Percolation transition of hydration water: from planar hydrophilic surfaces to proteins

The formation of a spanning hydrogen-bonded network of hydration water is found to occur via a 2D percolation transition in various systems: smooth hydrophilic surfaces, the surface of a single protein molecule, protein powder, and diluted peptide solution. The average number of water-water hydrogen bonds at the percolation threshold varies from 2.0 to 2.3, depending on temperature, system size, and surface properties. Calculation of nH allows an easy estimation of the percolation threshold of hydration water in various systems, including biomolecules.     

Percolation of polyatomic species on a square lattice

In this paper, the percolation of (a) linear segments of size k and(b) k-mers of different structures and forms deposited on a square lattice have been studied. In the latter case, site and bond percolation have been examined. The analysis of results obtained by using finite size scaling theory is performed in order to test the universality of the problem by determining the numerical values of the critical exponents of the phase transition occurring in the system. It is also determined that the percolation threshold exhibits a exponentially decreasing function when it is plotted as a function of the k-mer size. The characteristic parameters of that function are dependent not only on the form and structure of the k-mers but also on the properties of the lattice where they are deposited.Received: 3 September 2003, Published online: 23 December 2003PACS: 64.60.Ak Renormalization-group, fractal, and percolation studies of phase transitions - 68.35.Rh Phase transitions and critical phenomena - 68.35.Fx Diffusion; interface formation     

Percolation approach for atomic and molecular cluster formation

We apply a percolation approach for the theoretical analysis of mass spectra of molecular microclusters obtained by adiabatic expansion technique. The evolution of the shape of the experimental size distributions as function of stagnation pressure and stagnation temperature are theoretically reproduced by varying the percolation parameter. Remaining discrepancies between theory and experiment are discussed. In addition, the even-odd alternation as well as the “magic” shell structure within metallic, secondary ion mass spectra are investigated by introducing statistical weights for the cluster formation probabilities. Shell correction energies of atomic clusters as function of cluster size are deduced from the experimental data.     

Critical Behaviors and Universality Classes of Percolation Phase Transitions on Two-Dimensional Square Lattice

We have investigated both site and bond percolation on two-dimensional lattice under the random rule and the product rule respectively. With the random rule, sites or bonds are added randomly into the lattice. From two candidates picked randomly, the site or bond with the smaller size product of two connected clusters is added when the product rule is taken. Not only the size of the largest cluster but also its size jump are studied to characterize the universality class of percolation. The finite-size scaling forms of giant cluster size and size jump are proposed and used to determine the critical exponents of percolation from Monte Carlo data. It is found that the critical exponents of both size and size jump in random site percolation are equal to that in random bond percolation. With the random rule, site and bond percolation belong to the same universality class. We obtain the critical exponents of the site percolation under the product rule, which are different from that of both random percolation and the bond percolation under the product rule. The universality class of site percolation differs different from that of bond percolation when the product rule is used.     

Static and dynamic heterogeneities in a model for irreversible gelation

We study the structure and the dynamics in the formation of irreversible gels by means of molecular dynamics simulation of a model system where the gelation transition is due to the random percolation of permanent bonds between neighboring particles. We analyze the heterogeneities of the dynamics in terms of the fluctuations of the self-intermediate scattering functions: in the sol phase close to the percolation threshold, we find that this dynamic susceptibility increases with the time until it reaches a plateau. At the gelation threshold this plateau scales as a function of the wave vector k as k(eta-2), with eta being related to the decay of the percolation pair connectedness function. At the lowest wave vector, approaching the gelation threshold it diverges with the same exponent gamma as the mean cluster size. These findings suggest an alternative way of measuring critical exponents in a system undergoing chemical gelation.     

Dependency of percolation critical exponents on the exponent of power law size distribution

The standard percolation theory uses objects of the same size. Moreover, it has long been observed that the percolation properties of the systems with a finite distribution of sizes are controlled by an effective size and consequently, the universality of the percolation theory is still valid. In this study, the effect of power law size distribution on the critical exponents of the percolation theory of the two dimensional models is investigated. Two different object shapes i.e., stick-shaped and square are considered. These two shapes are the representative of the fractures in fracture reservoirs and the sandbodies in clastic reservoirs. The finite size scaling arguments are used for the connectivity to determine the dependency of the critical exponents on the power law exponent. In particular, the deviations of percolation exponents from their universal values as well as the connectivity behavior of such systems are investigated numerically. As a result, this extends the applicability of the conventional percolation approach to study the connectivity of systems with a very broad size distribution.     

Critical behavior in a model of correlated percolation

We study the critical behavior of certain two-parameter families of correlated percolation models related to the Ising model on the triangular and square lattices, respectively. These percolation models can be considered as interpolating between the percolation model given by the + and – clusters and the Fortuin-Kasteleyn correlated percolation model associated to the Ising model. We find numerically on both lattices a two-dimensional critical region in which the expected cluster size diverges, yet there is no percolation.     

