Continuum percolation thresholds in two dimensions

Effect of shape anisotropy on percolation of aligned and overlapping objects on lattices

We investigate the percolation transition of aligned, overlapping, anisotropic shapes on lattices. Using the recently proposed lattice version of excluded volume theory, we show that shape-anisotropy leads to some intriguing consequences regarding the percolation behavior of anisotropic shapes. We consider a prototypical anisotropic shape-rectangle-on a square lattice and show that, for rectangles of width unity (sticks), the percolation threshold is a monotonically decreasing function of the stick length, whereas, for rectangles of width greater than two, it is a monotonically increasing function. Interestingly, for rectangles of width two, the percolation threshold is independent of its length. We show that this independence of threshold on the length of a side holds for d-dimensional hypercubiods as well as for specific integer values for the lengths of the remaining sides. The limiting case of the length of the rectangles going to infinity shows that the limiting threshold value is finite and depends upon the width of the rectangle. This "continuum" limit with the lattice spacing tending to zero only along a subset of the possible directions in d dimensions results in a "semicontinuum" percolation system. We show that similar results hold for other anisotropic shapes and lattices in different dimensions. The critical properties of the aligned and overlapping rectangles are evaluated using Monte Carlo simulations. We find that the threshold values given by the lattice-excluded volume theory are in good agreement with the simulation results, especially for larger rectangles. We verify the isotropy of the percolation threshold and also compare our results with models where rectangles of mixed orientation are allowed. Our simulation results show that alignment increases the percolation threshold. The calculation of critical exponents places the model in the standard percolation universality class. Our results show that shape anisotropy of the aligned, overlapping percolating units has a marked influence on the percolation properties, especially when a subset of the dimensions of the percolation units is made to diverge.

Energy Formulation for Infinite Structures: Order Parameter for Percolation, Critical Bonds, and Power-Law Scaling of Contact-Based Transport

Investigating heterogeneous materials' microstructure, often simulated using periodic images, is crucial for understanding their physical traits. We propose a generic spring-based representation for periodic two-component structures. The equilibrium energy in this framework serves as an order parameter, offering an analytical expression for wrapping and introducing the concept of critical bonds. We show that these minimum bonds for depercolation can be efficiently detected. The number of critical bonds scales with system size, accurately capturing contact-based transport's scaling. This approach holds potential to analyze functional robustness of networks.

Synchronized rotations of active particles on chemical substrates

Many microorganisms use chemical 'signaling' - a quintessential self-organizing strategy in non-equilibrium - that can induce spontaneous aggregation and coordinated motion. Using synthetic signaling as a design principle, we construct a minimal model of active Brownian particles (ABPs) having soft repulsive interactions on a chemically quenched patterned substrate. The interplay between chemo-phoretic interactions and activity is numerically investigated for a proposed variant of the Keller-Segel model for chemotaxis. Such competition not only results in a chemo-motility-induced phase-separated state, but also results in a new cohesive clustering phase with synchronized rotations. Our results suggest that rotational order can emerge in systems by virtue of activity and repulsive interactions alone without an explicit alignment interaction. These rotations can also be exploited by designing mechanical devices that can generate reorienting torques using active particles.

Site percolation in distorted square and simple cubic lattices with flexible number of neighbors

This paper exhibits a Monte Carlo study on site percolation using the Newmann-Ziff algorithm in distorted square and simple cubic lattices where each site is allowed to be directly linked with any other site if the Euclidean separation between the pair is at most a certain distance d, called the connection threshold. Distorted lattices are formed from regular lattices by a random but controlled dislocation of the sites with the help of a parameter α, called the distortion parameter. The distinctive feature of this study is the relaxation of the restriction of forming bonds with only the nearest neighbors. Owing to this flexibility and the intricate interplay between the two parameters α and d, the site percolation threshold may either increase or decrease with distortion. The dependence of the percolation threshold on the average degree of a site has been explored to show that the obtained results are consistent with those on percolation in regular lattices with an extended neighborhood and continuum percolation.

A unifying criterion of the boiling crisis

We reveal and justify, both theoretically and experimentally, the existence of a unifying criterion of the boiling crisis. This criterion emerges from an instability in the near-wall interactions of bubbles, which can be described as a percolation process driven by three fundamental boiling parameters: nucleation site density, average bubble footprint radius and product of average bubble growth time and detachment frequency. Our analysis suggests that the boiling crisis occurs on a well-defined critical surface in the multidimensional space of these parameters. Our experiments confirm the existence of this unifying criterion for a wide variety of boiling surface geometries and textures, two boiling regimes (pool and flow boiling) and two fluids (water and liquid nitrogen). This criterion constitutes a simple mechanistic rule to predict the boiling crisis, also providing a guiding principle for designing boiling surfaces that would maximize the nucleate boiling performance.

Self-dual quasiperiodic percolation

How does the percolation transition behave in the absence of quenched randomness? To address this question, we study two nonrandom self-dual quasiperiodic models of square-lattice bond percolation. In both models, the critical point has emergent discrete scale invariance, but none of the additional emergent conformal symmetry of critical random percolation. From the discrete sequences of critical clusters, we find fractal dimensions of D_{f}=1.911943(1) and D_{f}=1.707234(40) for the two models, significantly different from D_{f}=91/48=1.89583... of random percolation. The critical exponents ν, determined through a numerical study of cluster sizes and wrapping probabilities on a torus, are also well below the ν=4/3 of random percolation. While these new models do not appear to belong to a universality class, they demonstrate how the removal of randomness can fundamentally change the critical behavior.

