Dynamic Percolation Research Articles

The role of conductive pathways in the conductivity and rheological behavior of poly(methyl methacrylate)–graphite composites

ABSTRACTUp to now, research on the dynamic process of conductive network formation has tended to focus on composite particles with one‐dimensional geometry, such as carbon black and carbon nanotubes. However, studies on this subject based on fillers with two‐dimensional structure, such as graphite, are rare in the literature. In this work, the dynamic percolation and rheological properties of poly(methyl methacrylate) (PMMA)–graphite composites under an electric field were investigated. The activation energies of conductive network formation and polymer matrix mobility were calculated from the temperature dependence of the percolation time and the zero‐shear viscosity. It was found that the activation energy calculated from the zero‐shear viscosity was not influenced by the electric field in the concentration range investigated, but the electric field had an effect on the activation energy calculated from the percolation time. This finding emphasizes that the electrical and rheological properties have different physical origins. © 2016 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2016, 133, 43810.

Universal aspects of hydrogel gelation kinetics, percolation and viscoelasticity from PA-hydrogel rheology

Polyacrylamide (PA) hydrogels have been studied extensively, but fundamental aspects of their gelation kinetics, percolation dynamics, and viscoelasticity are still not well understood. This paper focuses on the rheology of PA hydrogels having unusually low monomer concentrations (ca ≈ 3 w% equivalent to 0.42 mol l−1). These furnish loss tangents that span 4 orders of magnitude when varying the crosslinker concentration. An optimum crosslinker concentration (cbis/ca ≈ 2.5 mol. % equivalent to 5.3 w%) is identified, below which the storage modulus G′ increases almost linearly, and the loss modulus G″ acquires a local maximum. Above the optimum crosslinker concentration, G′ and G″ both plateau, accompanied by a notable decrease in the maximum strain (increase in brittleness) before breaking. The dynamic shear moduli reveal universal dynamics at the gel point, as indicated by (i) scaling exponents (y = 3.1 ± 0.1, z = 2.1 ± 0.1 and Δ = 0.70 ± 0.02) that are consistent with the de Gennes [“On a relation between percolation theory and the elasticity of gels,” J. Phys. Lett. 37, L1–L2 (1976)] electrical network analogy, and (ii) a critical relaxation exponent that is close to the Rouse limit Δ = 2/3 from the scaling theory of Martin. A close correspondence of the exponents with that of Adam and Delsanti [Macromolecules 18, 2285–2290 (1985)] for the radical copolymerization of a different material supports the long-standing hypothesis that dynamics at the gel point are universal for a prescribed gelation mechanism.

The large deviations of the whitening process in random constraint satisfaction problems

Random constraint satisfaction problems undergo several phase transitions as the ratio between the number of constraints and the number of variables is varied. When this ratio exceeds the satisfiability threshold no more solutions exist; the satisfiable phase, for less constrained problems, is itself divided in an unclustered regime and a clustered one. In the latter solutions are grouped in clusters of nearby solutions separated in configuration space from solutions of other clusters. In addition the rigidity transition signals the appearance of so-called frozen variables in typical solutions: beyond this threshold most solutions belong to clusters with an extensive number of variables taking the same values in all solutions of the cluster. In this paper we refine the description of this phenomenon by estimating the location of the freezing transition, corresponding to the disappearance of all unfrozen solutions (not only typical ones). We also unveil phase transitions for the existence and uniqueness of locked solutions, in which all variables are frozen. From a technical point of view we characterize atypical solutions with a number of frozen variables different from the typical value via a large deviation study of the dynamics of a stripping process (whitening) that unveils the frozen variables of a solution, building upon recent works on atypical trajectories of the bootstrap percolation dynamics. Our results also bear some relevance from an algorithmic perspective, previous numerical studies having shown that heuristic algorithms of various kinds usually output unfrozen solutions.

