Dependent Percolation Research Articles

Aggregation/agglomeration dependent percolation threshold of spherical nanoparticles in electrically conductive polymer nanocomposites

AbstractThis study evaluated the percolation threshold in polymer nanocomposites and its dependence on the aggregation/agglomeration phenomenon. The main objective of the investigation was to propose a particular method that besides revealing the impact of different involved parameters on the percolation threshold could also provide useful information regarding the system characteristics. In line with that, a specific geometrical structure was employed to represent the clusters and assess the variation of the percolation threshold and characteristics of the polymer/particle interphase region. The findings were applied in an improved form of the percolation theory to perform validation against the related experimental data. Improvement was conducted by involving the impact of the newly estimated percolation threshold, aggregations/agglomerations, interphase region, and so forth. Also, specific theories were employed to define the electrical conductivity of the nanoparticle clusters and interphase region. The results showed that the percolation threshold is directly affected by the content of the nanoparticles and the characteristics of the aggregates/agglomerates. Besides, it was revealed that parameters, such as the size of nanoparticles, physical characteristics of the polymer matrix and interphase region, dispersion quality, etc. may substantially affect the percolation threshold and subsequently other physical properties of the system, such as electrical conductivity.Highlights A structural approach toward understanding the percolation phenomenon. Defining the particular dependence of percolation threshold on dispersion quality. Estimating the content of aggregates/agglomerates using percolation theory. Improving the percolation theory to predict the electrical conductivity. Applying the scaling theory to define the electrical conductivity of the interphase.

Dependent percolation on Z2

Dependent percolation on Z2

Upper bounds on the one-arm exponent for dependent percolation models

We prove upper bounds on the one-arm exponent $$\eta _1$$ for a class of dependent percolation models which generalise Bernoulli percolation; while our main interest is level set percolation of Gaussian fields, the arguments apply to other models in the Bernoulli percolation universality class, including Poisson–Voronoi and Poisson–Boolean percolation. More precisely, in dimension $$d=2$$ we prove that $$\eta _1 \le 1/3$$ for continuous Gaussian fields with rapid correlation decay (e.g. the Bargmann–Fock field), and in $$d \ge 3$$ we prove $$\eta _1 \le d/3$$ for finite-range fields, both discrete and continuous, and $$\eta _1 \le d-2$$ for fields with rapid correlation decay. Although these results are classical for Bernoulli percolation (indeed they are best-known in general), existing proofs do not extend to dependent percolation models, and we develop a new approach based on exploration and relative entropy arguments. The proof also makes use of a new Russo-type inequality for Gaussian fields, which we apply to prove the sharpness of the phase transition and the mean-field bound for finite-range fields.

Size dependent percolation threshold and microwave absorption properties in nano carbon black/silicon rubber composites

Silicone rubber based lightweight composites have been prepared using nano carbon black (CB) as fillers to achieve broadband microwave absorption properties and electromagnetic interference (EMI) shielding over the 8–18 GHz frequency range. The effect of the filler material characteristics on percolation threshold and resultant electromagnetic (EM) properties of composites has been investigated. The composites are prepared by loading different filler fractions (wt. %) of three nano size CB having different characteristics, viz., CB1 (∼5–10 nm), CB2 (∼15–20 nm), and CB3 (∼30–40 nm) in a silicone rubber matrix. The volume resistivity measurements suggest low value of percolation threshold, i.e., 2 wt. % for CB1-rubber composite, 3 wt. % for CB2-rubber composites, as compared to 15 wt. % for CB3-rubber composite. Filler concentration dependent EM properties, i.e., dielectric constant and dielectric loss tangent (tan δe), are evaluated for all the CB-rubber composites. Furthermore, calculated reflection loss (RL) values for these composites indicate that 3 wt. % CB1-rubber, 5 wt. % CB2-rubber, and 18 wt. % CB3-rubber composites can provide more than 90% microwave absorption (RL > −10 dB) in X (8–12 GHz) and Ku (12–18 GHz) bands with thickness ∼2.7 and ∼1.9 mm, respectively. Interestingly, the composites with higher loading (15 wt. %) of CB1 in the rubber matrix are found to give EMI shielding effectiveness values of ∼ 42 dB over the 8–18 GHz frequency range. In conclusion, CB1-silicone rubber composites have been found to give the best performance among the three studied composites. This composite provides >90% microwave absorption over the X band with 2.7 mm thickness and over the Ku band with thickness 1.9 mm with lowest concentration of fillers, i.e., 3 wt. % CB1 in the rubber matrix, and hence found potential for development of lightweight microwave absorbers for stealth applications.

