Percolation Phenomenon Research Articles

Network Topology and Percolation in Model Covalent Adaptable Networks.

Incorporating dynamic covalent linkages into thermosets can endow previously unrecyclable materials with new functionality and reprocessing options. Recent work has shown that the properties of the resulting covalent adaptable networks (CANs) are highly dependent on network topology, specifically the phenomenon of percolation, when permanent linkages form a connected skeleton that spans the material. Here, we use a model glassy disulfide based CAN to assess the merits of mean-field percolation theory as a tool to describe the network topology of CANs. After challenging the theory with both experimental data and a coarse-grained molecular dynamics simulation, we find that the mean-field approach is surprisingly accurate, despite its simplifying assumptions. The theory is particularly well suited to the unique context of mixed-composition CANs and provides practical guidance on how to design for reprocessability.

Nanoparticle positioning effect on the percolation phenomenon of the polyethylene oxide/iron nanocomposite films

AbstractIn this research, we demonstrate the influence of compelled iron nanoparticles (FeNPs) up to the surface on the percolation phenomenon and the electrical conductivity of the PEO/FeNPs nanocomposite films by applying a uniform magnetic field. The electrical conductivity of the PEO/FeNPs nanocomposite films exhibits a sharp increase from to 390 S∙cm−1 as the FeNPs volume fraction rises from 0.12 to 0.24, where the FeNPs form percolating networks and interparticle contacts. Applying a magnetic field drives the FeNPs towards the surface of the PEO/FeNPs nanocomposite films and consequently increases the electrical conductivity of the PEO/FeNPs nanocomposite films up to 1960 S∙cm−1. The morphological features of the PEO/FeNPs nanocomposite films with different concentrations of FeNPs upon applying a magnetic field towards the surface were investigated. Finally, the corrosion protection efficiency of PEO/FeNPs nanocomposite films increases as FeNPs increase from an insulating zone to a conductive zone. On the other hand, applying a magnetic field drives the FeNPs towards the PEO/FeNPs nanocomposite surfaces, increasing the corrosion protection efficiency from 92.67% to 98.09%.Highlights The nanoparticle positioning effect on nanocomposite films is investigated. The electrical conductivity of PEO/FeNPs nanocomposite film reaches 390 S∙cm−1. Electrical conductivity increases to 1960 S∙cm−1 upon nanoparticle positioning. Corrosion protection efficiency increases to 98.09% after nanoparticle positioning.

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.

Abnormal resistivity behavior of Co–Cu alloys and responsible metastable liquid phase separation

The electrical resistivity of Co-Cu alloys with Cu content of 10, 20, 30, 40, 50, 60, 70, 80, and 90 at.% was measured in the temperature range from 1400 to 2100K. The experiments were performed during of heating and subsequent cooling of the samples. For alloys with Cu content of 30, 40, 60, 70, 80, and 90 at.%, a bend in the cooling curve specific to the liquid-liquid phase separation (LLPS) process was observed. LLPS is shown when the melt is undercooled to the binodal temperature T∗. The phase transition model of LLPS was proposed for to theoretically determine the temperatures T∗. The LLPS model demonstrates the possibility of percolation phenomena occurring in Cu-Co heterogeneous melts rich in cobalt. The LLPS model shows good agreement with experimental data and the abnormal resistivity behavior of Co–Cu alloys.

Efficient percolation simulations for lossy photonic fusion networks

The study of percolation phenomena has various applications ranging from social networks or materials science to quantum information. The most common percolation models are bond or site percolation for which the Newman-Ziff algorithm enables an efficient simulation. Here, we consider several nonstandard percolation models that appear in the context of measurement-based photonic quantum computing with so-called graph states and fusion networks. The associated percolation thresholds determine the tolerance to photon loss in such systems and we develop modifications of the Newman-Ziff algorithm to efficiently perform the corresponding percolation simulations. We demonstrate our algorithms by using them to characterize exemplary fusion networks and graph states. Published by the American Physical Society 2024

A hydrophilic–lipophilic deviation model in formulating microemulsion for topical application

The hydrophilic–lipophilic deviation (HLD) model is firmly established in oil recovery and can be potentially applied in various other fields. In this work, the HLD concept is systematically used to prepare microemulsion for topical application using isopropyl myristate (IPM), squalane, water, mixed nonionic surfactants and magnesium ascorbyl phosphate (MAP). The clear water-in-oil (W/O) microemulsion formed spontaneously and has a consistent submicron droplet size below 30 nm. The microemulsion was characterized by particle size, electrical conductivity, electrophoretic mobility, viscosity and in vitro diffusion. The addition of one part of squalane to three parts of IPM in the oil blend prevented the percolation phenomenon and reduced the electrophoretic mobility of MAP. The inclusion of squalane to IPM in the oil phase reduced the cumulative amount of MAP permeated from the microemulsion, reduced the permeation coefficient of MAP and increased the lag time, reduced the steady-state flux and permeability coefficient as well. The microemulsion provided an alternative approach to encapsulate the water-soluble MAP in an oil matrix as a cosmetic active ingredient for the manufacture of personal care products.

