Equilibrium Shape Research Articles

THE ROLE OF ELASTIC STRESS DURING THERMOMIGRATION OF LIQUID ZONES BASED ON ALUMINUM IN SILICON

The method of thermomigration of liquid zones based on aluminum makes it possible to create complex structures of closed channels with boundaries formed from p-n junctions in single-crystal silicon wafers. The channels are characterized by the uniformity of their properties, and the pn junctions are characterized by their sharpness. Such structures are used in high-current electronics, photovoltaics, and microelectromechanical converters. Volumetric deformation inside and outside the channel due to alloying of silicon with aluminum leads to the formation of mechanical stresses. The equilibrium shape of the channels formed at high temperatures is determined by the minimum elastic energy and depends on the material parameters and geometry of the structure. Within the framework of the linear theory of elasticity, the behavior of elastic energy during the formation of the structure of thermomigration channels at high temperatures doped with aluminum in a single-crystalline (001) cut disk was studied. The simulation was performed using the finite element method in the COMSOL Multiphysics mathematical package. The study was carried out for practically important structures in which the direction of the edges is oriented along the diagonal of the square. Only such structures do not have breaks during the process of thermomigration. Based on the calculation results, it was revealed that the minimum elastic energy corresponds to structures with different crystalline orientations inside the channel and outside it – in the main silicon matrix. The direction of the crystalline axes inside the channel, corresponding to the minimum elastic energy, is rotated by 45 degrees in the plane of the disk relative to the direction of the axes of the main matrix of the silicon crystal. In addition, calculations showed that with such a turn, the shape of the channels changes. The minimum elastic energy corresponds not to vertical structures, but to inclined ones. The angle of inclination of the pyramids depends on the width of the channels and the distance between them.

Joint calculation of mechanical and radar characteristics of calibration folding spherical reflectors

Combined modeling of the shape of the surface of the grid canvas belt stretched on the wireframe and statistical radar characteristics of the calibration reflector has been performed. To solve the problem, a complex computer model is proposed and implemented. The model provides analysis of the stress state of the grid canvas belt forming the surface of the folding spherical reflector, numerical electrodynamic calculation of the spatial backscattering diagram and estimation of statistical radar characteristics corresponding to the arbitrary location direction in the process of observation. Using the complex computer model, the equilibrium shape of the reflecting surface has been calculated for the selected design version of the folding spherical reflector, the facet 3D electrodynamic model has been developed, spatial backscattering diagrams have been calculated and statistical RCS characteristics for various wave sizes of the object have been evaluated. The presented complex computer model may be used in studies related to analysis of radar characteristics of calibration folding spherical radar reflectors with various versions of power wireframe design.

An energy-optimization method to study gel-swelling in confinement.

We recast the problem of hydrogel swelling under physical constraints as an energy optimization problem. We apply this approach to compute equilibrium shapes of hydrogel spheres confined within a jammed matrix of rigid beads and interpret the results to determine how confinement modifies the mechanics of swollen hydrogels. In contrast to the unconfined case, we find a spatial separation of strains within the bulk of the hydrogel as the strain becomes localized to an outer region. We also explore the contact mechanics of the gel, finding a transition from Hertzian behavior to non-Hertzian behavior as a function of swelling. Our model, implemented in the Morpho shape optimization environment and validated against an experimentally demonstrated prototypical scenario, can be applied in any dimension, readily adapted to diverse swelling scenarios and extended to use other energies in conjunction.