Enhancing the electrical conductivity of carbon black-filled immiscible polymer blends by tuning the morphology

Blends of carbon black (CB)-filled polystyrene (PS) and poly(methyl methacrylate) (PMMA) with different PS/PMMA ratios have been prepared by melt blending in order to obtain conductive polymer composites with a low electrical percolation threshold according to the concept of double percolation. The dependence of the electrical conductivity on the morphology of the composite was examined before and after annealing in the molten state. The electrical conductivity of the composites was highest when the PS and PMMA phases formed a co-continuous structure and CB particles were selectively located and percolated in the PS phase. PS/PMMA (30/70) blends exhibited the lowest electrical percolation threshold of 1.26vol% CB. A simple model is suggested to explain this effect. After annealing, the CB clearly affected the stabilization of the co-continuous structure. Moreover, the electrical conductivity increased after annealing and the percolation threshold decreased to below 1vol% CB regardless of PS content. The largest increase in electrical conductivity after annealing was found with PS/PMMA (70/30) blends filled with 1vol% CB. With increasing PMMA content, this effect became less pronounced. The different behaviors in the composites can be explained by dynamic percolation and decrease in continuity of the CB-filled PS phase induced by structure coarsening.

The Nanostructure Studies of Surfactant-Free-Microemulsions in Fragrance Tinctures.

As it was shown recently that nanostructures can exist in water-ethanol-citronellol tinctures, a deeper investigation of these media was performed using conductivity, UV-Vis and FT-IR spectroscopy techniques. Different regimes of conductivity, depending on the water content, an increase of the polarity of the polar pseudo-phase with increasing water content, and even the presence of free water molecules at higher water content are observed, just as in classical surfactant-based microemulsions. The percolation model, generally used to fit conductivity data in surfactant based microemulsion having a weak interfacial film, can be used to fit our conductivity data below a critical water content Φ(p)(w) with a critical exponent typical of dynamic percolation. In presence of higher water contents, superior to Φ(p)(w), obtained conductivity data cannot be fitted neither with a static nor a dynamic percolation model. As in surfactant-based microemulsions, an increase of polarity of the microenvironment with increasing water content can be postulated using respectively the UV-Vis wavelength absorption band (λ(max)) of methyl orange and performing FT-IR spectra.

Two-walker discrete-time quantum walks on the line with percolation.

One goal in the quantum-walk research is the exploitation of the intrinsic quantum nature of multiple walkers, in order to achieve the full computational power of the model. Here we study the behaviour of two non-interacting particles performing a quantum walk on the line when the possibility of lattice imperfections, in the form of missing links, is considered. We investigate two regimes, statical and dynamical percolation, that correspond to different time scales for the imperfections evolution with respect to the quantum-walk one. By studying the qualitative behaviour of three two-particle quantities for different probabilities of having missing bonds, we argue that the chosen symmetry under particle-exchange of the input state strongly affects the output of the walk, even in noisy and highly non-ideal regimes. We provide evidence against the possibility of gathering information about the walkers indistinguishability from the observation of bunching phenomena in the output distribution, in all those situations that require a comparison between averaged quantities. Although the spread of the walk is not substantially changed by the addition of a second particle, we show that the presence of multiple walkers can be beneficial for a procedure to estimate the probability of having a broken link.

Coarse-grained model of water diffusion and proton conductivity in hydrated polyelectrolyte membrane.

Using dissipative particle dynamics (DPD), we simulate nanoscale segregation, water diffusion, and proton conductivity in hydrated sulfonated polystyrene (sPS). We employ a novel model [Lee et al. J. Chem. Theory Comput. 11(9), 4395-4403 (2015)] that incorporates protonation/deprotonation equilibria into DPD simulations. The polymer and water are modeled by coarse-grained beads interacting via short-range soft repulsion and smeared charge electrostatic potentials. The proton is introduced as a separate charged bead that forms dissociable Morse bonds with the base beads representing water and sulfonate anions. Morse bond formation and breakup artificially mimics the Grotthuss mechanism of proton hopping between the bases. The DPD model is parameterized by matching the proton mobility in bulk water, dissociation constant of benzenesulfonic acid, and liquid-liquid equilibrium of water-ethylbenzene solutions. The DPD simulations semi-quantitatively predict nanoscale segregation in the hydrated sPS into hydrophobic and hydrophilic subphases, water self-diffusion, and proton mobility. As the hydration level increases, the hydrophilic subphase exhibits a percolation transition from isolated water clusters to a 3D network. The analysis of hydrophilic subphase connectivity and water diffusion demonstrates the importance of the dynamic percolation effect of formation and breakup of temporary junctions between water clusters. The proposed DPD model qualitatively predicts the ratio of proton to water self-diffusion and its dependence on the hydration level that is in reasonable agreement with experiments.

