Macroscopic Curvature Research Articles

Generalized propagation of light through optical systems. II. Numerical implications.

We present an algorithm implemented in a MATLAB toolbox that is able to compute the wave propagation of coherent visible light through macroscopic lenses. The mathematical operations that complete the status at the end of the first paper of this sequence, where only limited configurations of the propagation direction were allowed toward arbitrarily directed input beam computations, are provided. With their help, high numerical aperture (NA) field tracing is made possible that is based on fast Fourier routines and is Maxwell exact in the limit of macroscopic structures and large curvature radii, including reflection and transmission. Whereas the curvature-dependent terms in the Helmholtz equation are under analytical control through the first perturbation order in the curvature, they are only included in the propagation distance in the current investigation for the sake of reasonable time consumption. We give a number of examples that demonstrate the strengths of our approach, describe essential differences from other approaches that were not obvious when Paper 1 was written, and list a number of drawbacks and possible simplifications to overcome them.

Dynamical microstructure formation in 3D directional solidification of transparent model alloys: in situ characterization in DECLIC Directional Solidification Insert under diffusion transport in microgravity

To clarify and characterize the fundamental physical mechanisms active in the dynamical formation of three-dimensional (3D) arrays of cells and dendrites under diffusive growth conditions, in situ monitoring of series of experiments on transparent model alloy succinonitrile - 0.24 wt% camphor was carried out under low gravity in the DECLIC Directional Solidification Insert on-board the International Space Station. These experiments offered the very unique opportunity to in situ observe and characterize the whole development of the microstructure in extended 3D patterns. The experimental methods will be first briefly described, including in particular the observation modes and the image analysis procedures developed to quantitatively characterize the patterns. Microgravity environment provided the conditions to get quantitative benchmark data: homogeneous patterns corresponding to homogeneous values of control parameters along the whole interface were obtained. The sequence of microstructure formation will be presented as well as the evolution of the primary spacing which is one of the most important pattern characteristic. Time evolution of this primary spacing during the microstructure development will be analysed to identify the mechanisms of spacing selection and adjustment; the importance of the macroscopic interfacial curvature will be pointed out.

Disorder and broad-angle iridescence from Morpho-inspired structures.

The ordered, lamellae-structured ridges on the wing scales of Morpho butterflies give rise to their striking blue iridescence by multilayer interference and grating diffraction. At the same time, the random offsets among the ridges broaden the directional multilayer reflection peaks and the grating diffraction peaks that the color appears the same at various viewing angles, contrary to the very definition of iridescence. While the overall process is well understood, there has been little investigation into confirming the roles of each factor due to the difficulty of controllably reproducing such complex structures. Here we use a combination of self-assembly, selective etching, and directional deposition to fabricate Morpho-inspired structure with controlled random offsets. We find that while random offsets are necessary, it alone is not sufficient to produce the broad-angle reflection of Morpho butterflies. We identify diffraction as a critical factor for the bright, anisotropic broadening of the reflection peak of Morpho butterflies to a solid angle of 0.23 sr, and suggest random macroscopic surface curvature as a practical alternative, with an isotropic broad reflection peak whose solid angle can reach 0.11 sr at an incident angle of 60°.

X-ray diffraction imaging of metal-oxide epitaxial tunnel junctions made by optical lithography: use of focused and unfocused X-ray beams.

X-ray diffraction techniques are used in imaging mode in order to characterize micrometre-sized objects. The samples used as models are metal-oxide tunnel junctions made by optical lithography, with lateral sizes ranging from 150 µm down to 10 µm and various shapes: discs, squares and rectangles. Two approaches are described and compared, both using diffraction contrast: full-field imaging (topography) and raster imaging (scanning probe) using a micrometre-sized focused X-ray beam. It is shown that the full-field image gives access to macroscopic distortions (e.g. sample bending), while the local distortions, at the micrometre scale (e.g. tilts of the crystalline planes in the vicinity of the junction edges), can be accurately characterized only using focused X-ray beams. These local defects are dependent on the junction shape and larger by one order of magnitude than the macroscopic curvature of the sample.

