Concepts Of Percolation Theory Research Articles

Non‐Linearity in Mean Annual Peak Flow Scaling With Upstream Basin Area: Insights From Percolation Theory

ABSTRACTUnderstanding how annual peak flow, , relates to upstream basin area, , and their scaling have been one of the challenges in surface hydrology. Although a power‐law scaling relationship (i.e., ) has been widely applied in the literature, it is purely empirical, and due to its empiricism, the interpretation of its exponent, , and its variations from one basin to another is not clear. In the literature, different values of have been reported for various datasets and drainage basins of different areas. Invoking concepts of percolation theory as well as self‐affinity, we derived universal and non‐universal scaling laws to theoretically link to . In the universal scaling, we related the exponent to the fractal dimensionality of percolation, (i.e., ). In the non‐universal scaling, in addition to , the exponent was related to the Hurst exponent, , characterizing the boundaries of the drainage basin (i.e., ). The depends on the dimensionality of the drainage system (e.g., two or three dimensions) and percolation class (e.g., random or invasion percolation). We demonstrated that the theoretical universal and non‐universal bounds were in well agreement with experimental ranges of reported in the literature. More importantly, our theoretical framework revealed that greater values are theoretically expected when basins are more quasi two‐dimensional, while smaller values when basins are mainly quasi three‐dimensional. This is well consistent with the experimental data. We attributed it to the fact that small basins most probably display quasi‐two‐dimensional topography, while large basins quasi‐three‐dimensional topography.

Estimating the scale dependence of permeability at pore and core scales: Incorporating effects of porosity and finite size

Understanding the effect of sample size on hydraulic characteristics of porous media and estimating their scale-dependent properties have been challenging in subsurface hydrology. An empirical power-law relationship has been widely applied to link permeability, k, to sample volume. However, due to its empiricism, accurately estimating the scale dependence of permeability, k(L), in soils and rocks has remained unanswered. In this paper, we combine concepts of percolation theory and finite-size effect, develop a theoretical relationship to model k(L), and evaluate the proposed model using various types of porous media. To estimate k(L), one requires porosity, critical porosity, critical scaling exponent, a fundamental length scale, system length, and permeability determined at the smallest scale. Comparison with numerical simulations shows that incorporating effects of porosity and finite size results in accurate estimates of k(L). We find reasonable agreement between the theory and the simulations with average relative error ranged between -27.5% and 10.2%.

On Structural Rearrangements Near the Glass Transition Temperature in Amorphous Silica

The formation of clusters was analyzed in a topologically disordered network of bonds of amorphous silica (SiO2) based on the Angell model of broken bonds termed configurons. It was shown that a fractal-dimensional configuron phase was formed in the amorphous silica above the glass transition temperature Tg. The glass transition was described in terms of the concepts of configuron percolation theory (CPT) using the Kantor-Webman theorem, which states that the rigidity threshold of an elastic percolating network is identical to the percolation threshold. The account of configuron phase formation above Tg showed that (i) the glass transition was similar in nature to the second-order phase transformations within the Ehrenfest classification and that (ii) although being reversible, it occurred differently when heating through the glass–liquid transition to that when cooling down in the liquid phase via vitrification. In contrast to typical second-order transformations, such as the formation of ferromagnetic or superconducting phases when the more ordered phase is located below the transition threshold, the configuron phase was located above it.

Modeling and Characterization of the Ionic Conductivity of Proton-Exchange Membranes Based on Multiblock Copolymers of Sulfonated PSU/Ppsu Poly(Ether Sulfone)s

