Surface Energy Anisotropy Research Articles

A stabilized parametric finite element method for surface diffusion with an arbitrary surface energy

We proposed a structure-preserving stabilized parametric finite element method (SPFEM) for the evolution of closed curves under anisotropic surface diffusion with an arbitrary surface energy γˆ(θ). By introducing a non-negative stabilizing function k(θ) depending on γˆ(θ), we obtained a novel stabilized conservative weak formulation for the anisotropic surface diffusion. A SPFEM is presented for the discretization of this weak formulation. We construct a comprehensive framework to analyze and prove the unconditional energy stability of the SPFEM under a very mild condition on γˆ(θ), including the critical case where 3γˆ(θ⁎)=γˆ(θ⁎−π). This method can be applied to simulate solid-state dewetting of thin films with arbitrary surface energies, which are characterized by anisotropic surface diffusion and contact line migration. Extensive numerical results are reported to demonstrate the efficiency, accuracy and structure-preserving properties of the proposed SPFEM with anisotropic surface energies γˆ(θ) arising from different applications.

Hybrid CoFe2O4-CNTs-graphene: Synthesis and characterization for energy storage devices

Hybrid materials play a crucial role in a spectrum of energy storage devices. Among them CoFe2O4–CNT–graphene hybrid stands out as versatile magnetic materials with wide-ranging applications spanning electronics, magnetism, and sensor industries. In this study, we synthesized CoFe2O4–CNT–graphene hybrid utilizing ultrasonication method. This fabrication method resulted in the formation of CoFe2O4 nanoparticles, along with bamboo-like carbon nanotubes (CNTs) and graphene nanosheets which collectively establish an open three-dimensional structure. Thorough analyses were conducted on the synthesized nano-composites employing various characterization techniques such as XRD, FT-IR, Raman and FESEM. Further characterization through XPS confirmed the formation of spinel ferrites, detecting the presence of carbon sp2, C1s, carboxylates (O-C-OH), and sp3 carbon, indicating the presence of carbon‑carbon (CC) bonds and confirms the energy levels of Co2p1/2 and Co2p3/2 indicating the effective incorporation of CFO onto the CNT/graphene surface. Analysis of dielectric parameters revealed promising characteristics for high-frequency devices, attributed to low dielectric loss, high quality factor, short relaxation time, and diverse responses exhibited by these materials. The M–H loops of the composite samples displayed ferromagnetic hysteresis behavior due to the presence of ferrite in the matrices. The coercivity value shows a slight improvement in the hybrid samples, while saturation magnetization values decrease, indicating a 1:1 weight ratio of ferrite particles to the host matrix with the incorporation of nonmagnetic CNTs and graphene, and the Hc value increases with the addition of these carbon-based materials due to increased surface anisotropy energy. Upon evaluation of dielectric and magnetic properties the hybrid materials demonstrated an enhanced dielectric and magnetic properties which render these materials suitable for utilization across a spectrum of energy storage devices.

A unified structure-preserving parametric finite element method for anisotropic surface diffusion

We propose and analyze a unified structure-preserving parametric finite element method (SP-PFEM) for the anisotropic surface diffusion of curves in two dimensions ( d = 2 ) (d=2) and surfaces in three dimensions ( d = 3 ) (d=3) with an arbitrary anisotropic surface energy density γ ( n ) \gamma (\boldsymbol {n}) , where n ∈ S d − 1 \boldsymbol {n}\in \mathbb {S}^{d-1} represents the outward unit vector. By introducing a novel unified surface energy matrix G k ( n ) \boldsymbol {G}_k(\boldsymbol {n}) depending on γ ( n ) \gamma (\boldsymbol {n}) , the Cahn-Hoffman ξ \boldsymbol {\xi } -vector and a stabilizing function k ( n ) : S d − 1 → R k(\boldsymbol {n}):\ \mathbb {S}^{d-1}\to {\mathbb R} , we obtain a unified and conservative variational formulation for the anisotropic surface diffusion via different surface differential operators, including the surface gradient operator, the surface divergence operator and the surface Laplace-Beltrami operator. A SP-PFEM discretization is presented for the variational problem. In order to establish the unconditional energy stability of the proposed SP-PFEM under a very mild condition on γ ( n ) \gamma (\boldsymbol {n}) , we propose a new framework via local energy estimate for proving energy stability/structure-preserving properties of the parametric finite element method for the anisotropic surface diffusion. This framework sheds light on how to prove unconditional energy stability of other numerical methods for geometric partial differential equations. Extensive numerical results are reported to demonstrate the efficiency and accuracy as well as structure-preserving properties of the proposed SP-PFEM for the anisotropic surface diffusion with arbitrary anisotropic surface energy density γ ( n ) \gamma (\boldsymbol {n}) arising from different applications.