On the connectedness of arithmetic hyperplanes

Discrete geometry is a geometry specific to computers that studies Zd structures. It appears naturally in image analysis or 3D printing. Our goal is to find efficient algorithms to characterise these geometric structures and their properties.We are interested in a fundamental structure of discrete geometry, the arithmetic hyperplanes, and more particularly in their connectedness. Many works have studied a connectedness defined from the neighbourhood by faces and have allowed to observe a percolation phenomenon. These studies have also allowed to decide the connectedness of a plane in an efficient way. We propose an extension of these results in the case of connectivity defined from general neighbourhoods.Beyond the new concepts that this extension requires, the main contribution of the paper lies in the use of analysis to solve this arithmetic problem and in the design of an algorithm that decides the general connectedness problem. The study of the thickness of connectedness as a function reveals discontinuities at each rational point. However, a much more regular underlying structure appears in the irrational case. Thus, the consideration of irrational vectors allows a simpler approach to the connectedness of arithmetic hyperplanes.

Percolation-Triggered Negative Permittivity in Nano Carbon Powder/Polyvinylidene Fluoride Composites.

Percolating composites exhibiting negative permittivity have garnered considerable attention due to their promising applications in the realm of electromagnetic shielding, innovative capacitance devices, coil-less inductors, etc. Nano carbon powder/polyvinylidene fluoride (CP/PVDF) percolating composites were fabricated that exhibit Drude-type negative-permittivity behavior upon reaching the CP percolation threshold. This phenomenon is attributed to the formation of a plasmonic state within the interconnected CP network, enabling the delocalization of electrons under the alternating electric field. Furthermore, a significant (nearly two orders of magnitude) increase in the conductivity of sample is observed at a CP content of 12.5 wt%. This abrupt change coincides with the percolation phenomenon, suggesting a transition in the conduction mechanism. To elucidate this behavior, comprehensive analyses of the phase composition, microstructure, AC conductivity, and relative permittivity were performed. Additionally, the sample containing 5 wt% CP exhibits a remarkably high permittivity of 31.5, accompanied by a relatively low dielectric loss (tanδ < 0.2). The findings expand the potential applications of PVDF, while the fabricated percolating composites hold promise for electromagnetic shielding, antennas, and other electromagnetic devices.

Unveiling the impact of percolation phenomenon on the microstructure of polymer nanocomposites based on the combination of scaling and percolation theories

AbstractIn this study, a model was proposed to interpret the tensile strength of the nanocomposite systems based on the percolation and scaling theories. Using the Excluded Volumes method made it possible to characterize the reaction mechanism of the system by discretizing all sections of the system in the best way possible. Accordingly, the system was divided into two sections A and B, consisting of either fully dispersed, aggregated, or agglomerated nanoparticles using a percolation parameter (F). The stress concentration factor was introduced to the model based on the available surface area determining the share of each section in affecting the tensile strength of the system. A validation procedure was performed to verify the obtained results using the experimental results of our recent work for a High‐density polyethylene nanocomposite system containing 0.5, 0.75, and 2 wt% of the surface‐modified silica nanoparticles. The investigations indicate the capability of the proposed model to interpret the physical/mechanical characteristics of the nanocomposite systems. It was found that despite the negative impact of the agglomeration on the tensile strength of the system, the aggregation phenomenon enhanced the tensile strength due to the formation of strong networks in its structure.Highlights Interpreting the tensile strength of nanocomposites via excluded volume theory. Dividing the impacts of dispersed and clustered nanoparticles. Discretizing the effects of aggregation and agglomeration on final properties.

