Percolative transport

Balberg, I. (1998) Limits onthe continuum-percolation transport exponents. Phys. Rev., B57, 13351. 

Abstract Faradaic electron transfer in reverse microemulsions of water, AOT, and toluene is strongly influenced by cosurfactants such as primary amides. Cosurfactant concentration, as a field variable, drives redox electron transfer processes from a low-flux to a high-flux state. Thresholds in this electron-transport phenomenon correlate with percolation thresholds in electrical conductivity in the same microemulsions and are inversely proportional to the interfacial activity of the cosurfactants. The critical exponents derived from the scaling analyses of low-frequency conductivity and dielectric spectra suggest that this percolation is close to static percolation limits, implying that percolative transport is along the extended fractal clusters of swollen micellar droplets. and NMR spectra show that surfactant packing... 

From polarization curves the protectiveness of a passive film in a certain environment can be estimated from the passive current density in figure C2.8.4 which reflects the layer s resistance to ion transport tlirough the film, and chemical dissolution of the film. It is clear that a variety of factors can influence ion transport tlirough the film, such as the film s chemical composition, stmcture, number of grain boundaries and the extent of flaws and pores. The protectiveness and stability of passive films has, for instance, been based on percolation arguments [67, 681, stmctural arguments [69], ion/defect mobility [56, 57] and charge distribution [70, 71]. 

Eor pesticides to leach to groundwater, it may be necessary for preferential flow through macropores to dominate the sorption processes that control pesticide leaching to groundwater. Several studies have demonstrated that large continuous macropores exist in soil and provide pathways for rapid movement of water solutes. Increased permeabiUty, percolation, and solute transport can result from increased porosity, especially in no-tiUage systems where pore stmcture is stiU intact at the soil surface (70). Plant roots are important in creation and stabilization of soil macropores (71). 

This article addresses the synthesis, properties, and appHcations of redox dopable electronically conducting polymers and presents an overview of the field, drawing on specific examples to illustrate general concepts. There have been a number of excellent review articles (1—13). Metal particle-filled polymers, where electrical conductivity is the result of percolation of conducting filler particles in an insulating matrix (14) and ionically conducting polymers, where charge-transport is the result of the motion of ions and is thus a problem of mass transport (15), are not discussed. 

Composilion has a marked effect on p. Dilution can cause (i to drop by orders of magnitude. The functional dependence is often expressed in terms of an exponential dependence on intersite distance R=at m as suggested by the homogeneous lattice gas model, p c)exponential distance dependence of intersite coupling precludes observing a percolation threshold for transport. 

More detailed theoretical approaches which have merit are the configurational entropy model of Gibbs et al. [65, 66] and dynamic bond percolation (DBP) theory [67], a microscopic model specifically adapted by Ratner and co-workers to describe long-range ion transport in polymer electrolytes. 

The electron transport properties described earlier markedly differ when the particles are organized on the substrate. When particles are isolated on the substrate, the well-known Coulomb blockade behavior is observed. When particles are arranged in a close-packed hexagonal network, the electron tunneling transport between two adjacent particles competes with that of particle-substrate. This is enhanced when the number of layers made of particles increases and they form a FCC structure. Then ohmic behavior dominates, with the number of neighbor particles increasing. In the FCC structure, a direct electron tunneling process from the tip to the substrate occurs via an electrical percolation process. Hence a micro-crystal made of nanoparticles acts as a metal. 

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]. 

Electrical conductivity is an easily measured transport property, and percolation in electrical conductivity appears a sensitive probe for characterizing microstructural transformations. A variety of field (intensive) variables have been found to drive percolation in reverse microemulsions. Disperse phase volume fraction has been often reported as a driver of percolation in electrical conductivity in such microemulsions [17-20]. 

Lagues et al. [17] found that the percolation theory for hard spheres could be used to describe dramatic increases in electrical conductivity in reverse microemulsions as the volume fraction of water was increased. They also showed how certain scaling theoretical tools were applicable to the analysis of such percolation phenomena. Cazabat et al. [18] also examined percolation in reverse microemulsions with increasing disperse phase volume fraction. They reasoned the percolation came about as a result of formation of clusters of reverse microemulsion droplets. They envisioned increased transport as arising from a transformation of linear droplet clusters to tubular microstructures, to form wormlike reverse microemulsion tubules. 

A somewhat different water, decane, and AOT microemulsion system has been studied by Feldman and coworkers [25] where temperature was used as the field variable in driving microstructural transitions. This system had a composition (volume percent) of 21.30% water, 61.15% decane, and 17.55% AOT. Counterions (sodium ions) were assigned as the dominant charge transport carriers below and above the percolation threshold in electrical... 

Organic acid fluorescence. In a similar manner to trace constituents, such as Mg, Sr and P, concentrations of organic acids present in speleothem calcite are sufficient to observe variation at temporal scales of less than annual in some cases (e.g.. Baker et al. 1993, Shopov et al. 1994). Organic acids (humic and fulvic) are formed in the soil by humification, and transported to the cave void by percolating waters where they are entrapped in precipitating carbonates. Under certain circumstances, where precipitation patterns are strongly seasonal and the nature of vadose percolation is such that seasonal mixing is incomplete, bands with different luminescent intensities can be differentiated after excitation with UV radiation. In other cases, bands are not observable but secular... 

The advantages of this type of system are obvious the pore space is of sufficient complexity to represent any natural or technical pore network. As the model objects are based on computer generated clusters, the pore spaces are well defined so that point-by-point data sets describing the pore space are available. Because these data sets are known, they can be fed directly into finite element or finite volume computational fluid dynamics (CFD) programs in order to simulate transport properties [7]. The percolation model objects are taken as a transport paradigm for any pore network of major complexity. 

Figure 2.9.3 shows typical maps [31] recorded with proton spin density diffusometry in a model object fabricated based on a computer generated percolation cluster (for descriptions of the so-called percolation theory see Refs. [6, 32, 33]).The pore space model is a two-dimensional site percolation cluster sites on a square lattice were occupied with a probability p (also called porosity ). Neighboring occupied sites are thought to be connected by a pore. With increasing p, clusters of neighboring occupied sites, that is pore networks, begin to form. At a critical probability pc, the so-called percolation threshold, an infinite cluster appears. On a finite system, the infinite cluster connects opposite sides of the lattice, so that transport across the pore network becomes possible. For two-dimensional site percolation clusters on a square lattice, pc was numerically found to be 0.592746 [6]. 

Resulting maps of the current density in a random-site percolation cluster both of the experiments and simulations are represented by Figure 2.9.13(b2) and (bl), respectively. The transport patterns compare very well. It is also possible to study hydrodynamic flow patterns in the same model objects. Corresponding velocity maps are shown in Figure 2.9.13(d) and (c2). In spite of the similarity of the... 

The common disadvantage of both the free volume and configuration entropy models is their quasi-thermodynamic approach. The ion transport is better described on a microscopic level in terms of ion size, charge, and interactions with other ions and the host matrix. This makes a basis of the percolation theory, which describes formally the ion conductor as a random mixture of conductive islands (concentration c) interconnected by an essentially non-conductive matrix. (The mentioned formalism is applicable not only for ion conductors, but also for any insulator/conductor mixtures.)... 

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 ... 

For an argillic horizon to form, the (coagulated) clay must disperse in the horizon of eluviation before it is transported to the depth of accumulation by percolating water. 

Transport of clay through the soil body. Transport of peptized clay particles requires downward percolation of water through wide (>20 pm) pores and voids. Clay translocation is particularly prominent in soils that shrink and crack in the dry season but become wet during occasional downpours. 

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