Percolation threshold, interconnected

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. 

Reticulate composites comprising needles of TTT-TCNQ (tetrathiotetra-cene-tetracyaiioquinodimethane) in poly(bisphenol-A-carbonate) were first described in the early 1980s. The composites were prepared by casting thin films from a solution of the components in o-dichloromethane. The composite contained a network of fine interconnected needle crystals of the CT salt and had a percolation threshold of 2-3 mass% of the CT salt, as shown in Fig. 8.22. 

At concentrations near f,., the structure of the interconnected fibrillar network appears to be self-similar i.e. it looks the same at any degree of magnification [61,279]. The appearance of self-similarity is consistent with the well-known result that all systems are fractal near the percolation threshold on length scales below the correlation length of the percolating cluster [277]. 

For the 2- and 3-dimensional cases the channel interconnections allow a substantial fraction of the micropore capacity to be accessible below a percolation threshold in blocking probability. Above this percolation threshold, the accessible micropore volume is limited to the perimeter region of the ciystallite [82]. 

The percolation threshold is denoted as Xc. It depends on structural properties of the composite. For densely packed composite structures one finds Xc 0.19 (for the site percolation) [91]. Weakly interconnected random packings or pore spaces like those resembling a regular... 

Dependencies of the three main parameters on the electrolyte volume portion are depicted in Fig. 15. Here, a percolation threshold Xc = 0.1 is taken, implying a highly interconnected composite structure. The carbon/catalyst volume portion is kept fixed at Xm = 0.3, a value that lies well above the percolation threshold. Consistent with the small percolation threshold a large value M = 8-10 was presumed. The residual diffusivity coefficient was taken... 

For completeness, let us note that if the volume fraction of the dispersed phase becomes sufficiently large, it is expected that the interactions between the globules will affect their shape. As these Interactions become greater, a percolation threshold occurs. The electrical conductivity of the system increases steeply at the threshold because of the transient interconnections between an infinite number of globules of water. 

Network models are closely related to percolation models, which are dealt with under Emerging Areas . Sahimi (1995) and Berkowitz and Ewing (1998) have traced the development of both types of model, and have summarized the links between them. For the purposes of this review, a network is defined as a system of interconnected elements well above the percolation threshold (i.e., there are many connected paths through the network). Network models can be categorized as (i) uniform shape and uniform size distribution (Fig. 3-17A), (ii) uniform shape and variable size distribution (Fig. 3-17B), and (iii) variable shape and variable size distribution (Fig. 3-17C). 

Crete surface to the bulk of the concrete. Permeability is high (Figure 1.6) and transport processes like, e. g., capillary suction of (chloride-containing) water can take place rapidly. With decreasing porosity the capillary pore system loses its connectivity, thus transport processes are controlled by the small gel pores. As a result, water and chlorides will penetrate only a short distance into concrete. This influence of structure (geometry) on transport properties can be described with the percolation theory [8] below a critical porosity, p, the percolation threshold, the capillary pore system is not interconnected (only finite clusters are present) above p the capillary pore system is continuous (infinite clusters). The percolation theory has been used to design numerical experiments and apphed to transport processes in cement paste and mortars [9]. 

There are two types of conductive adhesives conventional materials that conduct electricity equally in all directions (isotropic conductors) and those materials that conduct in only one direction (anisotropic conductors). Isotropically conductive materials are typically formulated by adding silver particles to an adhesive matrix such that the percolation threshold is exceeded. Electrical currents are conducted throughout the composite via an extensive network of particle-particle contacts. Anisotropically conductive adhesives are prepared by randomly dispersing electrically conductive particles in an adhesive matrix at a concentration far below the percolation threshold. A schematic illustration of an anisotropically conductive adhesive interconnection is shown in Fig. 1. The concentration of particles is controlled such that enough particles are present to assure reliable electrical contacts between the substrate and the device (Z direction), while too few particles are present to achieve conduction in the X-Y plane. The materials become conductive in one direction only after they have been processed under pressure they do not inherently conduct in a preferred direction. Applications, electrical conduction mechanisms, and formulation of both isotropic and anisotropic conductive adhesives are discussed in detail in this chapter. 

