Asymmetry in the percolation thresholds of fully penetrable disks with two different radii

Continuum percolation of two-dimensional adaptive dynamics systems

The percolation phase transition of a continuum adaptive neuron system with homeostasis is investigated. In order to maintain their average activity at a particular level, each neuron (represented by a disk) varies its connection radius until the sum of overlapping areas with neighboring neurons (representing the overall connection strength in the network) has reached a fixed target area for each neuron. Tuning the two key parameters in the model, i.e., the density defined as the number of neurons (disks) per unit area and the sum of the overlapping area of each disk with its adjacent disks, can drive the system into the critical percolating state. These two parameters are inversely proportional to each other at the critical state, and the critical filling factors are fixed about 0.7157, which is much less than the case of the continuum percolation with uniform disks. It is also confirmed that the critical exponents in this model are the same as the two-dimensional standard lattice percolation. Although the critical state is relatively more sensitive and exhibits long-range spatial correlation, local fluctuations do not propagate in a long-range manner through the system by the adaptive dynamics, which renders the system overall robust against perturbations.

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.

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.

Elongation and percolation of defect motifs in anisotropic packing problems

We examine the regime between crystalline and amorphous packings of anisotropic objects on surfaces of different genus by continuously varying their size distribution or shape from monodispersed spheres to bidispersed mixtures or monodispersed ellipsoidal particles; we also consider an anisotropic variant of the Thomson problem with a mixture of charges. With increasing anisotropy, we first observe the disruption of translational order with an intermediate orientationally ordered hexatic phase as proposed by Nelson, Rubinstein and Spaepen, and then a transition to amorphous state. By analyzing the structure of the disclination motifs induced, we show that the hexatic-amorphous transition is caused by the growth and connection of disclination grain boundaries, suggesting this transition lies in the percolation universality class in the scenarios considered.

Postprocessing techniques for gradient percolation predictions on the square lattice

In this work, we revisit the classic problem of site percolation on a regular square lattice. In particular, we investigate the effect of quantization bias errors on percolation threshold predictions for large probability gradients and propose a mitigation strategy. We demonstrate through extensive computational experiments that the assumption of a linear relationship between probability gradient and percolation threshold used in previous investigations is invalid. Moreover, we demonstrate that, due to skewness in the distribution of occupation probabilities visited the average does not converge monotonically to the true percolation threshold. We identify several alternative metrics which do exhibit monotonic (albeit not linear) convergence and document their observed convergence rates.

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.

Ink synthesis and inkjet printing of electrostatically stabilized multilayer graphene nanoshells

HYPOTHESIS

Most functional inkjet inks are sterically stabilized nanoparticle dispersions that require a post-printing-process to remove stabilizing materials and gain functionality. This post-process limits material selection and increases fabrication time and complexity for printed devices. By optimizing the electrostatic stability of a carbon nanomaterial dispersed in water or ethylene glycol via pH adjustment, a stable and printable ink should be attainable without a steric stabilizing material and hence the post-process may be avoided.

EXPERIMENTS

The electrostatic stability of multilayer graphene nanoshells (MGNS)-an inexpensive and net carbon-negative nanomaterial-dispersed in water and ethylene glycol was studied by measuring zeta potential as a function of pH and modeling energetic potentials between particles. Requirements for electrical percolation of printed MGNS were analyzed and corroborated with electrical measurements.

FINDINGS

Electrostatic stability improved with increased zeta potential caused by an increased pH. Ionic strength also increased with pH, causing strong destabilization. By increasing zeta potential while minimizing ionic strength, the maximum solid-loading of MGNS in DI water and ethylene glycol was increased up to 20%. For the MGNS solid-loading achieved here, electrical percolation occurs with 20-30 consecutively printed layers producing a resistivity of 30 Ω-cm. The inexpensive, environmentally-friendly MGNS are a promising material for printed, flexible electronics.

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.

