Efficient measurement of the percolation threshold for fully penetrable discs

Domain convexification: A simple model for invasion processes

We propose an invasion model where domains grow up to their convex hulls and merge when they overlap. This model can be seen as a continuum and isotropic counterpart of bootstrap percolation models. From numerical investigations of the model starting with randomly deposited overlapping disks on a plane, we find an invasion transition that occurs via macroscopic avalanches. The disk concentration threshold and the width of the transition are found to decrease as the system size is increased. Our results are consistent with a vanishing threshold in the limit of infinitely large system sizes. However, this limit could not be investigated by simulations. For finite initial concentrations of disks, the cluster size distribution presents a power-law tail characterized by an exponent that varies approximately linearly with the initial concentration of disks. These results at finite initial concentration open novel directions for the understanding of the transition in systems of finite size. Furthermore, we find that the domain area distribution has oscillations with discontinuities. In addition, the deviation from circularity of large domains is constant. Finally, we compare our results to experimental observations on de-adhesion of graphene induced by the intercalation of nanoparticles.

Two-stage random sequential adsorption of discorectangles and disks on a two-dimensional surface

The different variants of two-stage random sequential adsorption (RSA) models for packing of disks and discorectangles on a two-dimensional (2D) surface were investigated. In the SD (sticks+disks) model, the discorectangles were first deposited and then the disks were added. In the DS (disks+sticks) model, the disks were first deposited and then discorectangles were added. At the first stage the particles were deposited up to the selected concentration and at the final (second) stage the particles were deposited up to the saturated (jamming) state. The main parameters of the models were the concentration of particles deposited at the first stage, aspect ratio of the discorectangles ɛ (length to diameter of ratio ɛ=l/d) and disk diameter D. All distances were measured using the value of d as a unit of measurement of linear dimensions, the disk diameter was varied in the interval D∈[1-10], and the aspect ratio value was varied in the interval ɛ∈[1-50]. The dependencies of the jamming coverage of particles deposited at the second stage versus the parameters of the models were analyzed. The presence of first deposited particles for both models regulated the maximum possible disk diameter D_{max} (SD model) or the maximum aspect ratio ɛ_{max} (DS model). This behavior was explained by the deposition of particles in the second stage into triangular (SD model) or elongated (DS model) pores formed by particles deposited at the first stage. The percolation connectivity of disks (SD model) and discorectangles (DS model) for the particles with a hard core and a soft shell structure was analyzed. The disconnectedness was ensured by overlapping of soft shells. The dependencies of connectivity versus the parameters of SD and DS models were also analyzed.

An explicit Dobrushin uniqueness region for Gibbs point processes with repulsive interactions

We present a uniqueness result for Gibbs point processes with interactions that come from a non-negative pair potential; in particular, we provide an explicit uniqueness region in terms of activity z and inverse temperature $\beta$ . The technique used relies on applying to the continuous setting the classical Dobrushin criterion. We also present a comparison to the two other uniqueness methods of cluster expansion and disagreement percolation, which can also be applied for this type of interaction.

The collective effect of finite-sized inhomogeneities on the spatial spread of populations in two dimensions

The dynamics of a population expanding into unoccupied habitat has been primarily studied for situations in which growth and dispersal parameters are uniform in space or vary in one dimension. Here, we study the influence of finite-sized individual inhomogeneities and their collective effect on front speed if randomly placed in a two-dimensional habitat. We use an individual-based model to investigate the front dynamics for a region in which dispersal or growth of individuals is reduced to zero (obstacles) or increased above the background (hotspots), respectively. In a regime where front dynamics is determined by a local front speed only, a principle of least time can be employed to predict front speed and shape. The resulting analytical solutions motivate an event-based algorithm illustrating the effects of several obstacles or hotspots. We finally apply the principle of least time to large heterogeneous environments by solving the Eikonal equation numerically. Obstacles lead to a slow-down that is dominated by the number density and width of obstacles, but not by their precise shape. Hotspots result in a speed-up, which we characterize as function of hotspot strength and density. Our findings emphasize the importance of taking the dimensionality of the environment into account.

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.

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.

