Klaus Mecke

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 model 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.

Minimal surface scaffold designs for tissue engineering

Triply-periodic minimal surfaces are shown to be a more versatile source of biomorphic scaffold designs than currently reported in the tissue engineering literature. A scaffold architecture with sheetlike morphology based on minimal surfaces is discussed, with significant structural and mechanical advantages over conventional designs. These sheet solids are porous solids obtained by inflation of cubic minimal surfaces to sheets of finite thickness, as opposed to the conventional network solids where the minimal surface forms the solid/void interface. Using a finite-element approach, the mechanical stiffness of sheet solids is shown to exceed that of conventional network solids for a wide range of volume fractions and material parameters. We further discuss structure-property relationships for mechanical properties useful for custom-designed fabrication by rapid prototyping. Transport properties of the scaffolds are analyzed using Lattice-Boltzmann computations of the fluid permeability. The large number of different minimal surfaces, each of which can be realized as sheet or network solids and at different volume fractions, provides design flexibility essential for the optimization of competing design targets.

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.

Robust Pore Size Analysis of Filamentous Networks from Three-Dimensional Confocal Microscopy

We describe a robust method for determining morphological properties of filamentous biopolymer networks, such as collagen or other connective tissue matrices, from confocal microscopy image stacks. Morphological properties including pore size distributions and percolation thresholds are important for transport processes, e.g., particle diffusion or cell migration through the extracellular matrix. The method is applied to fluorescently labeled fiber networks prepared from rat-tail tendon and calf-skin collagen, at concentrations of 1.2, 1.6, and 2.4 mg/ml. The collagen fibers form an entangled and branched network. The medial axes, or skeletons, representing the collagen fibers are extracted from the image stack by threshold intensity segmentation and distance-ordered homotopic thinning. The size of the fluid pores as defined by the radii of largest spheres that fit into the cavities between the collagen fibers is derived from Euclidean distance maps and maximal covering radius transforms of the fluid phase. The size of the largest sphere that can traverse the fluid phase between the collagen fibers across the entire probe, called the percolation threshold, was computed for both horizontal and vertical directions. We demonstrate that by representing the fibers as the medial axis the derived morphological network properties are both robust against changes of the value of the segmentation threshold intensity and robust to problems associated with the point-spread function of the imaging system. We also provide empirical support for a recent claim that the percolation threshold of a fiber network is close to the fiber diameter for which the Euler index of the networks becomes zero.

Euler Characteristic and Related Measures for Random Geometric Sets

By an elementary calculation we obtain the exact mean values of Minkowksi functionals for a standard model of percolating sets. In particular, a recurrence theorem for the mean Euler characteristic recently put forward is shown to be incorrect. Related previous mathematical work is mentioned. We also conjecture bounds for the threshold density of continuum percolation, which are associated with the Euler characteristic.

Additivity, Convexity, and Beyond: Applications of Minkowski Functionals in Statistical Physics

The aim of this paper is to point out the importance of geometric functionals in statistical physics. Integral geometry furnishes a suitable family of morphological descriptors, known as Minkowski functionals, which are related to curvature integrals and do not only characterize connectivity (topology) but also content and shape (geometry) of spatial patterns. Since many physical phenomena depend essentially on the geometry of spatial structures, integral geometry may provide useful tools to study physical systems, in particular, in combination with the Boolean model, well known in stochastic geometry. This model generates random structures by overlapping ‘grains’ (spheres, sticks) each with arbitrary location and orientation. The integral geometric approach to stochastic structures in physics is illustrated by applying morphological measures to such diverse topics as complex fluids, porous media and pattern formation in dissipative systems. It is not intended to cover these topics completely but to emphasize unsolved physical problems related to geometric features and to present ideas and proposals for future work in possible collaboration with spatial statisticians and statistical physicists.

Statistical physics and spatial statistics

Statistical physics and spatial statistics , Statistical physics and spatial statistics , کتابخانه دیجیتال جندی شاپور اهواز

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.

Morphological fluctuations of large{scale structure: The PSCz survey

In a follow-up study to a previous analysis of the IRAS 1.2Jy catalogue, we quantify the morphological fluctuations in the PSCz survey. We use a variety of measures, among them the family of scalar Minkowski functionals. We confirm the existence of significant fluctuations that are discernible in volume-limited samples out to 200Mpc/h. In contrast to earlier findings, comparisons with cosmological N-body simulations reveal that the observed fluctuations roughly agree with the cosmic variance found in corresponding mock samples. While two-point measures, e.g. the variance of count-in-cells, fluctuate only mildly, the fluctuations in the morphology on large scales indicate the presence of coherent structures that are at least as large as the sample.In a follow{up study to a previous analysis of the IRAS 1.2 Jy catalogue, we quantify the morphological fluctuations in the PSCz survey. We use a variety of measures, among them the family of scalar Minkowski functionals. We conrm the existence of signicant fluctuations that are discernible in volume{limited samples out to 200h 1 Mpc. In contrast to earlier ndings, comparisons with cosmological N{body simulations reveal that the observed fluctuations roughly agree with the cosmic variance found in corresponding mock samples. While two{point measures, e.g. the variance of count{in{cells, fluctuate only mildly, the fluctuations in the morphology on large scales indicate the presence of coherent structures that are at least as large as the sample.

Minkowski tensors of anisotropic spatial structure

This paper describes the theoretical foundation of and explicit algorithms for a novel approach to morphology and anisotropy analysis of complex spatial structure using tensor-valued Minkowski functionals, the so-called Minkowski tensors. Minkowski tensors are generalizations of the well-known scalar Minkowski functionals and are explicitly sensitive to anisotropic aspects of morphology, relevant for example for elastic moduli or permeability of microstructured materials. Here we derive explicit linear-time algorithms to compute these tensorial measures for three-dimensional shapes. These apply to representations of any object that can be represented by a triangulation of its bounding surface; their application is illustrated for the polyhedral Voronoi cellular complexes of jammed sphere configurations and for triangulations of a biopolymer fibre network obtained by confocal microscopy. The paper further bridges the substantial notational and conceptual gap between the different but equivalent approaches to scalar or tensorial Minkowski functionals in mathematics and in physics, hence making the mathematical measure theoretic formalism more readily accessible for future application in the physical sciences.