In this work, the ionic conductivity of proton-exchange membranes (PEMs) based on sulfonated multiblock copolymers composed of Polysulfone (PSU) and Polyphenylsulfone (PPSU) poly(ether sulfone) segments (SPSU/SPPSU) is evaluated. The copolymers were synthesized for the first time by polycondensation in a “one-pot two-step synthesis” of commercial monomers, followed by sulfonation reaction with trimethylsilyl chlorosulfonate (TMSCS) [1].The morphology of the membranes was examined by Field Emission Scanning Electron Microscopy (FE-SEM). Electrochemical Impedance Spectroscopy (EIS) was employed to measure the ionic conductivity of the membranes. The FE-SEM analysis of the samples revealed a random nanostructure of hydrophilic/hydrophobic domains with high and low content of ionic groups, respectively. Thus, no microphase separation was observed, even though the PSU block showed a greater affinity to be sulfonated. The ionic conductivity and water uptake of membranes with three different degrees of sulfonation, DS=0.45, 0.70 and 0.79, were characterized at 80 oC and relative humidities ranging from RH=10% to RH=100%. For a given RH, the ionic conductivity increased non-linearly with DS, showing a strong rise when DS was varied from 0.45 to 0.70, although the water uptake of the membranes remained nearly the same. In contrast, the increase of the ionic conductivity between DS=0.70 and DS=0.79 was significantly lower, but the water uptake increased sharply.The behavior of the membranes was modeled using percolation theory concepts. The numerical model was implemented in the finite volume-based code ANSYS Fluent [2,3]. The nanostructure of the membranes was divided into three types of sites in a random cubic network: sulfonated and well-hydrated sites, sulfonated and weakly hydrated sites and non-sulfonated sites (see Figure 1). The volume fraction of sulfonated sites was varied according to the DS of the samples, while the relative volume fraction of hydrated sites (i.e., the ratio between the volume fraction of hydrated sites and the volume fraction of sulfonated sites) was varied depending on RH. Isolated, hydrated sites not connected to the edges of the domain were removed from the network, since water was assumed to form a continuous network connected to the edges of the membrane according to the experimental conditions.Good agreement was found between the experimental data and numerical results, thus providing a fundamental explanation of the behavior of the multiblock copolymer membranes. The validated model will be used in future work to assist the design of high-performance and durable multiblock copolymer membranes for proton-exchange fuel cells, and related electrochemical devices.[1] N. Ureña, M.T. Pérez-Prior, C. del Río, A. Várez, J.Y. Sánchez, C. Iojoiu, B. Levenfeld, Multiblock copolymers of sulfonated PSU/PPSU Poly(ether sulfone)s as solid electrolytes for proton exchange membrane fuel cells, Electrochim. Acta 302 (2019) 428-440.[2] P.A. García-Salaberri, I.V. Zenyuk, J.T. Gostick, A.Z. Weber, Modeling Gas Diffusion Layers in Polymer Electrolyte Fuel Cells Using a Continuum-based Pore-network Formulation, ECS Trans. 97 (7) 615.[3] P.A. García-Salaberri, Modeling diffusion and convection in thin porous transport layers using a composite continuum-network model: Application to gas diffusion layers in polymer electrolyte fuel cells, Int. J. Heat Mass Trans. (2020), accepted.. Figure 1. Random cubic network composed of sulfonated and well-hydrated sites, sulfonated and weakly hydrated sites and non-sulfonated sites used to model the ionic conductivity of the multiblock copolymer membranes.. Figure 1

Comparing the power law constant (n) for mono- and bi-dispersed filled slurries: using percolation theory concepts

There is an enduring interest in concentrated particulate filled slurries in many industries for a wide variety of products and processes. In previous research, a collection of experimental procedures provided velocity profiles, pressure drops and power law constants for slurries flowing in a tube. A model previously developed for the power law, based on mono-disperse fillers and on percolation theory concepts, is reviewed and extended to bi-modal slurries. The model for the power law constant, n, is shown to be only a function of the volume of non-dissipating clusters in the slurry. Experimental results are used to compare the effects of mono-modal and bi-modal fillers on the viscosity and n in the composite flow of filled Newtonian carrier fluids. The data and analysis demonstrate that n can be related to the ratio of energy dissipated by the slurry divided by the dissipation of a Newtonian fluid flowing under similar conditions. The power law parameters were developed using Bingham model constants employed to approximate dissipation in tube flow. The power law constant was found to be smaller for the bi-modal systems than for mono-modal slurries at the same total filler concentration.

Modelling of metallic particle binders for increased part density in binder jet printed components

Binder jet printed components typically have low overall density in the green state and high shrinkage and deformation after heat treatment. It has previously been demonstrated that, by including nanoparticles of the same material in the binder, these properties can be improved as the nanoparticles can fill the interstices and pore throats between the bed particles. The beneficial effects from using these additive binder particles can be improved by maximising the binder particle size, enabling the space within the powder bed to be filled with a higher packing efficiency. The selection of maximum particle size for a binder requires detailed knowledge of the pores and pore throats between the powder bed particles. In this paper, a raindrop model is used to determine the critical radius at which binder particles can pass between pores and penetrate the bed. The model is validated against helium psychometry measurements and binder particle drop tests. It is found that the critical radius can be predicted, with acceptable accuracy, using a linear function of the mean and standard deviation of the particle radii. Percolation theory concepts have been employed in order to generalise the results for powder beds that have different mean particle sizes and size distributions. The results of this work can be employed to inform the selection of particle sizes required for binder formulations, to optimise density and reduce shrinkage in printed binder jet components.

