Equilibrium Shape Research Articles

Modelling of precipitate hardening in ferroelectric materials

AbstractThis investigation presents a mechanical model to determine energetically favourable particle shapes, motivated by precipitation hardening in ferroelectric materials. We examine Li-doped $$\text {NaNbO}_3$$ NaNbO 3 and propose a system of equations to derive the correct material constants for this material system. To model the elongated precipitate shape we introduce two approaches: An elliptic shape and a shape based on cubic B-spline-curves. We subject both definitions to an optimisation routine that yields equilibrium shapes. We further use the spline approach to derive configurational forces on the interface.

A Consistent Kinetic Fokker–Planck Model for Gas Mixtures

We propose a general multi-species Fokker–Planck model. We prove consistency of our model: conservation properties, positivity of all temperatures, H-Theorem and the shape of equilibrium as Maxwell distributions with the same mean velocity and temperature. Moreover, we derive the usual macroscopic equations from the kinetic two-species BGK model and compute explicitly the exchange terms of momentum and energy.

First Principles Unveiling Equilibrium Shapes for Understanding Morphological Growth Mechanisms in Alumina Polymorphs

Understanding the morphological evolution of crystals is crucial for industrial applications and surface engineering. This study employs a multifaceted analytical approach to comprehensively analyze gibbsite, boehmite, η, γ, δ, θ, κ, and α-Al2O3. Density Functional Theory (DFT) surface energy calculations provide essential viewpoints into facet exposure and stability, enriching our comprehension of growth patterns. Using the Bravais–Friedel–Donnay–Harker (BFDH) method, a meticulous investigation into morphological properties reveals equilibrium shapes for each structure. Crucially, our models unveil the inaugural equilibrium shapes for structures previously undocumented in the literature (e.g., η, δ, θ, and κ-Al2O3). The study explores morphological growth process during phase transitions from gibbsite and boehmite to alpha alumina, emphasizing nucleation and dissolution preferences for specific facets. A grain analysis, considering bulk energy, surface area, and volume, designates α-Al2O3 as the most stable phase, offering valuable standpoints on thermodynamics and indicating a direct correlation between facet area and volume. Additionally, a comprehensive 3D surface energy mapping offers intricate visualization of facet interactions, providing valuable viewpoints into the complex relationships between surface energy, facet area, and interplanar distance. Comparisons of Fourier-transform infrared spectroscopy (FTIR) spectra, exclusively within the 200–400 cm-1 range, underscore close agreement between theoretical and experimental spectra, consistent with the expected reduction in specific surface area during polymorphic transformation.

Phase-field modeling of facet growth of diamond using a modified anisotropic coefficient of interfacial energy

In the present work, a modified anisotropic coefficient of interfacial energy is used to simulate the anisotropic facet crystal growth along non-orthogonal directions by phase-field method. The modified anisotropic coefficient of interface anisotropy is simulated and evaluated against the theoretical prediction of equilibrium shape. The simulation results are compared between the modified form and classic form. Subsequently, the modified anisotropic coefficient is employed to simulate the Wulff construction of facet growth of diamond, encompassing both pristine diamond and diamond with reconstructed surfaces. The dynamics of diamond morphology and the reconstructed surface area and proportion are thoroughly evaluated, demonstrating agreement with experimental records. The effect of the reconstruction surface on the growth of diamond grain is analyzed in terms of the total energy reduction, and the range of interfacial energies for more stable diamond reconstructions is also summarized. By introducing the modified anisotropic coefficient, anisotropic facet growth can be described and the simulation results are verified both theoretically and experimentally.

Structural multistability for multi-speed wind energy harvesting from vortex-induced vibrations

