Percolation threshold concentration

Figure 4.15. Random square lattice of sites (A) and bonds (B), corresponding to the same concentration of passing bonds (solid lines). The concentration of sites is about 71% in lattice A, and that of bonds 50% in both lattices. Thus, the B lattice is at a percolation concentration, whereas the A lattice is above the percolation threshold.176-177 Note the aggregation of bonds in lattice A relative to lattice B.

This chapter will focus on infinitely-extended suspensions in which potential complications introduced by the presence of walls are avoided. The only wall-effect case that can be treated with relative ease is the interaction of a sphere with a plane wall (Goldman et ai, 1967a,b). The presence of walls can lead to relevant suspension rheological effects (Tozeren and Skalak, 1977 Brunn, 1981), which result from the existence of particle depeletion boundary layers (Cox and Brenner, 1971) in the proximity of the walls arising from the finite size of the suspended spheres. Going beyond the dilute and semidilute regions considered by the authors just mentioned is the ad hoc percolation approach, in which an infinite cluster—assumed to occur above some threshold particle concentration—necessarily interacts with the walls (cf. Section VI). 

On the other hand, higher SME resulted in all presented composites in a better CNT macrodispersion. At the same time, the CNTs were shortened significantly with increasing SME, leading to higher resistivity values or even crossing the percolation concentration. In composites with shorter nanotubes, an increase in electrical percolation threshold was shown. 

It is seen from the table that for the small effective mass p, = 0.1 and 82 > 3 corresponding critical percolation concentrations and are quite reasonable (less than 1 %, which is less than 10 cm (3D case) or 10 cm (2D case)), while for the case p = 1 m<. they may be very high (more than 20 % for 82 = 3 and more that 5 % for 82 = 10). Note, that in accordance with the results of Volnianska and Boguslawski [31] and Osorio-Guillen et al. [50] from the first principles calculations, the concentration of Ca vacancies in CaO percolation threshold is 4.9 % that agrees qualitatively with = 5.18 % in Table 4.1. We hope that for the values 82 and p characteristic for CaO the agreement could be better. 

The author together with V. Sukharev has shown [21] that the percolation behavior of nanocomposites conductivity is different from the one typical for composites containing larger particles (Fig. 4). It has been demonstrated that the threshold filler concentration values are lower for nanocomposites than for composites with micron-size particles, and the slope of the curve in the... 

From the viewpoint of their conduction and mechanical joining, ACAs are similar to ICAs, except that they have lower concentrations of conductive particles. This lower concentration provides unidirectional conductivity in the vertical or z-direction (perpendicular to the plane of the substrate), which is why they are called anisotropic conductive adhesives. In the same way, ACA materials are prepared by dispersing electrically conductive particles in an adhesive matrix at a concentration far below the percolation threshold. The concentration of particles is controlled, so that sufficient particles are present to ensure reliable electrical conductivity between the assembled parts in the z-direction, while insufficient particles are present to achieve percolation conduction in the x-y plane (Kim et al. 2008b). O Figure 50.6 shows a schematic description of an ACA interconnect, showing the electrical conductivity in the... 

The main notion of the percolation theory is the so-called percolation threshold Cp — minimal concentration of conducting particles C at which a continuous conducting chain of macroscopic length appears in the system. To determine this magnitude the Monte-Carlo method or the calculation of expansion coefficients of Cp by powers of C is used for different lattices in the knots of which the conducting par-... 

Experimental dependences of conductivity cr of the CPCM on conducting filler concentration have, as a rule, the form predicted by the percolation theory (Fig. 2, [24]). With small values of C, a of the composite is close to the conductivity of a pure polymer. In the threshold concentration region when a macroscopic conducting chain appears for the first time, the conductivity of a composite material (CM) drastically rises (resistivity Qv drops sharply) and then slowly increases practically according to the linear law due to an increase in the number of conducting chains. 

The maximum values of the percolation threshold are characteristic of matrix systems in which the filler does not form the chain-like structures till large concentrations are obtained. In practice, statistical or structurized systems are apparently preferable because they become conductive at considerably smaller concentrations of the filler. The deviation of the percolation threshold from the values of Cp to either side for a statistical system ( 0.15) can be used to judge the nature of filler distribution. 

The composites with the conducting fibers may also be considered as the structurized systems in their way. The fiber with diameter d and length 1 may be imagined as a chain of contacting spheres with diameter d and chain length 1. Thus, comparing the composites with dispersed and fiber fillers, we may say that N = 1/d particles of the dispersed filler are as if combined in a chain. From this qualitative analysis it follows that the lower the percolation threshold for the fiber composites the larger must be the value of 1/d. This conclusion is confirmed both by the calculations for model systems [27] and by the experimental data [8, 15]. So, for 1/d 103 the value of the threshold concentration can be reduced to between 0.1 and 0.3 per cent of the volume. 

