Phase Boundary Energy Research Articles

Stabilization of interconnected models with Nitsche's interface conditions using the two-grid approach: A finite element study

In this paper, a stabilized Stokes–Stokes system with Nitsche's type interface conditions is presented. These conditions are commonly employed in many multi-physical fields, including fluid–fluid interaction, fluid–structure interaction, oceanographic modeling, and atmospheric forecasting. For multi-physical domain modeling purposes, Nitsche's interface conditions provide useful benefits over classical conditions via addressing the complicated nature of fluid phase interface mathematical modeling, phase boundary tracking, interface interactions, and mass and energy transportation. It is not easy to find analytical and numerical solutions for models with these characteristics. We use more accurate interface conditions to solve the fluid–fluid interaction model to accomplish this numerically. This is achieved by including new terms at the interface and decoupling the domain through the two-grid technique, which ultimately reduces the main issue into several smaller problems. Comparing this method to existing models, we find that it is computationally feasible because it uses less memory and operates with a coarse grid instead of a fine grid and thus improves convergence rates for complex and nonlinear problems. Furthermore, it shows mesh independence, supports potential parallelization, and is crucial for advanced multigrid techniques. The optimality of the error is confirmed both theoretically and numerically. The numerical experimental section validates the model through three types of numerical experiments.

Two- and Three-Dimensional Modeling and Simulations of Grain Growth Behavior in Dual-Phase Steel Using Monte Carlo and Machine Learning.

Dual-phase (DP) steel has been widely used in automotive steel plates with a balance of excellent strength and ductility. Grain refinement in DP steel is important to improve the properties further; however, the factors affecting grain growth need to be well understood. The remaining problem is that acquiring data through experiments is still time-consuming and difficult to evaluate quantitatively. With the development of materials informatics in recent years, material development time and costs are expected to be significantly reduced through experimentation, simulation, and machine learning. In this study, grain growth behavior in DP steel was studied using two-dimensional (2D) and three-dimensional (3D) Monte Carlo modeling and simulation to estimate the effect of some key parameters. Grain growth can be suppressed when the grain boundary energy is greater than the phase boundary energy. When the volume fractions of the matrix and the second phase were equal, the suppression of grain growth became obvious. The long-distance diffuse frequency can promote grain growth significantly. The simulation results allow us to better understand the factors affecting grain growth behavior in DP steel. Machine learning was performed to conduct a sensitivity analysis of the affecting parameters and estimate the magnitude of each parameter's effects on grain growth in the model. Combining MC simulation and machine learning will provide one promising research strategy to gain deeper insights into grain growth behaviors in metallic materials and accelerate the research process.

Zero-dimensionality of a scaled-down VO2 metal-insulator transition via high-resolution electrostatic gating

An understanding of the phase transitions at the nanoscale is essential in state-of-the-art engineering1–5, instead of simply averaging the heterogeneous domains formed during phase transitions6,7. However, as materials are scaled down, the steepness of the phase transition rapidly increases8–13 and requires extremely high precision in the control method. Here, a three-terminal device, which could precisely control the phase transition electrically14–19, was applied for the first time to a scaled-down metal-insulator transition material VO2. The crossover from continuous to binary transitions with the scaled-down material was clarified, and the critical channel length was successfully elucidated via phase boundary energy. Notably, below the critical channel length, the spatial degrees of freedom degenerated, and the impact of drain voltage application disappeared in the phase transition, indicating zero-dimensionality of the VO2 channel. This zero-dimensionality could be the fundamental property in the scaled-down phase transition and have a significant impact on various fields that need nanoscale engineering.

Effect of TiC precipitation on the corrosion behavior of Monel K500 alloy in 3.5 wt.% NaCl solution

The microstructure of Monel K500 alloy was investigated consisting of (Ni-Cu-Al) solid solution matrix distributed with TiC precipitates. The semi-coherent interface was formed between the precipitates and the matrix, which reduced the phase boundary energy, resulting in the preferential corrosion of matrix instead of the interface in 3.5 wt.% NaCl solution. Immersion experiment combined with electrochemical test exhibited the uniform corrosion behavior of the alloy, covered with passivation film consisted of surface polycrystal layer and inner amorphous layer. Moreover, the accumulation of CuO in the film with time, led to generation of microcracks as well as deterioration of corrosion resistance.

