Percolation Research Articles

A percolation model of unsaturated hydraulic conductivity using three-parameter Weibull distribution

Accurate estimation of unsaturated hydraulic conductivity, K(Sw), is crucial for modeling water and solute transport in variably-saturated soils. In this study, a new model for the estimation of K(Sw) based on percolation theory and critical path analysis is presented, in which the pore-throat size distribution is described by the three-parameter Weibull probability density function. To evaluate the performance of the proposed K(Sw) model, two datasets including more than three hundred samples are analyzed. The first dataset includes 240 pore-network simulations, while the second dataset consists of 101 soil samples from the UNSODA database. For the pore-network simulations, for which both pore-throat size distributions and capillary pressure curves are available, we estimate K(Sw) from the capillary pressure curve with RMSLE = 0.54 slightly better than the estimations obtained from the pore-throat size distribution with RMSLE = 0.6. For the soil samples, since only the capillary pressure curve is measured, we estimate K(Sw) from the capillary pressure curve and evaluate three previously proposed approaches for determining the exponent, α, in Poiseuille's law. We find that RMSLE = 0.98 for α = 3, RMSLE = 0.87 for α = 2(4 − D) − (3 − D)/(2D − 3), and RMSLE = 1.06 for α = 4 − 0.74D, in which D is the fractal dimension characterizing the pore space. Comparing the results with previous studies in which the power-law probability density function was used to characterize capillary pressure curve and pore sizes shows that the proposed three-parameter Weibull distribution, in conjunction with the CPA analysis, may more accurately estimate K(Sw) for a wide variety of soils ranging from very fine to very coarse textures.

Predictable Synthesis of 3D Polymer Networks Using Crystal Component-Linking.

Controlled synthesis of 3D polymer networks presents a significant challenge because of the complexity of the polymerization reaction in solution. In this study, a polymerization system that facilitates the prediction of a polymer network structure via percolation simulations is realized. The most significant difference between general percolation simulations and experimental polymerization systems is the mobility of the molecules during the reaction. A crystal component-linking method that connects the precisely arranged monomer as a supramolecular crystalline state to imitate the simple percolation theory is adopted. The percolation simulation based on the crystal structure of the arranged monomers is used to accurately calculate the gelation point, gel fraction, degree of swelling, and atomic formula, which correspond with the experimental results. This suggests that the network structures polymerized via the crystal component-linking method can be predicted precisely by a simple percolation simulation. Further, the percolation simulation predicts the structures of the loop, branched polymer, and crosslinking point, which are difficult to measure experimentally. The polymerization of precisely-arranged immobilized monomers in supramolecular structures is promising in synthesizing precisely controlled polymer networks.

CMC/SWCNT biocomposites: A combined study on experiments, molecular simulations and continuum models

A comprehensive study is carried out including experimental, molecular dynamics (MD) simulations and continuum modelling of Carboxymethyl cellulose/Single walled carbon nanotube (CMC/SWCNT) biocomposites. The electrical, optical, and mechanical properties of CMC/SWCNT biocomposites were investigated in the experimental part of this work. In the result of measurements, it was determined that electrical conductivity (σdc), absorbance level (A) and tensile modulus (E) of the composites increased significantly with the increase of SWCNT content in the CMC matrix. These physical changes in the CMC/SWCNT composites were explained by the percolation theory and the electrical and optical percolation thresholds (Rσ and Rop) and the critical exponents (βσ and βop) of these composites were calculated. In addition, MD simulations were performed to estimate the material properties for the polymer composite structures. The results of the tensile test experiments were found to qualitatively overlap with the experiments at low concentration range. Moreover, a homogenous distribution of SWCNTs were observed in the CMC matrix together with a strong level of interactions in between. In the continuum modelling a two parameters augmentation model is used. A coupled Mori-Tanaka-self consistent method is utilized when obtaining effective properties of composites. Experimental, MD and continuum modelling results of composites were compared and reasonable agreement was obtained between results.

