Simple Percolation Research Articles

Percolation and fragmentation of excited atomic nuclei

We present an analysis which aims at explaining the similarities (and differences) which exist between a simple bond percolation process on a cubic lattice and the fragmentation of highly excited atomic nuclei. Emphasis is placed on discussing percolation in terms of concepts which are well known in nuclear physics such asQ-value and particle emission thresholds. Similarities and differences between the bond percolation process and nuclear fragmentation are discussed. An approximate expression for the microcanonical partition sum (number of microstates) corresponding to any given percolation partition is shown to provide a good starting point for predicting fragment size distributions.

Modelling moisture distribution and isothermal transfer in a heterogeneous porous material

This paper presents a study of moisture retention and isothermal transfer in a cement and lime mortar. The extended range of pore sizes and the intrinsic heterogeneity of the medium do not allow the use of a single-scale percolation network for numerically describing the topology and the physical processes related to fluid retention and transfer. Three different scales are used for the numerical description of the cement and lime mortar studied in this paper. Electron scanning pictures are used for the geometrical modelling of the material in the three different scales. The heterogeneity is associated with the fissures network, represented in the first scale, where the medium was conceived as a series of cubic blocks separated by straight channels. The second and third scale are used to represent the pores in the cement and lime paste. A 3D simple cubic percolation site network is used in the third scale to model imbibition and drainage and to predict the hydraulic conductivity in terms of the moisture content. In the second scale, pores are represented as embedded bodies in a continuous medium and the hydraulic conductivity is calculated using Maxwell-De Vries theory for composite media. At the first scale level, the isothermal mass diffusivity D 0 is modelled by studying an invasion process of liquid water into the fissures network, simulating an actual imbibition experiment, from one extremity of a sample column. Finally, the results of the simulation are compared with experimentally obtained values of D 0.

Ion-beam-induced transformation of diamond.

The ion-beam-induced transformation of insulating diamond to a conducting form of carbon is explored by performing measurements of the electrical conductivity of diamond subject to ion damage. A wide range of implantation temperatures (150--690 K) with both Xe (320 keV) and C (100 keV) ions are employed. The dose dependence of the conductivity, R(D), is found to scale with the nuclear energy deposited in the irradiated volume, thus demonstrating that it is the density of collisionally induced defects that governs the electrical conductivity. The data are analyzed in terms of a model that proposes that the passage of an ion through the solid leaves in its wake conducting spheres of varying radii. The average radius of these spheres is found to decrease from about 2.05 nm for irradiation at 150 K with 320-keV Xe ions to zero for implantation at about 815 K. At critical doses ${\mathit{D}}_{\mathit{c}}$, which depend on the implantation temperature and the ion species, these spheres overlap to form a continuous conductive pathway through the irradiated diamond. In some cases this transition is sharp enough to be well accounted for by a simple percolation theory. Below ${\mathit{D}}_{\mathit{c}}$, R(D) displays complicated nonmonotonic behavior, which is explained as being due to the competition of the contribution of different types of defects to the observed electrical conductivity. Remarkable similarities between the conductivity induced in diamond and fused quartz implanted with C ions under identical conditions are reported.

Simulating iceberg calving with a percolation model

The physics behind the iceberg calving process is poorly understood, but by using a simple fracturing criterion based on the accumulation and coupling of many microcracks, the complicated mechanics of calving can be simulated by a set of percolation rules. Calving simulations with this simple percolation model give results which are consistent with limited observations of iceberg size distributions and terminus morphologies. For steady state scenarios, the model suggests that the calving process can be characterized by a small set of scaling exponents.

Deexcitation of primary projectile-like fragments in the reaction 40Ca+natCu at 35 MeV/nucleon: Comparison with sequential binary decay and percolation models.

This article presents a new analysis of primary projectile-like fragments produced in peripheral and mid-peripheral collisions in the reaction $^{40}\mathrm{Ca}$${+}^{\mathrm{nat}}$Cu at 35 MeV/nucleon. The data were obtained using the charged particle amphora array. The distributions of multiplicity, inclusive charge, and heaviest fragment charge of the deexcitation products as well as the moments characterizing the single event charge partitions are remarkably well reproduced by a statistical sequential binary decay model even up to excitation energies of more than 8 MeV/nucleon. The results produced by the sequential binary decay model and a simple bond percolation model are qualitatively very similar. However the sequential binary decay mechanism leads to charge fluctuations which are in better agreement with the data than those predicted by the percolation model.

Fluctuations and the canonical model for fragmentation.

