Mean Curvature Surfaces Research Articles

A convergence result for the derivation of front propagation in nonlocal phase field models

We prove that the mean curvature of a smooth surface in \R^{n} , n\geq 2 , arises as the limit of a sequence of functions that are intrinsically related to the difference between an n - and 1 -dimensional fractional Laplacian of a phase transition. Depending on the order of the fractional Laplace operator, we recover the fractional mean curvature or the classical mean curvature of the surface. Moreover, we show that this is an essential ingredient for deriving the evolution of fronts in fractional reaction-diffusion equations such as those for atomic dislocations in crystals.

Mechanochemical modeling of morphogenesis in cell polarization for budding yeast

Cell polarization is a multiphysics problem resulting from coupling protein pathways and cell morphological evolution. The most common example is asexual reproduction in budding yeast through Cdc42 and septin signals. However, how the interaction between the Cdc42-septin systems and the mechanical deformation of the cell surface is not well understood. To explore the interaction, we build a three-dimensional mechanochemical coupling framework to investigate the Cdc42-septin systems and elucidate how morphogenesis at the cell scale enhances the robustness of cell polarization in budding yeast. We utilize the viscous active shell model to describe the moving boundary of the cell surface, and the mechanochemical coupling is achieved through the introduction of cell surface mean curvature and active contraction-growth effects. With a normal effect of septin, numerical results demonstrate the budding processes with different transition shapes of budding profiles. On the other hand, the mechanochemical coupling problem with a weak effect of septin drives the emergence of wrinkles on the cell surface. We also apply linear perturbation analysis of the cell surface deformation to show how cell shape evolution is subjected to active contraction-growth effects. Our results show that a 3D coupled mechanochemical model can reproduce the budding morphology in yeast. Such a model provides a more comprehensive portrait of cell polarity and can be used to characterize both wild-type and mutant phenotypes such as septin mutants.

Self-Detachment of Nanoporous Thin Films

Nanoporous metals consist of a network of interconnected ligaments with diameters tunable between 5 and 500 nm. The talk focusses on nanoporous metal thin films prepared by electrochemical dealloying of 300 nm thick Au25Ag75 films deposited on a pure Au base layer. This type of nanoporous gold film is the subject of many studies investigating the material as sensor, as catalyst, or as substrate for energy storage.The massive Au base layer is intended as an adhesion promoter and to prevent the corrosion from attacking the substrate. Yet, contrary to expectation, we observed that the as-dealloyed nanoporous films developed a gap near the base layer, with extremely loose connection. Screening electron micrographs in earlier publications revealed that the deficit in connectivity is ubiquitous – even though it has not been broached. We could readily reproduce the feature in kinetic Monte Carlo simulations of dealloying, with no special assumptions on the processes near the base layer. The talk will describe these observations, and we argue that the evolution of the connectivity in nanoporous thin films is sensitive to gradients, in the direction of the film thickness, of the laterally averaged surface mean curvature. Specifically, surface diffusion driven by the energetically favorable planar substrate results in a net current from ligaments connected to the substrate towards the substrate, leading to degradation and eventual pinch-off of these ligaments. A similar phenomenon occurs at the outer surface, causing densification and retraction of the outermost ligaments.Our findings suggest that this phenomenon may be relevant for all nanoscopic contact interfaces where surface diffusion is sufficiently fast. These results have significant implications for the design and implementation of nanoporous thin films in various applications, including sensor and actuator devices. Mitigation strategies will be discussed.Reference:G. Henkelmann, D. Waldow, M. Liu, L. Lührs, Y. Li and J. Weissmüller; Self-Detachment and Subsurface Densification of Dealloyed Nanoporous Thin Films. Nano Lett 22 (2022) 6787. Figure 1

Surfaces with Constant Negative Curvature

In this paper, we have considered surfaces with constant negative Gaussian curvature in the simply isotropic 3-Space by defined Sauer and Strubeckerr. Firstly, we have studied the isotropic II-flat, isotropic minimal and isotropic II-minimal, the constant second Gaussian curvature, and the constant mean curvature of surfaces with constant negative curvature (SCNC) in the simply isotropic 3-space. Surfaces with symmetry are obtained when the mean curvatures are equal. Further, we have investigated the constant Casorati, the tangential and the amalgamatic curvatures of SCNC.

PCL strut-like scaffolds appear superior to gyroid in terms of bone regeneration within a long bone large defect: An in silico study.