SOME PROBLEMS OF OPTIMIZATION OF ROD SYSTEMS CONTAINING COMPRESSED ELEMENTS USING ADDITIONAL CONSTRAINTS

The article is devoted to the problem of increasing the stability of rod systems containing longitudinally compressed elements. The influence of the imposition of constraints on the behavior of such systems is investigated in order to determine such places for imposing constraints that provide the maximum stability of the system reinforced by the constraint. To get generality, the consideration includes such rod systems that allow various equilibrium configurations, for example, having internal ideal hinges, as well as an arbitrary distribution of longitudinal compressive forces, including leaving some areas free from compression. For the same purpose, the constraints are considered as generalized, producing a reaction with an arbitrary spatial distribution. The paper formulates some general results related to the influence of the introduction of generalized constraints on the critical forces of a rod system with some generalizations related to the extension of the class of rod systems under consideration. Particular attention is paid to the buckling modes in view of their important role as a basis for describing various configurations of the structure. It has been established that the shape of these modes, in particular, the position of their nodes, is essential for finding the optimal position of the constraint. For the case of constraint in the form of a concentrated hinged support, analytical expressions are obtained that represent the derivatives of the critical forces of the system with respect to the coordinate of the support. The case of a multiple critical force, when this derivative, generally speaking, does not exist, is especially considered. These expressions make it possible to qualitatively characterize the optimal position of the support. The application of some of the obtained results is demonstrated by the example of the problem of finding the optimal position of an intermediate hinged support of a two-span rod supported at the ends by elastic hinged supports. These positions are qualitatively described for various values of the stiffness coefficients of the end supports. It has been established that under certain conditions, the optimal positions of the intermediate support correspond to a special semi-curved mode of buckling, in which one of the spans does not bend, but retains its rectilinear equilibrium shape.

О стабильности капель катализатора на границе контакта пар-жидкость-кристалл в процессе роста нитевидных нанокристаллов

The paper provides a physical justification for the wettability conditions for a crystalline surface of a limited area by a small-volume catalytic liquid at the end of a growing nanowires (NW) characterized by a contact angle β, which contributes to a fundamental understanding of the nature of the contact angle of catalyst drops at the top of the NW. It is shown that under the conditions of stationary growth of NWs with a transverse singular face, there are only values of the angles β and γ (the angle of inclination of the side surface of the crystal to this face), which correspond to the minimum increment of the free energy of the three-phase system αLVcosβ + αSL = αSVcosγ and determine the stability of the catalyst drop at the top of the NW. With the growth of cylindrical NWs, the conditions of indifferent equilibrium are realized at the drop wetting perimeter. A drop, due to the dissolution of a crystallizing substance or its separation from a liquid solution, can take an equilibrium shape with a contact angle β that does not satisfy the equilibrium contact angle condition θ in the Young equation. A concentric break (rib) at the NW tip should increase the observed wetting angle θ and lead to contact angle hysteresis. The restrictions imposed on the value of the contact angle of a stable catalyst drop at the top of the NW are determined. The catalyst drop will take an equilibrium shape if the hysteresis angle β is in the range θ < β ≤ θ' + γ (θ' is the wetting angle of the NW side walls). For the growth of semiconductor NWs in the form of a straight cylinder, γ = 90° and therefore always β > 90°. It is shown that the direction of displacement of the three-phase line relative to the droplet surface is determined by the growth angle φ0: for the nonwetting growth mode of NWs (with a transverse face) φ0 = β – γ; for the wetting growth mode (with the end surface curved near the three-phase line)

Collective Variables for Crystallization Simulations-from Early Developments to Recent Advances.

Crystallization is an important physicochemical process which has relevance in material science, biology, and the environment. Decades of experimental and theoretical efforts have been made to understand this fundamental symmetry-breaking transition. While experiments provide equilibrium structures and shapes of crystals, they are limited to unraveling how molecules aggregate to form crystal nuclei that subsequently transform into bulk crystals. Computer simulations, mainly molecular dynamics (MD), can provide such microscopic details during the early stage of a crystallization event. Crystallization is a rare event that takes place in time scales much longer than a typical equilibrium MD simulation can sample. This inadequate sampling of the MD method can be easily circumvented by the use of enhanced sampling (ES) simulations. In most of the ES methods, the fluctuations of a system's slow degrees of freedom, called collective variables (CVs), are enhanced by applying a bias potential. This transforms the system from one state to the other within a short time scale. The most crucial part of such CV-based ES methods is to find suitable CVs, which often needs intuition and several trial-and-error optimization steps. Over the years, a plethora of CVs has been developed and applied in the study of crystallization. In this review, we provide a brief overview of CVs that have been developed and used in ES simulations to study crystallization from melt or solution. These CVs can be categorized mainly into four types: (i) spherical particle-based, (ii) molecular template-based, (iii) physical property-based, and (iv) CVs obtained from dimensionality reduction techniques. We present the context-based evolution of CVs, discuss the current challenges, and propose future directions to further develop effective CVs for the study of crystallization of complex systems.

