Percolation Model Research Articles

Model development of aqueous diffusion softening transition in thermoplastic polyurethane cellulose nanocrystal composites

The mechanical adaptivity of polymer composites containing cellulose nanocrystals (CNCs) is of broad interest for the development of smart materials. The mechanism leading to diffusion-dependent mechanical softening in response to water uptake in these materials is well understood, however, the mass transport driving this phenomenon has not been completely resolved. Understanding such behavior would further enable the mechanism to be controlled and exploited for various applications. Here, we show that swelling of the bulk polymer composite during moisture uptake results in a decreased apparent diffusivity with increasing CNC concentration, despite the equilibrium water uptake increasing concurrently. Further, we demonstrate that a modification to the widely used percolation model predicts the full time-dependent evolution of storage modulus during the softening process by associating mass transport directly to the softening response. This model will be pertinent to studies of mechanical responses in CNC-polymer composites and is expected to be generally applicable to percolating networks which deactivate in response to a given stimulus.

An advection–diffusion–reaction model for coffee percolation

An advection–diffusion–reaction model for coffee percolation

Transfer learning of phase transitions in percolation and directed percolation.

The latest advances of statistical physics have shown remarkable performance of machine learning in identifying phase transitions. In this paper, we apply domain adversarial neural network (DANN) based on transfer learning to studying nonequilibrium and equilibrium phase transition models, which are percolation model and directed percolation (DP) model, respectively. With the DANN, only a small fraction of input configurations (two-dimensional images) needs to be labeled, which is automatically chosen, to capture the critical point. To learn the DP model, the method is refined by an iterative procedure in determining the critical point, which is a prerequisite for the data collapse in calculating the critical exponent ν_{⊥}. We then apply the DANN to a two-dimensional site percolation with configurations filtered to include only the largest cluster which may contain the information related to the order parameter. The DANN learning of both models yields reliable results which are comparable to the ones from Monte Carlo simulations. Our study also shows that the DANN can achieve quite high accuracy at much lower cost, compared to the supervised learning.

Multirange percolation on oriented trees: Critical curve and limit behavior

AbstractWe consider an inhomogeneous oriented percolation model introduced by de Lima, Rolla and Valesin. In this model, the underlying graph is an oriented rooted tree in which each vertex points to each of its children with “short” edges, and in addition, each vertex points to each of its descendant at a fixed distance with “long” edges. A bond percolation process is then considered on this graph, with the prescription that independently, short edges are open with probability and long edges are open with probability . We study the behavior of the critical curve : we find the first two terms in the expansion of as . We also prove limit theorems for the percolation cluster in the supercritical, subcritical and critical regimes.

Tailoring the surface energy and area surface resistance of solid-electrolyte polymer membrane for dendrite free, high-performance, and safe solid-state Li-batteries

Solid electrolyte polymer composite enabled solid-state lithium batteries have shown extensive viability for mass production and commerciality for advanced energy storage technology. Most studies of polymer composite electrolyte are produced by solution casting and free-standing methods, where ionic conductors are randomly distributed into the polymeric network, suffering from poor ionic conductivity and agglomeration during charge/discharge cycling. In this work, using the threshold percolation model and critical volume fraction requirement, we have designed a highly efficient garnet-type ion-conducting nanofibrous sheet where ionic conductors of different aspect ratios are dispersed to tune the surface roughness, area surface resistance, and wettability of the developed electrolyte membrane. Moreover, the impact of filler volume fraction and percolation threshold on the surface properties of the membrane, regulating the area interfacial resistance of the Li metal cell has also been examined, which has not been studied so far in the literature. The corresponding high potential cathode-based lithium assembled coin cells exhibit an excellent specific capacity of 170 mAh g −1 at 0.3C, with high capacity retention under ambient circumstances. By optimizing the critical aspect ratio of the fillers and the percolation threshold, a high-performing solid electrolyte membrane can be realized for advanced and safer Li-batteries technology.

