Anisotropy Of Interfacial Energy Research Articles

Dendritic tip selection during solidification of alloys: Insights from phase-field simulations

Abstract The effect of undercooling ΔT and the interface energy anisotropy parameter ε 4 on the shape of the equiaxed dendritic tip has been investigated by using a quantitative phase-field model for solidification of binary alloys. It was found that the tip radius ρ increases and the tip shape amplitude coefficient A 4 decreases with the increase of the fitting range for all cases. The dendrite tip shape selection parameter σ* decreases and then stabilizes with the increase of the fitting range, and σ* increases with the increase of ε 4. The relationship between σ* and ε 4 follows a power-law function σ ∗ ∝ ε 4 α , and α is independent of ΔT but dependent on the fitting range. Numerical results demonstrate that the predicted σ* is consistent with the curve of microscopic solvability theory (MST) for ε 4 < 0.02, and σ* obtained from our phase-field simulations is sensitive to the undercooling when ε 4 is fixed.

Imaging flow focusing and isolation of aqueous fluids in synthetic quartzite: implications for permeability and retained fluid fraction in deep-seated rocks

The microstructure of realistic fluid–rock systems evolves to minimize the overall interfacial energy, enabling local variations in fluid geometry beyond ideal models. Consequently, the permeability–porosity relationship and fluid distribution in these systems may deviate from theoretical expectations. Here, we aimed to better understand the permeability development and fluid retention in deep-seated rocks at low fluid fractions by using a combined approach of high-resolution synchrotron radiation X-ray computed microtomography imaging of synthesized rocks and numerical permeability computation. We first synthesized quartzite using a piston-cylinder apparatus at different fluid fractions and wetting properties (wetting and non-wetting systems with dihedral angles of 52° and 61°–71°, respectively) under conditions of efficient grain growth. Although all fluids should be connected along grain edges and tubules in the homogeneous isotropic wetting fluid–rock system enabling segregation by gravitational compaction in natural settings, the fluid connectivity rapidly decreased to ~ 0 when the total fluid fraction decreased to 0.030–0.037, as the non-ideality of quartzite, including the interfacial energy anisotropy (i.e., grain faceting), became critical. In non-wetting systems, where the minimum energy fluid fraction based solely on the dihedral angle is ~ 0.015–0.035, the isolated (disconnected) fractions was 0.048–0.062. A streamline computation in the non-wetting system revealed that with decreasing total porosity, flow focusing into fewer channels maintained permeability, allowing the effective segregation of the connected fluids. These results provide insight into how non-wetting fluids segregate from rocks and exemplify the fraction of retained fluids in non-wetting systems. Thus, the findings suggest a potential way for wetting system fluids to be transported into the deep Earth's interior, and the amount of fluids dragged down to the Earth’s interior could be higher than what was previously estimated.

Perpendicular magnetic anisotropy of Cr/CoFeB/MgO modulated by MgO thickness

The dependence of perpendicular magnetic anisotropy (PMA) on the MgO thickness in Cr/CoFeB/MgO/Ta films has been experimentally investigated. A clear PMA is observed in the as-deposited samples with 1.8 nm MgO while no as-deposited PMA is shown in those with 4.0 nm MgO. This may be attributed to the moderate oxidation degree of CoFeB and larger interfacial anisotropy energy density K i to overcome the volume magnetic anisotropy and demagnetization field. On the contrary, samples with 4.0 nm MgO demonstrate PMA only after annealing, which might be due to the oxygen and boron diffusion during the annealing process. These results would provide a method to optimize the design of CoFeB/MgO structures on 3d metals for future applications in perpendicular magnetic devices.

Ligament morphology and elastic modulus of porous structure formed by liquid metal dealloying

We report that the morphology of ligaments also governs the mechanical properties of dealloyed porous materials, in addition to the topology- and size-effects that have been extensively studied previously. Porous Fe–Cr with similar relative density but different Cr content were prepared by liquid metal dealloying. The mechanical efficiency of this material, which is quantified by relative elastic modulus, decreases dramatically with increasing Cr content, although the relative density and network connectivity do not vary significantly. This is linked to the more severe spheroidization of Fe–Cr ligaments at higher Cr, driven by the large excess energy of solid-liquid interfaces and interface energy anisotropy of Fe–Cr under dealloying environment. A shape parameter is introduced to quantitatively account for this ligament-morphology effect. Current study suggests that tailoring interfacial energy, which was largely overlooked in previous studies, is essential to improving the mechanical efficiency of porous or nanoporous materials self-organized in dealloying.