Integration of inter-ply electrical percolation phenomena in the multiphysics modelling of laminated composite materials

PurposeThis study aims to introduce a predictive homogenization model incorporating electrical percolation considerations to forecast the electrical characteristics of unidirectional carbon-epoxy laminate composites.Design/methodology/approachThis study presents a method for calculating the electrical conductivity tensor for various ply arrangement patterns to elucidate phenomena occurring around the interfaces between plies. These interface models are then integrated into a three-dimensional (3D) magneto-thermal model using the finite element method. A comparative study is conducted between different approaches, emphasizing the advantages of the new model through experimental measurements.FindingsThis research facilitates the innovative integration of electrical percolation considerations, resulting in substantial improvement in the prediction of electrical properties of composites. The validity of this improvement is established through comprehensive validation against existing approaches and experimentation.Research limitations/implicationsThe study primarily focuses on unidirectional carbon-epoxy laminate composites. Further research is needed to extend the model's applicability to other composite materials and configurations.Originality/valueThe proposed model offers a significant improvement in predicting the electrical properties of composite materials by incorporating electrical percolation considerations at inter-ply interfaces, which have not been addressed in previous studies. This research provides valuable information to improve the accuracy of predictions of the electrical properties of composites and offers a methodology for accounting for these properties in 3D magneto-thermal simulations.

Piezoresistive properties of well-dispersed carbon nanotubes (CNTs) modified cement paste

When developing smart cement-based sensors with carbon nanotubes (CNTs) as conductive fillers, CNTs with good dispersion can enhance the conductivity and piezoresistive properties of cement-based materials. In this study, a combination of sonication and dispersant was used to disperse CNTs, and then investigated the piezoresistive properties of well-dispersed CNTs modified cement paste. The experimental results showed that the dispersant used was polyvinylpyrrolidone (PVP), and when the mass ratio of PVP to CNTs was 4:1, the sonication time was 25 min, and the standing time did not exceed 3 h, CNTs had higher dispersion quality. The resistivity of the cement paste decreased by two orders of magnitude with increasing CNTs dosage and exhibited visible percolation phenomenon with percolation thresholds ranging from 0.1 % to 0.3 %. The fractional change in resistivity, stress sensitivity and strain sensitivity of cement paste under cyclic loading reached the maximum values of 10.95 %, 1.146 %/MPa and 123.450 %, respectively, with high repeatability (92.63 %) and low hysteresis (8.40 %) when the content of CNTs was 0.4 %. At the same time, CNTs improved the microstructure of the cement paste, which positively affected the mechanical, electrical conductivity and piezoresistive properties of the material.

Sokoban percolation on the Bethe lattice

Abstract `With persistence, a drop of water hollows out the stone' goes the ancient Greek proverb. Yet, canonical percolation models do not account for interactions between a moving tracer and its environment. Recently, we have introduced the Sokoban model, which differs from this convention by allowing a tracer to push single obstacles that block its path. To test how this newfound ability affects percolation, we hereby consider a Bethe lattice on which obstacles are scattered randomly and ask for the probability that the Sokoban percolates through this lattice, i.e., escapes to infinity. We present an exact solution to this problem and determine the escape probability as a function of obstacle density. Similar to regular percolation, we show that the escape probability undergoes a second-order phase transition. We exactly determine the critical obstacle density at which this transition occurs and show that it is higher than that of a tracer without obstacle-pushing abilities. Our findings assert that pushing facilitates percolation on the Bethe lattice, as intuitively expected. This result, however, sharply contrasts with our previous findings on the 2D square lattice, where the Sokoban cannot escape even at obstacle densities well below the regular percolation threshold. This indicates that the presence of a regular percolation transition does not guarantee a percolation transition for a pushy tracer. The stark contrast between the Bethe and 2D lattices also highlights the significant impact of network topology on the effects of obstacle pushing and underscores the necessity for a more comprehensive understanding of percolation phenomena in systems with tracer-media interactions.

Polyolefin elastomer/cyclic olefin copolymer/multi‐walled carbon nanotubes nanocomposites: Rheological, morphological, electrical and mechanical study

AbstractThe subject of this research is a relatively detailed investigation of the rheological, morphological and electrical properties of nanocomposites based on polyolefin elastomer (POE), cyclic olefin copolymer (COC) and multi‐walled carbon nanotubes (MWCNTs). To formulate a mixture of two polymers (COC and POE), the melt mixing method was used and their morphology was studied by SEM technique and it was found that the morphology is of co‐continuous type. When the weight percentage ratio of COC was between 40 and 60, the desired morphology (co‐continuous) appeared. Interfacial tension calculations and TEM images proved that the nanotube particles were stabilized both within the phase of POE as well as at the interface of the two polymer phases. The rheological behavior was analyzed for composite samples containing 0–3 wt % MWCNT. The electrical and rheological percolation thresholds appeared at 1 and a 0.5 percent by weight of nanotube particles, respectively, which may be better to consider as related to the phenomenon of dual percolation. The highest conductivity was obtained for the nanocomposite containing 3% MWCNT. The results of mechanical tests also showed a strong upward trend of properties due to the addition of nanotube.Highlights The effect of COC content on the morphology of POE/COC blend was investigated. The Morphology for the POE/COC blend at low COC content was droplet‐matrix type. MWCNT has a greater tendency to be in the POE phase than COC. Rheological behavior strongly depended on the amount of nanotube particles. The highest conductivity was obtained for the sample containing 3% MWCNT.