The influence of inhomogeneous properties of a system on the percolation process in two-dimensional space

The percolation process in a two-dimensional inhomogeneous lattice is studied by the Monte Carlo method. The inhomogeneous lattice is simulated by a random distribution of inhomogeneities differing in size and number. The influence of inhomogeneities on the parameters (critical concentration, average number of sites in finite clusters, percolation probability, critical exponents, and fractal dimension of an infinite cluster) characterizing the percolation in the system is analyzed. It is demonstrated that all these parameters essentially depend on the linear size of inhomogeneities and their relative area.     

Molecular dynamics simulations of jet breakup and ejecta production from a grooved Cu surface under shock loading

Large-scale non-equilibrium molecular dynamics simulations are performed to explore the jet breakup and ejecta production of single crystal Cu with a triangular grooved surface defect under shock loading. The morphology of the jet breakup and ejecta formation is obtained where the ejecta clusters remain spherical after a long simulation time. The effects of shock strength as well as groove size on the steady size distribution of ejecta clusters are investigated. It is shown that the size distribution of ejecta exhibits a scaling power law independent of the simulated shock strengths and groove sizes. This distribution, which has been observed in many fragmentation processes, can be well described by percolation theory.     

Study of the magnetic microstructure of Ni/NiO nanogranular samples above the electric percolation threshold by magnetoresistance measurements

Magnetoresistance measurements have been exploited to gain information on the magnetic microstructure of two Ni/NiO nanogranular materials consisting of Ni nanocrystallites (mean size of the order of 10 nm) embedded in a NiO matrix and differing in the amount of metallic Ni, ~33 and ~61 vol%. The overall conductance of both samples is metallic in character, indicating that the Ni content is above the percolation threshold for electric conductivity; the electric resistivity is two orders of magnitude smaller in the sample with higher Ni fraction (10(-5) Ωm against 10(-3) Ωm). An isotropic, spin-dependent magnetoresistance has been measured in the sample with lower Ni content, whereas both isotropic and anisotropic magnetoresistance phenomena coexist in the other material. This study, associated with magnetization loop measurements and the comparison with the exchange bias effect, allows one to conclude that in the sample with lower Ni content neither the physical percolation of the Ni nanocrystallites nor the magnetic percolation (i.e., formation of a homogeneous ferromagnetic network) are achieved; in the other sample physical percolation is reached while magnetic percolation is still absent. In both behaviors, a key role is played by the NiO matrix, which brings about a magnetic nanocrystallite/matrix interface exchange energy term and rules both the direct exchange interaction among Ni nanocrystallites and the magnetotransport properties of these nanogranular materials.     

Sharpness of the Phase Transition and Exponential Decay of the Subcritical Cluster Size for Percolation on Quasi-Transitive Graphs

We study homogeneous, independent percolation on general quasi-transitive graphs. We prove that in the disorder regime where all clusters are finite almost surely, in fact the expectation of the cluster size is finite. This extends a well-known theorem by Menshikov and Aizenman & Barsky to all quasi-transitive graphs. Moreover we deduce that in this disorder regime the cluster size distribution decays exponentially, extending a result of Aizenman & Newman. Our results apply to both edge and site percolation, as well as long range (edge) percolation. The proof is based on a modification of the Aizenman & Barsky method.     

The affection on the nature of percolation by concentration of pentagon–heptagon defects in graphene lattice

In this paper the percolation behavior with a specific concentration of the defects was discussed on the twodimensional graphene lattice. The percolation threshold is determined by a numerical method with a high degree of accuracy. This method is also suitable for locating the percolation critical point on other crystalline structures. Through investigating the evolution of the largest cluster size and the cluster sizes distribution, we find that under various lattice sizes and concentrations of pentagon-heptagon defects there is no apparent change for the percolation properties in graphene lattice.     

Iterated Fully Coordinated Percolation on a Square Lattice

We study, on a square lattice, an extension to fully coordinated percolation which we call iterated fully coordinated percolation. In fully coordinated percolation, sites become occupied if all four of its nearest neighbors are also occupied. Repeating this site selection process again yields the iterated fully coordinated percolation model. Our results show a large enhancement in the size of highly connected regions after each iteration (from ordinary to fully coordinated and then to iterated fully coordinated percolation); enhancements that are much larger than an extension of correlations by an extra lattice constant might suggest. We also study the universality among the three problems by determining the corresponding static and dynamic critical exponents. Specifically, a new method to directly calculate the walk dimension, d _w , using finite size scaling applied to normal mode analysis is used. This method is applicable to any geometry and requires significantly less computation than previously known calculations to determine d _w .     