Excluded volume effects in on- and off-lattice reaction-diffusion models

Mathematical models are important tools to study the excluded volume effects on reaction-diffusion systems, which are known to play an important role inside living cells. Detailed microscopic simulations with off-lattice Brownian dynamics become computationally expensive in crowded environments. In this study, the authors therefore investigate to which extent on-lattice approximations, the so-called cellular automata models, can be used to simulate reactions and diffusion in the presence of crowding molecules. They show that the diffusion is most severely slowed down in the off-lattice model, since randomly distributed obstacles effectively exclude more volume than those ordered on an artificial grid. Crowded reaction rates can be both increased and decreased by the grid structure and it proves important to model the molecules with realistic sizes when excluded volume is taken into account. The grid artefacts increase with increasing crowder density and they conclude that the computationally more efficient on-lattice simulations are accurate approximations only for low crowder densities.

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.

Finite-size effects and percolation properties of Poisson geometries

Random tessellations of the space represent a class of prototype models of heterogeneous media, which are central in several applications in physics, engineering, and life sciences. In this work, we investigate the statistical properties of d-dimensional isotropic Poisson geometries by resorting to Monte Carlo simulation, with special emphasis on the case d=3. We first analyze the behavior of the key features of these stochastic geometries as a function of the dimension d and the linear size L of the domain. Then, we consider the case of Poisson binary mixtures, where the polyhedra are assigned two labels with complementary probabilities. For this latter class of random geometries, we numerically characterize the percolation threshold, the strength of the percolating cluster, and the average cluster size.

Temperature and layer thickness dependent in situ investigations on epindolidione organic thin-film transistors

We report on in situ performance evaluations as a function of layer thickness and substrate temperature for bottom-gate, bottom-gold contact epindolidione organic thin-film transistors on various gate dielectrics. Experiments were carried out under ultra-high vacuum conditions, enabling quasi-simultaneous electrical and surface analysis. Auger electron spectroscopy and thermal desorption spectroscopy (TDS) were applied to characterize the quality of the substrate surface and the thermal stability of the organic films. Ex situ atomic force microscopy (AFM) was used to gain additional information on the layer formation and surface morphology of the hydrogen-bonded organic pigment. The examined gate dielectrics included SiO2, in its untreated and sputtered forms, as well as the spin-coated organic capping layers poly(vinyl-cinnamate) (PVCi) and poly((±)endo,exo-bicyclo[2.2.1]hept-5-ene-2,3-dicarboxylic acid, diphenylester) (PNDPE, from the class of polynorbornenes). TDS and AFM revealed Volmer-Weber island growth dominated film formation with no evidence of a subjacent wetting layer. This growth mode is responsible for the comparably high coverage required for transistor behavior at 90-95% of a monolayer composed of standing molecules. Surface sputtering and an increased sample temperature during epindolidione deposition augmented the surface diffusion of adsorbing molecules and therefore led to a lower number of better-ordered islands. Consequently, while the onset of charge transport was delayed, higher saturation mobility was obtained. The highest, bottom-contact configuration, mobilities of approximately 2.5 × 10-3cm2/Vs were found for high coverages (50 nm) on sputtered samples. The coverage dependence of the mobility showed very different characteristics for the different gate dielectrics, while the change of the threshold voltage with coverage was approximately the same for all systems. An apparent decrease of the mobility with increasing coverage on the less polar PNDPE was attributed to a change in molecular orientation from upright standing in the thin-film phase to tilted in the bulk phase. From temperature-dependent mobility measurements we calculated activation barriers for the charge transport between 110 meV and 160 meV, depending on the dielectric configuration.

Multifluorophore localization as a percolation problem: limits to density and precision

We show that the maximum desirable density of activated fluorophores in a superresolution experiment can be determined by treating the overlapping point spread functions as a problem in percolation theory. We derive a bound on the density of activated fluorophores, taking into account the desired localization accuracy and precision, as well as the number of photons emitted. Our bound on density is close to that reported in experimental work, suggesting that further increases in the density of imaged fluorophores will come at the expense of localization accuracy and precision.

