Macroscopic Curvature Research Articles

Macroscopic scalar curvature and codimension 2 width

We show that a complete [Formula: see text]-dimensional Riemannian manifold [Formula: see text] with finitely generated first homology has macroscopic dimension [Formula: see text] if it satisfies the following “macroscopic curvature” assumptions: every ball of radius [Formula: see text] in [Formula: see text] has volume at most [Formula: see text], and every loop in every ball of radius [Formula: see text] in [Formula: see text] is null-homologous in the concentric ball of radius [Formula: see text].

Influence of macroscopic interface curvature on dendritic patterns during directional solidification of bulk samples: Experimental and phase-field studies

The envelope of the solidification front naturally develops a macroscopic curvature in a multitude of solidification processes. However, its effect on dendritic microstructure formation remains poorly understood. Here we exploit a microgravity environment where convection is suppressed to investigate quantitatively the effect of curvature on dendritic array growth during directional solidification of a transparent succinonitrile – 0.46 wt% camphor alloy. In addition, we interpret the results using both theoretical analyses and phase-field simulations. In situ observations reveal that even a weak macroscopic interface curvature can have a major effect on both the array pattern evolution and grain structure. First, convex and concave interfaces lead to a continuous increase and decrease in time of the average primary spacing, respectively, which only attains a stationary value for a flat interface. We show that this results from the formation of drifting velocity gradients along the interface under the combined effect of the crystal misorientation and the interface curvature. Second, interface concavity is accompanied by “stray grains” of larger misorientation. These grains form at sample boundaries and overgrow pre-existing interior grains of smaller misorientation to the detriment of well-oriented crystal growth. In addition, they induce a change of array structure when the direction normal to the sample boundary is parallel to one of <100> axis of secondary branches. These results provide new insights for controlling and optimizing directionally solidified alloy microstructures.

Mirau interferometry of fluid interfaces deformed by colloids under the influence of external fields

The interfacial curvature surrounding colloidal particles pinned to fluid interfaces dictates their interparticle capillary interaction and assembly; however, it is a nontrivial function of particle anisotropy, surface roughness, external field conditions, macroscopic interfacial curvature, and the chemistry of each fluid phase. The prospect of dynamically modifying the pinning properties and interfacial organization of colloidal particles adhered to fluid interfaces via these approaches necessitates the development of experimental techniques capable of measuring changes in the interfacial deformation around particles in situ. Here, we describe a modified technique based on phase-shift Mirau interferometry to determine the relative height of the fluid interface surrounding adsorbed colloids while applying external electric fields. The technique is corrected for macroscopic curvature in the interface as well as in-plane motion of the particle in order to isolate the contribution of the particle to the interfacial deformation. Resultant height maps are produced with a maximum resolution of ±1nm along the height axis. The measured topography of the interface is used to identify the contact line where the two fluids meet the particle, along with the maximal interfacial deformation (Δumax) of the undulating contact line and the three-phase contact angle, θc. The technique is calibrated using anisotropic polymer ellipsoids of varying aspect ratio before the effect of external AC electric fields on the pinned particle contact angle is demonstrated. The results show promise for this new technique to measure and quantify dynamic changes in interfacial height deformation, which dictate interparticle capillary energy and assembly of colloids at fluid interfaces.

Macroscopic scalar curvature and local collapsing

Consider a closed Riemannian $n$-manifold $M$ admitting a negatively curved Riemannian metric. We show that for every Riemannian metric on $M$ of sufficiently small volume, there is a point in the universal cover of $M$ such that the volume of every ball of radius $r \geq 1$ centered at this point is greater or equal to the volume of the ball of the same radius in the hyperbolic $n$-space. We also give an interpretation of this result in terms of macroscopic scalar curvature. This result, which holds more generally in the context of polyhedral length spaces, is related to a question of Guth. Its proof relies on a generalization of recent progress in metric geometry about the Alexandrov/Urysohn width involving the volume of balls of radius in a certain range with collapsing at different scales.

Triple line destabilization: Tuning film thickness through meniscus curvature

The stability of a dynamic meniscus depends upon its macroscopic curvature. In particular, the critical velocity for the dynamic wetting transition is affected by the meniscus curvature which, as such, emerges as a thickness selection mechanism for liquid film deposition. These ideas have been demonstrated experimentally by flowing water in tubes at high velocities and rationalized by numerical modeling.

Electrostatically Driven Polarization Flop and Strain-Induced Curvature in Free-Standing Ferroelectric Superlattices.