Microstructure evolution in the hypo-eutectic alloy Al0.75CrFeNi2.1 manufactured by laser powder bed fusion and subsequent annealing

The hypo-eutectic medium entropy alloy Al0.75CrFeNi2.1 was processed by laser powder bed fusion (LPBF). The off-equilibrium solidification conditions prohibited coupled eutectic growth. Instead, the primary face centered cubic phase A1(FCC) solidified with a cellular morphology and the body centered, initially ordered B2(BCC) phase formed as a thin intercellular envelope. During post-build annealing an ultrafine quasi-lamellar pattern evolved following BCC growth and coarsening. The novel solid state transformation from cellular to lamellar morphology was attributed to a pronounced anisotropy of the FCC|BCC phase boundary energy. Microstructure evolution was also studied during continuous heating using in situ high-energy synchrotron X-ray diffraction (HEXRD) carried out at the beamline P07-HEMS of PETRA III (German Electron Synchrotron, DESY). The ultrafine and nano-scale features of the microstructure were quantitatively analyzed by atom probe tomography (APT) in the as-built condition and after isothermal annealing at 950 °C. The benefits of LPBF processing were discussed on the basis of mechanical properties measured by 3-point bending. The estimated tensile properties after annealing at 950 °C/6 h reached YS ≈ 860 MPa, UTS ≈1384 MPa with an elongation at fracture of ≈11%. Tensile properties in the as-built condition were comparable to martensitic steels.

Phase-field modelling: effect of an interface crack on precipitation kinetics in a multi-phase microstructure

Premature failures in metals can arise from the local reduction of the fracture toughness when brittle phases precipitate. Precipitation can be enhanced at the grain and phase boundaries and be promoted by stress concentration causing a shift of the terminal solid solubility. This paper provides the description of a model to predict stress-induced precipitation along phase interfaces in one-phase and two-phase metals. A phase-field approach is employed to describe the microstructural evolution. The combination between the system expansion caused by phase transformation, the stress field and the energy of the phase boundary is included in the model as the driving force for precipitate growth. In this study, the stress induced by an opening interface crack is modelled through the use of linear elastic fracture mechanics and the phase boundary energy by a single parameter in the Landau potential. The results of the simulations for a hydrogenated (alpha +beta ) titanium alloy display the formation of a precipitate, which overall decelerates in time. Outside the phase boundary, the precipitate mainly grows by following the isostress contours. In the phase boundary, the hydride grows faster and is elongated. Between the phase boundary and its surrounding, the matrix/hydride interface is smoothened. The present approach allows capturing crack-induced precipitation at phase interfaces with numerical efficiency by solving one equation only. The present model can be applied to other multi-phase metals and precipitates through the use of their physical properties and can also contribute to the efficiency of multi-scale crack propagation schemes.

One-pot construction of robust BiOCl/ZnO p-n heterojunctions with semi-coherent interfaces toward improving charge separation for photodegradation enhancement.

Heterojunction engineering is an effective strategy to enhance the photodegradation activity via improving the spatial charge separation. However, the poor interface interactions and stability limit the photocatalytic activity and stability of traditional heterojunctions. Herein, robust BiOCl/ZnO p–n heterojunctions with semi-coherent interfaces were prepared by a one-pot hydrothermal method to improve the activity and stability toward photocatalytic degradation than that of the counterpart, in which the semi-coherent interfaces exhibited lower phase boundary energy, resulting in highly-stable interfaces between BiOCl and ZnO as well as the formation of the built-in electric field in this robust p–n heterojunction for enhanced charge separation. The cycle test results verified that the BiOCl/ZnO heterojunctions with semi-coherent interfaces can maintain the photocatalytic degradation activity at the initial level even after 10 cycles, while deactivation of the sample without semi-coherent interfaces occurred after 3 cycles only. Optical and electrical properties revealed that BiOCl/ZnO heterojunctions with semi-coherent interfaces possessed the highest electron migration and charge separation efficiency, resulting in the highest photodegradation activity. Density functional theory (DFT) calculations and electron spin-resonance (ESR) results verified that the enhanced charge separation was assigned to the type-II photocatalytic mechanism, leading to the enhancement of ˙OH and ˙O2− reactive oxygen species. This work would provoke the development of one-step construction of new highly active BiOX (X = Cl, Br, and I)-based heterogeneous photocatalysts with stable semi-coherent interfaces.