Description of laminar-turbulent transition of an airfoil boundary layer measured by differential image thermography using directed percolation theory

Description of laminar-turbulent transition of an airfoil boundary layer measured by differential image thermography using directed percolation theory

Low percolation threshold polyvinylidene fluoride/multi‐walled carbon nanotube composites: A perspective of electrical conductivity, crystalline, rheological and mechanical properties

AbstractRecently, conductive polymer‐based composites (CPC) have gained considerable interest for their outstanding processing, mechanical, electrical properties, and resistance tunability. Multi‐walled carbon nanotube (MWCNT) was introduced into polyvinylidene fluoride (PVDF) as the conductive filler via means of melt blending in this work. Effects of MWCNT on crystalline structure, thermodynamic behavior, electrical conductivity, processing and mechanical properties of PVDF/MWCNT were under investigation. By measuring the volume resistivity of PVDF/MWCNT, it was observed to have declined from 1.54 × 1012 to 3.71 × 101 Ω∙m with the MWCNT content changed from 0.7 to 1 vol%, which plummeted by 11 orders of magnitude. Therefore, it can be inferred that the percolation threshold of MWCNT lies within the range of 0.7–1 vol%. Additionally, the percolation threshold is obtained as 0.81 vol% by fitting using the percolation theory equation. Around the percolation threshold, PVDF/MWCNT composites exhibit negligible alterations in their rheological properties when contrasted with pure PVDF, yet their mechanical properties are improved. Furthermore, MWCNT does not substantially impact the crystalline structure and thermodynamic behavior of PVDF/MWCNT.Highlights The percolation threshold of MWCNT in PVDF is 0.81 vol%. The volume resistivity of the 1 vol% MWCNT sample was only 3.71 Ω∙m. 1 vol% MWCNT exhibit favorable rheological and mechanical properties.

Robustness of interdependent directed higher-order networks against cascading failures

In the real world, directed networks are not just constructed as pairs of directed interactions, but also occur in groups of three or more nodes that form the higher-order structure of the network. From social networks to biological networks, there is growing evidence that real-world systems connect the functional relationships of multiple systems through interdependence. To understand the robustness of interdependent directed higher-order networks, we propose a new theoretical framework to model and analyze the robustness of such networks under random failures by percolation theory. We find that adding higher-order edges makes the network more vulnerable which quantifies and compares by two criteria: the size of the giant connected components and the percolation threshold. Increasing the hyperdegree distribution of heterogeneity or the hyperedge cardinality distribution of heterogeneity in interdependent directed higher-order networks will also make the network more vulnerable. Interestingly, the phase transition type changes from continuous to discontinuous with the increase of coupling strength, and partially interdependent directed higher-order networks exist hybrid phase transition. Moreover, by applying our theoretical analysis to real interdependent directed higher-order networks further validated our conclusion, it has implications for the design of flexible complex systems.

Predicting Ecosystem Net Primary Productivity by Percolation Theory and Optimality Principle

AbstractThe basic partitioning of precipitation P into evapotranspiration ET and run‐off Q is known as the “central problem of hydrology.” ET depends primarily on precipitation, P, and potential evapotranspiration, PET, which are connected by the biological process of photosynthesis. Photosynthesis is the fundamental step underlying the productivity of plant ecosystems. An important measure of plant productivity is the creation of plant biomass, which is quantified by net primary productivity, NPP. NPP is most parsimoniously related simply to the evapotranspiration, ET. The dependence of NPP (ET) appears to be in the form of a power law with an upper limit derived from the maximum plant‐available solar energy accessible on Earth. However, theoretical, rather than phenomenological, treatments relating ET and its complementary variable, run‐off (P − ET ≡ Q) to climate variables that measure the available energy (PET) and available water (P) (known as the water balance) are, at best, scarce, and their continued application to predicting NPP (P, PET) even more so. One theory developed recently to predict the water balance is based on determining how to divide P into Q and ET in such a way as to maximize NPP. Substitution of the resulting optimized ET (P, PET) into the function NPP (ET) then yields NPP (P, PET). We investigate the possibility that this new theoretical framework, which yields ET(P, PET) based on ecological optimality and percolation theory, can predict the dependence of NPP (P, PET). If the prediction of NPP (P, PET) is verified, this result may lead to significant progress in other areas, including the problem of the chief causes of geographical variability of species richness.

Resilience analysis of highway network under rainfall using a data-driven percolation theory-based method

This paper proposes a data-driven approach using percolation theory to analyze the resilience of highway networks under rainfall conditions. The proposed approach's main contribution is integrating real-world traffic data with percolation theory to evaluate the impact of rainfall on traffic flow and identify the critical links of highway networks. The resilience indicators, accounting for network topology and functionality, were formulated. To calculate these indicators under various rainfall intensities, the traffic flow fundamental diagrams were established using empirical rainfall and traffic data, and a probabilistic rainfall simulation model was developed. A case study of the East Midlands, UK highway network under a heavy rainfall event on September 27, 2019, validated the approach's feasibility. Furthermore, control experiments showed that the critical links identified by the proposed method enhance highway network resilience more effectively than traditional methods, thus validated the novelty of our approach.

