Percolation Clusters Research Articles

Polymers in Physics, Chemistry and Biology: Behavior of Linear Polymers in Fractal Structures.

We start presenting an overview on recent applications of linear polymers and networks in condensed matter physics, chemistry and biology by briefly discussing selected papers (published within 2022-2024) in some detail. They are organized into three main subsections: polymers in physics (further subdivided into simulations of coarse-grained models and structural properties of materials), chemistry (quantum mechanical calculations, environmental issues and rheological properties of viscoelastic composites) and biology (macromolecules, proteins and biomedical applications). The core of the work is devoted to a review of theoretical aspects of linear polymers, with emphasis on self-avoiding walk (SAW) chains, in regular lattices and in both deterministic and random fractal structures. Values of critical exponents describing the structure of SAWs in different environments are updated whenever available. The case of random fractal structures is modeled by percolation clusters at criticality, and the issue of multifractality, which is typical of these complex systems, is illustrated. Applications of these models are suggested, and references to known results in the literature are provided. A detailed discussion of the reptation method and its many interesting applications are provided. The problem of protein folding and protein evolution are also considered, and the key issues and open questions are highlighted. We include an experimental section on polymers which introduces the most relevant aspects of linear polymers relevant to this work. The last two sections are dedicated to applications, one in materials science, such as fractal features of plasma-treated polymeric materials surfaces and the growth of polymer thin films, and a second one in biology, by considering among others long linear polymers, such as DNA, confined within a finite domain.

Molecular condensation of the CO/NF-YB/NF-YC/FT complex gates floral transition in Arabidopsis.

The plant master photoperiodic regulator CONSTANS (CO) interacts with Nuclear Factor-Y subunits B2 (NF-YB2) and C9 (NF-YC9) and transcriptionally activates the florigen gene FLOWERING LOCUS T (FT), regulating floral transition. However, the molecular mechanism of the functional four-component complex assembly in the nucleus remains elusive. We report that co-phase separation of CO with NF-YB2/NF-YC9/FT precisely controls heterogeneous CO assembly and FT transcriptional activation. In response to light signals, CO proteins form functional percolation clusters from a diffuse distribution in a B-box-motif-dependent manner. Multivalent coassembly with NF-YC9 and NF-YB2 prevents inhibitory condensate formation and is necessary to maintain proper CO assembly and material properties. The intrinsically disordered region (IDR) of NF-YC9, containing a polyglutamine motif, fine-tunes the functional properties of CO/NF-YB/NF-YC condensates. Specific FT promoter recognition with polyelectrolyte partitioning also enables the fluidic functional properties of CO/NF-YB/NF-YC/FT condensates. Our findings offer novel insights into the tunable macromolecular condensation of the CO/NF-YB/NF-YC/FT complex in controlling flowering in the photoperiod control.

Statistics of random walks in geologically relevant conductance fields.

Statistics of diffusion, modeled by random walks, such as the mean number of distinct sites visited S(t) at time t, the mean probability P_{0}(t) of being at the origin of the walk, and the mean-squared displacements 〈R^{2}(t)〉 of the random walkers have been studied extensively in the past in both regular lattices and such disordered media as percolation clusters and other fractal structures, and universal power laws for such quantities have been derived. S(t) provides insight into reaction properties of geological formations, while P_{0}(t) is directly linked with the problem of back diffusion in remediation of groundwater aquifers. In all such studies, it was assumed that the conductances of the bonds that connect nearest-neighbor sites of the lattices are equal. Motivated by the problem of transport and reaction in large-scale porous media that are characterized by a broad spatial distribution of hydraulic conductances, we demonstrate, using extensive Monte Carlo simulations, that the statistics of random walks, when the conductances are broadly distributed, depend on the structure of the distribution. Five geologically relevant conductance distributions, namely, normal, log-normal, fractional Brownian motion (FBM), log-FBM, and stable distributions, are considered and random walks in a two-dimensional model with the five distributions are simulated. The first two distributions are uncorrelated, while the last three induce long-range correlations in the values of the conductance. We show that if S(t)∼t^{p} and P_{0}(t)∼t^{-ζ}, the exponents p and ζ may depend on the conductance distribution, in which case they are neither equal to those for homogeneous lattices, nor those for percolation clusters and other fractal structures. For at least three of the conductance distributions, diffusion is anomalous, with 〈R^{2}(t)〉 not growing linearly with the time t, even in the long-time limit. In addition to being of scientific interest, and the fact that transport processes in geomedia are simulated by random walks, the dependence of such statistics on the distribution of the conductances and their deviations from the statistics of random walks in homogeneous systems, and percolation and other types of fractal structures, indicate that diffusion in highly heterogeneous porous media is anomalous, and is described by fractional partial differential equationsin which the temporal and spatial fractional orders depend on the details of the conductance distribution.