Abstract Piezoelectric energy harvesters utilizing vortex-induced vibrations (VIV) have been extensively studied for converting wind energy into usable power for microelectronics. In this work, we explore the use of structural bistability to increase the range of flow speeds over which energy can be harvested without the need for complicated assemblies. We propose a harvester system featuring a piezoelectric transducer bonded to a cantilevered bistable composite laminate, which has two distinct equilibrium shapes at room temperature. To enhance the VIV, we attach a cylindrical bluff body to the free edge of the harvester. The structure’s inherent bistability allows for high power generation at two different flow speeds, contrasting with the single synchronization region typical of linear piezoelectric harvesters. We develop a reduced-order model to predict power output across varying flow speeds and validate these predictions through wind tunnel experiments, showing good agreement. Furthermore, we conduct a parametric study to optimize the model parameters for maximum power output. Our results demonstrate that the bistable harvester can generate up to 4.5 mW of power over a wind speed range of 9.3 m s−1–11.7 m s−1, outperforming the limited speed range of traditional linear VIV-based harvesters. This work underscores the potential to design VIV-based energy harvesters capable of operating efficiently across multiple flow speed ranges using a single structure with its dual stable configurations.

Topology‐preserving phase‐field modeling of elastic bending energy

AbstractIn vesicle membranes, the minimum of the elastic bending energy determines the equilibrium shape. We can reformulate the energy using a phase‐field function and optimize it using the standard gradient flow approach. The method, in its typical formulation, allows topological changes in the membrane; however, in certain events, like in the simulation of blood cells, maintaining the initial topology of the membrane is crucial. In this study, we add a constraint on the phase‐field method to preserve the topology. We note that even if the phase‐field method is formulated using a hyperbolic tangent function, during a change in topology, the membrane's profile deviates from the tanh function. Owing to this idea, the constraint imposes an additional requirement on the profile to maintain the tanh shape, thus preserving the topology. We perform extensive experiments in two‐dimensional and three‐dimensional scenarios to demonstrate that our method preserves the topology during the simulation.

Droplets can enhance microcapsule deformation in channel flow

The dynamics of soft microparticles enclosed in a droplet flowing in a channel is an unexplored fundamental problem that lies at the heart of numerous applications, including droplet-based microfluidics, tissue engineering and smart material synthesis. Here we show that enclosing a flexible capsule into a droplet can amplify the capsule’s deformation parameters in channel flow by up to two orders of magnitude. Previously unreported capsule equilibrium shapes in channel flow, including an oblate spheroid and a reversed bullet, have also been discovered. We propose two theoretical models to predict the equilibrium position of the capsule inside the droplet, and estimate the capsule deformation, respectively. The present study provides an effective but simple approach to enhance and control the deformation of soft particles in a flowing suspension, which may inspire widespread applications, from high-throughput single-cell mechanical phenotyping, enhanced cross-membrane drug delivery, to manufacturing shape-controlled non-spherical particles and artificial cells.

A Variational Model of Charged Drops in Dielectrically Matched Binary Fluids: The Effect of Charge Discreteness

This paper addresses the ill-posedness of the classical Rayleigh variational model of conducting charged liquid drops by incorporating the discreteness of the elementary charges. Introducing the model that describes two immiscible fluids with the same dielectric constant, with a drop of one fluid containing a fixed number of elementary charges together with their solvation spheres, we interpret the equilibrium shape of the drop as a global minimizer of the sum of its surface energy and the electrostatic repulsive energy between the charges under fixed drop volume. For all model parameters, we establish the existence of generalized minimizers that consist of at most a finite number of components “at infinity”. We also give several existence and non-existence results for classical minimizers consisting of only a single component. In particular, we identify an asymptotically sharp threshold for the number of charges to yield existence of minimizers in a regime corresponding to macroscopically large drops containing a large number of charges. The obtained non-trivial threshold is significantly below the corresponding threshold for the Rayleigh model, consistently with the ill-posedness of the latter and demonstrating a particular regularizing effect of the charge discreteness. However, when a minimizer does exist in this regime, it approaches a ball with the charge uniformly distributed on the surface as the number of charges goes to infinity, just as in the Rayleigh model. Finally, we provide an explicit solution for the problem with two charges and a macroscopically large drop.

