Site Percolation Theory Research Articles

An evaluation of the ionic conductivity in AgI-doped glasses: The graded-percolation model

Various percolation theories, Effective Medium Percolation Theory (EMPT), Site Percolation Theory (SPT) and Bond Percolation Theory (BPT), have been applied to the ionic conductivity of a series of AgI-doped glasses in order to reexamine the proposed “AgI-microdomain” model for these glasses. In contrast to previous reports the possibility that percolation effects do control the conductivity in these glasses, it is found here that SPT predicts a conductivity catastrophe at x ≈ 0.3 at the percolation threshold and the EMPT model predicts too sharp a threshold as well. Both of these effects are not observed in any of the AgI-doped glass studied to date. Variation from the percolation theories and experimental data, however, has been used to support two new features of the ionic conduction in these glasses. At low AgI contents, sharp increases in conductivity are proposed to arise from a “co-connectivity” of the Ag+ cations and at higher AgI contents failure of the conductivity to reach the value of α-AgI is associated with the distortion of the I-polyhedra away from those in α-AgI.

Percolation Phenomenon for Dissolution of Sodium Borosilicate Glasses in Aqueous Solutions

A percolation phenomenon has been discovered for the dissolution of sodium borosilicate glasses in aqueous solutions under low‐pH conditions (pH ∼2). The glasses fall into one of two categories which are markedly different in the dissolution behavior, in terms of whether the Si/B ratio in the glass network exceeds a threshold value or not. The transition from one of the categories to the other is very sharp when the Si/B ratio is changed as the key parameter. The qualitative aspects of the experimental observation can be explained very well by a simple model combined with the site percolation theory.

Deformation consolidation of metal powders containing steel inclusions

Uniaxial consolidation experiments have been conducted at room temperature for two deformable metal powders (1100 Al and Pb5%Sb) containing various amounts of spherical steel inclusions. The experiments illustrate that the inclusion phase offers little constraint to matrix deformation at volume fractions <0.20, but produces a rapidly increasing constraint at larger volume fractions. Two constraining mechanisms have been identified through microstructural observations: (1) the matrix must be deformed more within the composite because of the excluded volume associated with the packing of particles and inclusions of different sizes, and (2) the inclusions form a continuous touching network (predicted with site percolation theory and direct observations of deformation flats on the steel spheres) which supports a portion of the applied stress and thus, partially “shields” the deformable phase from the total applied consolidation pressure. An analysis is presented to separate the contributions of each mechanism and shows that both mechanisms contribute comparable amounts to the constraint of matrix consolidation. It is suggested that because the inclusion network supports a significant portion of the applied pressure and releases its elastic strain in a non-linear manner (as described by Hertzian theory), the matrix is placed in tension when the applied pressure is released after consolidation.

Sphere and distance models for binary disordered systems

Abstract First-neighbour geometrical models are presented and shown to provide a good approximation for the structural description of binary disordered systems with metallic, covalent or ionic bonding. Sphere mixture models taking into account atomic size and chemical order effects are first studied and analytical expressions for the partial coordination numbers are derived which only involve packing fraction, diameter ratio, concentration and chemical order parameter. Relations between the concentration thresholds separating different chemical order regimes and site percolation theory are established. Close-packed sphere mixtures are finally shown to be a good representation of liquid or glassy alloys. Hole sphere models are then introduced and shown to provide a satisfactory approximation to the structure of loose-packed covalent glasses of silicon and germanium. Finally, first-neighbour distance models are presented and the variations of their partial structure factors with non-additivity parameter are...

Densities of melts in the system CaMgSi2O6-CaAl2Si2O8 at low and high pressures, and their structural significance

Density measurements have been carried out on the melt system diopside-anorthite from room temperature to 1600° C at 1 atm, and from 1400° C to 1800° C at pressures up to 20 Kb. The densities were determined based on the dilatometric curve and density at 22° C for lowtemperatures, the double-bob Archimedean method for high-temperatures at 1 atm, and on the sinking and floating spheres method for high-pressure conditions. The results at 1 atm indicate that the thermal expansion coefficient of the glassy state is almost constant, while that of the liquid state decreases with increasing temperature. Density decreased with increasing anorthite content for both glassy and liquid states. Melts in the liquid-state mix ideally with respect to volume, while the glassy state exhibits a maximum excess volume at Di30An70. Density-pressure relations clearly show a density reversion between diopsiderich and anorthite-rich melts; the anorthite-rich melt becomes denser than diopside-rich melt at pressures above 8 kb. The free volumes of both the liquid and glassy states decreased with increasing anorthite content. Isothermal compressibilities and the hard-sphere diameter have been calculated based on the hard-sphere liquid model using thermal expansion coefficients and surface tension data. Calculated compressibilities for diopside-rich melt (Di:>Di60) agreed well with the experimental data, while calculated and observed compressibilities for anorthite-rich melt did not. This evidence indicates that diopside melt may be regarded as a discrete-melt composed of small constituent units (about 10 A in average diameter) and much interstitial space, while anorthite melt is a three-dimensional network melt with little interstitial space. The critical composition Di60An40 is similar to that of the eutectic and corresponds to breaks between composition and other physical properties. It is proposed that the composition may reflect a kind of “critical state” in the substitution of the “continuous structure” of anorthite melt for the “discrete structure” of diopside melt. The critical state may be interpreted based on the site-percolation theory.