Starting from a parametrized canonical expression for partition weights we investigate effects of uncorrelated stochastic fluctuations in the parameters. A special class of weights which are not affected by uncorrelated fluctuations is isolated and used as a reference for a systematic analysis of fluctuations. The technique is appled to partition weights obtained from a simple bond breaking percolation simulation. Strong evidence for the existence of uncorrelated fluctuations is obtained. The effects of fluctuations and of finite size effects on scaled factorial moments of the mass distribution are analyzed and discussed.

Seismic constraints on a model of partial melts under ridge axes

In a recent global scale seismic study, the correlation between S wave velocity under ridge axes and spreading rate was pointed out. The correlation is strong for depths to about 70 km, but it diminishes below this depth. We present the correlation plots at four depths, 38, 66, 90, and 110 km, for which correlation is strong at 38 and 66 km but is weak at 90 km and is almost nonexistent at 110 km. We present a model to explain this behavior, which includes a thermal conduction model for the development of lithosphere and a simple melt percolation. Thermal effects on S wave velocity are assumed to be accounted for entirely by the plate cooling (thermal conduction) model. We point out that the thermal model under this assumption predicts asymptotically no correlation between S wave velocity and spreading rate, specifically for spreading rate larger than about 3 cm yr−1. This contradicts the correlation observed in the data at shallow depths. The existence of partial melt is thus required to explain the observed behavior at 38 and 66 km depths. We start from four basic equations that govern the distribution of partial melt and derive the relation between the amount of partial melt and the spreading rate. We adopt a simple power law relation between permeability (k) and porosity (ƒ) by k(ƒ) = k0ƒn, where k0 and n are constants and assume that pores are filled with melt. We then set up an integral relation between S wave velocity and spreading rate. The final formula indicates that the gradient in the correlation plots is the inverse of the power (1/n) in the permeability‐porosity relation, thus enabling us to constrain n as well as k0 from seismic data. The data also have some sensitivity to the depth to solidus. We show that (1) the depth to solidus is probably within the range 60–100 km and (2) if the power n is n = 2–3, then k0 = 10−8 ‐ 10−10 m2. These parameters predict that porosity and fluid velocity are 1–2% and about 0.5 m yr−1, respectively. The depth to solidus is consistent with previous estimates by petrological data but is perhaps the first and direct seismological evidence of partial melt from surface wave data. Analytical forms for the dependence on depth and spreading rate of porosity, fluid velocity within permeable rocks, and ascent times of magma are also obtained.

Dissipation in a weak-link-limited superconductor as a problem of percolation theory.

Broadening of the resistive transition of sintered ${\mathrm{GdBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathit{x}}$ superconductor has been studied varying the measuring current over the range of five orders of magnitude. The observed well-defined branching points in the resistive transition coincide with the quasi-Ohmic saturation of V-I curves and scale with the average grain size. The results suggest the validity of a simple percolation model for dissipation in systems whose global superconductivity is realized by Josephson coupling of the superconducting grains. A common origin of structured resistive transition, observed in many superconducting systems, has been also discussed.

Mass diffusion of diatomic fluids in random micropore spaces using equilibrium molecular dynamics.

An equilibrium-molecular-dynamics study of diffusion of a heteronuclear diatomic fluid in random pore systems is reported. The pore space is generated by the use of a simple percolation technique on a tessellation of three-dimensional space with periodic boundary conditions. Simulations using the rattle algorithm [H. C. Andersen, J. Comput. Phys. 52, 24 (1983)] at constant temperature are performed on the percolating cluster only where the substrate atoms are explicitly represented and the substrate-fluid and fluid-fluid interactions are modeled using Lennard-Jones potentials. Using this technique, results are presented for a heteronuclear diatomic molecule (CO-like) within a graphitelike system for porosities between the percolation threshold and \ensuremath{\varphi}=0.99, and over the temperature range of T=100 K to T=1500 K at atmospheric pressure. The results, which include mean-square-displacement and velocity-autocorrelation functions (VACF's), indicate once again that mass diffusion within porous media is fundamentally different from that in the bulk phase. The degree of anomalous behavior tends to decrease with temperature and porosity. Although this is the case, the mean-square-displacement exponent in the long-time limit for the lower porosities (\ensuremath{\varphi}=0.312 and \ensuremath{\varphi}=0.4) increases with temperature to a limiting value much less than one in the vicinity of T=300 K. Trends in the VACF's seen here have been reported in past Lorentz gas simulation studies. Effective diffusion coefficients for our gas-in-pore system were calculated from the VACF's---these appear to vary exponentially with porosity and inverse temperature. This temperature variation is similar to that of a liquid in the bulk phase and hence brings into doubt the use of a temperature-independent tortuosity that relates the diffusion coefficient of a gas in the bulk phase to that within porous media.

