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.

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.

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).

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.

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.