Numerical study on the effects of tube inclination angle and heat conductive cover plate on freeze plug performance in molten salt reactors

The freeze plug is an important passive safety system used in the molten salt reactors (MSRs). However, although the inclination angle is one of the fundamental design parameters, its influence on the freeze plug performance has not been explored systematically. Multidimensional numerical simulations were hence carried out to analyze the effects of tube inclination on the solidification and melting processes of salt encountered in the freeze plugs of the MSRs in this work. The effects of placing a high thermal conductivity cover plate on the outer wall of the freeze plug were also investigated. In the solidification process, the tube inclination had a significant impact on the equilibrium shape of the frozen salt. A decrease in the inclination angle from the horizontal resulted in a reduction in the plug volume, and closing of the freeze plug became more difficult when the inclination angle was small and the tube diameter was large. In the melting process, natural convection induced in the liquid salt was intensified with a decrease in the inclination angle. This accelerated the melting of solid salt to shorten the opening time significantly. Optimizing the tube inclination angle was hence shown to be of great importance to improve the safety of MSRs. When a copper plate was placed on the freeze plug outer wall to facilitate the heat transfer to the frozen salt, its influence on the solidification process was not noticeable, while melting was accelerated and the opening time was reduced by about 20%.

Magnetic shaping effects on turbulence in ADITYA-U tokamak

The work reported in this paper addresses two aspects. In the first part, numerical simulations are conducted to examine the impact of magnetic equilibrium shaping (elongation and triangularity), on both conventional Ion Temperature Gradient (ITG) modes and Short Wavelength ITG modes. This analysis is performed considering the experimental profiles and parameters of the ADITYA-U tokamak, employing the nonlinear global gyrokinetic Particle-in-Cell code ORB5. The linear and nonlinear collisionless electrostatic simulation of these modes are carried out with kinetic ions and adiabatic electrons. From the linear results, it has been observed that the magnetic equilibrium shaping slighty reduced the growth rates and widened the spectrum, and the maxima of growth rate curve is shifted to higher toroidal wave number. We find that the heat flux is reduced by a significant ≃35% for the true circular magnetohydrodynamic magnetic equilibrium as compared to ad hoc concentric circular equilibrium reported in Singh et al (2023 Nucl. Fusion 63 086029). A further ≃10% reduction in the heat flux is seen with magnetic equilibrium shaping. In the second part, linear collisionless electrostatic simulation studies of ITG coupled with fully kinetic electrons (both trapped and passing electrons are treated kinetically) using a drift-kinetic ordering is performed. It can be seen from the linear results that, in presence of kinetic electrons, the growth rate and real frequency of the ITG mode are increased significantly for ADITYA-U parameters and a mode propagating in the electron diamagnetic direction is identified at high toroidal wavenumbers.

End-rotation-driven kinetic elastica structures: Concept and morphology design

Euler’s elastica, a family of planar curves describing the equilibrium shapes of a thin elastic beam subjected to end forces and moments, demonstrates both geometric versatility and mechanical functionality, leading to its diverse applications across multiple disciplines. Despite its roots in ancient structural engineering practices, the kinetic potential of Euler’s elastica remains largely unexplored in this domain. To address this gap, this paper proposes the concept of end-rotation-driven kinetic elastica (ERKE) structures. This novel kinetic form consists of a sequence of discrete elastica arches transformed from initially straight members by elastic bending. The kinetic behavior arises from manipulating the end rotations of each elastica arch, prompting a continuous flow of motions resulting from adaptations in the arch shape. This adaptive quality opens up limitless possibilities for creating visually captivating dynamic patterns in the overall structure by coordinating the motions of individual arches. As the inaugural exploration of ERKE structures, this paper centers around their conceptualization, with an emphasis on morphology design, which refers to the creation of desirable dynamic patterns. To this end, the paper provides an exposition of the general design procedure and criteria, along with demonstrative examples. The paper also envisions potential application scenarios and identifies research needs, paving the way for future investigations.

Real-time plasma equilibrium reconstruction and shape control for the MAST Upgrade tokamak

Real-time magnetic control has been developed to deliver precise control of multiple plasma shape parameters for advanced divertor configurations, including double-null, Super-X, X-point target and X-divertor for the first time on the MAST Upgrade (MAST-U) spherical tokamak. Successful real-time magnetic equilibrium control of different plasma shape variables has been accomplished in the 2022–2023 MAST-U experimental campaign for the advanced MAST-U divertor configurations. Application of the MAST-U boundary reconstruction algorithm, LEMUR, is described and compared with off-line equilibrium reconstruction and diagnostic measurements. The process of design and verification of the axisymmetric magnetic control schemes using a suite of control analysis tools (known collectively as TokSys) is also described.