Athermal behavior of core-shell particles in nanocrystalline Cu-Ta

Nanocrystalline Cu-Ta is among the most successfully developed nanostructured alloys involving microstructural design, synthesis, characterization, and testing. However, the interplay of direct and indirect thermal stability mechanisms is still unclear. In this work, mechanically alloyed Cu-10at.%Ta was characterized with atomic-resolution electron microscopy to identify thermal stability mechanisms. A new mechanism involving athermal core-shell particles that were coarsening resistant was discovered after annealing between 100–800 °C up to 1000 h. Evidence suggests that oxide amorphous shells decreased the phase boundary energy thereby reducing the thermodynamic driving force for particle growth and indirectly maximizing thermal stability of Cu(Ta) grains.

Grain size-dependent energy partition in phase transition of NiTi shape memory alloys studied by molecular dynamics simulation

Nanocrystalline NiTi shape memory alloys show fundamental changes in phase transition behavior when the grain size is reduced down to nano-scale (grain size <30 nm). However, due to the extreme difficulties of in-situ experimental observation, universally acknowledged physical mechanisms and detailed microstructures at atomic level are still not clear. In this paper, detailed microstructure images and quantified energy partition at atomic level during stress-induced phase transition are obtained by molecular dynamics simulation. Phase transition is gradually suppressed and incomplete phase transition occurs as grain size decreases. The potential energy landscape of the nanocrystalline system changes significantly from nonconvex to convex, which leads to corresponding changes in stress–strain response and microstructural evolution. With decreasing grain size, the interface (grain boundary and phase boundary) energy makes much more contributions in variation of the system potential energy than the crystallite (austenite and martensite) energy. The mechanism of energy dissipation changes from atomic interface friction in phase transition to plastic deformation caused by grain boundary sliding. It is the gradual dominance of interfacial energy terms in total potential energy that leads to the observed fundamental changes of phase transition behavior in nanocrystalline NiTi.

Al–Cu alloy fabricated by novel laser-tungsten inert gas hybrid additive manufacturing

Owing to its high strength to weight ratio, Al–Cu alloy is extensively used in the aeronautic and aerospace industries. However, there are some shortcomings in the additive manufacturing of Al–Cu alloy, such as cracks and poor strength. In this work, Al–Cu (2219-Al) specimens with excellent mechanical properties were fabricated by laser-Tungsten Inert Gas (TIG) hybrid additive manufacturing. From the microstructural studies, the average grain size in the laser zone (LZ) decreased to 14.4 μm, which was approximately 40.3 % smaller than that in the arc zone (AZ). Under the influence of laser stirring, Cu in the LZ was distributed more uniformly than in the AZ. An incoherent θ phase, at the nanoscale, was discovered in both the AZ and the LZ. Its crystal orientation relationship was described as [110]α∥[002]θ, (110)α∥(002)θ between the α-Al matrix and the θ phase. The semi-coherent θ′ phase was observed in the LZ. Meanwhile, the θ′ phase characterized a good coherent relationship with the α-Al matrix, which resulted in low phase boundary energy. Furthermore, the deposited specimens exhibited a yield strength of 155.5 ± 7.9 MPa and an ultimate tensile strength of 301.5 ± 16.7 MPa, with an elongation of 12.8 ± 2.8 %. These mechanical properties were higher than in specimens fabricated by TIG, CMT and SLM methods. The improved properties were predominately related to the smaller size of eutectics, the uniform distribution of Cu and the semi-coherent θ′ phases in the LZ. The combined effect of laser and arc can yield components with excellent mechanical properties, promoting the material for an expansive range of applications.

