Anisotropic Curvature Research Articles

Minimizing movements for forced anisotropic curvature flow of droplets

We study forced anisotropic curvature flow of droplets on an inhomogeneous horizontal hyperplane. As in Bellettini and Kholmatov [J. Math. Pures Appl. 117 (2018), 1–58], we establish the existence of smooth flow, starting from a regular droplet and satisfying the prescribed anisotropic Young’s law, and also the existence of a 1/2 -Hölder continuous in time minimizing movement solution starting from a set of finite perimeter. Furthermore, we investigate various properties of minimizing movements, including comparison principles, uniform boundedness, and the consistency with the smooth flow.

Stability of the Wulff shape with respect to anisotropic curvature functionals

For a function f which foliates a one-sided neighborhood of a closed hypersurface M, we give an estimate of the distance of M to a Wulff shape in terms of the Lp-norm of the traceless F-Hessian of f, where F is the support function of the Wulff shape. This theorem is applied to prove quantitative stability results for the anisotropic Heintze-Karcher inequality, the anisotropic Alexandrov problem, as well as for the anisotropic overdetermined boundary value problem of Serrin-type.

Three-Dimensional Metallic Surface Micropatterning through Tailored Photolithography-Transfer-Plating.

Precise micropatterning on three-dimensional (3D) surfaces is desired for a variety of applications, from microelectronics to metamaterials, which can be realized by transfer printing techniques. However, a nontrivial deficiency of this approach is that the transferred microstructures are adsorbed on the target surface with weak adhesion, limiting the applications to external force-free conditions. We propose a scalable "photolithography-transfer-plating" method to pattern stable and durable microstructures on 3D metallic surfaces with precise dimension and location control of the micropatterns. Surface patterning on metallic parts with different metals and isotropic and anisotropic curvatures is showcased. This method can also fabricate hierarchical structures with nanoscale vertical and microscale horizontal dimensions. The plated patterns are stable enough to mold soft materials, and the structure durability is validated by 24 h thermofluidic tests. We demonstrate micropatterned nickel electrodes for oxygen evolution reaction acceleration in hydrogen production, showing the potential of micropatterned 3D metallic surfaces for energy applications.

Efficient semianalytical investigation of a fractional model describing human cornea shape

Purpose: This study presents a novel application of the semianalytical residual power series method to investigate a one-dimensional fractional anisotropic curvature equation describing the human cornea, the outermost layer of the eye. The fractional boundary value problem, involving the fractional derivative of curvature, poses challenges that conventional methods struggle to address. Methods: The analytical results are obtained by utilizing the simple and efficient residual power series method. The proposed method is accessible to researchers in all medical fields and is extendable to various models in disease spread and control. Results: The derived solution is a crucial outcome of this study. Through the application of the proposed method to the corneal shape model, an explicit formula for the curvature profile is obtained. To validate the solution, direct comparisons are made with numerical solutions for the integer case and other analytical solutions available in the literature for the fractional case. Conclusion: Our findings highlight the potential of the proposed method to significantly contribute to the diagnosis and treatment of various ophthalmological conditions.

Uniqueness of solutions to some classes of anisotropic and isotropic curvature problems

In this paper, we apply various methods to establish the uniqueness of solutions to some classes of anisotropic and isotropic curvature problems. Firstly, by employing integral formulas derived by S. S. Chern [20], we obtain the uniqueness of smooth admissible solutions to a class of Orlicz-(Christoffel)-Minkowski problems. Secondly, inspired by Simon's uniqueness result [61], we then prove that the only smooth strictly convex solution to the following isotropic curvature problem(0.1)(Pk(W)Pl(W))1k−l=ψ(u,r)onSn must be an origin-centred sphere, where W=(∇2u+ug0), ∂1ψ≥0,∂2ψ≥0 and at least one of these inequalities is strict. As an application, we establish the uniqueness of solutions to the isotropic Gaussian-Minkowski problem. Finally, we derive the uniqueness result for the following isotropic Lp dual Minkowski problem(0.2)u1−prq−n−1det⁡(W)=1onSn, where −n−1<p≤−1 and n+1≤q≤n+12+14−(1+p)(n+1+p)n(n+2). This result utilizes the method developed by Ivaki and Milman [42] and generalizes a result due to Brendle, Choi and Daskalopoulos [10].

A self-referenced interferometer for in situ cryogenic wafer curvature measurements.