How do Ligands Tune the Assembly and Dissolution of Biomolecular Condensates?

Biomolecular condensates are thought to assemble via spontaneous or driven phase transitions of condensate-specific scaffold molecules. Scaffolds are multivalent protein or RNA molecules that enable concentration dependent phase separation and percolation transitions. Condensates also encompass non-scaffold molecules and the diversity of these molecules determines the compositional complexity of condensates. Non-scaffold molecules known as ligands can tune the phase behavior of scaffold molecules by preferentially binding to scaffolds in condensates or the coexisting dilute phase. Accordingly, cells can control condensate formation / dissolution by tuning the expression levels of ligands. Here, we show phase separation is destabilized when multivalent ligands bind to stickers, which are scaffold regions that drive phase separation. Phase separation is also destabilized by monovalent ligands, irrespective of whether they bind to sticker or spacer regions on scaffolds. In contrast, multivalent ligands that bind to spacer regions can promote scaffold phase separation. In this scenario, the binding of ligands adds valence to the scaffold molecules by enabling additional physical crosslinking of molecules within condensates. This suggests that the driving forces for condensate formation can be enhanced or diminished by controlling the expression levels of distinct categories of ligands - monovalent and/or multivalent ligands that bind to stickers vs. spacers. We show that modulatory effects of ligands cannot be discerned by measurements of partition coefficients of ligands. While these measurements are useful for compositional profiling of condensates, they have to be augmented by measuring the effects of ligands on the scaffold concentrations in coexisting dilute and dense phases. Overall, our studies provide a rigorous framework for modeling the effects of networks of ligands on the phase behavior of condensates, thereby enabling a deeper understanding of the compositional biases of condensates and ligand-mediated regulation of condensates in cells.

Effect of magnetic-field-induced restructuring on the elastic properties of magnetoactive elastomers

Composite materials where magnetic micrometer-sized particles are embedded into a compliant polymer matrix are known as magnetorheological (or magnetoactive) elastomers (MAEs). They are distinguished by huge variations in their physical properties, when in a magnetic field, which is commonly attributed to the restructuring of the filler. The process of the magnetic-field-induced restructuring in a magnetorheological elastomer is interpreted as progression towards percolation. Such a physical model was previously used to explain the dependence of the magnetic permeability and dielectric permittivity of MAEs on the magnetic field strength. Based on this hypothesis, the magnetorheological effect in MAEs is considered theoretically. The theoretical approach is built upon a self-consistent effective-medium theory for the elastic properties, extended to the variable (field dependent) percolation threshold. The proposed model allows one to describe the large variations (over several orders of magnitude) of the effective elastic moduli of these composite materials, known as the giant magnetorheological (MR) and field-stiffening effects. The existence of a giant magnetic Poisson effect is predicted. The relation of the proposed model to the existing theories of the MR effect in MAEs is discussed. The results can be useful for applications of MAEs in magnetic-field-controlled vibration dampers and isolators.

Time Dependent Percolation Analysis of the Degradation of Coherent Tunneling in Ultra-Thin CoFeB/MgO/CoFeB Magnetic Tunneling Junctions

Time Dependent Percolation Analysis of the Degradation of Coherent Tunneling in Ultra-Thin CoFeB/MgO/CoFeB Magnetic Tunneling Junctions

Synthesis, characterization and Hall-effect studies of highly conductive polyaniline/graphene nanocomposites

