988), Seaton (1991), Liu et al., (1993, 1994), Lopez-Ramon et al., (1997) and others (Zhdanov et al.,1987 Neimark 1991). One approach is to picture the pore space as a three-dimensional network (or lattice) of cavities and necks. If the total neck volume is relatively small, the location of the adsorption branch should be mainly determined by the cavity size distribution. On the other hand, if the evaporation process is controlled by percolation, the location of the desorption branch is determined by the network coordination number and neck size distribution. 

According to percolation theory, the effect of a reduction in the drug particle size should be similar to an increase in the excipient particle size in a binary system It may be expected that the relative particle size of the component, but not its absolute particle size, will determine the properties of the system. 

Since the development of the equation, it has been tried to derive further information from it. Rees and Rue [129] determined the area under the Heckel plot. Duberg and Nystrom [137] used the nonlinear part for characterization of particle fracture. Paronen [138] deduced elastic deformation from the appearance of the Heckel plot during decompression. Morris and Schwartz [139] analyzed different phases of the Heckel plot. Imbert et al. [134] used, in analogy to Leuenberger and Ineichen [14], percolation theory for the compression process as described by the Heckel equation. Based on the Heckel equation, Kuentz and Leuenberger [135,140] developed a new derived equation for the pressure sensitivity of tablets. 

The pore network connectivity is usually determined by gas sorption analysis [2-4] or mercury intrusion [5] based on percolation theory. Recently, Ismadji and Bhatia [6] have successfully employed the liquid phase adsorption isotherms to determine the pore network connectivity and the pore size distribution of three commercial activated carbons. In our recent study [7], the pore network connectivity of three commercial activated carbons was characterized using liquid phase adsorption isotherms of eight different compounds. In that study we used ester molecules with complex structure, as probe molecules. 

In the rest of this chapter, we will discuss briefly the theoretical ideas and the models employed for the study of failure of disordered solids, and other dynamical systems. In particular, we give a very brief summary of the percolation theory and the models (both lattice and continuum). The various lattice statistical exponents and the (fractal) dimensions are introduced here. We then give brief introduction to the concept of stress concentration around a sharp edge of a void or impurity cluster in a stressed solid. The concept is then extended to derive the extreme statistics of failure of randomly disordered solids. Here, we also discuss the competition between the percolation and the extreme statistics in determining the breakdown statistics of disordered solids. Finally, we discuss the self-organised criticality and some models showing such critical behaviour. 

However, as Everett (60) pointed out, the analogy of a pore as a narrownecked bottle is overspecialized, and in practice a series of interconnected pore spaces rather than discrete bottles is more likely. In the latter case, a void with the radius r > rp is empty during the desorption process only if it is connected with the outer surface by a chain of voids and necks with r > Kp. Thus, the emptying of a given pore is not solely determined by its immediate characteristics. Hence, a correct analysis of the desorption branch of the isotherm should take into account the three-dimensional interconnection of various voids. This problem can be solved by using percolation theory, as has been done by W ll and Brown (14), Kheifets and Nei-mark (15-17), Mason (18-21), Fenelonov et al. (22-25), Palar and Yortsos (26,27), Mayagoitia et al. (28-32), Yanuka (33), and Seaton (34). 

These necks correspond to unblocked bonds in percolation theory. Thus, the fraction of unblocked bonds for the sublattice of voids with r > rp is determined by the relationship... 

A great number of studies have been published to deal with relation of transport properties to structural characteristics. Pore network models [12,13,14] are engaged in determination of pore network connectivity that is known to have a crucial influence on the transport properties of a porous material. McGreavy and co-workers [15] developed model based on the equivalent pore network conceptualisation to account for diffusion and reaction processes in catalytic pore structures. Percolation models [16,17] are based on the use of percolation theory to analyse sorption hysteresis also the application of the effective medium approximation (EMA) [18,19,20] is widely used. 

The Soxhlet extraction method discussed in Section 6.6 can be used to separate the sol and gel fractions of a gel in the gelation regime, allowing direct determination of the gel fraction gel- Percolation theory expects the molar mass of a network strand M to be the same as the characteristic molar mass in the sol fraction. Hence, M can be determined by the size exclusion chromatography methods of Section 6.6, applied to the sol fraction. Equation (7.93) is tested in Fig. 7.19, where the shear modulus is shown to be proportional to Pgei/M. ... 

