Broadband dielectric spectroscopy of inhomogeneous and composite weak conductors

ABSTRACT

In this paper, we discuss broadband dielectric spectroscopy from mHz up to the infrared range mainly for materials with inhomogeneous weak conductivity, including conductor-dielectric nanocomposites. Our discussion is based on the effective medium approximation (EMA) and experiments modeled by this approach are reviewed. We discuss core–shell composites modeled by coated-spheres (Hashin–Shtrikman model) and normal composites with a possible percolation of the conductor component resulting in sharp or smeared percolation threshold of the DC conductivity and diverging static permittivity in the former case. The sharp percolation threshold is modeled by the Bruggeman EMA or by general EMA with arbitrary percolation threshold and arbitrary critical exponents of the DC conductivity and static permittivity. For the case of smeared percolation threshold in the case of complex topologies, we use the Lichtenecker model allowing for partial percolation of both the components. Finally, numerous papers reporting negative permittivity in weakly conducting materials are criticized and concluded to be due to spurious effects.     

Scaling relation for determining the critical threshold for continuum percolation of overlapping discs of two sizes

We study continuum percolation of overlapping circular discs of two sizes. We propose a phenomenological scaling equation for the increase in the effective size of the larger discs due to the presence of the smaller discs. The critical percolation threshold as a function of the ratio of sizes of discs, for different values of the relative areal densities of two discs, can be described in terms of a scaling function of only one variable. The recent accurate Monte Carlo estimates of critical threshold by Quintanilla and Ziff [Phys. Rev. E76, 051115 (2007)] are in very good agreement with the proposed scaling relation.     

Critical fields and currents in a weakly nonlinear medium near the percolation threshold

For fields above a critical value the expansion of the conductivity in powers of the field ceases to be valid and the weak-nonlinearity approximation no longer works. The density behavior of the critical fields in strongly inhomogeneous media near the percolation threshold is found on the basis of two criteria—an average criterion and a local criterion. The parameter values of the medium for which crossover—a change of the critical behavior—occurs are determined. Similar calculations are performed for the critical currents. Zh. Tekh. Fiz. 68, 5–8 (June 1998)     

Duality relations in the nonlinear percolation problem: Theory and numerical simulation

The problem of a nonlinear current flow in a heterophase medium formed by a random mixture of linear and nonlinear phases is investigated. The duality relation is derived for the critical indices describing the effective response of a heterogeneous system. The critical index is calculated at the percolation threshold (for equal concentrations of the phases). The nonlinear percolation problem is simulated numerically for degrees k = 3, 5, and 7 of the nonlinear phase. The existence of a duality relation for critical indices is established in a range of phase concentrations.     

A damage model based on failure threshold weakening

A variety of studies have modeled the physics of material deformation and damage as examples of generalized phase transitions, involving either critical phenomena or spinodal nucleation. Here we study a model for frictional sliding with long-range interactions and recurrent damage that is parameterized by a process of damage and partial healing during sliding. We introduce a failure threshold weakening parameter into the cellular automaton slider-block model which allows blocks to fail at a reduced failure threshold for all subsequent failures during an event. We show that a critical point is reached beyond which the probability of a system-wide event scales with this weakening parameter. We provide a mapping to the percolation transition, and show that the values of the scaling exponents approach the values for mean-field percolation (spinodal nucleation) as lattice size L is increased for fixed R. We also examine the effect of the weakening parameter on the frequency-magnitude scaling relationship and the ergodic behavior of the model.     

Quasi-one-dimensional transport of nondegenerate electrons in two-dimensional systems with a fluctuation potential

A threshold vanishing of the Hall emf with decreasing gate voltage is observed at ≈ 77 K in semiconductor systems which are disordered as a result of a high built-in charge density near the plane of the 2D-electron channel. The effect is observed at a channel conductivity σ ≈e ²/h and is due to a transition to nondegenerate-electron transport via a 2D percolation cluster having a quasi-1D character of the conduction. We have established that the conductance of “short” structures, having a length of the order of the correlation length of a percolation cluster, equals ≈e ²/h per electron and is determined by isolated percolation paths having a lowered percolation threshold. These phenomena are a general property of disordered 2D systems. Pis’ma Zh. éksp. Teor. Fiz. 66, No. 10, 633–638 (25 November 1997)     

Magnetic susceptibility and inelastic electrical resistivity of GdZnxCu1−x alloys