3D-Printed Sugar Scaffold for High-Precision and Highly Sensitive Active and Passive Wearable Sensors

In this study, a pairing of a previously unidentified 3D printing technique and soft materials is introduced in order to achieve not only high-resolution printed features and flexibility of the 3D-printed materials, but also its light-weight and electrical conductivity. Using the developed technique and materials, high-precision and highly sensitive patient-specific wearable active or passive devices are fabricated for personalized health monitoring. The fabricated biosensors show low density and substantial flexibility because of 3D microcellular network-type interconnected conductive materials that are readily printed using an inkjet head. Using high-resolution 3D scanned body-shape data, on-demand personalized wearable sensors made of the 3D-printed soft and conductive materials are fabricated. These sensors successfully detect both actively changing body strain signals and passively changing signals such as electromyography (EMG), electrodermal activity (EDA), and electroencephalogram EEG. The accurately tailored subject-specific shape of the developed sensors exhibits higher sensitivity and faster real-time sensing performances in the monitoring of rapidly changing human body signals. The newly developed 3D printing technique and materials can be widely applied to various types of wearable, flexible, and light-weight biosensors for use in a variety of inexpensive on-demand and personalized point-of-care diagnostics.

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.

Simple model of complex dynamics of activity patterns in developing networks of neuronal cultures

Living neuronal networks in dissociated neuronal cultures are widely known for their ability to generate highly robust spatiotemporal activity patterns in various experimental conditions. Such patterns are often treated as neuronal avalanches that satisfy the power scaling law and thereby exemplify self-organized criticality in living systems. A crucial question is how these patterns can be explained and modeled in a way that is biologically meaningful, mathematically tractable and yet broad enough to account for neuronal heterogeneity and complexity. Here we derive and analyse a simple network model that may constitute a response to this question. Our derivations are based on few basic phenomenological observations concerning the input-output behavior of an isolated neuron. A distinctive feature of the model is that at the simplest level of description it comprises of only two variables, the network activity variable and an exogenous variable corresponding to energy needed to sustain the activity, and few parameters such as network connectivity and efficacy of signal transmission. The efficacy of signal transmission is modulated by the phenomenological energy variable. Strikingly, this simple model is already capable of explaining emergence of network spikes and bursts in developing neuronal cultures. The model behavior and predictions are consistent with published experimental evidence on cultured neurons. At the larger, cellular automata scale, introduction of the energy-dependent regulatory mechanism results in the overall model behavior that can be characterized as balancing on the edge of the network percolation transition. Network activity in this state shows population bursts satisfying the scaling avalanche conditions. This network state is self-sustainable and represents energetic balance between global network-wide processes and spontaneous activity of individual elements.

On the critical threshold for continuum AB percolation

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ,μ. For any λ>0 we consider the percolation threshold μc(λ) associated to the parameter μ. Denoting by λc the percolation threshold for the standard Poisson Boolean model with radii r, we show the lower bound μc(λ)≥clog(c∕(λ−λc)) for any λ>λc with c>0 a fixed constant. In particular, there is no phase transition in μ at the critical value of λ, that is, μc(λc) =∞.

Determination of the Membrane Environment of CD59 in Living Cells

The organization and dynamics of proteins and lipids in the plasma membrane, and their role in membrane functionality, have been subject of a long-lasting debate. Specifically, it is unclear to what extent membrane proteins are affected by their immediate lipid environment and vice versa. Studies on model membranes and plasma membrane vesicles indicated preferences of proteins for lipid phases characterized by different acyl chain order; however, whether such phases do indeed exist in live cells is still not known. Here, we refine a previously developed micropatterning approach combined with single molecule tracking to quantify the influence of the glycosylphosphatidylinositol-anchored (GPI-anchored) protein CD59 on its molecular environment directly in the live cell plasma membrane. We find that locally enriched and immobilized CD59 presents obstacles to the diffusion of fluorescently labeled lipids with a different phase-partitioning behavior independent of cell cholesterol levels and type of lipid. Our results give no evidence for either specific binding of the lipids to CD59 or the existence of nanoscopic ordered membrane regions associated with CD59.