A self-referenced interferometer to measure time-varying curvature in mechanically unstable environments is needed in many applications. One application that demands this measurement technique with fast data acquisition, 2D sensitivity, and insensitivity to vibration is the measurement of thermal strain in thin films in operational environments. The diverging beam interferometer described here demonstrates an angular sensitivity to the local curvature using interferograms captured by a CMOS camera. Two-dimensional Fourier analysis is used to extract curvature changes. The interferometer demonstrates an experimental sensitivity to curvature changes on the order of 10-4 m-1 and is used to measure thermal stresses in a cryogenic environment of a polycrystalline titanium nitride thin film on a silicon wafer that exhibits anisotropic curvature.

Spontaneous Curvature Induction in an Artificial Bilayer Membrane.

Maintaining lipid asymmetry across membrane leaflets is critical for functions like vesicular traffic and organelle homeostasis. However, a lack of molecular-level understanding of the mechanisms underlying membrane fission and fusion processes in synthetic systems precludes their development as artificial analogs. Here, we report asymmetry induction of a bilayer membrane formed by an extended π-conjugated molecule with oxyalkylene side chains bearing terminal tertiary amine moieties (BA1) in water. Autogenous protonation of the tertiary amines in the periphery of the bilayer by water induces anisotropic curvature, resulting in membrane fission to form vesicles and can be monitored using time-dependent spectroscopy and microscopy. Interestingly, upon loss of the induced asymmetry by extensive protonation using an organic acid restored bilayer membrane. The mechanism leading to the compositional asymmetry in the leaflet and curvature induction in the membrane is validated by density functional theory (DFT) calculations. Studies extended to control molecules having changes in hydrophilic (BA2) and hydrophobic (BA3) segments provide insight into the delicate nature of molecular scale interactions in the dynamic transformation of supramolecular structures. The synergic effect of hydrophobic interaction and the hydrated state of BA1 aggregates provide dynamicity and unusual stability. Our study unveils mechanistic insight into the dynamic transformation of bilayer membranes into vesicles.

Spontaneous Curvature Induction in an Artificial Bilayer Membrane

AbstractMaintaining lipid asymmetry across membrane leaflets is critical for functions like vesicular traffic and organelle homeostasis. However, a lack of molecular‐level understanding of the mechanisms underlying membrane fission and fusion processes in synthetic systems precludes their development as artificial analogs. Here, we report asymmetry induction of a bilayer membrane formed by an extended π‐conjugated molecule with oxyalkylene side chains bearing terminal tertiary amine moieties (BA1) in water. Autogenous protonation of the tertiary amines in the periphery of the bilayer by water induces anisotropic curvature, resulting in membrane fission to form vesicles and can be monitored using time‐dependent spectroscopy and microscopy. Interestingly, upon loss of the induced asymmetry by extensive protonation using an organic acid restored bilayer membrane. The mechanism leading to the compositional asymmetry in the leaflet and curvature induction in the membrane is validated by density functional theory (DFT) calculations. Studies extended to control molecules having changes in hydrophilic (BA2) and hydrophobic (BA3) segments provide insight into the delicate nature of molecular scale interactions in the dynamic transformation of supramolecular structures. The synergic effect of hydrophobic interaction and the hydrated state of BA1 aggregates provide dynamicity and unusual stability. Our study unveils mechanistic insight into the dynamic transformation of bilayer membranes into vesicles.

Role of Shape in Particle-Lipid Membrane Interactions: From Surfing to Full Engulfment.

Understanding and manipulating the interactions between foreign bodies and cell membranes during endo- and phagocytosis is of paramount importance, not only for the fate of living cells but also for numerous biomedical applications. This study aims to elucidate the role of variables such as anisotropic particle shape, curvature, orientation, membrane tension, and adhesive strength in this essential process using a minimal experimental biomimetic system comprising giant unilamellar vesicles and rod-like particles with different curvatures and aspect ratios. We find that the particle wrapping process is dictated by the balance between the elastic free energy penalty and adhesion free energy gain, leading to two distinct engulfment pathways, tip-first and side-first, emphasizing the significance of the particle orientation in determining the pathway. Moreover, our experimental results are consistent with theoretical predictions in a state diagram, showcasing how to control the wrapping pathway from surfing to partial to complete wrapping by the interplay between membrane tension and adhesive strength. At moderate particle concentrations, we observed the formation of rod clusters, which exhibited cooperative and sequential wrapping. Our study contributes to a comprehensive understanding of the mechanistic intricacies of endocytosis by highlighting how the interplay between the anisotropic particle shape, curvature, orientation, membrane tension, and adhesive strength can influence the engulfment pathway.

