Percolation transitions

As described in the introduction, certain cosurfactants appear able to drive percolation transitions. Variations in the cosurfactant chemical potential, RT n (where is cosurfactant concentration or activity), holding other compositional features constant, provide the driving force for these percolation transitions. A water, toluene, and AOT microemulsion system using acrylamide as cosurfactant exhibited percolation type behavior for a variety of redox electron-transfer processes. The corresponding low-frequency electrical conductivity data for such a system is illustrated in Fig. 8, where the water, toluene, and AOT mole ratio (11.2 19.2 1.00) is held approximately constant, and the acrylamide concentration, is varied from 0 to 6% (w/w). At about = 1.2%, the arrow labeled in Fig. 8 indicates the onset of percolation in electrical conductivity. 

Perchlorotoluene, 6 327 Perchlorylation, 12 183 Perchloryl fluoride, 18 279 Percolation leaching, 16 153 Percolation processes of filled polymers, 11 303 for wood, 26 358-359 Percolation theory, 20 345 23 63 Percolation transition, 10 16 Percutaneous transluminal coronary angioplasty (PTCA), 3 712 -per- designation, 7 609t PE resins, applications of, 20 206t. See also Polyethylene (PE)... 

This oversimplified random network model proved to be rather useful for understanding water fluxes and proton transport properties of PEMs in fuel cells. - - - It helped rationalize the percolation transition in proton conductivity upon water uptake as a continuous reorganization of the cluster network due to swelling and merging of individual clusters and the emergence of new necks linking them. ... 

The influence of the various types of quantum effects and, in particular, of the quantum-size effect on the electronic transport in granular metals is described in the vicinity of the percolation transition. 

Attempts to take into account both localization and percolation or, in other words, to allow for quantum effects in percolation go back to Khmel-nitskii s pioneer paper [68]. The experimental attempts to study quantum effects in conductivity close to the percolation threshold have been undertaken in Refs. [69-71]. The physical sense of these results is stated in Ref. [71] and could be described as follows. The percolation cluster is non-uniform it includes both big conductive regions ( lakes ) and small regions (weak links or bottlenecks) which connect lakes to each other. On approaching the percolation threshold from the metallic side of the transition, these weak links become thinner and longer, and at x = xc the cluster breaks or tears into pieces just in such areas. As a result, exactly these conditions start to be sufficient for the electron localization. Thus, a percolation provokes an Anderson localization in bottlenecks of the percolation cluster. Sheng and collaborators [36,37,72] tried to take into account the influence of tunneling on conductivity for systems in the vicinity of the percolation transition. Similar attempts have been made in papers [38,56]. The obtained results prove that the possibility of tunneling shifts the percolation threshold toward smaller x values and affects material properties in its vicinity. 

The galvanomagnetic properties of nanocomposites and their conductivity, in particular, near the percolation transition can be described within the two-component model developed for the case by Efros and Shklovskii [73] on the basis of Dykhne theory [74]. This theory was developed just for the description of materials containing two different components with sharp distinction for conductivity values (Dykhne media) and describes well the concentration dependence of the effective conductivity in the case of the classical grain sizes and so in the absence of quantum effects. However, even if quantum effects do not play an essential role, the adequate description of the conductivity dependence on temperature has not been elaborated till now. The reason is that numerous experimental results for granules, with the metal contents x[Pg.612]

In particular, it is established for a granular system (NiFe)x/(Si02)i x [101-104] that in this case Rs is two orders of magnitude larger than R() and can exceed upto four orders of magnitude from Rs obtained in case of a homogeneous metal (x — 1). However, despite the high value of the Hall coefficient in these materials, experiments till now were performed mainly at the metallic side of the percolation transition [101-104]. 

Lagues, M., and Sauterey, C. (1980), Percolation transition in water in oil microemulsions. Electrical conductivity measurements, / Phys. Chem., 84,3503-3508. 