A Connectivity Metrics-Based Approach for the Prediction of Stress-Dependent Fracture Permeability

Rapid and accurate assessment of fracture permeability is critical for subsurface resource and energy development as well as rock engineering stability. Fracture permeability deviates from the classical cubic law under the effect of roughness, geological stress, as well as mining-induced stress. Conventional laboratory tests and numerical simulations are commonly costly and time-consuming, whereas the use of a connectivity metric based on percolation theory can quickly predict fracture permeability, but with relatively low accuracy. For this reason, we selected two static connectivity metrics with the highest and lowest prediction accuracy in previous studies, respectively, and proposed to revise and use them for fracture permeability estimation, considering the effect of isolated large-aperture regions within the fractures under increasing normal stress. Several hundred fractures with different fractal dimensions and mismatch lengths were numerically generated and deformed, and their permeability was calculated by the local cubic law (LCL). Based on the dataset, the connectivity metrics were counted using the revised approach, and the results show that, regardless of the connectivity metrics, the new model greatly improves the accuracy of permeability prediction compared to the pre-improved model, by at least 8% for different cutoff aperture thresholds.

Rapid formation of carbon nanotubes–natural rubber films cured with glutaraldehyde for reducing percolation threshold concentration

Carbon nanotubes (CNTs) filled natural rubber (NR) composites with various CNT contents at 0, 1, 2, 3, 4 and 5 phr were prepared by latex mixing method using glutaraldehyde as curing agent. This work aims to improve the electrical and mechanical properties of CNT filled NR vulcanizates. The CNT dispersion of NR composites was clarified using dispersion grader, optical microscopy and scanning electron microscopy. The electrical properties of NR composites in the existing of CNT networks were studied by following the well-known percolation theory. It was observed that the NR composites exhibited low percolation threshold at 0.98 phr of CNT. Moreover, a three-dimensional network formation of CNT in the NR composites was observed and it is indicated by the t-value of 1.67. The mechanical properties of NR composites in terms of modulus, tensile strength and hardness properties were increased upon the addition of CNT to the optimum mechanical properties at 1 phr of CNT. Therefore, the present work is found the novelty of the study that the conductive rubber latex film can be produced using GA as low-temperature curing agent which enhanced good electrical properties. Moreover, this work is found to be beneficial in case of conductive rubber latex film that requires high modulus at low strain. The additional advantage of this system is the curing process occurs at low-temperature using GA and it can be easily processed.Graphical abstract

Networks of Limited-Valency Patchy Particles.

Equilibrium gels provide physically attractive counterparts of nonequilibrium gels, allowing statistical understanding and design of the equilibrium gel structure. Here, we assemble two-dimensional equilibrium gels from limited-valency "patchy" colloidal particles and follow their evolution at the particle scale to elucidate cluster-size distributions and free energies. By finely adjusting the patch attraction with critical Casimir forces, we let a mixture of two-valent and pseudo-three-valent patchy particles approach the percolated network state through a set of equilibrium states. Comparing this equilibrium route with a deep quench, we find that both routes approach the percolated state via the same equilibrium states, revealing that the network topology is uniquely set by the particle bond angles, independent of the formation history. The limited-valency system follows percolation theory remarkably well, approaching the percolation point with the expected universal exponents.

The effects of increasing filler loading on the contact resistivity of interconnects based on silver–epoxied conductive adhesives and silver metallization pastes

AbstractOur previous work highlighted how microscopic structural effects influence the sheet and contact resistance of electrically conductive adhesives (ECAs). Herein, we delve further by investigating how the contact and bulk resistivity of several ECAs that are based on the same formulation, but with different filler content, are correlated with the filler content. Additionally, two different filler geometries — high and low surface area (HSA and LSA) fillers — are combined in different ratios to maintain a similar viscosity and therefore processability. Hence, contact and bulk resistivities are also correlated with the different geometry ratios of these two fillers. As expected, it was found that the contact and bulk resistivities decreased when the filler content was increased. However, the magnitude of the decrease was found to depend strongly on the filler geometry ratio. At extreme filler geometry ratios, when the bulk is either mostly loaded with HSA‐fillers or mostly with LSA‐fillers, the impact of changes in the filler content on the bulk and contact resistivities is markedly different. The measured data is interpreted within the context of percolation theory and it is determined that the optimum ratio of the LSA and HSA Ag‐fillers investigated in this study is approximately 60:40 (for an epoxy‐based adhesive). This work has important ramifications for the design of ECAs, where cost considerations and the need to reduce silver resource usage demand the lowest (silver) filler content, but the demands of product performance point to higher filler content.