Morphological diversity in directionally-solidified microstructures with varying anisotropy of solid-liquid interfacial free energy

Morphological diversity in directionally-solidified microstructures of fcc-based binary alloys was investigated using quantitative phase-field simulation. The growth morphology for each set of anisotropy parameters was identified from the degree of undercooling of the dendrite tip and a morphology map was constructed. We investigated the effects of solidification conditions and alloy systems on growth morphology and found that the pulling speed has a significant effect, while temperature gradient and partition coefficient have only small effects on the morphology selection. Furthermore, we examined the emergence of doublon and triplet dendrites under conditions of weak interfacial anisotropy.

Perpendicular magnetic anisotropy in permalloy ultrathin film grown on RuO2(101) surface

Permalloy (Py) films are commonly regarded as soft magnetic materials, wherein the magnetization aligns within the film plane. Our studies reveal the presence of perpendicular magnetic anisotropy in Py thin films deposited on the collinear antiferromagnetic RuO2(101) surface. By employing both the magneto-optical Kerr effect and the anomalous Hall effect, we identified the interfacial origin of the observed perpendicular anisotropy, quantifying it with an interfacial anisotropy energy of approximately 0.77 erg/cm2. Current-induced magnetization switching in Py/RuO2(101) has been achieved under an in-plane field, with the current applied along both [010] and [10 1¯] directions. Py films exhibiting perpendicular magnetic anisotropy offer an innovative material platform for investigating the spin–orbit effect, holding significant potential for spintronics applications.

Inclined θ′ precipitates with terraced Cu/Mn-rich multilayers in Al–Cu alloys

In this work, an inclined θ′ morphology with terraced Cu/Mn-rich multilayers is observed in Al–Cu based alloys. The GPA (geometric phase analysis) results indicate that the inclined (∼3.1°) θ′ precipitates could induce significant lattice distortion at the interface. The chemical analysis results demonstrate that Mn atoms could get into the coherent (001)θ′//(001)Al interface with large misfit strains, which may contribute to maintaining the coherent relationship of the (001)θ'//(001)Al interface. Additionally, the interfacial energy between coherent Cu/Mn-rich (001)θ'//(001)Al interface and semi-coherent (010)θ′//(010)Al interface is more closer than that of the normal (001)θ'//(001)Al interface, suggesting a weakness of the anisotropic interfacial energy of θ′ with Cu/Mn-rich layers. The phase field simulation results suggest the formation of inclined θ′ precipitates when elastic strain energy dominates the θ′ morphology. Therefore, the inclined morphology of θ′ precipitates with terraced Cu/Mn-rich multilayers is a plausible outcome in Al–Cu alloys.

Atomistic insight into dendrite growth orientation transition in Al-Cu alloy

In this study, we investigate the impact of solid-melt interfacial energy anisotropy on the microstructure of components. Our focus is on the Al-Cu system's Al-rich side, where we compute the solid-melt interfacial free energy and its anisotropy using Molecular Dynamics (MD) simulations. Our findings reveal that, as the Cu concentration in the melt increases, the preferred dendrite growth orientation shifts from <100> to <110>. Furthermore, atomic density profile analysis shows an improvement in ordering in the (110) plane at higher Cu concentrations. Our results suggest that the transition in dendrite growth orientation is a consequence of the weakened anisotropy between (100) and (110) interfacial energies at higher Cu concentrations and undercooling temperatures. We also observe displaced atomic density profiles between Al and Cu atoms in proximity to the interface. We propose that a displaced atomic density profile between solute and solvent atoms could be a characteristic feature of systems undergoing a dendrite orientation transition.

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.