Connection times in large ad-hoc mobile networks

We study connectivity properties in a probabilistic model for a large mobile ad-hoc network. We consider a large number of participants of the system moving randomly, independently and identically distributed in a large domain, with a space-dependent population density of finite, positive order and with a fixed time horizon. Messages are instantly transmitted according to a relay principle, that is, they are iteratively forwarded from participant to participant over distances smaller than the communication radius until they reach the recipient. In mathematical terms, this is a dynamic continuum percolation model. We consider the connection time of two sample participants, the amount of time over which these two are connected with each other. In the above thermodynamic limit, we find that the connectivity induced by the system can be described in terms of the counterplay of a local, random and a global, deterministic mechanism, and we give a formula for the limiting behaviour. A prime example of the movement schemes that we consider is the well-known random waypoint model. Here, we give a negative upper bound for the decay rate, in the limit of large time horizons, of the probability of the event that the portion of the connection time is less than the expectation.

Local time on the exceptional set of dynamical percolation and the incipient infinite cluster

In dynamical critical site percolation on the triangular lattice or bond percolation on Z2, we define and study a local time measure on the exceptional times at which the origin is in an infinite cluster. We show that at a typical time with respect to this measure, the percolation configuration has the law of Kesten’s incipient infinite cluster. In the most technical result of this paper, we show that, on the other hand, at the first exceptional time, the law of the configuration is different. We believe that the two laws are mutually singular, but do not show this. We also study the collapse of the infinite cluster near typical exceptional times and establish a relation between static and dynamic exponents, analogous to Kesten’s near-critical relation.

Chemically Specific Dynamic Bond Percolation Model for Ion Transport in Polymer Electrolytes

We introduce a coarse-grained approach for characterizing the long-timescale dynamics of ion diffusion in general polymer electrolytes using input from short molecular dynamics trajectories. The approach includes aspects of the dynamic bond percolation model [J. Chem. Phys. 1983, 79, 3133−3142] by treating ion diffusion in terms of hopping transitions on a fluctuating lattice. We extend this well-known approach by using short (i.e., 10 ns) molecular dynamics (MD) trajectories to predict the distribution of ion solvation sites that comprise the lattice and to predict the rate of hopping among the lattice sites. This yields a chemically specific dynamic bond percolation (CS-DBP) model that enables the description of long-timescale ion diffusion in polymer electrolytes at a computational cost that makes feasible the screening of candidate materials. We employ the new model to characterize lithium-ion diffusion properties in six polyethers that differ by oxygen content and backbone stiffness: poly(trimethylen...

Dynamic Percolation and Droplet Growth Behavior in Porous Electrodes of Polymer Electrolyte Fuel Cells

The percolating flow of liquid water in the gas-diffusion layer (GDL) of polymer electrolyte fuel cells (PEFCs) was studied ex situ using a simple water-injection experiment. Water was injected into the top of the sample so the droplet was free to detach from the bottom, allowing for uninterrupted study of the dynamic cycle of droplet appearance, growth, and detachment. Although droplets emerged from a single point on the GDL, the measured pressure response in the water phase was clearly not equivalent to a single needle. The behavior of the system was explained by the simultaneous filling and inflating of many menisci, resulting in extended periods with no droplet activity at the GDL surface, followed by the sudden eruption of a droplet at the breakthrough site as all interfaces deflated and their stored water was directed toward the droplet. A simple numerical model was presented that could qualitatively explain the observed behavior. Tests were performed on GDLs, with and without microporous layers (MPLs), and all observed behavior could be interpreted in terms of the proposed model. MPLs shifted the behavior to a more needlelike behavior which was consistent with the MPL reducing the number of invading liquid clusters in the system.

Quantum walk coherences on a dynamical percolation graph.