Unraveling How Antimicrobial Lipid Mixtures Disrupt Virus-Mimicking Lipid Vesicles: A QCM-D Study.

Single-chain lipid amphiphiles such as fatty acids and monoglycerides are promising antimicrobial alternatives to replace industrial surfactants for membrane-enveloped pathogen inhibition. Biomimetic lipid membrane platforms in combination with label-free biosensing techniques offer a promising route to compare the membrane-disruptive properties of different fatty acids and monoglycerides individually and within mixtures. Until recently, most related studies have utilized planar model membrane platforms, and there is an outstanding need to investigate how antimicrobial lipid mixtures disrupt curved model membrane platforms such as intact vesicle adlayers that are within the size range of membrane-enveloped virus particles. This need is especially evident because certain surfactants that completely disrupt planar/low-curvature membranes are appreciably less active against high-curvature membranes. Herein, we conducted quartz crystal microbalance-dissipation (QCM-D) measurements to investigate the membrane-disruptive properties of glycerol monolaurate (GML) monoglyceride and lauric acid (LA) fatty acid mixtures to rupture high-curvature, ~75 nm diameter lipid vesicle adlayers. We identified that the vesicle rupture activity of GML/LA mixtures mainly occurred above the respective critical micelle concentration (CMC) of each mixture, and that 25/75 mol% GML/LA micelles exhibited the greatest degree of vesicle rupture activity with ~100% efficiency that exceeded the rupture activity of other tested mixtures, individual compounds, and past reported values with industrial surfactants. Importantly, 25/75 GML/LA micelles outperformed 50/50 GML/LA micelles, which were previously reported to have the greatest membrane-disruptive activity towards planar model membranes. We discuss the mechanistic principles behind how antimicrobial lipid engineering can influence membrane-disruptive activity in terms of optimizing the balance between competitive membrane remodeling processes and inducing anisotropic vs. isotropic spontaneous curvature in lipid membrane systems.

Negative Gaussian curvature regulated pattern evolution on curved bilayer system

Diverse wrinkling patterns may occur on curved bilayer system under equi-biaxial compression. Here we study the wrinkling pattern evolution on curved bilayer system with negative Gaussian curvature via theoretical analysis and numerical simulations. Different with curved system with zero Gaussian curvature or positive Gaussian curvature, critical buckling analysis demonstrates that negative Gaussian curvature doesn't delay the occurrence of the surface wrinkling. Fourier spectral method, finite element simulations and post buckling analysis based on an upper-bound method have revealed three different evolutionary paths of the wrinkling patterns with the increase of the excess stress. Selection of the evolutionary path is determined by the dimensionless curvature and curvature anisotropy. The results in this work help understand the profound effect of curvature on surface wrinkling of bilayer systems, and provide a new means to regulate the wrinkling pattern and its evolutionary path on soft materials.

Curvature Heterogeneities Act as Singular Perturbations to Smooth Laplacian Fields: A Fluid Mechanics Demonstration.

In this Letter, we use a model fluid mechanics experiment to elucidate the impact of curvature heterogeneities on two-dimensional fields deriving from harmonic potential functions. This result is directly relevant to explain the smooth stationary structures in physical systems as diverse as curved liquid crystal and magnetic films, heat and Ohmic transport in wrinkled two-dimensional materials, and flows in confined channels. Combining microfluidic experiments and theory, we explain how curvature heterogeneities shape confined viscous flows. We show that isotropic bumps induce local distortions to Darcy's flows, whereas anisotropic curvature heterogeneities disturb them algebraically over system-spanning scales. Thanks to an electrostatic analogy, we gain insight into this singular geometric perturbation, and quantitatively explain it using both conformal mapping and numerical simulations. Altogether, our findings establish the robustness of our experimental observations and their broad relevance to all Laplacian problems beyond the specifics of our fluid mechanics experiment.

Flow by 𝜎_{𝑘} curvature to the Orlicz Christoffel-Minkowski problem

In this paper, we consider the anisotropic curvature flow of smooth, origin-symmetric, uniformly convex hypersurfaces in R n + 1 \mathbb {R}^{n+1} . The flow exists for all time and converges smoothly to a solution of the even Orlicz Christoffel-Minkowski problem. Our proof also gives an approach to the solution of the L p L_p Christoffel-Minkowski problem.

