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Results for “Multiscale analysis”

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 451 records · Page 25
All sourcesNASA NTRSDOE OSTI
DOE OSTI2019

Understanding Microstructural Evolution During Rapid Heat Treatment of Microalloyed Steels Through Computational Modeling, Advanced Physical Simulation, and Multiscale Characterization Techniques

An AISI 1045 steel modified with vanadium (V) and niobium (Nb) was studied to evaluate microstructural conditioning prior to and throughout a rapid heat treat process. In order to accomplish this, both computational and physical simulation techniques have been employed with the goal of assessing the microstructural evolution in a medium-carbon bar steel during the rapid austenitization and quenching procedures involved in an induction hardening process. The appropriate thermal profiles for induction hardening were obtained through finite element modeling using Flux 2D software. Physical simulations of the induction hardening process were carried out using a Gleeble{sup ®} 3500. Analysis of prior austenite grain size is complemented by observation of nanoscale carbonitride precipitation via transmission electron microscopy, scanning transmission electron microscopy, and high-energy synchrotron small-angle x-ray scattering. Through a combination of characterization techniques, this study presents a deeper understanding of nano- and microstructural changes occurring in a microalloyed steel during an induction hardening process.

36 MATERIALS SCIENCE↗
DOE OSTI2018

Investigating PID shunting in polycrystalline silicon modules via multiscale, multitechnique characterization

We investigated the potential-induced degradation (PID) shunting mechanism in multicrystalline-silicon photovoltaic modules by using a multiscale, multitechnique characterization approach. Both field-stressed modules and laboratory-stressed mini modules were studied. We used photoluminescence, electroluminescence, and dark lock-in thermography imaging to identify degraded areas at the module scale. Small samples were then removed from degraded areas, laser marked, and imaged by scanning electron microscopy. We used simultaneous electron-beam induced current imaging and focused ion beam milling to mark around PID shunts for chemical analysis by time-of-flight secondary-ion mass spectrometry or to isolate individual shunt defects for transmission electron microscopy and atom-probe tomography analysis. By spanning a range of 10 orders of magnitude in size, this approach enabled us to investigate the root-cause mechanisms for PID shunting. We observed a direct correlation between recombination active shunts and sodium content. The sodium content in shunted areas peaks at the SiNX/Si interface and is consistently observed at a concentration of 0.1% to 2% in shunted areas. Analysis of samples subjected to PID recovery, either activated by electron beam or thermal effects only, reveals that recovery of isolated shunts correlates with diffusion of sodium out of the structural defects to the silicon surface. We observed the role of oxygen and chlorine in PID shunting and found that those species - although sometimes present in structural defects where PID shunting was observed - do not play a consistent role in PID shunting.

14 SOLAR ENERGY↗
DOE OSTI2020

Deformation-enhanced hierarchical multiscale structure heterogeneity in a Pd-Si bulk metallic glass

Here, the multiscale structures in a Pd 82 Si 18 binary bulk metallic glass before and after deformation were studied using electron microscopies, high-energy synchrotron X-ray diffraction, and small-angle scattering techniques. The experimental results revealed an enhancement of hierarchical structure heterogeneities on multiple length scales after deformation. Hierarchical multiple shear bands of high number density were observed after bending, introducing complex but periodically distributed residual strain. Pair distribution function analysis revealed that the connectivity of the short-range clusters on the medium-range scale determines the packing density difference between the tension side and the compression side in the sample after bending. In-situ synchrotron X-ray diffraction study also revealed a transformation of connection modes among short-range clusters under uniaxial tension and compression, which is consistent with those of triaxial tension/compression parts upon bending in Pd 82 Si 18 glassy alloys. The nanoscale heterogeneities for metallic glasses after deformation observed by small-angle scattering and transmission electron microscopy may be attributed to the nanoscale amorphous phase separation and interacting multiple shear bands enhanced by plastic deformation. Our findings suggested that the enhancement of hierarchical heterogeneous structure on multiple length scales may explain the excellent plasticity of Pd-Si glassy alloys, deepening the understanding of structure-property relation during plastic deformation in metallic glasses.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Deep abiotic weathering of pyrite