Random transceiver networks

Consider randomly scattered radio transceivers in ℝd, each of which can transmit signals to all transceivers in a given randomly chosen region about itself. If a signal is retransmitted by every transceiver that receives it, under what circumstances will a signal propagate to a large distance from its starting point? Put more formally, place points {xi} in ℝd according to a Poisson process with intensity 1. Then, independently for each xi, choose a bounded region Axi from some fixed distribution and let be the random directed graph with vertex set whenever xj ∈ xi + Axi. We show that, for any will almost surely have an infinite directed path, provided the expected number of transceivers that can receive a signal directly from xi is at least 1 + η, and the regions xi + Axi do not overlap too much (in a sense that we shall make precise). One example where these conditions hold, and so gives rise to percolation, is in ℝd, with each Axi a ball of volume 1 + η centred at xi, where η → 0 as d → ∞. Another example is in two dimensions, where the Axi are sectors of angle ε γ and area 1 + η, uniformly randomly oriented within a fixed angle (1 + ε)θ. In this case we can let η → 0 as ε → 0 and still obtain percolation. The result is already known for the annulus, i.e. that the critical area tends to 1 as the ratio of the radii tends to 1, while it is known to be false for the square (l∞) annulus. Our results show that it does however hold for the randomly oriented square annulus.

Connectivity of random k-nearest-neighbour graphs

Let 𝓅 be a Poisson process of intensity one in a square S n of area n. We construct a random geometric graph G n,k by joining each point of 𝓅 to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that G n, k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that G n, k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1)) log n. In this paper we improve these lower and upper bounds to 0.3043 log n and 0.5139 log n, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209 log n and 0.9967 log n for the directed version of this problem. A related question concerns coverage. With G n, k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209 log n then the probability that these discs cover S n tends to 0 as n → ∞ while, if k ≥ 0.9967 log n, then the probability that the discs cover S n tends to 1 as n → ∞.

Connectivity of random k-nearest-neighbour graphs

Let 𝓅 be a Poisson process of intensity one in a square Sn of area n. We construct a random geometric graph Gn,k by joining each point of 𝓅 to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that Gn, k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that Gn, k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1)) log n. In this paper we improve these lower and upper bounds to 0.3043 log n and 0.5139 log n, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209 log n and 0.9967 log n for the directed version of this problem. A related question concerns coverage. With Gn, k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209 log n then the probability that these discs cover Sn tends to 0 as n → ∞ while, if k ≥ 0.9967 log n, then the probability that the discs cover Sn tends to 1 as n → ∞.

Random transceiver networks

Consider randomly scattered radio transceivers in ℝ d , each of which can transmit signals to all transceivers in a given randomly chosen region about itself. If a signal is retransmitted by every transceiver that receives it, under what circumstances will a signal propagate to a large distance from its starting point? Put more formally, place points {x i } in ℝ d according to a Poisson process with intensity 1. Then, independently for each x i , choose a bounded region A x i from some fixed distribution and let be the random directed graph with vertex set whenever x j ∈ x i + A x i . We show that, for any will almost surely have an infinite directed path, provided the expected number of transceivers that can receive a signal directly from x i is at least 1 + η, and the regions x i + A x i do not overlap too much (in a sense that we shall make precise). One example where these conditions hold, and so gives rise to percolation, is in ℝ d , with each A x i a ball of volume 1 + η centred at x i , where η → 0 as d → ∞. Another example is in two dimensions, where the A x i are sectors of angle ε γ and area 1 + η, uniformly randomly oriented within a fixed angle (1 + ε)θ. In this case we can let η → 0 as ε → 0 and still obtain percolation. The result is already known for the annulus, i.e. that the critical area tends to 1 as the ratio of the radii tends to 1, while it is known to be false for the square (l∞) annulus. Our results show that it does however hold for the randomly oriented square annulus.

