Spherical Cap Model Research Articles

On the kinetics of electrodeposition in a magnesium metal anode

Magnesium (Mg) metal batteries are promising for next-generation energy storage due to Mg’s abundance and potential for improved energy densities through two-electron redox in cathodes and thin film anodes supporting high current densities. However, reversible and uniform electrodeposition of Mg necessary for high-energy-density Mg batteries continues to be plagued by challenges traceable to energy barriers to desolvation in liquid electrolytes, the stabilization of passivation layers that impede Mg diffusion at electrified interfaces, and deleterious high-asperity electrodeposition morphologies. Thus, the specific impact and coupling of factors such as (i) composition of the salt solution, (ii) electrode overpotential, and (iii) reversible stripping/plating on the interfacial kinetics of electrodes remains an area of intense interest. This study investigates the effects of overpotential time-transients and electrolyte concentration using experimentally-guided phase-field methods and compares them with an electrochemical spherical cap model. Complex dynamics at the electrode/electrolyte interface involve transient changes in overpotential, influencing nucleation, diffusion, and crystallization, ultimately altering surface evolution. We find that the Damkohler number converges to approximately one for a symmetric Mg-Mg cell using 0.1 M Mg[TFSI]2 in 1:1 v/v diglyme/dimethoxyethane suggesting a subtle balance between reactive and diffusive forces. The results indicate the formation of dendritic structures is primarily driven by limitations in electrolyte transport. In addition, we determine the limiting current density of this cell under various electrolyte concentrations, which is an essential step in successful battery design and operation. Our findings underscore the critical role of overpotentials in underpinning reaction kinetics and non-equilibrium growth dynamics during metal electrodepositon.

Maximum spreading of an impacting drop

The maximum diameter of a drop impacting on a flat solid surface is studied theoretically assuming axi-symmetric spreading without splashing. The energy balance between the initial state of the drop (sphere diameter d0) and that a maximum spread (contact diameter dm) is closed by two novel concepts. For the gas-liquid surface area, an approximate spherical cap model is proposed. Energy loss by viscous dissipation is related to the total energy dissipation when the drop has come to rest. The fractional dissipation upon maximum spread is modelled as a function of an impact parameter (P) that combines the power laws of the capillary and viscous regimes depending on a regime discrimination parameter (A). Exponents of the Weber (We) and Reynolds (Re) numbers in P=WeRe−2/5 are determined by asymptotic analysis. The parameter A is determined from experimental data as a function of the advancing contact angle (θa). In this way, an explicit model for the maximum spread factor (βm=dm/d0) is proposed which includes the scaling laws βm∼We1/2, βm∼We1/4 and βm∼Re1/5 and is in good agreement with experimental data for wide ranges of We, Re and θa.

Static Wetting of a Liquid Droplet on a Soft Elastic Substrate

Abstract A refined spherical cap model, combined with an elastic foundation model for the elastic substrate, is proposed to study the static wetting of a liquid droplet on a soft elastic substrate. The strain energy of the substrate is evaluated by the Johnson–Kendall–Roberts (JKR) model, and the increase of the surface energy of the substrate outside the contact zone is calculated based on the elastic foundation model. The total potential energy of the droplet-substrate system is given in terms of four geometrical parameters: the contact radius, the contact angle of the droplet, the deflection angle inside the contact zone, and the maximum downward displacement of the substrate surface at the contact zone center. The equilibrium state is determined based on the stationary condition of total potential energy. The present model reduces to the Young’s equation for a rigid substrate and to the Neumann’s triangle for a liquid-like substrate. Three equations are given to determine the liquid droplet shape in terms of surface energies and substrate’s elastic modulus. Reasonable agreement with existing experimental data and simulation results shows that the present model with derived formulas has the potential to catch the role of substrate’s elastic deformation on static wetting and fill the gap between the Young’s equation and the Neumann’s triangle for a soft elastic substrate.