The treatment of large bone defects represents a major clinical challenge. 3D printed scaffolds appear as a promising strategy to support bone defect regeneration. The 3D design of such scaffolds impacts the healing path and thus defect regeneration potential. Among others, scaffold architecture has been shown to influence the healing outcome. Gyroid architecture, characterized by a zero mean surface curvature, has been discussed as a promising scaffold design for bone regeneration. However, whether gyroid scaffolds are favourable for bone regeneration in large bone defects over traditional strut-like architecture scaffolds remains unknown. Therefore, the aim of this study was to investigate whether gyroid scaffolds present advantages over more traditional strut-like scaffolds in terms of their bone regeneration potential. Validated bone defect regeneration principles were applied in an in silico modeling approach that allows to predict bone formation in defect regeneration. Towards this aim, the mechano-biological bone regeneration principles were adapted to allow simulating bone regeneration within both gyroid and strut-like scaffolds. We found that the large surface curvatures of the gyroid scaffold led to a slower tissue formation dynamic and conclusively reduced bone regeneration. The initial claim, that an overall reduced zero mean surface curvature would enhance bone formation, could not be confirmed. The here presented approach illustrates the potential of in silico tools to evaluate in pre-clinical studies scaffold designs and eventually lead to optimized architectures of 3D printed implants for bone regeneration.

Self-Detachment and Subsurface Densification of Dealloyed Nanoporous Thin Films.

Experiment shows thin films of dealloyed nanoporous gold (NPG) spontaneously detaching from massive gold base layers. NPG can also densify near its external surface. This is naturally reproduced by kinetic Monte Carlo (KMC) simulation of dealloying and coarsening and so appears generic for nanoscale network materials evolving by surface diffusion. Near the porous layer’s external surface and near its interface with the base layer, gradients in the depth-profile of a laterally averaged mean surface curvature provide driving forces for diffusion and cause divergences of the net fluxes of matter, leading to accretion/densification or to erosion/disconnection. As a toy model, the morphology evolution of substrate-supported nanopillars by surface diffusion illustrates and confirms our considerations. Contrary to cylindrical nanowires, the ligaments in nanoporous materials exhibit pre-existing gradients in the mean curvature. The Plateau-Rayleigh long-wavelength stability criterion is then not applicable and the disconnection accelerated.

Detection of Corneal Ectasia Using OCT Maps of Pachymetry and Posterior Surface Mean Curvature.

To quantify the abnormal corneal thinning and posterior surface steepening that is observed in keratoconus with an Ectasia Index. Optical coherence tomography (OCT) was used to image the corneas of normal individuals and patients with varying stages of keratoconus (manifest, subclinical, and forme fruste). Maps of corneal pachymetry and posterior surface mean curvature were generated, and an Ectasia Index was calculated by multiplying Gaussian fits obtained from the two types of maps. Repeated five-fold cross-validation was used to evaluate the ability of the Ectasia Index to differentiate between normal and keratoconic eyes. The classification performance of the Ectasia Index was compared to minimum pachymetry and maximum posterior mean curvature. Thirty-two eyes from 16 normal individuals, 89 eyes from 63 patients with manifest keratoconus, 16 eyes from 15 patients with subclinical keratoconus, and 26 eyes from 26 patients with forme fruste keratoconus were included in the study. During cross-validation, 100% of the eyes with manifest (89 of 89) and subclinical (16 of 16) keratoconus were correctly classified by the Ectasia Index. The average classification accuracy for the forme fruste keratoconus group was 63 ± 21% (16.4 of 26). The specificity for the normal group was 91 ± 10% (29.1 of 32). The Ectasia Index had a higher sensitivity for keratoconus detection and similar specificity in comparison to minimum pachymetry and maximum posterior mean curvature. The Ectasia Index could be a valuable additional metric for clinicians to consider when screening for keratoconus. [J Refract Surg. 2022;38(8):502-510.].

Wicking in Porous Polymeric Membranes: Determination of an Effective Capillary Radius to Predict the Flow Behavior in Lateral Flow Assays.

The working principle of lateral flow assays, such as the widely used COVID-19 rapid tests, is based on the capillary-driven liquid transport of a sample fluid to a test line using porous polymeric membranes as the conductive medium. In order to predict this wicking process by simplified analytical models, it is essential to determine an effective capillary radius for the highly porous and open-pored membranes. In this work, a parametric study is performed with selected simplified structures, representing the complex microstructure of the membrane. For this, a phase-field approach with a special wetting boundary condition to describe the meniscus formation and the corresponding mean surface curvature for each structure setup is used. As a main result, an analytical correlation between geometric structure parameters and an effective capillary radius, based on a correction factor, are obtained. The resulting correlation is verified by applying image analysis methods on reconstructed computer tomography scans of two different porous polymeric membranes and thus determining the geometric structure parameters. Subsequently, a macroscale flow model that includes the correlated effective pore size and geometrical capillary radius is applied, and the results are compared with wicking experiments. Based on the derived correction function, it is shown that the analytical prediction of the wicking process in highly porous polymeric membranes is possible without the fitting of experimental wicking data. Furthermore, it can be seen that the estimated effective pore radius of the two membranes is 8 to 10 times higher than their geometric mean pore radii.