Viable vector coherent oscillation dark matter

We construct a viable model of the vector coherent oscillation dark matter. The vector boson is coupled to the inflaton through the kinetic function so that the effective Hubble mass term is cancelled out. In order to avoid strong constraints from isocurvature perturbation and statistically anisotropic curvature perturbation, the inflaton is arranged so that it does not contribute to the observed large scale curvature perturbation and we introduce a curvaton. We found viable vector coherent oscillation dark matter scenario for the wide vector mass range from 10-21 eV to 1 eV.

Curvature-Induced Cell Rearrangements in Biological Tissues.

On a curved surface, epithelial cells can adapt to geometric constraints by tilting and by exchanging their neighbors from apical to basal sides, known as an apico-basal topological transition 1 (AB-T1). The relationship between cell tilt, AB-T1s, and tissue curvature still lacks a unified understanding. Here, we propose a general framework for cell packing in curved environments and explain the formation of AB-T1s from the perspective of strain anisotropy. We find that steep curvature gradients can lead to cell tilting and induce AB-T1s. Alternatively, pressure differences across the epithelial tissue can drive AB-T1s in regions of large curvature anisotropy. The two mechanisms compete to determine the impact of tissue geometry and mechanics on optimized cell rearrangements in three dimensions.

Formation of protein-mediated bilayer tubes is governed by a snapthrough transition.

Plasma membrane tubes are ubiquitous in cellular membranes and in the membranes of intracellular organelles. They play crucial roles in trafficking, ion transport, and cellular motility. These tubes can be formed due to localized forces acting on the membrane or by the curvature induced by membrane-bound proteins. Here, we present a mathematical framework to model cylindrical tubular protrusions formed by proteins that induce anisotropic spontaneous curvature. Our analysis revealed that the tube radius depends on an effective tension that includes contributions from the bare membrane tension and the protein-induced curvature. We also found that the length of the tube undergoes an abrupt transition from a short, dome-shaped membrane to a long cylinder and this transition is characteristic of a snapthrough instability. Finally, we show that the snapthrough instability depends on the different parameters including coat area, bending modulus, and extent of protein-induced curvature. Our findings have implications for tube formation due to BAR-domain proteins in processes such as endocytosis, t-tubule formation in myocytes, and cristae formation in mitochondria.

Some aspects of anisotropic curvature flow of planar partitions

Some aspects of anisotropic curvature flow of planar partitions

Uniqueness of solutions to the logarithmic Minkowski problem in [formula omitted

In this paper, we prove the uniqueness of solutions to the logarithmic Minkowski problem in R3 without symmetry condition, provided the density of the measure is close to 1 in Cα norm. This result also implies the uniqueness of self-similar solutions to the anisotropic Gauss curvature flow in R3 when the speed function is Cα close to a positive constant.

Benchmarking of different strategies to include anisotropy in a curvature-driven multi-phase-field model

Two benchmark problems for quantitative assessment of anisotropic curvature driving force in phase field method were developed and introduced. Both benchmarks contained an anisotropically shrinking grain in homogeneous matrix. The first benchmark was a shrinking Wulff shape and in the second, such inclination dependence of the kinetic coefficient was added so that the shrinkage was isotropic. In both cases the match to the expected shape was quantified by means of Hausdorff distance and the shrinkage rate was analytically expressed. Three different ways of interface energy anisotropy inclusion in a multi-phase field model were compared. Because their performance was comparable, they were tested in an additional benchmark problem, which concerned direct measurement of equilibrium triple junction angles. Based on this benchmark, only one of the three strategies to include anisotropy was reliable in strongly anisotropic systems.

Transformation of Supramolecular Membranes to Vesicles Driven by Spontaneous Gradual Deprotonation on Membrane Surfaces

The various proteins and asymmetric lipid bilayers present in cell membranes form curvatures, resulting in structural transformations to generate vesicles. Fission and fusion processes between vesicles and cell membranes are reversible in living organisms. Although the transformation of a two-dimensional membrane to a three-dimensional vesicle structure is a common natural phenomenon, the lack of a detailed understanding at the molecular level limits the development of synthetic systems for functional materials. Herein, we report a supramolecular membrane system through donor-acceptor interactions using a π-deficient acceptor and π-rich donor as building blocks. The reduced electrostatic repulsion between ammonium cations and the spontaneously deprotonated neutral amino group induced anisotropic membrane curvature, resulting in membrane fission to form vesicles with a detailed understanding at the molecular level. Furthermore, the reversible transformation of vesicles to membranes upon changing the pH provides a novel synthetic system exhibiting both fission and fusion processes.