A Homogenized Bending Theory for Prestrained Plates

The presence of prestrain can have a tremendous effect on the mechanical behavior of slender structures. Prestrained elastic plates show spontaneous bending in equilibrium—a property that makes such objects relevant for the fabrication of active and functional materials. In this paper we study microheterogeneous, prestrained plates that feature non-flat equilibrium shapes. Our goal is to understand the relation between the properties of the prestrained microstructure and the global shape of the plate in mechanical equilibrium. To this end, we consider a three-dimensional, nonlinear elasticity model that describes a periodic material that occupies a domain with small thickness. We consider a spatially periodic prestrain described in the form of a multiplicative decomposition of the deformation gradient. By simultaneous homogenization and dimension reduction, we rigorously derive an effective plate model as a Gamma -limit for vanishing thickness and period. That limit has the form of a nonlinear bending energy with an emergent spontaneous curvature term. The homogenized properties of the bending model (bending stiffness and spontaneous curvature) are characterized by corrector problems. For a model composite—a prestrained laminate composed of isotropic materials—we investigate the dependence of the homogenized properties on the parameters of the model composite. Secondly, we investigate the relation between the parameters of the model composite and the set of shapes with minimal bending energy. Our study reveals a rather complex dependence of these shapes on the composite parameters. For instance, the curvature and principal directions of these shapes depend on the parameters in a nonlinear and discontinuous way; for certain parameter regions we observe uniqueness and non-uniqueness of the shapes. We also observe size effects: The geometries of the shapes depend on the aspect ratio between the plate thickness and the composite period. As a second application of our theory, we study a problem of shape programming: We prove that any target shape (parametrized by a bending deformation) can be obtained (up to a small tolerance) as an energy minimizer of a composite plate, which is simple in the sense that the plate consists of only finitely many grains that are filled with a parametrized composite with a single degree of freedom.

On shape forming by contractile filaments in the surface of growing tissues.

Growing tissues are highly dynamic, and flow on sufficiently long timescales due to cell proliferation, migration, and tissue remodeling. As a consequence, growing tissues can often be approximated as viscous fluids. This means that the shape of microtissues growing in vitro is governed by their surface stress state, as in fluid droplets. Recent work showed that cells in the near-surface region of fibroblastic or osteoblastic microtissues contract with highly oriented actin filaments, thus making the surface properties highly anisotropic, in contrast to what is expected for an isotropic fluid. Here, we develop a model that includes mechanical anisotropy of the surface generated by contractile fibers and we show that mechanical equilibrium requires contractile filaments to follow geodesic lines on the surface. Constant pressure in the fluid forces these contractile filaments to be along geodesics with a constant normal curvature. We then take this into account to determine equilibrium shapes of rotationally symmetric bodies subjected to anisotropic surface stress states and derive a family of surfaces of revolution. A comparison with recently published shapes of microtissues shows that this theory accurately predicts both the surface shape and the direction of the actin filaments on the surface.

Effect of geometrical parameters on the nonlinear behavior of DE-based minimum energy structures: Numerical modeling and experimental investigation

This work presents a finite element framework for simulating the quasi-static response of dielectric elastomer-based minimum energy structure (DEMES). The DEMES is an actuator formed by combining an inextensible frame and pre-stretched dielectric membrane that exhibits the unique shape-morphing characteristics of the actuator. A continuum strain energy-based model is implemented to investigate the impact of the different geometrical parameters on the performance of the DEMES actuator. Finite element analyses are performed using user-defined element (UEL) in ABAQUS for determining the equilibrium shape of the actuators and further investigating their electromechanical response. Experiments are performed using the commercially available VHB-4910 acrylic tape and the PET frames. 3D-printed reinforcements are used to impart anisotropy in the specimen. The findings of the model solutions provide preliminary insights on the alteration of the initial and final configurations of the DEMES affected by different geometrical parameters. It is observed that the shape of the electrode (rectangular, circular and triangular), compliant frame (rectangular, circular and triangular) and implemented stiffeners appreciably alter the attained initial configuration, final configuration and actuation range of the DEMES actuator. In general, this investigation can find its potential use in designing the futuristic DEMES through topological optimization of the compliant electrode and frame geometry together with material anisotropy of the elastomer.