Percolation of binary disk systems: Modeling and theory

The dispersion and connectivity of particles with a high degree of polydispersity is relevant to problems involving composite material properties and reaction decomposition prediction and has been the subject of much study in the literature. This work utilizes Monte Carlo models to predict percolation thresholds for a two-dimensional systems containing disks of two different radii. Monte Carlo simulations and spanning probability are used to extend prior models into regions of higher polydispersity than those previously considered. A correlation to predict the percolation threshold for binary disk systems is proposed based on the extended dataset presented in this work and compared to previously published correlations. A set of boundary conditions necessary for a good fit is presented, and a condition for maximizing percolation threshold for binary disk systems is suggested.

High resolution scanning near field mapping of enhancement on SERS substrates: comparison with photoemission electron microscopy

The need for a dedicated spectroscopic technique with nanoscale resolution to characterize SERS substrates pushed us to develop a proof of concept of a functionalized tip-surface enhanced Raman scattering (FTERS) technique. We have been able to map hot spots on semi-continuous gold films; in order to validate our approach we compare our results with photoemission electron microscopy (PEEM) data, the complementary electron microscopy tool to map hot spots on random metallic surfaces. Enhanced Raman intensity maps at high spatial resolution reveal the localisation of hotspots at gaps for many neighboring nanostructures. Finally, we compare our findings with theoretical simulations of the enhancement factor distribution, which confirms a dimer effect as the dominant origin of hot spots.

Monte Carlo simulations of protein micropatterning in biomembranes: effects of immobile sticky obstacles

Single molecule trajectories of lipids and proteins can yield valuable information about the nanoscopic organization of the plasma membrane itself. The interpretation of such trajectories, however, is complicated, as the mobility of molecules can be affected by the presence of immobile obstacles, and the transient binding of the tracers to these obstacles. We have previously developed a micropatterning approach that allows for immobilizing a plasma membrane protein and probing the diffusional behavior of a putative interaction partner in living cells. Here, we provide guidelines on how this micropatterning approach can be extended to quantify interaction parameters between plasma membrane constituents in their natural environment. We simulated a patterned membrane system and evaluated the effect of different surface densities of patterned immobile obstacles on the relative mobility as well as the surface density of diffusing tracers. In the case of inert obstacles, the size of the obstacle can be assessed from its surface density at the percolation threshold, which in turn can be extracted from the diffusion behavior of the tracer. For sticky obstacles, two-dimensional dissociation constants can be determined from the tracer diffusion or surface density.

Thin Film Formation and Morphology of Electrosprayed Polydimethylsiloxane

Low-voltage dielectric actuators (DEAs) can be fabricated using submicrometer-thin polydimethylsiloxane (PDMS) films. The two established techniques, namely spin coating and molecular beam deposition, however, are inappropriate to produce multistack DEAs in an efficient way. Therefore, we propose an alternative deposition technique, i.e., the alternating current electrospray deposition (ACESD) of 5 vol % PDMS in ethyl acetate solution and subsequent ultraviolet light curing. Atomic force microscopy makes possible the three-dimensional analysis of cured droplet-like islands. These circular islands, prepared on 2 in. Si(100) wafers from four polymers with molecular masses between 800 and 62,700 g/mol, reveal a characteristic morphology with an increasing height-to-diameter ratio. Using the 6000 g/mol polymer for ACESD, the film morphology evolution was tracked by applying conventional optical microscopy and spectroscopic ellipsometry. When the deposition was terminated after 13 s, circular islands with a mean height of 30 nm were found, while terminating the deposition after about 155 s led to a confluent layer with a mean height of 91 ± 10 nm. Potential electrostatic interactions between the droplets could not be identified through the analysis of spatial island distribution. Nevertheless, ACESD is a budget-priced and competitive deposition technique that can be employed to fabricate submicrometer-thin PDMS films with true nanometer roughness.