Dynamics of phase separation of metastable crystal surfaces by surface diffusion: A phase-field study.

A new phase-field approach is designed to model surface diffusion of crystals with strongly anisotropic surface energy. The model can be shown to asymptotically converge toward the sharp-interface equationfor surface diffusion in the limit of vanishing interface width. It is employed to investigate the dynamical evolution of a thermodynamically metastable crystal surface. We find that nucleation and growth by surface diffusion of the newly formed surface induce the formation of additional stable surfaces at its wake. This induced nucleation mechanism is found to produce domains composed of several stable surfaces of prescribed width. The domains propagate on the crystal surface and then coalesce to form a hill-and-valley structure. The resulting morphology is more regular than the typical hill-and-valley surface produced by random thermal nucleation, i.e., when motion-by-curvature controls the phase separation dynamics. Moreover, the induced nucleation mechanism is found to be peculiar to surface diffusion and to dominate the phase separation at high degree of metastability. Once the hill-and-valley structure is formed, coarsening operates by motion and elimination of facet junctions, points where two facets merge to form one and we find the following scaling law L∼t^{1/6}, for the growth in time t of the characteristic length scale L during this coarsening stage. The density of junctions is found to exhibit a t^{-2/3} regime. Our results elucidate the role of the induced nucleation mechanism on the dynamics of interfacial phase separation and corroborate surface faceting experiments on ceramics.

A level-set method for simulating solid-state dewetting in systems with strong crystalline anisotropy

Accurately modeling solid-state dewetting in materials with strong crystalline anisotropy is an open problem in materials science with importance for both the manufacturing and reliability of nano-scale devices. In this work, we propose and demonstrate a level-set method framework for simulating solid-state dewetting which is capable of modeling systems with strongly anisotropic surface energies and diffusivities. Surface energy anisotropy is handled through the use of the Cahn-Hoffman vector construction, and surface self-diffusivity anisotropy is handled through the use of a diffusivity tensor. We benchmark our method against isotropic phenomena with analytical descriptions and go on to demonstrate that our method is capable of reproducing a host of experimentally observed behaviors in strongly anisotropic single-crystal materials.

Quasiconvex bulk and surface energies: C 1,α regularity

Abstract We establish regularity results for equilibrium configurations of vectorial multidimensional variational problems, involving bulk and surface energies. The bulk energy densities are uniformly strictly quasiconvex functions with p p -growth, p ≥ 2 p\ge 2 , without any further structure conditions. The anisotropic surface energy is defined by means of an elliptic integrand Φ \Phi not necessarily regular. For a minimal configuration ( u , E ) \left(u,E) , we prove partial Hölder continuity of the gradient ∇ u \nabla u of the deformation.

Predicting surface-energy anisotropy of metals with geometric properties of surfaces and atoms

Surface-energy anisotropy of metals is crucial for the stability and structure, however, its determining factors and structure-property relationship are still elusive. Herein, we identify three key factors for predicting surface-energy anisotropy of pure metals and alloys: the surface-atom density, coordination numbers and atomic radius. We find that the coupling rules of surface geometric determinants, which determining surface-energy anisotropy of face-centred-cubic (FCC), hexagonal-close-packed (HCP) and body-centred-cubic (BCC) metals, are essentially controlled by the crystal structures instead of chemical bonds, alloying or electronic structures. Furthermore, BCC metals exhibit material-dependent surface-energy anisotropy depending on the atomic radius, unlike FCC and HCP metals. The underlying mechanism can be understood from the bonding properties in the framework of the tight-binding model. Our scheme provides not only a new physical picture of surface stability but also a useful tool for material design.

