Using convex quadratic programming to model random media with Gaussian random fields

Three-dimensional reconstruction of liquid phases in disordered mesopores usingin situsmall-angle scattering

Small-angle scattering of X-rays (SAXS) or neutrons is one of the few experimental methods currently available for thein situanalysis of phenomena in mesoporous materials at the mesoscopic scale. In the case of disordered mesoporous materials, however, the main difficulty of the method lies in the data analysis. A stochastic model is presented, which enables one to reconstruct the three-dimensional nanostructure of liquids confined in disordered mesopores starting from small-angle scattering data. This so-called plurigaussian model is a multi-phase generalization of clipped Gaussian random field models. Its potential is illustrated through the synchrotron SAXS analysis of a gel permeated with a critical nitrobenzene/hexane solution that is progressively cooled below its consolute temperature. The reconstruction brings to light a wetting transition whereby the nanostructure of the pore-filling liquids passes from wetting layers that uniformly cover the solid phase of the gel to plugs that locally occlude the pores. Using the plurigaussian model, the dewetting phenomenon is analyzed quantitatively at the nanometre scale in terms of changing specific interface areas, contact angle and specific length of the triple line.

Fractal heterogeneous media

A method is presented for generating compact fractal disordered media by generalizing the random midpoint displacement algorithm. The obtained structures are invasive stochastic fractals, with the Hurst exponent varying as a continuous parameter, as opposed to lacunar deterministic fractals, such as the Menger sponge. By employing the detrending moving average algorithm [A. Carbone, Phys. Rev. E 76, 056703 (2007)], the Hurst exponent of the generated structure can be subsequently checked. The fractality of such a structure is referred to a property defined over a three-dimensional topology rather than to the topology itself. Consequently, in this framework, the Hurst exponent should be intended as an estimator of compactness rather than of roughness. Applications can be envisaged for simulating and quantifying complex systems characterized by self-similar heterogeneity across space. For example, exploitation areas range from the design and control of multifunctional self-assembled artificial nanostructures and microstructures to the analysis and modeling of complex pattern formation in biology, environmental sciences, geomorphological sciences, etc.

Morphological models of complex ordered materials based on inhomogeneously clipped Gaussian fields

Clipping a Gaussian random field at a level that is position-dependent yields statistically inhomogeneous morphologies, relevant to many ordered nanostructured materials. The one-point and two-point probability functions of the morphology are derived, as well as a general relation between the specific surface area and the gradient of the clipping function. The general results are particularized for the comprehensive analysis of small-angle x-ray scattering and nitrogen adsorption of SBA-15 ordered mesoporous silica.

Three-dimensional reconstruction of statistically optimal unit cells of polydisperse particulate composites from microtomography

In this paper, we present a systematic approach for characterization and reconstruction of statistically optimal representative unit cells of polydisperse particulate composites. Microtomography is used to gather rich three-dimensional data of a packed glass bead system. First-, second-, and third-order probability functions are used to characterize the morphology of the material, and the parallel augmented simulated annealing algorithm is employed for reconstruction of the statistically equivalent medium. Both the fully resolved probability spectrum and the geometrically exact particle shapes are considered in this study, rendering the optimization problem multidimensional with a highly complex objective function. A ten-phase particulate composite composed of packed glass beads in a cylindrical specimen is investigated, and a unit cell is reconstructed on massively parallel computers. Further, rigorous error analysis of the statistical descriptors (probability functions) is presented and a detailed comparison between statistics of the voxel-derived pack and the representative cell is made.

Necessary and sufficient conditions for the two-point phase probability function of two-phase random media

Constructing realizations of random media with a specified two-point phase probability function S 2 has attracted considerable attention in the recent literature. However, little is known about conditions under which a prescribed S 2 is realizable. The only known necessary and sufficient condition, due to McMillan, involves a class of square matrices, called corner-positive matrices, about which almost nothing is known except their definition. As a result, McMillan's theorem has gone mostly unused in the literature for over 50 years. In this paper, we present a general decomposition formula for corner-positive matrices, which allows McMillan's theorem to be written in a significantly more tractable and testable form. We also connect McMillan's theorem with many known but heretofore unrelated necessary conditions on S 2 , extending many of these conditions.