Percolation Model Research Articles

Percolation effects in the Fortuin-Kasteleyn Ising model on the complete graph.

The Fortuin-Kasteleyn (FK) random-cluster model, which can be exactly mapped from the q-state Potts spin model, is a correlated bond percolation model. By extensive Monte Carlo simulations, we study the FK bond representation of the critical Ising model (q=2) on a finite complete graph, i.e., the mean-field Ising model. We provide strong numerical evidence that the configuration space for q=2 contains an asymptotically vanishing sector in which quantities exhibit the same finite-size scaling as in the critical uncorrelated bond percolation (q=1) on the complete graph. Moreover, we observe that, in the full configuration space, the power-law behavior of the cluster-size distribution for the FK Ising clusters except the largest one is governed by a Fisher exponent taking the value for q=1 instead of q=2. This demonstrates the percolation effects in the FK Ising model on the complete graph.

An extensive study on the effect of percolation on thermo-physical characteristics of boron-nitride polyester composites

The present study aims at accomplishment of very high thermal conductivity due to outstanding properties exhibited at percolation threshold in BN-polyester composites. The model of heat transfer networks was used as a unified model for the parallel as well as series conduction model. The filler size BN particulate of 100microns-300microns has bimodal distribution which enhances the packing density in the polyester matrix. This fiber–matrix combination of BN-polyester for 20 wt% and above has a tendency to form conductive networks with low thermal resistance but very high jump in thermal conductivity as well as electrical conductivity which thereby becomes an added advantage for micro-electronic application areas. The characteristics like diffusion, electrical conductivity, dielectric constant and elasticity are also related to arrangement of these conductive networks which basically deals with disordered systems with random orientations. Hence, the percolation theory is studied in the present investigation and methods of determining the percolation threshold effectively with different weight fractions of the filler are discussed. The experimental results also showed a good agreement with the percolation model.

Quantification of the ion transport mechanism in protective polymer coatings on lithium metal anodes.

Protective Polymer Coatings (PPCs) have been proposed to protect lithium metal anodes in rechargeable batteries to stabilize the Li/electrolyte interface and to extend the cycle life by reducing parasitic reactions and improving the lithium deposition morphology. However, the ion transport mechanism in PPCs remains unclear. Specifically, the degree of polymer swelling in the electrolyte and the influence of polymer/solvent/ion interactions are never quantified. Here we use poly(acrylonitrile-co-butadiene) (PAN–PBD) with controlled cross-link densities to quantify how the swelling ratio of the PPC affects conductivity, Li+ ion selectivity, activation energy, and rheological properties. The large difference in polarities between PAN (polar) and PBD (non-polar) segments allows the comparison of PPC properties when swollen in carbonate (high polarity) and ether (low polarity) electrolytes, which are the two most common classes of electrolytes. We find that a low swelling ratio of the PPC increases the transference number of Li+ ions while decreasing the conductivity. The activation energy only increases when the PPC is swollen in the carbonate electrolyte because of the strong ion–dipole interaction in the PAN phase, which is absent in the non-polar PBD phase. Theoretical models using Hansen solubility parameters and a percolation model have been shown to be effective in predicting the swelling behavior of PPCs in organic solvents and to estimate the conductivity. The trade-off between conductivity and the transference number is the primary challenge for PPCs. Our study provides general guidelines for PPC design, which favors the use of non-polar polymers with low polarity organic electrolytes.

A new look at the interfaces in percolation

We propose a new definition of the interface in the context of the Bernoulli percolation model. We construct a coupling between two percolation configurations, one which is a standard percolation configuration, and one which is a percolation configuration conditioned on a disconnection event. We define the interface as the random set of the edges where these two configurations differ. We prove that, inside a cubic box $\Lambda$, the interface between the top and the bottom of the box is typically localised within a distance of order $(\ln |\Lambda|)^2$ of the set of the pivotal edges. We prove also that, in our dynamical coupling, the typical speed of the pivotal edges remains bounded as the box $\Lambda$ grows.

