Searcharxiv⌕ Search

arXiv subjects

Klaus Mecke

Publications and source records attributed to Klaus Mecke.

At least 37 records · Page 2Linked to original sources
arXiv2013

Group Theory of Circular-Polarization Effects in Chiral Photonic Crystals with Four-Fold Rotation Axes, Applied to the Eight-Fold Intergrowth of Gyroid Nets

We use group or representation theory and scattering matrix calculations to derive analytical results for the band structure topology and the scattering parameters, applicable to any chiral photonic crystal with body-centered cubic symmetry I432 for circularly-polarised incident light. We demonstrate in particular that all bands along the cubic [100] direction can be identified with the irreducible representations E+/-,A and B of the C4 point group. E+ and E- modes represent the only transmission channels for plane waves with wave vector along the ? line, and can be identified as non-interacting transmission channels for right- (E-) and left-circularly polarised light (E+), respectively. Scattering matrix calculations provide explicit relationships for the transmission and reflectance amplitudes through a finite slab which guarantee equal transmission rates for both polarisations and vanishing ellipticity below a critical frequency, yet allowing for finite rotation of the polarisation plane. All results are verified numerically for the so-called 8-srs geometry, consisting of eight interwoven equal-handed dielectric Gyroid networks embedded in air. The combination of vanishing losses, vanishing ellipticity, near-perfect transmission and optical activity comparable to that of metallic meta-materials makes this geometry an attractive design for nanofabricated photonic materials.

physics.optics↗
arXiv2013

Group Theory of Chiral Photonic Crystals with 4-fold Symmetry: Band Structure and S-Parameters of Eight-Fold Intergrown Gyroid Nets

The Single Gyroid, or srs, nanostructure has attracted interest as a circular-polarisation sensitive photonic material. We develop a group theoretical and scattering matrix method, applicable to any photonic crystal with symmetry I432, to demonstrate the remarkable chiral-optical properties of a generalised structure called 8-srs, obtained by intergrowth of eight equal-handed srs nets. Exploiting the presence of four-fold rotations, Bloch modes corresponding to the irreducible representations E- and E+ are identified as the sole and non-interacting transmission channels for right- and left-circularly polarised light, respectively. For plane waves incident on a finite slab of the 8-srs, the reflection rates for both circular polarisations are identical for all frequencies and transmission rates are identical up to a critical frequency below which scattering in the far field is restricted to zero grating order. Simulations show the optical activity of the lossless dielectric 8-srs to be large, comparable to metallic metamaterials, demonstrating its potential as a nanofabricated photonic material.

physics.optics↗
arXiv2013

Minkowski tensor density formulas for Boolean models

A stationary Boolean model is the union set of random compact particles which are attached to the points of a stationary Poisson point process. For a stationary Boolean model with convex grains we consider a recently developed collection of shape descriptors, the so called Minkowski tensors. By combining spatial and probabilistic averaging we define Minkowski tensor densities of a Boolean model. These densities are global characteristics of the union set which can be estimated from observations. In contrast local characteristics like the mean Minkowski tensor of a single random particle cannot be observed directly, since the particles overlap. We relate the global to the local properties by density formulas for the Minkowski tensors. These density formulas generalize the well known formulas for intrinsic volume densities and are obtained by applying results from translative integral geometry. For an isotropic Boolean model we observe that the Minkowski tensor densities are proportional to the intrinsic volume densities, whereas for a non-isotropic Boolean model this is usually not the case. Our results support the idea that the degree of anisotropy of a Boolean model may be expressed in terms of the Minkowski tensor densities. Furthermore we observe that for smooth grains the mean curvature radius function of a particle can be reconstructed from the Minkowski tensor densities. In a simulation study we determine numerically Minkowski tensor densities for non-isotropic Boolean models based on ellipses and on rectangles in two dimensions and find excellent agreement with the derived analytic density formulas. The tensor densities can be used to characterize the orientational distribution of the grains and to estimate model parameters for non-isotropic distributions.

math.PR↗
arXiv2013

Minkowski Tensors of Anisotropic Spatial Structure

This article 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 generalisations 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 article 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 method more readily accessible for future application in the physical sciences.

cond-mat.soft↗
arXiv2013

Network-like propagation of cell-level stress in random foams

Quasistatic simple shearing flow of random monodisperse soap froth is investigated by analyzing Surface Evolver simulations of spatially periodic foams. Elastic-plastic behavior is caused by irreversible topological rearrangements (T1s) that occur when Plateau's laws are violated; the first T1s occur at the elastic limit and at large strains frequent cascades of T1s, composed of one or more individual T1s, sustain the yield-stress plateau. The stress and shape anisotropy of individual cells is quantified by $Q$, a scalar measure derived from the interface tensor that gauges each cell's contribution to the global stress. During each T1 cascade, the connected set of cells with decreasing $Q$, called the \textit{stress release domain}, is network-like and highly non-local. Geometrically, the network-like nature of the stress release domains is corroborated through morphological analysis using the Euler characteristic. The stress release domain is distinctly different from the set of cells that change topology during a T1 cascade. Our results highlight the unique rheological behavior of foams, where complex large-scale cooperative rearrangements of foam cells are observed as a consequence of distinctly local events.