Automated ascending aorta delineation from ECG-gated computed tomography images.

The instrumental role of comprehensive geometrical quantification in contemporary, effective descriptions of aortic growth and disease is well established. General or specific purpose algorithms are being developed to provide automatic landmark detection and high accuracy measurements. In the present study, an objective method for automated delineation of the ascending aorta is introduced, based on geometrical properties of the aortic wall. In the proximal ascending aorta, the method identifies the sinotubular junction by tracing the mean surface curvature transition region from the origins of the coronary arteries to the location where the aorta acquires its tubular shape. In the distal ascending aorta, the brachiocephalic artery origin is defined by a split centreline cross section within the brachiocephalic artery-aortic arch bifurcation region. The method's accuracy of detection was quantified against the manual border identification performed by two experienced observers on 3D aortic reconstructions of 44 computed tomography examinations. Median (method, observer) distance and inclination measurements ranged from 0.89 [1.02] mm and 4.66 [5.07]°, respectively, in the proximal border, to 2.18 [2.39] mm and 7.13 [4.77]° in the distal. Accuracy of border detection was found to be high compared to interobserver variability and relevant automatic and manual methodology results previously reported in literature. Delineation of the ascending aorta on a three-dimensional aortic reconstruction with automated identification of the sinotubular junction (proximal border) and of the origin of the brachiocephalic artery (distal border).

Special Mean and Total Curvature of a Dual Surface in Isotropic Spaces

The study of the geometry surfaces in spaces with a degenerate metric is one of the urgent problems of modern geometry since its results find numerous applications in problems of mechanics and quantum mechanics.In this paper, we study the properties of the total and mean curvatures of a surface and its dual image in an isotropic space. We prove the equality of the mean curvature and the second quadratic forms. The relation of the mean curvature of a surface to its dual surface is found. The superimposed space method is used to investigate the geometric characteristics of a surface relative to the normal and special normal.

Differentiating Between Contact Lens Warpage and Keratoconus Using OCT Maps of Corneal Mean Curvature and Epithelial Thickness.

To formulate an Epithelial Modulation index to differentiate between eyes with contact lens warpage and keratoconus. Normal eyes and eyes with either contact lens warpage or keratoconus were scanned by a Fourier-domain optical coherence tomography (OCT) system. Maps of epithelial thickness and anterior surface mean curvature were generated and converted to deviation maps by subtracting the average maps from a healthy population. The Epithelial Modulation index was defined as the covariance between the two types of deviation maps. A logistic regression model was used to classify eyes as non-keratoconus (normal or warp-age) or keratoconus (manifest, subclinical, or forme fruste). The average Epithelial Modulation index value for normal eyes was -0.6 ± 1.0 µm/m. Eyes with keratoconus were characterized by coincident high anterior surface mean curvature and low epithelial thickness, resulting in a high Epithelial Modulation index (manifest: 103.0 ± 82.9 µm/m, subclinical: 37.0 ± 23.0 µm/m, forme fruste: 7.3 ± 13.2 µm/m). The Epithelial Modulation index was closer to normal for eyes with warpage (-1.9 ± 4.0 µm/m). The classification accuracy of the Epithelial Modulation index during five-fold cross-validation of the logistic regression model was 100 ± 0% for normal eyes and 99.0 ± 2.0% for eyes with warpage. The accuracy was 100 ± 0%, 100 ± 0%, and 53.1 ± 1.5% for the manifest, subclinical, and forme fruste keratoconus groups, respectively. The Epithelial Modulation index is useful in distinguishing eyes with secondary epithelial modulation (keratoconus) from those with primary epithelial deformation (contact lens-related warpage). [J Refract Surg. 2022;38(2):112-119.].

End-exclusion zones in strongly stretched, molten polymer brushes of arbitrary shape.