Correlation between macroscopic and microscopic stress fields: Application to the 3C–SiC/Si heteroepitaxy

Abstract

Curvature Entropy for Curved Profile Generation

In a curved surface design, the overall shape features that emerge from combinations of shape elements are important. However, controlling the features of the overall shape in curved profiles is difficult using conventional microscopic shape information such as dimension. Herein two types of macroscopic shape information, curvature entropy and quadrature curvature entropy, quantitatively represent the features of the overall shape. The curvature entropy is calculated by the curvature distribution, and represents the complexity of a shape (one of the overall shape features). The quadrature curvature entropy is an improvement of the curvature entropy by introducing a Markov process to evaluate the continuity of a curvature and to approximate human cognition of the shape. Additionally, a shape generation method using a genetic algorithm as a calculator and the entropy as a shape generation index is presented. Finally, the applicability of the proposed method is demonstrated using the side view of an automobile as a design example.

Droplet formation and scaling in dense suspensions

When a dense suspension is squeezed from a nozzle, droplet detachment can occur similar to that of pure liquids. While in pure liquids the process of droplet detachment is well characterized through self-similar profiles and known scaling laws, we show here the simple presence of particles causes suspensions to break up in a new fashion. Using high-speed imaging, we find that detachment of a suspension drop is described by a power law; specifically we find the neck minimum radius, r(m), scales like near breakup at time τ = 0. We demonstrate data collapse in a variety of particle/liquid combinations, packing fractions, solvent viscosities, and initial conditions. We argue that this scaling is a consequence of particles deforming the neck surface, thereby creating a pressure that is balanced by inertia, and show how it emerges from topological constraints that relate particle configurations with macroscopic Gaussian curvature. This new type of scaling, uniquely enforced by geometry and regulated by the particles, displays memory of its initial conditions, fails to be self-similar, and has implications for the pressure given at generic suspension interfaces.

The Nature of Membrane Curvature-Induction by Amphipathic α-Helices Relies upon Protein Length: Simulations of α-Synuclein and H0

Remodeling of membranes, for example the induction of membrane curvature, is a necessary step in vesicle fusion and fission. One mechanism for curvature induction relies upon the interfacial binding of amphipathic proteins, which act as a wedge in one leaflet of the membrane. In this study, we explore differences between the membrane remodeling effects of two biologically important amphipathic proteins, specifically α-Synuclein (αS) and the N-BAR helix H0. αS is an intrinsically disordered protein that adopts an amphipathic α-helical structure upon binding a membrane. Recently it has been shown that αS can induce tubulation and vesiculation of vesicles (Varkey et al. 2010, J Biol Chem, 285(42):32486). In contrast, the amphipathic helix H0 is unable, on its own, to induce tubulation, as N-BAR relies instead upon its large scaffolding domain (Fernandes et al. 2008, Biophys J, 94(8):3065). Using a combination of x-ray scattering and coarse-grained molecular dynamics simulations, we explored the membrane remodeling effects of αS. Our findings suggest that αS 1) causes a significant thinning of the bilayer; and 2) stabilizes an anisotropic curvature-field of both positive mean curvature and negative Gaussian curvature. Simulations of H0 show a similar magnitude of the local curvature as with αS, however owing to its molecular length the curvature-field around H0 is isotropic. We propose that the difference in curvature-field anisotropy explains the difference in the two proteins’ abilities to induce macroscopic curvature (i.e. tubulation) of lipid vesicles.

