Nonuniversal critical behavior in the critical current of superconducting composites

Quantum fluctuations in percolating superconductors: an evolution with effective dimensionality

We investigate percolating films of superconducting nanoparticles and observe an evolution from superconducting to metallic to insulating states as the surface coverage of the nanoparticles is decreased. We demonstrate that this evolution is correlated with a reduction in the effective/dominant dimensionality of the system, from 2D to 1D to 0D, and that the physics in each regime is dominated by vortices, phase slips and tunnelling respectively. Finally we construct phase diagrams that map the various observed states as a function of surface coverage (or, equivalently, normal state resistance), temperature and measurement current.

Direct relations between morphology and transport in Boolean models

We study the relation of permeability and morphology for porous structures composed of randomly placed overlapping circular or elliptical grains, so-called Boolean models. Microfluidic experiments and lattice Boltzmann simulations allow us to evaluate a power-law relation between the Euler characteristic of the conducting phase and its permeability. Moreover, this relation is so far only directly applicable to structures composed of overlapping grains where the grain density is known a priori. We develop a generalization to arbitrary structures modeled by Boolean models and characterized by Minkowski functionals. This generalization works well for the permeability of the void phase in systems with overlapping grains, but systematic deviations are found if the grain phase is transporting the fluid. In the latter case our analysis reveals a significant dependence on the spatial discretization of the porous structure, in particular the occurrence of single isolated pixels. To link the results to percolation theory we performed Monte Carlo simulations of the Euler characteristic of the open cluster, which reveals different regimes of applicability for our permeability-morphology relations close to and far away from the percolation threshold.

Permeability of porous materials determined from the Euler characteristic

We study the permeability of quasi-two-dimensional porous structures of randomly placed overlapping monodisperse circular and elliptical grains. Measurements in microfluidic devices and lattice Boltzmann simulations demonstrate that the permeability is determined by the Euler characteristic of the conducting phase. We obtain an expression for the permeability that is independent of the percolation threshold and shows agreement with experimental and simulated data over a wide range of porosities. Our approach suggests that the permeability explicitly depends on the overlapping probability of grains rather than their shape.

Statistical physics of grain-boundary engineering

Percolation theory is now standard in the analysis of polycrystalline materials where the grain boundaries can be divided into two distinct classes, namely "good" boundaries that have favorable properties and "bad" boundaries that seriously degrade the material performance. Grain-boundary engineering (GBE) strives to improve material behavior by engineering the volume fraction c and arrangement of good grain boundaries. Two key percolative processes in GBE materials are the onset of percolation of a strongly connected aggregate of grains, and the onset of a connected path of weak grain boundaries. Using realistic polycrystalline microstructures, we find that in two dimensions the threshold for strong aggregate percolation c(SAP) and the threshold for weak boundary percolation c(WBP) are equivalent and have the value c(SAP) = c(WBP) =0.38 (1) , which is slightly higher than the threshold found for regular hexagonal grain structures, c(RH) =2 sin (pi/18) =0.347... . In three dimensions strong aggregate percolation and weak boundary percolation occur at different locations and we find c(SAP) =0.12 (3) and c(WBP) =0.77 (3) . The critical current in high T(c) materials and the cohesive energy in structural systems are related to the critical manifold problem in statistical physics. We develop a theory of critical manifolds in GBE materials, which has three distinct regimes: (i) low concentrations, where random manifold theory applies, (ii) critical concentrations where percolative scaling theory applies, and (iii) high concentrations, c> c(SAP) , where the theory of periodic elastic media applies. Regime (iii) is perhaps most important practically and is characterized by a critical length L(c) , which is the size of cleavage regions on the critical manifold. In the limit of high contrast epsilon-->0 , we find that in two dimensions L(c) proportional, gc/ (1-c) , while in three dimensions L(c) proportional, g exp [ b(0) c/ (1-c) ] / [c (1-c) ](1/2) , where g is the average grain size, epsilon is the ratio of the bonding energy of the weak boundaries to that of the strong boundaries, and b(0) is a constant which is of order 1. Many of the properties of GBE materials can be related to L(c) , which diverges algebraically on approach to c=1 in two dimensions, but diverges exponentially in that limit in three dimensions. We emphasize that GBE percolation processes and critical manifold behavior are very different in two dimensions as compared to three dimensions. For this reason, the use of two dimensional models to understand the behavior of bulk GBE materials can be misleading.

Transport on percolation clusters with power-law distributed bond strengths

The simplest transport problem, namely finding the maximum flow of current, or maxflow, is investigated on critical percolation clusters in two and three dimensions, using a combination of extremal statistics arguments and exact numerical computations, for power-law distributed bond strengths of the type P(sigma) approximately sigma(-alpha). Assuming that only cutting bonds determine the flow, the maxflow critical exponent v is found to be v(alpha)=(d-1)nu+1/(1-alpha). This prediction is confirmed with excellent accuracy using large-scale numerical simulation in two and three dimensions. However, in the region of anomalous bond capacity distributions (0< or =alpha< or =1) we demonstrate that, due to cluster-structure fluctuations, it is not the cutting bonds but the blobs that set the transport properties of the backbone. This "blob dominance" avoids a crossover to a regime where structural details, the distribution of the number of red or cutting bonds, would set the scaling. The restored scaling exponents, however, still follow the simplistic red bond estimate. This is argued to be due to the existence of a hierarchy of so-called minimum cut configurations, for which cutting bonds form the lowest level, and whose transport properties scale all in the same way. We point out the relevance of our findings to other scalar transport problems (i.e., conductivity).