Percolation behaviour

Thus, in summary, self diffusion measurements by Lindman et a (29-34) have clearly indicated that the structure of microemulsions depends to a large extent on the chain length of the oosurfactant (alcohol), the surfactant and the type of system. With short chain alcohols (hydrophilic domains and the structure is best described by a bicontinuous solution with easily deformable and flexible interfaces. This picture is consistent with the percolative behaviour observed when the conductivity is measured as a function of water volume fraction (see above). With long chain alcohols (> Cg) on the other hand, well defined "cores" may be distinguished with a more pronounced separation into hydrophobic and hydrophilic regions. 

The electrical conductivity of two-phase, incompatible polymer blends containing carbon black has been shown to depend on the relative affinity of the conductive particles to each of the polymer components in the blend, the concentration of carbon black in the filler-rich phase, and the structural continuity of this phase [82]. Hence, by judicious manipulation of the phase microstructure, these three-phase filled composites can exhibit double percolation behaviour. 

Composites containing metal particles provide a clear example of the percolation behaviour described in Section 8.3. By way of illustration we can cite the results shown in Table 8.1 for a dispersion of approximately spherical nickel particles (about 10 pm in diameter) in a low-density polyethylene. Here conductivity is lost entirely at concentrations below about 20% by volume of metal, corresponding to about 70% by mass. Nevertheless, conductive paints, which are frequently used for painting electrodes on to electrical test specimens and devices, work in just this way. 

Percolation theory (the model supposes that asymmetrically structured carbon-black particles are statistically distributed and results in percolation in accordance with probability laws) [3,27,31,32,33,34,35], Although this theory is the most widespread one, it lacks important experimental fundamentals and cannot describe the multitude of factors affecting percolation behaviour,... 

Fig. 8.3 Schematic representation of the percolative behaviour of a nanocomposite, indicating the different regions as a function of the electrical conductivity and the total amount of filler present. A schematic representation of the network creation for artificial spherical particles can also be seen for the different conductivity regions...

For the understanding of the percolative behaviour of conductive fillers in immiscible polymer blends, the details of the filler distribution in the different phases are important. The critical condition for achieving the all-important co-continuity of the conductive filler-rich phase has been suggested to be determined by the viscosity of the phases as (Paul and Barlow 1980)... 

Percolation behaviour of ultrahigh molecular weight polyethylene/multi-walled carbon nanotubes composites. European Polymer Journal, 43, 949-958. 

R. Schuler, J. Petermann, K. Schulte, H.-P. Wentzel (1997) Agglomeration and electrical percolation behaviour of carbon black dispersed in epoxy resin, J. Appl. Polym. Sci. 63, 1741. 

With the U-Type systems (i.e. with the low chain alcohols) the trends in the conductivity - curve are consistent with percolative conduction originally proposed to explain the behaviour of conductance of conductor-insulator composite materials (27). In the latter model, the effective conductivity is practically zero as long as the conductive volume fraction is smaller than a critical value called the percolation threshold, beyond which k suddenly takes a non-zero value and rapidly increases with increase of Under these conditions. 

Present leaching behaviour can be characterized, although the applicability of many leach tests with fly ash is doubtful. Special tests are often more appropriate. For example, Fig. 5 shows a simple packed bed, through which lea-chant percolates and which acts like a lysimeter leachate can be sampled periodically through the... 

There is no adequate theory of the Neel temperature of a random distribution of centres in a dilute alloy, of indeed one exists. For higher concentrations of the magnetic matrix, with the assumption that only nearest neighbours interact, there is considerable theoretical work, giving a percolation limit , the concentration c0 at which long-range order disappears. The behaviour of TN is as (c —c0)12. For details see Brout (1965), Elliott and Heap (1962) and Klein and Brout (1963). 

At the time of writing, the only evidence for critical fluctuations near the consolute point known to us comes from the work of Damay (1973). The thermopower of Na-NH3 plotted against T at the critical concentration is shown in Fig. 10.21. We conjecture that this behaviour is due to long-range fluctuations between two metallic concentrations, and that near the critical point, where the fluctuations are wide enough to allow the use of classical percolation theory, the... 

We study here the A + 5B2 —> 0 reaction upon a disordered square lattice on which only a certain fraction S of lattice sites can be accessed by the particles (the so-called active sites). We study the system behaviour as a function of the mole fractions of A and B in the gas phase and as a function of a new parameter S. We obtain reactive states for S > Sq where Sq is the kinetically defined percolation threshold which means existence of an infinite cluster of active sites. For S < Sq we obtain only finite clusters of active sites exist. On such a lattice all active sites are covered by A and B and no reaction takes place as t —> 00. 

Caraballo, I., Fernandez-Arevalo, M., Holgado, M. A., and Rabasco, A. M. (1993), Percolation theory Application to the study of the release behaviour from inert matrix system, Int. I. Pharm., 96,175-181. 

The all or nothing feature of metal powder composites is very much a feature of conductive composite systems. In order to understand this behaviour, most theories borrow from percolation theory (Broadbent and Hamersley, 1957), which was originally developed as a model for predicting fluid permeation through porous media. The percolation model is based on having a medium... 

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