Pyrite is a ubiquitous iron sulfide mineral that is oxidized by trace oxygen. The mineral has been largely absent from global sediments since the rise in oxygen concentration in Earth’s early atmosphere. We analyzed weathering in shale, the most common rock exposed at Earth’s surface, with chemical and microscopic analysis. By looking across scales from 10 −9 to 10 2 meters, we determined the factors that control pyrite oxidation. Under the atmosphere today, pyrite oxidation is rate-limited by diffusion of oxygen to the grain surface and regulated by large-scale erosion and clast-scale fracturing. We determined that neither iron- nor sulfur-oxidizing microorganisms control global pyrite weathering fluxes despite their ability to catalyze the reaction. This multiscale picture emphasizes that fracturing and erosion are as important as atmospheric oxygen in limiting pyrite reactivity over Earth’s history.

58 GEOSCIENCES↗
DOE OSTI2024

Air quality and health impacts of the 2020 wildfires in California

Abstract Background Wildfires in 2020 ravaged California to set the annual record of area burned to date. Clusters of wildfires in Northern California surrounded the Bay Area covering the skies with smoke and raising the air pollutant concentrations to hazardous levels. This study uses the Fire Inventory from the National Center for Atmospheric Research database and the Community Multiscale Air Quality model to estimate the effects of wildfire emissions on air quality during the period from August 16 to October 28 of 2020. In addition, low-cost sensor data for fine particulate matter (PM 2.5 ) from the PurpleAir network is used to enhance modeled PM 2.5 concentrations. The resulting impacts on ozone and PM 2.5 are used to quantify the health impacts caused by wildfires using the Benefits Mapping and Analysis Program – Community Edition. Results Wildfire activity significantly increased direct PM 2.5 emissions and emissions of PM 2.5 and ozone precursors. Direct PM 2.5 emissions surged up to 38 times compared to an average day. Modeling results indicated that wildfires alone led to a rise in ozone daily maximum 8-h average by up to 10 ppb and exceeded PM 2.5 air quality standards in numerous locations by up to 10 times. While modeled PM 2.5 concentrations were lower than measurements, correcting these with PurpleAir data improved the accuracy. The correction using PurpleAir data increased estimates of wildfire-induced mortality due to PM 2.5 exposure by up to 16%. Conclusions The increased hospital admissions and premature mortality attributed to wildfires were found to be comparable to the health impacts avoided by strategies aimed at meeting ozone and PM2.5 air quality standards. This suggests that widespread wildfire emissions can negate years of efforts dedicated to controlling air pollution. The integration of low-cost sensor data proved invaluable in refining the estimates of health impacts from PM2.5 resulting from wildfires.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2016

solveME: fast and reliable solution of nonlinear ME models

Background Genome-scale models of metabolism and macromolecular expression (ME) significantly expand the scope and predictive capabilities of constraint-based modeling. ME models present considerable computational challenges: they are much (>30 times) larger than corresponding metabolic reconstructions (M models), are multiscale, and growth maximization is a nonlinear programming (NLP) problem, mainly due to macromolecule dilution constraints. Results Here, we address these computational challenges. We develop a fast and numerically reliable solution method for growth maximization in ME models using a quad-precision NLP solver (Quad MINOS). Our method was up to 45 % faster than binary search for six significant digits in growth rate. We also develop a fast, quad-precision flux variability analysis that is accelerated (up to 60× speedup) via solver warm-starts. Finally, we employ the tools developed to investigate growth-coupled succinate overproduction, accounting for proteome constraints. Conclusions Just as genome-scale metabolic reconstructions have become an invaluable tool for computational and systems biologists, we anticipate that these fast and numerically reliable ME solution methods will accelerate the wide-spread adoption of ME models for researchers in these fields.