The objective of this research is to prepare a series of conducting polyaniline/graphene composites by in situ chemical oxidative method through polymerization of aniline in the existence of graphene of various weight percentages. The prepared composites were examined through Fourier transform infrared spectroscopy, X-ray diffraction and scanning electron microscopy respectively. The conductivity and dielectric attributes exhibited composition dependent percolation behavior with enhanced properties for 15 wt% of graphene in polyaniline matrix. The Hall-effect studies were performed in the prepared composites and the investigations confirm that the composites behave as n-type semi-conductors. From the data obtained one can envisage remarkable improvement in the charge transport and dielectric properties of the composite at the percolation threshold due to the presence of nano-size additive in the composite. The temperature dependent conductivity in doped composites was increased by five fold in compare to pure PANI. The doping of graphene in PANI matrix, the improved dielectric parameters were achieved. The Hall effect studies were carried out in the prepared composites, the results shows that composites behaves like a n-type semiconductor. Further, the mechanical properties such as tensile stress and tensile strain were carried out in the prepared composites.

Percolation in majority dynamics

We consider two-dimensional dependent dynamical site percolation where sites perform majority dynamics. We introduce the critical percolation function at time t as the infimum density with which one needs to begin in order to obtain an infinite open component at time t. We prove that, for any fixed time t, there is no percolation at criticality and that the critical percolation function is continuous. We also prove that, for any positive time, the percolation threshold is strictly smaller than the critical probability for independent site percolation.

On macroscopic holes in some supercritical strongly dependent percolation models

We consider $Z^d$, with d bigger or equal to three. We investigate the vacant set of random interlacements in the strongly percolative regime, the vacant set of the simple random walk, and the excursion set above a given level of the Gaussian free field in the strongly percolative regime. We derive asymptotic upper and lower exponential bounds for the large deviation probability that the adequately thickened component of the boundary of a large box centered at the origin in the respective vacant sets or excursion set leaves in the box a macroscopic volume in its complement. We also derive geometric information on the shape of the left-out volume. It is plausible, but open at the moment, that certain critical levels coincide, both in the case of random interlacements and of the Gaussian free field. If this holds true, the asymptotic upper and lower bounds that we obtain are matching in principal order for all three models, and the macroscopic holes are nearly spherical. We heavily rely on the recent work arXiv:1706.07229 by Maximilian Nitzschner and the author for the coarse graining procedure, which we employ in the derivation of the upper bounds.

Strongly Correlated Random Interacting Processes

The focus of the workshop was to discuss the recent developments and future research directions in the area of large scale random interacting processes, with main emphasis in models where local microscopic interactions either produce strong correlations at macroscopic levels, or generate non-equilibrium dynamics. This report contains extended abstracts of the presentations, which featured research in several directions including self-interacting random walks, spatially growing processes, strongly dependent percolation, spin systems with long-range order, and random permutations.

Percolation and isoperimetry on roughly transitive graphs

Dans cet article, nous etudions la percolation sur un graphe grossierement transitif $G$ a croissance polynomiale et de dimension isoperimetrique plus grande que 1. Pour ces graphes, nous prouvons que $p_{c}<1$ ou, en d’autres termes, nous prouvons qu’il existe une phase de percolation. Les resultats principaux de l’article sont valables a la fois pour les processus de percolation dependants ou independants, car ils s’appuient sur des arguments de renormalisation assez robustes. Quand $G$ est transitif, le fait que $p_{c}<1$ etait deja connu. Mais meme dans ce cas notre preuve donne des resultats nouveaux et est entierement probabiliste, evitant l’utilisation du theoreme de Gromov sur les groupes a croissance polynomiale. Nous concluons l’article par quelques exemples de percolation dependante pour lesquels nos resultats s’appliquent.

Temperature dependent percolation mechanism for conductivity in TiO3 revealed by a microstructure study

We have performed optical microscopy, micro-photoelectron spectroscopy, and micro-Raman scattering measurements on TiO3 single crystals in order to clarify the interplay between the microstructure and the temperature dependent electronic transport mechanisms in this material. Optical microscopy observations reveal dark and bright domain patterns on the surface with length scales of the order of several to a hundred micrometers showing a pronounced temperature dependent evolution. Spatially resolved photoelectron spectroscopy measurements show the different electronic character of these domains. Using micro-Raman spectroscopy, we observe a distinct temperature dependence of the crystal structure of these domains. On the basis of these findings the different domains are assigned to insulating and metallic volume fractions, respectively. By decreasing the temperature, the volume fraction of the conducting domains increases, hence allowing the electrons to percolate through the sample at temperatures lower than ∼150 K.

