Percolation theory, single

The scaling of the relaxation modulus G(t) with time (Eq. 1-1) at the LST was first detected experimentally [5-7]. Subsequently, dynamic scaling based on percolation theory used the relation between diffusion coefficient and longest relaxation time of a single cluster to calculate a relaxation time spectrum for the sum of all clusters [39], This resulted in the same scaling relation for G(t) with an exponent n following Eq. 1-14. 

For either conventional polycrystalline semiconductors or nanotubes and nanowires to be successful, the development of model and simulation tools that can be used for device and circuit design as well as for predictive engineering must be available. Since these devices are not necessarily based on single wires or single crystals, but rather on an ensemble of particles, the aggregate behavior must be considered. Initial efforts to provide the necessary physical understanding and device models using percolation theory have been reported.64,65... 

A careful examination of the Stockmayer and the percolation distributions reveals that both theories gives the same type of distribution [110]. In terms of the two exponents in Eqs. 52 and 53, the percolation calculation yields t=2.2 and 0 0.44, and the Stockmayer distribution yields r 2.5 and o 0.50. These differences in the exponents appear to be small in a double logarithmic plot, but they cause significant differences in the absolute values for w(x) when 3-4 decades in the degree of polymerization are covered. Another point is that the cut-off function could be calculated analytically in the FS-theory to be a single exponential function [110], while the percolation theory could only make a guess about its shape [7]. 

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. 

In actual use for mobility control studies, the network might first be filled with oil and surfactant solution to give a porous medium with well-defined distributions of the fluids in the medium. This step can be performed according to well-developed procedures from network and percolation theory for nondispersion flow. The novel feature in the model, however, would be the presence of equations from single-capillary theory to describe the formation of lamellae at nodes where tubes of different radii meet and their subsequent flow, splitting at other pore throats, and destruction by film drainage. The result should be equations that meaningfully describe the droplet size population and flow rates as a function of pressure (both absolute and differential across the medium). 

The single most important concept in percolation theory is the percolation threshold, which may be understood as follows. First, assume that p (the probabihty of occupation) is very small. In this case, very few sites are occupied, and the clusters are all of finite size (each cluster is surrounded by many empty sites). Then assume that p is close to unity, so that almost all sites are occupied. Reversing the argument, it is realized that a few holes of finite size exist in an otherwise unbounded cluster that spans the... 

It should be noted that phase inversion prediction models focus on only a single composition, whereas in reality, co-continuous structures are observed over a composition range. Considering the definition of co-cmitinuous structure and equations based on the percolation theory, a model was proposed to correlate a continuity index (/) with the volume fraction at onset of co-continuity (0,- ) (see Table 7.3) (Lyngaae-Jorgensen et al. 1999). Numerical simulation predicted cr to be about 0.2 for classical percolation in three-dimensional systems (Dietrich and Airmon 1994 Potschke and Paul 2003). 

The approaches considered allow modeling of the primary texture of PS and the processes, limited by individual PBUs that mainly correspond to level III and partially to level IV in the hierarchical system of models (see Section 9.6.3). PBUs are identical in regular PSs, and simulation of numerous processes may be reduced to analysis of a process in a single PBU/C or PBU/P. An accurate modeling of the processes in irregular PSs requires the studies of the properties of structure and properties of the ensembles (clusters) of particles and pores (level IV of the system of models) and the lattices of such clusters (levels V to VII of the system of models). Let us consider the composition of clusters on the basis of fractal [127], and the lattices on the basis of percolation [8] theories. 

A novel and simple method for determination of micropore network connectivity of activated carbon using liquid phase adsorption is presented in this paper. The method is applied to three different commercial carbons with eight different liquid phase adsorptives as probes. The effect of the pore network connectivity on the prediction of multicomponent adsorption equilibria was also studied. For this purpose, the Ideal Adsorbed Solution Theory (lAST) was used in conjuction with the modified DR single component isotherm. The results of comparison with experimental data show that incorporation of the connectivity, and consideration of percolation processes associated with the different molecular sizes of the adsorptives in the mixture, can improve the performance of the lAST in predicting multicomponent adsorption equilibria. 

While the Choi and Schowalter [113] theory is fundamental in understanding the rheological behavior of Newtonian emulsions under steady-state flow, the Palierne equation [126], Eq. (2.23), and its numerous modifleations is the preferred model for the dynamic behavior of viscoelastic liquids under small oscillatory deformation. Thus, the linear viscoelastic behavior of such blends as PS with PMMA, PDMS with PEG, and PS with PEMA (poly(ethyl methacrylate))at <0.15 followed Palierne s equation [129]. From the single model parameter, R = R/vu, the extracted interfacial tension coefficient was in good agreement with the value measured directly. However, the theory (developed for dilute emulsions) fails at concentrations above the percolation limit, 0 > (p rc 0.19 0.09. 

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