Biochemistry & Molecular Biology↗
DOE OSTI2020

Applying the index of watershed integrity to the Matanuska–Susitna basin

The Matanuska-Susitna Borough is the fastest growing region in the State of Alaska and is impacted by a number of human activities. We conducted a multiscale assessment of the stressors facing the borough by developing and mapping the Index of Watershed Integrity (IWI) and Index of Catchment Integrity (the latter considers stressors in areas surrounding individual stream segments exclusive of upstream areas). The assessment coincided with the borough’s stormwater management planning. We adapted the list of anthropogenic stressors used in the original conterminous United States IWI application to reflect the borough’s geography, human activity, and data availability. This analysis also represents an early application of the NHDPlus High Resolution geospatial framework and the first use of the framework in an IWI study. We also explored how remediation of one important stressor, culverts, could impact watershed integrity at the catchment and watershed scales. Overall, we found that the integrity scores for the Matanuska– Susitna basin were high compared to the conterminous United States. Low integrity scores did occur in the rapidly developing Wasilla–Palmer core area. We also found that culvert remediation had a larger proportional impact in catchments with fewer stressors.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2019

Qualitative Analysis, Structure-Preserving Discretizations and Scalable Solvers for Multiphysics Problems. Final Technical Report

Many problems in physics, engineering, meteorology, computer science, and other fields involve a certain level of coupling between different physical quantities. With the development of advanced computational modeling, simulation, and hardware, simulations for many important multiphysics/multiscale coupling effects are now more tractable. In this project, we have developed robust, efficient, and practical numerical algorithms for such simulation software that are capable of tackling these complex mathematical models in many fundamental ways such as stability, wellposedness, and consistency. We expect that such an integrated theoretical and algorithmic framework will have a substantial impact on numerous multiscale and multiphysics modeling and simulations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2019

Quantitative Nanoscale 3D Imaging of Intergranular Corrosion of 304 Stainless Steel Using Hard X-Ray Nanoprobe

Multi-modal imaging, which visualizes changes in the structure and chemistry of the same region of interest in materials by combining various techniques, is becoming attractive to material scientists. Here, we describe a multi-modal approach to investigate intergranular (IG) corrosion of sensitized stainless steels using synchrotron-based nanoscale X-ray imaging with sensitivity to microstructure and chemical composition. Three-dimensional tomography of 304 stainless steel samples deteriorated by IG corrosion, was carried out using X-ray fluorescence and differential phase contrast imaging. These findings were further supported with surface imaging and chemical analysis using scanning electron microscopy and energy dispersive X-ray spectroscopy. The combined quantitative structural and chemical analysis indicates that chromium segregates along grain boundaries, cracking is due to IG corrosion, and Cr-enrichment occurs on the cracked surfaces. Such quantitative 3D imaging of Cr 23 C 6 nucleation at grain boundaries and Cr enriched oxides at cracked surfaces is a powerful tool to correlate IG corrosion mechanisms at nanoscale with buried grain boundary structure and alloy composition. Such understanding of IG corrosion mechanisms is critical for both the development of predictive multiscale corrosion models, and the engineering of corrosion resistant materials.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2017

Theoretical analysis of non‐ G aussian heterogeneity effects on subsurface flow and transport

Abstract Much of the stochastic groundwater literature is devoted to the analysis of flow and transport in Gaussian or multi‐Gaussian log hydraulic conductivity (or transmissivity) fields, ( x being a position vector), characterized by one or (less frequently) a multiplicity of spatial correlation scales. Yet Y and many other variables and their (spatial or temporal) increments, Δ Y , are known to be generally non‐Gaussian. One common manifestation of non‐Gaussianity is that whereas frequency distributions of Y often exhibit mild peaks and light tails, those of increments are generally symmetric with peaks that grow sharper, and tails that become heavier, as separation scale or lag between pairs of Y values decreases. A statistical model that captures these disparate, scale‐dependent distributions of Y and in a unified and consistent manner has been recently proposed by us. This new “generalized sub‐Gaussian (GSG)” model has the form where is (generally, but not necessarily) a multiscale Gaussian random field and is a nonnegative subordinator independent of G . The purpose of this paper is to explore analytically, in an elementary manner, lead‐order effects that non‐Gaussian heterogeneity described by the GSG model have on the stochastic description of flow and transport. Recognizing that perturbation expansion of hydraulic conductivity diverges when Y is sub‐Gaussian, we render the expansion convergent by truncating Y 's domain of definition. We then demonstrate theoretically and illustrate by way of numerical examples that, as the domain of truncation expands, (a) the variance of truncated Y (denoted by Y t ) approaches that of Y and (b) the pdf (and thereby moments) of Y t increments approach those of Y increments and, as a consequence, the variogram of approaches that of . This in turn guarantees that perturbing to second order in (the standard deviation of Y t ) yields results which approach those we obtain upon perturbing to second order in even as the corresponding series diverges. Our analysis is rendered mathematically tractable by considering mean‐uniform steady state flow in an unbounded, two‐dimensional domain of mildly heterogeneous Y with a single‐scale function G having an isotropic exponential covariance. Results consist of expressions for (a) lead‐order autocovariance and cross‐covariance functions of hydraulic head, velocity, and advective particle displacement and (b) analogues of preasymptotic as well as asymptotic Fickian dispersion coefficients. We compare these theoretically and graphically with corresponding expressions developed in the literature for Gaussian Y . We find the former to differ from the latter by a factor k = ( denoting ensemble expectation) and the GSG covariance of longitudinal velocity to contain an additional nugget term depending on this same factor. In the limit as Y becomes Gaussian, k reduces to one and the nugget term drops out.

