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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 1,081 records · Page 60
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Magnetic and electron transport properties of Co 2 Si nanomagnets

In this study, magnetotransport and ferromagnetism in thin films of Co 2 Si nanoclusters are investigated experimentally and theoretically. The nanoclusters are fabricated by an inert-gas condensation-type cluster-deposition method and have an average size of 11.3 nm. Unlike the bulk Co 2 Si that exhibits a very weak net magnetic moment only below 10 K, the nanoclusters exhibit room-temperature ferromagnetism with a substantial saturation magnetization. Key features of the system are its closeness to the Stoner transition, magnetic moments induced by spin polarization starting from surface atoms, and nonuniaxial anisotropy associated with the orthorhombic crystal structure of Co 2 Si . A method is introduced to determine the effective anisotropy using the experimental magnetization data of this complex system and its relationship with the two lowest-order nonuniaxial anisotropy constants. On decreasing temperature from 300 K, the nanoclusters show electron-transport properties unusual for a ferromagnetic metal, including an increase of Hall resistivity and a nonmonotonic change of negative magnetoresistance with a peak at around 100 K. The underlying physics is explained on the basis of the large polarization of surface spins and variation in the degree of their misalignments due to temperature-dependent effective anisotropy.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Atomic resolution scanning transmission electron microscopy at liquid helium temperatures for quantum materials

Fundamental quantum phenomena in condensed matter, ranging from correlated electron systems to quantum information processors, manifest their emergent characteristics and behaviors predominantly at low temperatures. This necessitates the use of liquid helium (LHe) cooling for experimental observation. Atomic resolution scanning transmission electron microscopy combined with LHe cooling (cryo-STEM) provides a powerful characterization technique to probe local atomic structural modulations and their coupling with charge, spin and orbital degrees-of-freedom in quantum materials. However, achieving atomic resolution in cryo-STEM is exceptionally challenging, primarily due to sample drifts arising from temperature changes and noises associated with LHe bubbling, turbulent gas flow, etc. In this work, we demonstrate atomic resolution cryo-STEM imaging at LHe temperatures using a commercial side-entry LHe cooling holder. Firstly, we examine STEM imaging performance as a function of He gas flow rate, identifying two primary noise sources: He-gas pulsing and He-gas bubbling. Secondly, we propose two strategies to achieve low noise conditions for atomic resolution STEM imaging: either by temporarily suppressing He gas flow rate using the needle valve or by acquiring images during the natural warming process. Lastly, we show the applications of image acquisition methods and image processing techniques in investigating structural phase transitions in Cr 2 Ge 2 Te 6 , CuIr 2 S 4 , and CrCl 3 . In conclusion, our findings represent an advance in the field of atomic resolution electron microscopy imaging for quantum materials and devices at LHe temperatures, which can be applied to other commercial side-entry LHe cooling TEM holders.

36 MATERIALS SCIENCE↗
DOE OSTI2021

H 2 O-assisted O 2 reduction by H 2 on Pt and PtAu bimetallic nanoparticles: Influences of composition and reactant coverages on kinetic regimes, rates, and selectivities