Predicting Water Cycle Characteristics from Percolation Theory and Observational Data

The fate of water and water-soluble toxic wastes in the subsurface is of high importance for many scientific and practical applications. Although solute transport is proportional to water flow rates, theoretical and experimental studies show that heavy-tailed (power-law) solute transport distribution can cause chemical transport retardation, prolonging clean-up time-scales greatly. However, no consensus exists as to the physical basis of such transport laws. In percolation theory, the scaling behavior of such transport rarely relates to specific medium characteristics, but strongly to the dimensionality of the connectivity of the flow paths (for example, two- or three-dimensional, as in fractured-porous media or heterogeneous sediments), as well as to the saturation characteristics (i.e., wetting, drying, and entrapped air). In accordance with the proposed relevance of percolation models of solute transport to environmental clean-up, these predictions also prove relevant to transport-limited chemical weathering and soil formation, where the heavy-tailed distributions slow chemical weathering over time. The predictions of percolation theory have been tested in laboratory and field experiments on reactive solute transport, chemical weathering, and soil formation and found accurate. Recently, this theoretical framework has also been applied to the water partitioning at the Earth’s surface between evapotranspiration, ET, and run-off, Q, known as the water balance. A well-known phenomenological model by Budyko addressed the relationship between the ratio of the actual evapotranspiration (ET) and precipitation, ET/P, versus the aridity index, ET0/P, with P being the precipitation and ET0 being the potential evapotranspiration. Existing work was able to predict the global fractions of P represented by Q and ET through an optimization of plant productivity, in which downward water fluxes affect soil depth, and upward fluxes plant growth. In the present work, based likewise on the concepts of percolation theory, we extend Budyko’s model, and address the partitioning of run-off Q into its surface and subsurface components, as well as the contribution of interception to ET. Using various published data sources on the magnitudes of interception and information regarding the partitioning of Q, we address the variability in ET resulting from these processes. The global success of this prediction demonstrated here provides additional support for the universal applicability of percolation theory for solute transport as well as guidance in predicting the component of subsurface run-off, important for predicting natural flow rates through contaminated aquifers.

Investigation of the Effect of Filler Concentration on the Flow Characteristics of Filled Polyethylene Melts

Abstract All polymeric slurries that have a high concentration of filler are shear thinning. Shear thinning is an important characteristic of polymers, filled and unfilled, because it enables an increase in the throughput, shear rate in a die or an injection molding system without having to use substantially more power to increase the flow rate. Newtonian fluid-based slurries show an increase in shear thinning as the concentration of “filler” increases above the percolation threshold. As particle maximum packing concentration is approached the slurries begin to approach a perfect pseudoplastic fluid. In some cases, the shear thinning characteristics of a filled polymer do not increase substantially as the filler loading is increased. This is a quite different response than in Newtonian fluid-based slurry. Therefore, it is important to understand the materials engineering interactions that control shear thinning so that process flow models can better predict the performance of filled polymer systems. Highly filled polymers can have processing issues, including high screw shaft torque, energy consumption, die pressure and melt temperature rise. Previous theoretical developments and experimental evaluations of highly filled polymer melts showed that the rheology can be effectively described with a percolation model. In this work, capillary rheometer measurements using several low-density polyethylene resins, calcium carbonate and titanium dioxide fillers are reported using percolation theory concepts. The theoretical treatment of the rheology as a particulate percolating system with power-law behavior is used to analyze capillary rheometer data. The observed effects of resin molecular weight, filler type and size on rheology are described. Engineers that design and debottleneck polymer processes need to utilize the polymer viscosity at the minimum process shear rate to determine the smallest motor that will allow the process to run; in addition, the shear thinning characteristics of the polymer are used to indicate how much increased production may be possible with a given motor size. Thus, some examples of expected effects on melt processing are also presented.