Influence of (0.19HfO2-0.81ZrO2) ceramics filler content on structural and dielectric properties of PVDF polymer

The rapid evolution of modern technology for the sustainable development of human beings demands eco-friendly, flexible, cost-effective, and lightweight smart materials. Ceramics-polymer composites are those novel materials with improved features, which fulfill the prerequisite conditions for application in flexible devices. Pristine PVDF is one of such electroactive flexible polymers which exhibits good dielectric, ferroelectric and piezoelectric properties. But, its functional properties in the perspective of device application still need improvement and this can be achieved by adding filler to the pure PVDF. In this communication, we have reported the influence of Hafnia-Zirconia (HZ) ceramics (0.19HfO2-0.81ZrO2) filler on structural, microstructural, dielectric, and electrical properties of pure PVDF matrix. The polymer-ceramics flexible composites are synthesized by an eco-friendly and cost-effective method i.e., solution casting method. The phase formation of flexible composites is confirmed from the room temperature XRD spectra. The degree of crystallinity increases with an increase in filler content. The degree of β-phase crystallinity is found to be maximum for 5 wt% of the ceramics filler. An enhanced dielectric constant of ~53 and 20 was observed (as expected) for 5 and 10 wt% of HZ ceramics filler respectively at 100 Hz due to the enrichment of polar β-phase for these two ceramics-polymer composites. The influence of filler content on the dielectric properties was explained based on the percolation model. Relaxation and conduction mechanisms of flexible composites are analyzed from the impedance spectroscopy and ac conductivity study, respectively.

Random walk on the random connection model

We study the behavior of the random walk in a continuum independent long-range percolation model, in which two given vertices x and y are connected with probability that asymptotically behaves like |x−y|−α with α>d, where d denotes the dimension of the underlying Euclidean space. More precisely, focus is on the random connection model in which the vertex set is given by the realization of a homogeneous Poisson point process. We show that this random graph exhibits similar properties as classical discrete long-range percolation models studied by Berger (2002) with regard to recurrence and transience of the random walk. Moreover, we address a question which is related to a conjecture by Heydenreich, Hulshof and Jorritsma (2017) for this graph.

New bounds for the site percolation threshold of the hexagonal lattice

The site percolation threshold of the hexagonal lattice satisfies 0.656 246 < p c < 0.739 695. For comparison, the largest previous lower bound of 0.652 703… was established in 1981, and the smallest previous upper bound of 0.743 359 was derived in 2007. The bound is obtained by using the substitution method to compare the hexagonal lattice site model to an exactly-solved two-parameter site percolation model on the martini lattice. Computational reductions involving graph-welding, symmetry, non-crossing partitions, and network flow computations overcome challenges to establishing stochastic ordering between the models.

Integrating acoustic emission into a percolation model to evaluate crack distribution characteristics of heated granite subjected to rapid cooling

To compare the distribution of cracks formed in the failure process of heated granite subjected to rapid cooling, compression tests (0–60 MPa) and acoustic emission (AE) monitoring were performed on heated granite (25, 200, and 400 °C) after water cooling and liquid nitrogen (LN2) cooling. Integrating AE events into the percolation model and counting the AE events with a cube covering method, the main AE events with the highest spatial connectivity were identified among the whole set of AE events. The count, density, and fractal dimension of the whole set of AE events and the main AE events were discussed to indirectly evaluate the crack distribution in the failure process. The results show that the most frequent, maximum, and minimum neighbor distances between AE events can be evaluated by the covering cubes with various lengths. The main AE events with the highest spatial connectivity should be characterized by short neighbor distances and most frequent neighbor distances. When the length of covering cubes is large enough to exceed the most frequent neighbor distances, the corresponding length is defined as the characteristic length. When the characteristic length is determined as the length of the covering cubes, the main AE events can be identified among the whole set of AE events. According to the distributions of all the AE events and the main AE events, LN2 cooling is proven to result in a more number, density, and complexity of cracks than water cooling, under high temperatures and high confining pressures.

Simulación de la percolación eléctrica de nanotubos de carbono a través de una matriz de polimetilmetacrilato: análisis bidimensional y tridimensional usando el efecto de la tortuosidad de los nanoaditivos

The present work aims to study the simulation of the electrical percolation of carbon nanotubes (CNTs) in polymethylmethacrylate matrix (PMMA). In order to do this, simulations were employed in two-dimensional and three-dimensional analyses through a program developed by the Monte Carlo method, supported by the excluded volume model in order to analyze the geometric tortuosity in the polymeric matrix. The percolation threshold was analyzed by different aspect ratios, volumetric fractions and conductive load geometries. In the simulations, a decrease in the percolation threshold was observed when the aspect ratio increased and the particles became more tortuous, being these characteristics the parameters to evaluate both two-dimensional and three-dimensional systems. The results show percolation threshold values of 0.625% vol and aspect ratio of 1000 for two-dimensional analyses and, 0.08% vol and aspect ratio of 250 for three-dimensional analyses, being the adequacy to the main models for electrical percolation with tortuosity effect of the nanocarriers proposed by literature.