Electroluminescence of organic thin films based on blends of polystyrene and fluorescent dyes

Blue and red light-emitting diodes were constructed as devices in which a polystyrene/dye blend was sandwiched between a calcium electrode and a transparent indium-tin oxide electrode. As dye molecules, four alkoxy-substituted derivatives of 2,5-dialkoxy-1,4-bis(2-phenylvinyl)benzene (i.e. p-phenylene vinylene oligomers), a derivative of bisanthracene (2,2′,3,3′-tetrachloro-9,9′-dimethoxy-10,10′-bianthracene) and the laser dye 4-dicyanomethylene-2-methyl-6-(p-dimethylaminophenylvinyl)-4H-pyran (DCM) were used. The critical field for light emission in devices using 20 wt.% PPV oligomers in polystyrene was found to be of the order of 2 × 108 V m–1. The dependence of the device conductivity on the concentration of dye molecules in the polystyrene film is described by simple percolation theory. The synthesis of 2,2′,3,3′-tetrachloro-9,9′-dimethoxy-10,10′-bianthracene is reported.

Intranuclear cascade-percolation approach for protons and light fragments production in neon-niobium reactions at 400 and 800 MeV per nucleon.

The results of intranuclear cascade calculations (ideal gas with two-body collisions and no mean field), complemented by a simple percolation procedure, are compared with experimental data on protons and light nuclear fragments ([ital d], [ital t], [sup 3]He, and [sup 4]He) measured in 400 and 800 MeV/nucleon Ne+Nb collisions using a large solid angle detector. The model reproduces quite well global experimental observables like nuclear fragment multiplicity distributions or production cross sections, and nuclear fragment to proton ratios. For rapidity distributions the best agreement occurs for peripheral reactions. Transverse momentum analysis confirms once again that the cascade, although being a microscopic approach, gives too small a collective flow, the best agreement being reached for [ital Z]=2 nuclear fragments. Nevertheless these comparisons are encouraging for further improvements of the model. Moreover, such an approach is easy to extend to any other models that could calculate the nucleon phase space distribution after the compression stage of the reaction, when light nuclear fragments emitted at large angles are constructed from percolation.

Mass diffusion of atomic fluids in random micropore spaces using equilibrium molecular dynamics.

An equilibrium-molecular-dynamics study of the diffusion of atomic fluids in random pore systems is reported. The pore space is generated by the use of a simple percolation technique on a tessellation of three-dimensional space with periodic boundary conditions. Simulations are performed on the percolating cluster only. The substrate atoms are explicitly represented and the substrate-fluid and fluid-fluid interactions are modeled using Lennard-Jones potentials. Using this technique, results are presented for an argon-in-graphite system for different porosities at 140 K and a fluid density of 3.5 k ${\mathrm{gm}}^{\mathrm{\ensuremath{-}}3}$. The results, which include mean-square displacement, velocity-autocorrelation functions and their associated memory kernels, and the total-force autocorrelation functions, clearly indicate that mass diffusion in completely random pore systems is different from that in the bulk phase. Trends shown in the results here are similar to those reported in Monte Carlo, random-walk, and theoretical works elsewhere. Restriction of the technique to the study of atomic fluids and random pore systems is made here for simplicity; it is presently being extended to molecular fluids and mixtures in real micropore spaces. The substrate is assumed to be rigid; however, it is felt that the technique can be extended to include substrate-fluid energy exchange.

Percolation approach to proton conductivity in hydrated protein powders

Abstract The dielectric response of hydrated myoglobin powder has been measured in the range 106-109Hz. The data reveal at least two distinct processes which are explained by reorientation of water dipoles and conduction of protons along connected patches of water molecules. It is shown that proton hopping may be approximated by a simple percolation model, which predicts an algebraic low-frequency divergence ∊(Ω)αΩ−0.5 at the threshold.

Competing two-species directed percolation

The authors study an extension of conventional directed percolation (DP) with two species (A and B) using mean-field and Monte Carlo methods. The densities of the two species in the steady-state exhibit phase transitions which are due both to simple percolation and to competition between the species. In 1+1 dimensions as well as a simple DP transition, there is a line of first-order transitions between pure-A and pure-B phases. The phase diagram in a 2+1 dimensions agrees qualitatively with that obtained from mean-field calculations with second-order transitions between pure and mixed phases. Preliminary studies suggest that the critical exponents are in the same universality class as (one-species) DP.

New universality classes in ?Percolative? dynamics

We study a site analogue of directed percolation. Random trajectories are generated and their critical behavior is studied. The critical behavior corresponds to that of simple percolation in some of the parameter space, but elsewhere the exponents reveal new universality classes. As a byproduct, we use the model to make an improved estimate of the percolation hull exponents and to calculate the site percolation probability for the square lattice.