Continuum AB percolation and AB random geometric graphs

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ and μ. We show for d ≥ 2 that if λ is supercritical for the one-type random geometric graph with distance parameter 2r, there exists μ such that (λ, μ) is supercritical (this was previously known for d = 2). For d = 2, we also consider the restriction of this graph to points in the unit square. Taking μ = τ λ for fixed τ, we give a strong law of large numbers as λ → ∞ for the connectivity threshold of this graph.

Effects of shape and flexibility of conductive fillers in nanocomposites on percolating network formation and electrical conductivity

Nanocomposites consist of nanofillers and matrices, thus allowing one to design novel materials with desirable properties of both nanofillers and matrices. The percolating network formation of nanofillers in matrices is critical to such desired properties. Some nanofillers such as carbon nanotubes and graphene nanosheets are so flexible that they become either wavy or crumpled. Such a variability in the nanofiller conformation may affect the percolating network formation but has been often (but not always) ignored in the theoretical and computational investigation. In this work, we investigate how the flexibility of different kinds of nanofillers influences the formation of the percolating network by performing extensive Langevin dynamics simulations. We consider three kinds of nanofillers of different shape: nanospheres, nanorods, and nanoplates. When the sizes of nanofillers (or the radius of gyration, R(g)) are comparable, nanorods form a percolating network at a lower volume fraction than nanoplates while nanofillers require the highest volume fraction to form the percolating network, which is consistent with previous experiments. The percolation threshold concentration (ϕ(c)) of nanospheres increases with an increase in their R(g), while ϕ(c) of nanorods and nanoplates decrease with R(g). However, the effect of flexibility on the percolation threshold volume fraction is much more significant for nanoplates than nanorods. We also estimate the electric conductivity and find that the electric conductivity follows a scaling relation faithfully but with different critical exponents depending on the shape and flexibility.

Anomalous Magnetotransport in Disordered Structures: Classical Edge-State Percolation

By event-driven molecular dynamics simulations we investigate magnetotransport in a two-dimensional model with randomly distributed scatterers close to the field-induced localization transition. This transition is generated by percolating skipping orbits along the edges of obstacle clusters. The dynamic exponents differ significantly from those of the conventional transport problem on percolating systems, thus establishing a new dynamic universality class. This difference is tentatively attributed to a weak-link scenario, which emerges naturally due to barely overlapping edge trajectories. We make predictions for the frequency-dependent conductivity and discuss implications for active colloidal circle swimmers in a hetegogeneous environment.

Direct relations between morphology and transport in Boolean models

We study the relation of permeability and morphology for porous structures composed of randomly placed overlapping circular or elliptical grains, so-called Boolean models. Microfluidic experiments and lattice Boltzmann simulations allow us to evaluate a power-law relation between the Euler characteristic of the conducting phase and its permeability. Moreover, this relation is so far only directly applicable to structures composed of overlapping grains where the grain density is known a priori. We develop a generalization to arbitrary structures modeled by Boolean models and characterized by Minkowski functionals. This generalization works well for the permeability of the void phase in systems with overlapping grains, but systematic deviations are found if the grain phase is transporting the fluid. In the latter case our analysis reveals a significant dependence on the spatial discretization of the porous structure, in particular the occurrence of single isolated pixels. To link the results to percolation theory we performed Monte Carlo simulations of the Euler characteristic of the open cluster, which reveals different regimes of applicability for our permeability-morphology relations close to and far away from the percolation threshold.

Continuum percolation of polydisperse hyperspheres in infinite dimensions

We analyze the critical connectivity of systems of penetrable d-dimensional spheres having size distributions in terms of weighed random geometrical graphs, in which vertex coordinates correspond to random positions of the sphere centers, and edges are formed between any two overlapping spheres. Edge weights naturally arise from the different radii of two overlapping spheres. For the case in which the spheres have bounded size distributions, we show that clusters of connected spheres are treelike for d→∞ and they contain no closed loops. In this case, we find that the mean cluster size diverges at the percolation threshold density η(c)→2(-d), independently of the particular size distribution. We also show that the mean number of overlaps for a particle at criticality z(c) is smaller than unity, while z(c)→1 only for spheres with fixed radii. We explain these features by showing that in the large dimensionality limit, the critical connectivity is dominated by the spheres with the largest size. Assuming that closed loops can be neglected also for unbounded radii distributions, we find that the asymptotic critical threshold for systems of spheres with radii following a log-normal distribution is no longer universal, and that it can be smaller than 2(-d) for d→∞.