High-precision anomalous dimension of three-dimensional percolation and spatial profile of the critical giant cluster

In three-dimensional percolation, we apply and test the critical geometry approach for bounded critical phenomena based on the fractional Yamabe equation. The method predicts the functional shape of the order parameter profile ϕ, which is obtained by raising the solution of the Yamabe equation to the scaling dimension Δ_{ϕ}. The latter can be fixed from outcomes of numerical simulations, from which we obtain Δ_{ϕ}=0.47846(71) and the corresponding value of the anomalous dimension η=-0.0431(14). The comparison with values of η determined by using scaling relations is discussed. A test of hyperscaling is also performed.

Class of models for random hypergraphs

Despite the recently exhibited importance of higher-order interactions for various processes, few flexible (null) models are available. In particular, most studies on hypergraphs focus on a small set of theoretical models. Here, we introduce a class of models for random hypergraphs which displays a similar level of flexibility of complex network models and where the main ingredient is the probability that a node belongs to a hyperedge. When this probability is a constant, we obtain a random hypergraph in the same spirit as the Erdos-Renyi graph. This framework also allows us to introduce different ingredients such as the preferential attachment for hypergraphs, or spatial random hypergraphs. In particular, we show that for the Erdos-Renyi case there is a transition threshold scaling as 1/sqrt[EN] where N is the number of nodes and E the number of hyperedges. We also discuss a random geometric hypergraph which displays a percolation transition for a threshold distance scaling as r_{c}^{*}∼1/sqrt[E]. For these various models, we provide results for the most interesting measures, and also introduce new ones in the spatial case for characterizing the geometrical properties of hyperedges. These different models might serve as benchmarks useful for analyzing empirical data.

Percolation Effects in Mixed Matrix Membranes with Embedded Carbon Nanotubes

Polymeric membranes with embedded nanoparticles, e.g., nanotubes, show a significant increase in permeability of the target component while maintaining selectivity. However, the question of the reasons for this behavior of the composite membrane has not been unequivocally answered to date. In the present work, based on experimental data on the permeability of polymer membranes based on Poly(vinyl trimethylsilane) (PVTMS) with embedded CNTs, an approach to explain the abnormal behavior of such composite membranes is proposed. The presented model considered the mass transfer of gases and liquids through polymeric membranes with embedded CNTs as a parallel transport of gases through the polymeric matrix and a "percolation" cluster-bound regions around the embedded CNTs. The proposed algorithm for modeling parameters of a percolation cluster of embedded tubular particles takes into account an agglomeration and makes it possible to describe the threshold increase and subsequent decrease permeability with increasing concentration of embedded particles. The numerical simulation of such structures showed: an increase in the particle length leads to a decrease in the percolation concentration in a matrix of finite size, the power of the percolation cluster decreases significantly, but the combination of these effects leads to a decrease in the influence of the introduced particles on the properties of the matrix in the vicinity of the percolation threshold; an increase in the concentration of embedded particles leads to an increase in the probability of the formation of agglomerates and the characteristic size of the elements that make up the percolation cluster, the influence of individual particles decreases and the characteristics of the percolation transition determine the ratio of the sizes of agglomerates and matrix; and an increase in the lateral linear dimensions of the matrix leads to a nonlinear decrease in the proportion of the matrix, which is affected by the introduced particles, and the transport characteristics of such MMMs deteriorate.

Normal and anomalous diffusion in the disordered wind-tree model

Ehrenfests' wind-tree model (EWTM) refers to a two-dimensional system where noninteracting point tracer particles move through a random arrangement of overlapping or nonoverlapping square-shaped scatterers. Here, extensive event-driven molecular dynamics simulations of the EWTM at different reduced scatterer densities ρ are presented. For nonoverlapping scatterers, the asymptotic motion of the tracer particles is diffusive. We compare their diffusion coefficient D, as obtained from the simulation, with that predicted by kinetic theory where D^{-1} is expanded up to the second order in the scatterer density. While at low density quantitative agreement between theory and simulation is found, we show that beyond the low-density regime deviations to the theory are associated with the emergence of a maximum in the non-Gaussian parameter at intermediate times. For the case of overlapping scatterers, in agreement with a theoretical prediction, the asymptotic motion of the tracer particles is subdiffusive, i.e., the mean-squared displacement at long times t grows like t^{1-2ρ/3}. We propose a model of the van Hove correlation function that describes the density dependence of the tracer particles' asymptotic subdiffusive transport on a quantitative level.

Molecular hints of two-step transition to convective flow via streamline percolation

Convection is a key transport phenomenon important in many different areas, from hydrodynamics and ocean circulation to planetary atmospheres or stellar physics. However, its microscopic understanding still remains challenging. Here we numerically investigate the onset of convective flow in a compressible (non-Oberbeck-Boussinesq) hard disk fluid under a temperature gradient in a gravitational field. We uncover a surprising two-step transition scenario with two different critical temperatures. When the bottom plate temperature reaches a first threshold, convection kicks in (as shown by a structured velocity field) but gravity results in hindered heat transport as compared to the gravity-free case. It is at a second (higher) temperature that a percolation transition of advection zones connecting the hot and cold plates triggers efficient convective heat transport. Interestingly, this picture for the convection instability opens the door to unknown piecewise-continuous solutions to the Navier-Stokes equations.