Most commercially available anisotropically conductive adhesives are formulated on the bridging concept, as illustrated in Fig. 1. A concentration of conductive particles far below the percolation threshold is dispersed in an adhesive. The composite is applied to the surface either by screen printing a paste or laminating a film. When a device is attached to a PWB, the placement force displaces the adhesive composite such that a layer the thickness of a single particle remains. Individual particles span the gap between device and PWB and form an electrical interconnection. For successful implementation of anisotropically conductive adhesives, the concentration of metal particles must be carefully controlled such that a sufficient number of particles is present to assure reliable electrical conductivity between the PWB and the device (Z direction) while electrical isolation is maintained between adjacent pads (X,Y directions). 

The transition from a system of isolated droplets to an interconnected bicontinuous structure is often denoted as a percolation process. This phenomenon has been studied in much detail for the systems AOT-water-hydrocarbon [91-94] and didodecyldimethylammonium bromide-water-hydrocarbon [95]. The clustering of the droplets at the percolation threshold typically leads to an increase in viscosity [96] (as already discussed above). The percolation threshold is typically located at the point where in a logarithmic plot of viscosity versus volume fraction of droplets the slope attains a maximum. This means that a plot of (1///) versus O displays a maximum that coincides with the percolation... 

The physical model can be used to describe trends seen in experimental data. For example, the interconnectivity of the cluster network is predicted to have a profound effect on a membrane s transport properties. The percolation threshold for conductivity should increase when the clusters become smaller, which could be due to a stiflfer and/or more crystalline polymer matrix. These smaller clusters would also mean that the membrane would exhibit lower electro-osmotic coefficients, larger liquid water uptakes, and a greater dependence of the various properties on water content than in Nafion . In fact, these predictions are what is seen in such systems as sulfonated polyetherketones [19, 72] and Dow membranes [73, 74] or when the equivalent weight [22] or drying temperature [4, 6] of Nafion is increased. 

Isotropic adhesives conduct current equally in all directions and are the most common and widely used in industry. The anisotropic types, also referred to as z-direction adhesives or anisotropic-conductive adhesives (ACA), although filled with metal particles, are filled at much lower levels (0.5%-5% by volume) than isotropic types (filled 25%-30% by volume). The volume Iraction of filler is well below the percolation threshold at which the adhesive becomes highly conductive in all directions. Because of the low volume Iraction of metal particles, there are no continuous electrically conductive paths in the x-y plane. During the connection process, the anisotropic adhesive, either as a film or paste, is positioned between a flip-chip bumped die or a tape-automated bonded (TAB) die and the corresponding pads of an interconnect substrate. Pressure and heat are... 

Metal sulfides have been incorporated into zeolites X, Y, and A. [222, 223] It was concluded that superclusters made up of quantized semiconductors can be formed by comjdetely filling the internal zeolite cavities to the percolation threshold. The supercluster is created by the three dimensional interconnection of... 

Atomic and electronic processes that occur at the polymer-nanoparticle interface largely determine the unique properties of nanocomposite. These materials become electrical conductors only at definite component ratios when conducting chain-type coagulated structures are formed instead of matrix systems. In other words, the fractal clusters formed upon cohesion of nanoparticles serve as ciurent-conducting channels. The highest conductivity is attained when the metallopoly-meric material is permeated by interconnected chains of conducting particles that are in contact. This forms an electrical percolation network that exceeds the percolation threshold. As a rule, this is achieved at a nanoparticles content of 50 vol%. 

Other notions that are very important in consideration of pore systems are percolation and percolation threshold. The permeability of the system composed of strongly disordered pores is non-linearly related to pore number and total volume, and that relation may be characterized by a threshold - as long as a certain pore density is not reached, the system remains impermeable. When the density becomes higher than that critical value, then permeability appears and increases rapidly (Stauffer 1986). As this concerns the connections between pores, these may be distinguished interconnected pores, closed pores and pores closed at one end. The roles of these kinds of pores in the flow of fluids and gases across material are different. 

Both CGMD and DPD simulations indicate a percolation threshold for hydrophilic domains at A 4, which corresponds to a 10% volume fraction of water. The low value of the conductivity percolation threshold in Nation PEMs has been concluded as well from proton conductivity studies (Cappadonia et al., 1994, 1995). The high interconnectivity of water channels and the peculiarities of swelling and reorganization of the polymer matrix upon water uptake promote percolation at low water content (Eikerling et al., 1997, 2008). 

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