Hydrogen peroxide (H 2 O 2 ) can replace hazardous oxidants in industrial processes but is currently too expensive for many such applications. While direct synthesis of H 2 O 2 (H 2 + O 2 → H 2 O 2 ) may reduce costs in comparison to incumbent technology, current catalysts lack the requisite stability and selectivity. Here, we examine the direct synthesis of H 2 O 2 on bimetallic Pt 1 Au x (0 ≤ x ≤ 230) and Pt catalysts at steady-state in pure water and relate kinetic parameters for H 2 O 2 and H 2 O formation to possible active site structures informed by complementary characterization methods. X-ray photoelectron spectra show significant Pt surface enrichment compared to the bulk composition. Analysis of infrared spectra of mixed monolayers of 12 CO* and 13 CO* indicate that Pt and Au form substitutional surface alloys. The Pt 1 Au x nanoparticles with the greatest mole fractions of Au predominantly expose Pt monomers (i.e., isolated Pt atoms), yet Pt atoms exposed upon all these nanoparticles possess electronic structures distinct from bulk Pt. Despite these differences, rate measurements are consistent with product formation through proton-electron transfer pathways for all Pt 1 Au x catalysts. In situ XAS indicate that Pt remains metallic during H 2 O 2 synthesis. Under the most oxidizing conditions, selectivities toward H 2 O 2 increase strongly with the Au to Pt ratio from 2% for monometallic Pt to 85% for Pt 1 Au 170 . However, selectivities are similar among all catalysts within reducing conditions. Comparisons of apparent activation enthalpies for the formation of H 2 O 2 and H 2 O across these catalysts and the range of conditions suggest that Pt monomers within Au provide the greatest selectivities for H 2 O 2 formation, because these active sites present high barriers for O-O bond rupture. Further, selectivities decrease with increasing ratios of H 2 to O 2 pressures, because Pt atoms aggregate and form oligomers that readily dissociate dioxygen intermediates. The combined use of spectroscopy, kinetics, and concepts employed in reaching these conclusions take inspiration from the legacy of Prof. Michel Boudart, and specifically his elegant methods for interrogating bimetallic catalysts.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Predicting band gaps and band-edge positions of oxide perovskites using density functional theory and machine learning

Density functional theory (DFT) within the local or semilocal density approximations, i.e., the local density approximation (LDA) or generalized gradient approximation (GGA), has become a workhorse in the electronic structure theory of solids, being extremely fast and reliable for energetics and structural properties, yet remaining highly inaccurate for predicting band gaps of semiconductors and insulators. The accurate prediction of band gaps using first-principles methods is time consuming, requiring hybrid functionals, quasiparticle GW, or quantum Monte Carlo methods. Efficiently correcting DFT-LDA/GGA band gaps and unveiling the main chemical and structural factors involved in this correction is desirable for discovering novel materials in high-throughput calculations. In this direction, we, in this study, use DFT and machine learning techniques to correct band gaps and band-edge positions of a representative subset of ABO 3 perovskite oxides. Relying on the results of HSE06 hybrid functional calculations as target values of band gaps, we find a systematic band-gap correction of ~1.5 eV for this class of materials, where ~1 eV comes from downward shifting the valence band and ~0.5 eV from uplifting the conduction band. The main chemical and structural factors determining the band-gap correction are determined through a feature selection procedure.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2020

A method for mapping submicron-scale crystallographic order/disorder applied to human tooth enamel

Tooth enamel, the outermost layer of human teeth, is a complex, hierarchically structured biocomposite. The details of this structure are important in multiple human health contexts, from understanding the progression of dental caries (tooth decay) to understanding the process of amelogenesis and related developmental defects. Enamel is composed primarily of long, nanoscale crystallites of hydroxyapatite that are bundled by the thousands to form micron-scale rods. Studies with transmission electron microscopy show the relationships between small groups of crystallites and X-ray diffraction characterize averages over many rods, but the direct measurement of variations in local crystallographic structure across and between enamel rods has been missing. Here, we describe a synchrotron X-ray-based experimental approach and a novel analysis method developed to address this gap in knowledge. A ~500-nm-wide beam of monochromatic X-rays in conjunction with a sample section only 1 μm in thickness enables 2D diffraction patterns to be collected from small well-separated volumes within the enamel microstructure but still probes enough crystallites (~300 per pattern) to extract population-level statistics on crystallographic features like lattice parameter, crystallite size, and orientation distributions. Furthermore, the development of a quantitative metric to characterize relative order and disorder based on the azimuthal autocorrelation of diffracted intensity enables these crystallographic measurements to be correlated with their location within the enamel microstructure (e.g., between rod and interrod regions). These methods represent a step forward in the characterization of human enamel and will elucidate the variation of the crystallographic structure across and between enamel rods for the first time.

crystallographic order/disorder↗
DOE OSTI2022

X-Ray Induced Chemical Reaction Revealed by In Situ X-Ray Diffraction and Scanning X-Ray Microscopy in 15 nm Resolution

