Percolation network

It has been possible to directly image the percolation network at the surface of a CB-polymer composite. An early report is that of Viswanathan and Heaney [24] on CB in HOPE in which it was shown that there are three regions of conductivity as a function of the length L, used as a metric for the image analysis. Below I = 0.6pm, the fractal dimension D of the CB aggregates is 1.9 0.1. Between 0.8 and 2 pm, the data exhibit D = 2.6 0.1 while above 3 pm, D = 3 corresponding to homogeneous behavior. Theory predicts D = 2.53. It is not obvious that the carbon black-polymer system should be explainable in terms of standard percolation theory, or that it should be in the same universality class as three-dimensional lattice percolation problems [24]. Subsequent experiments of this kind were made by Carmona [25, 26]. 

It has been correcdy realized that most microscopies - including AFM and electrical probes - render only a two-dimensional section of a three-dimensional conducting network. Gubbels [27, 28] applied methods of image analysis to blends consisting of two polymers plus CB. The oft-observed segregation of CB to the phase boundaries or interphase was confirmed. 

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Viswanathan, R. and Heaney, M.B., Direct imaging of the percolation network in a three-dimensional disordered conductor-insulator composite, Phys. Rev. Lett., 75, 4433, 1995. 

Arbabi, S Sahimi, M, Elastic Properties of Three-Dimensional Percolation Networks with Stretching and Bond-Bending Eorces, Physical Review B 38, 7173, 1988. 

Arbabi, S Sahimi, M, Critical Properties of Viscoelasticity of Gels and Elastic Percolation Networks, Physical Review Letters 65, 725, 1990. 

It was shown, that the conception of reactive medium heterogeneity is connected with free volume representations, that it was to be expected for diffusion-controlled sohd phase reactions. If free volume microvoids were not connected with one another, then medium is heterogeneous, and in case of formation of percolation network of such microvoids - homogeneous. To obtain such definition is possible only within the framework of the fractal free volume conception. 

Martin CA, Sandler JKW, Shaffer MSP, Schwarz M-K, Bauhofer W, Schulte K, et al. Formation of percolating networks in multi-wall carbon-nanotube-epoxy composites. Composites Science and Technology. 2004 Nov 64(15) 2309-16. 

O2 diffusion through the membrane seems to be limited by the percolation network of the diffusion path, which is not only defined by the amount of water in the membrane, but also by the different chemical structure of the membranes. It is difficult to make comparisons of gaseous diffusion behavior among polymers with different structures because polymer morphology can change drastically without appreciable changes in density, and the presence of water and the hydrogen bonds formed between polymer-water moieties also has major effects on system properties. However, some points can be made from these particular studies. 

In addition to the amount of filler content, the shape, size and size distribution, surface wettability, interface bonding, and compatibility with the matrix resin of the filler can all influence electrical conductivity, mechanical properties, and other performance characteristics of the composite plates. As mentioned previously, to achieve higher electrical conductivity, the conductive graphite or carbon fillers must form an interconnected or percolated network in the dielectrical matrix like that in GrafTech plates. The interface bonding and compatibility between... 

Veerman, C. (2004) Properties of fibrillar protein assemblies and their percolating networks. PhD Thesis, Wageningen University, the Netherlands. 

This behavior can be understood if a superimposed kinetic aggregation process of primary carbon black aggregates in the rubber matrix is considered that alters the local structure of the percolation network. A corresponding model for the percolation behavior of carbon black filled rubbers that includes kinetic aggregation effects is developed in [22], where the filler concentrations and c are replaced by effective concentrations. In a simplified approach, not considering dispersion effects, the effective filler concentration is given by ... 

CNT nanocomposites, even at low CNT volume fractions, provided they form a percolating network. In such cases, it appears that SEM observations show not only the nanocomposite surface topology, but also the CNT arrangement near the surface within a thickness of even few pm. On the other hand, as for other electron microscopy methods, spectroscopy analysis can be used for imaging purposes. 

Benoit et al. 2001 (41) SWCNT Arc Discharge As-synthesized Casting CNT Loading levels 0.1 to 8 wt% Film Electrical conductivity increases with CNT content by 9 orders of magnitude from 0.1 to 8% mass fraction with percolation threshold 0.33 wt%. Temperature dependence of resistivity for 0.2 to 8 wt% SWCNT composite films showed that percolating network is affected at low temperature, enhancing the relative resistivity ... 

It is interesting to note that the (L-N-B)-model leads to similar expressions for the moduli like the VTG-model apart from the first summand of Eq. (38). However, contrary to the semi-empirical weighting functions W(y6) of the VTG-model, the corresponding density distribution function/la(y) in the (L-N-B)-model is related to the morphological structure of the filler network, i.e., the distribution of singly connected bonds in a percolation network. Unfortunately, this distribution function is not known, exactly. Therefore, a simple exponential... 

Here, yc is the strain amplitude, where half of the clusters are broken, m being an empirical exponent. Then, the storage modulus AG (y0) = G (y0) - G, at a given strain y0 results from Eqs. (73) and (74) and the power law expression for the modulus of a percolation network [62,64,91] as follows ... 

Here, pc 0.2 is the critical occupation number where a percolation network is formed and r = 3.6 is the elasticity exponent of percolation [91 ]. For large values of p (p >p<), Eq. (75) can be approximated by a function of the Havriliak-Negami type ... 

Internal protein motion (H exchange) increases from 1/1000 at 0.04 A to full solution rate at 0.15 A At 0.1-0.15 A chymotrypsin and some other enzymes develop activity At 0.15 A long-range proton movements along percolative networks, seen in dielectric measurements... 

The percolation model suggests that it may not be necessary to have a rigid geometry and definite pathway for conduction, as implied by the proton-wire model of membrane transport (Nagle and Mille, 1981). For proton pumps the fluctuating random percolation networks would serve for diffusion of the ion across the water-poor protein surface, to where the active site would apply a vectorial kick. In this view the special nonrandom structure of the active site would be limited in size to a dimension commensurate with that found for active sites of proteins such as enzymes. Control is possible conduction could be switched on or off by the addition or subtraction of a few elements, shifting the fractional occupancy up or down across the percolation threshold. Statistical assemblies of conducting elements need only partially fill a surface or volume to obtain conduction. For a surface the percolation threshold is at half-saturation of the sites. For a three-dimensional pore only one-sixth of the sites need be filled. 

In cases in which a sharp question can be formulated and so suggest a test of a mechanism of solvent participation, site-directed mutagenesis provides an experimental tool, perhaps appropriately guided by computer simulation of the protein—solvent system. The participation of fixed or random (percolative) network structures in proton movement is... 

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