Riva, Monica↗
DOE OSTI2025

Selenium Migration and Local Structures in Cu‐Doped CdSeTe Solar Cells after Aging

Selenium grading plays a critical role in state-of-the-art Cadmium Telluride photovoltaic cells by enhancing long-wavelength absorption and extending minority carrier lifetimes —key to enabling the current performance record of 23.08%. However, very little is understood about selenium motion. In this study, a comprehensive, multimodal, and multiscale approach is employed to investigate Se migration and local structural changes in copper (Cu)-doped CdSeTe solar cells subjected to accelerated stress. X-ray fluorescence (XRF) microscopy shows unexpected levels of Se diffusion after 500 h under heat (75°C) and light (0.8 suns, 80 mW/cm 2 ), suggesting the coexistence of fast and slow diffusion channels even at low temperatures, with unexpectedly low activation energies (<0.85 eV). X-ray Absorption Near Edge Structure (XANES) analysis indicates a preferential migration of Se atoms to anionic lattice sites and a reduction in Se-Cl co-passivation at Te-terminated dislocation cores. Furthermore, these findings point to a reconfiguration of Se local environments and highlight the potential role of extended structural defects in enabling Se transport at low temperatures. Additionally, XANES results suggest that the presence of metallic Cu across the absorber layer may contribute to back-contact degradation and reduced hole density in both fresh and aged devices.

14 SOLAR ENERGY↗
DOE OSTI2025

Impacts of the Conductive Networks on Solid‐State Battery Operation

The micromorphology of composite cathodes is known to play a vital role in determining all-solid-state battery (ASSB) performance. However, much of our current understanding is derived from empirical observations, lacking a deeper mechanistic foundation. The “rocking chair” concept of battery chemistry requires maintaining charge neutrality, emphasizing the necessity of examining electrode micromorphology from the perspective of conductive networks. This study systematically investigates the microscopic electrochemical impacts of conductive network micromorphology by varying the Li + -to-e − channel ratio in cathodes comprising LiNbO 3 -coated LiNi 0.8 Co 0.1 Mn 0.1 O 2 , Li 6 PS 5 Cl, and carbon fibers. Utilizing multiscale synchrotron-based spectro-microscopy, we unravel that unbalanced Li + and e − conducting channels intensify charge polarization within active cathode particles and accelerate their degradation. A further model system with X-ray nano-tomography resolved e − and Li + channels indicates that spatially uniform and well-paired Li + and e − conducting channels are highly desirable as they could promote more uniform lithiation/delithiation, mitigating microscopic electrochemical polarization. Electrode-scale X-ray holotomography analysis reveals that the impact of conductive networks is particle-size-dependent, with smaller cathode particles being more significantly affected. These findings provide mechanistic insights into the interplay between conductive networks and all-solid-state battery operation, laying the groundwork for rational design and optimization of cathode architectures in future solid-state battery technologies.