Monitoring of the elements stability of building constructions by means of example of vertical elastic rod of high flexibility

The article is related to the field of geodetic and engineering construction monitoring of building structures in the form of rod systems. It investigates the measurement accuracy of the compression deformation of the rod. As an example, there is considered a vertical rectilinear rod of high flexibility, loaded with a longitudinal axial force. Experience demonstrates that the rod, under the influence of an axial compressive load not exceeding the permissible design values, retains its original form of equilibrium and stability. At the same time, it works in the field of elastic compression deformations. As the load increases to its critical values, there occurs a particular state of the element when the form of equilibrium becomes unstable, and at any moment, it can turn from a straight line into a curvilinear one with a slight external influence or as a result of position errors. This is accompanied by large displacements and permanent deformations. The importance of monitoring is noted when a building structure with similar constructions, for example, is situated in the area of development of negative natural and anthropogenic processes. As a result, the load on some elements may increase to critical values due to uneven horizontal and vertical displacement and inclination of the structure, including vibrations, etc. Attention is focused on the fact that the complex of engineering and technical measurements should show, with the necessary completeness, reliability and accuracy of the value, the nature and dynamics of changes in the external load on the elements of the building structure, changes in internal forces and stresses in them, the development of deformations (changes in the size and shape of equilibrium). There is explained the accuracy with which it is necessary to measure the displacement of the rod end during the deformation. There was derived the dependence with the help of which it is possible to see the root-mean-square error of the displacement measuring that depends on the choice of the maximum stress deviation in the sections of the rod made of a certain material. There is laid out the calculation of the maximum stress deviation based on the stability and flexibility of the rod.

Equilibrium shapes of two- and three-dimensional two-phase rotating fluid drops with surface tension: Effects of inner drop displacement

The shapes of rotating fluid drops held together by surface tension are an important field of study in fluid mechanics. Recently, experiments with micrometer-scale droplets of liquid helium have been undertaken and it has proven useful to compare the shapes of the resultant superfluid droplets with classical analogs. If the helium is a mixture of He3 and He4, two phases are present. In a recent paper, the shapes of rotating two-phase fluid droplets were calculated where the inner drop was constrained to stay at the drop center. The outer shapes and dimensionless rotation rate–angular momentum relationships were shown to be similar to single-phase drops, providing that the density and surface tension scales were chosen appropriately. In the current paper, I investigate models in which the inner drop can displace from the center. In order to simplify the analyses, two-dimensional drops are first investigated. I show that the inner drop is unstable in the center position if its density is greater than the outer density and that the inner drop will move toward the outer boundary of the drop in these cases. When the inner drop has a higher density than the outer drop, the moment of inertia of displaced inner drops is increased relative to centered drops, and hence, the kinetic energy is decreased. Shapes of two- and three-dimensional drops, rotation rate–angular momentum, and kinetic and surface energy relationships are investigated for off-axis inner drops with parameters relevant to recent liquid He experiments.

Defining shapes of two-dimensional crystals with undefinable edge energies

The equilibrium shape of crystals is a fundamental property of both aesthetic appeal and practical importance: the shape and its facets control the catalytic, light-emitting, sensing, magnetic and plasmonic behaviors. It is also a visible macro-manifestation of the underlying atomic-scale forces and chemical makeup, most conspicuous in two-dimensional (2D) materials of keen current interest. If the crystal surface/edge energy is known for different directions, its shape can be obtained by the geometric Wulff construction, a tenet of crystal physics; however, if symmetry is lacking, the crystal edge energy cannot be defined or calculated and thus its shape becomes elusive, presenting an insurmountable problem for theory. Here we show how one can proceed with auxiliary edge energies towards a constructive prediction, through well-planned computations, of a unique crystal shape. We demonstrate it for challenging materials such as SnSe, which is of C2v symmetry, and even AgNO2 of C1, which has no symmetry at all.

