Percolation basic properties

The research on nanocarbons dispersed in polymer matrices in recent years has shown that this route is very efficient at small volume fractions above electrical percolation, where it can be the basis for new composite functionalities in terms of processing and properties. It is also clear that there is an inherent difficulty in dispersing these nanoscopic objects at high volume fractions, which therefore limits composite absolute properties to a very small fraction of those of the filler. Independent of their absolute properties, composites based on dispersed nanocarbons have served as a test ground to understand better the basic interaction between nanocarbons and polymer matrices, often setting the foundation to study more complex composite structures, such as those discussed in the following sections. 

Since percolation is a property of macroscopic many-particle systems, it can be analyzed in terms of statistical mechanics. The basic idea of statistical mechanics is the relaxation of the perturbed system to the equilibrium state. In general the distribution function p(p,q t) of a statistical ensemble depends on the generalized coordinates q, momentum p, and time t. However, in the equilibrium state it does not depend explicitly on time [226-230] and obeys the equation... 

It will thus be seen that there is a strong relationship between the critical volume concentration and the properties of the matrix. But was it not one of the basic premises of percolation theory that no interactions took place between matrix and carbon-black particles How else could one explain a random distribution ... 

In the previous section, we have briefly reviewed the basics of the lattice models of percolation where the systems have sites (or bonds) that are occupied or empty. In continuum percolation, the systems are composed of objects (or members) that are randomly placed in space. These objects may be of various sizes and shapes. If the latter are nonisotropic, one also considers the distribution of their orientations. Correspondingly, the values of the physical parameters that determine the bonding between two objects may vary from bond to bond, depending, say, on the local geometry and/or properties of the bond. 

In Section 23.2 was discussed the theory of reinforcement of polymer and elastomers which refers to the Guth-Gold-Smallwood equation (Equation (23.1)) to correlate the compound initial modulus (E ) with the filler volume fraction ( ). Moreover, it was already commented on the key roles played by the surface area and by the aspect ratio (/). Basic feature of nanofillers, such as clays, CNTs and nanographites, is the nano-dimension of primary particles and thus their high surface area. This allows creating filler networks at low concentrations, much lower than those typical of nanostructured fillers, such as CB and silica, provided that they are evenly distributed and dispersed in the rubber matrix. In this case, low contents of nanofiller particles are required to mutually disturb each other and to get to percolation. Moreover, said nanofillers are characterized by an aspect ratio /that can be remarkably higher than 1. Barrier properties are improved when fillers (such as clays and nanographites) made by... 

For materials with random morphologies, the effect of microscopic composition, i.e., of size, shape, and random distribution of phase domains and of their coimectedness, on its macroscopic properties is the topic addressed by percolation theory [129, 130], Here, we briefly describe the basic concepts of this theory and outline their application to determine the morphology and effective properties of random composite materials. We refer the interested reader to Stauffer and Aharony [46] and Sahimi [129] for detailed discussion of percolation theory and its applications. 

Electrically conductive polymer nanocomposites are widely used especially due to their superior properties and competitive prices. It is expected that as the level of control of the overall morphology and associated properties increases we will see an even wider commercialisation on traditional and totally novel applications. In this section we have discussed the basic principles of the percolation theory and the different types of conduction mechanisms, outlined some of the critical parameters of controlling primarily the electrical performance and we have provided some indications on the effect such conductive fillers have on the overall morphology and crystallisation of the nanocomposite. The latter becomes even more critical if we take into consideration that modem nanosized fillers offer unique potential for superior properties at low loadings (low percolation thresholds) but have a more direct impact on the morphology of the system. Furthermore we have indicated that similar systems can have totally different behaviour as the preparation methods, the chain conformation and the surface chemistry of the fillers will have a massive... 

So far our description of the random-coil chain basically assumes a dilute solution and we have not yet defined the term dilute solution. It has been discovered that when the concentration increases to a certain point, interesting phenomena occur chain crossover and chain entanglement. Chain crossover refers to the transition in configuration from randomness to some kind of order, and chain entanglement refers to the new statistical discovery of the self-similar property of the random coil (e.g., supercritical conductance and percolation theory in physics). Such phenomena also occur to the chain near the theta temperature. In this section, we describe the concentration effect on chain configurations on the basis of the theories advanced by Edwards (1965) and de Gennes (1979). In the next section, we describe the temperature effect, which is parallel to the concentration effect. 

Percolation theory represents the most advanced and most widely used statistical framework to describe structural correlations and effective transport properties of random heterogeneous media (Sahimi, 2003 Torquato, 2002). Here, briefly described are the basic concepts of this theory (Sahimi, 2003 Stauffer and Aharony, 1994) and its application to catalyst layers in PEFCs. 

Discussion Some years ago it was pointed out that the connectivity properties at the sol-gel transition can be described by a percolation model. The basic concept of this theory is that the mass distribution... 

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