Water cooling assisted laser dissimilar welding with filler wire of nickel-based alloy/austenitic stainless steel

To clarify the effects of water cooling on solidification behavior of pulsed laser welding of Nickel-based alloy/austenitic stainless steel with filler wire, and to improve the mechanical properties of laser welded joints, the analyses of morphology, microstructure, anti-peel property and micro-hardness were carried out. The results indicated that adding water cooling increased cooling rate and temperature gradient of the weld. And the addition of water cooling made constitutional supercooling of the weld easier, so that the solidification rate increased with the increase of the temperature gradient. Therefore, the weld grain was refined by about 28%, and the average width of unmixed zone decreased by 35%, with a low volume fraction of Mo-rich precipitated phase consisting of p phase and μ phase. The μ phase was discovered in laser welding Nickel-based alloy/austenitic stainless. Both p phase and μ phase had a good coherent relationship with the matrix, which resulted in the low phase boundary energy between precipitated phase and matrix. The coherent relationship is described as [1 4 3] p // [5 3 1] γ between the p phase and the matrix. And the coherent relationship is described as [1 2 0] μ // [2 2 1] γ between the μ phase and the matrix. There are the stacking faults in the μ phase along the direction of [1 1 0], which is one of the slip system in μ phase. At the same time coherent relationship between p phase and matrix is better than that of μ phase. The micro-hardness, load capacity per length, and anti-peel stress were improved by 10.5%, 8.1%, and 11.8%, respectively.

Phase boundary anisotropy and its effects on the maze-to-lamellar transition in a directionally solidified Al[sbnd]Al2Cu eutectic

Solid-solid phase boundary anisotropy is a key factor controlling the selection and evolution of non-faceted eutectic patterns during directional solidification. This is most remarkably observed during the so-called maze-to-lamellar transition. By using serial sectioning, we followed the spatio-temporal evolution of a maze pattern over long times in a large AlAl2Cu eutectic grain with known crystal orientation of the Al and Al2Cu phases, hence known crystal orientation relationship (OR). The corresponding phase boundary energy anisotropy (γ-plot) was also known, as being previously estimated from molecular-dynamics computations. The experimental observations reveal the time-scale of the maze-to-lamellar transition and shed light on the processes involved in the gradual alignment of the phase boundaries to one distinct energy minimum which nearly corresponds to one distinct plane from the family {120}Al||{110}Al2Cu. This particular plane is selected due to a crystallographic bias induced by a small disorientation of the crystals relative to the perfect OR. The symmetry of the OR is thus slightly broken, which promotes lamellar alignment. Finally, the maze-to-lamellar transition leaves behind a network of fault lines inherited from the phase boundary alignment process. In the maze pattern, the fault lines align along the corners of the Wulff shape, thus allowing us to propose a link between the pattern defects and missing orientations in the Wulff shape.

Pinning effect of coherent particles on moving planar grain boundary: Theoretical models and molecular dynamics simulations

Pinning effect of second-phase particles has been demonstrated as an extremely effective way to impede grain boundary (GB) migration to increase strength or thermal stability, particularly for nano-scale materials. Previous studies on the particle pinning mainly focus on the incoherent particle; the coherent particle pinning, more effective in retarding the GB migration, has not yet been completely understood, owing to varied phase boundary (PB) energy essentially and additionally considered. In this paper, the maximum and the average pinning forces of the coherent particle are modeled, considering the evolved GB shape. A critical criterion is deduced to judge the condition that the pinning force from multiple particle can be obtained by a linear summation of the pinning force from the individual particle. Using Cu–Ag as the model system, molecular dynamics (MD) simulations are performed to study the interaction between the coherent Ag particles and the planar Cu GBs, where the maximum and the average pinning forces are estimated as functions of particle parameters (e.g. the size, the volume fraction, the shape and the orientation of particle). By combination of models and MD simulations, the pinning effect of coherent particles is attributed to a change in Gibbs free energy ascribed to the GB energy and the PB energy, while the maximum and the average pinning forces are enhanced with an increase in volume fraction or a decrease in size of particle.