Constrained network model for predicting densification behavior of composite powders

Inert particles that do not contribute to the densification of a composite powder compact are visualized as located on network sites; the network is defined by the distribution of the particles in the powder matrix. Because the distances between neighboring network sites are not identical, the strain produced by the sintering powder between all inert particle pairs cannot be the same as that for the powder compact without the inert particles. The constrained network model is based on the hypothesis that the densification of the composite will be constrained by the network and will mimic that of the network. The shrinkage of the network, and thus the densification of the composite, is estimated with a periodic network. A distance between the minimum and maximum site pairs within the unit cell defines the distance between site pairs in the random network where the powder between the particles densifies in the same manner as that for the powder without the inert particle. When the particles form a continuous touching network, composite shrinkage and densification is nil. The chosen lattice must also conform to this condition. A simple relation was developed relating the densification behavior of the composite to that of the matrix without the inert particles and the parameter associated with the chosen lattice. By choosing the lattice formed by the tetrakaidecahedron unit cell (volume fraction of particles for a touching network = 0.277), remarkable agreement was achieved for the experimental data concerning the densification behavior of the ZnO/SiC composite system reported by De Jonghe et al. [L. C. De Jonghe, M. N. Rahaman, and C. H. Hsueh, Acta Metall. 34, 1467 (1986)]. The universal nature of this lattice for other composites is discussed with respect to site percolation theory. The application of this concept to powder compacts containing either whiskers or agglomerates is briefly discussed.

Superconductivity in Cu-V-Si alloys

The superconducting properties of copper ternary alloys containing small amounts of vanadium and silicon have been investigated. The wire samples of three alloys: Cu/sub 90/V/sub 7.5/Si/sub 2.5/, Cu/sub 88/V/sub 9/Si/sub 3/, and Cu/sub 84/V/sub 12/Si/sub 4/: show a resistive superconducting transition around 17 K, the characteristic transition temperature of the A15 compound V/sub 3/Si, followed by a plateau and a second transition between 5 and 10 K. The resistivity, however, does not drop to zero down to 2.5 K. The maximum drop in resistivity is observed for the alloy containing minimum amounts of vanadium and silicon. Complete superconductivity is absent in these alloys for two reasons. First the effective superconducting volume fraction in these alloys is well below the threshold value obtained by either the effective medium theory or the site percolation theory for solid conduction. Second, the superconducting phase does not precipitate in a fine structure in the matrix and a strong proximity effect does not operate. Experimental evidence confirms the formation of brittle phase V/sub 3/Si in the cast alloys which does not get elongated to the same extent as the matrix when rolled or drawn. This reduces the effective volume fraction of V/sub 3/Si in the drawnmore » wires. Annealing of these wires causes the superconducting particles to coalesce and grow in size, increasing the interparticle distances to the detriment of superconductivity in these alloys.« less

Two-step superconducting transition in Cu-V-Si alloys

Copper ternary alloys containing small amounts of vanadium and silicon exhibit a two-step superconducting resistive transition. The first transition occurs around 17K, the transition temperature of beta -W V3Si, followed by a plateau and a second transition around 10K. The resistivity, however, does not drop to zero down to 2.5K. Reduction of the wire diameter causes the two transitions to shift to lower temperatures. Complete superconductivity in these specimens is absent for two reasons. Firstly, the superconducting volume fraction present in these alloy-wires is below the threshold given by either the effective-medium theory or the site percolation theory. Secondly, the superconducting phase V3Si does not precipitate in copper matrix in a fine structure and the proximity effect does not operate strongly. Annealing causes the superconducting particles to coalesce and grow in size and suppresses the proximity effect and superconductivity further in these alloy wires.

Origin of superconductivity in copper niobium alloys

Superconductivity in copper-rich niobium alloys has been investigated. A controlled cooling technique was developed to provide homogeneous distributions of niobium in the copper. Two distinct microstructures were observed depending on the cooling rate. For low cooling rates (&amp;lt;700 °C/sec) and low niobium concentrations a subgrain structure of niobium is evident. These samples are superconducting with low critical current density (&amp;lt;10 A/cm2). For faster cooling rates (≳1000 °C/sec) the subgrain structure is eliminated and a critical concentration for the observation of superconductivity is observed experimentally at 10–12 wt% Nb (7–9 at.% Nb). The experimentally determined critical concentration is explained by a model based on site percolation theory modified for the shape of the precipitate. The model suggests that continuous niobium chains in the copper matrix lead to the observed superconducting properties. The proximity effect, previously suggested as the basis for superconductivity in these alloys, appears to have little contribution to the appearance of superconductivity in these as-cast samples. The only contribution, if any, is to bridge gaps in the otherwise continuous niobium network.