Detecting phase transitions based on siamese neural network

Machine learning has been widely applied in physics research. Although unsupervised learning can extract the critical points of phase transitions, the percolation model remains a challenge. Unsupervised learning using the raw configurations of the percolation model fails to capture the critical points. To capture the configuration characteristics of the percolation model, this paper proposes using the maximum cluster as input to the neural network. It is well understood that the order parameter of the percolation model is not simply the particle density, but rather the probability that a given site or bond belongs to the percolating cluster. Additionally, we introduce the use of a Siamese Neural Network (SNN) to detect percolation phase transitions. Unlike unsupervised dimensionality reduction methods or supervised binary classification outputs, the SNN produces a scalar output referred to as similarity. By combining the maximum cluster and the SNN, we not only successfully extract the critical value of the percolation model, but also calculate the correlation exponent via data collapse. We believe that the SNN has great potential in handling phase transition classification problems and can serve as a reference for studying other phase transition systems.

Relation between critical exponent of the conductivity and the morphological exponents of percolation theory.

A central unsolved problem in percolation theory over the past five decades has been whether there is a direct relationship between the critical exponents that characterize the power-law behavior of the transport properties near the percolation threshold, particularly the effective electrical conductivity σ_{e}, and the exponents that describe the morphology of percolation clusters. The problem is also relevant to the relation between the static exponents of percolation clusters and the critical dynamics of spin waves in dilute ferromagnets, the elasticity of gels and composite solids, hopping conductivity in semiconductors, solute transport in porous media, and many others. We propose an approach to address the problem by showing that the contributions to σ_{e} can be decomposed into several groups representing the structure of percolation networks, including their mass and tortuosity, as well as constrictivity that describes the fluctuations in the driving potential gradient along the transport paths. The decomposition leads to a relationship between the critical exponent t of σ_{e} and other percolation exponents in d dimensions, t/ν=(d-D_{bb})+2(D_{op}-1)+d_{C}, where ν, D_{bb}, D_{op}, and d_{C} are, respectively, the correlation length exponent, the fractal dimensions of the backbones and the optimal paths, and the exponent that characterizes the constrictivity. Numerical simulations in two and three dimensions, as well as analytical results in d=1 and d=6, the upper critical dimension of percolation, validate the relationship. We, therefore, believe that the solution to the 50-year-old problem has been derived.

EIS Behavior of Polyethylene + Graphite Composite Considered as an Approximation to an Ensemble of Microelectrodes

The electrical percolation of alternating current through two-phase polyethylene/graphite composite electrodes with different contents of graphite microparticles immersed in aqueous KCl solutions has been studied. Above the graphite content of the first percolation threshold, the electrochemical impedance response of this electrode is associated with an equivalent circuit of resistance Ru in series with a constant phase element (CPE). An insulator material + conducting filler model is proposed in which the electroactive surface is considered as the intersection of the percolation cluster through the solid and the cluster associated with the interfacial region. CPE is analyzed assuming a distribution of microcapacitors of the graphite particles in contact with the dielectric solution and inside the dielectric polymeric phase.