The mechanism of reinforcement of nanocomposites polyurethane/graphene

The percolation model of reinforcement was used for the theoretical analysis of reinforcement mechanism for nanocomposites polyurethane/graphene. This model allows to elucidate influence of main factors (level of interfacial adhesion, nanofiller content, interfacial regions) on the degree of reinforcement or modulus of elasticity of the considered nanocomposites. It has been shown that for these nanocomposites actually nanofiller (graphene) serves as main reinforcing element. The sharp increasing of modulus of elasticity of nanocomposite occurs at achievement of critical content of nanofiller (∼9 % mas.). The same effect of increasing the level of interfacial adhesion is obtained by a polymer matrix-nanofiller, characterized by a transition from perfect adhesion to nanoadhesion. The structure type of nanofiller in polymer matrix (exfoliated or intercalated one) in one more factor. The proposed model is universal one for all nanocomposites polymer/2D-nanofiller.

Correction: Elliptic percolation model for predicting the electrical conductivity of graphene–polymer composites

Correction for 'Elliptic percolation model for predicting the electrical conductivity of graphene-polymer composites' by Asghar Aryanfar et al., Soft Matter, 2021, 17, 2081-2089, DOI: 10.1039/D0SM01950J.

Elliptic percolation model for predicting the electrical conductivity of graphene–polymer composites

Graphene-based polymers exhibit a conductive microstructure formed by aggregates in a matrix which drastically enhances their transmitting properties. We develop a new numerical framework for predicting the electrical conductivity based on continuum percolation theory in a two dimensional stochastically-generated medium. We analyze the role of the flake shape and its aspect ratio and consequently predict the onset of percolation based on the particle density and the domain scale. Simultaneously, we have performed experiments and have achieved very high electrical conductivity for such composites compared to other film fabrication techniques, which have verified the results of computing the homogenized electrical conductivity. As well, the proximity to and a comparison with other analytical models and other experimental techniques are presented. The numerical model can predict the composite transmitting conductivity in a larger range of particle geometry. Such quantification is exceedingly useful for effective utilization and optimization of graphene filler densities and their spatial distribution during manufacturing.

Fractal geometry of Airy$_{2}$ processes coupled via the Airy sheet

In last passage percolation models lying in the Kardar–Parisi–Zhang universality class, maximizing paths that travel over distances of order $n$ accrue energy that fluctuates on scale $n^{1/3}$; and these paths deviate from the linear interpolation of their endpoints on scale $n^{2/3}$. These maximizing paths and their energies may be viewed via a coordinate system that respects these scalings. What emerges by doing so is a system indexed by $x,y\in \mathbb{R}$ and $s,t\in \mathbb{R}$ with $s<t$ of unit order quantities $W_{n}(x,s;y,t)$ specifying the scaled energy of the maximizing path that moves in scaled coordinates between $(x,s)$ and $(y,t)$. The space-time Airy sheet is, after a parabolic adjustment, the putative distributional limit $W_{\infty }$ of this system as $n\to \infty $. The Airy sheet has recently been constructed in (Dauvergne, Ortmann and Virág (2020)) as such a limit of Brownian last passage percolation. In this article, we initiate the study of fractal geometry in the Airy sheet. We prove that the scaled energy difference profile given by $\mathbb{R}\to \mathbb{R}:z\to W_{\infty }(1,0;z,1)-W_{\infty}(-1,0;z,1)$ is a nondecreasing process that is constant in a random neighbourhood of almost every $z\in \mathbb{R}$; and that the exceptional set of $z\in \mathbb{R}$ that violate this condition almost surely has Hausdorff dimension one-half. Points of violation correspond to special behaviour for scaled maximizing paths, and we prove the result by investigating this behaviour, making use of two inputs from recent studies of scaled Brownian LPP; namely, Brownian regularity of profiles, and estimates on the rarity of pairs of disjoint scaled maximizing paths that begin and end close to each other.

Simple Percolation Model of Cracking in Textile Composites with a Woven Matrix

Simple Percolation Model of Cracking in Textile Composites with a Woven Matrix

A positivity phenomenon in Elser's Gaussian-cluster percolation model

Veit Elser proposed a random graph model for percolation in which physical dimension appears as a parameter. Studying this model combinatorially leads naturally to the consideration of numerical graph invariants which we call Elser numberselsk(G), where G is a connected graph and k a nonnegative integer. Elser had proven that els1(G)=0 for all G. By interpreting the Elser numbers as reduced Euler characteristics of appropriate simplicial complexes called nucleus complexes, we prove that for all graphs G, they are nonpositive when k=0 and nonnegative for k⩾2. The last result confirms a conjecture of Elser. Furthermore, we give necessary and sufficient conditions, in terms of the 2-connected structure of G, for the nonvanishing of the Elser numbers.