Electrical conductivity of random metallic nanowire networks: an analytical consideration along with computer simulation

The current interest in the study of the 2D systems of randomly deposited metallic nanowires is inspired by a combination of their high electrical conductivity with excellent optical transparency. Metallic nanowire networks show great potential for use in numerous technological applications. Although there are models that describe the electrical conductivity of the random nanowire networks through wire resistance, junction resistance, and number density of nanowires, they are either not rigorously justified or contain fitting parameters. We have proposed a model for the electrical conductivity in random metallic nanowire networks. We have mimicked such random nanowire networks as random resistor networks (RRN) produced by the homogeneous, isotropic, and random deposition of conductive zero-width sticks onto an insulating substrate. We studied the electrical conductivity of these RRNs using a mean-field approximation. An analytical dependency of the electrical conductivity on the main physical parameters (the number density and electrical resistances of these wires and of the junctions between them) has been derived. Computer simulations have been performed to validate our theoretical predictions. We computed the electrical conductivity of the RRNs against the number density of the conductive fillers for the junction-resistance-dominated case and for the case where the wire resistance and the junction resistance were equal. The results of the computations were compared with this mean-field approximation. Our computations demonstrated that our analytical expression correctly predicts the electrical conductivity across a wide range of number densities.

Suppression of discontinuous phase transitions by particle diffusion

We investigate the phase transitions of the q-state Brownian Potts model in two dimensions (2D) comprising Potts spins that diffuse like Brownian particles and interact ferromagnetically with other spins within a fixed distance. With extensive Monte Carlo simulations we find a continuous phase transition from a paramagnetic to a ferromagnetic phase even for q>4. This is in sharp contrast to the existence of a discontinuous phase transition in the equilibrium q-state Potts model in 2D with q>4. We present detailed numerical evidence for a continuous phase transition and argue that diffusion generated dynamical positional disorder suppresses phase coexistence leading to a continuous transition.

Electrical conductivity of nanorod-based transparent electrodes: Comparison of mean-field approaches

We mimic nanorod-based transparent electrodes as random resistor networks (RRNs) produced by the homogeneous, isotropic, and random deposition of conductive zero-width sticks onto an insulating substrate. We suppose that the number density (the number of objects per unit area of the surface) of these sticks exceeds the percolation threshold, i.e., the system under consideration is a conductor. We computed the electrical conductivity of random resistor networks versus the number density of conductive fillers for the wire-resistance-dominated case, for the junction-resistance-dominated case, and for an intermediate case. We also offer a consistent continuous variant of the mean-field approach. The results of the RRN computations were compared with this mean-field approach. Our computations suggest that, for a qualitative description of the behavior of the electrical conductivity in relation to the number density of conductive wires, the mean-field approximation can be successfully applied when the number density of the fillers n>2n_{c}, where n_{c} is the percolation threshold. However, note the mean-field approach slightly overestimates the electrical conductivity. We demonstrate that this overestimate is caused by the junction potential distribution.

Aggregation of self-propelled particles with sensitivity to local order

We study a system of self-propelled particles (SPPs) in which individual particles are allowed to switch between a fast aligning and a slow nonaligning state depending upon the degree of the alignment in the neighborhood. The switching is modeled using a threshold for the local order parameter. This additional attribute gives rise to a mixed phase, in contrast to the ordered phases found in clean SPP systems. As the threshold is increased from zero, we find the sudden appearance of clusters of nonaligners. Clusters of nonaligners coexist with moving clusters of aligners with continual coalescence and fragmentation. The behavior of the system with respect to the clustering of nonaligners appears to be very different for values of low and high global densities. In the low density regime, for an optimal value of the threshold, the largest cluster of nonaligners grows in size up to a maximum that varies logarithmically with the total number of particles. However, on further increasing the threshold the size decreases. In contrast, for the high density regime, an initial abrupt rise is followed by the appearance of a giant cluster of nonaligners. The latter growth can be characterized as a continuous percolation transition. In addition, we find that the speed differences between aligners and nonaligners is necessary for the segregation of aligners and nonaligners.

Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions

Extended-range percolation on various regular lattices, including all 11 Archimedean lattices in two dimensions and the simple cubic (sc), body-centered cubic (bcc), and face-centered cubic (fcc) lattices in three dimensions, is investigated. In two dimensions, correlations between coordination number z and site thresholds p_{c} for Archimedean lattices up to 10th nearest neighbors (NN) are seen by plotting z versus 1/p_{c} and z versus -1/ln(1-p_{c}) using the data of d'Iribarne et al. [J. Phys. A 32, 2611 (1999)JPHAC50305-447010.1088/0305-4470/32/14/002] and others. The results show that all the plots overlap on a line with a slope consistent with the theoretically predicted asymptotic value of zp_{c}∼4η_{c}=4.51235, where η_{c} is the continuum threshold for disks. In three dimensions, precise site and bond thresholds for bcc and fcc lattices with 2nd and 3rd NN, and bond thresholds for the sc lattice with up to the 13th NN, are obtained by Monte Carlo simulations, using an efficient single-cluster growth method. For site percolation, the values of thresholds for different types of lattices with compact neighborhoods also collapse together, and linear fitting is consistent with the predicted value of zp_{c}∼8η_{c}=2.7351, where η_{c} is the continuum threshold for spheres. For bond percolation, Bethe-lattice behavior p_{c}=1/(z-1) is expected to hold for large z, and the finite-z correction is confirmed to satisfy zp_{c}-1∼a_{1}z^{-x}, with x=2/3 for three dimensions as predicted by Frei and Perkins [Electron. J. Probab. 21, 56 (2016)1083-648910.1214/16-EJP6] and by Xu et al. [Phys. Rev. E 103, 022127 (2021)2470-004510.1103/PhysRevE.103.022127]. Our analysis indicates that for compact neighborhoods, the asymptotic behavior of zp_{c} has universal properties, depending only on the dimension of the system and whether site or bond percolation but not on the type of lattice.

High-dimensional percolation criticality and hints of mean-field-like caging of the random Lorentz gas

The random Lorentz gas (RLG) is a minimal model for transport in disordered media. Despite the broad relevance of the model, theoretical grasp over its properties remains weak. For instance, the scaling with dimension d of its localization transition at the void percolation threshold is not well controlled analytically nor computationally. A recent study [Biroli et al., Phys. Rev. E 103, L030104 (2021)2470-004510.1103/PhysRevE.103.L030104] of the caging behavior of the RLG motivated by the mean-field theory of glasses has uncovered physical inconsistencies in that scaling that heighten the need for guidance. Here we first extend analytical expectations for asymptotic high-d bounds on the void percolation threshold and then computationally evaluate both the threshold and its criticality in various d. In high-d systems, we observe that the standard percolation physics is complemented by a dynamical slowdown of the tracer dynamics reminiscent of mean-field caging. A simple modification of the RLG is found to bring the interplay between percolation and mean-field-like caging down to d=3.

Random nanowire networks: Identification of a current-carrying subset of wires using a modified wall follower algorithm

We mimic random nanowire networks by the homogeneous, isotropic, and random deposition of conductive zero-width sticks onto an insulating substrate. The number density (the number of objects per unit area of the surface) of these sticks is supposed to exceed the percolation threshold, i.e., the system under consideration is a conductor. To identify any current-carrying part (the backbone) of the percolation cluster, we have proposed and implemented a modification of the well-known wall follower algorithm-one type of maze solving algorithm. The advantage of the modified algorithm is its identification of the whole backbone without visiting all the edges. The complexity of the algorithm depends significantly on the structure of the graph and varies from O(sqrt[N_{V}]) to Θ(N_{V}). The algorithm has been applied to backbone identification in networks with different number densities of conducting sticks. We have found that (i) for number densities of sticks above the percolation threshold, the strength of the percolation cluster quickly approaches unity as the number density of the sticks increases; (ii) simultaneously, the percolation cluster becomes identical to its backbone plus simplest dead ends, i.e., edges that are incident to vertices of degree 1. This behavior is consistent with the presented analytical evaluations.

Interplay between percolation and glassiness in the random Lorentz gas

The random Lorentz gas (RLG) is a minimal model of transport in heterogeneous media that exhibits a continuous localization transition controlled by void space percolation. The RLG also provides a toy model of particle caging, which is known to be relevant for describing the discontinuous dynamical transition of glasses. In order to clarify the interplay between the seemingly incompatible percolation and caging descriptions of the RLG, we consider its exact mean-field solution in the infinite-dimensional d→∞ limit and perform numerics in d=2...20. We find that for sufficiently high d the mean-field caging transition precedes and prevents the percolation transition, which only happens on timescales diverging with d. We further show that activated processes related to rare cage escapes destroy the glass transition in finite dimensions, leading to a rich interplay between glassiness and percolation physics. This advance suggests that the RLG can be used as a toy model to develop a first-principle description of particle hopping in structural glasses.

Nearest-neighbor connectedness theory: A general approach to continuum percolation

We introduce a method to estimate continuum percolation thresholds and illustrate its usefulness by investigating geometric percolation of noninteracting line segments and disks in two spatial dimensions. These examples serve as models for electrical percolation of elongated and flat nanofillers in thin film composites. While the standard contact volume argument and extensions thereof in connectedness percolation theory yield accurate predictions for slender nanofillers in three dimensions, they fail to do so in two dimensions, making our test a stringent one. In fact, neither a systematic order-by-order correction to the standard argument nor invoking the connectedness version of the Percus-Yevick approximation yield significant improvements for either type of particle. Making use of simple geometric considerations, our new method predicts a percolation threshold of ρ_{c}l^{2}≈5.83 for segments of length l, which is close to the ρ_{c}l^{2}≈5.64 found in Monte Carlo simulations. For disks of area a we find ρ_{c}a≈1.00, close to the Monte Carlo result of ρ_{c}a≈1.13. We discuss the shortcomings of the conventional approaches and explain how usage of the nearest-neighbor distribution in our method bypasses those complications.

