Percolation Conditions Research Articles

Weathering processes in a lignite mine spoil treated with a CaCO3-rich waste slurry under two moisture programmes.

Chemical processes affecting the acidity of asulphide-rich lignite mine spoil treated with high orlow doses of a CaCO3-rich industrial waste slurrywere studied in the laboratory under two moistureprogrammes (cycles of alternate waterlogging anddrying, and percolation) so as to determine the slurrydose neutralizing spoil acidity and the possibledetrimental effects of high doses. The most importantacidity-reducing process was the dissolution ofapplied calcium carbonate, and CaCO3 consumptionwas greater under percolation conditions than underwaterlogging and drying conditions. The most importantacidity-increasing process was the oxidation ofsulphides, which was again more intense underpercolation. Under waterlogging and drying conditions,the formation of hydroxysulphates may also havecontributed to acidification.

The Pitting Behavior of Iron‐Chromium Thin Film Alloys in Hydrochloric Acid

The pitting behavior of a range of iron‐chromium thin film alloys which are sulfide‐free has been studied by electro‐chemical methods coupled with in situ optical microscopy. In 0.5 M HCl, iron‐chromium alloys with > ∼16 atom % chromium do not pit at any potential, whereas alloys with < ∼16 atom % chromium pit at approximately +500 mV standard calomel electrode. The sharp transition is believed to be associated with the existence of a critical size of iron cluster which act as pit nucleation sites. This is modeled by the achievement of a high density percolation condition for iron in the body‐centered cubic alloy. The pits propagate under remnants of the passive film as two‐dimensional disks, with current densities up to and simultaneous hydrogen evolution.

Critical exponents for the contact process under the triangle condition

We show that a continuous-time version of the triangle condition for percolation implies mean-field values for several contact process critical exponents. Our results support the belief that the upper critical (spatial) dimension for the contact process is four.

Atom interaction in expanded metals

For a virtual atomic structure of expanded metals near the metal-nonmetal transition point the interatomic interaction energy has been calculated. An atomic cell model has been developed describing atoms in percolation clusters. The pseudowave equation is solved with a quasiclassical percolation condition for the nearly bound electron states. The interaction energy has been shown to be nearly one third of the OCP Madelung energy. An application to the equation of state near the critical point of metallic fluids has been made.

Laser-induced bursts of subsurface liquid Mo at transition from planar to volume vaporization: ballistic and percolation surface aggregation of ejected particles

The quality of laser-driven processes associated with vaporization strongly depends on the vaporization regime. Transition from planar to ‘volume’ vaporization is connected with superheating which causes significant topological changes in the surface itself. A study of molybdenum surface has shown that the inrush and ejection of liquid molybdenum, caused by laser-induced subsurface heating in the periphery zone of the spot, occurs in the form of small droplets. Corresponding Reynolds numbers are estimated to be Re ≈ 10 3, similar to that for the turbulence in the boundary layer. Similarity with the morphology of the subsonic jets, which under strong disturbance break into droplets at the same value of the Reynolds number, is also established. Droplets of liquid molybdenum aggregate into two-dimensional structures of the columnar form which grow by ballistic aggregation, and into irregular, lace-clusters which grow by percolation. In the near-periphery zone a high radial momentum of ejected particles causes ballistic aggregation, while in a distant-periphery zone (near the edge of the spot), a random particle momentum causes growth of random aggregates. Numerical simulation of the random lace-clusters indicates that percolation conditions (number of particles, percolation probability, and the lattice) are different in different local basins, as a consequence of inhomogenous momentum distribution in the subsurface liquid molybdenum. As a consequence, these processes lead to loss of precise control of either laser film deposition, or laser-material removal.

Release of the GRF29NH 2 analog of human GRF44NH 2 from a PLA/GA matrix

The GRF29NH 2 fragment of human growth hormone releasing hormone (GHRH or GRF44NH 2) is currently regarded as of interest to increase body growth by stimulation of the release of growth hormone (GH). Because this analog has a short lifetime in body fluids and because it is usually injected i.v. repeatedly, a biodegradable pellet system was selected in an attempt to apply the concept of controlled drug delivery to the administration of GRF29NH 2 according to therapeutic requirements. Pellets were made by compression moulding of powdered GRF29NH 2 mixed with a lactic/glycolic acid copolymer, namely PLA37.5GA25 synthesized by bulk polymerization of a feed containing 75% dllactide and 25% glycolide. This protocol was supposed to yield pellets releasing the oligopeptide under percolation conditions. The effects of pH and of ionic strength on the release of GRF29NH 2 in water were investigated in vitro. GRF29NH 2 appeared unexpectedly insoluble in salt-containing neutral water and clotted the pellet porosity under static conditions. A simple device based on the use of a cotton dental tampon entrapped within a glass tube supplied with a regular flow of aqueous medium was used to show that release was achievable under dynamic conditions mimicking the situation at the administration site after s.c. injections or pellet implantations.