Percolation and cluster distribution. III. Algorithms for the site-bond problem

Algorithms for estimating the percolation probabilities and cluster size distribution are given in the framework of a Monte Carlo simulation for disordered lattices for the generalized site-bond problem. The site-bond approach is useful when a percolation process cannot be exclusively described in the context of pure site or pure bond percolation. An extended multiple labeling technique (ECMLT) is introduced for the generalized problem. The ECMLT is applied to the site-bond percolation problem for square and triangular lattices. Numerical data are given for lattices containing up to 16 million sites. An application to polymer gelation is suggested.Supported by NSF Grant DMR76-07832 and NIH Grant NS08116-9.     

Simulating percolating behavior of conductive particles in anisotropically conductive composite films

A numerical scheme is developed to simulate the percolating behavior of conductive particles within a non-conductive matrix film with a preferential alignment of particles achieved via externally imposed deterministic driving forces. The sharp transition from non-conducting to conducting of the composite film is successfully revealed with the model. The percolation behavior is studied in terms of four percolation parameters, including the percolation probability, the normalized shortest percolation path, normalized gyration radius and density of the percolation cluster, subject to variation in five important system parameters. These include particle concentration, relative importance of the externally applied force, film thickness, film width and particle size. The threshold particle concentration can be reduced by increasing the strength of the deterministic driving force, decreasing film thickness, increasing film width or using smaller size particles. Our study suggests that using stronger applied force for wider and thinner films containing smaller particles may be a good practice to obtain anisotropically conductive films with a light particle loading that possess good conduction capability in the thickness direction and good insulation in the planar direction. Received: 19 February 2001 / Accepted: 30 May 2001 / Published online: 30 August 2001     

Explosive percolation transition is actually continuous

Recently a discontinuous percolation transition was reported in a new "explosive percolation" problem for irreversible systems [D. Achlioptas, R. M. D'Souza, and J. Spencer, Science 323, 1453 (2009)] in striking contrast to ordinary percolation. We consider a representative model which shows that the explosive percolation transition is actually a continuous, second order phase transition though with a uniquely small critical exponent of the percolation cluster size. We describe the unusual scaling properties of this transition and find its critical exponents and dimensions.     

Percolation and fragmentation of excited atomic nuclei

We present an analysis which aims at explaining the similarities (and differences) which exist between a simple bond percolation process on a cubic lattice and the fragmentation of highly excited atomic nuclei. Emphasis is placed on discussing percolation in terms of concepts which are well known in nuclear physics such asQ-value and particle emission thresholds. Similarities and differences between the bond percolation process and nuclear fragmentation are discussed. An approximate expression for the microcanonical partition sum (number of microstates) corresponding to any given percolation partition is shown to provide a good starting point for predicting fragment size distributions.Communicated by: X. Campi     

Percolation phenomenon of calcium bis(2-ethylhexyl) sulfosuccinate water-in-oil microemulsions by conductivity and dielectric spectroscopy measurements

The sodium counterion (Na+) of the sodium bis(2-ethylhexyl) sulfosuccinate (AOT) surfactant was exchanged with calcium Ca2+ to investigate the counterion charge effect on the structure of water in normal decane microemulsions. Ohmic conductivity and dielectric permittivity measurements were performed on samples at constant water to surfactant mole ratio [water]/[Ca(AOT)(2)]=26.6. Increasing the volume fraction of the dispersed phase phi, a percolation phenomenon was observed at the constant temperature of 25 degrees C. The percolation threshold was found at phi approximately 15% by Ohmic conductivity and static dielectric permittivity studied as a function of phi, and by the frequency dependence of the complex permittivity. Critical exponents typical of the static percolation mechanism (formation of bicontinuous microemulsions) were found below and above threshold. The comparison of these results obtained for the two different counterions, Ca2+ and Na+, in AOT surfactant water in normal decane microemulsions allows detection of an important difference. The percolation below threshold is dynamic for the sodium-based microemulsions, accounting for the formation of clusters of droplets, whereas calcium-based microemulsions show a static percolation. For this system, the coalescence of droplets begins to occur below threshold at phi approximately 12%.     

The September 11 attack: A percolation of individual passive support

A model to terrorism is presented using the theory of percolation. Terrorism power is related to the spontaneous formation of random backbones of people who are sympathetic to terrorism but without being directly involved in it. They just don't oppose in case they could. In the past such friendly-to-terrorism backbones have been always existing but were of finite size and localized to a given geographical area. The September 11 terrorist attack on the US has revealed for the first time the existence of a world wide spread extension. It is argued to have result from a sudden world percolation of otherwise unconnected and dormant world spread backbones of passive supporters. The associated strategic question is then to determine if collecting ground information could have predict and thus avoid such a transition. Our results show the answer is no, voiding the major criticism against intelligence services. To conclude the impact of military action is discussed. Received 7 February 2002