A random hermitian matrix representation for two-dimensional percolation model‎

In this letter, the random matrix theory representation of a bond-percolation model on square lattice is presented. The behavior of random matrix model can be determined only by its two largest eigenvalues. The second largest eigenvalue sits exactly on the edge of semicircle part of eigenvalue’s distribution and its position is a function of p, on the other hand the first largest eigenvalue is disjointed from other eigenvalues and its distribution is Gaussian. Also the first largest eigenvalue is responsible for scaling properties near criticality. Numerical simulations show power-law divergences emerged from coalescence of two largest eigenvalues near critical point at the thermodynamic limit. In this letter a scaling formalism is presented which describes complete scaling behavior of largest eigenvalue’s fluctuations with a set of scaling exponents in finite-size systems.

The Influence of Pore Distribution of Coal Char in the Char Fragmentation and Included Minerals Partitioning: A Percolation Modeling

The processes of char fragmentation, including mineral partitioning and particulate matter (PM) formation during dense and porous char combustion, were observed by a site percolation model. This model simulated the diffusion-controlled regime of char combustion, and the size distributions of included minerals in typical bituminous coal were determined by the computer-controlled scanning electron microscope (CCSEM), and the data were put into the char matrix randomly. The model presents the influence of initial pore distribution on char oxidation and fragmentation, the impact of the char conversion process on the extent of fragmentation, the change of ash distributions with the char conversion, and the particulate matters (PM) size distribution, which is derived from the consequence of the competition between char fragmentation and included minerals partitioning and coalescence. The results indicate that with increasing initial char porosity (φ), the number of large size pores increases but the number of pores decreases, which leads to open pores increasing, close pores decreasing, and the surface reaction area increasing. While φ ≥ 0.4, char fragmentation obviously occurs during the stage in which the rates of char conversion are 0.4–0.6, and it looks as though the maximum value of fragmentation will transfer to an earlier conversion stage if it has a larger φ. The enhanced φ shows a positive effect on the increase in the number and concentration of PM &lt; 10 μm (nominally aerodynamic diameter), this is attributed to char fragments more drastically, and the probability of mineral coalescence reduces a lot.

Bootstrap Percolation, Probabilistic Cellular Automata and Sharpness

We establish new connections between percolation, bootstrap percolation, probabilistic cellular automata and deterministic ones. Surprisingly, by juggling with these in various directions, we effortlessly obtain a number of new results in these fields. In particular, we prove the sharpness of the phase transition of attractive absorbing probabilistic cellular automata, a class of bootstrap percolation models and kinetically constrained models. We further show how to recover a classical result of Toom on the stability of cellular automata w.r.t. noise and, inversely, how to deduce new results in bootstrap percolation universality from his work.

Percolation of interdependent networks with limited knowledge.

Real-world networks are often not isolated and the interdependence between different networks in a complex system is as important as the topological connectivity within individual networks. We develop a theoretical framework to study the robustness of interdependent networks under attacks with limited knowledge. A node may be attacked if it is the most connected node among a given number of potential victims. This number is referred to as the attacker's knowledge level, which joins the two ends, namely, the random failure with zero knowledge and the intentional attack with full knowledge of the network. We introduce percolation models with attacks over one layer and two layers as well as mixed site-bond percolation. Along with the discontinuous phase transition, we show the existence of a critical knowledge level, which indicates a transition of network robustness under the competition between connectivity and interdependence. It is unraveled that interdependent networks can be extremely fragile to the extent that a random failure on two layers would be more deleterious than a targeted attack with full knowledge over one layer. Moreover, we find that a balanced distribution of attack knowledge on both layers tends to be most destructive if the total knowledge is a conserved quantity.

DEM study and machine learning model of particle percolation under vibration

This paper aims to understand model the effect of vibration on particle percolation. The percolation of small particles in a vibrated bed of big particles is studied by DEM. It is found the percolation velocity (Vp) decreases with increasing vibration amplitude (A) and frequency (f) when the size ratio of small to large particles (d/D) is smaller than the spontaneous percolation threshold of 0.154. Vibration can enable percolation when the size ratio is larger than 0.154, while Vp increases with increasing A and f first and then decreases. Vp can be correlated to the vibration velocity amplitude under a given size ratio. Previous radial dispersion model can still be applied while the dispersion coefficient is affected by vibration conditions and size ratio. Furthermore, a machine learning model is trained to predict Vp as a function of A, f and d/D, and is then used to obtain the percolation threshold size ratio as a function of vibration conditions.