A Study of Concentrated Suspensions in Polyethylene Melts and the Impact on Viscosity and Polymer Processing Operations

AbstractFilled polymer compounds can present processing challenges, including poor dispersive mixing, high screw shaft torque and energy consumption, flow instabilities, and significant increases in melt pressures and temperatures. Previous theoretical development and experimental evaluations of polymer melts filled with particulate solids showed that the viscosity can be described with a model based on percolation theory concepts. This paper describes a batch mixer characterization method developed to measure the effects of filler concentration and operating conditions on the melt viscosity and process response. Experimental results are compared with capillary rheometer measurements using several low-density polyethylene resins, calcium carbonate and titanium dioxide. The theoretical treatment of the viscosity as a particulate percolating system with power-law behavior was used to analyze and compare rheometer and batch mixer data. The effects of resin molecular weight, filler type and size on viscosity and melt processing were examined and there was good agreement between rheometer and mixer viscosity trends, and percolation model predictions. The laboratory experiments and model estimates provide quantitative information that can be used to develop and troubleshoot extrusion compounding, single screw extrusion and low-shear forming processes.

Newtonian, power law, and infinite shear flow characteristics of concentrated slurries using percolation theory concepts

There is a strong interest today in concentrated particulate-filled dispersion and slurries in both polymeric and Newtonian fluids. This paper reviews and extends theoretical approaches using percolation theory concepts to characterize the rheological behavior of fluids filled with particulate solids. First, a previously proposed limiting, zero shear viscosity model based on percolation theory concepts is reviewed. This model has been primarily tested with rigid fillers in a Newtonian carrier and polymeric fluids. Second, all Newtonian fluid-based slurries that have a high concentration of filler become pseudoplastic, shear-thinning slurries at some threshold shear rate. A new theory is reviewed and new data are evaluated that correlate the power law constant, n, to cluster formation of the fillers suspended in the fluids in shear flow. Slurry systems reported here cover a size range from 58 nm to 200 μm. Third, this cluster percolation-based rheological analysis is then extended to a newly proposed model for the calculation of the ratio of infinite shear, η∞, to the zero shear viscosity, η0. Using literature data, it is demonstrated that measurements of the viscosity ratio, η∞/η0, correlate with the power law through the use of an energy dissipation-based model for Bingham rheological fluids.

Development of a predictive power law relationship for polymer composites based on newtonial carrier concentrated slurries

A new theory that predicts the power law “n” for Newtonian carrier particulate composites has been developed using percolation theory concepts. Many polymer composites, such as sheet molding compound, polyurethanes based on polymer‐polyols, and highly filled epoxy compounds, are generated using highly filled Newtonian carriers. Essentially, all of slurries that have a high concentration of filler become shear thinning at some shear rate. Several sizes of calcium carbonate were initially investigated because of their extensive use in many industries as fillers. A new function was found that linearly correlates the power law constant, n, to the concentration of the filler above the percolation concentration. The behavior of this function suggests that the Newtonian to power law behavior may be dominated by percolation processes. A new percolation theory based model was developed that theoretically predicted the power law from the volume of the nondissipating clusters. A theory is proposed that relates the plug volume in tube flow to the power law, n. The central plug in tube flow was experimentally shown to be a nondissipating volume in the flowing slurry. Data were taken in Poiseuille flow that correlates the power law, n, calculated in the classical manner using pressure drop along the tube with the power law, n, calculated from the percolation‐based theory. POLYM. COMPOS., 39:1172–1191, 2018. © 2016 Society of Plastics Engineers

Invited: Pore Network Modeling of the Full Membrane Electrode Assembly of a Polymer Electrolyte Membrane Fuel Cell

(PEMFCs) are a major component of a sustainable energy economy. Their high energy and power density make them uniquely capable of replacing the internal combustion engine. The porous electrode in the PEMFC must be designed to withstand the presence of liquid water, which is a by-product of the electrochemical reaction. Ensuring high rates of gaseous reactant transport to the catalyst layer is essential to producing high power density, efficient and cost effective cells. Consequently, the study of liquid water behavior in the porous electrodes is of extreme interest. A very large number of numerical models have been published, attempting to use multiphase flow models in PEMFC and computational fluid dynamics packages based on continuum mechanics to optimize the electrode. There are several limitations to the continuum approach that will be discussed. The most obvious is that the transport properties must be measured experimentally, and then input into the computation as constitutive relationships. This is problematic for properties are difficult to measure, such as effective diffusivity in partially water saturated media [1, 2], gas-liquid surface area, etc. An alternative modeling approach that is receiving increased interest is pore network modeling (PNM). In PNMs, the media is mapped as a set of interconnected pores and throats, transport is modeled as a resistor network and capillary behavior is modeled using percolation theory concepts. In this paradigm there is no difficulty modeling the impact of multiphase flow, and importantly, PNMs do not require experimentally measured transport parameters are constitutive relationships. A pore network model produced using the OpenPNM package has been developed to simulate the impact of mass transfer through the GDL network on the electrochemical kinetics and fuel cell operation. OpenPNM is an open source framework implemented in Python. This framework is capable of building a porous structure, applying pore-scale physics, and simulating numerous algorithms on pore network models. In this work, constant voltage boundary conditions have been applied to the catalyst layer to predict overvoltage in proton exchange fuel cell systems, as shown by the polarization curve in Figure 1. As can be seen, the mass transfer losses place an upper limit on the maximum current that can be generated in a cell. Only mass transfer losses in the GDL are considered, but there is no theoretical reason why the catalyst layer can cannot be included in future work. This approach can be extended to include ionic losses in the membrane phase as well. The eventual goal is to provide a fully viable alternative to the continuum model approach.