Tunnelling and Thermally Stimulated Phenomena in Highly Filled PMMA Composites

Abstract Films of polymethylmethacrylate composites, filled with high volume fractions (P ≥ 26%) of Fe fine powder, were prepared by casting method. The electrical resistivity was measured in the temperature range of 305–406 K. Composites of P = 26–34% exhibited a temperature (T) independent resistivity in the range of T = 305–400 K indicating the predominance of a tunnelling conduction. Relatively sharp resistivity peaks were observed at T > 400K, due to the thermally stimulated discharging current (TSDC). Composites of P > 34% were characterised by broad TSDC peaks without tunnelling region. The results were interpreted on the basis of a previously suggested theoretical model which considers the discharge process in a dielectric with two bipolar deeply trapped charge layers. It is implied that the present system exhibits a high effective surface charge density and a low value of the average electric field inside the layers. These results ruled out the application of simple percolation models and confirm...

Charge transport properties of cofacially joined [GaPcF(BF 4) x] n macrocycles

Stacked, fluorine-bridged gallium phthalocyanine [GaPcF] n polymers are prepared for use in controlling the crystal architecture in cofacial assemblies. The UV-Vis transmission spectra of these partially oxidized materials show the progression of the phthalocyanine π radical cation species. Static magnetic susceptibility of [GaPcF(BF 4) x ] n reveals a temperature-dependent Curie component and a temperature-independent Pauli-like contribution. The magnitude of the latter is proportional to the square of the doping level. These properties indicate that π-electrons of the phthalocyanine (Pc) rings mainly transport the charge. The crystal structure analysis of [GaPcF] n after BF 4 − doping by the X-ray powder diffraction method reveals that the doping process is inhomogeneous and the GaPc unit stacks perpendicularly to the molecular plane. The unit cell parameters of the doped crystals are a = 18.94 A ̊ , c = 3.730 A ̊ with space group P4/ n and Z = 2. The Pc intervals along and perpendicular to the polymer chain vary by only several percent. It is also revealed that the eclipsed stacking of the Pc molecule persists in [GaPcF(BF 4) 0.87] n . Four-probe electrical conductivity measurements of polycrystalline samples of the polymers reveal a sharp increase in conductivity with BF 4 − doping. The partially oxidized form of [GaPcF(BF 4) 0.87] n gives conductivity as high as 5.4 × 10 −2 S cm −1 at 20 °C. The plot of the functional form of the conductivity versus the doping level can be fitted to a simple percolation model in accord with the inhomogeneous character of doping. It is clarified that the polymerization of GaPc by fluorine is a useful technique to reduce the change in crystal structure after doping, without any detectable change in charge transport and chemical properties.

Mobility edges for the quantum percolation problem in two and three dimensions.

Numerical results are reported for the quantum site-percolation problem. For the square lattice and the simple cubic lattice quantum percolation thresholds ${\mathit{p}}_{\mathit{q}}$ are calculated by studying the sensitivity of eigenvalues to a change in boundary conditions. Observing the energy dependence of the transition from localized to extended states, mobility edge trajectories are calculated. We obtain ${\mathit{p}}_{\mathit{q}}$=0.45 in three dimensions and ${\mathit{p}}_{\mathit{q}}$=0.70 in two dimensions. The latter value of ${\mathit{p}}_{\mathit{q}}$ is identified with a transition from weakly to strongly localized states, according to a similar localization behavior observed for the Anderson problem.

Percolation Theory and Fire Spread

Abstract Percolation theory deals with the statistics of random arrays. Two-dimensional percolation theory has been used as a model of forest fires. The important result is that if burnable elements are randomly placed on a two-dimensional array with a packing ratio p, then there is a value pc below which fires will not propagate, and above which a fire will propagate from one end of the array to the other. At p = pc,. the system exhibits greatest fluctuations. Furthermore, at p equals; pc fire properties such as the mean position of the fire-front and the total number of burnt sites should vary as power-laws. The critical exponent for these power-laws, as determined from simple percolation theory, were compared with those experimentally delermined by burning random arrays of matchsticks (with ignitable heads) in a square lattice. The most notable discrepancy is that the theory predicts that at critical percolation a fire-front decelerates, whereas the experiments indicate acceleration Although simple per...

Structural granularity and ionic conduction mechanism in glass

Ionic conduction processes in glass are discussed in terms of a simple “granular” (cluster bypass) model in which the preferred pathways for ionic migration (the percolation channels) are located within a connective tissue surrounding more densely packed islands of material (or clusters). The connectivity of the conduction pathways has been calculated by simple percolation theory from measured values of the “limiting slopes” to the conduction isotherms obtained in mixed cation glasses. In general, the connectivity parameter is small (Z<2.5) but there is apparently a connection between how Z varies with temperature and Angell's concept of “strong” and “fragile” glasses. Structural evidence for the cluster/tissue model has been reported elsewhere.