Iterative site percolation on triangular lattice

The site percolation on the triangular lattice stands out as one of the few exactly solved statistical systems. By initially configuring critical percolation clusters of this model and randomly reassigning the color of each percolation cluster, we obtain coarse-grained configurations by merging adjacent clusters that share the same color. It is shown that the process can be infinitely iterated in the infinite-lattice limit, leading to an iterative site percolation model. We conjecture from the self-matching argument that percolation clusters remain fractal for any finite generation, which can even take any positive real number by a generalized process. Extensive simulations are performed, and, from the generation-dependent fractal dimension, a continuous family of previously unknown universalities is revealed. Published by the American Physical Society 2024

Chemical short-range order increases the phonon heat conductivity in a refractory high-entropy alloy

We study the effects of the chemical short-range order (SRO) on the thermal conductivity of the refractory high-entropy alloy HfNbTaTiZr using atomistic simulation. Samples with different degrees of chemical SRO are prepared by a Monte Carlo scheme. With increasing SRO, a tendency of forming HfTi and TiZr clusters is found. The phonon density of states is determined from the velocity auto-correlation function and chemical SRO modifies the high-frequency part of the phonon density of states. Lattice heat conductivity is calculated by non-equilibrium molecular dynamics simulations. The heat conductivity of the random alloy is lower than that of the segregated binary alloys. Phonon scattering by SRO precipitates might be expected to reduce scattering times and, therefore, decrease thermal conductivity. We find that, in contrast, due to the increase of the conductivity alongside SRO cluster percolation pathways, SRO increases the lattice heat conductivity by around 12 %. This is expected to be a general result, extending to other HEAs.

Percolation of conditional dependency clusters based on edge-coupled interdependent networks

Considering the existence of multiple edge dependencies in realistic interdependent networks, we propose a model of edge-coupled interdependent networks with conditional dependency clusters (EINCDCs). In this model, the edges in network A depend on the edges in dependency clusters of size m in network B. If the failure rate of edges within the dependency clusters in network B exceeds the failure tolerance α, the corresponding edges in network A that depend on those clusters in network B will fail accordingly. By adopting the self-consistent probabilities approach, a theoretical analytical framework is established to quantitatively address this model. Specifically, we study the robustness of the system verified with numerical simulations in the effect of the cluster size and failure tolerance under random attacks on systems composed of two networks A and B constructed with Random Regular (RR), Erdös-Rényi (ER) and Scale Free (SF) models. Our results show that both networks A and B undergo a first-order or hybrid phase transition when the dependency cluster size does not exceed 2. However, when the cluster size of dependency clusters exceeds 2, a non-monotonic behavior is observed. In particular, when the failure tolerance is in the range from 0 to 0.5, the robustness of the system weakens with the growing in the number of dependency clusters of size 2. While, this tendency reverses when the failure tolerance is in the range from 0.5 to 1. Moreover, we observe that due to the asymmetric interdependency between the two networks, network B always undergoes first-order phase transition, whereas network A could exhibit different types of phase transitions, which depends on the size of dependency clusters. In addition, the failure tolerance may have opposite effects on the two networks with the growing of dependency cluster sizes. The conclusions of the study may provide useful implications and enrich the understanding in the robustness of edge-coupled interdependent networks.

A percolation model for numerical simulations of 2D non-gravity impregnation in porous media

The aim of this work is to propose a new percolation model for numerical simulation of impregnation in porous media. Indeed, the impregnation phenomena concern many engineering fields, and could affect mechanical behavior of the medium, specifically when it is coupled with other physics such as heat or chemistry. The understanding of this phenomenon requires sophisticated numerical models that take into account the coupling between different physics. Classical numerical methods present known problems such as high computational costs, spurious oscillations and are not well adapted to such a complex coupling problem. To avoid these computational difficulties, Self-organized Gradient Percolation (SGP) model could be a good compromise. SGP is a numerical model based on a probabilistic approach, namely gradient percolation, already introduced in the 1D case, [1]. In this work we explore the non-reactive case. Instead of using finite element methods to solve Richards’ equation, we model the propagation of the saturation in the medium by the evolution of gradient percolation clusters. We hence delimitate the “wet” zone of the porous media to which we associate macroscopic saturation using interpolation techniques. The obtained results are free from spurious oscillation, and computational cost is drastically reduced compared with the classical approaches based on the finite element method. We believe that the 2D-SGP method can be enhanced by a futur 3D extension, but also to include the chemical and thermo-mechanical aspects.

Quenched invariance principle for a reflecting diffusion in a continuum percolation cluster

Quenched invariance principle for a reflecting diffusion in a continuum percolation cluster

Emergence of biconnected clusters in explosive percolation.