Effects of gravity on the capillary flow of a molten metal

Repair and construction in space by means of brazing requires understanding of the effect of gravity on the capillary flow of a molten metal. We perform two sets of experiments: (1) spreading of a sessile drop of liquid aluminum-silicon-flux (Al-Si-KxAlyFz) composite braze on a horizontal alumina (Al2O3) substrate (a non-wetting surface), and (2) capillary flow of the same braze alloy on an inclined aluminum-manganese (AA3003) substrate (a wetting surface). We vary the mass of the brazing liquid and the inclination of the substrate (i.e., the relative direction of gravity).In the composite metal/flux sessile drop experiments, we observe a secondary liquid flux meniscus forming at the contact line between the liquid Al-Si and the alumina substrate. We demonstrate that the equilibrium contact angle appears to be close to 180°, while the apparent contact angle depends on the mass of the braze. In the second set of experiments, we study the molten braze alloy on different inclinations of AA3003 in a wedge-T wetting/non-wetting assembly. As the inclination angle decreases, the wetting distance increases and the surface profile changes from a non-symmetric bag-like shape to a more symmetric pancake shape. With the mass decreasing, the surface profile on the vertical substrate approaches a symmetric shape when the wetting distance is equal or smaller than the capillary length. While the non-homogeneous melting and hence, the non-homogeneous microstructure of the melt, may prevent a full symmetry. Computational predictions of equilibrium shapes are in good agreement with experimental results.

Rapid Geometric Screening of Low-Energy Surfaces in Crystals

A high-throughput approach to determine the equilibrium shape of a crystal by brute force is impractical due to the vast number of density functional theory (DFT) calculations required along just a single crystallographic direction. We propose a screening method that allows the bypass of performing DFT calculations for each candidate surface. Using a series of physically-motivated simplifications, we are able to consider the relative surface energy of each of the large number of candidate surfaces required to solve the surface energy minimization problem in 3 dimensions. Application of this technique to calculate the surface energy landscapes of a set of well-known crystal structures demonstrates high accuracy in the prediction of stable planes and validates its potential as a valuable tool in ab initio determination of equilibrium crystal shapes.

Equilibrium Manipulation Planning for a Soft Elastic Rod Considering an External Distributed Force and Intrinsic Curvature

Previous work has shown that the set of equilibrium shapes of a rod is a smooth 6-dimensional manifold. We develop the Kirchhoff rod model in a differential equation form using Darboux vector that easily adds distributed force and intrinsic curvature. It can be seen that in this letter, the manifold of the equilibrium set is the same. Furthermore, we address the rod manipulation planning problem which has the certain point on rod following a predefined spatial track. We propose a novel rod manipulation planning algorithm that considers the spatial configuration distance between the current and the goal, and follows its quickest descent direction in each planning step. The experiments show that it has a much higher successful rate than straight-line path planning in the Euclidean space of the gripper and significantly less planning time than the commonly used sampling method. The hardware experiments are also conducted on a platform that consists of a manipulator, an F/T sensor and two cameras. We choose soft rubber rods in the experiments, and the results validate the proposed model and methods.

Optimal design of planar shapes with active materials

Active materials have emerged as valuable candidates for shape morphing applications, where a body reconfiguration is achieved upon triggering its active response. Given a desired shape change, a natural question is to compare different morphing mechanisms to select the most effective one with respect to an optimality criterion. We introduce an optimal control problem to determine the active strains suitable to attain a target equilibrium shape while minimizing the complexity of the activation. Specifically, we discuss the planar morphing of active, hyperelastic bodies in the absence of external forces and exploit the notion of target metric to encompass a broad set of active materials in a unifying approach. For the case of affine shape changes, we derive explicit conditions on the body reference configuration for the optimality of homogeneous target metrics. More complex shape changes are analysed via numerical simulations to explore the impact on optimal solutions of different objective functionals inspired by features of existing materials. We show how stresses arising from incompatibilities contribute to reduce the complexity of the controls. We believe that our approach may be exploited for the optimal design of active systems and may contribute to gather insight into the morphing strategies of biological systems.