Methane production and its fate in paddy fields

Special attention is being paid to methane fluxes from paddy fields in relation to the greenhouse effect. However, the flux to the atmosphere is not the sole flux direction of methane produced in paddy soils. A pot experiment where rice plants were grown was conducted to analyze the effects of the percolation rate and rice straw application on the methane fluxes both to the atmosphere and subsoil. The amount of methane carbon percolated to the subsoil was evaluated in comparison with other forms of organic carbon in the leachate. The increase of the percolation rate accelerated the methane fluxes to the subsoil, while it did not affect the methane fluxes to the atmosphere. Application of rice straw increased the methane fluxes in both directions until the panicle initiation stage of rice plants. The total amount of methane flux to the subsoil accounted for 8.6% of the sum of the amount of emitted and leached methane under 15 mm/d percolation condition. The amount of methane carbon in the leachate...

Monte-carlo simulation of combustion-induced percolative fragmentation of carbons

Percolative fragmentation of carbon particles during combustion is modelled by means of two Monte-Carlo simulation models on a 3-D lattice: the first applies, to combustion in absence of oxygen concentration gradients within the particle; the second applies to percolative fragmentation in the intraparticle diffusion-limited combustion regime. The latter simulates oxygen diffusion and reaction by a random walk technique. Model results include the ratio of the mass rate of percolated fragments to the rate of carbon removal by combustion and percolative fragmentation, the penetration depth of oxygen in the porous matrix and the average size of fragments generated by the percolation process. The correspondence is established between model parameters and variables of the actual physical process. In particular it is shown that fragmentation rate and average fragment size and functions of a dimensionless group (the Thiele modulus) which expresses the ratio between the average size of the cavities in the lattice and the oxygen penetration depth. In absence of porosity gradients within the lattice (that is as Thiele modulus tends to 0) the rate of percolative fragmentation turns out to be smaller than that corresponding to the voidage at the percolation of the solid phase in the classical percolation theory framework. Non-uniform distribution of porosity in the lattice, like that established in diffusion limited combustion conditions, further reduces the rate of fragmentation in respect to that observed under uniform percolation conditions. In particular it is shown that fragmentation rate decreases with increasing combustion temperature, in agreement with experimental results of previous authors. The average size of fragments generated in the percolation process increases as porosity and average size of pores increase. For given lattice properties, the fragment sizes decrease when moving from uniform to diffusion limited combustion conditions, consistent with the parallel decrease of oxygen penetration depth within the lattice.

Modeling fragmentation by percolation in combustion of carbons

A 3-D model has been developed to investigate the process of combined combustion and percolative fragmentation of carbons in the intermediate regime between intraparticle diffusion and chemical kinetics control. The model is based on the discretization of the field into a cubic structure. Vacancies can be originated in the structure to build a lattice having a given original voidage and void size distribution. A Monte Carlo simulation is used to remove cells from such a lattice by combustion and percolative fragmentation. The fraction of cells removed by percolation with respect to the total number of cells removed by combustion and percolation and the distribution of volumes of percolated fragments have been calculated as functions of model variables. These include: the lattice original voidage and void size distribution, the depth of penetration of combustion and the size of the unit cell of the lattice. The simulation successfully describes carbon combustion efficiency and size distribution of carbon fines as functions of porosity and pore size distribution under conditions for both peripheral and uniform percolation, provided that an appropriate size of the unit cell is used. The choice of this variable is a difficult point in the simulation. The size of the unit cell, in fact, cannot be predicted on a general basis due to its dependence on the physical and chemical properties of the specific carbon being considered.

Multiscale models of failure and percolation

Abstract Multiscale hierarchical percolation and failure models are introduced, and a wide class of discrete hierarchical models and their asymptotical properties are studied. This asymptotical analysis is carried out by the help of the ideas of scaling. The necessary and sufficient conditions for failure and percolation are discovered. For continuous spherical hierarchical percolation models the classical percolation theory is developed. It is shown that by increasing the number of scales and the scale step the critical volume concentration can be made to approach unity.