Rounding of the localization transition in model porous media

The generic mechanisms of anomalous transport in porous media are investigated by computer simulations of two-dimensional model systems. In order to bridge the gap between the strongly idealized Lorentz model and realistic models of porous media, two models of increasing complexity are considered: a cherry-pit model with hard-core correlations as well as a soft-potential model. An ideal gas of tracer particles inserted into these structures is found to exhibit anomalous transport which extends up to several decades in time. Also, the self-diffusion of the tracers becomes suppressed upon increasing the density of the systems. These phenomena are attributed to an underlying percolation transition. In the soft potential model the transition is rounded, since each tracer encounters its own critical density according to its energy. Therefore, the rounding of the transition is a generic occurrence in realistic, soft systems.

Tunneling Conductivity and Piezoresistivity of Composites Containing Randomly Dispersed Conductive Nano-Platelets

In this study, a three-dimensional continuum percolation model was developed based on a Monte Carlo simulation approach to investigate the percolation behavior of an electrically insulating matrix reinforced with conductive nano-platelet fillers. The conductivity behavior of composites rendered conductive by randomly dispersed conductive platelets was modeled by developing a three-dimensional finite element resistor network. Parameters related to the percolation threshold and a power-low describing the conductivity behavior were determined. The piezoresistivity behavior of conductive composites was studied employing a reoriented resistor network emulating a conductive composite subjected to mechanical strain. The effects of the governing parameters, i.e., electron tunneling distance, conductive particle aspect ratio and size effects on conductivity behavior were examined.

Continuum percolation of overlapping disks with a distribution of radii having a power-law tail

We study the continuum percolation problem of overlapping disks with a distribution of radii having a power-law tail; the probability that a given disk has a radius between R and R+dR is proportional to R(-(a+1)), where a>2. We show that in the low-density nonpercolating phase, the two-point function shows a power-law decay with distance, even at arbitrarily low densities of the disks, unlike the exponential decay in the usual percolation problem. As in the problem of fluids with long-range interaction, we argue that in our problem, the critical exponents take their short-range values for a>3-η(sr) whereas they depend on a for a<3-η(sr) where η(sr) is the anomalous dimension for the usual percolation problem. The mean-field regime obtained in the fluid problem corresponds to the fully covered regime, a≤2, in the percolation problem. We propose an approximate renormalization scheme to determine the correlation length exponent ν and the percolation threshold. We carry out Monte Carlo simulations and determine the exponent ν as a function of a. The determined values of ν show that it is independent of the parameter a for a>3-η(sr) and is equal to that for the lattice percolation problem, whereas ν varies with a for 2<a<3-η(sr). We also determine the percolation threshold of the system as a function of the parameter a.

Percolation thresholds of two-dimensional continuum systems of rectangles

The present paper introduces an efficient Monte Carlo algorithm for continuum percolation composed of randomly oriented rectangles. By conducting extensive simulations, we report high-precision percolation thresholds for a variety of homogeneous systems with different rectangle aspect ratios. This paper verifies and extends the excluded area theory. It is confirmed that percolation thresholds are dominated by the average excluded areas for both homogeneous and heterogeneous rectangle systems (except for some special heterogeneous systems where the rectangle lengths differ too much from one another). In terms of the excluded areas, generalized formulas are proposed to effectively predict precise percolation thresholds for all these rectangle systems. This paper is, therefore, helpful for both practical applications and theoretical studies concerning relevant systems.