Bending a graphene cantilever by a diamagnetic force

The application of a magnetic field perpendicular to the surface of a graphene cantilever generates a bending force owing to the strong anisotropy of the magnetic susceptibility. We calculate the mechanically stable equilibrium shape of a graphene cantilever in the presence of a magnetic field by minimizing the magnetic and bending energies, which are calculated using the tight-binding model and the Tersoff–Brenner potential, respectively. Furthermore, the introduction of a continuous model enables the size-dependence of the displacement by bending to be considered.

Bulk and decay properties of neutron-deficient odd-mass Hg isotopes near A=185

Ground and isomeric states of the neutron-deficient odd-$A$ isotopes $^{183}$Hg, $^{185}$Hg, and $^{187}$Hg are described from a microscopic calculation based on a self-consistent, axially-deformed Hartree-Fock mean field with the Skyrme functional and pairing within BCS approximation. For each equilibrium shape and different odd-neutron states, results on mean square charge radii and magnetic dipole moments are given and analyzed in the context of their sensitivity to the nuclear deformation and to the spin and parity. Spin-isospin correlations within proton-neutron quasiparticle random phase approximation are then introduced in the nuclear states to obtain the distributions of Gamow-Teller strength and the $\beta^+/EC$ half-lives of these isotopes, whose measurements are planned at ISOLDE-CERN using total absorption gamma-ray spectroscopy techniques.

Unsteady convective flow of a preheated water-in-oil emulsion droplet impinging on a heated wall

This work proposes a mechanism of deformation of an emulsion droplet upon collision with a wall, considering the vortex motion of a liquid inside the droplet. This motion leads to an increase in dissipative energy losses, affects spreading, corona splashing, and droplet relaxation at different liquid and wall temperatures, ranging from 20 °C to 80 °C, and influences the equilibrium shape of the drop during the liquid relaxation. For We = 100–900 and Re = 100–4000, a physical model is presented for the maximum spreading diameter of the emulsion droplet; it takes into account the heating of the boundary viscous layer and the development of temperature gradients along the droplet height, convective mixing of the liquid layers, and translational and vortex flow motion along the radius and height of the droplet. The process of corona splashing of the emulsion droplet has been studied, and the influence of the viscosity gradient due to the intermittent near-wall water film formation on the dynamics of the “corona” has been revealed. These differences led to the formation of an air gap, which in the case of an emulsion drop caused the development of a corona at lower We compared to homogeneous liquids. The duration of the liquid relaxation before capillary wetting was affected by the potential barrier of the contact line of the droplet, which depended on the vortex component of the velocity field as well as on the temperatures of the interacting media. Altering the initial thermal boundary conditions changed the relaxation time up to 60%.

Surface tension of model tissues during malignant transformation and epithelial–mesenchymal transition

Epithelial–mesenchymal transition is associated with migration, invasion, and metastasis. The translation at the tissue scale of these changes has not yet been enlightened while being essential in the understanding of tumor progression. Thus, biophysical tools dedicated to measurements on model tumor systems are needed to reveal the impact of epithelial–mesenchymal transition at the collective cell scale. Herein, using an original biophysical approach based on magnetic nanoparticle insertion inside cells, we formed and flattened multicellular aggregates to explore the consequences of the loss of the metastasis suppressor NME1 on the mechanical properties at the tissue scale. Multicellular spheroids behave as viscoelastic fluids, and their equilibrium shape is driven by surface tension as measured by their deformation upon magnetic field application. In a model of breast tumor cells genetically modified for NME1, we correlated tumor invasion, migration, and adhesion modifications with shape maintenance properties by measuring surface tension and exploring both invasive and migratory potential as well as adhesion characteristics.