Evaluation of the equilibrium shape of the soft shell of an arched tent structure subjected to external load

Evaluation of the equilibrium shape of the soft shell of an arched tent structure subjected to external load

Impact-induced deformation away from the impact point on small asteroids

Abstract We investigated the propagation of energy and momentum inside a small gravitational aggregate asteroid following a hyper-velocity impact, comparable to NASA’s Double Asteroid Redirection Test on asteroid Dimorphos. We show that the impact energy damps rapidly inside different kinds of granular structures, unable to reach the antipodal hemisphere of the impact. However, global reshaping of the asteroid after the formation of a sizeable (>1/3 of target size) crater causes meter-range displacement of boulders on the antipodal hemisphere due to mass rearrangement to achieve a new equilibrium shape. As a result, a surface depression opposite to the crater is formed, which may produce some surface refreshment. The boulder mass ejection following the synthetic DART-like impact is estimated to be at least $0.1\, {{\%}}$ of the mass of the Dimorphos-like target, in agreement with recent detections of boulders ejected from the actual DART impact.

Theoretical study of adsorption properties and CO oxidation reaction on surfaces of higher tungsten boride

Most modern catalysts are based on precious metals and rear-earth elements, making some of organic synthesis reactions economically insolvent. Density functional theory calculations are used here to describe several differently oriented surfaces of the higher tungsten boride WB5-x, together with their catalytic activity for the CO oxidation reaction. Based on our findings, WB5-x appears to be an efficient alternative catalyst for CO oxidation. Calculated surface energies allow the use of the Wulff construction to determine the equilibrium shape of WB5-x particles. It is found that the (010) and (101) facets terminated by boron and tungsten, respectively, are the most exposed surfaces for which the adsorption of different gaseous agents (CO, CO2, H2, N2, O2, NO, NO2, H2O, NH3, SO2) is evaluated to reveal promising prospects for applications. CO oxidation on B-rich (010) and W-rich (101) surfaces is further investigated by analyzing the charge redistribution during the adsorption of CO and O2 molecules. It is found that CO oxidation has relatively low energy barriers. The implications of the present results, the effects of WB5-x on CO oxidation and potential application in the automotive, chemical, and mining industries are discussed.

A coupled VOF/embedded boundary method to model two-phase flows on arbitrary solid surfaces

We present an hybrid VOF/embedded boundary method allowing to model two-phase flows in presence of solids with arbitrary shapes. The method relies on the coupling of existing methods: a geometric Volume of fluid (VOF) method to tackle the two-phase flow and an embedded boundary method to sharply resolve arbitrary solid geometries. Coupling these approaches consistently is not trivial and we present in detail a quad/octree spatial discretization for solving the corresponding partial differential equations. Modelling contact angle dynamics is a complex physical and numerical problem. We present a Navier-slip boundary condition compatible with the present cut cell method, validated through a Taylor–Couette test case. To impose the boundary condition when the fluid–fluid interface intersects a solid surface, a geometrical contact angle approach is developed. Our method is validated for several test cases including the spreading of a droplet on a cylinder, and the equilibrium shape of a droplet on a flat or tilted plane in 2D and 3D. The temporal evolution and convergence of the droplet spreading on a flat plane is also discussed for the moving contact line given the boundary condition (Dirichlet or Navier) used. The ability of our numerical methodology to resolve contact line statics and dynamics for different solid geometries is thus demonstrated.

Boom-Membrane Space Structure’s Incomplete Deployment After Storage in OrigamiSat-1 Ground Demonstrator

This study investigates the deployment reliability of multifunctional membrane deployable space structures that can be compactly stored and deployed in a two-dimensional direction using carbon composite booms to realize the construction of thin-film solar arrays and membrane antennas. Herein, two aspects are experimentally clarified using the structural models of a nano-satellite OrigamiSat-1, which is a 1 m-by-1 m square multifunctional deployable membrane structure demonstrator. First, a measurement system is developed to quasi-statically quantify the deployment torque of the structural models that failed to complete deployment after long-term storage and fell into an undesirable equilibrium shape. The measurement results confirm that the deployment torque decreases with long-term storage of the structure, and the torque in the deployment direction becomes negative at the deployment angle of the undesirable equilibrium shape. Therefore, as the second contribution, design modifications are made to the boom and membrane, and quasi-static and dynamic deployment tests are employed to evaluate their effects. The deployment reliability evaluation method and two design updates proposed herein will contribute to future space demonstrations and the utilization of lightweight membrane deployment structures with high packaging efficiency.

