Percolation behavior

D-TEM gave 3D images of nano-filler dispersion in NR, which clearly indicated aggregates and agglomerates of carbon black leading to a kind of network structure in NR vulcanizates. That is, filled rubbers may have double networks, one of rubber by covalent bonding and the other of nanofiller by physical interaction. The revealed 3D network structure was in conformity with many physical properties, e.g., percolation behavior of electron conductivity. 

The morphology of the agglomerates has been problematic, although some forms of network-like structures have been assumed on the basis of percolation behavior of conductivity and some mechanical properties, e.g., the Payne effect. These network stmctures are assumed to be determining the electrical and mechanical properties of the carbon-black-filled vulcanizates. In tire industries also, it plays an important role for the macroscopic properties of soft nano-composites, e.g., tear. 

Two system-dependent interpretative pictures have been proposed to rationalize this percolative behavior. One attributes percolation to the formation of a bicontinuous structure [270,271], and the other it to the formation of very large, transient aggregates of reversed micelles [249,263,272], In both cases, percolation leads to the formation of a network (static or dynamic) extending over all the system and able to enhance mass, momentum, and charge transport through the system. This network could arise from an increase in the intermicellar interactions or for topological reasons. Then all the variations of external parameters, such as temperature and micellar concentration leading to an extensive intermicellar connectivity, are expected to induce percolation [273]. 

When the conductivities of "dry" pores and channels vanish, the true percolation behavior is obtained ... 

A value = 24 reproduces well the quasi-percolation behavior of Nation with equivalent weight of l,100g moH. 

At the time of this writing, it must be conceded that there have been no fundamental principles-based mathematical model for Nafion that has predicted significantly new phenomena or caused property improvements in a significant way. Models that capture the essence of percolation behavior ignore chemical identity. The more ab initio methods that do embrace chemical structure are limited by the number of molecular fragments that the computer can accommodate. Other models are semiempirical in nature, which limits their predictive flexibility. Nonetheless, the diversity of these interesting approaches offers structural perspectives that can serve as guides toward further experimental inquiry. 

In the temperature interval of —70 to 0°C and in the low-frequency range, an unexpected dielectric relaxation process for polymers is detected. This process is observed clearly in the sample PPX with metal Cu nanoparticles. In sample PPX + Zn only traces of this process can be observed, and in the PPX + PbS as well as in pure PPX matrix the process completely vanishes. The amplitude of this process essentially decreases, when the frequency increases, and the maximum of dielectric losses have almost no temperature dependence [104]. This is a typical dielectric response for percolation behavior [105]. This process may relate to electron transfer between the metal nanoparticles through the polymer matrix. Data on electrical conductivity of metal containing PPX films (see above) show that at metal concentrations higher than 5 vol.% there is an essential probability for electron transfer from one particle to another and thus such particles become involved in the percolation process. The minor appearance of this peak in PPX + Zn can be explained by oxidation of Zn nanoparticles. 

The electrical percolation behavior for a series of carbon black filled rubbers is depicted in Fig. 26 and Fig. 27. The inserted solid lines are least square fits to the predicted critical behavior of percolation theory, where only the filled symbols are considered that are assumed to lie above the percolation threshold. According to percolation theory, the d.c.-conductivity Odc increases with the net concentration 0-0c of carbon black according to a power law [6,128] ... 

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 ... 

This behavior becomes more transparent in Fig. 30a,b, where the a.c.-con-ductivity a and relative dielectric constant (permittivity e ), respectively, for a series of less polar S-SBR-samples filled with various amounts of the coarse black N550 are show at 20 °C in a broader frequency range up to 107 Hz. For filler concentrations below the percolation threshold (O<0.15), the conductivity behaves essentially as that of an isolator and increases almost linearly with frequency. Above the percolation threshold (5>>0.2), it shows a characteristic conductivity plateau in the small frequency regime. Since at low frequencies the value of the conductivity a agrees fairly well with the d.c.-con-ductivity, the plateau value exhibits the characteristic percolation behavior considered above. In the high frequency regime the conductivity depicted in... 

For a quantitative analysis of the percolation behavior of R, considered in Eq. (20), the permittivity data shown in Fig. 30b have been fitted to an empirical Cole-Cole function [131,132]. The fits are quite good, yielding relatively small broadness parameters between 0.36 and 0.5. The obtained relaxation times rR= R 1 are depicted in Fig. 31 in dependence of filler concentration O. Furthermore, the cross-over times T =a)fl and the limiting low frequency plateau values of the conductivity o ( —>0), obtained from the data in Fig. 30a, are represented in Fig. 31. They have been evaluated by a simple shifting procedure for constructing a conductivity master curve, as... 