Superdiffusion, large-scale synchronization, and topological defects

We study an ensemble of random walkers carrying internal noisy phase oscillators which are synchronized among the walkers by local interactions. Due to individual mobility, the interaction partners of every walker change randomly, hereby introducing an additional, independent source of fluctuations, thus constituting the intrinsic nonequilibrium nature of the temporal dynamics. We employ this paradigmatic model system to discuss how the emergence of order is affected by the motion of individual entities. In particular, we consider both normal diffusive motion and superdiffusion. A non-Hamiltonian field theory including multiplicative noise terms is derived which describes the nonequilibrium dynamics at the macroscale. This theory reveals a defect-mediated transition from incoherence to quasi-long-range order for normal diffusion of oscillators in two dimensions, implying a power-law dependence of all synchronization properties on system size. In contrast, superdiffusive transport suppresses the emergence of topological defects, thereby inducing a continuous synchronization transition to long-range order in two dimensions. These results are consistent with particle-based simulations.

Splitting of the Universality Class of Anomalous Transport in Crowded Media

We investigate the emergence of subdiffusive transport by obstruction in continuum models for molecular crowding. While the underlying percolation transition for the accessible space displays universal behavior, the dynamic properties depend in a subtle nonuniversal way on the transport through narrow channels. At the same time, the different universality classes are robust with respect to introducing correlations in the obstacle matrix as we demonstrate for quenched hard-sphere liquids as underlying structures. Our results confirm that the microscopic dynamics can dominate the relaxational behavior even at long times, in striking contrast to glassy dynamics.

Optimizing pentacene thin-film transistor performance: Temperature and surface condition induced layer growth modification

In this work we present in situ electrical and surface analytical, as well as ex situ atomic force microscopy (AFM) studies on temperature and surface condition induced pentacene layer growth modifications, leading to the selection of optimized deposition conditions and entailing performance improvements. We prepared p++-silicon/silicon dioxide bottom-gate, gold bottom-contact transistor samples and evaluated the pentacene layer growth for three different surface conditions (sputtered, sputtered + carbon and unsputtered + carbon) at sample temperatures during deposition of 200 K, 300 K and 350 K. The AFM investigations focused on the gold contacts, the silicon dioxide channel region and the highly critical transition area. Evaluations of coverage dependent saturation mobilities, threshold voltages and corresponding AFM analysis were able to confirm that the first 3-4 full monolayers contribute to the majority of charge transport within the channel region. At high temperatures and on sputtered surfaces uniform layer formation in the contact-channel transition area is limited by dewetting, leading to the formation of trenches and the partial development of double layer islands within the channel region instead of full wetting layers. By combining the advantages of an initial high temperature deposition (well-ordered islands in the channel) and a subsequent low temperature deposition (continuous film formation for low contact resistance) we were able to prepare very thin (8 ML) pentacene transistors of comparably high mobility.

In situ preparation, electrical and surface analytical characterization of pentacene thin film transistors

The fabrication of organic thin film transistors with highly reproducible characteristics presents a very challenging task. We have prepared and analyzed model pentacene thin film transistors under ultra-high vacuum conditions, employing surface analytical tools and methods. Intentionally contaminating the gold contacts and SiO2 channel area with carbon through repeated adsorption, dissociation, and desorption of pentacene proved to be very advantageous in the creation of devices with stable and reproducible parameters. We mainly focused on the device properties, such as mobility and threshold voltage, as a function of film morphology and preparation temperature. At 300 K, pentacene displays Stranski-Krastanov growth, whereas at 200 K fine-grained, layer-like film growth takes place, which predominantly influences the threshold voltage. Temperature dependent mobility measurements demonstrate good agreement with the established multiple trapping and release model, which in turn indicates a predominant concentration of shallow traps in the crystal grains and at the oxide-semiconductor interface. Mobility and threshold voltage measurements as a function of coverage reveal that up to four full monolayers contribute to the overall charge transport. A significant influence on the effective mobility also stems from the access resistance at the gold contact-semiconductor interface, which is again strongly influenced by the temperature dependent, characteristic film growth mode.