Three-dimensional phase-field simulation of microstructural evolution in three-phase materials with different interfacial energies and different diffusivities

The coarsening behavior of three-phase materials, such as eutectic alloys, is of high technological interest. In this study, 3D ternary three-phase polycrystalline materials were modeled to study the effect of bulk diffusion and phase arrangement on the coarsening kinetics. The diffusion mobilities were defined to be different in the three phases. By varying the phase boundary and grain boundary energies, microstructures with different phase arrangements were obtained, in which the different types of grains had a tendency to alternate or cluster. In all cases, a regime was reached where the average grain size follows a power growth law with growth exponent $$n=3$$ , indicating bulk diffusion-controlled coarsening. The overall growth rate and that of the individual phases were clearly affected by the phase arrangement, the magnitude of the phase boundary energy and the diffusion mobilities of the different phases. In all cases, the phase with the lowest diffusion mobility showed the highest growth rate and on average a larger number of grain faces. While the average number of grain faces became constant in time in systems with constant grain boundary energy, the average number of grain faces continued to increase during the whole simulation time when the grain boundary energy was misorientation dependent.

Recrystallization and Grain Growth in Accumulative Roll-Bonded Metal Composites

We examine recrystallization and grain growth during processing of accumulative roll-bonded (ARB) Cu-Nb and Zr-Nb composites. Throughout the ARB process, from initial millimeter thick layers down to nanometer thick layers, the mechanism for recrystallization and grain growth is the motion of high-angle grain boundaries (HAGBs). However, the driving forces for these phenomena change as the densities of different types of defects evolve during the process. The creation and redistribution of dislocations, grain boundaries, and phase boundaries has significant effects on recrystallization and grain growth and, thus, on microstructural evolution. Both Cu-Nb and Zr-Nb exhibit a distinct transition in recrystallization and growth behavior at around 500-nm average layer thicknesses. For the thicker layered materials, the microstructure evolution during recrystallization and growth is determined by the density and distribution of dislocations and HAGBs. For layers less than 500 nm, the layers are largely one-grain thick and the grains are nearly dislocation free; coarsening of grains within layers at the nanoscale is due to reduction in phase boundary energy.

Molecular dynamics simulations of Al–Al2Cu phase boundaries

Potential based molecular dynamics simulations were performed for Al–Al2Cu phase boundaries (PBs) at a temperature of 50K using a newly designed computation geometry that enables modeling hetero-interface configurations for an arbitrary pair of phases. The computational method and geometry were validated for symmetric grain boundaries in Al, both [001] and [110] tilt boundaries, followed by extensive simulations of Al–Al2Cu PBs. For randomly oriented Al and Al2Cu as well as randomly oriented phase boundary planes the average PB energy reaches γav.=0.456±0.002J/m2. Two special orientation relationships (ORs), known to prevail in Al–Al2Cu eutectics after unidirectional solidification, were characterized in detail. For each OR the 3D phase boundary energy surface was mapped and the energy minima were carefully analyzed. Computational results were compared to experimental observations based on EBSD measurements and allowed concluding that lamellar eutectic interfaces select a shallow energy minimum γα4=0.407J/m2 for one OR, but a deep, cusp-like energy minimum γβ6=0.253J/m2 for the second OR. The calculations thus substantiate the experimentally observed behavior of lamellar eutectic interfaces, being nearly isotropic in the first case, but strongly anisotropic in the latter.

Characterization of an anomaly in the crystallographic orientation of plate-like carbides precipitated in a wrought Ni-base superalloy

Face-centred cubic Cr-rich carbide is known to precipitate in a face-centred cubic matrix with a cube–cube orientation relationship, thereby minimizing the elastic strain energy. In the present study, for the first time, the precipitation was observed of an abnormal Cr-rich carbide, which did not have the cube–cube orientation relationship in its face-centred cubic matrix. The abnormally oriented carbides nucleated and grew around random grain boundaries, and were observed to have a lamellar or plate-like morphology. The crystallographic orientation anomaly was characterized by measuring the tilt angles of the three crystal poles of the matrices, carbides and adjacent grains, using a transmission electron microscope to find the closest coincidence site lattice boundary. The carbides showed a slight deviation from a cube–cube orientation with adjacent grains and did not present any particular orientational relationship with the matrix. The deviation angles from coincidence site lattice boundaries between the matrices and carbides were smaller than those between matrices and adjacent grains. The abnormally oriented carbides appeared to nucleate on adjacent grains, and underwent a rotation within the matrix during the initial stage of growth to release the phase boundary energy between the carbides and the matrix.