Strong cooperation growth of two phases in the ternary eutectic

The growth mechanism and tensile performance of ternary Ni40.6Fe36.4Ti23 alloy were investigated using directional solidification technique. The ternary Fe2Ti+γ-Fe(Ni)+Ni3Ti eutectic formed a lamellar structure under the steady state growth. After quenching, the Fe2Ti phase transformed into independently grown fiber while the γ-Fe(Ni) and Ni3Ti phases kept lamellar structure. It was confirmed that the solid/solid interfacial energy anisotropy contributed to the strong cooperation growth of the two phases in the ternary eutectic. The γ-Fe(Ni)/Ni3Ti interface preferred to be parallel to the common planes of {111}γ and {0001}Ni3Ti with low interfacial energy, leading to the formation of the stable lamellar structure under different solidification conditions. Owing to the reduction of transverse grain boundaries and grain refinement, the alloy solidified at 10 μm/s growth velocity showed the highest tensile strength and Young's modulus with translamellar fracture mechanism.

Multivariant microstructure evolution in Ti-alloys: insights from a quantitative phase-field study

Abstract In the present work, we employ a CALPHAD-integrated physics-based quantitative multi-phase field model to study the microstructure evolution in a multi-phase Ti-alloy. In this model we use the free energies obtained from TTTI3 thermodynamic database for Ti-based alloys using a CALPHAD approach. For numerical implementation of the model, we solve the governing equations in their weak form using the open-source MOOSE finite element framework. Our model is successfully able to capture all the possible orientational variants in Ti- alloy system. Moreover, we elucidate the complex interaction between the variants using two- and multi-particle simulations. We show that the elastic energies associated with the variants play significant role in their growth behavior. We also explore the effect of anisotropy in interfacial energy on the variant evolution.

Experimental determination of solid-liquid interfacial free energy anisotropy in Al-Zn via equilibrium droplet method

The increasing Zn content causes dendrite orientation transition during solidification of Al-Zn alloys, which is supposed to be related to the change of solid-liquid interfacial energy anisotropy. In this paper, experimental measurements of the interfacial energy anisotropy were performed in Al-17, 30 and 75wt.%Zn alloys with equilibrium droplet method. Droplets shape in relation to crystal orientation and droplets morphology are obtained by Electron Back-Scatter Diffraction and X-ray Computed Tomography, respectively. The result indicates that anisotropic parameters ε1 and ε2 both decreases with increasing Zn concentration and the directions of maximum interfacial energy transform from 〈100〉 directions in Al-17wt.%Zn and Al-30wt.%Zn to 〈110〉 directions in Al-75wt.%Zn. In addition, the Wulff plot and normalized interfacial stiffness are quantified and compared with the corresponding Al-Zn dendrites orientation selection.

A combined simulation and experimental study of the equilibrium shapes of η′ and α precipitates in Mn-containing 7xxx Al-alloys

The addition of Mn to 7xxx series Al-alloys helps improve tensile properties owed to the presence of α precipitates (Al(Mn,Fe)Si, simple cubic, 50–200 nm, non-shearable) in addition to the η′ precipitates (Mg2Zn5−xAl2+x, hexagonal, 4–6 nm, shearable) commonly observed in the 7xxx series Al-alloys. The coexistence of multiple α and η′ variants, 24 for α and 4 for η′, coupled with their small size-scale, makes the experimental determination by x-ray diffraction (XRD) and scanning transmission electron microscopy (STEM) challenging. This work formulates a phase-field model to predict precipitate shapes in three-dimension (3D) using experimental characterization as inputs to create virtual precipitate cross-sections for use in morphological comparison with experimental 2D (S)TEM images. Based on the atomic structures and orientation relationships observed in the experiments, the lattice site correspondences between the matrix and the two precipitate phases are derived, which are required for the prediction of shapes of coherent precipitates. Systematic phase-field simulations are then carried out to document the 3D shapes for all possible variants, as well as the corresponding 2D cross-sectional shapes when observed parallel to low-index matrix planes. These 2D cross-sectional shapes are compared with high-angle annular dark-field STEM (HAADF-STEM) images, which demonstrates that the identification of Mn precipitate type is possible by such comparisons. This helps unravel some of the complications encountered when characterizing precipitate microstructures in Mn containing Al-alloys by STEM. Finally, the equilibrium shapes as a function of precipitate size and the effect of interfacial energy anisotropy are investigated.