Elliptic percolation model for predicting the electrical conductivity of graphene-polymer composites

Graphene-based polymers exhibit a conductive microstructure formed by aggregates in a matrix which drastically enhances their transmitting properties. We develop a new numerical framework for predicting the electrical conductivity based on continuum percolation theory in a two dimensional stochastically-generated medium. We analyze the role of the flake shape and its aspect ratio and consequently predict the onset of percolation based on the particle density and the domain scale. Simultaneously, we have performed experiments and have achieved very high electrical conductivity for such composites compared to other film fabrication techniques, which have verified the results of computing the homogenized electrical conductivity. As well, the proximity to and a comparison with other analytical models and other experimental techniques are presented. The numerical model can predict the composite transmitting conductivity in a larger range of particle geometry. Such quantification is exceedingly useful for effective utilization and optimization of graphene filler densities and their spatial distribution during manufacturing.

Critical polynomials in the nonplanar and continuum percolation models

Exact or precise thresholds have been intensively studied since the introduction of the percolation model. Recently, the critical polynomial P_{B}(p,L) was introduced for planar-lattice percolation models, where p is the occupation probability and L is the linear system size. The solution of P_{B}=0 can reproduce all known exact thresholds and leads to unprecedented estimates for thresholds of unsolved planar-lattice models. In two dimensions, assuming the universality of P_{B}, we use it to study a nonplanar lattice model, i.e., the equivalent-neighbor lattice bond percolation, and the continuum percolation of identical penetrable disks, by Monte Carlo simulations and finite-size scaling analysis. It is found that, in comparison with other quantities, P_{B} suffers much less from finite-size corrections. As a result, we obtain a series of high-precision thresholds p_{c}(z) as a function of coordination number z for equivalent-neighbor percolation with z up to O(10^{5}) and clearly confirm the asymptotic behavior zp_{c}-1∼1/sqrt[z] for z→∞. For the continuum percolation model, we surprisingly observe that the finite-size correction in P_{B} is unobservable within uncertainty O(10^{-5}) as long as L≥3. The estimated threshold number density of disks is ρ_{c}=1.43632505(10), slightly below the most recent result ρ_{c}=1.43632545(8) of Mertens and Moore obtained by other means. Our work suggests that the critical polynomial method can be a powerful tool for studying nonplanar and continuum systems in statistical mechanics.

Site percolation on square and simple cubic lattices with extended neighborhoods and their continuum limit

By means of extensive Monte Carlo simulation, we study extended-range site percolation on square and simple cubic lattices with various combinations of nearest neighbors up to the eighth nearest neighbors for the square lattice and the ninth nearest neighbors for the simple cubic lattice. We find precise thresholds for 23 systems using a single-cluster growth algorithm. Site percolation on lattices with compact neighborhoods of connected sites can be mapped to problems of lattice percolation of extended objects of a given shape, such as disks and spheres, and the thresholds can be related to the continuum thresholds η_{c} for objects of those shapes. This mapping implies zp_{c}∼4η_{c}=4.51235 in two dimensions and zp_{c}∼8η_{c}=2.7351 in three dimensions for large z for circular and spherical neighborhoods, respectively, where z is the coordination number. Fitting our data for compact neighborhoods to the form p_{c}=c/(z+b) we find good agreement with this prediction, c=2^{d}η_{c}, with the constant b representing a finite-z correction term. We also examined results from other studies using this fitting formula. A good fit of the large but finite-z behavior can also be made using the formula p_{c}=1-exp(-2^{d}η_{c}/z), a generalization of a formula of Koza, Kondrat, and Suszcayński [J. Stat. Mech.: Theor. Exp. (2014) P110051742-546810.1088/1742-5468/2014/11/P11005]. We also study power-law fits which are applicable for the range of values of z considered here.

Confined diffusion in a random Lorentz gas environment

We study the diffusive behavior of biased Brownian particles in a two dimensional confined geometry filled with the freezing obstacles. The transport properties of these particles are investigated for various values of the obstacle density η and the scaling parameter f, which is the ratio of work done to the particles to available thermal energy. We show that, when the thermal fluctuations dominate over the external force, i.e., small f regime, particles get trapped in the given environment when the system percolates at the critical obstacle density η_{c}≈1.2. However, as f increases, we observe that particle trapping occurs prior to η_{c}. In particular, we find a relation between η and f which provides an estimate of the minimum η up to a critical scaling parameter f_{c} beyond which the Fick-Jacobs description is invalid. Prominent transport features like nonmonotonic behavior of the nonlinear mobility, anomalous diffusion, and greatly enhanced effective diffusion coefficient are explained for various strengths of f and η. Also, it is interesting to observe that particles exhibit different kinds of diffusive behaviors, i.e., subdiffusion, normal diffusion, and superdiffusion. These findings, which are genuine to the confined and random Lorentz gas environment, can be useful to understand the transport of small particles or molecules in systems such as molecular sieves and porous media, which have a complex heterogeneous environment of the freezing obstacles.

Site-bond percolation solution to preventing the propagation of Phytophthora zoospores on plantations

We propose a strategy based on the site-bond percolation to minimize the propagation of Phytophthora zoospores on plantations, consisting in introducing physical barriers between neighboring plants. Two clustering processes are distinguished: (i) one of cells with the presence of the pathogen, detected on soil analysis, and (ii) that of diseased plants, revealed from a visual inspection of the plantation. The former is well described by the standard site-bond percolation. In the latter, the percolation threshold is fitted by a Tsallis distribution when no barriers are introduced. We provide, for both cases, the formulas for the minimal barrier density to prevent the emergence of the spanning cluster. Though this work is focused on a specific pathogen, the model presented here can also be applied to prevent the spreading of other pathogens that disseminate, by other means, from one plant to the neighboring ones. Finally, the application of this strategy to three types of commercially important Mexican chili plants is also shown.