Porous structure design and mechanical behavior analysis based on TPMS for customized root analogue implant

Compared with the traditional dental implant with screw structure, the root analogue implant (RAI) is customized to fit with the wall of the alveolar bone, which helps to accelerate the clinical implantation process. However, a solid RAI made of Ti6Al4V material has a much higher Young's modulus than the surrounding bone tissue, which can cause a stress shielding effect and thereby lead to implant failure. Also, a solid RAI is not conducive to the growth of osteoblasts. To overcome these problems, a porous structure design and optimization method for customized RAIs is proposed. A triply periodic minimal surface (TPMS) offers a smooth surface with pore interconnectivity, which can satisfy the biological/mechanical implantation requirements and efficiently construct many complex bone scaffolds. P and G structures with four degrees of porosity (30%, 40%, 50%, and 60%) were designed and prepared as cubic samples. The Young's modulus, Poisson's ratio, and yield strength of each sample were measured through compression experiments. Additionally, the stress distribution at the interface between the customized RAI and surrounding bone tissue under different pore structures and porosities was evaluated by finite element analysis (FEA). It was found that the quantitative relationships between the Young's modulus/Poisson's ratio and porosity of the P and G structures were consistent with the rules of the percolation model. The yield strengths of the P and G structures with four different porosities were all greater than the yield strength of cortical bone, which satisfies the implantation conditions. Furthermore, the P and G structures with 30% and 40% porosity were proved by FEA to have no stress shielding effect, promote the growth of surrounding bone tissue, and form long-term and stable osseointegration. It can be concluded that the porous RAI constructed with a TPMS can reduce the stress shielding effect, which is beneficial for accelerating the clinical implantation process.

Homological Percolation: The Formation of Giant k-Cycles

Abstract In this paper we introduce and study a higher dimensional analogue of the giant component in continuum percolation. Using the language of algebraic topology, we define the notion of giant $k$-dimensional cycles (with $0$-cycles being connected components). Considering a continuum percolation model in the flat $d$-dimensional torus, we show that all the giant $k$-cycles ($1\le k \le d-1$) appear in the regime known as the thermodynamic limit. We also prove that the thresholds for the emergence of the giant $k$-cycles are increasing in $k$ and are tightly related to the critical values in continuum percolation. Finally, we provide bounds for the exponential decay of the probabilities of giant cycles appearing.

Geometric functionals of fractal percolation

AbstractFractal percolation exhibits a dramatic topological phase transition, changing abruptly from a dust-like set to a system-spanning cluster. The transition points are unknown and difficult to estimate. In many classical percolation models the percolation thresholds have been approximated well using additive geometric functionals, known as intrinsic volumes. Motivated by the question of whether a similar approach is possible for fractal models, we introduce corresponding geometric functionals for the fractal percolation process F. They arise as limits of expected functionals of finite approximations of F. We establish the existence of these limit functionals and obtain explicit formulas for them as well as for their finite approximations.

Atomic-Scale Coordination Effects on the Initial Stage of Passive Film Formation in Fe-Cr Alloys