Numerical modeling of faceted crystal growth using a lattice Boltzmann-phase field model with a new interfacial energy function

A lattice Boltzmann-phase field model is proposed to study faceted crystal growth in a supersaturated solution, taking into account a new anisotropic interfacial energy function to reproduce the facet characteristics. In this model, a flow field solver and a concentration field solver are coupled with the phase field equation to study faceted crystal growth with fluid convection and solute transfer. Qualitative and quantitative comparisons are carried out with the experimental observation and theoretical analysis of snowflakes to demonstrate the validity and accuracy of the present numerical method. Then crystallization of sodium chloride in a supersaturated solution is investigated in a 3D space at a constant temperature. The results show that the initial supersaturation, solute depletion rate, and solution flow can greatly change the faceted crystal growth by affecting solute transport and distribution. The mechanism of the formation of morphologies and interface velocity evolution is studied and concluded, which shows that the present model has instructional significance for studying and understanding the regulation mechanism of faceted crystal growth.

High order accurate and convergent numerical scheme for the strongly anisotropic Cahn–Hilliard model

AbstractWe propose and analyze a second order accurate in time, energy stable numerical scheme for the strongly anisotropic Cahn–Hilliard system, in which a biharmonic regularization has to be introduced to make the equation well‐posed. A convexity analysis on the anisotropic interfacial energy is necessary to overcome an essential difficulty associated with its highly nonlinear and singular nature. The second order backward differentiation formula temporal approximation is applied, combined with Fourier pseudo‐spectral spatial discretization. The nonlinear surface energy part is updated by an explicit extrapolation formula. Meanwhile, the energy stability analysis is enforced by the fact that all the second order functional derivatives of the energy stay uniformly bounded by a global constant. A Douglas‐Dupont type regularization is added to stabilize the numerical scheme, and a careful estimate ensures a modified energy stability with a uniform constraint for the regularization parameter . In turn, the combination with an appropriate treatment for the nonlinear double well potential terms leads to a weakly nonlinear scheme. More importantly, such an energy stability is in terms of the interfacial energy with respect to the original phase variable, which enables us to derive an optimal rate convergence analysis.

Mechanistic Exploration of Dendrite Growth and Inhibition for Lithium Metal Batteries

Li metal has been considered an ideal anode in lithium batteries due to its high theoretical capacity of 3860 mAh·g−1 and lowest negative reduction potential of −3.040 V among the standard hydrogen electrodes. However, lithium dendrites can easily grow on the surface of the negative electrode during charging, which results in a short circuit of the battery and reduces its efficiency. This paper investigated dendrite growth and inhibition mechanisms in lithium metal batteries to improve battery life. The impacts of the initial nucleation spacing, surface energy anisotropy strength, and interfacial electrochemical driving force on lithium dendrite growth were analyzed with electrochemical experiments and mathematical models. The results showed that the smaller nucleation spacing inhibits the growth of dendrite side branches and reduces the roughness of lithium metal deposition on the negative electrode. A lower interfacial energy anisotropy strength can slow down the growth of dendrite tips and improve the dendrite growth structure. The growth of the dendrites is influenced by the interfacial electrochemical driving force. Reducing the nucleation overpotential can effectively inhibit the growth of lithium dendrites.

Strain-Driven Faceting of Graphene-Catalyst Interfaces.

The interfacial interaction of 2D materials with the substrate leads to striking surface faceting affecting its electronic properties. Here, we quantitatively study the orientation-dependent facet topographies observed on the catalyst under graphene using electron backscatter diffraction and atomic force microscopy. The original flat catalyst surface transforms into two facets: a low-energy low-index surface, e.g. (111), and a vicinal (high-index) surface. The critical role of graphene strain, besides anisotropic interfacial energy, in forming the observed topographies is revealed by molecular simulations. These insights are applicable to other 2D/3D heterostructures.