cond-mat.soft↗
arXiv2013

Time-Dependent Hartree-Fock Approach to Nuclear Pasta at Finite Temperature

We present simulations of neutron-rich matter at subnuclear densities, like supernova matter, with the time-dependent Hartree-Fock approximation at temperatures of several MeV. The initial state consists of $α$ particles randomly distributed in space that have a Maxwell-Boltzmann distribution in momentum space. Adding a neutron background initialized with Fermi distributed plane waves the calculations reflect a reasonable approximation of astrophysical matter. This matter evolves into spherical, rod-like, and slab-like shapes and mixtures thereof. The simulations employ a full Skyrme interaction in a periodic three-dimensional grid. By an improved morphological analysis based on Minkowski functionals, all eight pasta shapes can be uniquely identified by the sign of only two valuations, namely the Euler characteristic and the integral mean curvature. In addition, we propose the variance in the cell density distribution as a measure to distinguish pasta matter from uniform matter.

nucl-th↗
arXiv2012

Shortcomings of the Bond Orientational Order Parameters for the Analysis of Disordered Particulate Matter

Local structure characterization with the bond-orientational order parameters q4, q6, ... introduced by Steinhardt et al. has become a standard tool in condensed matter physics, with applications including glass, jamming, melting or crystallization transitions and cluster formation. Here we discuss two fundamental flaws in the definition of these parameters that significantly affect their interpretation for studies of disordered systems, and offer a remedy. First, the definition of the bond-orientational order parameters considers the geometrical arrangement of a set of neighboring spheres NN(p) around a given central particle p; we show that procedure to select the spheres constituting the neighborhood NN(p) can have greater influence on both the numerical values and qualitative trend of ql than a change of the physical parameters, such as packing fraction. Second, the discrete nature of neighborhood implies that NN(p) is not a continuous function of the particle coordinates; this discontinuity, inherited by ql, leads to a lack of robustness of the ql as structure metrics. Both issues can be avoided by a morphometric approach leading to the robust Minkowski structure metrics ql'. These ql' are of a similar mathematical form as the conventional bond-orientational order parameters and are mathematically equivalent to the recently introduced Minkowski tensors [Europhys. Lett. 90, 34001 (2010); Phys. Rev. E. 85, 030301 (2012)].

cond-mat.soft↗
arXiv2012

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.

physics.flu-dyn↗
arXiv2012

Jammed Spheres: Minkowski Tensors Reveal Onset of Local Crystallinity

The local structure of disordered jammed packings of monodisperse spheres without friction, generated by the Lubachevsky-Stillinger algorithm, is studied for packing fractions above and below 64%. The structural similarity of the particle environments to fcc or hcp crystalline packings (local crystallinity) is quantified by order metrics based on rank-four Minkowski tensors. We find a critical packing fraction ϕ_c \approx 0.649, distinctly higher than previously reported values for the contested random close packing limit. At ϕ_c, the probability of finding local crystalline configurations first becomes finite and, for larger packing fractions, increases by several orders of magnitude. This provides quantitative evidence of an abrupt onset of local crystallinity at ϕ_c. We demonstrate that the identification of local crystallinity by the frequently used local bond-orientational order metric q_6 produces false positives, and thus conceals the abrupt onset of local crystallinity. Since the critical packing fraction is significantly above results from mean-field analysis of the mechanical contacts for frictionless spheres, it is suggested that dynamic arrest due to isostaticity and the alleged geometric phase transition in the Edwards framework may be disconnected phenomena.

cond-mat.soft↗
arXiv2011

All good things come in threes - Three beads learn to swim with lattice Boltzmann and a rigid body solver

We simulate the self-propulsion of devices in a fluid in the regime of low Reynolds numbers. Each device consists of three bodies (spheres or capsules) connected with two damped harmonic springs. Sinusoidal driving forces compress the springs which are resolved within a rigid body physics engine. The latter is consistently coupled to a 3D lattice Boltzmann framework for the fluid dynamics. In simulations of three-sphere devices, we find that the propulsion velocity agrees well with theoretical predictions. In simulations where some or all spheres are replaced by capsules, we find that the asymmetry of the design strongly affects the propelling efficiency.

cs.CE↗
arXiv2010

Anomalous transport of a tracer on percolating clusters

We investigate the dynamics of a single tracer exploring a course of fixed obstacles in the vicinity of the percolation transition for particles confined to the infinite cluster. The mean-square displacement displays anomalous transport, which extends to infinite times precisely at the critical obstacle density. The slowing down of the diffusion coefficient exhibits power-law behavior for densities close to the critical point and we show that the mean-square displacement fulfills a scaling hypothesis. Furthermore, we calculate the dynamic conductivity as response to an alternating electric field. Last, we discuss the non-gaussian parameter as an indicator for heterogeneous dynamics.