Photoconductivity of explosive percolation in conductive polymer/graphene oxide nanocomposite films

AbstractIn this study, we explored the behavior of protonated polyaniline/graphene oxide (PANI‐CSA/GO) nanocomposite films with varying GO concentrations, focusing on the novel phenomenon of explosive percolation. We observed a significant increase in electrical conductivity at the explosive percolation threshold, attributed to the emergence of a percolating metallic pathway. This discovery positions PANI‐CSA/GO films as promising materials for various electronic and electrical engineering applications. Additionally, the films demonstrated consistent and repeatable photoconductivity, showing potential for use in high‐performance UV‐photodetectors, photoactive layers in solar cells, light‐emitting diodes, and energy storage devices. Structural analyses using fourier transform infrared spectroscopy (FTIR) and x‐ray diffraction (XRD) confirmed successful GO incorporation within the PANI‐CSA matrix. Different morphological features were observed depending on the GO volume fraction, with increased GO enhancing thermal stability in the conductive zone. Our findings highlight the immense potential of PANI‐CSA/GO nanocomposite films in advanced electronic applications, emphasizing their novel conductive and photoconductive properties and improved thermal stability.

Design of interconnected graphene loaded thermoplastic elastomeric blend composite films for minimizing electromagnetic radiation and efficient heat management

AbstractIn this work, electrically conductive thermoplastic elastomeric blend composite films based on polystyrene (PS)/ethylene‐co‐methyl acrylate (EMA) filled with functionalized graphene were developed via the solution mixing technique. Morphological analysis revealed that selective localization of amine‐functionalized reduced graphene oxide (G‐ODA) sheets in the EMA phase of co‐continuous binary blend formed a well‐connected dense conductive pathway by graphene sheets ultimately facilitating the double percolation phenomenon. The electrical percolation threshold was achieved at ~2 wt% of G‐ODA loading which was much lower than that for both single polymer composites. An electrical conductivity of 0.9 S/cm was obtained for blend composite film with 10 wt% of graphene concentration whereas for the same filler loading, PS and EMA composites exhibited electrical conductivity of 1.9 × 10−1 and 2.3 × 10−1 S/cm, respectively. The obtained thermal conductivity of the blend composite with 10 wt% of G‐ODA loading was 0.95 W/m K with 400% enhancement compared to the neat blend system. The same composite exhibited increased real and imaginary permittivity of 92 and 83, respectively. The electrical percolation threshold is well‐correlated with the percolation concentration found from storage modulus and thermal conductivity data. The fabricated PS/EMA blend composite film exhibited absorption‐dominant electromagnetic interference SE of −25 and − 35 dB in X‐band frequency (8.2–12.4 GHz) for 10 wt% of graphene loading with a sample thickness of 0.5 and 1 mm, respectively.

DC assisted electric field on the improvement of the nonlinear electrical conductivity of SiC/LDPE field grading composites

Nonlinear resistive field grading material presents nonlinear electrical conductivity, and self-homogenizes the electric field distribution. The nonlinear conductivity is closely associated to the filler distribution, which can be modulated by the electric field assist during the material preparation. In this paper, the effects of the direct current (DC) electric field assist on the filler distribution and nonlinear conductivity of the nonlinear resistive field grading materials were studied for the first time. The silicon carbide (SiC)/low-density polyethylene (LDPE) composites were prepared under different DC assisted electric field. The composites’ microstructure, dielectric spectra and DC conductivity were characterized, respectively. Results illustrate that the DC electric field assist leads to the gradient distribution of SiC filler in the matrix, and also contributes to the SiC particles’ chains. Compared with the untreated SiC/LDPE composite, the electric-field assisted composites show more distinct nonlinear conductivity characteristics when the DC assisted electric field exceeds 0.3 kV mm−1, which is attributed to the percolation phenomenon of SiC particles. This work is helpful to design the dielectric functionally graded materials, and to self-homogenize the electric field distribution in power apparatus and electronic devices.