The deviation from the Nordheim-Kurnakov rule and the anomalous behavior of spin-disordered electrical resistivity in quasi-binary GdZn (T _C =268 K)-GdCu (T _N =142 K) solid solutions is explained in effective medium approximation within percolation theory for the case of three phases, viz., ferro-, antiferro-, and paramagnetic. The strong increase of ρ at zinc concentrations x∼0.45 is attributed to the closeness of the system to the percolation threshold. The phase volumes calculated for the random-distribution case fit well to the concentration dependence of magnetic susceptibility. Fiz. Tverd. Tela (St. Petersburg) 40, 974–979 (June 1998)     

Percolation analysis of stochastic models of galactic evolution

The stochastic star formation model of galactic evolution can be cast as a problem of directed percolation, the time dimension being that along which the directed bonds exist. We study various aspects of this percolation, those of general interest for the percolation phase transition and those of particular importance for the astrophysical application. Both analytical calculations and computer simulations are provided and the results compared. Among the properties are: value of the percolation threshold, critical indices, percolation probability (star density) near and away from the critical point, local density, cluster sizes, effects of rotation (for disk galaxy models) on the percolation threshold. Astrophysical consequences of some of these properties are discussed, in particular the way in which general phase transition behavior contributes to spiral arm morphology. We look at 1 (space) + 1 (time), 2 + 1 and + 1 dimensions, the 2 + 1 case being of interest for disk galaxies.     

Fluctuation states and optical spectra of solid solutions with a strong isoelectronic perturbation

We propose an approach to describing the density of fluctuation states in a disordered solid solution with a strong perturbation introduced by isoelectronic substitution in the range of attraction-center concentrations below the threshold of percolation along the sites of a disordered sublattice. To estimate the number of localized states we use the results of lattice percolation theory. We describe a method for distinguishing, within the continuum percolation theory, among the various “radiating” states of the fluctuation-induced tail, states that form the luminescence band at weak excitation. We also establish the position of the band of radiating states in relation to the absorption band of the excitonic ground state and the mobility edge of the system. The approach is used to describe the optical spectra of the solid solution ZnSe_1−c Te_c, which at low Te concentrations can be interpreted as a system with strong scattering. We take into account the exciton-phonon interaction and show that the calculated and observed luminescence spectra of localized excitons are in good agreement with each other. Zh. éksp. Teor. Fiz. 115, 1039–1062 (March 1999)     

Ti-Al2O3金属陶瓷的渗流模型

用渗流理论讨论了Ti-Al₂O₃金属陶瓷的显微结构同宏观电导率的关系,发现渗流模型结果与实验结果是一致的。该Ti-Al₂O₃金属陶瓷的渗流阈值为(Ti的临界百分含量)ω^*=0.16,其临界指数即为三维连续渗流的临界指数值。 关键词：     

Long-range connections, quantum magnets and dilute contact processes

In this article, we briefly review the critical behaviour of a long-range percolation model in which any two sites are connected with a probability that falls off algebraically with the distance. The results of this percolation transition are used to describe the quantum phase transitions in a dilute transverse Ising model at the percolation threshold p_c of the long-range connected lattice. In the similar spirit, we propose a new model of a contact process defined on the same long-range diluted lattice and explore the transitions at p_c. The long-range nature of the percolation transition allows us to evaluate some critical exponents exactly in both the above models. Moreover, mean field theory is valid for a wide region of parameter space. In either case, the strength of Griffiths McCoy singularities are tunable as the range parameter is varied.     

Percolating granular superconductor in a magnetic field

An array of Josephson tunnel junctions is studied as a model for granular superconductors. We determine the critical line in the (H, T)-plane, separating the phase-coherent and incoherent region as a function of the magnetic field. Both positional disorder and junction-dilution are treated. Approaching the percolation threshold from above, the critical exponent of the upper critical field is found to be 0.60±0.03.     

Field percolation and high current density in 80/20 DyBa2Cu3O7 − x/Dy2BaCuO5 bulk magnetically textured composite ceramics

We measured the AC susceptibility of magnetically textured (123) 80%/211(20%) DyBaCuO composite in a special set-up in order to enhance the intergrain contribution. The synthesis process led to very clean “weak links” at grain boundaries. At the percolation threshold bulk shielding paths were such that the intergrain critical current density J_c was above 10⁵ A/cm². The field dependence of J_c was understood through an analytical form indicating a distribution of currents similar to the law of clusters at fracture/percolation thresholds.     