In pressing, the threshold concentration of the filler amounts to about 0.5% of volume. The resulting distribution of the filler corresponds, apparently, to the model of mixing of spherical particles of the polymer (with radius Rp) and filler (with radius Rm) for Rp > Rm as the size of carbon black particles is usually about 1000 A [19]. During this mixing, the filler, because of electrostatical interaction, is distributed mainly on the surface of polymer particles which facilitates the forming of conducting chains and entails low values of the percolation threshold. 

For the second method the threshold concentration of the filler in a composite material amounts to about 5 volume %, i.e. below the percolation threshold for statistical mixtures. It is bound up with the fact that carbon black particles are capable (in terms of energy) of being used to form conducting chain structures, because of the availability of functional groups on their surfaces. This relatively sparing method of composite material manufacture like film moulding by solvent evaporation facilitates the forming of chain structures. 

In the case of the filler localization in one of the polymer components of the mixture, an increase of the portion of the second unfilled polymer component in it entails sharp (by a factor of lO10) rise of a in the conducting polymer composite. In this case the filled phase should be continuous, i.e. its concentration should be higher than the percolation threshold. 

As already noted, the main merit of fibers used as a filler for conducting composite materials is that only low threshold concentrations are necessary to reach the desired level of composite conductivity. However, introduction of fiber fillers into a polymer with the help of ordinary plastic materials processing equipment presents certain difficulties which are bound up mainly with significant shearing deformations entailing fiber destruction and, thereby, a decrease of parameter 1/d which determines the value of the percolation threshold. 

Above a critical hller concentration, the percolation threshold, the properties of the reinforced rubber material change drastically, because a hller-hUer network is estabhshed. This results, for example, in an overproportional increase of electrical conductivity of a carbon black-hUed compound. The continuous disruption and restorahon of this hller network upon deformation is well visible in the so-called Payne effect [20,21], as represented in Figure 29.5. It illustrates the strain-dependence of the modulus and the strain-independent contributions to the complex shear or tensUe moduli for carbon black-hlled compounds and sUica-hUed compounds. 

The configuration of a system in which percolation may occur is classically treated as one in which the ingredients do not move. Considerable work has been devoted to these static models, leading to numerical solutions of the critical concentrations and cluster sizes associated with a percolation threshold. 

The main conclusion of the percolation theory is that there exists a critical concentration of the conductive fraction (percolation threshold, c0), below which the ion (charge) transport is very difficult because of a lack of pathways between conductive islands. Above and near the threshold, the conductivity can be expressed as ... 

Thus, the whole area of fillings for the large lattices considered is concentrated into a narrow range of threshold values of ZB, and ZBpercolation processes, since the sites bound to them are filled through wider bonds. One should expect that nonpercolation bonds have a relatively small effect on diffusion mass transfer that extends from the external surface of a lattice to its bulk. 

One of the characteristics of weak flocculation is that the system is reversible. At low volume fractions the system will form some clusters and some single particles. The clusters can be easily disrupted by gentle shaking. As the concentration is increased the system will reach a percolation threshold . The number of nearest neighbours around any test particle reaches 3 at about (p — 0.25 and the attractive forces between... 

Proton conductivities of 0.1 S cm at high excess water contents in current PEMs stem from the concerted effect of a high concentration of free protons, high liquid-like proton mobility, and a well-connected cluster network of hydrated pathways. i i i i Correspondingly, the detrimental effects of membrane dehydration are multifold. It triggers morphological transitions that have been studied recently in experiment and theory.2 .i29.i ,i62 water contents below the percolation threshold, the well-hydrated pathways cease to span the complete sample, and poorly hydrated channels control the overall transports ll Moreover, the structure of water and the molecular mechanisms of proton transport change at low water contents. 

The physical mechanism of membrane water balance and the formal structure of modeling approaches are straightforward. Under stationary operation, the inevitable electro-osmotic flux has to be compensated by a back flux of water from cathode to anode, driven by gradients in concentration, activity, or liquid pressure of water. The water distribution in PEMs that is generated in response to these driving forces decreases from cathode to anode. With increasing/o, the water distribution becomes more nonuniform. the water content near the anode falls below the percolation threshold of proton conduction, X < X. This leaves only a small conductivity due to surface transport of water. As a consequence, increases dramatically this can lead to failure of the complete cell. 

Mixing different cross-linkers (19b and 19c) yielded systems with a strong gel-weak gel transition, rather than a distinct sol-gel shift. When the concentration of each of the cross-linkers was above the critical percolation threshold, the kinetically slower cross-linker (19c) dictated the gel properties. Upon addition of enough of a competitive binding additive, such as DMAP, to drop the concentration of the active cross-linking units below their individual percolation thresholds but still allowing the total amount of both active cross-linkers (19b and 19c) to be above the percolation threshold results in a gel whose properties are now controlled by the kinetically faster cross-linker (19c). 

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