A molecular dynamics model of melting and glass transition in an idealized two-dimensional material I

In a preparatory study of structural relaxations and plastic flow in a two-dimensional idealized atomic glass, the process of melting and quenching through a glass transition has been studied by computer simulation using a molecular dynamics model. In this model, the transition from a solid to a melt was observed to take place when liquid-like structural elements composed of dipoles of five- and seven-sided Voronoi polygons percolate through the two-dimensional structure of distorted hexagons in the form of strings. Such dipoles constitute discrete elements of excess free volume within which liquid like behaviour is established in the sense of reduced cohesion or local elastic moduli. Upon quenching the melt, the percolation condition of liquid-like regions is retained for under-cooled melts between the melting point and a glass transition temperature below which the percolation condition is broken and the thermal expansion is sharply reduced. The simulation that has used empirical pair potentials characteristic of Cu and Zr has substantially underpredicted the melting and glass transition temperatures and overpredicted the thermal expansion of C uxZr1-xtype glasses. These defects of the model can be partly attributed to the two-dimensional nature of the material, which stores larger concentrations of free volume than a corresponding three-dimensional material. In spite of these quantitative shortcomings, the model gives valuable insight into the topological features of the local atomic configurations at melting and upon vitrification.

Idiotypic networks incorporating T-B cell co-operation. The conditions for percolation

Previous work was concerned with symmetric immune networks of idiotypic interactions amongst B cell clones. The behaviour of these networks was contrary to expectations. This was caused by an extensive percolation of idiotypic signals. Idiotypic activation was thus expected to affect almost all (greater than 10(7] B cell clones. We here analyse whether the incorporation of helper T cells (Th) into these B cell models could cause a reduction in the percolation. Empirical work on idiotypic interactions between Th and B cells however, would suggest that two different idiotypic Th models should be developed: (1) a Th which recognises native B cell idiotypes, i.e. a non-MHC-restricted "ThId" model, and (2) a "classical" MHC-restricted helper T cell model. In the ThId model, the Th-B cell interaction is symmetric. A 2-D model of a Th and a B cell clone that interact idiotypically with each other accounts for various equilibria (i.e. one virgin and two immune states). Introduction of antigen does indeed lead to a state switch from the virgin to the immune state; such a system is thus able to "remember" its exposure to antigen. Idiotypic signals do however, percolate in ThId models via these "B-Th-B-Th" pathways: proliferating Th and B cell clones that interact idiotypically, will always activate each other reciprocally. In the MHC-restricted Th model, Th-B interactions are asymmetric. Because the B cell idiotypes are processed and subsequently presented by MHC molecules, the Th receptor and the native B cell receptor are not expected to be complementary. Thus the Th and the B cells are unable to activate each other reciprocally, and a 2-D Th-B cell model cannot account for idiotypic memory. In contrast to the ThId model, idiotypic activation cannot percolate via "B-Th-B-Th" interactions. Due to the assymmetry idiotypic activation stops at the first Th level. A Th clone cannot activate a subsequent B cell clone: if the B cells recognise the Th cells, they see idiotype but get no help; if the Th cells see the B cells, the B cells are helped but see no idiotype. The percolation along "B-B-B" pathways in these two models is next analysed. Two B cells clones, each helped by one Th clone, are connected by a symmetric idiotypic interaction. It turns out that in both models the second (i.e. anti-idiotypic) B cells (B2) never proliferate.(ABSTRACT TRUNCATED AT 400 WORDS)

Low voltage electrical properties of polypropylene filled with stainless steel fibres and a model of sample conductivity based on fibre geometry

The d.c, electrical properties of 80×80×3 mm3 polypropylene plaques filled with 6.5 μm diameter stainless steel fibres have been studied for volume fractions in the vicinity of a critical threshold at which the volume resistivity changes very rapidly with filler concentration. By the use of very low power inputs to eliminate any possibility of local temperature changes, the samples have been established to be ohmic conductors with resistivities ranging from 12 to 0.61 Ω cm for fibre volume fractions of 1 to 3%. It is suggested that percolation conditions i.e. continuous chains of metal fibres are produced at low volume fraction of filler because of a special fibre geometry i.e. a substantial proportion of the fibres are three dimensionally folded into shapes of roughly helical form, thus enhancing the probabilities of contact between adjacent fibres. For simplicity a model structure of perfect helices with identical diameters and pitch has been examined. The model leads to a critical volume fraction at the percolation threshold, which is in good agreement with experiment and proportional to the square of the ratio of fibre diameter to helix diameter. The threshold resistivity range predicted by the model is also a function of fibre and helix diameter and this resistivity also decreases with mean fibre length. It is argued further that there exists an optimum value of fibre aspect ratio for which the critical volume fraction is a minimum. The predicted threshold resistivity is in good agree ment with experiment providing that a small amount of the size coating is assumed to have been removed during manufacture of the plaques, thus allowing a small fraction of the fibre fibre contacts to be conducting.