Connectivity and percolation in simulated grain-boundary networks

Random percolation theory is a common basis for modelling intergranular phenomena such as cracking, corrosion or diffusion. However, crystallographic constraints in real microstructures dictate that grain boundaries are not assembled at random. In this work a Monte Carlo method is used to construct physically realistic networks composed of high-angle grain boundaries that are susceptible to intergranular attack, as well as twin-variant boundaries that are damage resistant. When crystallographic constraints are enforced, the simulated networks exhibit triple-junction distributions that agree with experiment and reveal the non-random nature of grain-boundary connectivity. The percolation threshold has been determined for several constrained boundary networks and is substantially different from the classical result of percolation theory; compared with a randomly assembled network, about 50-75% more resistant boundaries are required to break up the network of susceptible boundaries. Triple-junction distributions are also shown to capture many details of the correlated percolation problem and to provide a simple means of ranking microstructures.     

Negative Poisson coefficient of fractal structures

On the basis of a fractal model the macroscopic elastic properties of an inhomogeneous medium with random structure have been determined. It is shown that if the ratio of the bulk moduli of the phases K ₂/K ₁→0, then the percolation threshold p _c the Poisson coefficient is equal to 0.2. A study of the behavior of a two-phase medium with negative Poisson coefficient is carried out. Fiz. Tverd. Tela (St. Petersburg) 41, 2147–2153 (December 1999)     

Multiplicity of Phase Transitions and Mean-Field Criticality on Highly Non-Amenable Graphs

We consider independent percolation, Ising and Potts models, and the contact process, on infinite, locally finite, connected graphs. It is shown that on graphs with edge-isoperimetric Cheeger constant sufficiently large, in terms of the degrees of the vertices of the graph, each of the models exhibits more than one critical point, separating qualitatively distinct regimes. For unimodular transitive graphs of this type, the critical behaviour in independent percolation, the Ising model and the contact process are shown to be mean-field type. For Potts models on unimodular transitive graphs, we prove the monotonicity in the temperature of the property that the free Gibbs measure is extremal in the set of automorphism invariant Gibbs measures, and show that the corresponding critical temperature is positive if and only if the threshold for uniqueness of the infinite cluster in independent bond percolation on the graph is less than 1. We establish conditions which imply the finite-island property for independent percolation at large densities, and use those to show that for a large class of graphs the q-state Potts model has a low temperature regime in which the free Gibbs measure decomposes as the uniform mixture of the q ordered phases. In the case of non-amenable transitive planar graphs with one end, we show that the q-state Potts model has a critical point separating a regime of high temperatures in which the free Gibbs measure is extremal in the set of automorphism-invariant Gibbs measures from a regime of low temperatures in which the free Gibbs measure decomposes as the uniform mixture of the q ordered phases. Received: 27 March 2000 / Accepted: 7 December 2000     

Connectedness percolation of elongated hard particles in an external field

A theory is presented of how orienting fields and steric interactions conspire against the formation of a percolating network of, in some sense, connected elongated colloidal particles in fluid dispersions. We find that the network that forms above a critical loading breaks up again at higher loadings due to interaction-induced enhancement of the particle alignment. Upon approach of the percolation threshold, the cluster dimensions diverge with the same critical exponent parallel and perpendicular to the field direction, implying that connectedness percolation is not in the universality class of directed percolation.     

The electrical conductivity of binary disordered systems,percolation clusters,fractals and related models

We review theoretical and experimental studies of the AC dielectric response of inhomogeneous materials, modelled as bond percolation networks, with a binary (conductor-dielectric) distribution of bond conductances. We first summarize the key results of percolation theory, concerning mostly geometrical and static (DC) transport properties, with emphasis on the scaling properties of the critical region around the percolation threshold. The frequency-dependent (AC) response of a general binary model is then studied by means of various approaches, including the effective-medium approximation, a scaling theory of the critical region, numerical computations using the transfer-matrix algorithm, and several exactly solvable deterministic fractal models. Transient regimes, related to singularities in the complex-frequency plane, are also investigated. Theoretical predictions are made more explicit in two specific cases, namely R-C and RL-C networks, and compared with a broad variety of experimental results, concerning, for example, granular composites, thin films, powders, microemulsions, cermets, porous ceramics and the viscoelastic properties of gels.     

Viscosity critical behaviour at the gel point in a 3d lattice model

Within a recently introduced model based on the bond-fluctuation dynamics, we study the viscoelastic behaviour of a polymer solution at the gelation threshold. We here present the results of the numerical simulation of the model on a cubic lattice: the percolation transition, the diffusion properties and the time autocorrelation functions have been studied. From both the diffusion coefficients and the relaxation times critical behaviour a critical exponent k for the viscosity coefficient has been extracted: the two results are comparable within the errors giving , in close agreement with the Rouse model prediction and with some experimental results. In the critical region below the transition threshold the time autocorrelation functions show a long-time tail which is well fitted by a stretched exponential decay. Received 20 December 1999 and Received in final form 18 February 2000