Reconstructing complex materials via effective grain shapes

Fast Fourier transform and support-shift techniques for pore-scale microstructure classification using additive morphological measures

The Minkowski functionals, as the full set of additive morphological measures in three dimensions (3D) consisting of volume, surface area, mean curvature, and total curvature, can be calculated directly by evaluating the local contributions of vertices of a discrete structure. They are sensitive measures of microstructure, and for microstructures generated by a Boolean process, relate to their physical properties. In this work we introduce fast numerical techniques based on the additivity of the Minkowski functionals to derive fields of regional Minkowski measures over large regional support for large 3D data sets as generated, e.g., from x-ray tomography techniques. We demonstrate the application of these 3D feature fields to microstructure classification for a set of heterogeneous microstructures using a multivariate Gaussian mixture model and a thin-bedded sandstone. It is shown that for the case of a spatially heterogeneous Boolean process the internal boundaries of the generating process are recovered with high accuracy, while for the thin-bedded sandstone, compact partitions with clear layering are extracted.

Inferring statistical properties of 3D cell geometry from 2D slices

Although cell shape can reflect the mechanical and biochemical properties of the cell and its environment, quantification of 3D cell shapes within 3D tissues remains difficult, typically requiring digital reconstruction from a stack of 2D images. We investigate a simple alternative technique to extract information about the 3D shapes of cells in a tissue; this technique connects the ensemble of 3D shapes in the tissue with the distribution of 2D shapes observed in independent 2D slices. Using cell vertex model geometries, we find that the distribution of 2D shapes allows clear determination of the mean value of a 3D shape index. We analyze the errors that may arise in practice in the estimation of the mean 3D shape index from 2D imagery and find that typically only a few dozen cells in 2D imagery are required to reduce uncertainty below 2%. Even though we developed the method for isotropic animal tissues, we demonstrate it on an anisotropic plant tissue. This framework could also be naturally extended to estimate additional 3D geometric features and quantify their uncertainty in other materials.

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.

Grain sizing in porous media using Bayesian magnetic resonance

We introduce a Bayesian inference approach to analyze magnetic resonance data of granular solids. To characterize structure using magnetic resonance, it is usual to acquire data in k space which are then Fourier transformed to obtain an image. An alternative approach, adopted here, is to utilize the Rayleigh distribution observed in the signal intensity for a given k when a random selection of grains is measured in k space, to define a likelihood function for Bayesian analysis. This Bayesian likelihood function is used to noninvasively characterize grains within a porous medium on length scales below the practical resolution of magnetic resonance imaging. A pore size distribution is then calculated from the measured grain size distribution using a Monte Carlo approach. We demonstrate this general technique with specific examples of water-saturated rock cores.

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.

Minkowski tensor shape analysis of cellular, granular and porous structures

Predicting physical properties of materials with spatially complex structures is one of the most challenging problems in material science. One key to a better understanding of such materials is the geometric characterization of their spatial structure. Minkowski tensors are tensorial shape indices that allow quantitative characterization of the anisotropy of complex materials and are particularly well suited for developing structure-property relationships for tensor-valued or orientation-dependent physical properties. They are fundamental shape indices, in some sense being the simplest generalization of the concepts of volume, surface and integral curvatures to tensor-valued quantities. Minkowski tensors are based on a solid mathematical foundation provided by integral and stochastic geometry, and are endowed with strong robustness and completeness theorems. The versatile definition of Minkowski tensors applies widely to different types of morphologies, including ordered and disordered structures. Fast linear-time algorithms are available for their computation. This article provides a practical overview of the different uses of Minkowski tensors to extract quantitative physically-relevant spatial structure information from experimental and simulated data, both in 2D and 3D. Applications are presented that quantify (a) alignment of co-polymer films by an electric field imaged by surface force microscopy; (b) local cell anisotropy of spherical bead pack models for granular matter and of closed-cell liquid foam models; (c) surface orientation in open-cell solid foams studied by X-ray tomography; and (d) defect densities and locations in molecular dynamics simulations of crystalline copper.

Diffusion amid random overlapping obstacles: similarities, invariants, approximations

Efficient and accurate numerical techniques are used to examine similarities of effective diffusion in a void between random overlapping obstacles: essential invariance of effective diffusion coefficients (D(eff)) with respect to obstacle shapes and applicability of a two-parameter power law over nearly entire range of excluded volume fractions (φ), except for a small vicinity of a percolation threshold. It is shown that while neither of the properties is exact, deviations from them are remarkably small. This allows for quick estimation of void percolation thresholds and approximate reconstruction of D(eff) (φ) for obstacles of any given shape. In 3D, the similarities of effective diffusion yield a simple multiplication "rule" that provides a fast means of estimating D(eff) for a mixture of overlapping obstacles of different shapes with comparable sizes.

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.

Boolean reconstructions of complex materials: Integral geometric approach

We show that for the Boolean model of random composite media one can, from a single image of a system at any particle fraction, define a set of parameters which allows one to accurately reconstruct the medium for all other phase fractions. The morphological characterization is based on a family of measures known in integral geometry which provides powerful formulas for the Boolean model. The percolation thresholds of either phase are obtained with good accuracy. From the reconstructions one can subsequently predict property curves for the material across all phase fractions from the single three-dimensional image. We illustrate this for transport and mechanical properties of complex Boolean systems and for experimental sandstone samples.