Anomalous transport in the crowded world of biological cells

A ubiquitous observation in cell biology is that the diffusive motion of macromolecules and organelles is anomalous, and a description simply based on the conventional diffusion equation with diffusion constants measured in dilute solution fails. This is commonly attributed to macromolecular crowding in the interior of cells and in cellular membranes, summarizing their densely packed and heterogeneous structures. The most familiar phenomenon is a sublinear, power-law increase of the mean-square displacement (MSD) as a function of the lag time, but there are other manifestations like strongly reduced and time-dependent diffusion coefficients, persistent correlations in time, non-Gaussian distributions of spatial displacements, heterogeneous diffusion and a fraction of immobile particles. After a general introduction to the statistical description of slow, anomalous transport, we summarize some widely used theoretical models: Gaussian models like fractional Brownian motion and Langevin equations for visco-elastic media, the continuous-time random walk model, and the Lorentz model describing obstructed transport in a heterogeneous environment. Particular emphasis is put on the spatio-temporal properties of the transport in terms of two-point correlation functions, dynamic scaling behaviour, and how the models are distinguished by their propagators even if the MSDs are identical. Then, we review the theory underlying commonly applied experimental techniques in the presence of anomalous transport like single-particle tracking, fluorescence correlation spectroscopy (FCS) and fluorescence recovery after photobleaching (FRAP). We report on the large body of recent experimental evidence for anomalous transport in crowded biological media: in cyto- and nucleoplasm as well as in cellular membranes, complemented by in vitro experiments where a variety of model systems mimic physiological crowding conditions. Finally, computer simulations are discussed which play an important role in testing the theoretical models and corroborating the experimental findings. The review is completed by a synthesis of the theoretical and experimental progress identifying open questions for future investigation.

Effect of dimensionality on the percolation threshold of overlapping nonspherical hyperparticles

We study the effect of dimensionality on the percolation threshold η(c) of identical overlapping nonspherical convex hyperparticles in d-dimensional Euclidean space R(d). This is done by formulating a scaling relation for η(c) that is based on a rigorous lower bound [Torquato, J. Chem. Phys. 136, 054106 (2012)] and a conjecture that hyperspheres provide the highest threshold, for any d, among all convex hyperparticle shapes (that are not a trivial affine transformation of a hypersphere). This scaling relation also exploits the recently discovered principle that low-dimensional continuum percolation behavior encodes high-dimensional information. We derive an explicit formula for the exclusion volume v(ex) of a hyperparticle of arbitrary shape in terms of its d-dimensional volume v, surface area s, and radius of mean curvature R[over ¯] (or, equivalently, mean width). These basic geometrical properties are computed for a wide variety of nonspherical hyperparticle shapes with random orientations across all dimensions, including, among other shapes, various polygons for d=2, Platonic solids, spherocylinders, parallepipeds, and zero-volume plates for d=3 and their appropriate generalizations for d≥4. Using this information, we compute the lower bound and scaling relation for η(c) for this comprehensive set of continuum percolation models across dimensions. We demonstrate that the scaling relation provides accurate upper-bound estimates of the threshold η(c) across dimensions and becomes increasingly accurate as the space dimension increases.

Permeability of porous materials determined from the Euler characteristic

We study the permeability of quasi-two-dimensional porous structures of randomly placed overlapping monodisperse circular and elliptical grains. Measurements in microfluidic devices and lattice Boltzmann simulations demonstrate that the permeability is determined by the Euler characteristic of the conducting phase. We obtain an expression for the permeability that is independent of the percolation threshold and shows agreement with experimental and simulated data over a wide range of porosities. Our approach suggests that the permeability explicitly depends on the overlapping probability of grains rather than their shape.