Low-Energy Ion Scattering Intensities from Supported Nanoparticles: The Spherical Cap Model

Supported nanoparticles are of great importance to many technologies like fuel processing and chemical synthesis using catalysts and electrocatalysts, energy storage and generation using fuel cells and batteries, electrochemistry, magnetic information storage, and more. Low-energy ion scattering spectroscopy (LEIS) with noble gas ions like He+ is a powerful tool for the characterization of nanoparticles dispersed across flat support surfaces due to its ability to probe the elemental composition in the topmost atomic layer of a surface, providing quantitative information regarding the size and number density of nanoparticles. In this work, we present a derivation of the LEIS intensities expected from nanoparticles and the support material as a function of the average particle size, their number per unit area, and their contact angle with the support when modeled as spherical caps of the nanoparticle material dispersed over the surface of a flat support. The model assumes that the ion intensities are determined only by the physical blocking of linear ion trajectories and independent of the tilt angle of the local surface relative to the incident and scattered ion directions, an assumption we support by quantitative modeling of published data which tested tilt-angle effects. The model is a generalization to arbitrary contact angles of the hemispherical cap model, which assumes 90° contact angle and has been widely used to model spectroscopic signals in LEIS (and also in Auger and photoelectron spectroscopies) during nanoparticle growth. This new model quantitatively reveals how LEIS signals are sensitive not only to the diameter and number density of the nanoparticle but also to their contact angle (or height/diameter ratio). With the use of additional data (e.g., from microscopy or adsorption microcalorimetry), the model presented here will enable more accurate determination of the average size, shape, and number density of supported nanoparticles based on LEIS intensity measurements.

Monitoring the evaporation of a sessile water droplet with a chromatic confocal measurement system.

In this Letter, a chromatic confocal measurement system with high stability and accuracy is presented to monitor the evaporation of a sessile water droplet. The stability and accuracy of the system are tested by measuring the thickness of a cover glass. To compensate for the measurement error caused by the lensing effect of the sessile water droplet, a spherical cap model is proposed. Together with the parallel plate model, the contact angle of the water droplet can also be obtained. The evaporation process of sessile water droplet under different environment is monitored experimentally in this work, which demonstrates the potential application of chromatic confocal measurement system in the field of experimental fluid dynamics.

Elastic properties and shape of the Piezo dome underlying its mechanosensory function

We show in the companion paper that the free membrane shape of lipid bilayer vesicles containing the mechanosensitive ion channel Piezo can be predicted, with no free parameters, from membrane elasticity theory together with measurements of the protein geometry and vesicle size [C. A. Haselwandter, Y. R. Guo, Z. Fu, R. MacKinnon, Proc. Natl. Acad. Sci. U.S.A., 10.1073/pnas.2208027119 (2022)]. Here we use these results to determine the force that the Piezo dome exerts on the free membrane and hence, that the free membrane exerts on the Piezo dome, for a range of vesicle sizes. From vesicle shape measurements alone, we thus obtain a force-distortion relationship for the Piezo dome, from which we deduce the Piezo dome's intrinsic radius of curvature, [Formula: see text] nm, and bending stiffness, [Formula: see text], in freestanding lipid bilayer membranes mimicking cell membranes. Applying these estimates to a spherical cap model of Piezo embedded in a lipid bilayer, we suggest that Piezo's intrinsic curvature, surrounding membrane footprint, small stiffness, and large area are the key properties of Piezo that give rise to low-threshold, high-sensitivity mechanical gating.

A Ziegler-type spherical cap model reveals early stage ethylene polymerization growth versus catalyst fragmentation relationships

Polyolefin catalysts are characterized by their hierarchically complex nature, which complicates studies on the interplay between the catalyst and formed polymer phases. Here, the missing link in the morphology gap between planar model systems and industrially relevant spherical catalyst particles is introduced through the use of a spherical cap Ziegler-type catalyst model system for the polymerization of ethylene. More specifically, a moisture-stable LaOCl framework with enhanced imaging contrast has been designed to support the TiCl4 pre-active site, which could mimic the behaviour of the highly hygroscopic and industrially used MgCl2 framework. As a function of polymerization time, the fragmentation behaviour of the LaOCl framework changed from a mixture of the shrinking core (i.e., peeling off small polyethylene fragments at the surface) and continuous bisection (i.e., internal cleavage of the framework) into dominantly a continuous bisection model, which is linked to the evolution of the estimated polyethylene volume and the fraction of crystalline polyethylene formed. The combination of the spherical cap model system and the used advanced micro-spectroscopy toolbox, opens the route for high-throughput screening of catalyst functions with industrially relevant morphologies on the nano-scale.