Molecular dynamics simulations on heterogeneity and percolation of epoxy nanofilm during glass transition process

This study uses an all-atom computational model to investigate the temperature dependent heterogeneity and percolation in a nanofilm system of short linear epoxy chains on a solid graphene surface. The heterogeneity, which indicates having physical characters that vary within the nanofilm, is mainly manifested in distributions of volume, energy and the dynamic properties. Local glass transition temperatures, Tgs, from above properties depend largely on the separation to the graphene surface, and the glass transition of the nanofilm is asynchronous along the film normal. Distinct Tg increases and decreases are particularly observed in the solid and free interfaces, respectively, compared with the bulk. From the dynamic heterogeneity, percolation effect, which indicates the connectivity of mobile and immobile domains, of the nanofilm is also observed during glass transition by plotting internal atomic mobility distribution diagrams. A multi-stage percolation mechanism based on the glass transition state and the connection state of immobile and mobile domains of the nanofilm is developed. A relatively immobile domain near the graphene surface is observed, even at temperatures much higher than Tg, and it initiates the dynamic percolation. The interconnection of immobile domains after percolation accelerate the transition from the rubbery to the glassy state.

A new computation of the critical point for the planar random-cluster model with $q\ge1$

Nous proposons une nouvelle preuve du fait que le point critique du modele de percolation de Fortuin–Kasteleyn sur le reseau carre vaut $\sqrt{q}/(1+\sqrt{q})$ lorsque $q\ge1$. Cette preuve est une alternative a la strategie implementee dans (Probab. Theory Related Fields 153 (2012) 511–542). Cette approche a plusieurs avantages. Tout d’abord, la grande majorite des arguments peuvent etre generalises aux autres graphes planaires (ayant suffisamment de symetries). De plus, elle n’invoque pas d’argument de type RSW bases sur l’auto-dualite du modele. Enfin, elle repose sur une nouvelle facon d’appliquer les theoremes de seuil qui n’utilise pas la symmetrie des evǹements de croisement et les conditions de bord periodiques. Certaines de ces nouvelles methodes ont un champs d’application qui depasse largement le cas du modele de percolation de Fortuin–Kasteleyn et pourrait etre utile pour prouver la decroissance exponentielles d’autres modeles de percolation dependante.

Electrical, Mechanical and Electromechanical Properties of Graphene-Thermoset Polymer Composites Produced Using Acetone-DMF Solvents.

Recently, graphene-polymer composites gained a central role in advanced stress and strain sensing. A fundamental step in the production of epoxy-composites filled with graphene nanoplatelets (GNPs) consists in the exfoliation and dispersion of expanded graphite in a proper solvent, in the mixing of the resulting GNP suspension with the polymer matrix, and in the final removal of the solvent from the composite before curing through evaporation. The effects of traces of residual solvent on polymer curing process are usually overlooked, even if it has been found that even a small amount of residual solvent can affect the mechanical properties of the final composite. In this paper, we show that residual traces of N,N′-Dimethylformamide (DMF) in vinylester epoxy composites can induce relevant variations of the electrical, mechanical and electromechanical properties of the cured GNP-composite. To this purpose, a complete analysis of the morphological and structural characteristics of the composite samples produced using different solvent mixtures (combining acetone and DMF) is performed. Moreover, electrical, mechanical and electromechanical properties of the produced composites are assessed. In particular, the effect on the piezoresistive response of the use of DMF in the solvent mixture is analyzed using an experimental strain dependent percolation law to fit the measured electromechanical data. It is shown that the composites realized using a higher amount of DMF are characterized by a higher electrical conductivity and by a strong reduction of Young’s Modulus.