Low voltage electrical properties of polypropylene filled with stainless steel fibres and a model of sample conductivity based on fibre geometry

The d.c, electrical properties of 80×80×3 mm3 polypropylene plaques filled with 6.5 μm diameter stainless steel fibres have been studied for volume fractions in the vicinity of a critical threshold at which the volume resistivity changes very rapidly with filler concentration. By the use of very low power inputs to eliminate any possibility of local temperature changes, the samples have been established to be ohmic conductors with resistivities ranging from 12 to 0.61 Ω cm for fibre volume fractions of 1 to 3%. It is suggested that percolation conditions i.e. continuous chains of metal fibres are produced at low volume fraction of filler because of a special fibre geometry i.e. a substantial proportion of the fibres are three dimensionally folded into shapes of roughly helical form, thus enhancing the probabilities of contact between adjacent fibres. For simplicity a model structure of perfect helices with identical diameters and pitch has been examined. The model leads to a critical volume fraction at the percolation threshold, which is in good agreement with experiment and proportional to the square of the ratio of fibre diameter to helix diameter. The threshold resistivity range predicted by the model is also a function of fibre and helix diameter and this resistivity also decreases with mean fibre length. It is argued further that there exists an optimum value of fibre aspect ratio for which the critical volume fraction is a minimum. The predicted threshold resistivity is in good agree ment with experiment providing that a small amount of the size coating is assumed to have been removed during manufacture of the plaques, thus allowing a small fraction of the fibre fibre contacts to be conducting.

The triangle condition for percolation

Aizenman and Newman introduced an unverified condition, the triangle condition, which has been shown to imply that a number of percolation critical exponents take their mean field values, and which is expected to hold above six dimensions for nearest neighbour percolation. We prove that the triangle condition is satisfied in sufficiently high dimensions for the nearest neighbour model, and above six dimensions for a class of spread-out models. The proof uses an expansion which is related to the lace expansion for self-avoiding walk. Percolation is a simple probabilistic model for which many interesting problems remain unsolved. The basic objects in percolation are random graphs on an infinite lattice. Typically there is a critical density for the graph edges (known as bonds) below which there is zero probability that a fixed point in the lattice is part of an infinite connected graph, but above which this probability is strictly positive. This abrupt change in behaviour has been used in chemistry and statistical physics to model phase transitions in a variety of physical systems, such as fluid flow through a porous medium (hence the name percolation), random resistor networks, and the gelation of branched polymers. For an introduction, see [15, 20]. To define the models we are interested in, we consider the of-dimensional integer lattice Z^ (whose elements are referred to as sites) and the set of pairs b = {x,y} of distinct sites (the bonds). To each bond b is associated an independent Bernoulli random variable nb9 with nb = 1 with probability p • Jb and nb — 0 with probability p Jh. The Jb are fixed and Z invariant, and p is a parameter. The nearest-neighbour model is defined by taking Jb = 1 for those b = {x,y} such that \\x y\\i = 1, and Jb = 0 otherwise. Expectation with respect to the joint distribution of the {nb} is denoted by (•••)/?• If nb = 1 we say b is occupied, and otherwise b is vacant. Given a bond configuration {nb}, two sites x and y are said to be connected if there is a path from x to y which consists of occupied bonds. The set of sites which are connected to x is called the connected cluster of x and is denoted C(x). The probability that x is connected to y is written Tp(x,y), (1) ip(x,y) — (I[x and y are connected])^ = {I[y e C(x)])p, where I denotes the indicator function. Denoting by \C(x) the number of sites in C(x), the expected cluster size or susceptibility is given by

Continuous time random walk approach to dynamic percolation

We present an approximate solution for time (frequency) dependent response under conditions of dynamic percolation which may be related to excitation transfer in some random structures. In particular, we investigate the dynamics of structures where one random component blocks a second (carrier) component. Finite concentrations of the former create a percolation network for the latter. When the blockers are allowed to move in time, the network seen by the carriers changes with time, allowing for long-range transport even if the instantaneous carrier site availability is less than pc, the critical percolation concentration. A specific example of this situation is electrical transport in sodium β″-alumina. The carriers are Na + ions which can hop on a two-dimensional honeycomb lattice. The blockers are ions of much higher activation energy, such as Ba 2+. We study the frequency dependence of the conductivity for such a system. Given a fixed Ba 2+ hopping rate, 1/τ, the Na + ions experience a frozen site percolation environment for frequencies ω > 1/τ. At frequencies ω < 1/τ, the Na + ions experience a dynamic environment which allows long-range transport, even below pc. A continuous time random walk model combined with an effective medium approximation allows us to arrive at a numerical solution for the frequency-dependent Na+ conductivity σ(ω) which clearly exhibits the crossover from frozen to dynamic environment.