Percolation thresholds for discorectangles: Numerical estimation for a range of aspect ratios

Using Monte Carlo simulation, we have studied the percolation of discorectangles. Also known as stadiums or two-dimensional spherocylinders, a discorectangle is a rectangle with semicircles at a pair of opposite sides. Scaling analysis was performed to obtain the percolation thresholds in the thermodynamic limits. We found that (i) for the two marginal aspect ratios ɛ=1 (disc) and ɛ→∞ (stick) the percolation thresholds coincide with known values within the statistical error and (ii) for intermediate values of ɛ the percolation threshold lies between the percolation thresholds for ellipses and rectangles and approaches the latter as the aspect ratio increases.

Room temperature single electron transistor based on a size-selected aluminium cluster

Single electron transistors (SETs) are powerful devices to study the properties of nanoscale objects. However, the capabilities of placing a nano-object between electrical contacts under pristine conditions are lacking. Here, we developed a versatile two point contacting approach that tackles this challenge, which is demonstrated by constructing in situ a prototypical SET device consisting of a single aluminium cluster of 66 ± 5 atoms, deposited directly in a gold nanogap using an innovative cluster beam deposition technique. The gate driven conductance measurements demonstrate Coulomb blockade oscillations at room temperature correlating with an extracted charging energy of 0.14 eV, which is five times larger than k_BT at 300 K. Our work provides a model SET device platform to probe the quantum features of nano-objects with high precision.

Modeling Electrical Percolation to optimize the Electromechanical Properties of CNT/Polymer Composites in Highly Stretchable Fiber Strain Sensors

A simulation model of electrical percolation through a three-dimensional network of curved CNTs is developed in order to analyze the electromechanical properties of a highly stretchable fiber strain sensor made of a CNT/polymer composite. Rigid-body movement of the curved CNTs within the polymer matrix is described analytically. Random arrangements of CNTs within the composite are generated by a Monte-Carlo simulation method and a union-find algorithm is utilized to investigate the network percolation. Consequently, the strain-induced resistance change curves are obtained in a wide strain range of the composite. In order to compare our model with experimental results, two CNT/polymer composite fibers were fabricated and tested as strain sensors. Their effective CNT volume fractions are estimated by comparing the experimental data with our simulation model. The results confirm that the proposed simulation model reproduces well the experimental data and is useful for predicting and optimizing the electromechanical characteristics of highly stretchable fiber strain sensors based on CNT/polymer composites.

Mechanical Properties of DPPC-POPE Mixed Langmuir Monolayers

The mechanical properties of lipid monolayers and their responses to shear and compression stresses play an important role in processes such as breathing and eye blinking. We studied the mechanical properties of Langmuir monolayers of a model mixture, composed of an unsaturated lipid, 1-palmitoyl-2-oleoyl-sn-glycero-phosphoethanolamine (POPE), and a saturated lipid, 1,2-dipalmitoyl-sn-glycero-phosphocholine (DPPC). We performed isothermal compressions and sinusoidal shear deformations of these mixed monolayers. Also, the different phases were observed with Brewster angle microscopy. We found that the mechanical behavior is affected by the miscibility of both lipids. In the two-phase region, the compression elastic modulus increases with the amount of the LC phase but does not follow the predictions of a simple effective medium model. The discrepancies arise from the fact that, upon compression, the domains grow at a rate faster than the compression rate but not fast enough to reach thermodynamic equilibrium. Before reaching the LC phase, domain percolation is observed and compression and shear moduli become equal to those of the pure LC phase. Most of the monolayers behave as viscoelastic fluids at the frequencies investigated. A minimum in the compression modulus and shear viscosity was observed for mixtures close to equimolar composition, with the minimum being accompanied by a change in domain shapes.

From early nucleation past the percolation threshold: Status of the Kolmogorov-Avrami theory on a cold Ising lattice

A high-precision test of the Kolmogorov-Avrami (KA) model of "geometric" interactions between growing nuclei is performed for an Ising ferromagnet driven by Metropolis dynamics. As long as the interface boundaries of interacting nuclei are rough, the KA model is adequate for arbitrary large densities of the new phase, well beyond the percolation transition, as in the case of homogeneous nucleation. For a prenucleated system the KA description breaks down at sufficiently low temperature, when the length of a kinetically smooth interface exceeds the distance between seeds.

Clustering and anisotropic correlated percolation in polar flocks

We study clustering and percolation phenomena in the Vicsek model, taken here in its capacity of prototypical model for dry aligning active matter. Our results show that the order-disorder transition is not related in any way to a percolation transition, contrary to some earlier claims. We study geometric percolation in each of the phases at play, but we mostly focus on the ordered Toner-Tu phase, where we find that the long-range correlations of density fluctuations give rise to an anisotropic percolation transition.