A spherical cap model of the geomagnetic field over southeast Asia from CHAMP and Swarm satellite observations

In this paper, Spherical Cap Harmonic Analysis (SCHA) method was applied to model the geomagnetic field over Vietnam and adjacent area between 15°S and 25°N latitude and 90°E and 130°E in longitude by using magnetic data recorded on CHAMP and Swarm satellites. The characteristic parameters of the method were set at the maximum index Kint = 8 for internal fields, the spherical cap half-angle θ0 = 20°. The regional geomagnetic field over Vietnam and adjacent areas are modelled for the two epochs (2007.0 and 2015.0). Comparison between the SCHA regional geomagnetic field intensity and its time variation with those from IGRF was carried out. The geomagnetic field intensity $$ (E_{F}^{SCHA} ) $$ from SCHA model varies between −90 and 98 nT for epoch 2007.0 and between −139 and 143 nT for epoch 2015.0; however, the trends of their time variations are the same over Vietnam. The RMS between the magnetic components from SCHA model and ground observations are in the same order. The amplitude of time variation of total field intensity from SCHA model is about tens nT greater than from IGRF over Vietnam.

A New Approach in Numerical Modeling of Inoculation of Primary Silicon in a Hypereutectic Al-Si Alloy

The applicability of classical heterogeneous nucleation theory for an inoculated Al-18.6Si (wt pct) alloy was investigated. Nucleation model proposed by Perepezko was used to study heterogenous nucleation of primary silicon in phosphorus-inoculated alloys. For this system, the nucleation temperature was found to be the most crucial variable in the model. If a spherical cap model is assumed for heterogenous nucleation, then the contact angle changes only by the interfacial energy. However, the data applied to Perepezko’s model showed it changed by undercooling. Therefore, it is suggested that the Perepezko’s nucleation model is not applicable for analyzing data in inoculated hypereutectic Al-Si alloy. Instead, for the first time, the free growth model developed by Greer to study the inoculation of Aluminum by Al-Ti-B was used for the Al-18.6Si (wt pct) alloy inoculated with Al-Fe-P. The results of modeling compared with the experimental data showed that the free growth model gives a closer approximation when predicting the size of the primary silicon in the investigated alloy. Mechanical properties of the as-cast hypereutectic alloys are influenced by size and shape of the primary silicon and eutectic silicon, type, size, and frequency of entrainment defects and residual stresses. Finer silicon particles lead to higher tensile strength in the cast components. Being able to predict the size of primary silicon particles will facilitate control of the inoculation process, to enhance mechanical properties such as tensile strength. Developed model here provides a basis to predict the size of primary silicon in hypereutectic Al-Si alloys treated with phosphorous-containing inoculants.

A pilot study on the influence of mouth configuration and torso on singing voice directivity

Directivity of speech and singing is determined primarily by the morphology of a person, i.e., head size, torso dimensions, posture, and vocal tract. Previous works have suggested from measurements that voice directivity in singing is controlled unintentionally by spectral emphasis in the range of 2-4 kHz. The attempt is made to try to identify to what extent voice directivity is affected by the mouth configuration and the torso. Therefore, simulations, together with measurements that investigate voice directivity in more detail, are presented. Simulations are presented for a piston in an infinite baffle, a radiating spherical cap, and an extended spherical cap model, taking into account transverse propagation modes. Measurements of a classical singer, an amateur singer, and a head and torso simulator are undertaken simultaneously in the horizontal and vertical planes. In order to assess differences of voice directivity common metrics, e.g., horizontal and vertical directivity indexes, are discussed and compared to improved alternatives. The measurements and simulations reveal that voice directivity in singing is affected if the mouth opening is changed significantly. The measurements show that the torso generates side lobes due to diffraction and reflections at frequencies related to the torso's dimensions.

Population and Cultural Disasters Caused by Sea Level Rise Based on Regression Model

With the worsening environment, more people become EDPs since their homeland become inhabitable. Leaving the place they once called home, their rights may not be guaranteed. Meanwhile, their unique culture are also facing the edge of extinction. To determine the scope of the issue, we discuss the number of people at risk and specify the cultural loss. When predicting EDP population, we start by using the regression model to predict the trend of sea level rise due to the greenhouse effect and find out by 2120, the world’s average sea level will probably rise by 60mm. Considering the effect of greenhouse gas makes this model more persuasive than if we only use sea level data to predict. Then, we build a spherical cap model to resemble to geographical features of tropical islands. It greatly simplifies the complexity of island topography and extends our prospective from a few islands to the whole world. With the help of the spherical cap model, on conservative estimate, the current number of EDP is about 67 million; by 2050, the potential EDPs will reach 130 million and over 400,000 square kilometers of land will be submerged. In the aspect of culture, with island residents becoming refugees, the potential lost of religious rituals, folk stories, traditional techniques, intangible arts and unique languages lead to the huge loss of cultural heritage.