Percolation and cooperation with mobile agents: geometric and strategy clusters

We study the conditions for persistent cooperation in an off-lattice model of mobile agents playing the Prisoner's Dilemma game with pure, unconditional strategies. Each agent has an exclusion radius r(P), which accounts for the population viscosity, and an interaction radius r(int), which defines the instantaneous contact network for the game dynamics. We show that, differently from the r(P)=0 case, the model with finite-sized agents presents a coexistence phase with both cooperators and defectors, besides the two absorbing phases, in which either cooperators or defectors dominate. We provide, in addition, a geometric interpretation of the transitions between phases. In analogy with lattice models, the geometric percolation of the contact network (i.e., irrespective of the strategy) enhances cooperation. More importantly, we show that the percolation of defectors is an essential condition for their survival. Differently from compact clusters of cooperators, isolated groups of defectors will eventually become extinct if not percolating, independently of their size.

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.

Connecting the random organization transition and jamming within a unifying model system

While the random organization transition describes the change from reversible to irreversible dynamics in a nonequilibrium system, the athermal jamming transition at zero shear rate occurs when particles can no longer avoid overlaps. Despite the obvious differences between these two transitions, we show that they both occur within the same model packing problem. In this unifying model system the particles are first randomly distributed and then displaced in each step if they overlap. For random displacements we obtain a random organization transition, while jamming occurs in the case of deterministic shifts. We also analyze the critical behavior of random organization. Our results show that random organization and jamming are opposite limits of random sphere packings, and we expect that various equilibrium and nonequilibrium transitions can be formulated as related intermediate packing problems.

Analysis of the local organization and dynamics of cellular actin networks

Live cell imaging, high-resolution microscopy, and computational modeling show that dynamic formin-filamin-actin asters self-organize into an actomyosin contractile network that may maintain mechanical coherence of cytoplasm.

A ctin filaments, with the aid of multiple accessory proteins, self-assemble into a variety of network patterns. We studied the organization and dynamics of the actin network in nonadhesive regions of cells bridging fibronectin-coated adhesive strips. The network was formed by actin nodes associated with and linked by myosin II and containing the formin disheveled-associated activator of morphogenesis 1 (DAAM1) and the cross-linker filamin A (FlnA). After Latrunculin A (LatA) addition, actin nodes appeared to be more prominent and demonstrated drift-diffusion motion. Superresolution microscopy revealed that, in untreated cells, DAAM1 formed patches with a similar spatial arrangement to the actin nodes. Node movement (diffusion coefficient and velocity) in LatA-treated cells was dependent on the level and activity of myosin IIA, DAAM1, and FlnA. Based on our results, we developed a computational model of the dynamic formin-filamin-actin asters that can self-organize into a contractile actomyosin network. We suggest that such networks are critical for connecting distant parts of the cell to maintain the mechanical coherence of the cytoplasm.

Quantized conductance and switching in percolating nanoparticle films

We demonstrate switching behavior and quantized conductance at room temperature in percolating films of nanoparticles. Our experiments and complementary simulations show that switching and quantization result from the formation of atomic-scale wires in the gaps between particles. These effects occur only when tunnel gaps are present in the film, close to the percolation threshold.

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.

Molecular reorganization in organic field-effect transistors and its effect on two-dimensional charge transport pathways

Charge transport in organic thin film transistors takes place in the first few molecular layers in contact with the gate dielectric. Here we demonstrate that the charge transport pathways in these devices are extremely sensitive to the orientational defects of the first monolayers, which arise from specific growth conditions. Although these defects partially heal during the growth, they cause depletion of charge carriers in the first monolayer, and drive the current to flow in the monolayers above the first one. Moreover, the residual defects induce lower crystalline order and charge mobility. These results, which are not intuitively explained by electrostatics arguments, have been obtained by combining in situ real time structural and electrical characterization together with ex situ AFM measurements, on thin films of a relevant n-type organic semiconductor, N,N'-bis(n-octyl)-dicyanoperylene-3,4:9,10-bis dicarboximide grown by sublimation in a quasi-layer-by-layer mode at different substrate temperatures.

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.

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.