The detection sensitivity of synchrotron-based X-ray techniques has been largely improved due to the ever-increasing source brightness, which has significantly advanced ex situ and in situ research for energy materials such as lithium-ion batteries. However, the strong beam–material interaction arising from the high beam flux can substantially modify the material structure. The beam-induced parasitic effect inevitably interferes with the intrinsic material property, making the interpretation of the experimental results difficult and requiring comprehensive assessments. Here, we present a quantitative study of the beam effect on an electrode material Ag 2 VO 2 PO 4 using four different X-ray characterization methods with different radiation dose rates. The material system exhibits interesting and reversible radiation-induced thermal and chemical reactions, further evaluated under electron microscopy to illustrate the underlying mechanism. Further, the work will provide a guideline for using synchrotron X-rays to distinguish the intrinsic behavior from extrinsic structure change of materials induced by X-rays.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Phase-Controlled Synthesis of Pd–Sn Nanocrystal Catalysts of Defined Size and Shape

Bimetallic Pd-based nanoparticles (NPs) are of interest as electrocatalysts for formic acid electrooxidation (FAEO) because of their higher initial catalytic activity and CO tolerance when compared to Pt. Intermetallic NPs (i-NPs) with specific geometric and electronic structures generally exhibit superior catalytic activity, selectivity, and durability when compared to their disordered (random alloy) counterparts; however, the colloidal synthesis of i-NPs remains a challenge. Here, a one-pot method was demonstrated as a facile route to obtain monodisperse Pd–Sn NPs with phase control, including intermetallic hexagonal Pd 3 Sn 2 (P6 3 /mmc), intermetallic orthorhombic Pd 2 Sn (Pnma), and alloy cubic Pd 3 Sn (FCC, Fm 3 ¯m) as size-controlled NPs with quasi-spherical shapes. Initial metal precursor ratios and reaction temperature were critical parameters to achieving phase control. Also, slight modifications of synthetic conditions resulted in either Pd 2 Sn nanorhombohedra or nanorods with tunable aspect ratios. A systematic evaluation of the Pd–Sn NPs for FAEO showed that most presented higher specific activities when compared to commercial Pd/C, in which Pd 2 Sn quasi-spheres and nanorhombohedra showed the highest catalytic activity for FAEO. Furthermore, these results highlight the benefits of phase-controlled Pd-based nanocatalysts with defined nanocrystal size and shape, with use of trioctylphospine (TOP) and oleic acid (OA) central to shape and size control.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

CryoTRANS: predicting high-resolution maps of rare conformations from self-supervised trajectories in cryo-EM

Cryogenic electron microscopy (cryo-EM) has revolutionized structural biology, enabling efficient determination of structures at near-atomic resolutions. However, a common challenge arises from the severe imbalance among various conformations of vitrified particles, leading to low-resolution reconstructions in rare conformations due to a lack of particle images in these quasi-stable states. We introduce CryoTRANS, a method that predicts high-resolution maps of rare conformations by constructing a self-supervised pseudo-trajectory between density maps of varying resolutions. This trajectory is represented by an ordinary differential equation parameterized by a deep neural network, ensuring retention of detailed structures from high-resolution density maps. By leveraging a single high-resolution density map, CryoTRANS significantly improves the reconstruction of rare conformations and has been validated on four real-world datasets: alpha-2-macroglobulin, actin-binding protein complexes, SARS-CoV-2 spike glycoprotein, and the 70S ribosome. CryoTRANS can also predict high-resolution structures in cryogenic electron tomography maps using a high-resolution cryo-EM map.Cryogenic electron microscopy (cryo-EM) has revolutionized structural biology, enabling efficient determination of structures at near-atomic resolutions. However, a common challenge arises from the severe imbalance among various conformations of vitrified particles, leading to low-resolution reconstructions in rare conformations due to a lack of particle images in these quasi-stable states. We introduce CryoTRANS, a method that predicts high-resolution maps of rare conformations by constructing a self-supervised pseudo-trajectory between density maps of varying resolutions. This trajectory is represented by an ordinary differential equation parameterized by a deep neural network, ensuring retention of detailed structures from high-resolution density maps. By leveraging a single high-resolution density map, CryoTRANS significantly improves the reconstruction of rare conformations and has been validated on four real-world datasets: alpha-2-macroglobulin, actin-binding protein complexes, SARS-CoV-2 spike glycoprotein, and the 70S ribosome. CryoTRANS can also predict high-resolution structures in cryogenic electron tomography maps using a high-resolution cryo-EM map.