Continuum percolation thresholds in two dimensions

A wide variety of methods have been used to compute percolation thresholds. In lattice percolation, the most powerful of these methods consists of microcanonical simulations using the union-find algorithm to efficiently determine the connected clusters, and (in two dimensions) using exact values from conformal field theory for the probability, at the phase transition, that various kinds of wrapping clusters exist on the torus. We apply this approach to percolation in continuum models, finding overlaps between objects with real-valued positions and orientations. In particular, we find precise values of the percolation transition for disks, squares, rotated squares, and rotated sticks in two dimensions and confirm that these transitions behave as conformal field theory predicts. The running time and memory use of our algorithm are essentially linear as a function of the number of objects at criticality.

Percolation scaling in composites of exfoliated MoS2 filled with nanotubes and graphene

Applications of films of exfoliated layered compounds in many areas will be limited by their relatively low electrical conductivity. To address this, we have prepared and characterised composites of a nano-conductor (nanotubes or graphene) embedded in a matrix of exfoliated MoS(2) nanosheets. Solvent exfoliation of MoS(2) nanosheets, followed by blending with dispersions of graphene or nanotubes allowed the formation of such composite films by vacuum filtration. This gave spatially uniform mixtures with fully tuneable nano-conductor content. By addition of the nano-conducting phase, it was possible to vary the electrical conductivity of the composite over nine orders of magnitude. For both filler types the conductivity followed percolation scaling laws both above and below the percolation threshold. In the case of SWNT-filled composites, conductivities as high as ~40 S m(-1) were achieved at volume fractions as low as ~4%.

Connectivity percolation in suspensions of hard platelets

We present a study on connectivity percolation in suspensions of hard platelets by means of Monte Carlo simulation. We interpret our results using a contact-volume argument based on an effective single-particle cell model. It is commonly assumed that the percolation threshold of anisotropic objects scales as their inverse aspect ratio. While this rule has been shown to hold for rodlike particles, we find that for hard platelike particles the percolation threshold is nonmonotonic in the aspect ratio. It exhibits a shallow minimum at intermediate aspect ratios and then saturates to a constant value. This effect is caused by the isotropic-nematic transition preempting the percolation transition. Hence the common strategy to use highly anisotropic, conductive particles as fillers in composite materials in order to produce conduction at low filler concentration is expected to fail for platelike fillers such as graphene and graphite nanoplatelets.

Critical condition of the water-retention model

We study how much water can be retained without leaking through boundaries when each unit square of a two-dimensional lattice is randomly assigned a block of unit bottom area but with different heights from zero to n-1. As more blocks are put into the system, there exists a phase transition beyond which the system retains a macroscopic volume of water. We locate the critical points and verify that the criticality belongs to the two-dimensional percolation universality class. If the height distribution can be approximated as continuous for large n, the system is always close to a critical point and the fraction of the area below the resulting water level is given by the percolation threshold. This provides a universal upper bound of areas that can be covered by water in a random landscape.

Diffusion amid random overlapping obstacles: similarities, invariants, approximations

Efficient and accurate numerical techniques are used to examine similarities of effective diffusion in a void between random overlapping obstacles: essential invariance of effective diffusion coefficients (D(eff)) with respect to obstacle shapes and applicability of a two-parameter power law over nearly entire range of excluded volume fractions (φ), except for a small vicinity of a percolation threshold. It is shown that while neither of the properties is exact, deviations from them are remarkably small. This allows for quick estimation of void percolation thresholds and approximate reconstruction of D(eff) (φ) for obstacles of any given shape. In 3D, the similarities of effective diffusion yield a simple multiplication "rule" that provides a fast means of estimating D(eff) for a mixture of overlapping obstacles of different shapes with comparable sizes.