Simple SQP approach for out-of-plane loaded homogenized brickwork panels, accounting for softening

A simple homogenized model for the non linear analysis of masonry walls out-of-plane loaded is presented. In the model, the panels are assumed to behave as Kirchhoff-Love plates. A rectangular running bond elementary cell (RVE) is subdivided into several layers along the thickness and, for each layer, a discretization where bricks are meshed with plane-stress three-noded triangular elements and joints are reduced to interfaces is assumed. Non linearity is concentrated on brick-brick and joint interfaces, which exhibit a frictional behavior with limited tensile and compressive strength with softening. Finally, macroscopic curvature bending moment diagrams are obtained integrating along the thickness in-plane micro-stresses of each layer. Homogenized masonry flexural response is then implemented at a structural level in a FE non linear code based on a discretization with three-noded elements and elasto-damaging interfaces. Three different models of increasing accuracy are presented. The first (EPP) relies in assuming an elastic-perfectly plastic behavior for the interfaces. The incremental problem is solved at a structural level through a well known quadratic-programming approach. The second (ED) accounts in an approximate way for the softening behavior and consists in a preliminary homogenized limit analysis of the structure, which allows to identify the failure mechanism and in the subsequent FE non linear analysis of the whole structure assuming that all the non linearity is concentrated on the yield line defining the failure mechanism. The last (EPD) is a sequential quadratic programming approach. Here, deteriorating bending moment curvature curves obtained through homogenization are approximated through a linear piecewise constant discontinuous function. At each load step, all interfaces are assumed to behave as an elastic-perfectly plastic material and the discretized non linear problem is solved by means of the quadratic programming algorithm used for the EPP model. The two step model proposed is validated both a cell level and at a structural level comparing results provided with both experimental data and existing macroscopic numerical approaches available in the literature.

Vapor-Liquid Steady Meniscus at a Superheated Wall: Asymptotics in an Intermediate Zone Near the Contact Line

The study concerns steady configurations of a perfectly wetting liquid in contact with its pure vapor and a superheated substrate/wall maintained at a constant temperature. Despite the perfect wetting, the system is characterized by a finite apparent contact angle formed at a microscale, within a steady microstructure of the contact line, the finiteness owing itself to an actually dynamic situation caused by the evaporation process. The angle is assumed to be small here, which is the case for sufficiently small superheats. When macroscopically treating a steady meniscus, one typically implies that the wall is met at the contact angle given by the microstructure. This remains somewhat an intuitive, heuristic approach unless a more rigorous asymptotic matching is carried out between the meniscus and the microstructure, which is accomplished in the present paper by studying an intermediate zone connecting these two regions. The analysis, based upon a standard one-sided planar model of an evaporating liquid layer in the lubrication approximation, confirms the validity of the mentioned approach. A possible uncertainty in the definition of the contact angle is shown to be small given that the macroscopic curvature (i.e. that of the meniscus and of the wall) is small on the scale of the contact line microstructure.

The hydrodynamics of water-walking arthropods

We present the results of a combined experimental and theoretical investigation of the dynamics of water-walking insects and spiders. Using high-speed videography, we describe their numerous gaits, some analogous to those of their terrestrial counterparts, others specialized for life at the interface. The critical role of the rough surface of these water walkers in both floatation and propulsion is demonstrated. Their waxy, hairy surface ensures that their legs remain in a water-repellent state, that the bulk of their leg is not wetted, but rather contact with the water arises exclusively through individual hairs. Maintaining this water-repellent state requires that the speed of their driving legs does not exceed a critical wetting speed. Flow visualization reveals that the wakes of most water walkers are characterized by a series of coherent subsurface vortices shed by the driving stroke. A theoretical framework is developed in order to describe the propulsion in terms of the transfer of forces and momentum between the creature and its environment. The application of the conservation of momentum to biolocomotion at the interface confirms that the propulsion of water walkers may be rationalized in terms of the subsurface flows generated by their driving stroke. The two principal modes of propulsion available to small water walkers are elucidated. At driving leg speeds in excess of the capillary wave speed, macroscopic curvature forces are generated by deforming the meniscus, and the surface behaves effectively as a trampoline. For slower speeds, the driving legs need not substantially deform the surface but may instead simply brush it: the resulting contact or viscous forces acting on the leg hairs crossing the interface serve to propel the creature forward.

