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.

Circular dichroism in biological photonic crystals and cubic chiral nets.

Nature provides impressive examples of chiral photonic crystals, with the notable example of the cubic so-called srs network (the label for the chiral degree-three network modeled on SrSi2) or gyroid structure realized in wing scales of several butterfly species. By a circular polarization analysis of the band structure of such networks, we demonstrate strong circular dichroism effects: The butterfly srs microstructure, of cubic I4(1)32 symmetry, shows significant circular dichroism for blue to ultraviolet light, that warrants a search for biological receptors sensitive to circular polarization. A derived synthetic structure based on four like-handed silicon srs nets exhibits a large circular polarization stop band of a width exceeding 30%. These findings offer design principles for chiral photonic devices.

3D structural analysis: sensitivity of Minkowski functionals.

The Minkowski functionals, a family of statistical measures based on the Euler-Poincaré characteristic of n-dimensional space, are the complete set of additive morphological measures and can be simply calculated from local contributions. As such, they have found a wide range of applications. We consider the effects of distortions (drift, noise and blurring) on the morphological properties of complex random models, representative of a wide range of structure. Such distortions arise experimentally in imaging techniques due to diffraction, absorption and sample drift. The question is asked, how critically these distortions effect image quality as measured by the Minkowski functionals. Defining a length scale based on the two-point correlation function, we consider how distortion at different scales can lead to quantitative errors in morphological measures.

Tensorial Minkowski functionals and anisotropy measures for planar patterns.

Quantitative measures for anisotropic characteristics of spatial structure are needed when relating the morphology of microstructured heterogeneous materials to tensorial physical properties such as elasticity, permeability and conductance. Tensor-valued Minkowski functionals, defined in the framework of integral geometry, provide a concise set of descriptors of anisotropic morphology. In this article, we describe the robust computation of these measures for microscopy images and polygonal shapes. We demonstrate their relevance for shape description, their versatility and their robustness by applying them to experimental data sets, specifically microscopy data sets of non-equilibrium stationary Turing patterns and the shapes of ice grains from Antarctic cores.

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.

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.

Thermal noise influences fluid flow in thin films during spinodal dewetting.

Experiments on dewetting thin polymer films confirm the theoretical prediction that thermal noise can strongly influence characteristic time scales of fluid flow and cause coarsening of typical length scales. Comparing the experiments with deterministic simulations, we show that the Navier-Stokes equation has to be extended by a conserved bulk noise term to accomplish the observed spectrum of capillary waves. Because of thermal fluctuations the spectrum changes from an exponential to a power law decay for large wave vectors. Also the time evolution of the typical wave vector of unstable perturbations exhibits noise-induced coarsening that is absent in deterministic hydrodynamic flow.

Morphological characterization of point patterns.

A triplet of function s for the statistical characterization of planar point patterns is introduced. They are related to the integral-geometric quantities area, boundary length and Euler number of patterns of discs centred at the given points. These functions are able to give information on the distribution of a given point pattern which the traditional summary statistics of point process theory do not offer and so can lead to an improved statistical description. The paper describes the statistical estimation of the new characteristics. Some examples illustrate their application in the exploratory analysis of point patterns of tree positions in forests, in comparison to results obtained by means of second-order and distance characteristics.

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.

Reconstructing complex materials via effective grain shapes.

We introduce a powerful method based on integral geometry and the Kac theorem for the spectrum of the Laplace operator to define the effective shape of an inclusion in a system made up of a distribution of arbitrarily shaped constituents. Reconstructing the microstructure using the effective inclusion shape leads to an excellent match to the percolation thresholds and to the mechanical and transport properties across all phase fractions. Use of the equivalent shape in effective medium formulations leads to good predictions. The method is verified for a sedimentary rock sample.

Hard-sphere fluids in contact with curved substrates.

The properties of a hard-sphere fluid in contact with hard-spherical and cylindrical walls are studied. Rosenfeld's density functional theory (DFT) is applied to determine the density profile and surface tension gamma for wide ranges of radii of the curved walls and densities of the hard-sphere fluid. Particular attention is paid to investigate the curvature dependence and the possible existence of a contribution to gamma which is proportional to the logarithm of the radius of curvature. Moreover, by treating the curved wall as a second component at infinite dilution, we provide an analytical expression for the surface tension of a hard-sphere fluid close to arbitrary hard convex walls. The agreement between the analytical expression and DFT is good. Our results show no signs for the existence of a logarithmic term in the curvature dependence of gamma.