Phase field model for coupled displacive and diffusive microstructural processes under thermal loading

A non-isothermal phase field model that captures both displacive and diffusive phase transformations in a unified framework is presented. The model is developed in a formal thermodynamic setting, which provides guidance on admissible constitutive relationships and on the coupling of the numerous physical processes that are active. Phase changes are driven by temperature-dependent free-energy functions that become non-convex below a transition temperature. Higher-order spatial gradients are present in the model to account for phase boundary energy, and these terms necessitate the introduction of non-standard terms in the energy balance equation in order to satisfy the classical entropy inequality point-wise. To solve the resulting balance equations, a Galerkin finite element scheme is elaborated. To deal rigorously with the presence of high-order spatial derivatives associated with surface energies at phase boundaries in both the momentum and mass balance equations, some novel numerical approaches are used. Numerical examples are presented that consider boundary cooling of a domain at different rates, and these results demonstrate that the model can qualitatively reproduce the evolution of microstructural features that are observed in some alloys, especially steels. The proposed model opens a number of interesting possibilities for simulating and controlling microstructure pattern development under combinations of thermal and mechanical loading.

A phase field model of deformation twinning: Nonlinear theory and numerical simulations

A continuum phase field theory and corresponding numerical solution methods are developed to describe deformation twinning in crystalline solids. An order parameter is associated with the magnitude of twinning shear, i.e., the lattice transformation associated with twinning. The general theory addresses the following physics: large deformations, nonlinear anisotropic elastic behavior, and anisotropic phase boundary energy. The theory is applied towards prediction of equilibrium phenomena in the athermal and non-dissipative limit, whereby equilibrium configurations of an externally stressed crystal are obtained via incremental minimization of a free energy functional. Outcomes of such calculations are elastic fields (e.g., displacement, strain, stress, and strain energy density) and the order parameter field that describes the size and shape of energetically stable twin(s). Numerical simulations of homogeneous twin nucleation in magnesium single crystals demonstrate fair agreement between phase field solutions and available analytical elasticity solutions. Results suggest that critical far-field displacement gradients associated with nucleation of a twin embryo of minimum realistic size are 4.5%–5.0%, with particular values of applied shear strain and equilibrium shapes of the twin somewhat sensitive to far-field boundary conditions and anisotropy of twin boundary surface energy.

Binary-Component Self-Assembled Monolayer Comprising Tetrathiafulvalene and n-Tetradecane Molecules with Periodic Ordered Phase Separation Structures on a Highly Oriented Pyrolytic Graphite Surface

A binary-component self-assembled monolayer (SAM) comprising tetrathiafulvalene (TTF) and n-tetradecane (n-C(14)H(30)) molecules has been investigated by a scanning tunneling microscope (STM) on a highly oriented pyrolytic graphite (HOPG) surface at room temperature. High-resolution STM images of the SAM reveal that the two different kinds of molecules spontaneously form periodic ordered strip-like phase separation structure on the HOPG substrate. The phenomenon can be qualitatively understood in terms of a phase field model, in which the interplay of three ingredients, including free energy of the binary component solution monolayer, phase boundary energy, and surface stress, is suggested to play an important role in determining the equilibrium sizes of strip-like domains of the TTF and n-C(14)H(30) in the ordered phase separation structure. In addition, scanning tunneling spectrum (STS) measurements have also been performed by operating an STM tip to locate on the individual TTF or n-C(14)H(30) molecule in the SAM on the HOPG substrate. The STS at the TTF molecule shows a distinct rectifying behavior, while at the n-C(14)H(30) molecule it shows an intrinsic small current increase with the change of bias voltage and a slight asymmetry.