The self-healing passivation behavior Fe-Cr alloys serve as a model for the development of corrosion resistant alloys and coatings. Both first principles-based calculations [1,2], neutron scattering [3] and Mössbauer spectroscopy [4] studies demonstrated that in FepCr1-p alloys, the atomic-scale distribution of Cr atoms in the BCC lattice depends on both composition and thermal history. These studies showed that there is an inversion in the Warren Cowley short range order (SRO) parameter [5] that occurs for 1-p ≅ 0.11 At. Cr compositions, less than this value, the alloys show a tendency toward ordering while at higher Cr compositions the alloys show a tendency for clustering. In terms of a simple regular solution model, this behavior is determined by the interaction parameter, ΔE = [EFe-Cr - 1/2(EFe-Fe + ECr-Cr)], where the E’s are nearest neighbor bond energies taken to be negative. Then for ΔE < 0 the system tends to short-range order, while for ΔE > 0, clustering behavior is expected. In order to study these atomic scale coordination effects on passivation, FepCr1-p alloys were heat-treated according to the protocols used in the neutron scattering and Mössbauer spectroscopy studies. The behaviors of these alloys were compared to that of their random solid solution counterparts. Linear sweep voltammetry and chronoamperometry were used to characterize the initial stage of passivation in these two sets of alloys. Our results reveal some subtle yet significant differences in their passivation behaviors [6]. These results are interpreted within the context of the percolation conjecture suggested by Sieradzki and Newman [7]. Klaver, T. P. C., Drautz, R. & Finnis, M. W. “Magnetism and Thermodynamics of Defect-Free Fe-Cr Alloys", Rev. B. 74, 94435 (2006).Olsson, P., Abrikosov, I. A., Wallenius, J. “Electronic Origin Of The Anomalous Stability of Fe-Rich Bcc Fe-Cr Alloys", Rev. B. 73, 104416 (2006).Mirebeau, I., Parette, G. “Neutron Study of The Short Range Order Inversion in Fe1− x Crx" Rev. B. 82, 104203 (2010).Dubiel, S. M., Cieslak, J.“ Short-Range Order in Iron-Rich Fe-Cr Alloys as Revealed by Mössbauer Spectroscopy", Rev. B. 83, 180202 (2011).Cowley, J.M. “Short-Range Order and Long-Range Order Parameters, Rev., 138, A1384 – 1389 (1965).Liu, M. Aiello, A., Xie, Y. Sieradzki, K., “The Effect of Short-Range Order on Passivation of Fe-Cr Alloys”, Electrochem. Soc., 165, C830-C834 (2018).Sieradzki, K., Newman, R. C.“ A. Percolation Model for Passivation in Stainless Steels", Electrochem. Soc. 133, pp. 1979–1980 (1986).

Charge transport mechanism in the metal–nitride–oxide–silicon forming-free memristor structure

Abstract Metal-nitride-oxide-silicon structures that exhibit memristor properties were obtained using the low-pressure chemical vapor deposition at 700° C. The fabricated metal-nitride-oxide-silicon memristor structure does not require a forming procedure. In addition, the metal-nitride-oxide-silicon memristor has a memory window of about 3 orders of magnitude. In our work, the charge transport of high and low resistive states in a metal-nitride-oxide-silicon memristor is analyzed with two contact-limited models and six bulk-limited charge transport models. It is established that the Schottky effect model, thermally assisted tunneling model, Frenkel model of Coulomb traps ionization, Hill-Adachi model of overlapping Coulomb traps, Shklovskii-Efros percolation model, Makram-Ebeid and Lannoo model of multiphonon isolated traps ionization and the Nasyrov-Gritsenko model of phonon-assisted tunneling between traps, quantitatively, do not describe the charge transport of metal-nitride-oxide-silicon memristor. We found that the main charge transport mechanism in the metal-nitride-oxide-silicon memristor in a high resistive state is the model of space-charge-limited current with traps. In a low resistive state, the charge transport mechanism is described by the space-charge-limited current model with filled traps.

Effective use of nano-carbons in controlling the electrical conductivity of epoxy composites

Glass fiber reinforced epoxy composites were fabricated using MWCNT and GNP fillers, individually and in combination. The objective of combining the two conductive fillers was to enhance the electrical conductivity of the composites. The DC and AC conductivity measurements were carried out to assess the influence of individual fillers and their combinations on the electrical conductivity. High DC conductivity of 3.1 × 10−3 S/m and AC conductivity of 1.31 × 10−2 S/m at 8 MHz were observed, at 2 wt% of MWCNT and 3 wt% of GNP and the values were significantly reduced below 3 wt% of GNP. The observed increase in electrical conductivity is attributed to low contact resistance (299.85Ω) of fillers, reduced tunneling resistance (5.35 × 109Ω) and tunneling distance (1.65 nm) of the composites. The DC and AC conductivity, contact resistance, tunneling resistance and tunneling distance showed reduced dependence on temperature. The AC conductivity of the hybrid composites follows power law. The cut-off frequency of composites was 200 kHz at 3 wt% of GNP and 2 KHz with reduction in weight percentage of GNP. In composite with 2 wt% of MWCNT and 3 wt% of GNP, validity of the percolation model is established.