Coherent evolution governs the behaviour of all quantum systems, but in nature it is often subjected to influence of a classical environment. For analysing quantum transport phenomena quantum walks emerge as suitable model systems. In particular, quantum walks on percolation structures constitute an attractive platform for studying open system dynamics of random media. Here, we present an implementation of quantum walks differing from the previous experiments by achieving dynamical control of the underlying graph structure. We demonstrate the evolution of an optical time-multiplexed quantum walk over six double steps, revealing the intricate interplay between the internal and external degrees of freedom. The observation of clear non-Markovian signatures in the coin space testifies the high coherence of the implementation and the extraordinary degree of control of all system parameters. Our work is the proof-of-principle experiment of a quantum walk on a dynamical percolation graph, paving the way towards complex simulation of quantum transport in random media.

Morphology evolution, conductive and viscoelastic behaviors of chemically reduced graphene oxide filled poly(methyl methacrylate)/poly(styrene-co-acrylonitrile) nanocomposites during annealing

The effect of chemically reduced graphene oxide (CRGO) on the phase separation behavior of poly(methyl methacrylate)/poly(styrene-co-acrylonitrile) (PMMA/SAN) blends and the simultaneous response of rheological and conductive behavior of PMMA/SAN/CRGO nanocomposites upon annealing above the phase-separation temperatures were investigated. The introduction of CRGO causes the decrease of binodal temperature and the increase of spinodal temperature for PMMA/SAN blends and then enlarges their metastable regime. During annealing, the well-dispersed CRGO in the homogeneous blend matrix tends to be selectively located in the SAN-rich phase with the evolution of phase separation and then the CRGO further agglomerates effectively in the SAN-rich phase to form the conductive pathway. Thermal-induced dynamic percolation is observed for both the resistivity ρ and dynamic storage modulus G′ as a function of annealing time. The resistivity variation is ascribed to the agglomeration of CRGO in the SAN-rich phase, while the modulus evolution is attributed to the combined contribution of phase separation for blend matrix and the agglomeration of CRGO in the SAN-rich phase.

Static and dynamic percolation of phenoxy/carbon nanotube nanocomposites

The electrical conductivity of a carbon nanotube/phenoxy nanocomposite fabricated by melt-processing is presented. A low percolation threshold (pc∼0.20wt.%) for carbon nanotubes in phenoxy, a thermoplastic epoxy, is reported. The dispersion of the nanofillers within phenoxy matrix was examined by scanning electron microscopy, showing good levels of dispersion without aggregates. The temperature dependence of nanotube network formation is described and explained by a dynamic percolation process. Suitable processing and annealing conditions are provided to achieve low percolation thresholds based on observed electrical and rheological results. Since these phenoxy resins are dissolvable into epoxy, these findings open up possibilities to deliver carbon nanotubes into composites based on such epoxy resin, creating hierarchical nanoengineered composites.

Critical properties of the susceptible-exposed-infected model on a square lattice

The critical properties of the stochastic susceptible-exposed-infected model on a square lattice is studied by numerical simulations and by the use of scaling relations. In the presence of an infected individual, a susceptible becomes either infected or exposed. Once infected or exposed, the individual remains forever in this state. The stationary properties are shown to be the same as those of isotropic percolation so that the critical behavior puts the model into the universality class of dynamic percolation.

Invulnerability of grown Peer-to-Peer networks under progressive targeted attacks

Security issues of Peer-to-Peer (P2P) networks have attracted more and more research in recent years. In this paper, using complex features of P2P networks, we shift the focus to the study of invulnerability of grown P2P networks under progressive targeted attacks. Based on dynamic process and reverse percolation theory, we present several mechanisms that attacked P2P networks can adopt to minimize the disasters aftermath progressive targeted attacks. In this process, we proposed: (i) the dynamics of grown P2P networks under targeted attacks can make sure an attacked P2P network restore a power-law (PL) characteristic to a normal level; (ii) a global degree restoring process from the aftermath of progressive targeted attacks can restore the status of set of high degree peers to normal; (iii) a reverse percolation process glues the fragmented small connected component of a destroyed grown P2P network into a giant connected component (GCC). Experimental results show that an attacked grown P2P network can restore the key characteristics, such as power-law characteristic of original P2P network, the set of high degree peers and the giant connected component, to a regular status. In this way, we can illustrate the invulnerability of progressive targeted attacks on grown P2P networks which is particularly useful in designing complex P2P networks.