The combination of strain and electrostatic engineering in epitaxial heterostructures of ferroelectric oxides offers many possibilities for inducing new phases, complex polar topologies, and enhanced electrical properties. However, the dominant effect of substrate clamping can also limit the electromechanical response and often leaves electrostatics to play a secondary role. Releasing the mechanical constraint imposed by the substrate can not only dramatically alter the balance between elastic and electrostatic forces, enabling them to compete on par with each other, but also activates new mechanical degrees of freedom, such as the macroscopic curvature of the heterostructure. In this work, an electrostatically driven transition from a predominantly out-of-plane polarized to an in-plane polarized state is observed when a PbTiO3 /SrTiO3 superlattice with a SrRuO3 bottom electrode is released from its substrate. In turn, this polarization rotation modifies the lattice parameter mismatch between the superlattice and the thin SrRuO3 layer, causing the heterostructure to curl up into microtubes. Through a combination of synchrotron-based scanning X-ray diffraction imaging, Raman scattering, piezoresponse force microscopy, and scanning transmission electron microscopy, the crystalline structure and domain patterns of the curved superlattices are investigated, revealing a strong anisotropy in the domain structure and a complex mechanism for strain accommodation.

Macroscopic stability and simplicial norms of hypersurfaces

We introduce a $\mathbb{Z}$--coefficient version of Guth's macroscopic stability inequality for almost-minimizing hypersurfaces. In manifolds with a lower bound on macroscopic scalar curvature, we use the inequality to prove a lower bound on areas of hypersurfaces in terms of the Gromov simplicial norm of their homology classes. We give examples to show that a very positive lower bound on macroscopic scalar curvature does not necessarily imply an upper bound on the areas of minimizing hypersurfaces.

Phenomenology of curvature-induced quantum-gravity effects

Several studies have been devoted to the possibility that quantum gravity might tangibly affect relativistic kinematics for particles propagating from distant astrophysical sources to our telescopes, but the relevant literature has so far focused exclusively on a subclass of scenarios such that the quantum-gravity effects are independent of (macroscopic) curvature. It was assumed that a phenomenology for quantum-gravity effects that are triggered by curvature might be a dead end because of a double suppression: by the smallness of the characteristic quantum-gravity length scale and by the smallness of curvature. This state of affairs is becoming increasingly unsatisfactory in light of some recent quantum-gravity studies providing evidence of the fact that the presence of curvature might be required in order to have the novel relativistic properties. We here analyze an explicit scenario for curvature-induced quantum-gravity effects, and show that the smallness of curvature does not pose a challenge for phenomenology since it is compensated by the large distances traveled by the particles considered in the relevant phenomenological studies. We also observe that the present data situation for particles propagating from distant astrophysical sources to our telescopes, while inconclusive, provides more encouragement for our curvature-induced effects than for the curvature-independent effects that were so far studied.

Double-stranded RNA bending by AU-tract sequences.

Sequence-dependent structural deformations of the DNA double helix (dsDNA) have been extensively studied, where adenine tracts (A-tracts) provide a striking example for global bending in the molecule. However, in contrast to dsDNA, sequence-dependent structural features of dsRNA have received little attention. In this work, we demonstrate that the nucleotide sequence can induce a bend in a canonical Watson-Crick base-paired dsRNA helix. Using all-atom molecular dynamics simulations, we identified a sequence motif consisting of alternating adenines and uracils, or AU-tracts, that strongly bend the RNA double-helix. This finding was experimentally validated using atomic force microscopy imaging of dsRNA molecules designed to display macroscopic curvature via repetitions of phased AU-tract motifs. At the atomic level, this novel phenomenon originates from a localized compression of the dsRNA major groove and a large propeller twist at the position of the AU-tract. Moreover, the magnitude of the bending can be modulated by changing the length of the AU-tract. Altogether, our results demonstrate the possibility of modifying the dsRNA curvature by means of its nucleotide sequence, which may be exploited in the emerging field of RNA nanotechnology and might also constitute a natural mechanism for proteins to achieve recognition of specific dsRNA sequences.

Lipid-Composition-Mediated Forces Can Stabilize Tubular Assemblies of I-BAR Proteins

Collective action by inverse-Bin/Amphiphysin/Rvs (I-BAR) domains drive micron-scale membrane remodeling. The macroscopic curvature sensing and generation behavior of I-BAR domains is well characterized, and computational models have suggested various mechanisms on simplified membrane systems, but there remain missing connections between the complex environment of the cell and the models proposed thus far. Here, we show a connection between the role of protein curvature and lipid clustering in the relaxation of large membrane deformations. When we include phosphatidylinositol 4,5-bisphosphate-like lipids that preferentially interact with the charged ends of an I-BAR domain, we find clustering of phosphatidylinositol 4,5-bisphosphate-like lipids that induce a directional membrane-mediated interaction between membrane-bound I-BAR domains. Lipid clusters mediate I-BAR domain interactions and cause I-BAR domain aggregates that would not arise through membrane fluctuation-based or curvature-based interactions. Inside of membrane protrusions, lipid cluster-mediated interaction draws long side-by-side aggregates together, resulting in more cylindrical protrusions as opposed to bulbous, irregularly shaped protrusions.