47 OTHER INSTRUMENTATION↗
DOE OSTI2022

Bending as a control knob for the electronic and optical properties of phosphorene nanoribbons

We have assessed mechanical bending as a powerful controlling tool for the electronic structure and optical properties of phosphorene nanoribbons. We use state-of-the-art density functional approximations in our work. The overall performance of the recently developed meta-generalized gradient approximation (meta-GGA) mTASK [Phys. Rev. Materials 5, 063803 (2021)] functional establishes the method as a useful alternative to the screened hybrid HSE06 for better band gaps of phosphorene nanoribbons. We present a detailed analysis to interpret the optical absorption of bent phosphorene nanoribbons using the GW-Bethe-Salpeter approximation. Finally, we demonstrate the important role of the unoccupied in-gap state and conclude that this in-gap state in armchair nanoribbons introduced by bending can significantly affect the properties of low-energy excitons and add some useful opportunities for applications in optoelectronic devices.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Characterizing crystalline defects in single nanoparticles from angular correlations of single-shot diffracted X-rays

Characterizing and controlling the uniformity of nanoparticles is crucial for their application in science and technology because crystalline defects in the nanoparticles strongly affect their unique properties. Recently, ultra-short and ultra-bright X-ray pulses provided by X-ray free-electron lasers (XFELs) opened up the possibility of structure determination of nanometre-scale matter with Å spatial resolution. However, it is often difficult to reconstruct the 3D structural information from single-shot X-ray diffraction patterns owing to the random orientation of the particles. This report proposes an analysis approach for characterizing defects in nanoparticles using wide-angle X-ray scattering (WAXS) data from free-flying single nanoparticles. The analysis method is based on the concept of correlated X-ray scattering, in which correlations of scattered X-ray are used to recover detailed structural information. WAXS experiments of xenon nanoparticles, or clusters, were conducted at an XFEL facility in Japan by using the SPring-8 Ångstrom compact free-electron laser (SACLA). Bragg spots in the recorded single-shot X-ray diffraction patterns showed clear angular correlations, which offered significant structural information on the nanoparticles. The experimental angular correlations were reproduced by numerical simulation in which kinematical theory of diffraction was combined with geometric calculations. We also explain the diffuse scattering intensity as being due to the stacking faults in the xenon clusters.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2023

Structures and membrane interactions of native serotonin transporter in complexes with psychostimulants

The serotonin transporter (SERT) is a member of the SLC6 neurotransmitter transporter family that mediates serotonin reuptake at presynaptic nerve terminals. SERT is the target of both therapeutic antidepressant drugs and psychostimulant substances such as cocaine and methamphetamines, which are small molecules that perturb normal serotonergic transmission by interfering with serotonin transport. Despite decades of studies, important functional aspects of SERT such as the oligomerization state of native SERT and its interactions with potential proteins remain unresolved. Here, we develop methods to isolate SERT from porcine brain (pSERT) using a mild, nonionic detergent, utilize fluorescence-detection size-exclusion chromatography to investigate its oligomerization state and interactions with other proteins, and employ single-particle cryo-electron microscopy to elucidate the structures of pSERT in complexes with methamphetamine or cocaine, providing structural insights into psychostimulant recognition and accompanying pSERT conformations. Methamphetamine and cocaine both bind to the central site, stabilizing the transporter in an outward open conformation. We also identify densities attributable to multiple cholesterol or cholesteryl hemisuccinate (CHS) molecules, as well as to a detergent molecule bound to the pSERT allosteric site. Under our conditions of isolation, we find that pSERT is best described as a monomeric entity, isolated without interacting proteins, and is ensconced by multiple cholesterol or CHS molecules.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2020