By introducing a simple competition mechanism for bond insertion in random graphs, explosive percolation exhibits a sharp phase transition with rich critical phenomena. We investigate high-order connectivity in explosive percolation using an event-based ensemble, focusing on biconnected clusters, where any two sites are connected by at least two independent paths. Our numerical analysis confirms that explosive percolation with different intracluster bond competition rules shares the same percolation threshold and universality, with biconnected clusters percolating simultaneously with simply connected clusters. However, the volume fractal dimension d_{f}^{'} of biconnected clusters varies depending on the competition rules of intracluster bonds. The size distribution of biconnected clusters exhibits double-scaling behavior: large clusters follow the standard Fisher exponent derived from the hyperscaling relation τ^{'}=1+1/d_{f}^{'}, while small clusters display a modified Fisher exponent τ_{0}<τ^{'}. These findings provide insights into the intricate nature of connectivity in explosive percolation.

A Macroscopic Interpretation of the Correlation between Electrical Percolation and Mechanical Properties of Poly-(Ethylene Vinyl Acetate)/Zn Composites.

Elastic composites were prepared using a procedure involving hot plates and zinc powder that was directly dispersed into an EVA matrix. The correlation between the zinc content and the conductive properties of the material was studied via impedance spectroscopy, the thermal properties of the material were studied via differential calorimetry and the mechanical properties of the composites were studied via tensile strength curves, representing an important advancement in the characterization of this type of composite material. The composites' tensile strength and elongation at break decrease with the addition of filler since zinc particles act as stress-concentrating centres, while the composites' hardness and Young's modulus increase because of an increase in the stiffness of the material. The AC perturbation across the EVA/Zn composites was characterized using an RC parallel equivalent circuit that allowed us to easily measure their resistivity (ρp) and permittivity (εp). The dependence of these electrical magnitudes on the zinc content is correlated with their mechanical properties across the characteristic time constant τp = ρp·εp of this equivalent circuit. The dependence of the mechanical and electrical magnitudes on the zinc content is consistent with the formation of percolation clusters. The addition of graphite particles increases their potential performance. Three possible mechanisms for the electrical transport of the ac-perturbation across the EVA/Zn composites have been identified. Chemical corrosion in acid media causes the loss of zinc surface particles, but their bulk physical properties practically remain constant.

Soft Set Decision and Cluster Percolation Method-Based Policy Clustering and Encryption Optimization for CP-ABE

In ciphertext-policy attribute-based encryption, there might be different levels of overlapping in the access policies of different data objects outsourced by the same data owner. This paper proposes a soft set decision-making method and cluster percolation method-based policy clustering by using policy similarity for CP-ABE, aiming to merge the duplicated access policy pieces to reduce repeated computations during the encryption process of corresponding data objects. Firstly, the access policies are clustered using either the soft set decision-making or the cluster percolation method. Secondly, the access policies within the same cluster are integrated for further encryption of corresponding data objects as a whole, thereby preventing redundant computations during the encryption process and thus reducing computational overhead. Theoretical analysis and experimental results demonstrate the feasibility and effectiveness of the proposed approach in this paper.

Моделирование распространения сетевого вируса в локальной компьютерной сети методами теории перколяции

A net virus spreading in a local computer network is investigated in this paper. Two percolation models were proposed to describe two types of networks: wired and wireless. The percolation threshold corresponds to the fraction of infected computers in the network at which the network loses its operability. Algorithms for lattice filling by occupied nodes, for distributing occupied nodes into clusters, for searching a percolation cluster and for the percolation threshold determining were developed and implemented for the models. A numerical experiment was conducted to estimate the percolation threshold and its dependence on various virus characteristics.

Scaling of the cumulative weights of the invasion percolation cluster on a branching process tree

Scaling of the cumulative weights of the invasion percolation cluster on a branching process tree