Towards precise prediction of transport properties from synthetic computer tomography of reconstructed porous media

Transport properties of a multiscale carbonate rock are predicted from pore scale models, reconstructed using a continuum geometrical modeling technique. The method combines crystallite information from two-dimensional high-resolution images with sedimentary correlations from a three-dimensional low-resolution microcomputed tomography ( micro-CT) image to produce a rock sample with calibrated porosity, structural correlation, and transport properties at arbitrary resolutions. Synthetic micro-CT images of the reconstructed model match well with experimental micro-CT images at different resolutions, making it possible to predict physical transport parameters at higher resolutions.

Characterization of the dynamics of block copolymer microdomains with local morphological measures

We investigate the structure formation in thin films of cylinder forming block copolymers. With in situ scanning probe microscopy image sequences can be recorded with high temporal (2 min per frame) and spatial (10 nm) resolution. We compare different image processing methods for quantitative analysis of the large amount of data. Computing local Minkowski functionals yields local geometrical and morphological information about the observed structures and enables us to track their evolution with time. An alternative characterization method is to reduce the gray scale images to their skeleton and to classify and count the branching points of the skeletonized structure. We tracked the temporal evolution of these measures and computed correlation functions.

Stochastic multiscale model for carbonate rocks

A multiscale model for the diagenesis of carbonate rocks is proposed. It captures important pore scale characteristics of carbonate rocks: wide range of length scales in the pore diameters; large variability in the permeability; and strong dependence of the geometrical and transport parameters on the resolution. A pore scale microstructure of an oolithic dolostone with generic diagenetic features is successfully generated. The continuum representation of a reconstructed cubic sample of side length 2mm contains roughly 42 x 10{6} crystallites and pore diameters varying over many decades. Petrophysical parameters are computed on discretized samples of sizes up to 1000{3}. The model can be easily adapted to represent the multiscale microstructure of a wide variety of carbonate rocks.

Void structure and cage fluctuations in simulations of concentrated suspensions

We analyse the mesoscopic structure and structural fluctuations in simulated high-concentration hard sphere colloidal suspensions by means of calculations based on the void space. We show that remoteness, a quantity measuring the scale of spaces, is useful in studying crystallization, since ordering of the particles involves a change in the way empty space is distributed. Calculation of remoteness also allows breakdown of the system into mesoscopic neighbour sets. In the case of crystallizing systems, statistics of mean remoteness and local volume fraction in these neighbour sets are consistent with nuclei forming at locally higher concentration, nucleation involves increased heterogeneity of the system, as previously demonstrated in experiments with model colloidal particles. Meanwhile in dense amorphous systems, local volume fractions in neighbour sets reveal significant details of mesoscale structural fluctuations, indicating that abrupt dilations and compressions of local regions may be important physical components of cage fluctuations in the colloidal glass.

Morphological thermodynamics of fluids: shape dependence of free energies

We examine the dependence of a thermodynamic potential of a fluid on the geometry of its container. If motion invariance, continuity, and additivity of the potential are satisfied, only four morphometric measures are needed to describe fully the influence of an arbitrarily shaped container on the fluid. These three constraints can be understood as a more precise definition for the conventional term extensive and have as a consequence that the surface tension and other thermodynamic quantities contain, aside from a constant term, only contributions linear in the mean and Gaussian curvature of the container and not an infinite number of curvatures as generally assumed before. We verify this numerically in the entropic system of hard spheres bounded by a curved wall.

Analysis of 3D bone ingrowth into polymer scaffolds via micro-computed tomography imaging

This paper illustrates the utility of micro-computed tomography (micro-CT) to study the process of tissue engineered bone growth. A micro-CT facility for imaging and visualising biomaterials in three dimensions (3D) is described. The facility is capable of acquiring 3D images made up of 2000(3) voxels on specimens up to 60mm in extent with resolutions down to 2 microm. This allows the 3D structure of tissue engineered materials to be imaged across three orders of magnitude of detail. The capabilities of micro-CT are demonstrated by imaging the Haversian network within human femoral cortical bone (distal diaphysis) and bone ingrowth into a porous scaffold at varying resolutions. Phase identification combined with 3D visualisation enables one to observe the complex topology of the canalicular system of the cortical bone. Imaging of the tissue engineered bone at a scale of 1cm and resolutions of 10 microm allows visualisation of the complex ingrowth of bone into the polymer scaffold. Further imaging at 2 microm resolution allows observation of bone ultra-structure. These observations illustrate the benefits of tomography over traditional techniques for the characterisation of bone morphology and interconnectivity and performs a complimentary role to current histomorphometric techniques.

Diffusion of hard sphere fluids in disordered media: a molecular dynamics simulation study

Molecular dynamic simulations are reported for the static and dynamic properties of hard sphere fluids in matrices (or media) composed of quenched hard spheres. The effect of fluid and matrix density, matrix structure, and fluid to matrix sphere size ratio on the static and dynamic properties is studied using discontinuous molecular dynamics. The matrix density has a stronger effect on the self-diffusion coefficient than the fluid density, especially at high matrix densities where the geometric constraints due to the quenched spheres are significant. When the ratio of the size of the fluid spheres to that of the matrix spheres is equal to or greater than one, the diffusion increases as the fluid density is increased, at constant total volume fraction. This trend is however reversed if the ratio is smaller than one. Different methods of generating the matrix have a very strong effect on the dynamic properties even though the static correlations are similar. An analysis of the single-chain structure factor of the particle trajectories shows a change in the particle diffusive behavior at different time scales, suggestive of a hopping mechanism, although normal diffusion is recovered at long times. At high matrix densities, there is considerable heterogeneity in the diffusion of the fluid particles. The simulations demonstrate that the correlations in the matrix play a significant role on the diffusion of fluid spheres. For example, the diffusion constant in matrices constructed by different methods can be an order of magnitude different even though the pair correlation functions are almost identical.