Fractal approach to the description of the auroral region

The plasma of the auroral region, where energetic particles precipitate from the magnetosphere into the ionosphere, is highly inhomogeneous and nonstationary. In this case, traditional methods of classical plasma physics turn out to be inapplicable. In order to correctly describe the dynamic regimes, transition processes, fluctuations, and self-similar scalings in this region, nonlinear dynamics methods based of the concepts of fractal geometry and percolation theory can be used. In this work, the fractal geometry and percolation theory are used to describe the spatial structure of the ionospheric conductivity. The topological properties, fractal dimensions, and connective indices characterizing the structure of the Pedersen and Hall conductivities on the nightside auroral zone are investigated theoretically. The restrictions imposed on the fractal estimates by the condition of ionospheric current percolation are analyzed. It is shown that the fluctuation scalings of the electric fields and auroral glow observed in the auroral zone fit well the restrictions imposed by the critical condition on the percolation of the Pedersen current. Thus, it is demonstrated that the fractal approach is a promising and convenient method for studying the properties of the ionosphere.

Relationship between percolation–fractal properties and permeability of 2-D fracture networks

The concepts of percolation theory and fractal geometry are combined to define the connectivity characteristics of 2-D fracture networks and a new approach to estimate the equivalent fracture network permeability (EFNP) is introduced. In this exercise, the fractal dimensions of different fracture network features (intersection points, fracture lines, connectivity index, and also fractal dimensions of scanning lines in X- and Y-directions), and the dimensionless percolation density of fracture networks are required. The method is based on the proposed correlations between the EFNP and a percolation term, (ρ′–ρ′c). The first parameter in this term is the dimensionless density, and the second one is the percolation threshold (a constant number). This term is obtained using the relationships with the properties of fracture networks mentioned above. It was found that the highest correlation coefficient between the actual and the predicted EFNP could be obtained using the percolation term, (ρ′–ρ′c), calculated from the fractal dimension of fracture lines by the box-counting technique. The method introduced is validated using different fracture patterns representing a wide range of fracture and length values. In addition, a correlation between the number of fractures in the domain and the minimum size of the fracture length is presented to estimate the shortest or minimum fracture length required for percolation for a given number of fractures in the domain.

Release behaviour of clozapine matrix pellets based on percolation theory

The release behaviour of clozapine matrix pellets was studied in order to investigate if it is possible to explain it applying the concepts of percolation theory, previously used in the understanding of the release process of inert and hydrophilic matrix tablets. Thirteen batches of pellets with different proportions of clozapine/microcrystalline cellulose (MCC)/hydroxypropylmethyl cellulose (HPMC) and different clozapine particle size fractions were prepared by extrusion–spheronisation and the release profiles were studied. It has been observed that the distance to the excipient (HPMC) percolation threshold is important to control the release rate. Furthermore, the drug percolation threshold has a big influence in these systems. Batches very close to the drug percolation threshold, show a clear effect of the drug particle size in the release rate. However, this effect is much less evident when there is a bigger distance to the drug percolation threshold, so the release behaviour of clozapine matrix pellets is possible to be explained based on the percolation theory.