Gel transition of active triblock copolymers

The self-propulsion of active matter leads to many non-equilibrium self-organization phenomena, and the conformational freedom of polymer chains can produce unique equilibrium self-assembly behaviors, which stimulates cross-disciplinary research between active matter and polymer physics. In this work, we use molecular dynamics simulations to investigate the modulation of self-propulsion activity on the gel transition of ABA triblock copolymers. The research results indicate that under different active forces and attractive strengths, the gel states formed by ABA copolymers can be divided into three types: stable polymer gels with stable percolation paths and uniform spatial distribution, dynamic polymer gels with constantly changing percolation path and strand conformation, and collapsed polymer gels aggregating into large percolating clusters. The spatial uniformity of active gels is related not only to the concentration fluctuation during the formation of the network, but also to the inconsistent movement of the network chains caused by the activity, which is manifested in the rotation of crosslinking points in the flexible system and the directional movement of the bundles along their contour directions in the semi-flexible and rigid systems. In terms of topological conformation of polymer networks, when the attractive strength between A blocks is strong, the proportion of loop increases with the active force increasing. When attractive strength is weak, inter- and intra-chain binding are unstable, and the conformation is easily changed by the activity drive, noise and other chain collisions, so the proportion of loop decreases with the active force increasing. The branching number of crosslinking points varies with active force, which is not only affected by the attraction strength, but also related to the rigidity of the network chain. Generally, the branch number of crosslinking points in semi-flexible networks is larger than that in flexible and rigid networks. In addition, the directional motion of active polymers induces anomalous diffusion in stable polymer gels. This study contributes to the understanding of the collective behavior of active polymers and serves as a guide for designing and implementing active polymeric materials.

Decomposing the percolation backbone reveals novel scaling laws of the current distribution

The distribution of currents on critical percolation clusters is the fundamental quantity describing the transport properties of weakly connected systems. Nevertheless, its finite-size extrapolation is still one of the outstanding open questions concerning disordered media. By hierarchically decomposing the 3-connected components of the backbone, we disclose that the current distribution is determined from two distributions, namely, the one corresponding to the number of bonds in each level and another one corresponding to the factors by which the current is reduced, when going from one level to the next. The first distribution follows a finite-size scaling, while the second is a power law with an exponent consistent with 3/4 in two dimensions. The standard hierarchical model for the backbone is too simple to reproduce this complex scenario. Our new decomposition method of the backbone also allows to calculate much smaller currents than before, attaining a precision of 10−35 and systems of size L = 81922. Moreover, our method is not restricted to electric currents on critical percolation clusters but could also be applied to other transport problems on sparse graphs including fluid flow and car traffic.

Theoretical aspects of the genesis of composites: Dependence of properties on the history of formation

AbstractStudies of the last decades have brought the results of modern natural science into technology of composite materials of the widest range. The methods of percolation theory have become constructive tools for studying the structure of composites, for the purposeful creation of materials with desired properties. In this paper, a computer model of the processes of self‐organization of percolation clusters has been created. The model considers such problems with self‐organization, in which the properties of a percolation cluster are also determined by the history of the system's development. To study the structure and properties of such systems, a three‐dimensional continuum percolation problem with interacting elements is solved in the model. Iterative algorithms for the motion and interaction of particles and clusters, leading to the formation of percolation systems, have been developed. Their structure and properties have been studied, a number of standard parameters have been calculated, and the dependence of the structure and properties of clusters on the rate of system generation, degree of self‐organization, and characteristic values of the correlation length has been studied; analytical dependences of a number of parameters of percolation systems are obtained.

Structure of tert-butyl alcohol in carbon tetrachloride solution. Molecular dynamics simulation study

The structure of tert-butyl alcohol (TBA) solution in carbon tetrachloride (CCl4) is studied over the entire concentration range using molecular dynamics simulation (MD). It is shown that at low alcohol concentrations (within 5% molar fraction), separate small associates of TBA appear as a result of hydrogen bonding between alcohol molecules, and cyclic structures of four TBA molecules appear among these associates. With increasing concentration, the formation of hydrogen-bonded associates quickly reaches its limit corresponding to 1.7 bonds per molecule as in pure TBA. At high alcohol concentrations, both linear and cyclic associates of TBA molecules are observed with a noticeable fraction of cyclic tetramers. Using the Voronoi method, aggregates of the nearest TBA molecules are considered. The number and size of such aggregates grows rapidly with concentration due to the presence of “nuclei” formed by hydrogen bonding. At an alcohol mole fraction of 15%, a percolation cluster of TBA molecules begins to form in the solution. On the contrary, in the local environment of CCl4 molecules no significant structural rearrangements occur, only a gradual replacement of the molecules is observed upon addition of alcohol. However at low content of CCl4 in alcohol, we can see some tendency to clustering of CCl4 molecules, which may indicate on a slight solvophobic effect in this solution.