Percolation phase transition

At 0.15 A long-range connectivity of the surface water is established, in 2-dimensional percolative phase transition. Network of H-bonded water spans protein surface the network has fluctuating and random connectivity, richness of connections increasing with hydration level... 

The probability P(p) plays the role of the order parameter in the percolation phase transition. 

Ojovan, M.I. (2004). Glass Formation in Amorphous Si02 as a Percolation Phase Transition in a System of Network Defects. JETP Letters, 79(12) 632-634. 

In general, percolation is one of the principal tools to analyze disordered media. It has been used extensively to study, for example, random electrical networks, diffusion in disordered media, or phase transitions. Percolation models usually require approximate solution methods such as Monte Carlo simulations, series expansions, and phenomenological renormalization [16]. While some exact results are known (for the Bethe lattice, for instance), they are very rare because of the complexity of the problem. Monte Carlo simulations are very versatile but lack the accuracy of the other methods. The above solution methods were employed in determining the critical exponents given in the following section. 

We have studied the system (9.1.39) to (9.1.41) by means of the Monte Carlo method on a disordered surface where the active sites form a percolation cluster built at the percolation threshold and also above this threshold [25]. Finite clusters of active sites were removed from the surface to study only the effect of the ramification of the infinite cluster. The phase transition points show strong dependence on the fraction of active sites and on the... 

For S > Sc we obtain an infinite cluster for which in principle a reactive state exists. We use this fact to define the percolation threshold in a kinetic way for the particular reaction at hand as the transition point from the reactive (Rco2 > 0) to the non-reactive (Rco2 — 0) state. As we have shown above, this transition happens in such a way that the kinetic phase transition points of 2/1 and are approaching each other if S —> Sc [25]. At S = Sc the... 

For S = 0.70 which is well above the percolation threshold we observe the second-order phase transition at y = 0.200 and the first-order phase transition at y2 = 0.324. For Yco < y all active sites are covered by B particles (Cb = S ). A complete occupation with A does not exist (CA < S) thus showing that clusters which are poisoned by B are still remaining on the lattice (otherwise these clusters would be occupied by A). We obtain a reactive interval between y and yi with Rco2 > 0. 

We have studied above a model for the surface reaction A + 5B2 -> 0 on a disordered surface. For the case when the density of active sites S is smaller than the kinetically defined percolation threshold So, a system has no reactive state, the production rate is zero and all sites are covered by A or B particles. This is quite understandable because the active sites form finite clusters which can be completely covered by one-kind species. Due to the natural boundaries of the clusters of active sites and the irreversible character of the studied system (no desorption) the system cannot escape from this case. If one allows desorption of the A particles a reactive state arises, it exists also for the case S > Sq. Here an infinite cluster of active sites exists from which a reactive state of the system can be obtained. If S approaches So from above we observe a smooth change of the values of the phase-transition points which approach each other. At S = So the phase transition points coincide (y 1 = t/2) and no reactive state occurs. This condition defines kinetically the percolation threshold for the present reaction (which is found to be 0.63). The difference with the percolation threshold of Sc = 0.59275 is attributed to the reduced adsorption probability of the B2 particles on percolation clusters compared to the square lattice arising from the two site requirement for adsorption, to balance this effect more compact clusters are needed which means So exceeds Sc. The correlation functions reveal the strong correlations in the reactive state as well as segregation effects. 

Due to the second effect the system may reach a percolation threshold and consequently a metal-insulator phase transition may be induced by the magnetic field. It has to be stressed that the contributions to the effective magnetostriction of cobaltites mentioned above have a different dependence on temperature. An increase of temperature induces low-spin to intermediate-spin transitions. At the same time the volume of ferromagnetic clusters decreases with increasing temperature. The competition of these mechanisms leads to the unusual dependence of the effective magnetostriction of cobaltites on temperature. 

The onset of percolation and the conditions that produce this phase-transition phenomenon have been of considerable interest to many disciplines of science. The classical studies have focused on immobile ingredients in a system that increase in concentration by randomly adding to the collection of particles. It is possible to estimate the conditions leading to the onset of percolation under these circumstances. When the ingredients are in motion, this estimation is far more difficult. It is an obvious challenge that was tackled using cellular automata. 

According to Stauffer (1979), A complete understanding of percolation would require [one] to calculate these exponents exactly and rigorously. This aim has not yet been accomplished, even in general for other phase transitions. The aim of a scaling theory as reviewed here is more modest than complete understanding We want merely to derive relations between critical exponents. Three principal methods currently employed to derive critical exponents are (i) series expansions, (ii) Monte Carlo simulation, and... 

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. 

P is the key function characterizing a percolation process, and here it plays the role of the order parameter used to describe order-disorder phenomena and phase transitions. Its behavior for a square lattice is show in Fig. 39. 

Percolation. Application of local interactions to the problem of generating long-range order, including phase transitions and morphogenesis, to large-scale discrete models. 

Water is well known for its unusual properties, which are the so-called "anomalies" of the pure liquid, as well as for its special behavior as solvent, such as the hydrophobic hydration effects. During the past few years, a wealth of new insights into the origin of these features has been obtained by various experimental approaches and from computer simulation studies. In this review, we discuss points of special interest in the current water research. These points comprise the unusual properties of supercooled water, including the occurrence of liquid-liquid phase transitions, the related structural changes, and the onset of the unusual temperature dependence of the dynamics of the water molecules. The problem of the hydrogen-bond network in the pure liquid, in aqueous mixtures and in solutions, can be approached by percolation theory. The properties of ionic and hydrophobic solvation are discussed in detail. 

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