X-ray synchrotron study of liquid-vapor interfaces at short length scales: effect of long-range forces and bending energies.

We have investigated the small-scale structure of the liquid-vapor interface using synchrotron x-ray scattering for liquids with different molecular structures and interactions. The effective momentum-dependent surface energy first decreases from its macroscopic value due to the effect of long-range forces, and then increases with increasing wave vector. The results are analyzed using a recent density functional theory. The large wave-vector increase is attributed to a bending energy for which local and nonlocal contributions are equally important.

Entropic torque.

Quantitative predictions are presented of a depletion-induced torque and force acting on a single colloidal hard rod immersed in a solvent of hard spheres close to a planar hard wall. This torque and force, which are entirely of entropic origin, may play an important role for the key-lock principle, according to which a biological macromolecule (the key) is functional only in a particular orientation with respect to a cavity (the lock).

Adsorption isotherms of hydrogen: the role of thermal fluctuations.

It is shown that experimentally obtained isotherms of adsorption on solid substrates may be completely reconciled with Lifshitz theory when thermal fluctuations of the free film surface are taken into account. This is demonstrated for hydrogen adsorbed on gold as a model system. Analysis of the fluctuation contributions allows one to determine the surface tension of the free hydrogen film as a function of film thickness. It is found to decrease sharply for film thicknesses below seven atomic layers.

Euler-Poincaré characteristics of classes of disordered media.

We consider a family of statistical measures based on the Euler-Poincaré characteristic of n-dimensional space that are sensitive to the morphology of disordered structures. These measures embody information from every order of the correlation function but can be calculated simply by summing over local contributions. We compute the evolution of the measures with density for a range of disordered microstructural models; particle-based models, amorphous microstructures, and cellular and foamlike structures. Analytic results for the particle-based models are given and the computational algorithm verified. Computational results for the different microstructures exhibit a range of qualitative behavior. A length scale is derived based on two-point autocorrelation functions to allow qualitative comparison between the different structures. We compute the morphological parameters for the experimental microstructure of a sandstone sample and compare them to three common stochastic model systems for porous media. None of the statistical models are able to accurately reproduce the morphology of the sandstone.

Reduction in the surface energy of liquid interfaces at short length scales

Liquid-vapour interfaces, particularly those involving water, are common in both natural and artificial environments. They were first described as regions of continuous variation of density, caused by density fluctuations within the bulk phases. In contrast, the more recent capillary-wave models assumes a step-like local density profile across the liquid-vapour interface, whose width is the result of the propagation of thermally excited capillary waves. The model has been validated for length scales of tenths of micrometres and larger, but the structure of liquid surfaces on submicrometre length scales--where the capillary theory is expected to break down--remains poorly understood. Here we report grazing-incidence X-ray scattering experiments that allow for a complete determination of the free surface structure and surface energy for water and a range of organic liquids. We observe a large decrease of up to 75% in the surface energy of submicrometre waves that cannot be explained by capillary theory, but is in accord with the effects arising from the non-locality of attractive intermolecule interactions as predicted by a recent density functional theory. Our data, and the results of comparable measurements on liquid solutions, metallic alloys, surfactants, lipids and wetting films should thus provide a stringent test for any new theories that attempt to describe the structure of liquid interfaces with nanometre-scale resolution.

Effective Hamiltonian for liquid-vapor interfaces.

Starting from a density functional theory for inhomogeneous fluids we derive an effective Hamiltonian for liquid-vapor interfaces of simple fluids which goes beyond the common phenomenological capillary-wave description. In contrast to other approaches we take into account the long-ranged power-law decay of the dispersion forces between the fluid particles which changes the functional form of the wave-vector-dependent surface tension qualitatively. In particular, we find two different forms of the bending rigidity for the capillary waves, a negative one for small wave vectors determined by the long-ranged dispersion forces and a positive rigidity for large wave vectors due to the distortions of the intrinsic density profile in the vicinity of the locally curved interface. The differences to the standard capillary-wave theory and the relevance of these results for the interpretation of scattering experiments are discussed.

Spinodal dewetting in liquid crystal and liquid metal films

Theory predicts that dewetting of a homogeneous liquid film from a solid surface may proceed via unstable surface waves on the liquid. This phenomenon, usually termed spinodal dewetting, has been sought after in many systems. Observations in liquid crystal and liquid metal films showed that, as expected, the emerging structures were similar to those found for spinodal decomposition in mixtures. Certain differences, however, could be attributed to peculiarities of the wetting forces in these two dissimilar systems, thereby demonstrating the role of nonlinearities inherent in the wetting forces.