Theories of strongly stretched polymer brushes, particularly the parabolic brush theory, are valuable for providing analytically tractable predictions for the thermodynamic behavior of surface-grafted polymers in a wide range of settings. However, the parabolic brush limit fails to describe polymers grafted to convex curved substrates, such as the surfaces of spherical nanoparticles or the interfaces of strongly segregated block copolymers. It has previously been shown that strongly stretched curved brushes require a boundary layer devoid of free chain ends, requiring modifications of the theoretical analysis. While this "end-exclusion zone" has been successfully incorporated into the descriptions of brushes grafted onto the outer surfaces of cylinders and spheres, the behavior of brushes on surfaces of arbitrary curvature has not yet been studied. We present a formulation of the strong-stretching theory for molten brushes on the surfaces of arbitrary curvature and identify four distinct regimes of interest for which brushes are predicted to possess end-exclusion zones, notably including regimes of positive mean curvature but negative Gaussian curvature. Through numerical solutions of the strong-stretching brush equations, we report predicted scaling of the size of the end-exclusion zone, the chain end distribution, the chain polarization, and the free energy of stretching with mean and Gaussian surface curvatures. Through these results, we present a comprehensive picture of how the brush geometry influences the end-exclusion zones and exact strong-stretching free energies, which can be applied, for example, to model the full spectrum of brush geometries encountered in block copolymer melt assembly.

Canonical Coordinates and Natural Equation for Lorentz Surfaces in R13

We consider Lorentz surfaces in R13 satisfying the condition H2−K≠0, where K and H are the Gaussian curvature and the mean curvature, respectively, and call them Lorentz surfaces of general type. For this class of surfaces, we introduce special isotropic coordinates, which we call canonical, and show that the coefficient F of the first fundamental form and the mean curvature H, expressed in terms of the canonical coordinates, satisfy a special integro-differential equation which we call a natural equation of the Lorentz surfaces of a general type. Using this natural equation, we prove a fundamental theorem of Bonnet type for Lorentz surfaces of a general type. We consider the special cases of Lorentz surfaces of constant non-zero mean curvature and minimal Lorentz surfaces. Finally, we give examples of Lorentz surfaces illustrating the developed theory.

Integrability of the sub-Riemannian mean curvature at degenerate characteristic points in the Heisenberg group

Abstract We address the problem of integrability of the sub-Riemannian mean curvature of an embedded hypersurface around isolated characteristic points. The main contribution of this paper is the introduction of a concept of a mildly degenerate characteristic point for a smooth surface of the Heisenberg group, in a neighborhood of which the sub-Riemannian mean curvature is integrable (with respect to the perimeter measure induced by the Euclidean structure). As a consequence, we partially answer to a question posed by Danielli, Garofalo and Nhieu in [D. Danielli, N. Garofalo and D. M. Nhieu, Integrability of the sub-Riemannian mean curvature of surfaces in the Heisenberg group, Proc. Amer. Math. Soc. 140 2012, 3, 811–821], proving that the mean curvature of a real-analytic surface with discrete characteristic set is locally integrable.

An improved mean curvature-based bending model for cloth simulation

In cloth animation, the bending behaviour of a cloth is important for cloth simulation effects. The presentation of cloth bending properties plays a key role in cloth animation research because cloth is characterised by a strong resistance to stretch while weak resistance to bending. This paper proposed an improved approximate nonlinear bending model based on local geometric information. In the dynamic simulation, a cloth was divided into several regions according to mean curvature of surface. The bending force was updated according to the changes of the mean curvature in each region. Based on differential geometry and the dynamic simulation of the cloth, the calculation of bending force was simple and accurate with the proposed model. Experimental results show that wrinkles and folds generated in a natural way with the improved model and the efficiency of simulation is improved compared with the original algorithm.

Elastic wave propagation in structures with triply periodic minimal surfaces

We study the elastic wave propagation behavior of triply periodic minimal surface structures. Triply periodic minimal surfaces (TPMS) are surfaces with periodicity in three independent directions and a mean surface curvature of zero. Here, we apply the finite element method to obtain the dispersion curves of six different TPMS structures: Schwarz' P, Schwarz' D, Gyroid, Fischer-Koch's S, Schoen's I-WP, and Schoen's F-RD. The dispersion curves are obtained by modeling their respective unit cells, appropriately applying the Floquet-Bloch periodic boundary condition, and then performing an eigenfrequency analysis while sweeping over the boundaries of their respective irreducible Brillouin zones. We then classify each band according to their respective propagation modes (P- or S-waves) and study the effect of surface topology on each mode. Finally, we discuss the generation of directional and polarized bandgaps in these structures.