36 MATERIALS SCIENCE↗
DOE OSTI2026

Group-Additivity–Embedded Multiscale Modeling for Electric Field-Enhanced Nanocatalysis

Elucidating structure-performance relationships remains a central challenge in field-enhanced catalysis, where nanoparticles exhibit nonuniform surface sites with site-dependent responses to electric fields. Low-coordination sites (edges, corners, and tips) are particularly electric field-sensitive (EF), leading to nonuniform charge distribution, adsorption energies, and catalytic activity. Here, using ammonia decomposition on a ruthenium cluster as a model system, we develop a transferable multiscale framework integrating density functional theory, group additivity (GA), Brønsted-Evans-Polanyi scaling, and microkinetic modeling to predict EF-dependent activity across nonuniform cluster sites. Across sites and fields, the nitrogen adsorption energy (E N ) emerges as the governing descriptor, yielding robust volcano relationships whose optimum shifts systematically with field: negative fields strengthen N binding via electron accumulation, while positive fields weaken N binding via charge depletion, moving the optimal E N toward weaker binding. Microkinetic analysis shows that N≡N bond formation remains the key kinetic bottleneck over most conditions; positive fields lower the effective barrier and, critically, increase the fraction of near-optimal active sites, leading to a net enhancement in overall activity relative to zero-field and negative-field cases. By capturing EF- and site-dependent energetics with high accuracy and low computational cost, this GA-embedded multi-scale simulation workflow provides a physically interpretable route to predict and design field-enhanced nanocatalysis.

ammonia decomposition↗
DOE OSTI2021

Design nanoporous metal thin films via solid state interfacial dealloying

Thin-film solid-state interfacial dealloying (thin-film SSID) is an emerging technique to design nanoarchitecture thin films. The resulting controllable 3D bicontinuous nanostructure is promising for a range of applications including catalysis, sensing, and energy storage. Using a multiscale microscopy approach, we combine X-ray and electron nano-tomography to demonstrate that besides dense bicontinuous nanocomposites, thin-film SSID can create a very fine (5–15 nm) nanoporous structure. Not only is such a fine feature among one of the finest fabrications by metal-agent dealloying, but a multilayer thin-film design enables creating nanoporous films on a wider range of substrates for functional applications. Through multimodal synchrotron diffraction and spectroscopy analysis with which the materials’ chemical and structural evolution in this novel approach is characterized in details, we further deduce that the contribution of change in entropy should be considered to explain the phase evolution in metal-agent dealloying, in addition to the commonly used enthalpy term in prior studies. Here, the discussion is an important step leading towards better explaining the underlying design principles for controllable 3D nanoarchitecture, as well as exploring a wider range of elemental and substrate selections for new applications.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Multiscale simulation of integrated energy system and electricity market interactions

Accelerating the deep decarbonization of the world's electric grids requires the coordination of complex energy systems and infrastructures across timescales from seconds to decades. Here, we present a new multiscale simulation framework that integrates process- and grid-centric modeling paradigms to better design, operate, and control integrated energy systems (IESs), which combine multiple technologies, in wholesale energy markets. Traditionally, IESs are analyzed with a process-centric paradigm such as levelized cost of electricity (LCOE) or annualized net revenue, ignoring important interactions with electricity markets. This framework explicitly models the complex interactions between an IES's bidding, scheduling, and control decisions and the energy market's clearing and settlement processes, while incorporating operational uncertainties. Through two case studies, we show the importance of understanding and quantifying complex resource-grid interactions. In case study 1, we demonstrate that optimized bidding from one resource shifts the profit distribution for all energy systems in the market. This result suggests new and more flexible IES technologies can disrupt the economics of all market participants, possibly leading to accelerated retirements of less flexible resources. Interestingly, the optimized bidding has little impact on grid-level aggregate statistics, such as total generation costs and renewable penetration rate. While aggregate modeling strategies may remain valid under some IES adoption scenarios for analysis focused on regional outcomes, direct comparisons of IES technologies at specific locations without considering these interactions may lead to misleading or incorrect conclusions. In case study 2, we consider the design and flexible operation of IESs that hybridize conventional generators with energy storage. Through a sensitivity analysis, we find that as the size of the storage system increases, the total number of start-ups for coal- and natural gas-based IESs reduced by 25% and 33.6%, and the total thermal generator ramping (i.e., mileage) reduced by 86.5% and 62.5%, respectively. This shows the primary benefit of storage may not be reduced operational costs (which do not change significantly) but fewer start-ups and less ramping, which may greatly simplify the design, operation, and control of carbon capture systems. The new modeling and optimization capabilities from this work enable the coupling of rigorous, dynamic process models with grid-level production cost models to quantitatively identify the nuanced interdependencies across these vast timescales that must be addressed to realize clean, safe, and secure energy production. Moreover, the proposed general multiscale simulation framework is applicable to all IES technologies and can be easily extended to consider other energy carriers (e.g., hydrogen, ammonia) and energy infrastructures (e.g., natural gas pipelines).