On the heterogeneous nucleation pressure for hydrogen pores in liquid aluminium

ABSTRACT Heterogeneous nucleation based on the spherical cap model has been used in this study to evaluate nucleation pressure and the number of hydrogen atoms necessary for nucleation. Calculations have shown that the minimum heterogeneous nucleation pressure that can be achieved is in the order of 1 GPa, which is four orders of magnitude larger than the values assumed in the literature. Implications of the results are discussed.

Stick-slip behavior of sessile drop on the surfaces with irregular roughnesses

In this work, sessile drop and low-bond axisymmetric drop shape analysis methods were coupled to provide some new aspects on stick-slip behavior as well as stick time of a drop on calcite surfaces. Slightly hydrophobic calcite surfaces typified with three irregular roughnesses were used to create irregular surfaces to mimic defects for the water-calcite-air systems. Polishing papers of 200, 600, and 1200 grit and a polishing machine were used to prepare surfaces. X-ray diffraction, energy dispersive X-ray spectroscopy, Fourier transform infrared, and atomic force microscopy techniques were employed to evaluate the chemical and physical properties of surfaces. A model was developed to predict stick time based on the Laplace equation to take into account the effect of gravity which is neglected in the spherical cap model. An irregular stick–slip behavior of the three-phase contact line was observed for different roughness levels. Increasing roughness level from nono-metric to micrometric scale induced variation of 77.4°, 9.0e−03 mN/m, 1.23 mN/m to 67°, 293.0e−03 mN/m, 7.20 mN/m for the datum contact angle, scaled energy barrier, and the unbalanced Young force per unit length, respectively. Comparison of the models predictions and the obtained experimental data of stick times in this work, showed that the developed model is more accurate than the spherical cap model. This work could provide new insights into the wetting phenomenon of surfaces with irregular roughnesses.

Volumetric estimate of bordered pits in Pinus sylvestris based on X-ray tomography and light microscopy imaging

Bordered pits are a major determinant for the hydraulic function of wood tissues. Unlike microscopic imaging (e.g. light and electron microscopy) that is constrained to two-dimensional (2D) information, X-ray micro-computed tomography (XμCT) contributes to three-dimensional (3D) analysis. This advantage was used to estimate the volume of bordered pits in Pinus sylvestris. The 3D data obtained by XμCT were compared with two mathematical models (ellipsoid model and spherical cap model) using 2D data obtained by transmission light microscopy and XμCT. The findings of this study showed that the volume approximation using the ellipsoid model revealed values close to the volumes, which were three-dimensionally obtained by XμCT. This trend, however, is more pronounced for pits in earlywood than in latewood. Nevertheless, this study demonstrated that microscopic images can also be used for the approximation of pit volumes to some extent. Researchers should be aware of limitations that come with the 3D method (e.g. resolution, image analysis) and 2D method (unknown location of the section in the pit) as well as the natural variation of the pit morphology.

Understanding the temperature and size dependence of the contact angle of Cu/Si(1 1 1): A molecular dynamics study

Thin film dewetting is a low-cost approach to self-assembling nanoparticles on solid substrates. To pave the way for future dewetting engineering, studies of the parameters that influence the morphology of the final dewetted state are necessary. The dewetted contact angle, which is traditionally used for surface energy evaluation, is of importance in both application and fundamental studies. By introducing a spherical cap model, we found that the contact angle is quantitatively determined by the average surface free energy (SFE) of the nanoparticle (NP), the surface free energy of the substrate and the interfacial free energy (IFE) of the interface. The contact angle, SFE and IFE of the Cu/Si(1 1 1) system were calculated by molecular dynamics. The average SFE of solid Cu NPs and the IFE were found to decrease with increasing temperature and decreasing particle size, which accounted for the decrease of the contact angle with increasing temperature and decreasing particle size. This work suggests an approach to quantitatively determining the contact angle, which is significant for the controlled preparation of metal decorated surfaces.