The techniques given in Table 4.2 are well established and have been sub-divided into those which are described as either static or dynamic. We feel this distinction is of particular importance in the characterisation of the porous structure of membranes. Here the performance is determined by the complex link between the structural texture and transport behaviour. An insight into this complexity is frequently provided by dynamic techniques, which are not restricted by the limited quantity of membrane material and are sensitive to the active pathways through the porous structure. Further developments are required in this area both in the improvement of existing techniques and introduction of new techniques. Progress will also come from advances in the theory and modelling of flow behaviour in such porous media, which involve percolation theory and fractal geometry for example. With the refinement of such... 

A method of using GCMC simulation in conjunction with percolation theory [74,75] has been suggested for simultaneous determination of the PSD and network connectivity of a porous solid [76]. In this method, isotherms are measured for a battery of adsorbate probe molecules of different sizes, e.g., CH4, CF4, and SFg. As illustrated in Fig. 9a, the smaller probe molecules are able to access regions of the pore volume that exclude the larger adsorbates. Consequently, each adsorbate samples a different portion of the adsorbent PSD, as shown in Fig. 9b. By combining the PSD results for the individual probe gases with a percolation model, an estimate of the mean connectivity number of the network can be obtained [76]. 

Wall, G.C. and Brown, R.J.C. (1981). The determination of pore-size distributions from sorption isotherms and mercury penetration in interconnected pores the application of percolation theory. J. Colloid Interface Sci., 82, 141—9. 

Statistical network models were first developed by Flory (Flory and Rehner, 1943, Flory, 1953) and Stockmayer (1943, 1944), who developed a gelation theory (sometimes referred to as mean-field theory of network formation) that is used to determine the gel-point conversions in systems with relatively low crosslink densities, by the use of probability to determine network parameters. They developed their classical theory of network development by considering the build-up of thermoset networks following this random, percolation theory. 

In each of the above analyses, the pores were considered as parallel sets of large and small pores without interconnection between the separate sets. However, most void structures comprise a network in interconnected void spaces and "network effects" will diaate the potential implications of changes in pore structure. The generic influence of pore networks were analyzed by Beeckman and Froment22 based on modified Bethe tree two-dimensional networks. Based on this simulated analyses, the authors concluded that the nature of the deactivation does depend on the nature of the network structure. Sahami and Tsotsis employed percolation theory to analyze a three-dimensional network of interconnected pores and concluded that the void interconnectivity is crucial in determining the influence of network structure on the deactivation phenomena. 

The idea of novolac as an amphiphilic polymer into which the developer penetrates through a chain of hydrophilic sites has led to a mathematical treatment of resist dissolution as a percolative diffusion process. The basic ideas of the percolation theory of dissolution include the following six key facts, (i) The dissolution rate is determined by transport of ions through the percolation zone. 

The critical indices estimated from these relations fall into the admissible ranges of variation P = 0.39-0.40, V = 0.8-0.9, and t = 1.6-1.8, determined in terms of the percolation model for three-dimensional systems. The researchers [7] noted that not only numerical values but also the meanings of these values coincide. Thus the index P characterises the chain structure of a percolation cluster. The 1/p value, which serves as the index of the first subset of the fractal percolation cluster in the model considered [7], also determines the chain structure of the cluster. The index v is related to the cellular texture of the percolation cluster. The 2/df index of the second subset of the fractal percolation cluster is also associated with the cellular structure. By analogy, the index t defines the large-cellular skeleton of the fractal percolation cluster. The relationship between the critical percolation indices and the fractal dimension of the percolation cluster for three-dimensional systems and examples of determination of these values for filled polymers are considered in more detail in the book cited [7]. Thus, these critical indices are universal and significant for analysis of complex systems, the behaviour of which can be interpreted in terms of the percolation theory. 

Methods to determine the electronic conductivity of powdered battery materials and their mixtures have been studied intensively. To mathematically describe the electronic conductivity of the active electrode material ntixed with different amounts of conductive carbon, logarithmic equations, the percolation theory (PT), and the effective medium theory (EMT) " may be considered to be rules. 

There are several exact results available in this model to serve as checks on approximate calculations of and L(E). In the limit of small x or 6, all averaged quantities may be expanded in powers of a small parameter. For arbitrary x and 5, a number of moments of the density of states can be calculated exactly (Velicky, 1968). When 5 > 2, the density of states is split into two separated sub-bands, centered about and e , each of width B. Thus in the limit 5 -> a site containing a B atom is forbidden to an electron with energy near and percolation theory (Frisch, 1963) may be used to determine the probability that such an electron is trapped or free to move across the crystal. When x is greater than x, the criticd value for the onset of percolation, there will be extended states in the A sub-band. Since x, is less than 1/2 for all three-dimensional lattices, we observe that at least one of the strongly split sub-bands will always contain extended states, in contrast to the complete localization observed for r>Fg in the Lorentzian model of the preceding section. 

---