Spatial point pattern and urban morphology: Perspectives from entropy, complexity, and networks

Spatial organization of physical form of an urban system, or city, both manifests and influences the way its social form functions. Mathematical quantification of the spatial pattern of a city is, therefore, important for understanding various aspects of the system. In this work, a framework to characterize the spatial pattern of urban locations based on the idea of entropy maximization is proposed. Three spatial length scales in the system with discerning interpretations in terms of the spatial arrangement of the locations are calculated. Using these length scales, two quantities are introduced to quantify the system's spatial pattern, namely, mass decoherence and space decoherence, whose combination enables the comparison of different cities in the world. The comparison reveals different types of urban morphology that could be attributed to the cities' geographical background and development status.

Relaxation in two-dimensional suspensions of rods as driven by Brownian diffusion

Relaxation in a two-dimensional suspensions containing rods was studied by using dynamic Monte Carlo simulations. An off-lattice model with continuous positional and orientational degrees of freedom was considered. The initial state was produced by using a random sequential adsorption model. During the relaxation, the rods underwent translational and rotational Brownian motions. The simulations were run at different values of number density ρ (the number of rods per unit area) and of the initial orientation order parameter S_{i}. The rods were assumed to have core-shell structures. The evolutions of both the connectivity and the order parameter have revealed different relaxation behavior.

Area covered by disks in small-bounded continuum percolating systems: An application to the string percolation model

In string percolation model, the study of colliding systems at high energies is based on a continuum percolation theory in two dimensions, where the number of strings distributed in the surface of interest is strongly determined by the size and energy of the colliding particles. It is also expected that the surface where the disks are lying be finite, defining a system without periodic boundary conditions. In this work, we report modifications to the fraction of the area covered by disks in continuum percolating systems due to a finite number of disks and bounded by different geometries: circle, ellipse, triangle, square, and pentagon, which correspond to the first Fourier modes of the shape fluctuation of the initial state after the particle collision. We find that the deviation of the fraction of area covered by disks from its corresponding value in the thermodynamic limit satisfies a universal behavior, where the free parameters depend on the density profile, number of disks, and shape of the boundary. Consequently, it is also found that the color suppression factor of the string percolation model is modified by a damping function related to the small-bounded effects. Corrections to the temperature and the speed of sound defined in string systems are also shown for small and elliptically bounded systems.

Modeling Nanostructure in Graphene Oxide: Inhomogeneity and the Percolation Threshold

Graphene oxide (GO) is an amorphous 2D material, which has found widespread use in the fields of chemistry, physics, and materials science due to its similarity to graphene with the benefit of being far easier to synthesize and process. However, the standard of GO characterization is very poor because its structure is irregular, being sensitive to the preparation method, and it has a propensity to transform due to its reactive nature. Atomistic simulations of GO are common, but the nanostructure in these simulations is often based on little evidence or thought. We have written a computer program to generate graphene oxide nanostructures for general purpose atomistic simulation based on theoretical and experimental evidence. The structures generated offer a significant improvement to the current standard of randomly placed oxidized functional groups and successfully recreate the two-phase nature of oxidized and unoxidized graphene domains observed in microscopy experiments. Using this model, we reveal new features of GO structure and predict that a critical point in the oxidation reaction exists as the oxidized region reaches a percolation threshold. Even by a conservative estimate, we show that, if the carbon to oxygen ratio is kept above 6, a continuous aromatic network will remain, preserving many of graphene's desirable properties, irrespective of the oxidation method or the size distribution of graphene sheets. This is an experimentally achievable degree of oxidation and should aid better GO synthesis for many applications.

Percolative Scale-Free Behavior in the Boiling Crisis

We present the first experimental observations of scale-free behavior in the bubble footprint distribution during the boiling crisis of water, in pool and flow boiling conditions. We formulate a continuum percolation model that elucidates how the scale-free behavior emerges from the near-wall stochastic interaction of bubbles and provides a criterion to predict the boiling crisis. It also offers useful insights on how to engineer surfaces that enhance the critical heat flux limit.

Characterizing percolative materials by straining

Carrier transport in a wide range of nanomaterial assemblies proceeds by percolation through discontinuous networks of constituents. Improving percolative nanomaterials could enhance transparent conductors, sensors, and electronic devices. A significant obstacle in optimizing percolative materials is the challenge in their characterization. The critical connection pathways which determine a percolative material's conductivity are not easily accessible with existing metrology tools and traditional investigation approaches rely on indirect methods based on many samples and on simplifying assumptions. We here demonstrate the direct extraction of characteristic parameters from a single sample by analyzing the strain-dependent resistance of percolative materials. An analytical model is derived that can explain experimental data for various percolative materials, morphologies, and straining conditions. The relationship of the extracted parameters with previously introduced figures of merit allows us to compare nanostructures of diverse dimensionalities and compositions for applications such as strain gauges and transparent conductors.

Geometric quenching of orbital pair breaking in a single crystalline superconducting nanomesh network

In a superconductor Cooper pairs condense into a single state and in so doing support dissipation free charge flow and perfect diamagnetism. In a magnetic field the minimum kinetic energy of the Cooper pairs increases, producing an orbital pair breaking effect. We show that it is possible to significantly quench the orbital pair breaking effect for both parallel and perpendicular magnetic fields in a thin film superconductor with lateral nanostructure on a length scale smaller than the magnetic length. By growing an ultra-thin (2 nm thick) single crystalline Pb nanowire network, we establish nm scale lateral structure without introducing weak links. Our network suppresses orbital pair breaking for both perpendicular and in-plane fields with a negligible reduction in zero-field resistive critical temperatures. Our study opens a frontier in nanoscale superconductivity by providing a strategy for maintaining pairing in strong field environments in all directions with important technological implications.