One of the most interesting aspects of energy transport is the excitation percolation transition (, and its similarity (10) to magnetic phase transitions and other critical phenomena (, 8). In its simplest form the problem is one of connectivity. In a binary system, made only of hosts and donors, the question is can the excitation travel from one side of the material to the other The implicit assumption is that there are excitation-transfer-bonds only between two donors that are "close enough", where "close enough" has a practical aspect (e.g. defined by the excitation transfer probability or time). Obviously, if there is a succession of excitation-bonds from one edge of the material to the other, one has "percolation", i.e. a connected chain of donors forming an excitation conduit. We note that the excitation-bonds seldom correspond to real chemical bonds rather more often they correspond to van-der-Walls type bonds and most often they correspond to a dipole-dipole or equivalent quantum-mechanical interaction. 

The SSE phase-space portrait shown in Fig. 6.5 reminds us of the phase-space portraits of the kicked rotor presented in Chapter 5. In Fig. 6.5 we can identify resonances and sealing invariant curves. In Chapter 5 we saw that resonance overlap in the standard mapping defines a sudden percolation transition when for K > Kc the seahng invariant... 

The numerical results reviewed above were obtained for infinite lattices. How do the various quantities of interest behave near the percolation threshold in a large but finite lattice This problem has been studied by renormalization methods, which are essentially equivalent to finite-size scaling. For finite lattices the percolation transition is smeared out over a range of p, and one must expect a similar trend in other functions, including the conductivity. Computer simulations by the Monte Carlo method have been carried out for bond percolation on a three-dimensional simple cubic lattice by Kirkpatrick (1979). Five such experimental curves are shown in Fig. 40, each of which corresponds to a cube of size b, containing bonds. In Fig. 40 the vertical axis gives the fraction p of such samples that percolate (i.e., have opposite faces con-... 

A percolating network forms an uninterrupted path between opposite boundaries of a system. The word spanning is used when the system has no boundary, like the surface of a single sphere. In this case, the degree of connectivity, at which a spanning network appears, is detected by the distribution of finite clusters in analogy to a percolation transition. 

Oleinikova A, Brovchenko I. Percolation transition of hydration water in bio-systems. Mol. Phys. 2006 104 3841-3855. Oleinikova A, Brovchenko I. Percolating networks and liquid-liquid transitions in supercooled water. J. Phys. Cond. Matter 2006 18 S2247-S2259. 

Partay LB, ledlowsky P, Brovchenko I, Oleinikova A. Percolation transition in supercritical water A Monte Carlo simulation study. J. Phys. Chem. B 2007 111 7603-7609. 

Figure 7.4 Phase diagram for adhesive hard spheres as a function of Baxter temperature rg. The solid line is the spinodal line for liquid-liquid phase separation (the dense liquid phase is probably metastable), the dot-dashed line is the freezing line for appearance of an ordered packing of spheres, and the dashed line is the percolation transition. (Adapted from Grant and Russel 1993, reprinted with permission from the American Physical Society.)...

Below the percolation line, there is predicted to be a sample-spanning cluster of contacting spheres. Woutersen et al. (1994) found that the gel point for 47-nm octadecyl-grafted silica spheres in benzene is in reasonable agreement with the predicted percolation transition. However, Grant and Russel (1993) found that the gelation line is below the percolation ... 

Pc q q qm r t t fraction of allowed bonds at percolation transition (Chapter 5) electric charge scattering vector wavenumber at maximum scattering intensity distance of center-of-mass separation time past time... 

If a unimodal pore network of arbitrary size is considered then, if the spatial distribution of pore sizes is non-random, the desorption percolation transition would be apparently smeared out (in addition to any finite size effect). It is possible that particular pores occupied by liquid-like phase might gain access to the vapour phase before would be expected to be the case for a purely random system because the actual layout of the pores might provide a convenient access route that would not have existed at that bond occupation level in a random system. The simulations of the nitrogen sorption... 

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