Possible formation of a perfect fluid in pp, p–Pb, Xe–Xe and Pb–Pb collisions at the Large Hadron Collider energies: a color string percolation approach

Isothermal compressibility ($\kappa_{\rm T}$) is an important thermodynamic observable which gives information about the deviation of a system from perfect fluid behavior. In this work, for the first time we have estimated the isothermal compressibility of QCD matter formed in high energy hadronic and nuclear collisions using color string percolation model (CSPM), where we investigate the change in $\kappa_{\rm T}$ as a function of final state charged particle multiplicity and initial percolation temperature across various collision species. The estimated initial percolation temperature for different collision systems at different collision energies helps us to have a better understanding of the system at the initial phase of evolution. The comparison of the CSPM results for isothermal compressibility with that for the well known fluids, indicates that the matter formed in heavy-ion collisions might be the {\it closest perfect fluid} found in nature. This estimation complements the well-known observation of minimum shear viscosity to entropy density ratio for a possible QGP medium created in heavy-ion collision experiments. A threshold of pseudorapidity density of charged particles, $\langle dN_{\rm ch}/d\eta \rangle \geq 20 $ in the final state event multiplicity is observed, after which one may look for a possible QGP formation at the LHC energies.

Two-dimensional continuum percolation models with disks under the generalized Achlioptas process

We investigate the disk percolation phase transitions in continuum models in two dimensions under four kinds of generalized product rule and four kinds of generalized sum rule, respectively. We study the critical behaviors of the largest, second largest clusters and their size ratio to characterize the universality class of percolation transition. Using the finite size scaling analysis and the Monte Carlo simulation, we calculate the critical exponents of the percolation transitions under the above rules. We find that the universality classes of continuum percolation under the generalized product rule, under the generalized sum rule, and the lattice percolation are different.

Kinetics and percolation: coke in heterogeneous catalysts

In the conventional lattice percolation models, bonds or sites are open at random, whereas in reality there is often interplay of percolation and the kinetics under consideration. An interesting and practically important example is hydrocarbon conversion occurring in a reactor containing mm-sized porous pellets with catalytic nanoparticles deposited at walls of the nanopores. Such catalytic heterogeneous reactions are often accompanied by coke formation deactivating catalytic nanoparticles and blocking pores for reactant diffusion, so that one needs to remove coke from time to time e.g. via reaction with oxygen. Herein, I first present generic coarse-grained Monte Carlo simulations of coke formation in a single pellet with the emphasis on the reaction regime influenced by reactant diffusion in pores. Then, the obtained coke distributions are used for similar simulations of coke removal. This combination of the models has allowed me to illustrate qualitatively new spatio-temporal regimes of the processes under consideration. For example, the removal of coke can be slow in the beginning, due to blocking of oxygen diffusion near the external pellet-gas interface and preventing its penetration to the central part of a pellet, and then fast when the pathways for diffusion to the center become to be open.

A novel temperature-dependent percolation model for the electrical conductivity and piezoresistive sensitivity of carbon nanotube-filled nanocomposites

Monte Carlo simulation is paired with a percolation model to analytically investigate the temperature effect and carbon nanotube (CNT) dispersion state on the temperature-dependent piezoresistive conductive network nanocomposite. The charge transport switch from electrical tunneling to elevated temperature quantum tunneling is studied based on a temperature-dependent tunneling model. The percolation network theory searches for the shortest distance of any CNT line segments scattered in the representative volume element to form possible interconnections and tunneling resistance approach evaluates resistivity by recognizing the conductive percolating paths. The effect of different factors such as the height of barrier potential, CNT length, degree of orientation and the cutoff distance on the resistance of the nanocomposite which is mainly caused by the inter-tube tunneling resistance is studied through analytical parametric studies. It is shown that the strain sensitivity of reinforced nanocomposite is increased as a result of the uniform dispersion of short aligned CNTs.

New Generation of MOF-Monoliths Based on Metal Foams

Herein, it has been developed a method to prepare metallic foams starting from Zamak5 (ZnAlCu alloy) with different pore sizes. The Zamak5 metallic foam is designed to serve as a support and metallic precursor of ZIF-8. In this way, composite materials MOF-metal can be prepared, these composites have a large number of application in energy exchange processe such as: adsorption or chemical reactions. Additionally, this method of sythesizing MOFs is environmentally friendly thanks to absence of solvents. Hanerssing the low melting point of the linker, the linker is infiltrated into the foam where the foam and the linker react to form the ZIF-8. In this way we have managed to transform part of the foam into ZIF-8 crystals that remain adhered to the foam. The foams have been characterized and modeled studying the mechanical and electrical properties, finding that both can be predected by various models. Among these, Ashby and Mortensen models for mechanical properties and Ashby and Percolation model for electrical properties stand.