Nanometer-Thick Sr 2 IrO 4 Freestanding Films for Flexible Electronics

In this article, we report the structural and optical properties of nanoscale Sr 2 IrO 4 freestanding thin-films fabricated using a water-soluble Sr 3 Al 2 O 6 layer. The coherent lattice structure, phonon modes, two-magnon Raman scattering, and optical absorption spectra of the Sr 2 IrO 4 nanomembrane are analogous to those of the layered iridate epitaxial thin-films and single crystals. Remarkably, the formation of 3-unit-cell-thick SrIrO 3 and interfacial composite layers alleviates the antiphase boundaries at the Sr 2 IrO 4 /Sr 3 Al 2 O 6 interface, resulting in structurally-robust nanomembranes. Our experimental results show that this freestanding thin-film approach of layered oxides can provide techniques for tuning or realizing unprecedented states beyond conventional thin-film methods, suggesting a pathway in achieving flexible layered-oxide electronics.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Machine learning for deep elastic strain engineering of semiconductor electronic band structure and effective mass

Abstract The controlled introduction of elastic strains is an appealing strategy for modulating the physical properties of semiconductor materials. With the recent discovery of large elastic deformation in nanoscale specimens as diverse as silicon and diamond, employing this strategy to improve device performance necessitates first-principles computations of the fundamental electronic band structure and target figures-of-merit, through the design of an optimal straining pathway. Such simulations, however, call for approaches that combine deep learning algorithms and physics of deformation with band structure calculations to custom-design electronic and optical properties. Motivated by this challenge, we present here details of a machine learning framework involving convolutional neural networks to represent the topology and curvature of band structures in k -space. These calculations enable us to identify ways in which the physical properties can be altered through “deep” elastic strain engineering up to a large fraction of the ideal strain. Algorithms capable of active learning and informed by the underlying physics were presented here for predicting the bandgap and the band structure. By training a surrogate model with ab initio computational data, our method can identify the most efficient strain energy pathway to realize physical property changes. The power of this method is further demonstrated with results from the prediction of strain states that influence the effective electron mass. We illustrate the applications of the method with specific results for diamonds, although the general deep learning technique presented here is potentially useful for optimizing the physical properties of a wide variety of semiconductor materials.

Tsymbalov, Evgenii↗
DOE OSTI2020

Valence-to-core X-ray emission spectroscopy of vanadium oxide and lithiated vanadyl phosphate materials

In this work, we report valence-to-core (VTC) X-ray emission spectroscopy (XES) measurements and theoretical calculations of the electrochemical sequence ε-VOPO 4 , ε-LiVOPO 4 , ε-Li 2 VOPO 4 and the reference oxides V 2 O 3 , VO 2 , and V 2 O 5 . In our analysis of these results, we establish a framework for interrogating chemical bonding that is generally applicable to a wide range of systems, including complex, extended inorganic compounds. While this latter regime has garnered less focused application than, e.g. , metalloenzymes in many excellent catalysis studies, we show that the technique provides high utility in materials-focused energy storage research. Here, sensitivities to the local atomic structure and hybridization schemes are discussed in detail. Similarly, the effect of lithiation on oxidation, delocalization, and shifts in ligand valence energy levels are all readily apparent in the analyzed results. Finally, the TDDFT projections clearly reveal the directional dependencies of the valence band at each of the vanadium sites. Our results demonstrate laboratory-based X-ray spectroscopy instrumentation is a viable route for attaining well-resolved VTC-XES features for inorganic compounds of 3d transition metals, even for samples of limited quantity or suffering from sensitivity to the atmosphere. The experimental results are in good agreement with results produced by real-space Green's function and time-dependent density functional theory (TDDFT) methods, respectively. Hence, we propose that VTC-XES, when equipped with appropriate theoretical support, can be a valuable complement to X-ray absorption pre-edge features for more detailed characterization of a compound's electronic structure. We expect similar analyses will find application in a broad range of materials chemistry research and provide both fundamental and applied insights.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Hybridization effect on the x-ray absorption spectra for actinide materials: Application to PuB 4