Hydrophobicity of curved microstructured surfaces

This paper presents measurements and models for how the macroscopic curvature of microstructured polymers affects hydrophobicity. Flexible polymer substrates were fabricated with arrays of regular microstructures. The interaction of liquid drops with these surfaces was analyzed for flat substrates and substrates flexed into either positive or negative cylindrical shapes. Liquid droplet static contact angle and dynamic slide angle were measured for a range of surfaces. An increase in substrate curvature corresponded with decreased slide angle for liquid droplets suspended on the surface asperities. This phenomenon is investigated in terms of solid–liquid contact line and the periodicity of surface microstructures. We present a model that can be used to understand the observed phenomena and to design microstructure geometries for hydrophobicity.

X-ray diffraction micro-imaging of strain in laterally overgrown GaAs layers. Part I: analysis of a single GaAs stripe

Spatially resolved X-ray diffraction (SRXRD) is used for micro-imaging of strain in GaAs:Si layers grown by liquid phase epitaxial lateral overgrowth (ELO) on SiO2-masked GaAs substrates. We show that laterally overgrown parts of the layers (wings) are tilted towards the underlying mask. By SRXRD mapping local wing tilt is easily distinguished from macroscopic sample curvature. The direction of the tilt and distribution of tilt magnitude across the width of each layer can also be readily determined. This allows measuring of the shape of the lattice planes in individual ELO stripes. Downward wing tilt disappears completely when the mask is removed by selective etching. Then residual strain in ELO layers is exposed. In particular, upward tilt is found in free-standing ELO wings. Numerical simulations show that this phenomenon is caused by different concentrations of silicon dopant in vertically and laterally grown parts of the layer.

Modulated Pattern Formation of Phospholipid Monolayers on Curved Surfaces

Langmuir-Blodgett transfer of a dipalmitoylphosphatidylcholine monolayer onto macroscopically curved mica surfaces results in microscopic patterns of the transferred monolayer that differ from those of films transferred onto a flat mica substrate. On curved surfaces a modulated horizontal striped pattern evolves that has a zigzag boundary at the liquid condensed front of the stripe and a continuous straight boundary at the liquid condensed rear. We propose that the sensitivity of the pattern to the macroscopic curvature of the sample is due to a flow-controlled hydrodynamic instability caused by the subphase flow close to the three-phase contact line.

Second order homogenization method based on higher order finite elements

AbstractModeling materials with lattice‐like microstructures like open‐cell foams requires an extended continuum mechanical setting on the macroscopic scale, e. g. a micropolar or micromorphic theory. In order to avoid the formulation of constitutive equations a higher order numerical homogenization scheme (FE2) is proposed. Therefore, each integration point possesses its own microstructure which, in the present case, consists of beam‐like elements representing the cell walls. In this paper, the microstructures are discretized by continuum‐based higher order locking free finite elements with high aspect ratios, leading to a numerically efficient treatment of a local displacement‐driven boundary value problem according to the macroscopic strain and curvature. The resulting stress distributions in the microstructures are homogenized to macroscopic stresses and couple stresses. The approach is demonstrated by a numerical example. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Capillary Forces between Spherical Particles Floating at a Liquid−Liquid Interface

We study the capillary forces acting on sub-millimeter particles (0.02-0.6 mm) trapped at a liquid-liquid interface due to gravity-induced interface deformations. An analytical procedure is developed to solve the linearized capillary (Young-Laplace) equation and calculate the forces for an arbitrary number of particles, allowing also for a background curvature of the interface. The full solution is expressed in a series of Bessel functions with coefficients determined by the contact angle at the particle surface. For sub-millimeter spherical particles, it is shown that the forces calculated using the lowest order term of the full solution (linear superposition approximation; LSA) are accurate to within a few percents. Consequently the many particle capillary force is simply the sum of the isolated pair interactions. To test these theoretical results, we use video microscopy to follow the motion of individual particles and pairs of interacting particles at a liquid-liquid interface with a slight macroscopic background curvature. Particle velocities are determined by the balance of capillary forces and viscous drag. The measured velocities (and thus the capillary forces) are well described by the LSA solution with a single fitting parameter.