Thermodynamically consistent Cahn–Hilliard–Navier–Stokes equations using the metriplectic dynamics formalism

Cahn–Hilliard–Navier–Stokes (CHNS) systems describe flows with two-phases, e.g., a liquid with bubbles. Obtaining constitutive relations for general dissipative processes for such systems, which are thermodynamically consistent, can be a challenge. We show how the metriplectic 4-bracket formalism (Morrison and Updike, 2024) achieves this in a straightforward, in fact algorithmic, manner. First, from the noncanonical Hamiltonian formulation for the ideal part of a CHNS system we obtain an appropriate Casimir to serve as the entropy in the metriplectic formalism that describes the dissipation (e.g. viscosity, heat conductivity and diffusion effects). General thermodynamics with the concentration variable and its thermodynamics conjugate, the chemical potential, are included. Having expressions for the Hamiltonian (energy), entropy, and Poisson bracket, we describe a procedure for obtaining a metriplectic 4-bracket that describes thermodynamically consistent dissipative effects. The 4-bracket formalism leads naturally to a general CHNS system that allows for anisotropic surface energy effects. This general CHNS system reduces to cases in the literature, to which we can compare.

A $$\theta $$-L Approach for the Simulation of Solid-State Dewetting Problems with Strongly Anisotropic Surface Energies

A $$\theta $$-L Approach for the Simulation of Solid-State Dewetting Problems with Strongly Anisotropic Surface Energies

Phase-field simulation of magnesium-metal anode interface growth in different electrolytes during electrodeposition

Various types of Mg electrolytes have emerged with the deepening and development of research on Rechargeable Magnesium-Metal Batteries. In order to assess the potential of these electrolytes to inhibit the growth of Mg dendrites, we apply a Mg-electrodeposition phase-field model with surface energy anisotropy to simulate the interface growth processes of Mg-metal anodes in different electrolytes during electrodeposition. The results show that the conductivity and Mg-ion diffusivity of the Mg electrolyte are key factors affecting the interface growth, with the diffusivity having a particularly significant effect. In the designed 1.0 M MgX/THF electrolyte with an applied overpotential of −0.30 V, by profoundly analyzing the fluctuation of the interface contour, we find that when the diffusivity reaches 1.4 × 10−10 m2/s, the Mg-metal anode interface can achieve stable growth.

Heterogeneous nucleation and dendritic growth of niobium under containerless electrostatic levitation

With the containerless electrostatic levitation method, the undercooling level and nucleation solidification mechanism of niobium samples of 99.7% and 99.95% in purity were comprehensively studied. The classical nucleation theory was used to measure and calculate the thermophysical and thermodynamic characteristics associated with the nucleation and solidification procedure. The measurement results show that the maximum undercooling of experimental samples is 455.7 K, while the hyper cooling limit of the niobium is derived at 739 K. The most probable nucleation undercooling, pre-exponential factor, nucleation activation energy, solid/liquid interface free energy, and the critical nucleus radius are determined. The dendrite growth velocity has a powerful relationship with undercooling, and the dendritic growth velocity of liquid niobium reaches 42.1 m/s at the undercooling of 454 K. The effect of the anisotropy of surface energy is taken into account, and the forecast results display excellent consistency with the experimental ones.

Mg incorporation induced microstructural evolution of reactively sputtered GaN epitaxial films to Mg-doped GaN nanorods

Mg-doped GaN films/nanorods were grown epitaxially on c-sapphire by reactive co-sputtering of GaAs and Mg at different N2 percentages in Ar–N2 sputtering atmosphere. Energy dispersive x-ray spectroscopy revealed that the Mg incorporation increases with increase of Mg area coverage of GaAs target, but does not depend on N2 percentage. In comparison to undoped GaN films, Mg-doped GaN displayed substantial decrease of lateral conductivity and electron concentration with the initial incorporation of Mg, indicating p-type doping, but revealed insulating behaviour at larger Mg content. Morphological investigations by scanning electron microscopy have shown that the films grown with 2%–4% Mg area coverages displayed substantially improved columnar structure, compared to undoped GaN films, along with rough and voided surface features at lower N2 percentages. With increase of Mg area coverage to 6%, the growth of vertically aligned and well-separated nanorods, terminating with smooth hexagonal faces was observed in the range of 50%–75% N2 in sputtering atmosphere. High-resolution x-ray diffraction studies confirmed the epitaxial character of Mg-doped GaN films and nanorods, which displayed complete c-axis orientation of crystallites and a mosaic structure, aligned laterally with the c-sapphire lattice. The catalyst-free growth of self-assembled Mg-doped GaN nanorods is attributed to increase of surface energy anisotropy due to the incorporation of Mg. However, with further increase of Mg area coverage to 8%, the nanorods revealed lateral merger, suggesting enhanced radial growth at larger Mg content.