Vacuum fluctuation, microcyclic universes, and the cosmological constant problem

We point out that the standard formulation of the cosmological constant problem itself is problematic since it is trying to apply the very large scale homogeneous cosmological model to very small (Planck) scale phenomenon. At small scales, both the spacetime and the vacuum stress energy are highly inhomogeneous and wildly fluctuating. This is a version of Wheeler's "spacetime foam". We show that this "foamy" structure would produce a large positive contribution to the average macroscopic spatial curvature of the Universe. In order to cancel this contribution to match the observation, the usually defined effective cosmological constant $\lambda_{\mathrm{eff}}=\lambda_B+8\pi G\langle\rho\rangle$ has to take a large negative value. The spacetime dynamics sourced by this large negative $\lambda_{\mathrm{eff}}$ would be similar to the cyclic model of the universe in the sense that at small scales every point in space is a "micro-cyclic universe" which is following an eternal series of oscillations between expansions and contractions. Moreover, if the bare cosmological constant $\lambda_B$ is dominant, the size of each "micro-universe" would increase a tiny bit at a slowly accelerating rate during each micro-cycle of the oscillation due to the weak parametric resonance effect produced by the fluctuations of the quantum vacuum stress energy tensor. In this way, the large cosmological constant generated at small scales is hidden at observable scale and no fine-tuning of $\lambda_B$ to the accuracy of $10^{-122}$ is needed. This at least resolves the old cosmological constant problem and suggests that it is the quantum vacuum fluctuations serve as the dark energy which is accelerating the expansion of our Universe.

Theoretical approaches to study the optical response of the red-legged honeycreeper's plumage (Cyanerpes cyaneus).

In this paper, we investigate the unusual color effect exhibited by the plumage of the heads of Cyanerpes cyaneus males, whose color turns from green to turquoise as the angle between the illumination and observation directions is increased. This singular color effect is characteristic of species that have quasi-ordered nanostructures of short-range order within the feather barbs. However, among species of the same family and even within feather patches of the same individual, one can find barbs with different characteristics, both macroscopic (curvature, shape, cross-sectional area) and in their internal microstructure. We apply the Korringa-Kohn-Rostoker method with the averaging technique to model the reflectance spectra for different angles of incidence and explain the dependence of the observed color with the incidence-collection angle. To investigate the influence of the disorder in the optical response of the spongy matrix, we apply the integral method for a two-dimensional cylinder system that simulates the distribution of air cavities within the $ \beta $β-keratin medium. The experimental reflectance was interpreted as the result of multiple reflections in the internal interfaces generated by large air voids present within the spongy matrix. The application of rigorous methods to the study of natural photonic structures is of fundamental relevance for the design of efficient bioinspired artificial materials.

Transient domains of ordered water induced by divalent ions lead to lipid membrane curvature fluctuations

Cell membranes are composed of a hydrated lipid bilayer that is molecularly complex and diverse, and the link between molecular hydration structure and membrane macroscopic properties is not well understood, due to a lack of technology that can probe and relate molecular level hydration information to micro- and macroscopic properties. Here, we demonstrate a direct link between lipid hydration structure and macroscopic dynamic curvature fluctuations. Using high-throughput wide-field second harmonic (SH) microscopy, we observe the formation of transient domains of ordered water at the interface of freestanding lipid membranes. These domains are induced by the binding of divalent ions and their structure is ion specific. Using nonlinear optical theory, we convert the spatiotemporal SH intensity into maps of membrane potential, surface charge density, and binding free energy. Using an electromechanical theory of membrane bending, we show that transient electric field gradients across the membrane induce spatiotemporal membrane curvature fluctuations.

The effective tensile and bending stiffness of nanotube fibers

We analyze the pure bending state, preliminarily considering the pure traction problem, for a fiber with circular cross-section composed by monodispersed Nano Tube (NT) segments, whose length is much lower than the length of the fiber, arranged in a cross-sectional square lattice, possibly inclined with respect to the bending plane. The proposed model accounts for the compliant shear coupling of the constituent NTs along their lateral contact surfaces, and considers that the position of each NT is offset with respect to the neighboring ones by a fixed distance. The shearing of the NTs permits their mobility in longitudinal direction, inducing an internal rearrangement, calculated through energy minimization, that affects the macroscopic response. With a micro-macro approach, the internal actions in the whole fiber are calculated. We find that uniform axial macroscopic strain is associated with constant tensile normal force in the fiber and null moment; correspondingly, constant macroscopic curvature provides a constant bending moment in the plane of bending and null axial force. It is then possible to consider the NT fiber as an equivalent homogeneous beam, whose effective tensile and bending stiffness can be calculated, finding a dependence upon the aspect ratio of the fiber, the aspect ratio of the individual NTs and, more important, the offset of the NTs. The model permits to interpret recent experimental results.