Flow behaviour and drag coefficients of spherical bubbles in surfactant-laden Carreau model fluids

The flow and drag phenomena of contaminated spherical bubbles in columns filled with surfactant-laden Carreau model fluids is numerically investigated using a computational fluid dynamics based commercial solver, COMSOL Multiphysics 4.3b. The effect of contaminants is incorporated in the solver by the use of the spherical stagnant cap model. The numerical solver is thoroughly benchmarked through extensive validation with the existing literature results. Further new simulations are performed over wide range of the conditions as the Reynolds number (Re) varying in the range of (0-100), the power law index (n) ranging between (0.2-0.8), the Carreau number (Λ) varying in the range of (1-100) and the degree of contamination (α) ranging between (0-180°). Briefly, the results indicate that recirculation wakes behind the bubbles are observed for all values of Carreau number ranging between 1-100 if the bubble is at least partially contaminated, i.e., for α > 30°. The total drag coefficient decreases with the increasing Carreau and/or Reynolds numbers and/or with the decreasing power-law index and/or with the decreasing cap angle.

Statistical modelling of coating layer thickness distributions: Influence of overspray on coating quality

This paper investigates the layer formation in spray coating processes. Based on a Monte-Carlo simulation, a stochastic model of the coating layer thickness distribution was derived. It couples the stochastic process of droplet deposition on the particle surface with the droplet shape constructed from a spherical cap model and the droplets wetting properties (contact angle). The model was successfully shown to be able to replace the simulation. A parameter study revealed recommendations for designing a coating process, which were in agreement with the works from other authors. The model was then used to investigate the influence of overspray on the coating quality in comparison with experiments. It was found that the presence of overspray not only reduces the process efficiency but also increases the coefficient of variation of the resulting layer thickness distribution. This was caused by an increase in droplet size due to a predominant drying of small drops. It was also found, that a higher solid content of the spray solution increases the coefficient of variation, not only due to a decreased number of droplets, but also due to a greater variability in the layer thickness each droplet introduces.

Measurement of contact angles of microscopic droplets by focal length method

We present a method to measure contact angles of microscopic droplets with a conventional microscope that possesses a precision focus adjustment stage. The droplets are modeled as spherical caps that act as lenses. Their focal length is determined by measuring the distance from the substrate surface to the level where a sharp image of the aperture stop is observed. The lens diameter is found by edge detection of a microscope image of the microdroplets. The spherical cap model relates the focal length and diameter of such lenses to the contact angle of the used liquid with known refractive index. The measurement procedure was applied to condensed water droplets on a silicon substrate covered by its native oxide layer. The results are found to be in good agreement with conventional, goniometric sessile drop measurements of the advancing contact angle.

Range space super spherical cap discriminant analysis

To overcome the separability problem caused by sample fusion in the process of sample vectors normalization, this paper presents a unit super spherical cap discriminant analysis in the range space of the total scatter matrix. It is proved that the unit super spherical cap model can maintain the topological invariability of the structural characteristics of sample vectors. Furthermore, a sufficient condition is derived for improving the separability of sample data under the proposed model. The proposed algorithm projects sample data to the range space of the total scatter matrix, and then adds one dimension to each sample of the range space and nonlinearly maps it on the surface of the unit super spherical cap. We put forth a new classifier called the “spherical inner product nearest neighbor classifier’’ for the transformed data. It is designed for the deviation problem of the discriminant vector and the separability problem caused by sample vectors normalization when different sub-classes are located in different low-dimensional subspaces or manifolds. Experimental results on different databases show that our method outperforms other methods in terms of recognition accuracy and numerical stability.

Light Trapping in Isotextured Silicon Wafers

We present an experimental investigation into light trapping within isotextured wafers. Measurements of reflectance and transmittance are performed on Si wafers both before and after the deposition of thin films on their surfaces. The wafers have a variety of surface morphologies that were created by varying the isotexture etch duration. We find that, unlike external reflectance, light trapping in isotextured wafers depends only weakly on the isotexture etch time (for practical etch durations). We also find that the light trapping induced by isotexture is similar to that induced by random pyramids. Moreover, we conclude that the spherical-cap model, which is extended to account for internal rays within the isotexture, can be applied to predict the transmittance and escape reflectance and, hence, the light trapping in isotextured wafers. By way of quantification, the model predicts the AM1.5g generation current to within ±1% and the reflectance and transmittance to within ±2% absolute for λ < 1100 nm (and to within ±4% for λ < 1200 nm for most samples). With an established light-trapping model, photovoltaic researchers can more accurately predict the behavior of isotextured solar cells for various antireflection coatings, encapsulants, and light sources.