A structure-preserving parametric finite element method for geometric flows with anisotropic surface energy

A structure-preserving parametric finite element method for geometric flows with anisotropic surface energy

First-principles study of the surface energies and electronic structures of γ-CsSnI3 surfaces.

All-inorganic perovskite CsSnI3 has attracted intense research interests due to its prominent optoelectronic properties, high thermal stability, and environmentally friendly character. The surface energies and electronic structures of black orthorhombic (γ) CsSnI3 surfaces are investigated by using first-principles methods. The anisotropic and termination-dependent surface energies of low-index surfaces (i.e., the (110), (001), (100) and (101) surfaces) are obtained, providing important data for CsSnI3, since these values are difficult to be measured in experiments. The CsI-terminated (110) and (001) surfaces are predicted to be the most stable and their surface energies are close, making the cube-shape of nanocrystals favorable at thermodynamic equilibrium, which is consistent with the experimental observations. Calculated surface electronic structures show that the quantum confinement effect is orientation dependent. The band gaps of the (100) and (101) surfaces are significantly larger than those of the (110) and (001) surfaces by using slabs with similar thickness. This distinction can be attributed to different 'electronic dimensionalities' of these surfaces. Our results provide physical insights into the thermodynamic stability and electronic properties of CsSnI3 surfaces.

Study on the recrystallization microstructure and texture evolution of a rolled 0.075 mm ultra-thin grain-oriented silicon steel

This paper investigated the recrystallization behavior of a rolled 0.075 mm ultra-thin grain-oriented silicon steel, and microstructure and texture evolutions were studied. At the early stage, dominant {hk0}<001> recrystallized grains were ascribed to the preferential {hk0}<001> nucleation, then their growth was reduced by texture pinning effect. In contrast, a few ‘random’ grains grew excessively to consume surrounding {hk0}<001> fine-grained matrix, meanwhile nuclei originating from initial non-Goss grains also showed size advantage and further grew. Despite of quasi-two-dimensional microstructure of ultra-thin sheet, the anisotropic surface energy did not show marked effect on grain growth difference. Those coarse grains, which were 10 times larger than the matrix grains, were surrounded by high energy (HE) boundaries, indicating that their growth was driven by high grain boundary mobility. With the increasing annealing time, the formation of {hk0}<001> oriented assemblies and the growth of non-{hk0}<001> oriented grains contributed to the overall grain size increase. Regarding the relationship between microstructure, texture and magnetic properties, {hk0}<001> texture was weakened and reduced magnetic induction accordingly, and the combination of growing while heterogeneous grain size and changing texture made iron loss a first decrease and then increase tendency. It is essential to control annealing time to obtain suitable combination of microstructure and texture.

Influence of surface energy anisotropy on nucleation and crystallographic texture of polycrystalline deposits

This paper aims to elucidate the role of interface energy anisotropy in orientation selection during nucleation of new grains in a polycrystalline film growth. An assessment of (heterogeneous) nucleation probability as function of orientation of both the bottom grain and of the nucleus was developed (using the concepts of classical nucleation theory). Novel solutions to the generalized Winterbottom construction were described in cases of very strong anisotropy and arbitrary orientations. In order to demonstrate the effect on the film crystallographic texture, a 2D Monte Carlo algorithm for anisotropic polycrystalline growth was used to simulate growth of films with columnar microstructure. The effect of strength of anisotropy, the deposition rate and initial texture were investigated. Results showed that with larger strength of anisotropy, the nucleation rate is less dependent on the driving force, but more dependent on the initial texture. With certain initial textures, the anisotropic nucleation may even be either impossible or having probability close to one irrespective of the driving force. Depending on the conditions, the anisotropic nucleation could hasten the evolution towards the interface-energy minimizing texture or retard it. Based on these insights, a hypothesis was offered to explain a peculiar texture evolution in electrodeposited nickel.