Effects of macroscopic curvature of substrate surface on the cell orientation along nano grooves

Effects of macroscopic curvature of substrate surface on the cell orientation along nano grooves

Nanostructured Free-Form Objects via a Synergy of 3D Printing and Thermal Nanoimprinting.

High‐resolution surface patterning has garnered interests as a nonchemical‐based surface engineering approach for creating functional surfaces. Applications in consumer products, parts for transportation vehicles, optics, and biomedical technologies demand topographic patterning on 3D net shape objects. Through a hybrid approach, high‐resolution surface texture is incorporated onto 3D‐printed polymers via direct thermal nanoimprinting process. The synergy of geometry design freedom in 3D printing and the high spatial resolution in nanoimprinting is demonstrated to be a versatile fabrication of high‐fidelity surface pattern (from 2 µm to 200 nm resolution) on convex, concave semicylindrical, and hemispherical objects spanning a range of surface curvatures. The novel hybrid fabrication is further extended to achieve a high‐resolution curved mold insert for rapid prototyping via injection molding. The versatility of the fabrication strategies reported here not only provides a post‐3D printing process that enhances the surface properties of 3D‐printed objects but also opens a new pathway to enable future study on the effects of combining microscale and nanoscale surface texture with macroscopic curvature. Both have been known, individually, as an effective approach to tune surface functionalities.

Multiphase SPH modeling of water boiling on hydrophilic and hydrophobic surfaces

A multiphase smoothed particle hydrodynamics (SPH) model was established to study the boiling phenomenon of a water film heated on a hydrophilic or hydrophobic surface. For the two-phase fluids of water vapor and liquid with continuous density gradient, conservation equations in Lagrangian formulation were discretized by SPH fluid particles. While the solid is treated by a new fluid-solid wetting surface model, which considered both the macroscopic interface curvature and the microscopic attractive forces. This SPH model enabled us to simulate the water boiling directly without empirical assumptions. After this model was well validated by three benchmark cases, we studied water boiling using two types of heating strategies. One strategy was partial heating with static or dynamic spots. The other was complete heating on an overall heating surface. Several fluid ρ-T phase diagrams were plotted with a detailed insight of the boiling development. We further simulated the critical heat flux (CHF) for both partial and total heating. The results illustrated the influences of the fluid thickness and the heating condition on the CHF value. The effects of wettability on boiling were also calculated. We found that a jumping boiling phenomenon would be observed on a relatively nonwettable surfaces, enhancing the heat exchange capability.

On the solid-state dewetting of polycrystalline thin films: Capillary versus grain growth approach

Solid-state dewetting of polycrystalline silver thin films was investigated with in situ and real time Environmental Scanning Electron Microscopy at High Temperature (HT-ESEM) in different annealing atmospheres: secondary vacuum or oxygen-rich (partial pressure ≥100 Pa) environment. A model where oxygen plays a key role is proposed to explain the very different observed morphologies; oxygen favours hole creation and isotropic hole propagation as well as grain selection. But, whatever the atmosphere, dewetting does not proceed through the propagation of a rim but instead involves the growth of specific grains and shrinkage of others. Models based on macroscopic curvature to account for the propagation speed of the dewetting front fail to fit the present observations. This points to a paramount role of the grain size and stability in the dewetting morphology.

Macroscopic scalar curvature and areas of cycles

In this paper we prove the following. Let $${\Sigma}$$ be an n–dimensional closed hyperbolic manifold and let g be a Riemannian metric on $${\Sigma \times \mathbb{S}^1}$$ . Given an upper bound on the volumes of unit balls in the Riemannian universal cover $${(\widetilde{\Sigma\times \mathbb{S}^1},\widetilde{g})}$$ , we get a lower bound on the area of the $${\mathbb{Z}_2}$$ –homology class $${[\Sigma \times \ast]}$$ on $${\Sigma \times \mathbb{S}^1}$$ , proportional to the hyperbolic area of $${\Sigma}$$ . The theorem is based on a theorem of Guth and is analogous to a theorem of Kronheimer and Mrowka involving scalar curvature.

Water evaporation in parallel plates

The influence of macroscopic curvature on evaporation of the meniscus between parallel glass plates standing on purified water pool has been quantified experimentally. The gap distance between the plates was short compared to the capillary length, so the surface curvature increased with decrease of the gap distance. The meniscus was not heated and the thermal relaxation time of the plates estimated to be small. The evaporation rate and flux, obtained from time-evolution of vapor pressure in an experimental enclosure, could increase with increase of the curvature between the plates. Based on the comparison among the existing data including ours, it was clear the evaporation rate and flux could be organized by a curvature, not a perimeter (a contact-line length).