Growth Mechanism of Ice Crystals with Five‐Fold Symmetry by the Phase‐Field Method

In order to simulate the ice crystal growth with five‐fold symmetry, a new phase‐field model was established. The finite difference method was used to solve the governing equation, reproduce the growth morphology of ice crystals with five‐fold symmetry, and study the influence of interfacial energy anisotropy and undercooling on the growth mechanism of ice crystals. The results indicate that when the strength of interfacial energy anisotropy (ε5) is greater than the critical value (1/24), the interface discontinuity and corners form at the tips of the main stem and side branches. The characteristic parameters of the ice crystal tip in the horizontal direction have an extreme value at ε5 equal to 0.1. With an increase in undercooling, the main branch becomes slender and side branches appear, which is consistent with the mechanism of crystal continuous growth. When undercooling reaches 0.4, ice crystal growth changes from diffusion control to dynamic control. Therefore, by selecting the quick freezing conditions, undercooling is controlled at around 0.4 and the interfacial energy anisotropy at around 0.1, and the tip radius of curvature is minimized so as to prevent large ice crystals from damaging the cell wall and leading to the decline in the quality of quick‐frozen food. The findings of this study can help us have a better understanding of ice crystal growth in the process of quick‐frozen food preservation.

Influence of La addition on Fe-rich intermetallic phases formation and mechanical properties of Al-7Si-4Cu-0.35Mg-0.2Fe alloys prepared by squeeze casting

Brittle β-Al5FeSi (β-Fe) intermetallic phases directly affect the mechanical properties of Al–Si alloys; thus, it is essential to modify these phases. Scanning electron microscope analysis showed that rare earth Lanthanum (La) additions (0, 0.15, 0.30 and 0.50 wt%) had a refining effect on β-Fe phases in Al–7Si–4Cu-0.35Mg-0.2Fe (wt.%) alloys prepared by squeeze casting. The underlying refinement mechanisms of rare earth La on the β-Fe were discussed in terms of experimental observations, nucleation thermodynamics and growth kinetics. For nucleation thermodynamics, La additions could induce the formation of La-rich intermetallic phases Al4Cu2SiLa with orthorhombic structure. Because β-Fe nuclei had a good lattice matching and epitaxial relationship with the Al4Cu2SiLa phases, the Al4Cu2SiLa provided energetically favorable nucleation sites for β-Fe and greatly promoted the nucleation of β-Fe in alloys. Meanwhile, the added La atoms could restrain the diffusion of atoms to the solid/liquid interface front and alter the β-Fe interfacial structure and energy, therefore restricting the β-Fe growth. Furthermore, β-Fe/Al4Cu2SiLa grew into side-by-side and cross-like morphologies, and its growth behaviors were controlled by solute concentration and anisotropy of the solid-liquid interfacial energy in a cooperative eutectic reaction. In addition, different fracture behaviors from brittle fracture to mixed ductile fracture and quasi-cleavage brittle fracture were also illustrated with increasing La addition, and these fracture behaviors were influenced by the characteristics of brittle La-rich and β-Fe intermetallic phases. When the addition of La was increased to 0.3 wt% and 0.5 wt%, it occurred quasi-cleavage brittle fracture, which was caused by the precipitation of coarse plate-like La-rich phases.

Phase-field computations of anisotropic ice crystal growth on a spherical surface

In this paper, we present a numerical method for the phase-field model of anisotropic ice crystal growth on a spherical surface. The mathematical model includes terms related to the anisotropic interfacial energy, which is defined by the interface angle with respect to a reference angle. One of the natural numerical methods on curved surfaces is a computational technique based on a triangular mesh for the surface in a three-dimensional space. However, it is difficult to compute terms with the interface angle on a triangular mesh. To resolve this problem, we solve the governing equation in Cartesian coordinates after rotating each vertex and the 1-ring neighborhood of the vertex on the triangular mesh. After rotation and interpolation, we numerically solve the phase-field model using a standard finite difference method. We present the results of several tests to demonstrate that the proposed algorithm can recover anisotropic ice crystal growth on a spherical surface.