Zirconia-strengthened yttria ceramics for plasma chamber applications

Yttria (Y2O3) is the most popular lining material for the inner walls of plasma chambers, owing to its well-known resistance against corrosion by reactive fluorinated plasma. Yttria has a relatively lower toughness and strength compared to alumina (Al2O3) and zirconia (ZrO2). The present work aims to explore the strengthening and toughening of yttria by doping with zirconia, while retaining its good corrosion resistance. Ceramics with zirconia to yttria molar ratios of 2:8 and 5:5 (named 20ZY and 50ZY, respectively) were fabricated by pressure sintering in Ar atmosphere. The corrosion of the prepared ceramics in reactive fluorinated plasma (C4F8/CHF3/Ar) was investigated and compared with that of yttria as a control. The results indicated that 20ZY exhibited excellent corrosion resistance, comparable to that of yttria, while the fracture toughness and flexural strength showed increases of 87% and 44%, respectively. 50ZY exhibited a further improvement in fracture toughness and flexural strength, but at the price of a much lower resistance against plasma corrosion. A percolation model was proposed to interpret the observed plasma corrosion behaviors.

Investigation of Ce0.9Gd0.1O2-δ dispersed Sm1.5Sr0.5NiO4+δ: Cathode for intermediate temperature solid oxide fuel cell applications

Dispersion of nanocrystalline (94–350 nm) Ce0.9Gd0.1O2-δ in superfine (260–312 nm) Sm1.5Sr0.5NiO4+δ using modified precipitation technique is established using X-ray powder diffraction, scanning electron microscopy and transmission electron microscopy. Presence of Ce0.9Gd0.1O2-δ grains inhibits grain growth of Sm1.5Sr0.5NiO4+δ, which provides morphological stability (up to 1100 °C). Ce0.9Gd0.1O2-δ concentration dependent behaviours of ionic conductivity, surface exchange rate and electrode polarization resistance (Rp) of composites (determined using electrochemical impedance spectroscopy) are comprehended using percolation model. Three oxygen reduction reaction mechanisms are considered to understand electrochemical performance. Minimum Rp (0.81 Ω cm2 at 700 °C) for 70Sm1.5Sr0.5NiO4+δ:30Ce0.9Gd0.1O2-δ is correlated to percolation threshold (optimum (i) electrochemically active sites (ii) oxygen reduction reaction kinetics, (iii) O2- conductivity and (iv) charge transfer rate). Nano crystallite size of Ce0.9Gd0.1O2-δ is crucial for enhancement in electrochemical performance. Oxygen partial pressure dependent electrochemical impedance spectroscopy studies reveal dominance of coexisting non-charge transfer oxygen adsorption/desorption and bulk O2- diffusion.

Distribution law of the percolation threshold in one-dimensional bond problems

A one-dimensional lattice percolation model is constructed for the bond problem at flowing along non-nearest neighbors. Arbitrary parameters were taken (percolation radius, number of nodes in a one-dimensional lattice and number of experiments). Based on original algorithms operating on a computer faster than standard ones, the values of the percolation threshold were obtained with the corresponding error. Based on these data, the hypothesis about the normal distribution of the percolation threshold is tested. Using Pearson’s criterion it was shown for the first time that there is no reason to reject this hypothesis for one-dimensional problems of bonds and sites with an arbitrary percolation radius.

Existence of phase transition for percolation using the Gaussian free field

In this paper, we prove that Bernoulli percolation on bounded degree graphs with isoperimetric dimension $d>4$ undergoes a non-trivial phase transition (in the sense that $p_c<1$). As a corollary, we obtain that the critical point of Bernoulli percolation on infinite quasi-transitive graphs (in particular, Cayley graphs) with super-linear growth is strictly smaller than 1, thus answering a conjecture of Benjamini and Schramm. The proof relies on a new technique consisting in expressing certain functionals of the Gaussian Free Field (GFF) in terms of connectivity probabilities for percolation model in a random environment. Then, we integrate out the randomness in the edge-parameters using a multi-scale decomposition of the GFF. We believe that a similar strategy could lead to proofs of the existence of a phase transition for various other models.