Polymer Percolation Threshold in HPMC Extended Release Formulation of Carbamazepine and Verapamil HCl

The principles of the percolation theory were applied to further understand and design hydroxypropyl methylcellulose (HPMC) extended release matrix tablets containing carbamazepine and verapamil HCl. This statistical theory studies disordered or chaotic systems where the components are randomly distributed in a lattice. The application of this theory to study the hydration and drug release of hydrophilic matrices allows describing the changes in hydration and drug release kinetics of swellable matrices. The aim of this work was to study and develop extended release matrix formulations for carbamazepine and verapamil HCl, containing hypromellose (HPMC, METHOCEL Premium K100M CR) as rate controlling polymer using the concepts of percolation theory. The knowledge of the percolation threshold of the components of the matrix formulations contributes to improve their design. First, reducing the time to market and second, avoiding to formulate in the nearby of the percolation threshold, which will result in a lower variability. Therefore these formulations will be more robust when they are prepared at industrial scale. The HPMC percolation threshold for drugs with very different water solubilities was determined and it was shown that there was no significant influence of drug solubility on the HPMC critical concentration threshold (excipient percolation threshold). This may be related to the versatility and broad functionality of the swelling hydrophilic matrices.

Identifying the cluster structure of blast-furnace batch by resistance measurements

In accordance with the basic concepts of percola� tion theory, a mixture of friable components is a cluster structure. The formation of clusters (their size, shape, etc.) is described by mathematical laws [1–3]. A mixture of friable materials with different ratios of the components is characterized by a specific prob� ability of cluster structure, from which the mixture composition may be established. To determine the 〈〉 �������� � , =

Critical Density for Coverage and Connectivity in Three-Dimensional Wireless Sensor Networks Using Continuum Percolation

Although most of the studies on coverage and connectivity in wireless sensor networks (WSNs) considered two-dimensional (2D) settings, such networks can in reality be accurately modeled in a three-dimensional (3D) space. The concepts of continuum percolation theory best fit the problem of connectivity in WSNs to find out whether the network provides long-distance multihop communication. In this paper, we focus on percolation in coverage and connectivity in 3D WSNs. We say that the network exhibits a coverage percolation (respectively, connectivity percolation) when a giant covered region (respectively, giant connected component) almost surely spans the entire network for the first time. Because of the dependency between coverage and connectivity, the problem is not only a continuum percolation problem but also an integrated continuum percolation problem. Thus, we propose an integrated-concentric-sphere model to address coverage and connectivity in 3D WSNs in an integrated way. First, we compute the critical density lambdaC con above which coverage percolation in 3D WSNs will almost surely occur. Second, we compute the critical density lambdac con above which connectivity percolation in 3D WSNs will almost surely occur. Third, we compute the critical density lambdac cov-con above which both coverage and connectivity percolation in 3D WSNs will almost surely occur. For each of these three problems, we also compute their corresponding critical network degree. Our results can be helpful in the design of energy-efficient topology control protocols for 3D WSNs in terms of coverage and connectivity.

Determination of the percolation thresholds for polyethylene oxide and polyacrylic acid matrix tablets

Abstract The purpose of this study was to implement the concepts of percolation theory in the characterization of drug release from hydrophilic matrix tablets. Percolation theory is a powerful statistical tool that enables mathematical insight into geometrically complex and disordered systems. Matrix tablets are effective substrate for the implementation of percolation theory because of their inherent disordered structure. The objective was to predict percolation thresholds of polyethylene oxide and polyacrylic polymers in diclofenac sodium hydrophilic matrices. Matrix tablets were prepared using polyethylene oxide or polyacrylic acid as matrix forming materials and diclofenac sodium was used as a model drug substance. Ten formulations with different drug/excipient ratios were prepared using the direct compression method. Dissolution studies were performed using the paddle apparatus method. For estimating percolation threshold the change of the kinetic parameters in aspect to the volumetric fraction of excipient plus initial porosity of the tablets was studied. Observed critical points with sudden changes in behavior of kinetic parameters can be attributed to the percolation thresholds. Percolation threshold is found to be 60.22% v/v polyethylene oxide + initial porosity and 39.94% v/v polyacrylic acid + initial porosity. The results obtained demonstrate that percolation theory can be used to design and develop matrix tablet formulations. Determination of percolation threshold is a useful tool for preparing robust formulations.

Fundamentals of Percolation Phenomenon with Emphasis on Its Concept in Disordered Electrochemical Systems

This article covers the fundamentals of percolation phenomenon giving emphasis to the percolation concept involved in disordered electrochemical systems. After a brief discourse on the basic concepts of percolation theory, the geometrical properties and fractality of percolation clusters were presented. Then, anomalous behaviours of diffusion in percolation clusters were explained in terms of the fractal structures of the infinite percolation clusters. Finally, the conductivity-related properties of composite ionic materials were shortly discussed on the basis of percolation theory from practical points of view.