Fabrication of a novel high electrical conductivity Si3N4 matrix composite with Cu three-dimensional network structure by spark plasma sintering

Silicon nitride (Si3N4) ceramic matrix conductive composite materials have shown great promise as conductive layer materials for electrical transmission components. However, existing conductive phases struggle to form a three-dimensional (3D) interconnected network in Si3N4 matrix, resulting in poor electrical conductivity. This study proposed a spark plasma sintering (SPS) process utilizing Si3N4 as the substrate and Cu particles as the reinforcement phase to fabricate a novel electrical conductivity Si3N4/Cu composite material. The results indicated that the diffusion of Si atoms in Si3N4 facilitated the formation of copper silicide (CuiSi) interface between the two constituents during sintering, creating a strong chemical bonding for high conductivity. Simultaneously, composite materials with optimized Cu content formed a 3D interconnect network structure, providing a continuous path for electrical conduction. At Cu content of 30 vol%, the Si3N4/Cu composite exhibited a satisfying electrical conductivity of 295.37 S/m, which was 14 orders of magnitude higher than that of Si3N4. The composites also demonstrated a percolation phenomenon, with a theoretical percolation threshold of Cu particles at just 0.1 vol%, an order of magnitude lower than that of carbon reinforcement particles. Furthermore, an integrated design featuring external insulation and internal conductivity was achieved by wrapping a Si3N4 insulation layer around the conductive Si3N4/Cu composites. The fabricated insulation layer exhibited higher resistivity (1.12×1014 Ω) and a lower wear rate (1.3×10−6 mm3/N·m) compared to some contemporary insulation ceramics, while the conductive layer had a lower calorific value, making them excellent candidates for electrical transmission component materials.

Rational design of high‐performance epoxy/expandable microsphere foam with outstanding mechanical, thermal, and dielectric properties

AbstractPrevious studies on epoxy foams were subjected to the specific property, and improvements in thermal insulation or dielectric property were usually accompanied by the sacrifice of mechanical strength or thermal stability. The current research reported a high‐performance epoxy foam by the pre‐curing process using E‐51/DDS and expandable microsphere (EMP) as polymer matrix and physical blowing agent, respectively. The effect of EMP loading on the structure &amp; property of epoxy foam was reported. The mean diameter of E‐51/EMP foam was in the range of 19–21 μm. The compression property of E‐51/EMP foam was fit for the power‐law model and its integrity was well maintained during compressive test. Compared with bulky epoxy, a significant decrease of thermal conductivity of 62.93% was observed in 10 wt% EMP‐filled epoxy foam. E‐51/EMP foam showed superior thermal stabilities, with a T5% of over 300°C. The dielectric constant was decreased by 54.5% from 4.92 in bulky epoxy to 2.24 in E‐51/EMP foam (10 wt% EMP). Notably, there was a percolation phenomenon of EMP (6 wt%) regarding on the structure &amp; property of E‐51/EMP foam. This study provides lightweight, robust, high‐temperature‐resistant, and low‐dielectric‐constant epoxy foams with the wide application prospects in buoyant, electronic packaging, and energy‐saving industries.

Percolation connectivity in deposits obtained usingcompetitive random sequential adsorption of binarydisk mixtures

Connectedness percolation phenomena in the two-dimensional (2D) packing of binary mixtures of disks with different diameters were studied numerically. The packings were produced using random sequential adsorption (RSA) model with simultaneous deposition of disks. The ratio of the particle diameters was varied within the range D=1-10, and the selection probability of the small disks was varied within the range 0-1. A core-shell structure of the particles was assumed for the analysis of connectivity. The packing coverages in a jamming state for different components, connectivities through small, large and both types of disks, the behavior of electrical conductivity were analyzed. The observed complex effects were explained accounting for the formation of conductive "bridges" from small disks in pores between large disks.

Universality of random-site percolation thresholds for two-dimensional complex noncompact neighborhoods.

The phenomenon of percolation is one of the core topics in statistical mechanics. It allows one to study the phase transition known in real physical systems only in a purely geometrical way. In this paper, we determine thresholds p_{c} for random-site percolation in triangular and honeycomb lattices for all available neighborhoods containing sites from the sixth coordination zone. The results obtained (together with the percolation thresholds gathered from the literature also for other complex neighborhoods and also for a square lattice) show the power-law dependence p_{c}∝(ζ/K)^{-γ} with γ=0.526(11), 0.5439(63), and 0.5932(47), for a honeycomb, square, and triangular lattice, respectively, and p_{c}∝ζ^{-γ} with γ=0.5546(67) independently on the underlying lattice. The index ζ=∑_{i}z_{i}r_{i} stands for an average coordination number weighted by distance, that is, depending on the coordination zone number i, the neighborhood coordination number z_{i}, and the distance r_{i} to sites in the ith coordination zone from the central site. The number K indicates lattice connectivity, that is, K=3, 4, and 6 for the honeycomb, square, and triangular lattice, respectively.