A symmetrized parametric finite element method for simulating solid-state dewetting problems

We propose an accurate, area/mass-conservative and energy-dissipative parametric finite element method for solving the sharp-interface continuum model of solid-state dewetting with arbitrary anisotropic surface energy. The model is governed by the anisotropic surface diffusion equation with suitable boundary conditions at the contact points, and could be used to describe the kinetic evolution of the film/vapor interface with contact line migration. The sharp-interface continuum model is firstly reformulated into a conservative form by introducing a symmetric positive definite matrix Zk(n) that depends on the Cahn–Hoffman ξ-vector and a stabilizing function k(n). A new symmetrized variational formulation with arbitrary anisotropic surface energy is built and then its area/mass conservation and energy dissipation properties are proved. By using piecewise linear elements in space, we propose the spatial semi-discretized scheme, and using backward Euler in time with an appropriate approximation of the unit normal vector, we further derive the fully discretized parametric finite element approximation. We show, theoretically, that the semi-discretized and fully discretized schemes are both area/mass-conservative and energy-dissipative. Finally, numerical experiments are reported to show the accuracy and efficiency of the numerical method as well as the anisotropic effects on the morphological evolution processes for solid-state dewetting of thin films.

Effect of surface energy anisotropy on hole growth in a single-crystalline thin film: A phase-field study

Prediction of the temporal evolution of a pre-existing hole on a solid-state thin film has important implications for self-organized nanoscale pattern formation. In this work, we employ a three-dimensional phase-field model to elucidate the mechanism of corner instability during capillary-driven hole growth in a single-crystalline free-standing thin film. We perform a systematic study to demonstrate the effect of crystalline anisotropy in surface energy, surface diffusivity, and film thickness on hole growth. Our simulations reveal that the instability during hole growth arises due to saturation of the rim height at the corner of the hole that is directly related to the arc length of the corner, specified by the anisotropy in surface energy. Our investigation further reveals the presence of a time-invariant shape of the corner with the onset of corner instability.

Phase‐Field Simulation of Texture Evolution in Magmatic Rocks

AbstractThe tool of phase‐field modeling for the prediction of chemical as well as microstructural evolution during crystallization from a melt in a mineralogical system has been developed in this work. We provide a compact theoretical background and introduce new aspects such as the treatment of anisotropic surface energies that are essential for modeling mineralogical systems. These are then applied to two simple model systems—the binary olivine‐melt and plagioclase‐melt systems ‐ to illustrate the application of the developed tools. In one case crystallization is modeled at a constant temperature and undercooling while in the other the process of crystallization is tracked for a constant cooling rate. These two examples serve to illustrate the capabilities of the modeling tool. The results are analyzed in terms of crystal size distributions (CSD) and with a view toward applications in diffusion chronometry; future possibilities are discussed. The modeling results demonstrate that growth at constant rates may be expected only for limited extents of crystallization, that breaks in slopes of CSD‐plots should be common, and that the lifetime of a given crystal of a phase is different from the lifetime of this phase in a magmatic system. The last aspect imposes an inherent limit to timescales that may be accessed by diffusion chronometry. Most significantly, this tool provides a bridge between CSD analysis and diffusion chronometry—two common tools that are used to study timescales of magmatic processes.

Calculating the Surface Layer Thickness and Surface Energy of Aircraft Materials

The surface layer determines the physical properties of aviation materials and, based on these properties, the calculation of surface energy anisotropy can be implemented. Moreover, the value of the surface energy determines the service time and the destruction of aircraft structures surface layer, while the surface layer thickness determines the distance at which this process usually takes place. In this work, a new atomically smooth crystal empirical model is built without considering the surface roughness. This model can be used to theoretically predict the surface energy anisotropy and surface layer thickness of metals and other compounds, in particular the aviation materials. The work shows that the surface layer of an atomically smooth metal, like other compounds, consists of two nanostructured layers: d(I) and d(II). Having sufficient accuracy, the proposed model would allow the prediction of aviation materials performance properties without the need for ultrahigh vacuum or other complicated theoretical methods to analyze the surfaces of nanosystem atomic structures.