Raindrop Deformation in Turbulence

AbstractThe physical behavior of a falling raindrop is governed by delicate fluid dynamics and thermodynamics, and oscillates with time. Despite this time‐variant nature, past observational and simulation studies have aimed to generalize parameterizations for describing rain microphysics bearing the assumption that raindrops fall at terminal speeds with an equilibrium shape. However, the applicability of this hypothesis in a realistic atmosphere that is inherently turbulent remains an open question. Here, we employ novel retrieval techniques to quantify the impact of turbulence on raindrop microphysics using long‐term in situ observations with careful assessment of the wind effect. We find that raindrop microphysics increasingly deviate from the equilibrium state as the turbulence dissipation rate increases, and this effect is more pronounced for large raindrops. We present turbulence‐invoked rain microphysical parameterizations which shed light on the complex interactions between turbulence dynamics and raindrop microphysics.

Confinement-induced drift in Marangoni-driven transport of surfactant: a Lagrangian perspective

Successive drops of coloured ink mixed with surfactant are deposited onto a thin film of water to create marbling patterns in the Japanese art technique of Suminagashi. To understand the physics behind this and other applications where surfactant transports adsorbed passive matter at gas–liquid interfaces, we investigate the Lagrangian trajectories of material particles on the surface of a thin film of a confined viscous liquid under Marangoni-driven spreading by an insoluble surfactant. We study a model problem in which several deposits of exogenous surfactant simultaneously spread on a bounded rectangular surface containing a pre-existing endogenous surfactant. We derive Eulerian and Lagrangian formulations of the equations governing the Marangoni-driven surface flow. Both descriptions show how confinement can induce drift and flow reversal during spreading. The Lagrangian formulation captures trajectories without the need to calculate surfactant concentrations; however, concentrations can still be inferred from the Jacobian of the map from initial to current particle position. We explore a link between thin-film surfactant dynamics and optimal transport theory to find the approximate equilibrium locations of material particles for any given initial condition by solving a Monge–Ampère equation. We find that as the endogenous surfactant concentration $\delta$ vanishes, the equilibrium shapes of deposits using the Monge–Ampère approximation approach polygons with corners curving in a self-similar manner over lengths scaling as $\delta ^{1/2}$ . We explore how Suminagashi patterns may be produced by using computationally efficient successive solutions of the Monge–Ampère equation.

Equilibrium structure and shape of Ag and Pt nanoparticles grown on silica surfaces: From experimental investigations to the determination of a metal-silica potential.

A combination of experimental and numerical investigations on metallic silver and platinum nanoparticles deposited on silica substrates is presented, with a focus on metal-substrate interactions. Experimentally, the nanoparticles, obtained by ultra-high vacuum atom deposition, are characterized by grazing-incidence small-angle x-ray scattering and high resolution transmission electronic microscopy to determine their structure and morphology and, in particular, their aspect ratio (height/diameter), which quantifies the metal-substrate interaction, from the as-grown to equilibrium state. Numerically, the interactions between the metal and the silica species are modeled with the Lennard-Jones (12, 6) potential, with two parameters for each metal and silica species. The geometric parameters were found in the literature, while the energetic parameters were determined from our experimental measurements of the aspect ratio. The parameters are as follows: σAg-O = 0.278nm, σAg-Si = 0.329nm, ɛAg-O = 75meV, and ɛAg-Si = 13meV for Ag-silica and σPt-O = 0.273nm, σPt-Si = 0.324nm, ɛPt-O = 110meV, and ɛPt-Si = 18 meV for Pt-silica. The proposed Ag-silica potential reproduces quantitatively the unexpected experimental observation of the variation of the aspect ratio for Ag nanoparticles larger than 5nm, which has been interpreted as a consequence of the silica roughness. The nanoparticle orientation, structure, and disorder are also considered. This metal-silica potential for Ag and Pt should be helpful for further studies on pure metals as well as their alloys.