cond-mat.stat-mech↗
arXiv2010

Local Anisotropy of Fluids using Minkowski Tensors

Statistics of the free volume available to individual particles have previously been studied for simple and complex fluids, granular matter, amorphous solids, and structural glasses. Minkowski tensors provide a set of shape measures that are based on strong mathematical theorems and easily computed for polygonal and polyhedral bodies such as free volume cells (Voronoi cells). They characterize the local structure beyond the two-point correlation function and are suitable to define indices $0\leq β_ν^{a,b}\leq 1$ of local anisotropy. Here, we analyze the statistics of Minkowski tensors for configurations of simple liquid models, including the ideal gas (Poisson point process), the hard disks and hard spheres ensemble, and the Lennard-Jones fluid. We show that Minkowski tensors provide a robust characterization of local anisotropy, which ranges from $β_ν^{a,b}\approx 0.3$ for vapor phases to $β_ν^{a,b}\to 1$ for ordered solids. We find that for fluids, local anisotropy decreases monotonously with increasing free volume and randomness of particle positions. Furthermore, the local anisotropy indices $β_ν^{a,b}$ are sensitive to structural transitions in these simple fluids, as has been previously shown in granular systems for the transition from loose to jammed bead packs.

cond-mat.stat-mech↗
arXiv2008

Topological estimation of percolation thresholds

Global physical properties of random media change qualitatively at a percolation threshold, where isolated clusters merge to form one infinite connected component. The precise knowledge of percolation thresholds is thus of paramount importance. For two dimensional lattice graphs, we use the universal scaling form of the cluster size distributions to derive a relation between the mean Euler characteristic of the critical percolation patterns and the threshold density $p_c$. From this relation, we deduce a simple rule to estimate $p_c$, which is remarkably accurate. We present some evidence that similar relations might hold for continuum percolation and percolation in higher dimensions.

cond-mat.stat-mech↗
arXiv2007

Motion by Stopping: Rectifying Brownian Motion of Non-spherical Particles

We show that Brownian motion is spatially not symmetric for mesoscopic particles embedded in a fluid if the particle is not in thermal equilibrium and its shape is not spherical. In view of applications on molecular motors in biological cells, we sustain non-equilibrium by stopping a non-spherical particle at periodic sites along a filament. Molecular dynamics simulations in a Lennard-Jones fluid demonstrate that directed motion is possible without a ratchet potential or temperature gradients if the asymmetric non-equilibrium relaxation process is hindered by external stopping. Analytic calculations in the ideal gas limit show that motion even against a fluid drift is possible and that the direction of motion can be controlled by the shape of the particle, which is completely characterized by tensorial Minkowski functionals.

cond-mat.soft↗
arXiv2006

Radial Distribution Function for Semiflexible Polymers Confined in Microchannels

An analytic expression is derived for the distribution $G(\vec{R})$ of the end-to-end distance $\vec{R}$ of semiflexible polymers in external potentials to elucidate the effect of confinement on the mechanical and statistical properties of biomolecules. For parabolic confinement the result is exact whereas for realistic potentials a self-consistent ansatz is developed, so that $G(\vec{R})$ is given explicitly even for hard wall confinement. The theoretical result is in excellent quantitative agreement with fluorescence microscopy data for actin filaments confined in rectangularly shaped microchannels. This allows an unambiguous determination of persistence length $L_P$ and the dependence of statistical properties such as Odijk's deflection length $λ$ on the channel width $D$. It is shown that neglecting the effect of confinement leads to a significant overestimation of bending rigidities for filaments.

cond-mat.soft↗
arXiv2006

Microscopic theory for interface fluctuations in binary liquid mixtures

Thermally excited capillary waves at fluid interfaces in binary liquid mixtures exhibit simultaneously both density and composition fluctuations. Based on a density functional theory for inhomogeneous binary liquid mixtures we derive an effective wavelength dependent Hamiltonian for fluid interfaces in these systems beyond the standard capillary-wave model. Explicit expressions are obtained for the surface tension, the bending rigidities, and the coupling constants of compositional capillary waves in terms of the profiles of the two number densities characterizing the mixture. These results lead to predictions for grazing-incidence x-ray scattering experiments at such interfaces.

cond-mat.soft↗
arXiv2005

Local orientations of fluctuating fluid interfaces

Thermal fluctuations cause the local normal vectors of fluid interfaces to deviate from the vertical direction defined by the flat mean interface position. This leads to a nonzero mean value of the corresponding polar tilt angle which renders a characterization of the thermal state of an interface. Based on the concept of an effective interface Hamiltonian we determine the variances of the local interface position and of its lateral derivatives. This leads to the probability distribution functions for the metric of the interface and for the tilt angle which allows us to calculate its mean value and its mean square deviation. We compare the temperature dependences of these quantities as predicted by the simple capillary wave model, by an improved phenomenological model, and by the microscopic effective interface Hamiltonian derived from density functional theory. The mean tilt angle discriminates clearly between these theoretical approaches and emphasizes the importance of the variation of the surface tension at small wave lengths. Also the tilt angle two-point correlation function is determined which renders an additional structural characterization of interfacial fluctuations. Various experimental accesses to measure the local orientational fluctuations are discussed.

cond-mat.soft↗