Complexity of Economical Systems

In this study new theoretical concepts are described concerning the interpretation of economical complex dynamics. In addition a summary of an extended algorithm of nonlinear time series analysis is provided which is applied not only in economical time series but also in other physical complex systems (e.g. [22, 24]). In general, Economy is a vast and complicated set of arrangements and actions wherein agents—consumers, firms, banks, investors, government agencies—buy and sell, speculate, trade, oversee, bring products into being, offer services, invest in companies, strategize, explore, forecast, compete, learn, innovate, and adapt. As a result the economic and financial variables such as foreign exchange rates, gross domestic product, interest rates, production, stock market prices and unemployment exhibit large-amplitude and aperiodic fluctuations evident in complex systems. Thus, the Economics can be considered as spatially distributed non-equilibrium complex system, for which new theoretical concepts, such as Tsallis non extensive statistical mechanics and strange dynamics, percolation, nonGaussian, multifractal and multiscale dynamics related to fractional Langevin equations can be used for modeling and understanding of the economical complexity locally or globally. Kewwords: Economical Dynamics, Time series analysis, Tsallis non-extensive statistics, Fractal Dynamics, Non-equilibrium dynamics, Turbulence.

Random walks on dynamical percolation: mixing times, mean squared displacement and hitting times

We study the behavior of random walk on dynamical percolation. In this model, the edges of a graph $$G$$ are either open or closed and refresh their status at rate $$\mu $$ while at the same time a random walker moves on $$G$$ at rate 1 but only along edges which are open. On the $$d$$ -dimensional torus with side length $$n$$ , we prove that in the subcritical regime, the mixing times for both the full system and the random walker are $$n^2/\mu $$ up to constants. We also obtain results concerning mean squared displacement and hitting times. Finally, we show that the usual recurrence transience dichotomy for the lattice $${\mathbb {Z}}^d$$ holds for this model as well.

Form fluctuations of polymer loaded spherical microemulsions studied by neutron scattering and dielectric spectroscopy.

We investigate the structure and shell dynamics of the droplet phase in water/AOT/octane microemulsions with polyethyleneglycol (MW = 1500) molecules loaded in the droplets. Size and polydispersity of the droplets is determined with small angle X-ray scattering and small angle neutron scattering experiments. Shell fluctuations are measured with neutron spin echo spectroscopy and related to the dynamic percolation seen in dielectric spectroscopy. Shell fluctuations are found to be well described by the bending modulus of the shell and the viscosities inside and outside the droplets. Addition of the polymer decreases the modulus for small droplets. For large droplets the opposite is found as percolation temperature shifts to higher values.

Specific, trace gas induced phase transition in copper(II)oxide for highly selective gas sensing

Here, we present results on the investigation of the percolation phase transition in copper(II)oxide (CuO) and show how it may be used to determine trace gas concentrations. This approach provides a highly selective sensing mechanism for the detection of hydrogen sulfide even in oxygen depleted atmospheres. In real-world applications, this scenario is encountered in biogas plants and natural gas facilities, where reliable H2S sensing and filtering are important because of the destructive effects H2S has on machinery. As opposed to gas detection via standard metal-oxide reaction routes, the percolation dynamics are demonstrated to be independent of the surface morphology in accordance with the universality of phase transitions. The sensing behavior of ink-jet printed CuO layers was tested for a large set of parameters including layer temperature, hydrogen sulfide (H2S) and oxygen concentration, as well as the sensitivity towards other gas species. The electrical percolation of the sensing layer is heralded by a dramatic drop in the overall resistivity of the CuO layer for temperatures below 200 °C. The observed percolation phenomena in this temperature regime are unique to H2S even in comparison with related volatile thio-compounds making the sensing mechanism highly selective. At elevated temperatures above 300 °C, the phase transition does not occur. This enables two distinct operational modes which are tunable via the sensor temperature and also allows for resetting the sensing layer after an electrical breakthrough.