Finite-size effects in nanocomposite thin films and fibers

Monte Carlo simulations of finite-size effects for continuum percolation in three-dimensional, rectangular sample spaces filled with spherical particles were performed. For samples with any dimension less than 10-20 times the particle diameter, finite-size effects were observed. For thin films in the finite-size regime, percolation across the thin direction of the film gave critical volume fraction (p(c)) values that differed from those along the plane of the film. Simulations perpendicular to the film for very thin samples resulted in p(c) values lower than the classical limit of ∼29% (for spheres in a three-dimensional matrix) which increased with film thickness. For percolation along thin films, while holding film thickness constant, p(c) increased with increasing sample size, which is a modification of the finite-sized scaling effect for cubic samples. For samples with a large aspect ratio (fibers) and a finite-sized cross-sectional area, the critical volume fraction increased with sample length, as the sample became quasi-one-dimensional. The results are discussed in the context of adding volume along or perpendicular to the percolation direction. From an experimental perspective, these findings indicate that sample shape, as well as relative size, influences percolation in the finite-size regime.

Sampling-based algorithms for optimal motion planning

During the last decade, sampling-based path planning algorithms, such as probabilistic roadmaps (PRM) and rapidly exploring random trees (RRT), have been shown to work well in practice and possess theoretical guarantees such as probabilistic completeness. However, little effort has been devoted to the formal analysis of the quality of the solution returned by such algorithms, e.g. as a function of the number of samples. The purpose of this paper is to fill this gap, by rigorously analyzing the asymptotic behavior of the cost of the solution returned by stochastic sampling-based algorithms as the number of samples increases. A number of negative results are provided, characterizing existing algorithms, e.g. showing that, under mild technical conditions, the cost of the solution returned by broadly used sampling-based algorithms converges almost surely to a non-optimal value. The main contribution of the paper is the introduction of new algorithms, namely, PRM* and RRT* , which are provably asymptotically optimal, i.e. such that the cost of the returned solution converges almost surely to the optimum. Moreover, it is shown that the computational complexity of the new algorithms is within a constant factor of that of their probabilistically complete (but not asymptotically optimal) counterparts. The analysis in this paper hinges on novel connections between stochastic sampling-based path planning algorithms and the theory of random geometric graphs.

Diffusion in cytoplasm: effects of excluded volume due to internal membranes and cytoskeletal structures

The intricate geometry of cytoskeletal networks and internal membranes causes the space available for diffusion in cytoplasm to be convoluted, thereby affecting macromolecule diffusivity. We present a first systematic computational study of this effect by approximating intracellular structures as mixtures of random overlapping obstacles of various shapes. Effective diffusion coefficients are computed using a fast homogenization technique. It is found that a simple two-parameter power law provides a remarkably accurate description of effective diffusion over the entire range of volume fractions and for any given composition of structures. This universality allows for fast computation of diffusion coefficients, once the obstacle shapes and volume fractions are specified. We demonstrate that the excluded volume effect alone can account for a four-to-sixfold reduction in diffusive transport in cells, relative to diffusion in vitro. The study lays the foundation for an accurate coarse-grain formulation that would account for cytoplasm heterogeneity on a micron scale and binding of tracers to intracellular structures.

Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations

The site percolation threshold for the random Voronoi network is determined numerically, with the result pc=0.714 10+/-0.000,02 , using Monte Carlo simulation on periodic systems of up to 40,000 sites. The result is very close to the recent theoretical estimate pc approximately 0.7151 of Neher For the bond threshold on the Voronoi network, we find pc=0.666, 931+/-0.000,005 implying that, for its dual, the Delaunay triangulation pc=0.333 069+/-0.000 005 . These results rule out the conjecture by Hsu and Huang that the bond thresholds are 2/3 and 1/3, respectively, but support the conjecture of Wierman that, for fully triangulated lattices other than the regular triangular lattice, the bond threshold is less than 2 sin pi/18 approximately 0.3473 .

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.

Connectivity, coverage and placement in wireless sensor networks

Wireless communication between sensors allows the formation of flexible sensor networks, which can be deployed rapidly over wide or inaccessible areas. However, the need to gather data from all sensors in the network imposes constraints on the distances between sensors. This survey describes the state of the art in techniques for determining the minimum density and optimal locations of relay nodes and ordinary sensors to ensure connectivity, subject to various degrees of uncertainty in the locations of the nodes.

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.