Adjustable selective profiles using cermet absorbing films

The optical and electrical properties of metal-dielectric composites (cermets) show large variations as a function of volume fraction, size, shape and the statistical distribution of metallic grains in the dielectric. These variations are of particular importance around the percolation threshold, where the so-called dielectric anomaly, attributed to the excitation of a resonant mode of the conduction electrons in the metallic grains, is observed. Therefore, cermet films close to, but below, the percolation condition are strongly absorbing in the visible range. In the infrared, they behave like a dielectric and are thus transparent. The flexibility of the optical properties around the percolation threshold can be used to develop new spectrally selective coatings totally absorbing in the visible range and highly reflecting in the IR, with an adjustable cut-off wavelength. We present optical simulations of some specially adjusted spectral profiles and their experimental realization using a three-layer coating consisting of a Pt-Al 2O 3 cermet absorbing film, an Al 2O 3 antireflecting film and an Al 2O 3 underlayer, deposited on a metallic substrate. These coatings are stable at high temperatures and exhibit a high mechanical resistance. They can be used as selective absorbers for many specific applications as a result of their cut-off wavelength: selective solar absorber surfaces, surface thermometers and flux meters made of thin film thermocouples, IR detectors, etc.

Percolation threshold and interface optimization in a HUP/C doublelayer supercapacitor

Energy consumption of electronic devices has decreased gradually with progress in integrated circuits; thus, the durable voltage required for a capacitor has become lower and use of the electric double layer developed at a highly polarizable blocking electrode/electrolyte interface is possible. The demand for miniaturized high energy density and low leakage current capacitors as, for example, a stand-by power source for RAM devices or for actuators offers new opportunities for these devices. The first attempts to achieve such a device were made in the seventies [1], but electrical performances were poor and potential applications were restricted. More recently, a double layer capacitor using a liquid electrolyte (H2SO4) and a carbon electrode was developed and commercialized by NEC [2]. Use of a solid electrolyte opens new possibilities: (i) manufacture of microdevices using a collective process, (ii) a better long-time stability and, owing to the use of pure ionic conductor as separator, a high charge capability, (iii) the slow rate of electrochemical reactions in the solid state prevents degradation due to transient overvoltages. All these potential advantages require a high interface area between the electrolyte and the electrode. The double layer capacity of a typical electrode/electrolyte interface falls into the 10 to 40/~F cm 2 range per effective interface area, indeed. Use of a liquid electrolyte facilitates: contacts with area of carbon. On the other hand, a double percolation (between the solid electrolyte itself and the carbon particulates itself) it necessary to combine in the all solid system a high interface area and a (relatively) low serial resistance [3]. Here we present results concerning the optimization of percolation conditions for two kinds of black activated carbon in the case of an all solid HUP (H3OUO2PO 4. 3H20)/C double layer capacitor. H 3 O U O z P O 4 ° 3H20 is among the best known-fast proton solid conductors [4] and is used in microionic devices such as electrochromic displays [5], sensors [6] or batteries [7]. The HUP structure consists in a quasitwo-dimensional network of water molecules and H3 O+ ions, alternately with layers of uranyl and phosphate ions. The high protonic conductivity occurs via the formation of a quasi-liquid state of H30 + and H20 species between the ( U O z P O 4 ) n slabs [81. Fig. 1 shows a schematic representation of the *To whom all correspondence should be addressed.

A comparison between ‘mixed phase electrode’ and percolation models for composite electrodes in solid state cells

A statistical computer model of mixed phase electrode systems has been reformulated to extend its applicability to include standard percolation conditions, and the effects of changing composition and lattice dimensions have been investigated. When both phases are present at volume percentages above a certain critical value, the effective interfacial area of the mixed phase region can be raised indefinitely by increasing the thickness of the region. Below this limit a steady reduction in the ‘efficiency’ of the mixed phase region, rather than the abrupt cut-off suggested by percolation theory, is predicted.

The effect of correlations on the conductivity of the small-polaron regime in disordered systems

The authors apply percolation theory to the high-temperature small-polaron hopping regime including correlation between bonds due to the energy in the common site. This drastically affects the temperature dependence of the conductivity. Thus, despite the fact that the average transition rates, and consequently the percolation condition, for the few-phonon low-T and the multiphonon high-T regimes are different, they obtain the same temperature dependence for the conductivity, namely Mott's law sigma approximately exp(-(T0/T)14/).