Infinite and Giant Components in the Layers Percolation Model

In this work we continue the investigation launched in [FHR16] of the structural properties of the structural properties of the Layers model, a dependent percolation model. Given an undirected graph $G=(V,E)$ and an integer $k$, let $T_k(G)$ denote the random vertex-induced subgraph of $G$, generated by ordering $V$ according to Uniform$[0,1]$ $\mathrm{i.i.d.}$ clocks and including in $T_k(G)$ those vertices with at most $k-1$ of their neighbors having a faster clock. The distribution of subgraphs sampled in this manner is called the layers model with parameter $k$. The layers model has found applications in the study of $\ell$-degenerate subgraphs, the design of algorithms for the maximum independent set problem and in the study of bootstrap percolation. We prove that every infinite locally finite tree $T$ with no leaves, satisfying that the degree of the vertices grow sub-exponentially in their distance from the root, $T_3(T)$ $\mathrm{a.s.}$ has an infinite connected component. In contrast, we show that for any locally finite graph $G$, $\mathrm{a.s.}$ every connected component of $T_2(G)$ is finite. We also consider random graphs with a given degree sequence and show that if the minimal degree is at least 3 and the maximal degree is bounded, then $\mathrm{w.h.p.}$ $T_3$ has a giant component. Finally, we also consider ${\mathbb{Z}}^{d}$ and show that if $d$ is sufficiently large, then $\mathrm{a.s.}$ $T_4(\mathbb{Z}^d)$ contains an infinite cluster.

Contact formation of front side metallization in p-type, single crystalline Si solar cells: Microstructure, temperature dependent series resistance and percolation model

Screen printed front side contacts were investigated in single-crystalline (planar and textured) Si solar cells with n-type emitters, yielding maximum efficiencies of 18.0%. The crystallographic orientation of the Si surface and the paste strongly affect the contact formation as well as the contact resistance of the cells. For textured cells a continuous glass layer together with the formation of Ag colloids yielded a small contact resistance. Planar 〈111〉oriented Si yielded specifically lower contact resistance (〈5mΩcm2) as compared to planar 〈100〉 orientation (〉10–40mΩcm2) for different pastes. Pyramidal Ag crystals are formed only on 〈100〉 oriented Si, whereas lens shaped Ag crystals are grown on 〈111〉surfaces. From this it was concluded that the shape of the Ag nanocrystals determines the contact resistance, pyramidal Ag crystals formed on 〈100〉 planar surfaces yielded cells with large contact resistance and are, therefore, not considered to be necessary for a low contact resistance.Temperature dependent series resistance measurements yielded metallic behavior for cells with the lowest contact resistance bound to a certain paste. For other pastes and processing conditions a semiconducting behavior of the series resistance was found. However, cells with significant density of colloids in the glass layer yielded a small series and contact resistance. By considering the above arguments, a percolation model has been introduced in which metallic Ag colloids generate current filaments across the glass layer. This reduces the resistivity of the glass layer and thereby introduces a percolative nature of the current via Ag nanocolloids. The percolation limit for the 2d case was calculated for periodically arranged colloids with equal size and yields a minimum volume fraction of 15% for the Ag colloids in the glass layer.

Percolation and local isoperimetric inequalities

In this paper we establish some relations between percolation on a given graph G and its geometry. Our main result shows that, if G has polynomial growth and satisfies what we call the local isoperimetric inequality of dimension $$d > 1$$ , then $$p_c(G) < 1$$ . This gives a partial answer to a question of Benjamini and Schramm (Electron Commun Probab 1(8):71–82 1996). As a consequence of this result and Haggstrom (Adv Appl Probab 32(1):39–66 2000) we derive, that these graphs also undergo a non-trivial phase transition for the Ising-Model, the Widom-Rowlinson model and the beach model. Our techniques are also applied to dependent percolation processes with long range correlations. We provide results on the uniqueness of the infinite percolation cluster and quantitative estimates on the size of finite components. Finally we leave some remarks and questions that arise naturally from this work.

Clairvoyant embedding in one dimension

Let v, w be infinite 0-1 sequences, and mi¾? a positive integer. We say that w is mi¾?-embeddable in v, if there exists an increasing sequence ni:ii¾?0 of integers with n0=0, such that 1i¾?ni-ni-1i¾?mi¾?, wi=vni for all ii¾?1. Let X and Y be coin-tossing sequences. We will show that there is an mi¾? with the property that Y is mi¾?-embeddable into X with positive probability. This answers a question that was open for a while. The proof generalizes somewhat the hierarchical method of an earlier paper of the author on dependent percolation. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 520-560, 2015