Tin oxide nanocluster hydrogen and ammonia sensors

We have prepared sensitive hydrogen and ammonia sensors from thin films of tin nanoclusters with diameters between 3 and 10 nm. By baking the samples at 200 °C in ambient air the clusters were oxidized, resulting in very stable films of tin oxide clusters with similar diameters to the original Sn clusters. By monitoring the electrical resistance, it is shown that the cluster films are highly responsive to hydrogen and ammonia at relatively low temperatures, thereby making them attractive for commercial applications in which low power consumption is required. Doping of the films by depositing Pd on top of the clusters resulted in much improved sensor response and response times. It is shown that optimal sensor properties are achieved for very thin cluster films (a few monolayers of clusters).

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

We perform simulations of continuum gradient percolation to measure the percolation threshold of systems of fully penetrable disks of two different radii. As a function of volumetric proportion v, the percolation threshold phic(v,lambda) is approximately symmetric about v=0.5 for a fixed ratio lambda of the radii. However, the difference from symmetry is strongly statistically significant. We also improve, by an order of magnitude, the measurement of the percolation threshold for disks of equal radius: phic*=0.676 347 5(6).

Crossover in the slow decay of dynamic correlations in the lorentz model

The long-time behavior of transport coefficients in a model for spatially heterogeneous media in two and three dimensions is investigated by molecular dynamics simulations. The behavior of the velocity autocorrelation function is rationalized in terms of a competition of the critical relaxation due to the underlying percolation transition and the hydrodynamic power-law anomalies. In two dimensions and in the absence of a diffusive mode, another power-law anomaly due to trapping is found with an exponent -3 instead of -2. Further, the logarithmic divergence of the Burnett coefficient is corroborated in the dilute limit; at finite density, however, it is dominated by stronger divergences.

Betti number signatures of homogeneous Poisson point processes

The Betti numbers are fundamental topological quantities that describe the k-dimensional connectivity of an object: beta{0} is the number of connected components and beta{k} effectively counts the number of k-dimensional holes. Although they are appealing natural descriptors of shape, the higher-order Betti numbers are more difficult to compute than other measures and so have not previously been studied per se in the context of stochastic geometry or statistical physics. As a mathematically tractable model, we consider the expected Betti numbers per unit volume of Poisson-centered spheres with radius alpha. We present results from simulations and derive analytic expressions for the low intensity, small radius limits of Betti numbers in one, two, and three dimensions. The algorithms and analysis depend on alpha shapes, a construction from computational geometry that deserves to be more widely known in the physics community.

Power law tails of time correlations in a mesoscopic fluid model

In a quenched mesoscopic fluid, modeling transport processes at high densities, we perform computer simulations of the single particle energy autocorrelation function C(e) (t) , which is essentially a return probability. This is done to test the predictions for power law tails, obtained from mode coupling theory. We study both off and on-lattice systems in one- and two-dimensions. The predicted long time tail approximately t(-d/2) is in excellent agreement with the results of computer simulations. We also account for finite size effects, such that smaller systems are fully covered by the present theory as well.

Model system for classical fluids out of equilibrium

A model system for classical fluids out of equilibrium, referred to as a dissipative particles dynamics (DPD) solid, is studied by analytical and simulation methods. The time evolution of a DPD particle is described by a fluctuating heat equation. This DPD solid with transport based on collisional transfer (high-density mechanism) is complementary to the Lorentz gas with only kinetic transport (low-density mechanism). Combination of both models covers the qualitative behavior of transport properties of classical fluids over the full-density range. The heat diffusivity is calculated using a mean-field theory, leading to a linear-density dependence of this transport coefficient, which is exact at high densities. Subleading density corrections are obtained as well. At lower densities the model has a conductivity threshold below which heat conduction is absent. The observed threshold is explained in terms of percolation diffusion on a random proximity network. The geometrical structure of this network is the same as in continuum percolation of completely overlapping spheres, but the dynamics on this network differs from continuum percolation diffusion. Furthermore, the kinetic theory for DPD is extended to the generalized hydrodynamic regime, where the wave-number-dependent decay rates of the Fourier modes of the energy and temperature fields are calculated.