Anisotropy in electrical conductivity of films of aligned intersecting conducting rods

Numerical simulations by means of the Monte Carlo method have been performed to study the electrical properties of a two-dimensional composite filled with rodlike particles. The main goal was to study the effect of the alignment of such rods on the anisotropy of its electrical conductivity. A continuous model was used. In this model, the rods have zero-width (i.e., infinite aspect ratio) and they may intersect each other. To involve both the low conductive host matrix and highly conductive fillers (rods) in the consideration, a discretization algorithm based on the use of a supporting mesh was applied. The discretization is equivalent to the substitution of rods with the polyominoes. Once discretized, the Frank-Lobb algorithm was applied to evaluate the electrical conductivity. Our main findings are (i) the alignment of the rods essentially affects the electrical conductivity and its anisotropy, (ii) the discrete nature of computer simulations is crucial. For slightly disordered system, high electrical anisotropy was observed at small filler content, suggesting a method to enable the production of optically transparent and highly anisotropic conducting films.

Topological data analysis of continuum percolation with disks

We study continuum percolation with disks, a variant of continuum percolation in two-dimensional Euclidean space, by applying tools from topological data analysis. We interpret each realization of continuum percolation with disks as a topological subspace of [0,1]^{2} and investigate its topological features across many realizations. Specifically, we apply persistent homology to investigate topological changes as we vary the number and radius of disks, and we observe evidence that the longest persisting invariant is born at or near the percolation transition.

Anisotropy in electrical conductivity of two-dimensional films containing aligned nonintersecting rodlike particles: Continuous and lattice models

The electrical conductivity of two-dimensional films filled with rodlike particles (rods) was simulated by the Monte Carlo method. The main attention has been paid to the investigation of the effect of the rod alignment on the electrical properties of the films. Both continuous and lattice approaches were used. Intersections of particles were forbidden. Our main findings are (i) both models demonstrate similar behaviors, (ii) at low concentration of rods, both approaches lead to the same dependencies of the electrical conductivity on the concentration of the rods, (iii) the alignment of the rods essentially affects the electrical conductivity, (iv) at some concentrations of partially aligned rods, the films may be conducting only in one direction, and (v) the films may simultaneously be both highly transparent and electrically anisotropic.

Monte Carlo simulations of electrical percolation in multicomponent thin films with nanofillers

We developed a 2D disk-stick percolation model to investigate the electrical percolation behavior of an insulating thin film reinforced with 1D and 2D conductive nanofillers via Monte Carlo simulation. Numerical predictions of the percolation threshold in single component thin films showed good agreement with the previous published work, validating our model for investigating the characteristics of the percolation phenomena. Parametric studies of size effect, i.e., length of 1D nanofiller and diameter of 2D nanofiller, were carried out to predict the electrical percolation threshold for hybrid systems. The relationships between the nanofillers in two hybrid systems was established, which showed differences from previous linear assumption. The effective electrical conductance was evaluated through Kirchhoff's current law by transforming it into a resistor network. The equivalent resistance was obtained from the distribution of nodal voltages by solving a system of linear equations with a Gaussian elimination method. We examined the effects of stick length, relative concentration, and contact patterns of 1D/2D inclusions on electrical performance. One novel aspect of our study is its ability to investigate the effective conductance of nanocomposites as a function of relative concentrations, which shows there is a synergistic effect when nanofillers with different dimensionalities combine properly. Our work provides an important theoretical basis for designing the conductive networks and predicting the percolation properties of multicomponent nanocomposites.

Selective Mechanical Transfer Deposition of Langmuir Graphene Films for High-Performance Silver Nanowire Hybrid Electrodes

In this work, we present silver nanowire hybrid electrodes prepared through the addition of small quantities of pristine graphene by mechanical transfer deposition from surface-assembled Langmuir films. This technique is a fast, efficient, and facile method for modifying the optoelectronic performance of AgNW films. We demonstrate that it is possible to use this technique to perform two-step device production by selective patterning of the stamp used, leading to controlled variation in the local sheet resistance across a device. This is particularly attractive for producing extremely low cost sensors on arbitrarily large scales. Our aim is to address some of the concerns surrounding the use of AgNW films as replacements for indium tin oxide (ITO), namely, the use of scarce materials and poor stability of AgNWs against flexural and environmental degradation.

Percolation thresholds in hyperbolic lattices

We use invasion percolation to compute numerical values for bond and site percolation thresholds p_{c} (existence of an infinite cluster) and p_{u} (uniqueness of the infinite cluster) of tesselations {P,Q} of the hyperbolic plane, where Q faces meet at each vertex and each face is a P-gon. Our values are accurate to six or seven decimal places, allowing us to explore their functional dependency on P and Q and to numerically compute critical exponents. We also prove rigorous upper and lower bounds for p_{c} and p_{u} that can be used to find the scaling of both thresholds as a function of P and Q.