Crystallization induced segregated structure for outstanding conductive property with low percolation threshold

Conductive polymer composites (CPCs) are promising alternatives to metal conductive materials due to their light-weight and low-cost. This work investigates the effect of the crystallization behavior of the matrix on the electrical property of CPCs, where semi-crystalline polypropylene (PP) and amorphous styrene-butadiene-styrene tri-block copolymer (SBS) are selected as a matrix, respectively, and conductive carbon black (CB) particles are introduced into matrixes by melting blending. Based on the electrical properties of CPCs with various CB contents and percolation theory, the percolation threshold of 1.8 vol% in PP matrix composites is obtained, much lower than that of SBS matrix composites (5.4 vol%). The analysis with a polarizing microscope and differential scanning calorimeter confirms that such a significant difference in percolation threshold between two composites is due to the crystallization behavior while the crystallization induced segregated structure is proposed. Finally, the rheology behaviors and mechanical properties are characterized for microstructures and applications, respectively.

A cross-scaled simulation on cell inactivation efficacy of pulsed electric fields by leveraging percolation theory

From the microscopic electroporation to the irregular distribution of cell populations, the inactivation efficacy of pulsed electric fields (PEFs) from in vitro experiments has lacked a unified physical model due to its cross-scale complexity. Inspired by a coarse-grained approach from the percolation theory, the inactivation process is simulated from a simple yet robust lattice model, where the spatiotemporal heterogeneity of the collective structure and the stochastic PEF strike are portrayed as random matrices, while also accounting for the rules of single-cell electroporation and subsequent death. Beyond successfully simulating the inactivation of monolayer adherent cells and suspended cells, which are in good agreement with in vitro results, our model reveals that (1) macroscopically three-staged inactivation pattern originates from the “accelerate–uniform–decelerate” transition of inactivation velocity, and (2) the inactivation patterns obey a universal scaling law under varied field strength, which is not satisfied under varied pulsed widths. The simulation not only sheds light on the PEF inactivation of the macroscopic cell collectives but also provides a simple and generalized numerical method for predicting PEF efficacy in experiments or engineering.

A performance assessment method for urban rail transit last train network based on percolation theory

A performance assessment method for urban rail transit last train network based on percolation theory

Asymmetric response emerges between creation and disintegration of force-bearing subcellular structures as revealed by percolation analysis.

Cells dynamically remodel their internal structures by modulating the arrangement of actin filaments (AFs). In this process, individual AFs exhibit stochastic behavior without knowing the macroscopic higher-order structures they are meant to create or disintegrate, but the mechanism allowing for such stochastic process-driven remodeling of subcellular structures remains incompletely understood. Here we employ percolation theory to explore how AFs interacting only with neighboring ones without recognizing the overall configuration can nonetheless create a substantial structure referred to as stress fibers (SFs) at particular locations. We determined the interaction probabilities of AFs undergoing cellular tensional homeostasis, a fundamental property maintaining intracellular tension. We showed that the duration required for the creation of SFs is shortened by the increased amount of preexisting actin meshwork, while the disintegration occurs independently of the presence of actin meshwork, suggesting that the coexistence of tension-bearing and non-bearing elements allows cells to promptly transition to new states in accordance with transient environmental changes. The origin of this asymmetry between creation and disintegration, consistently observed in actual cells, is elucidated through a minimal model analysis by examining the intrinsic nature of mechano-signal transmission. Specifically, unlike the symmetric case involving biochemical communication, physical communication to sense environmental changes is facilitated via AFs under tension, while other free AFs dissociated from tension-bearing structures exhibit stochastic behavior. Thus, both the numerical and minimal models demonstrate the essence of intracellular percolation, in which macroscopic asymmetry observed at the cellular level emerges not from microscopic asymmetry in the interaction probabilities of individual molecules, but rather only as a consequence of the manner of the mechano-signal transmission. These results provide novel insights into the role of the mutual interplay between distinct subcellular structures with and without tension-bearing capability. Insight: Cells continuously remodel their internal elements or structural proteins in response to environmental changes. Despite the stochastic behavior of individual structural proteins, which lack awareness of the larger subcellular structures they are meant to create or disintegrate, this self-assembly process somehow occurs to enable adaptation to the environment. Here we demonstrated through percolation simulations and minimal model analyses that there is an asymmetry in the response between the creation and disintegration of subcellular structures, which can aid environmental adaptation. This asymmetry inherently arises from the nature of mechano-signal transmission through structural proteins, namely tension-mediated information exchange within cells, despite the stochastic behavior of individual proteins lacking asymmetric characters in themselves.