Two-dimensional continuum percolation threshold for diffusing particles of nonzero radius

Lateral diffusion in the plasma membrane is obstructed by proteins bound to the cytoskeleton. The most important parameter describing obstructed diffusion is the percolation threshold. The thresholds are well known for point tracers, but for tracers of nonzero radius, the threshold depends on the excluded area, not just the obstacle concentration. Here thresholds are obtained for circular obstacles on the continuum. Random obstacle configurations are generated by Brownian dynamics or Monte Carlo methods, the obstacles are immobilized, and the percolation threshold is obtained by solving a bond percolation problem on the Voronoi diagram of the obstacles. The percolation threshold is expressed as the diameter of the largest tracer that can cross a set of immobile obstacles at a prescribed number density. For random overlapping obstacles, the results agree with the known analytical solution quantitatively. When the obstacles are soft disks with a 1/r(12) repulsion, the percolating diameter is approximately 20% lower than for overlapping obstacles. A percolation model predicts that the threshold is highly sensitive to the tracer radius. To our knowledge, such a strong dependence has so far not been reported for the plasma membrane, suggesting that percolation is not the factor controlling lateral diffusion. A definitive experiment is proposed.

Conductivity of ruthenate nanosheets prepared via electrostatic self-assembly: characterization of isolated single nanosheet crystallite to mono- and multilayer electrodes

Ultrathin films composed of ruthenate nanosheets (RuO(2)ns) were fabricated via electrostatic self-assembly of unilamellar RuO(2)ns crystallites derived by total exfoliation of an ion-exchangeable layered ruthenate. Ultrathin films with submonolayer to monolayer RuO(2)ns coverage and multilayered RuO(2)ns thin films were prepared by controlled electrostatic self-assembly and layer-by-layer deposition using a cationic copolymer as the counterion. Electrical properties of a single RuO(2)ns crystallite were successfully measured by means of scanning probe microscopy. The sheet resistance of an isolated single RuO(2)ns crystallite was 12 kΩ sq(-1). Self-assembled submonolayer films behaved as a continuous conducting film for coverage above 70%, which was discussed based on a two-dimensional percolation model. Low sheet resistance was attained for multilayered films with values less than 1 kΩ sq(-1). Interestingly, the grain boundary resistance between nanosheets seems to contribute only slightly to the sheet resistance of self-assembled films.

Finite-size scaling in stick percolation

This work presents the generalization of the concept of universal finite-size scaling functions to continuum percolation. A high-efficiency algorithm for Monte Carlo simulations is developed to investigate, with extensive realizations, the finite-size scaling behavior of stick percolation in large-size systems. The percolation threshold of high precision is determined for isotropic widthless stick systems as Ncl2=5.637 26+/-0.000 02 , with Nc as the critical density and l as the stick length. Simulation results indicate that by introducing a nonuniversal metric factor A=0.106 910+/-0.000 009 , the spanning probability of stick percolation on square systems with free boundary conditions falls on the same universal scaling function as that for lattice percolation.

Geometric percolation thresholds of interpenetrating plates in three-dimensional space

The geometric percolation thresholds for circular, elliptical, square, and triangular plates in the three-dimensional space are determined precisely by Monte Carlo simulations. These geometries represent oblate particles in the limit of zero thickness. The normalized percolation points, which are estimated by extrapolating the data to zero radius, are etac=0.961 4+/-0.000 5, 0.864 7+/-0.000 6, and 0.729 5+/-0.000 6 for circles, squares, and equilateral triangles, respectively. These results show that the noncircular shapes and corner angles in the plate geometry tend to increase the interparticle connectivity and therefore reduce the percolation point. For elliptical plates, the percolation threshold is found to decrease moderately, when the aspect ratio epsilon is between 1 and 1.5, but decrease rapidly for epsilon greater than 1.5. For the binary dispersion of circular plates with two different radii, etac is consistently larger than that of equisized plates, with the maximum value located at around r1/r2=0.5.

Universal condition for critical percolation thresholds of kagomé-like lattices

Lattices that can be represented in a kagomé-like form are shown to satisfy a universal percolation criticality condition, expressed as a relation between P3 , the probability that all three vertices in the triangle connect, and P0 , the probability that none connect. A linear approximation for P3(P0) is derived and appears to provide a rigorous upper bound for critical thresholds. A numerically determined relation for P3(P0) gives thresholds for the kagomé, site-bond honeycomb, (3-12;{2}) lattice, and "stack-of-triangle" lattices that compare favorably with numerical results.