IN SITU MEASUREMENTS OF MACROSCOPIC FILM STRESS DURING GROWTH, COOLING, AND THERMAL CYCLING OF THIN FILM PTiO3

ASTRACT Coherent Gradient Sensing is a full-field optical technique that produces real-time images of macroscopic wafer curvature that, for thin films, can be related to stress through Stoney's equation. Here we descrie the use of CGS as an in situ diagnostic to oserve film stress during chemical vapor deposition. We discuss the oserved changes in wafer curvature as a thin film of PTiO3 is deposited by chemical vapor deposition on single crystal MgO, cooled to room temperature, and then thermally cycled.

Compensating bends in a 16‐base‐pair DNA oligomer containing a T3A3 segment: A NMR study of global DNA curvature

In-phase ligated DNA containing T(n)A(n) segments fail to exhibit the retarded polyacrylamide gel electrophoresis (PAGE) migration observed for in-phase ligated A(n)T(n) segments, a behavior thought to be correlated with macroscopic DNA curvature. The lack of macroscopic curvature in ligated T(n)A(n) segments is thought to be due to cancellation of bending in regions flanking the TpA steps. To address this issue, solution-state NMR, including residual dipolar coupling (RDC) restraints, was used to determine a high-resolution structure of [d(CGAGGTTTAAACCTCG)2], a DNA oligomer containing a T3A3 tract. The overall magnitude and direction of bending, including the regions flanking the central TpA step, was measured using a radius of curvature, Rc, analysis. The Rc for the overall molecule indicated a small magnitude of global bending (Rc = 138 +/- 23 nm) towards the major groove, whereas the Rc for the two halves (72 +/- 33 nm and 69 +/- 14 nm) indicated greater localized bending into the minor groove. The direction of bending in the regions flanking the TpA step is in partial opposition (109 degrees), contributing to cancellation of bending. The cancellation of bending did not correlate with a pattern of roll values at the TpA step, or at the 5' and 3' junctions, of the T3A3 segment, suggesting a simple junction/roll model is insufficient to predict cancellation of DNA bending in all T(n)A(n) junction sequence contexts. Importantly, Rc analysis of structures refined without RDC restraints lacked the precision and accuracy needed to reliably measure bending.

Neutron diffraction study of the contribution of grain contacts to nonlinear stress‐strain behavior

Repeatable, hysteretic loops in quasi‐static loading measurements on rocks are well known; the fundamental processes responsible for them are not. The grain contact region is usually treated as the site of these processes, but there is little supporting experimental evidence. We have performed simultaneous neutron diffraction and quasi‐static loading experiments on a selection of rocks to experimentally isolate the response of these contact regions. Neutron diffraction measures strain in the lattice planes of the bulk of the grain material, so differences between this strain and the macroscopic response yield information about grain contact behavior. We find the lattice responds linearly to stress in all cases, oblivious to the macroscopic unrecoverable strains, curvature, and hysteresis, localizing these effects to the contacts. Neutron diffraction shows that the more granular rocks appear to distribute stresses so that the same strain appears in all the grains, independent of crystallographic orientation.

A particle center based homogenization strategy for granular assemblies

In this paper, a new homogenization technique for the determination of dynamic and kinematic quantities of representative elementary volumes (REVs) in granular assemblies is presented. Based on the definition of volume averages, expressions for macroscopic stress, couple stress, strain and curvature tensors are derived for an arbitrary REV. Discrete element model simulations of two different test set‐ups including cohesionless and cohesive granular assemblies are used as a validation of the proposed homogenization technique. A non‐symmetric macroscopic stress tensor, as well as couple stresses are obtained following the proposed procedure, even if a single particle is described as a standard continuum on the microscopic scale.