Reynolds analogy and turbulent heat transfer on rotating curved surfaces

To evaluate turbulent heat transfer on rotating turbine blades, knowledge of the Reynolds stress and heat conduction moments on the blades is required. This involves solving the mean flow equations of momentum and thermal energy on the rotating blades. In this paper, the turbine blades are represented by rotating curved surfaces, and turbulent heat transfer calculations are carried out by invoking a valid Reynolds analogy because of its simplicity. The classic Reynolds analogy is defined as the ratio between the eddy and thermal diffusivity in a stationary plane flow; however, its counterpart for rotating curved flows is still not known. To derive a Reynolds analogy for rotating curved flows, appropriate turbulence models to close the Reynolds transport equations for momentum and thermal energy are required. Assuming that, in the constant flux region, advection and diffusion of Reynolds stress and heat conduction moments are negligible compared to the production of turbulent energy, the Reynolds shear stresses and heat conduction moments can be solved in terms of the mean flow gradients, a turbulent Prandtl number (Prt)rc and a gradient Richardson number for rotating curved flows Rirc. These dimensionless numbers are dependent on mean flow properties, surface curvature, and system rotational speed. Thus derived, the classic Reynolds analogy can be recovered only when Rirc << 1 and (Prt)rc = 1. For turbine blade heat transfer, Rirc is not necessary very small and (Prt)rc differs from unity in the whole flow field. Therefore, using the classic Reynolds analogy to estimate turbine blade heat transfer will lead to incorrect result. The new Reynolds analogy, formulated for turbulent thermal flows on rotating curved surfaces, takes all pertinent external body force effects into account; thus, turbulent heat transfer evaluation on turbine blade surfaces can be substantially improved.

Development of a novel method to characterize mean surface peak curvature as a signature of tribological performance of dental implant surfaces

The integrity of dental implant surfaces might be compromised during surgical insertion. Wear and topographical modifications can occur during implant insertion, which can potentially have clinical implications. Accordingly, the analysis and prediction of the wear behaviour of dental implant surfaces is fundamental. Surface mean curvature is a key topographical parameter in contact mechanics, related to wear and plastic deformation; however, there is not a consensus on its characterization. In this paper, a critical analysis of the feature parameter Spc from ISO 25178-2 is carried out, which addresses its limited ability to characterize dental implant surfaces. A novel alternative parameter (named ‘relative mean peak curvature’ ΔSdq) is presented and evaluated in representative dental implant treatments. The obtained results suggest a positive correlation between the surface peak curvature and its tribological signature in biomedical implants. Although biomedical experiments are needed to validate this correlation, the novel method is presented as a tool for future studies.

Modification of the Extended Advanced IEM for Scattering From Randomly Rough Surfaces

In this letter, we modify the extended advanced integral equation model (EAIEM) for electromagnetic backscattering and bistatic scattering from rough surfaces with small to moderate heights. We extend the first-order approximation of the error function as introduced in the EAIEM model to the second order, in a hope to be more suitable for large roughness and high frequency. In addition, a new transition model for the reflection coefficient is proposed to make the dependences explicit on the mean surface curvature, frequency, and dielectric constant, whereas making no use of the complementary term, the effect of inadequate evaluation of this term is mitigated. Comparison with POLARSCAT data for backscattering and with European Microwave Signature Laboratory (EMSL) measurements for bistatic scattering demonstrates the validity of the updated model.

Analytical modeling of micelle growth. 2. Molecular thermodynamics of mixed aggregates and scission energy in wormlike micelles

HypothesesQuantitative molecular-thermodynamic theory of the growth of giant wormlike micelles in mixed nonionic surfactant solutions can be developed on the basis of a generalized model, which includes the classical “phase separation” and “mass action” models as special cases. The generalized model describes spherocylindrical micelles, which are simultaneously multicomponent and polydisperse in size. TheoryThe model is based on explicit analytical expressions for the four components of the free energy of mixed nonionic micelles: interfacial-tension, headgroup-steric, chain-conformation components and free energy of mixing. The radii of the cylindrical part and the spherical endcaps, as well as the chemical composition of the endcaps, are determined by minimization of the free energy. FindingsIn the case of multicomponent micelles, an additional term appears in the expression for the micelle growth parameter (scission free energy), which takes into account the fact that the micelle endcaps and cylindrical part have different compositions. The model accurately predicts the mean mass aggregation number of wormlike micelles in mixed nonionic surfactant solutions without using any adjustable parameters. The endcaps are enriched in the surfactant with smaller packing parameter that is better accommodated in regions of higher mean surface curvature. The model can be further extended to mixed solutions of nonionic, ionic and zwitterionic surfactants used in personal-care and house-hold detergency.