Studying the local moment and 5 f -electron occupations sheds insight into the electronic behavior in actinide materials. X-ray absorption spectroscopy (XAS) has been a powerful tool to reveal the valence electronic structure when assisted with theoretical calculations. However, the analysis currently taken in the community on the branching ratio of the XAS spectra generally does not account for the hybridization effects between local f orbitals and conduction states. In this paper, we discuss an approach which employs the density functional theory plus Gutzwiller rotationally invariant slave boson method to obtain a local Hamiltonian for the single-impurity Anderson model, and calculates the XAS spectra by the exact diagonalization (ED) method. A customized numerical routine was implemented for the ED XAS part of the calculation. By applying this technique to the recently discovered 5 f -electron topological Kondo insulator Pu B 4 , we determined the signature of 5 f -electronic correlation effects in the theoretical x-ray spectra. Furthermore, we found that the Pu 5 f - 6 d hybridization effect provides an extra channel to mix the j = 5 / 2 and 7 / 2 orbitals in the 5 f valence. As a consequence, the resulting electron occupation number and spin-orbit coupling strength deviate from the intermediate-coupling regime.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Importance of Dispersion in the Molecular Geometries of Mn(III) Spin-Crossover Complexes

The computational investigation of the molecular geometries of a pair of manganese(III) spin-crossover complexes is reported. For the geometry of the quintet high-spin state, density functionals significantly overestimate Mn–Namine bond distances, although the geometry for the triplet intermediate-spin state is well described. Here, comparisons with several wave function-based methods demonstrate that this error is due to the limited ability of commonly used density functionals to recover dispersion beyond a certain extent. Among the methods employed for geometry optimization, restricted open-shell Møller–Plesset perturbation theory (MP2) appropriately describes the high-spin geometry but results in a slightly shorter Mn–O distance in both spin states. On the other hand, extended multistate complete active space second-order perturbation theory (XMS-CASPT2) provides a good description of the geometry for the intermediate-spin state but also sufficiently recovers dispersion, performing well for the high-spin state. Despite the fact that the electronic structure of both spin states is dominated by one-electron configuration, XMS-CASPT2 offers a balanced approach, leading to molecular geometries with much better agreement with experiment than MP2 and DFT. A scan along the Mn–N amine bond demonstrates that for these complexes coupled cluster methods (i.e., DLPNO-CCSD(T)) also yield bond distances in agreement with experiment while multiconfiguration pair density functional theory (MC-PDFT) is unable to recover dispersion well enough, analogous to single-reference DFT.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Prediction of Above-Room-Temperature Superconductivity in Lanthanide/Actinide Extreme Superhydrides

Achieving superconductivity at or above room temperature has been a century long held dream for physicists since the discovery of superconductivity in mercury in 1911. Following the recent predictions and ensuing synthesis of clathrate superhydride LaH 10 under pressure exhibiting extraordinary superconducting critical temperatures (T c ) of 250 260 K, we predict via advanced crystal structure search methods a new class of extremely hydrogen rich clathrate superhydrides. These MH 18 (M: rare earth/actinide metal atom) stoichiometric compounds consisting of H36 cage networks are predicted to host T c values above room temperature up to 330 K at pressures of 350 GPa. The bonding and electronic properties of these MH 18 clathrate superhydrides parallel those of atomic metallic hydrogen, giving rise to the highest superconducting temperatures predicted thus far for a thermodynamically stable hydride compound. In depth examination of these extreme superhydrides offers key insights for elucidating and further exploring phonon mediated superconductivity above room temperature in hydrogen rich and other low Z materials.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Bifurcation Analysis Reveals Solution Structures of Phase Field Models

The phase field method is playing an increasingly important role in understanding and predicting morphological evolution in materials and biological systems. Here, in this study, we develop a new analytical approach based on the bifurcation analysis to explore the mathematical solution structure of phase field models. Revealing such solution structures not only is of great mathematical interest but also may provide guidance to experimentally or computationally uncover new morphological evolution phenomena in materials undergoing electronic and structural phase transitions. To elucidate the idea, we apply this analytical approach to three representative phase field equations: the Allen-Cahn equation, the Cahn-Hilliard equation, and the Allen-Cahn-Ohta-Kawasaki system. The solution structures of these three phase field equations are also verified numerically by the homotopy continuation method.

97 MATHEMATICS AND COMPUTING↗