If the estimated fitting parameters are compared to the predicted values of percolation theory, one finds that all three exponents are much larger than expected. The value of the conductivity exponent ji=7A is in line with the data obtained in Sect. 3.3.2, confirming the non-universal percolation behavior of the conductivity of carbon black filled rubber composites. However, the values of the critical exponents q=m= 10.1 also seem to be influenced by the same mechanism, i.e., the superimposed kinetic aggregation process considered above (Eq. 16). This is not surprising, since both characteristic time scales of the system depend on the diffusion of the charge carriers characterized by the conductivity. 

The percolation behavior is manifested by the rapid increase in the dc electrical conductivity a and the static dielectric permittivity ss as the system approaches the percolation threshold (Fig. 7). 

Figure 7. The percolation behavior in AOT-water-decane microemulsion (17.5 21.3 61.2 vol%) is manifested by the temperature dependences of the static dielectric permittivity es (A left axis) and conductivity r (Q right axis). Toa is the temperature of the percolation onset Tp is the temperature of the percolation threshold. Insets are schematic presentations of the microemulsion structure far below percolation and at the percolation onset. (Reproduced with permission from Ref. 149. Copyright 1998, Elsevier Science B.V.)...

Zhang et al. studied the effect of conductive network formation in a polymer melt on the conductivity of MWNT/TPU composite systems (91). An extremely low percolation threshold of 0.13 wt% was achieved in hot-pressed composite film samples, whereas a much higher CNT concentration (3-4 wt%) is needed to form a conductive network in extruded composite strands. This was explained in terms of the dynamic percolation behavior of the CNT network in the polymer melt. The conductivity of extruded strand showed a hopping resistivity dominated behavior at low concentrations and a dynamic percolation induced network dominated behavior at higher concentrations. It was shown that a higher temperature can reduce the filler concentration required for the dynamic percolation to take effect. 

The studies on electrical properties of CNT-PMMA composites show that electrical conductivity of these composites follow the classical percolating behavior though values of percolation threshold vary from one study to another (22,24,41,57). 

Connectivity has a central role in biochemistry and biology, and one imagines that the percolation model, with its focus on connectivity, should have wide application. Percolative behavior is to be expected for the coordinate functioning of systems of proteins in metabolic pathways, for functional interactions between proteins embedded in a membrane, for the interactions between domains in the folding of a polypeptide, or for the onset of function in anhydrobiotic organisms, seeds, and spores. 

It has been recently reported " that Nafion membranes show an ohmic behavior in 5 M NaOH, while in 10 M NaOH solution the specific conductance of the membranes increases with increasing current density. It is suggested that the passage of high currents at a severely dehydrated membrane may produce morphological changes that alter the character of the ionic conduction paths in the polymer. Hsu et have observed that the membrane conductivity of Nafion in alkaline electrolyte exhibits ion percolation behavior and can be described by... 

Fordedal H, Sjoblom J. Percolation behavior in W/O emulsions stabilized by interfacially active fractions from crude oils in high external electric fields. J Colloid Interface Sci 1996 181 589-594. 

However, unlike the behavior expected from the percolation theory, charge propagation can start slowly from low concentration regions in actual systems [48,49] because of a dynamic percolation behavior [50] or bounded motion in which limited site motion exists. 

Figure 40 Static permittivity of an asphaltene-stabilized model emulsion vs. the applied external electric field. The percolation behavior is clearly seen. (From Ref 170.)...

Let us first assume that the observed structural stability of the nanostructure, the absence of Ostwald ripening, is due to a slow diffusion [153,154]. In other terms, we assume, in contradiction to the experimentally found percolation behavior of these systems (see Figs. 6, 7, Fig. 10 in [155], and the discussion above), that the nanostructure is thermodynamically unstable and it forms due to the lack of diffusion... 

The percolation behavior seen in (zero crossings (Figure 15.44a). The dispersion of the localized electrons at high energy (>0.06 eV) and the Drude dispersion in the lar-IR are hoth evident. For PPy(TsO), the... 

An electrically conductive polymer composite of polypyrrole and poly(ethyl methacrylate) has been prepared by an emulsion polymerization procedure [ 142]. In this case, the relation between conductivity and the polypyrrole content of the composite exhibited a percolation behavior, with conductivities as high as 6-7 S/ctn. Such composites might be amenable to melt processing for coating formation. Composite films consisting of polypyrrole or poly(N-ethylaniline) filler dispersed in a polyimide matrix have been described for potential use as corrosion control coatings for the A1 alloy AA 2024-T3 [143]. 

Yeon Yi, J., Man Choi, G. Percolation behavior of conductor-insulator composites with varying aspect ratio of conductive fiber. J. Electroceram. 3(4), 361-369 (1999)... 

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