Theory of percolation on hypergraphs.

Hypergraphs capture the higher-order interactions in complex systems and always admit a factor graph representation, consisting of a bipartite network of nodes and hyperedges. As hypegraphs are ubiquitous, investigating hypergraph robustness is a problem of major research interest. In the literature the robustness of hypergraphs so far only has been treated adopting factor-graph percolation, which describes well higher-order interactions which remain functional even after the removal of one of more of their nodes. This approach, however, fall short to describe situations in which higher-order interactions fail when any one of their nodes is removed, this latter scenario applying, for instance, to supply chains, catalytic networks, protein-interaction networks, networks of chemical reactions, etc. Here we show that in these cases the correct process to investigate is hypergraph percolation, with is distinct from factor graph percolation. We build a message-passing theory of hypergraph percolation, and we investigate its critical behavior using generating function formalism supported by Monte Carlo simulations on random graph and real data. Notably, we show that the node percolation threshold on hypergraphs exceeds node percolation threshold on factor graphs. Furthermore we show that differently from what happens in ordinary graphs, on hypergraphs the node percolation threshold and hyperedge percolation threshold do not coincide, with the node percolation threshold exceeding the hyperedge percolation threshold. These results demonstrate that any fat-tailed cardinality distribution of hyperedges cannot lead to the hyper-resilience phenomenon in hypergraphs in contrast to their factor graphs, where the divergent second moment of a cardinality distribution guarantees zero percolation threshold.

Nature of hypergraph k-core percolation problems.

Hypergraphs are higher-order networks that capture the interactions between two or more nodes. Hypergraphs can always be represented by factor graphs, i.e., bipartite networks between nodes and factor nodes (representing groups of nodes). Despite this universal representation, here we reveal that k-core percolation on hypergraphs can be significantly distinct from k-core percolation on factor graphs. We formulate the theory of hypergraph k-core percolation based on the assumption that a hyperedge can be intact only if all its nodes are intact. This scenario applies, for instance, to supply chains where the production of a product requires all raw materials and all processing steps; in biology it applies to protein-interaction networks where protein complexes can function only if all the proteins are present; and it applies as well to chemical reaction networks where a chemical reaction can take place only when all the reactants are present. Formulating a message-passing theory for hypergraph k-core percolation, and combining it with the theory of critical phenomena on networks, we demonstrate sharp differences with previously studied factor graph k-core percolation processes where it is allowed for hyperedges to have one or more damaged nodes and still be intact. To solve the dichotomy between k-core percolation on hypegraphs and on factor graphs, we define a set of pruning processes that act either exclusively on nodes or exclusively on hyperedges and depend on their second-neighborhood connectivity. We show that the resulting second-neighbor k-core percolation problems are significantly distinct from each other. Moreover we reveal that although these processes remain distinct from factor graphs k-core processes, when the pruning process acts exclusively on hyperedges the phase diagram is reduced to the one of factor graph k-cores.

Robustness of interdependent hypergraphs: A bipartite network framework

In this paper, we develop a bipartite network framework to study the robustness of interdependent hypergraphs. From such a perspective, nodes and hyperedges of a hypergraph are equivalent to each other, a property that largely simplifies their mathematical treatment. We develop a general percolation theory based on this representation and apply it to study the robustness of interdependent hypergraphs against random damage, which we verify with numerical simulations. We analyze a variety of interacting patterns, from heterogeneous to correlated hyperstructures, and from full- to partial-dependency couplings between an arbitrary number of hypergraphs, and characterize their structural stability via their phase diagrams. Given its generality, we expect that our framework will provide useful insights for the development of more realistic venues to characterize cascading failures in interdependent higher-order systems. Published by the American Physical Society 2024

Mathematical modelling of epidemic processes in the case of the contact stepwise infection pattern

Herein we consider mathematical models of infection in a population consisting of two types of people: those who transmit infection to others (type 1) and those who do not participate in the spread of infection (type 2). On the basis of the percolation theory and a model of the urn test type, a critical value of the proportion of infected persons in the population is determined, after which the infection process may become explosive. The probabilities of continuous infection and the interruption of its transmission are investigated. On the basis of Feigenbaum logistic mapping for the epidemic process, it is possible to estimate the change in the value of the parameter of the number of contacts and the bifurcations arising in this case, which are modelled in accordance with the scenario of transition to deterministic chaos through the doubling of the cycle period. In modes of stochasticity there are local modes of periodicity, the identification of which, if the model is adequate to the real situation, allows predicting and controlling the epidemic process, translating it or keeping the process in a stable cyclic state.