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2024

Structural characterization of high-protein system through ultra-small and small-angle X-ray scattering

High-protein systems exhibit a hierarchical structure consisting of interconnected multiscale length assemblies. Recent investigations are directed toward identifying these structural units under various processing and environmental conditions to establish structure–function correlations. Ultra-small and small-angle X-ray scattering (USAXS/SAXS) has become a crucial tool for characterizing the structures of proteins and their clusters/aggregates, ranging from nanometers to micrometers, with minimal disruption to their original state. Here, this review first describes the facilities and principles of X-ray scattering, followed by discussions on the analysis of scattering data, including the interpretation of fitting models. It then delves into the main applications of USAXS/SAXS in plant and dairy protein-rich systems. Future strategies to enhance the utilization of scattering techniques for elucidating the structure of high-protein systems are also included.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2023

Quantification of the Impact of Acidified Brine on Fracture-Matrix Transport in a Naturally Fractured Shale Using in Situ Imaging and Modeling

Understanding flow, transport, chemical reactions, and hydromechanical processes in fractured geologic materials is key for optimizing a range of subsurface processes including carbon dioxide and hydrogen storage, unconventional energy resource extraction, and geothermal energy recovery. Flow and transport processes in naturally fractured shale rocks have been challenging to characterize due to experimental complexity and the multiscale nature of quantifying continuum scale descriptions of mass exchange between micrometer-scale fractures and nanometer-scale pores. In this study, we use positron emission tomography (PET) to image the transport of a conservative tracer in a naturally fractured Wolfcamp shale core before and after the core was exposed to low pH brine conditions. Image-based experimental observations are interpreted by fitting an analytical transport model to fracture-containing voxels in the core. Results of this analysis indicate subtle increases in matrix diffusivity and a slightly more uniform fracture velocity distribution following exposure to low pH conditions. These observations are compared with a multicomponent one-dimensional reactive transport model that indicates the capacity for a 10% increase in porosity at the fracture-matrix interface as a result of the low pH brine exposure. This porosity change is the result of the dissolution of carbonate minerals in the shale matrix to low pH conditions. Further, this image-based workflow represents a new approach for quantifying spatially resolved fracture-matrix transport processes and provides a foundation for future work to better understand the role of coupled transport, reaction, and mechanical processes in naturally fractured rocks.

58 GEOSCIENCES↗
DOE OSTI2016

A multiscale coupled finite-element and phase-field framework to modeling stressed grain growth in polycrystalline thin films

A previously-developed finite-deformation- and crystal-elasticity-based constitutive theory for stressed grain growth in cubic polycrystalline bodies has been augmented to include a description of excess surface energy and grain-growth stagnation mechanisms through the use of surface effect state variables in a thermodynamically-consistent manner. The constitutive theory was also implemented into a multiscale coupled finite-element and phase-field computational framework. With the material parameters in the constitutive theory suitably calibrated, our three-dimensional numerical simulations show that the constitutive model is able to accurately predict the experimentally-determined evolution of crystallographic texture and grain size statistics in polycrystalline copper thin films deposited on polyimide substrate and annealed at high-homologous temperatures. In particular, our numerical analyses show that the broad texture transition observed in the annealing experiments of polycrystalline thin films is caused by grain growth stagnation mechanisms. - Graphical abstract: - Highlights: • Developing a theory for stressed grain growth in polycrystalline thin films. • Implementation into a multiscale coupled finite-element and phase-field framework. • Quantitative reproduction of the experimental grain growth data by simulations. • Revealing the cause of texture transition to be due to the stagnation mechanisms.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