Percolation is the simplest fundamental model in statistical mechanics that exhibits phase transitions signaled by the emergence of a giant connected component. Despite its very simple rules, ...percolation theory has successfully been applied to describe a large variety of natural, technological and social systems. Percolation models serve as important universality classes in critical phenomena characterized by a set of critical exponents which correspond to a rich fractal and scaling structure of their geometric features. We will first outline the basic features of the ordinary model.
Over the years a variety of percolation models has been introduced some of which with completely different scaling and universal properties from the original model with either continuous or discontinuous transitions depending on the control parameter, dimensionality and the type of the underlying rules and networks. We will try to take a glimpse at a number of selective variations including Achlioptas process, half-restricted process and spanning cluster-avoiding process as examples of the so-called explosive percolation. We will also introduce non-self-averaging percolation and discuss correlated percolation and bootstrap percolation with special emphasis on their recent progress. Directed percolation process will be also discussed as a prototype of systems displaying a nonequilibrium phase transition into an absorbing state.
In the past decade, after the invention of stochastic Löwner evolution (SLE) by Oded Schramm, two-dimensional (2D) percolation has become a central problem in probability theory leading to the two recent Fields medals. After a short review on SLE, we will provide an overview on existence of the scaling limit and conformal invariance of the critical percolation. We will also establish a connection with the magnetic models based on the percolation properties of the Fortuin–Kasteleyn and geometric spin clusters. As an application we will discuss how percolation theory leads to the reduction of the 3D criticality in a 3D Ising model to a 2D critical behavior.
Another recent application is to apply percolation theory to study the properties of natural and artificial landscapes. We will review the statistical properties of the coastlines and watersheds and their relations with percolation. Their fractal structure and compatibility with the theory of SLE will also be discussed. The present mean sea level on Earth will be shown to coincide with the critical threshold in a percolation description of the global topography.
A new theoretical model and its computational implementation in two dimensions (2D) for the study of continuum percolation phenomena is presented. The aim was the development of a model which has ...inherent similarity with lattice percolation. The physical medium is simulated as an (infinite) grid comprising of representative surface elements (RSEs). Assuming medium’s homogeneity the RSEs average propagation probability can be interpreted and generalized as the occupation probability for the infinite medium. The RSE’s resulting from a Monte Carlo iterative process involving the creation of the relative small samples and their propagation ability checked individually from their top to bottom. The propagation in the actual physical medium takes place when the calculated probability (p) is higher than the critical propagation probability (pc≈0.5927). The proposed method treats the low dimensional material system as a 2D infinite homogenized medium which can be further reduced leading to a mapping on a square lattice with site occupation. The proposed numerical algorithm considers the particles in the RSE as digitized using sites-pixels without contacts. Following the digitization procedure, traditional computational methods like Depth First Search are involved for the detection of possible propagation paths in the randomly selected square samples. For the confirmation of the theoretical model as well as the algorithm, problems known from the literature were used and it was found that regardless of microstructure at the critical concentration Φc the percolation probability on the RSE converges to the anticipated pc≈0.5927 value. In addition, the results obtained from the proposed methodology compare very well with available predictions in the literature. New results are reported covering a wide range of particle geometrical types (circular, elliptical, rectangular) and surface fractions in matrix-filler or matrix-fillers systems proving the robustness and applicability of the proposed methodology.
•A new computational method is developed in 2D to study continuum percolation•Tunnel effect was taken into account in the modeling approach•The proposed methodology compares well with available predictions in the literature•The results cover different particles and surface fractions in matrix-filler systems•The method is computational affordable requiring moderate computational resources
This study examines the frequency and temperature dependence of conductivity, phase shift angle and capacity of the (FeCoZr)x (CaF2)(100−x) nanocomposites, with the metal phase content in the range ...of 45 at.% < x < 90 at.%, produced by pure argon ion-beam sputtering. It was found that for the metallic phase content lower than 67 at.% there are temperature dependencies of the dielectric type of conductivity, for which the derivative dσ/dT > 0. Activation energy of conductivity, determined from the Arrhenius plot, is about ΔE ≈ 0.061 eV. For a metallic phase content above 67 at.%, the metallic type of conductivity occurs, for which the derivative dσ/dT < 0. At the frequency of 100 Hz and at liquid nitrogen temperature (LNT) a dependence of conductivity on the content of the metallic phase was shown. In the area of low metallic phase content a rapid increase of conductivity occurs (over six orders of magnitude) and temperature dependence exhibits the dielectric type of conduction. For higher contents of the metallic phase the metallic type of conductivity occurs, which decreases with increasing temperature. Based on the analysis of conductivity depending on the content of the metallic phase, the percolation threshold was determined for the (FeCoZr)x (CaF2)(100−x) nanocomposite, the value of which is xc ≈ (65.8 ± 2) at.%. It was agreed that in nanocomposites below the percolation threshold there are resonance phenomena of currents and voltages similar to the resonance phenomena in RLC circuits containing conventional inductance coils and capacitors.
•Percolation threshold for (FeCoZr)x (CaF2)(100−x) produced by argon ion-beam sputtering.•Frequency and temperature dependence of conductivity, phase shift angle and capacity.•Activation energy of conductivity is about ΔE ≈ 0.061 eV.•Phenomena of current and voltage resonance similar to resonance phenomena of RLC circuits.
Morphological transformation only occurs in polymer blends via changing their components, which can be utilized to form double percolation structures. However, the weight percentage of either polymer ...in the blend generally ranges from 40 wt% to 60 wt%, which does not favor the construction of a densely conductive path. Herein, a double percolation structure had been successfully constructed in a polylactide-polycaprolactone (PLA/PCL) blend with only 30 wt% PCL by adding silicon dioxide (SiO2) to the PLA phase. The added SiO2 effectively strengthens the viscosity and elasticity of the PLA phase, which results in the morphological transformation of the PCL phase from segregated to continuous. In the blend, multi-wall carbon nanotubes (MWCNTs) were selectively distributed in the PCL phase to construct a more efficient conductive network, which brings about an enormous increase in conductivity for the nanocomposites. The electrical conductivity of PLA/PCL nanocomposite increases from 1.19 × 10−14 S/m to 4.57 × 10−4 S/m with the addition of 1.0 wt% MWCNTs and 6.0 wt% SiO2. The percolation threshold of the MWCNTs in the nanocomposite (0.11 wt%, i.e. 0.06 vol%) is 125% lower than that without SiO2 (0.28 wt%, i.e. 0.14 vol%). Furthermore, the nanocomposites with SiO2 were found to exhibit an electromagnetic interference (EMI) of approximately 26 dB with only 3.0 wt% MWCNTs, which means it can block 99.997% of microwaves. This study would provide a critical method to design eco-friendly nanocomposites with ultralow percolate threshold and excellent EMI shielding.
The addition of SiO2 can change the viscosity ratio of PLA:PCL, then the morphology inversion occurs. The PCL (30 wt%) phase change from segregated to continue, and the MWCNTs was selectively dispersed in the PCL phase so that the conductive polymer nanocomposites with low percolation threshold and excellent EMI was achieved. Display omitted
•A low percolation threshold of CPCs with excellent EMI was achieved.•It is first time to use the SiO2 to alter the morphology and form the conductive network.•The MWCNTS was selectively dispersed in the PCL phase (30 wt%).
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